ANALYSIS OF HIGHWAY BRIDGES USING
COMPUTER ASSISTED MODELING, NEURAL NETWORKS,
AND DATA COMPRESSION TECHNIQUES
By
GARY RAPH CONSOLAZIO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Copyright 1995
by
Gary Raph Consolazio
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Professor Marc I. Hoit for his
guidance in both research and professional issues, for his generous support, and for his
enthusiastic encouragement throughout the duration of my doctoral program. In
addition I would like to express my gratitude to Professor Clifford O. Hays, Jr., for his
guidance and support, especially during the initial part of my doctoral program. I
would also like to thank Professors Ron A. Cook, John M. Lybas, W. J. Sullivan, and
Loc VuQuoc for serving on my committee.
I would especially like to thank my wife, Lori, for the enduring patience and
support she has shown during my graduate educationone could not possibly hope for
a more supportive spouse. I would like to thank my parents, Lynne and Bruce, for
instilling in me the importance of an education and my grandfather, William V.
Consolazio, for encouraging my interest in science and making it possible for me to
pursue an advanced degree.
Finally, I would like to thank my friend and colleague Petros Christou and all
my other fellow graduate studentsespecially Wilson Moy and Prashant Andradefor
their friendship and encouragement.
The work presented in this dissertation was partially sponsored by the Florida
Department of Transportation.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ..................................... ............... iii
ABSTRACT ..................... .............. .......... ...............viii
CHAPTERS
1 INTRODUCTION ........................................ .................... 1
1.1 Background..................................... ............................ 1
1.1.1 Computer Assisted Bridge Modeling ...................................... 1
1.1.2 Computational Aspects of Highway Bridge Analysis ................. 3
1.2 Present Research..................................... ....................... 6
1.2.1 Computer Assisted Bridge Modeling ....................................... 7
1.2.2 RealTime Data Compression ............................................. 9
1.2.3 Neural Network Equation Solver .........................................12
1.3 Literature Review .........................................................15
1.3.1 Computer Assisted Bridge Modeling .................................... 15
1.3.2 Data Compression in FEA .................................................18
1.3.3 Neural Network Applications in Structural Engineering...............18
2 A PREPROCESSOR FOR BRIDGE MODELING ................................21
2.1 Introduction ........................................................... 21
2.2 Overview of the Bridge Modeling Preprocessor ...............................22
2.3 Design Philosophy of the Preprocessor .........................................25
2.3.1 Internal Preprocessor Databases.................... ....................25
2.3.2 The Basic Model and Extra Members...................................26
2.3.3 Generation................................ ............................ 28
2.3.4 The Preprocessor History File..........................................29
2.4 Common Modeling Features and Concepts......................................30
2.4.1 Bridge Directions................................... ......................31
2.4.2 Zero Skew, Constant Skew, and Variable Skew Bridge Geometry...32
2.4.3 Live Load Models and Full Load Models ..............................33
2.4.4 Live Loads ...........................................................34
2.4.5 Line Loads and Overlay Loads........................ ...................36
2.4.6 Prismatic and Nonprismatic Girders......................................37
2.4.7 Composite Action ..........................................................38
2.5 Modeling Features Specific to Prestressed Concrete Girder Bridges .........40
2.5.1 Cross Sectional Property Databases ....................................40
2.5.2 Pretensioning and PostTensioning .....................................41
2.5.3 Shielding of Pretensioning ........................... ...................42
2.5.4 PostTensioning Termination .........................................43
2.5.5 End Blocks ....................................... .............. 43
2.5.6 Temporary Shoring ........................................................44
2.5.7 Stiffening of the Deck Slab Over the Girder Flanges.................45
2.6 Modeling Features Specific to Steel Girder Bridges ...........................46
2.6.1 Diaphragms ........................ ......... ................... 46
2.6.2 Hinges.................................................... .............. 47
2.6.3 Concrete Creep and Composite Action...................................48
2.7 Modeling Features Specific to Reinforced Concrete TBeam Bridges........49
2.8 Modeling Features Specific to FlatSlab Bridges ...............................50
3 MODELING BRIDGE COMPONENTS .........................................51
3.1 Introduction ........................................ ...................... 51
3.2 Modeling the Common Structural Components.................................51
3.2.1 Modeling the Deck Slab............................ ......................51
3.2.2 Modeling the Girders and Stiffeners......................................54
3.2.3 Modeling the Diaphragms..........................................55
3.2.4 Modeling the Supports..................... .......................57
3.3 Modeling Composite Action................................................ .......58
3.3.1 Modeling Composite Action with the Composite Girder Model ......60
3.3.2 Modeling Composite Action with the Eccentric Girder Model........61
3.4 Modeling Prestressed Concrete Girder Bridge Components .................65
3.4.1 Modeling Prestressing Tendons ...........................................65
3.4.2 Increased Stiffening of the Slab Over the Concrete Girders ...........68
3.5 Modeling Steel Girder Bridge Components ....................................70
3.5.1 Modeling Hinges ..................................... ....................70
3.5.2 Accounting for Creep in the Concrete Deck Slab .....................72
3.6 Modeling Reinforced Concrete TBeam Bridge Components.................74
3.7 Modeling FlatSlab Bridge Components ........................................ 75
3.8 Modeling the Construction Stages of Bridges...................................76
3.9 Modeling Vehicle Loads ..................................................80
4 DATA COMPRESSION IN FINITE ELEMENT ANALYSIS ..................83
4.1 Introduction .................................. ..... .................... 83
4.2 Background............................. ........ ...........................84
4.3 Data Compression in Finite Element Software..................................86
4.4 Compressed I/O Library Overview ....................... .....................91
4.5 Compressed I/O Library Operation.......................................92
4.6 Data Compression Algorithm.................... .......................95
4.7 Fortran Interface to the Compressed I/O Library...............................99
4.8 Data Compression Parameter Study and Testing ............................ 101
4.8.1 Data Compression in FEA Software Coded in C..................... 102
4.8.2 Data Compression in FEA Software Coded in Fortran.............. 112
5 NEURAL NETWORKS.................................. ............ 119
5.1 Introduction ............................ ............ ..... .......... 119
5.2 Network Architecture and Operation................... ..................... 120
5.3 Problem Solving Using Neural Networks .................................... 124
5.4 Network Learning ......................... .... ...................... 125
5.5 The NetSim Neural Network Package ....................................... 128
5.6 Supervised Training Techniques ........................... .................. 130
5.7 Gradient Descent and Stochastic Training Techniques...................... 133
5.8 Backpropagation Neural Network Training................................... 137
5.8.1 ExampleByExample Training and Batching ......................... 141
5.8.2 Momentum ......................... ..... .................. 143
5.8.3 Adaptive Learning Rates ........................... ................... 146
6 NEURAL NETWORKS FOR HIGHWAY BRIDGE ANALYSIS ............. 151
6.1 Introduction ............................................................. 151
6.2 Encoding Structural Behavior .............................. .................. 151
6.3 Separation of Shape and Magnitude ........................................ 153
6.3.1 Generating Network Training Data.................................... 157
6.3.2 Using Trained Shape and Scaling Networks.......................... 160
6.4 Generating Analytical Training Data........................................ 163
6.5 Encoding Bridge Coordinates....................... ..................... 168
6.6 Shape Neural Networks ....................................... 172
6.7 Scaling Neural Networks.................................... 175
6.8 Implementation and Testing ............................. .............. 183
7 ITERATIVE EQUATION SOLVERS FOR HIGHWAY BRIDGE
ANALYSIS ......................................... ................... 185
7.1 Introduction ........................................ .. .................... 185
7.2 Exploiting Domain Knowledge...................... ................. 186
7.3 Iterative FEA Equation Solving Schemes...................................... 188
7.4 Preconditioning in Highway Bridge Analysis ................................. 194
7.4.1 Diagonal and Band Preconditioning ................................... 195
7.4.2 Incomplete Cholesky Decomposition Preconditioning ............... 201
7.5 A Domain Specific Equation Solver...................... ................... 205
7.6 Implementation and Results...................................................... 212
7.6.1 Seeding the Solution Vector Using Neural Networks .............. 213
7.6.2 Preconditioning Using Neural Networks............................... 229
8 CONCLUSIONS AND RECOMMENDATIONS................................ 234
8.1 Computer Assisted Modeling ............................. .............. 234
8.2 Data Compression in FEA ........................ .................... 236
8.3 Neural Networks and Iterative Equation Solvers ............................ 238
REFERENCES........................ ....... ............ 242
BIOGRAPHICAL SKETCH ..................... ........... ....................... 247
vii
Abstract of Dissertation Presented to the Graduate
School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANALYSIS OF HIGHWAY BRIDGES USING
COMPUTER ASSISTED MODELING, NEURAL NETWORKS,
AND DATA COMPRESSION TECHNIQUES
By
GARY RAPH CONSOLAZIO
August 1995
Chairman: Marc I. Hoit
Major Department: Civil Engineering
By making use of modern computing facilities, it is now possible to routinely
apply finite element analysis (FEA) techniques to the analysis of complex structural
systems. While these techniques may be successfully applied to the area of highway
bridge analysis, there arise certain considerations specific to bridge analysis that must
be addressed.
To properly analyze bridge systems for rating purposes, it is necessary to model
each distinct structural stage of construction. Also, due to the nature of moving
vehicular loading, the modeling of such loads is complex and cumbersome. To address
these issues, computer assisted modeling software has been developed that allows an
engineer to easily model both the construction stages of a bridge and complex vehicular
loading conditions.
Using the modeling software an engineer can create large, refined FEA models
that otherwise would have required prohibitively large quantities of time to prepare
manually. However, as the size of these models increases so does the demand on the
computing facilities used to perform the analysis. This is especially true in regard to
temporary storage requirements and required execution time.
To address these issues a real time lossless data compression strategy suitable
for FEA software has been developed, implemented, and tested. The use of this data
compression strategy has resulted in dramatically reduced storage requirements and, in
many cases, also a significant reduction in the analysis execution time. The latter result
can be attributed to the reduced quantity of physical data transfer which must be
performed during the analysis.
In a further attempt to reduce the analysis execution time, a neural network has
been employed to create a domain specific equation solver. The chosen domain is that
of twospan flatslab bridges. A neural network has been trained to predict
displacement patterns for these bridges under various loading conditions. Subsequently,
a preconditioned conjugate gradient equation solver was constructed using the neural
network both to seed the solution vector and to act as a preconditioner. Results are
promising but further network training is needed to fully realize the potential of the
application.
CHAPTER 1
INTRODUCTION
1.1 Background
In spite of the widespread success with which finite element analysis (FEA)
techniques have been applied to problems in solid mechanics and structural analysis,
the use of FEA in highway bridge analysis has suffered from a lack of requisite pre
and postprocessing tools. Without question, the finite element method (FEM) affords
engineers a powerful and flexible tool with which to solve problems ranging in
complexity from static linear elastic analyses to dynamic nonlinear analyses. During the
past few decades, numerous high quality FEA software packages have been developed
both in the form of commercial products and research codes.
1.1.1 Computer Assisted Bridge Modeling
In addition to containing a core FEA engine many of these packagesespecially
the commercial onesinclude or can be linked to separate pre and postprocessing
modules to aid the engineer in preparing and interpreting FEA data. Modeling
preprocessors for building structures that allow the engineer to accurately and
efficiently prepare FEA models are common. Once the core FEA engine has been
executed, postprocessing packages facilitate interpretation of the often voluminous
quantities of analysis results generated by such software.
Whereas the development of such packages for the analysis and design of
buildingtype structures has roughly kept pace with the demands of industry, the same
is not true for the case of highway bridge analysis. This is probably attributable to the
fact that there are simply many more buildingtype structures constructed than there are
highway bridge structures, and therefore a greater demand exists. However, this is not
to say that there is not a demand for such software in bridge analysis. With an
inventory of more than half a million bridges in the United States alone, and roughly
20 percent of those bridges considered structurally deficient and in need of evaluation,
the demand for computer assisted bridge analysis packages exists.
Modeling highway bridges for FEA presents certain challenges that are not
present in the analysis of building structures. For example, in addition to being
subjected to the usual fixed location loads, bridges are also subjected moving vehicular
loads which are often complex and cumbersome to describe with the level of detail
needed for FEA. Also, because moving vehicle loads are typically represented using a
large number of discrete vehicle locations, bridge analyses often contain a large number
of load cases. As a direct result, the engineer is faced not only with the daunting task of
describing the loads, but also of interpreting the vast quantity of results that will be
generated by the analysis.
In order to properly analyze bridge systems for evaluation purposes, as in a
design verification or rating of an existing bridge, each distinct structural stage of
construction should be represented in the model. This is because the bridge has a
distinct structural configuration at each stage of construction, and it is that structural
configuration that resists loads at that point in the construction sequence. Stresses and
forces developed at each of these stages will be locked into the structure in subsequent
stages of construction. Such conditions cannot simply be modeled by applying all dead
loads to the final structural configuration of the bridge. Modeling of the distinct
construction stages is important in the analysis of steel girder bridges and is very
important in prestressed concrete girder bridges.
Therefore, in addition to describing complex vehicular loading conditions the
engineer is also faced with preparing multiple FEA models to represent each distinct
stage of the bridge construction. Thus, the need for the development of computer
assisted bridge modeling software is clear.
1.1.2 Computational Aspects of Highway Bridge Analysis
Assuming that modeling software for highway bridges exists, an actual analysis
must still be performed. As a result of advances in computational hardware and decades
of refinement of FEA code, is it now possible to perform analyses of complex
structural systems on a more or less routine basis. However, there arise certain
considerations specific to bridge analysis that must still be addressed if the full potential
of computer assisted modeling is to be realized.
In the FEA of bridge structures, the computational demands imposed by the
analysis generally fall into one of two categoriesrequired storage and required
execution time. Required storage can be subdivided into incore storage, also referred
to as primary or high speed storage, and outofcore storage, also referred to variously
as secondary storage, low speed storage, and backing store. Incore storage generally
refers to the amount of physical random access memory (RAM) available on a
computer, although on computers running virtual memory operating systems there can
also be virtual incore memory. Outofcore storage generally refers to available space
on hard disks, also called fixed disks.
Optimizing the use of available incore storage has been an area of considerable
research during the past few decades. In contrast, little research has been performed
that addresses the large outofcore storage requirements often imposed by FEA. Out
ofcore storage is used for three primary purposes in FEA :
1. To hold temporary data such as element stiffness, load, and stress recovery
matrices (collectively referred to as element matrices) that exist only for the
duration of the analysis.
2. To hold analysis results such as global displacements and element stresses
that will later be read by postprocessing software.
3. To perform blocked, outofcore equation solutions in cases where the
global stiffness or global load matrices are too large to be contained incore
as a single contiguous unit.
In cases 1 and 3, once the analysis is complete the storage is no longer needed, i.e. the
storage is temporary in nature. In case 2, the storage will be required at least until the
analysis results have been read by postprocessing software.
In the analysis of highway bridges, the amount of outofcore storage that is
available to hold element matrices can frequently become a constraint on the size of
model that can be analyzed. It is not uncommon for a bridge analysis to require
hundreds of megabytes of outofcore storage during an analysis. Also, as a result of
the proliferation of low cost personal computers (PCs), there has been a migration of
analysis software away from the large mainframe computers of the past toward the
smaller PC and workstation platforms of today. This migration has resulted in greater
demands being placed on smaller computerscomputers that often have only moderate
amounts of incore and outofcore storage.
Although the development of preprocessing tools is necessary to make routine
use of FEA in bridge analysis a reality, it also introduces a new problem. Using
computer assisted modeling software, it becomes quite simple for an engineer to create
very large FEA bridge modelsmodels that would otherwise would be too tedious to
prepare manually. While this is generally regarded as desirable from the standpoint of
analysis accuracy it also has the adverse effect of greatly increasing the demand for out
ofcore storage. It is clear then that the issue of outofcore storage optimization must
addressed in conjunction with the development of computer assisted modeling software
if the full potential of the latter is to be realized.
While the size of FEA bridge models may be physically constrained by the
availability of outofcore storage, these same models may also be practically
constrained by the amount of execution time required to perform the analysis. When
moving vehicle loads are modeled using a large number of discrete vehicle positions,
the number of load cases that must be analyzed can quickly reach into the hundreds.
Combine this fact with the aforementioned ease with which complex FEA models can
be createdusing preprocessing softwareand the result is the need to analyze large
bridge models for potentially hundreds of load cases. In such situations, the execution
time required to perform the analysis may diminish the usefulness of the overall
system. This is especially true in situations where multiple analyses will need to be
performed, as in an iterative designevaluation cycle or within a nonlinear analysis
scheme.
Thus, it is evident that in order for a computer assisted bridge modeling system
to be practical and useful, the FEA analysis component must be as numerically efficient
as possible so as to minimize the required analysis time and minimize the use of outof
core storage.
1.2 Present Research
The research reported on in this dissertation focuses on achieving three primary
objectives with respect to FEA bridge modeling. They are :
1. Developing an interactive bridge modeling preprocessor capable of
generating FEA models that can account for bridge construction stages and
vehicular loading conditions.
2. Developing a realtime data compression strategy that, once installed into
the FEA engine of a bridge analysis package, will reduce the computational
demands of the analysis.
3. Developing a domain specific equation solver based on neural network
technology and the subsequent installation of that solver into the FEA engine
of a bridge analysis package.
Each of these objectives attempts to address and overcome a specific difficulty
encountered when applying FEA techniques to the analysis of highway bridge systems.
The following sections describein greater detaileach objective and the methods used
to attain those objectives.
1.2.1 Computer Assisted Bridge Modeling
Widespread use of FEA techniques in highway bridge analysis has been
curtailed by a lack of requisite pre and postprocessing tools. Routine use of FEA in
bridge analysis can only occur when computer assisted modeling software has been
developed specifically with the highway bridge engineer in mind. To address this issue,
an interactive bridge modeling program has been developed as part of the research
reported on herein. The resulting bridge modeling preprocessor, called BRUFEM1, is
one component of the overall BRUFEM system (Hays et al. 1990, 1991, 1994).
BRUFEM, which is an acronym for Bridge Rating Using the Finite Element Method, is
a software package consisting of a series of Fortran 77 programs which, when working
collectively as a system, is capable of modeling, analyzing, and rating many types of
highway bridge structures.
The BRUFEM preprocessor, which will hereafter be referred to simply as the
preprocessor, allows an engineer to create detailed FEA bridge models by specifying
interactivelya minimal amount of bridge data. Information needed specifically for the
modeling of bridge structures and bridge loading is embedded directly into the
software. Thus the usual barriers that would prevent an engineer from manually
constructing the FEA bridge model are overcome. The primary barriers are :
1. Discretizing each and every structural component of the bridge into discrete
finite elements and subsequently specifying the characteristicsgeometry,
material properties, connectivities, eccentricities, etc.of each of those
elements.
2. Modeling the structural configuration and the appropriate dead loads at each
distinct stage of construction.
3. Computing potentially hundreds of discrete vehicle positions and
subsequently computing and specifying the load data required for FEA.
All of these barriers are overcome through the use of the preprocessor because it
handles these tasks in a semiautomated fashion. The term semiautomated, which is
used synonymously with computer assisted in this dissertation, alludes to the fact that
there is an interaction between the engineer and the modeling software. Neither has
complete responsibility for controlling the creation of the bridge model. General
characteristics of bridge structures and bridge loading are built into the preprocessor so
as to allow rapid modeling of such structures. However, the engineer retains the right
to introduce engineering judgmentwhere appropriateinto the creation of the model
by interacting with the software. Thus, the engineer is freed from the tedium of
manually preparing all of the data needed for FEA and allowed to focus on more
important aspects of the rating or design process.
In addition to handling the primary modeling tasks discussed above, the
preprocessor handles numerous other tasks which are required in bridge modeling. The
most important of these are listed here.
1. Modeling composite action between the girders and slab, in some cases
including the calculation of composite girder section properties based on the
recommended AASHTO (AASHTO 1992) procedure.
2. Modeling pretensioning and posttensioning tendons, including the
specification of finite element end eccentricities.
3. Modeling variable cross section girders, including the generation and
calculation of all necessary cross sectional properties and eccentricities.
4. Modeling complex bridge geometry such as variable skew.
5. Modeling live loading conditions considering not only a single standard
vehicle but often several different standard vehicles.
These features facilitate the rapid development of FEA bridge models by alleviating
the user of manually performing these tasks. Detailed descriptions of the capabilities of
the preprocessor will be given in subsequent chapters.
1.2.2 RealTime Data Compression
To address the issue of the large storage and execution time requirements arising
from the analysis of bridge structures, a realtime data compression strategy suitable for
FEA software has been developed and implemented. In the of discretization stage of
FEA modeling, any repetition or regularity in either structural geometry or
configuration is usually exploited to the fullest possible extent. This exploitation of
regularity has the advantage of not only minimizing the effort needed to prepare the
model but also of generally leading to a model that is desirable from the standpoint of
accuracy. An additional yet largely unexploited benefit of this regularity is that because
the model itself is highly repetitive, the data generated by the analysis software will
also be highly repetitive. Such conditions are ideal for the use of data compression.
Data compression is the process of taking one representation of set of data and
translating it into a different representation that requires less space to store while
preserving the information content of the original data set. Since compressed data
cannot be directly used in its compressed format, it must be decompressed at some later
stage in the life cycle of the data. This process is called either decompression or
uncompression of the data. However, the term data compression is also used to refer to
the overall process of compressing and subsequently decompressing a data set. It should
be clear from context which meaning is intended.
Data compression techniques may be broadly divided into two categories
lossless data compression and lossy data compression. In lossless data compression, the
data set may be translated from its original format into a compressed format and
subsequently back to the original format without any loss, corruption, or distortion of
the data. In contrast, lossy data compression techniques allow some distortion of the
data to occur during the translation process. This can result in greater compression than
that which can be achieved using lossless techniques. Lossy compression methods are
widely used in image compression where a modest amount of distortion of the data can
be tolerated.
In the compression of numeric FEA data such as finite element matrices it is
necessary to utilize lossless data compression methods since corruption of the data to
any extent would invalidate the analysis. Thus, in the present work, in order to
capitalize on the repetitive nature of FEA data, a realtime lossless data compression
strategy has been developed, implemented, and tested in bridge FEA software.
The term realtime is used to indicate that the FEA data is not created and then
subsequently compressed as a separate step but instead is compressed in realtime as the
data is being created. Thus the compression may be looked upon as a filter through
which a stream of numeric FEA data is passed in, and a stream of compressed data
emerges. This type of compression is also more loosely referred to as onthefly data
compression. Of course, the direction of the data stream must eventually be reversed so
that the numeric FEA data can be obtained by decompressing the compressed data. This
reversed process is also performed in realtime with the data being decompressed and
retrieved on demand as required by the FEA software.
The compression strategy developed in the present work consists of the
combination of a file input/output (i/o) library and a buffering algorithm both wrapped
around a core data compression algorithm called Ross Data Compression (RDC). RDC
is a sequential data compression algorithm that utilizes run length encoding (RLE) and
pattern matching to compress sequential data streams. Once developed, the technique
was implemented into two FEA programs used in the analysis of highway bridge
structures and tested using several realistic FEA bridge models.
Due to the repetitive nature of FEA bridge models, the data compression
strategy of the present work has been shown to greatly reduce the storage requirements
of FEA software. In the bridge models tested, the storage requirements for FEA
software equipped with data compression were roughly an order of magnitude smaller
than the storage requirements of the same FEA software lacking data compression.
Also, the use of data compression was shown to substantially decrease the
analysis execution time in many cases. This is due to the fact that when using data
compression, the quantity of disk i/o that must be performed by the FEA software is
greatly decreased often resulting in decreased execution time. This benefit has been
shown to be especially advantageous on workstation and personal computer platforms
running FEA software written in Fortran 77. Under such circumstances, the execution
time required for the bridge analysis was shown to decrease to as little as approximately
one third of the execution time needed when compression was not utilized.
1.2.3 Neural Network Equation Solver
In order for a computer assisted bridge modeling system to be effective, the
time required to perform each FEA analysis must be minimized. To address this issue,
an application of Artificial Neural Networks (ANNs) has been used to create a domain
specific equation solver. Since the equation solving stage of a FEA accounts for a large
portion of the total time required to perform an analysis, increasing the speed of this
stage will have a significant effect on the speed of the overall analysis.
In the present work, the approach taken to minimize the analysis execution time
is to implicitly embed, using ANNs, domain knowledge related to bridge analysis into
the equation solver itself. In this way a domain specific equation solver, i.e. an
equation solver constructed to solve problems within the specific problem domain of
bridge analysis, is created. The concept behind such an equation solver is that by
exploiting knowledge of the problem, e.g. knowing displacement characteristics of
bridge structures, the solver will be able to more rapidly arrive at the solution.
In the present application ANNs have been trained to learn displacement
characteristics of twospan flatslab bridges under generalized loading conditions. Using
analytical displacement data generated by numerous finite element analyses, a set of
network training data was created with which to train the ANNs. Next, using ANN
training software that was developed as part of the present research, several neural
networks were trained to predict displacement patterns in flatslab bridge under
generalized loading conditions. Once the networks were trained, a preconditioned
conjugate gradient (PCG) equation solver was implemented using the neural networks
both to seed the solution vector and to act as an implicit preconditioner.
In the case of seeding the solution vector, the networks attempt to predict the
actual set of displacements that would occur in the bridge under the given loading
condition. These displacements are then used as the initial estimate of the solution
vector in the equation solving process. Conceptually, the idea here is to make use of
the domain knowledge embedded in the ANNs to allow for the computation of a very
good initial guess at the solution vector. Clearly, for any iterative method, the ideal
initial solution estimate would be the exact solution since in that case no iteration would
be required.
Since the exact solution is obviously not known, it is typically necessary to use
a simplified scheme to estimate the solution vector. Such schemes include seeding the
solution vector with random numbers, zeros, or values based on the assumption of
diagonal dominance. None of these methods works particularly well for bridge
structures. In the present research, these simplistic methods are replaced by a
sophisticated set of neural networks that can predict very good initial estimates by
exploiting their 'knowledge' of the problem.
In general, the neural networks are not be able to predict the exact set of
displacements that occur in the bridge. Therefore it will be necessary to perform
iterations within the PCG algorithm in order to converge on the exact solution. The
PCG algorithm was specifically chosen for this application because one component of
that algorithm involves the use of an approximate stiffness matrix to precondition the
problem. Preconditioning reduces the effective condition number of the system and thus
increases the rate of convergence of the iterative process. A more detailed discussion of
this phenomenon will be presented later in this work.
Implicitly embodied in the connection weights of the neural networks is the
relationship between applied loads and resulting displacements in flatslab bridge
structures. This is precisely the same relationship that is captured in the more
traditional stiffness matrix of FEA. Since the PCG algorithm calls for an approximation
of the stiffness matrix to precondition the problem, what is actually needed is an
approximation of the relationship between loads and displacements. While that
approximate relationship is usually expressed explicitly in terms of an approximate
stiffness matrix, in the present research it is expressed implicitly within the neural
networks.
Thus, the current application of neural networks seeks to accelerate the equation
solving process by
1. Using the embedded domain knowledge to yield very accurate initial
estimates of the solution.
2. Using the implicit relationship between loads and displacements embodied in
the networks to precondition, and thus accelerate, the convergence of the
PCG solution process.
Detailed descriptions of neural network theory, the representation of bridge data,
network training, and implementation of the trained networks into a PCG solver will be
presented in later chapters.
1.3 Literature Review
The research being reported on herein focuses on three distinct yet strongly
linked topics related to FEA of highway bridge structures. In the following sections the
work of previous researchers in each of these three areas will be surveyed.
1.3.1 Computer Assisted Bridge Modeling
The widespread proliferation of FEA as the tool of choice for solid mechanics
analysis has resulted in the demand for and creation of numerous computer assisted
modeling packages during the past few decades. In the area of structural analysis, these
modeling packages generally fall into one of three general classificationsgeneral
purpose, building oriented, or bridge oriented. Computer assisted preprocessors that are
intended for use in the modeling and analysis of highway bridge structures can be
further classified as commercial packages or research packages.
Software packages for the modeling, analysis, and postprocessing of bridge
structures are often bundled together and distributed or sold as a single system. For this
reason, bundled packages falling into the general category of bridge analysis will be
considered here along with packages which belong to the narrower category of bridge
modeling. Also, because the determination of wheel load distributions on highway
bridges is often needed during both design and evaluation phases, packages that are
aimed at determining such distribution characteristics are also considered here.
Zokaie (1992) performed an extensive review and evaluation of software
capable of predicting wheel load distributions on highway bridges. Included in the
review were general purpose analysis programs such as SAP and STRUDL as well as
specialized bridge analysis programs such as GENDEK, CURVBRG, and MUPDI. In
addition, simplified analysis packages such as SALOD (Hays and Miller 1987) were
also reviewed. Each of the various programs were evaluated and compared primarily
on the basis of the analysis accuracy. However, the modeling capabilities of the
software were not of primary concern in the review.
At present, there are several commercial packages available for bridge modeling
and analysis, however, their modeling capabilities and analysis accuracy vary widely.
The commercial program BARS is widely used by state departments of transportation
(DOTs) throughout the United States. However BARS utilizes a simplified one
dimensional beam model to represent the behavior of the bridge and therefore cannot
accurately account for lateral load distribution between adjacent girders or skewed
bridge geometry.
Another commercial package is CBRIDGE. CBRIDGEand its design
counterpart CBDesignhave the ability to model and analyze straight and curved
girder bridges as well as generate bridge geometry and vehicular loading conditions.
Although CBRIDGE is now a commercial package, the original analysis methods were
developed under funded research programs at Syracuse University. A limitation of the
CBRIDGE package is that the bridge models created do not account for the individual
construction stages of the bridge.
Public domain (noncommercial) programs for finite element modeling and
analysis include the CSTRUCT and XBUILD packages developed by Austin.
CSTRUCT (Austin et al. 1989) is an interactive program developed for the design,
modeling, and analysis of planar steel frames under both static and seismic loading
conditions. Although CSTRUCT is not capable of modeling highway bridges, the
general approach to usersoftware interaction developed in that package was later
extended in the development of the XBUILD bridge modeling system (Austin et al.
1993, Creighton et al. 1990).
The XBUILD package allows a user to interactively build, and simultaneously
view via a graphical interface, finite element models of steel girder highway bridge
structures. XBUILD also allows the user to interactively specify the location and type
of vehicle loading present on the bridge. However XBUILD also has several important
limitations.
1. It can only model steel girder bridges. Thus, the modeling of other types of
bridges such as prestressed concrete girder, reinforced concrete Tbeam, and
flatslab bridges cannot be accomplished.
2. It can only model right (90 degree) bridges having a rectangular finite
element mesh. Thus, neither constant skew nor variable skew bridges can be
modeled.
3. It cannot model the construction stages of the bridge.
In summary, although the XBUILD package provides a user friendly environment for
bridge modeling as well as some powerful graphical features, it is still limited in scope.
While additional bridge modeling packages do exist which have not been
mentioned here, the vast majority of these packages never appear in the literature. This
is due to the fact that such modeling systems are often informal projects developed by
engineering firms strictly for inhouse use.
1.3.2 Data Compression in FEA
During the past few decades a great deal of effort by FEA researchers has been
directed at both optimizing the use of available incore storage in FEA software and
optimizing the numerical efficiency of matrix equation solvers. However, relatively
little attention has been focused on the optimization of outofcore storage
requirements. It is true that researchers have developed various special purpose
bookkeeping strategies that can moderately reduce outofcore storage demands in
specific situations. However, aside from his own work (Consolazio and Hoit 1994), the
author has been unable to find any references in the literature regarding general
purpose strategies directly incorporating the use of data compression techniques in FEA
software.
In contrast, the development of advanced data compression techniques has been
an active area of research in the Computer and Information Science (CIS) field for at
least two decades. In recent years, system software developers have realized the many
fold benefits of using realtime data compression and have begun embedding data
compression directly into the computer operating systems they develop. However, no
such applications of data compression in FEA have appeared in the engineering
literature.
1.3.3 Neural Network Applications in Structural Engineering
During the past five to ten years, there has been a steadily increasing interest in
applying the neural network solution paradigm to structural engineering problems.
VanLuchene and Sun (1990) illustrated some potential uses of neural networks in
structural engineering applications by training networks to perform simple concrete
beam selection and concrete slab analysis. Since that time, researchers have begun
using neural networks in many areas of structural engineering.
Ghaboussi et al. (1991) utilized neural networks for material modeling of
concrete under static and cyclic loading conditions. Neural networks were employed to
capture the materiallevel behavior characteristics (constitutive laws) of concrete using
experimentally collected results as network training data. In this way, the constitutive
laws of the material were derived directly from experimental data and implicitly
embedded in the networks. This is in contrast to the traditional method of formulating a
set of explicit mathematical rules that collectively form the constitutive laws of a
material.
Wu et al. (1992) explored the use of neural networks in carrying out damage
detection in structures subjected to seismic loading. This was accomplished by training
a network to recognize the displacement behavior of a frame structure under various
damage states each of which represented the damage of a single structural component.
Elkordy and Chang (1993) refined this concept by using analytically generated training
data to train networks to detect changes in the dynamic properties of structures.
Accurate prediction of the absolute dynamic properties of a structure by analytical
techniques such as FEA can be very difficult. Instead, Elkordy and Chang used
analytical models to identify changes in dynamic properties. In this way they were able
to train neural networks to detect structural damage by recognizing shifts in the
vibrational signature of a structure. Szewczyk and Hajela (1994) extended the concept
once again by utilizing counterpropagation neural networks instead of the more often
used backpropagation neural networks. Counterpropagation networks can be trained
much more rapidly than traditional "plain vanilla" backpropagation networks and are
therefore well suited for damage detection applications where a large number of
training sets need to be learned.
Several other diverse applications of neural networks in structural engineering
have also appeared in the literature. Garcelon and Nevill (1990) explored the use of
neural networks in the qualitative assessment of preliminary structural designs. Hoit
et al. (1994) investigated the use of neural networks in renumbering the equilibrium
equations that must be solved during a structural analysis. Gagarin et al. (1994) used
neural networks to determine truck attributes (velocity, axle spacings, axle loads) of in
motion vehicles on highway bridges using only girder strain data. Rogers (1994)
illustrated how neural network based structural analyses can be combined with
optimization software to produce efficient structural optimization systems.
CHAPTER 2
A PREPROCESSOR FOR BRIDGE MODELING
2.1 Introduction
This chapter will describe the development and capabilities of an interactive
bridge modeling preprocessor that has been created to facilitate rapid computer assisted
modeling of highway bridges. This preprocessor is one component of a larger system of
programs collectively called the BRUFEM system (Hays et al. 1994). BRUFEM,
which is an acronym for Bridge Rating Using the Finite Element Method, is a software
package consisting of a series of Fortran 77 programs capable of rapidly and accurately
modeling, analyzing, and rating most typical highway bridge structures.
The development of the BRUFEM system was funded by the Florida
Department of Transportation (FDOT) with the goal of creating a computer assisted
system for rating highway bridges in the state of Florida. Bridge rating is the process
of evaluating the structural fitness of a bridge under routine and overload vehicle
loading conditions. With a significant portion of existing highway bridges in the United
States nearing or exceeding their design life, the need for engineers to be able to
accurately and efficiently evaluate the health of such bridges is evident.
Development of the complete BRUFEM system was accomplished in
incremental stages of progress spanning several years and involving the efforts of
several researchers. Early work on the BRUFEM preprocessor was performed by
Selvappalam and Hays (Hays et al. 1990). Subsequently, the author took over
responsibility for the development of the preprocessor and took this portion of the
BRUFEM project to completion.
The four primary component programs that make up the BRUFEM system are
the following.
1. BRUFEM1. An interactive bridge modeling preprocessor.
2. SIMPAL. A core finite element analysis engine.
3. BRUFEM3. An interactive bridge rating postprocessor.
4. SIMPLOT. A graphical postprocessor for displaying analysis results.
The modeling capabilities of the preprocessor, BRUFEM1, will be the focus of this
chapter and also Chapter 3. Enhancements to the FEA program, SIMPAL, using data
compression techniques will be discussed later in Chapter 4. For complete descriptions
of the other component programs, see Hays et al. (1994).
2.2 Overview of the Bridge Modeling Preprocessor
The primary design goal in developing the preprocessor has been to create an
easy to use, interactive tool with which engineers can model complete bridge systems
for later finite element analysis. By using a computer assisted modeling preprocessor,
the usual barriers that would prevent an engineer from manually constructing an FEA
bridge model are overcome. These barriers, which were first introduced in Chapter 1,
are listed below.
1. Discretizing each and every structural component of the bridge into discrete
finite elements and subsequently specifying the characteristicsgeometry,
material properties, connectivities, eccentricities, etc.of each of those
elements.
2. Modeling the structural configuration and the appropriate dead loads at each
distinct stage of construction.
3. Computing potentially hundreds of discrete vehicle positions and
subsequently computing and specifying the load data required for FEA.
Each of these obstacles is overcome through the use of the preprocessor because
it handles these tasks in a semiautomated fashion in which the engineer and the
software both contribute to the creation of the model.
Bridge modeling is accomplished using the preprocessor in an interactive
manner in which the user is asked a series of questions regarding the characteristics of
the bridge being modeled. Each response given by the user determines which questions
will be asked subsequently. For example, assume that the user is asked to specify the
number of the spans in the bridge and a response of '2' is given. Then the user may
later be askeddepending on the type of bridge being modeledto specify the amount
of deck steel effective in negative moment regionsi.e. a parameter that is only
applicable to bridges having more than one span.
Both girderslab bridges and flatslab bridges may be modeled using the
preprocessor. Girderslab bridges are those characterized as having large flexural
elements, called girders, that run in the longitudinal direction of the bridge and which
are the primary means of applied bridge loads. In a girderslab bridge, the girders and
deck slab are often joined together in such a way that they act compositely in resisting
loads through flexure. This type of structural behavior is called composite action and
will be discussed in detail later. Flatslab bridges are constructed as thick slabs lacking
girders and resisting loads directly through longitudinal flexure of the slab.
The preprocessor has been developed so as to allow maximum flexibility with
respect to the types of bridges that can be modeled. Each of the following bridge types
can be modeled using the preprocessor.
1. Prestressed concrete girder. Bridges consisting of precast prestressed
concrete girders, optional reinforced concrete edge stiffeners, a reinforced
concrete deck slab, and reinforced concrete diaphragms.
2. Steel girder. Bridges consisting of steel girders, optional reinforced concrete
edge stiffeners, a reinforced concrete deck slab, and steel diaphragms.
3. Reinforced concrete Tbeam. Bridges consisting of reinforced concrete T
beam girders, optional reinforced concrete edge stiffeners, a reinforced
concrete deck slab, and reinforced concrete diaphragms.
4. Reinforced concrete flatslab. Bridges consisting of a thick reinforced
concrete deck slab and optional reinforced concrete edge stiffeners.
The general characteristics of each these bridge types are built into the preprocessor so
as to allow rapid modeling. Information regarding the construction sequence of each
type of bridge is also embedded in the preprocessor. This information includes not only
the structural configuration of the bridge at each stage of construction but also the
sequence in which dead loads of various types are applied to the bridge.
Finally, the preprocessor allows the engineer to rapidly and easily model live
loading conditions consisting of combinations of vehicle loads and lane loads. Vehicle
data, such as wheel loads and axle spacing, for a wide range of standard vehiclesfor
example the HS20are embedded in the preprocessor. In addition, there are a variety
of methods available to the user for specifying vehicle locations and shifting.
2.3 Design Philosophy of the Preprocessor
In the design of the preprocessor, the basic philosophy has been to exploit
regularity and repetition whenever and wherever possible in the creation of the bridge
model. This idea applies to bridge layout, bridge geometry, girder cross sectional
shape, vehicle configuration, and vehicle movement as well as several other bridge
variables.
2.3.1 Internal Preprocessor Databases
Regularity in the form of standardized bridge components and loading has been
accounted for by using databases. Standard girder cross sectional shapes, such as the
AASHTO girder cross sections, and standard vehicle descriptions are included in
internal databases that make up part of the preprocessor. Thus, instead of having to
completely describe the configuration of, say for example, an HS20 truck, the user
simply specifies an identification symbol, in this case 'HS20', and the preprocessor
retrieves all of the relevant information from an internal database.
The vehicle database embedded in the preprocessor contains all of the
information necessary for modeling loads due to the standard vehicles H20, HS20,
HS25, ST5, SFT, SU2, SU3, SU4, C3, C4, C5 TC95, T112, T137, T150, and T174.
In addition to these standard vehicle specifications, the user may create specifications
for customi.e. nonstandardvehicles by specifying all of the relevant information in
a text data file.
The cross sectional shape databases embedded in the preprocessor contain
complete descriptions of the following standard cross sections used for girders and
parapets.
1. AASHTO prestressed concrete girder types I, II, III, IV, V, and VI
2. Florida DOT prestressed concrete girder bulbT types I, II, III, and IV
3. Standard parapetsold and new standards
In addition to these standard cross sectional shapes, the user may describe nonstandard
cross sectional shapes interactively to the preprocessor.
2.3.2 The Basic Model and Extra Members
Girderslab bridges typically contain a central core of equally spaced girders
that is referred to as the basic model when discussing the preprocessor. In addition to
this central core the bridge may have extra girders at unequal spacings, parapets, and
railings near the bridge edges. The basic model and extra edge members are depicted in
Figure 2.1. Equal girder spacing arises because it simplifies the design, analysis, and
construction of the bridge. Flatslab bridges also contain a central core, or basic model,
in which the deck slab has a uniform reinforcing pattern. While there are no girders in
flatslab bridges, these bridges may have edge elements such as parapets or railings just
as girderslab bridges may.
Almost all of the bridge types considered by the preprocessor utilize the concept
of a basic model to simplify the specification of bridge data. Exceptions to this rule are
the variable skew bridge types in which the concept of a basic model is not applicable.
Within the basic model all bridge parameters are assumed to be constant and therefore
Parape Deck Slab Diaphragm Girder
I I II I I I
bI 2 bsic basic basic S3 S4
Extra Extra Extr Extra
Left Left Basic Model Right Right
Parapet Girder Girder Parape
Figure 2.1 Cross Section of a Girderslab Bridge Illustrating the Basic
Model and Extra Left and Right Edge Members
only need to be specified once by the user. For example, in the bridge shown in
Figure 2.1 notice that the girder spacing Sbaic is constant within the basic model and
that the cross sectional shape of each of the girders in the basic model is the same. In
this case, the user would only need to specify Sbasic once and describe the girder cross
sectional shape once for all four of the girders in the basic model of this bridge.
While the technique of using a basic model to describe a bridge can greatly
speed the process of gathering input from the user, most bridges possess additional
members near the edges that do not fit into the basic model scheme. In the preprocessor
these edge members are termed extra members and are appended to either side of the
basic model to complete the description of the bridge. For example, on each side of the
bridge in Figure 2.1 there is an extra girder and an extra stiffener. In this case the extra
girders have different cross sectional shapes and spacings than the girders of the basic
model. In addition, edge stiffening parapets are present which clearly are different from
the girders of the basic model. In the example shown, the bridge is symmetric but this
need not be the case. By specifying some of the girders in a bridge as extra girders,
unsymmetrical bridges can be modeled. A limit of three extra left members and three
extra right members is enforced by the preprocessor.
2.3.3 Generation
In order to further reduce the amount of time that an engineer must spend in
describing a bridge, the preprocessor performs many types of generation automatically.
Generation in this context means that the user needs only to specify a small set of data
that the preprocessor will use to generate, or create, a much larger set of data needed
for complete bridge modeling. To illustrate the types of generation that the
preprocessor performs, consider the following example.
Bridges containing nonprismatic girders, i.e. girders that have varying cross
sectional shape, can be easily modeled using the preprocessor. To describe a non
prismatic girder, the user only needs to define the shape of the girder cross section at
key definition points. Definition points are the unique locations along the girder that
completely describe the cross sectional variation of the girder. In the example steel
girder illustrated in Figure 2.2, the user only needs to specify the cross sectional shapes
Al through A6 at the six definition points. Using this data, the preprocessor will auto
matically generate cross sectional descriptions of the girder at each of the finite element
nodal location in the model. Also, the preprocessor will generate cross sectional
properties at each of these nodes and assign those properties to the correct elements in
the final model. Thus, the amount of data that must be manually prepared by the
engineer is kept to a minimum.
SI D2 D 3 D 4 D
A, G CCross Section
Figure 2.2 Nonpismatc Steel Girder Bridg WitDefinition
Paints
AI AI A2 A A A A3 A A A
Finite Element
Nodal Location
D = Distances Between Definition Points Nonprismatic
A i Unique Girder Cross Sections Gi
Figure 2.2 Nonprismatic Steel Girder Bridge With User Specified Definition
Points and Finite Element Nodes
The methods by which a user positions vehicles on a bridge provides another
illustration of the types of generation performed by the preprocessor. As will be seen
later, the user needs only to provide a minimal amount of information in order to
generate potentially hundreds of discrete vehicle positions.
2.3.4 The Preprocessor History File
When using a primarily interactive program such as the preprocessor, the
majority of required data is gathered directly from the user, as opposed to being
gathered from an input data file as in a batch program. An interactive approach to data
collection generally results in easier program operation from the viewpoint of the user.
However, one disadvantage of this approach is that because the user has not prepared
an input data file in advance, as is the case in batch programs, there is no record of the
data given by the user. This is undesirable for two reasons. First, there is no permanent
record of what data was specified by the user and therefore there is no 'paper trail' that
can be used to trace the source of an error should one be detected at some later date.
Second, if the user wishes to recreate the bridge model at a future date, all of the
necessary data must again be reentered exactly as before. Similarly, if the user wishes
to recreate the model but with a small variation in some parameter, all of the data must
again be reentered including the modified parameter.
To circumvent these problems, the preprocessor maintains a history file
containing each of the responses interactively entered by the user. Thus, there is a
permanentand commentedrecord of what data was in fact entered by the user
should this ever become a matter of dispute in the future. Since the history file contains
all of the data provided by the user, it may also be used to recreate an entire bridge
model. The user simply tells the preprocessor to read input data from the history file
instead of interactively from the user.
In addition to the uses mentioned above, the history file may also be used to
resume a suspended input session, revise selected bridge parameters, or revise the
vehicle loading conditions imposed on a bridge. Thus, the combination of an interactive
program interface and a reusableand editablehistory file results in a program that
exhibits the advantages of both the interactive and batch approaches without the
exhibiting the disadvantage of each.
2.4 Common Modeling Features and Concepts
Many of the modeling features available in the preprocessor are common to
several of the types of bridges that can modeled. Recall that the preprocessor is capable
of modeling prestressed girder, steel girder, reinforced concrete Tbeam, and flab slab
bridges. The features and concepts discussed below are common to manyor allof
these bridges types.
2.4.1 Bridge Directions
In discussing the preprocessor, the meaning of certain terminology regarding
bridge directions must be established. In this context, the longitudinal direction of a
bridge is the direction along which traffic moves. The lateral direction of the bridge is
the direction perpendicular to and ninety degrees clockwise from the longitudinal
direction. Finally, the transverse direction is taken as the direction perpendicular to the
bridge deck and positive upward from the bridge. These directions are illustrated in
Figure 2.3. The lateral, longitudinal, and transverse bridge directions correspond to the
global x, y, and zdirections respectively in the global coordinate system of the finite
element model.
Transverse
Direction
(ZDirection)
\Lateral
SDirection
w Lateral t (XDirection)
g Direction Longitudinal
Direction
(YDirection)
Figure 2.3 Lateral, Longitudinal, and Transverse Bridge Directions
2.4.2 Zero Skew. Constant Skew. and Variable Skew Bridge Geometry
Bridges modeled using the preprocessor may be broadly divided into two
categories based on the bridge geometryconstant skew and variable skew. A constant
skew bridge is one in which all of the support lines form the same angle with the lateral
direction of the bridge. The constant skew category includes right bridges as a
particular case since the skew angle in a right bridge is a constant value of zero. Right
bridges are those bridges in which the support lines are at right angles to the direction
of vehicle movement. Constant skew geometry, including zero skew, can be modeled
for all of the bridge types treated by the preprocessor.
Variable skew geometry may also be modeled using the preprocessor but only
for steel girder bridges and single span prestressed girder bridges. In a variable skew
bridge, each support line may form a different angle with the lateral direction of the
bridge. Each of the bridge skew cases considered is illustrated in Figure 2.4.
End Support Lines
Interior Support Line
0e=
Right Bidge
(Zero Skew) Constant Skew Bridge Variable Skew Bridge
I I
Constant Skew Geometry
Figure 2.4 Zero Skew, Constant Skew, and Variable Skew Bridge Geometry
2.4.3 Live Load Models and Full Load Models
Broadly speaking, there are two basic classes of bridge models that can be
created by the preprocessorlive load models and full load models. Live load models
are used primarily to compute lateral load distribution factors (LLDFs) for girderslab
bridges (see Hays et al. 1994 for more information regarding LLDFs). A live load
model represents only the final structural configuration of a bridgethat is the bridge
configuration that is subjected to live vehicle loads.
By contrast, a full load model is actually not a model at all but rather a series of
models that represent the different stages of construction of a single bridge. Full load
models are analyzed so that a bridge rating can subsequently be performed using the
analysis results. Each of the individual construction stage models, which collectively
constitute a full load model, simulates a particular stage of construction and the dead or
live loads associated with that stage. After all of the construction stage models have
been analyzed a rating may be performed by superimposing the forcet results from
each of the analyses. This is a very important pointeach analysis considers only
incremental loads, not accumulated loads. In fact, this procedure must be used in order
to account for locked in forces, i.e. forces that are developed at a particular stage of
construction and locked into the structure from that point forward.
The last construction stage model in any series of full load models is always a
live load model, i.e. a model representing the final structural configuration of the
t In this context, the term force is used in a general sense to mean either a shear force,
axial force, bending moment, shear stress, axial stress, or bending stress.
bridge and live loading. When analyzed, the force results from this analysis do not
represent the true forces in the structure but rather the increment of forces due only to
applied live loading. These force results must be combined with the force results from
the other construction stage modelsi.e. the stages that contain dead loadsin order to
determine the actual forces present in the structure.
In the BRUFEM bridge rating system, the superposition of analysis results is
performed automatically by the postprocessor. The analysis results are also factored
according to the type of loading that produced thembefore they are superimposed.
Thus, the preprocessor always creates bridge models that are subjected to unfactored
loads. Load factoring is then performed later in the rating process when the post
processor reads the analysis results.
2.4.4 Live Loads
The term live load is applied to loads that are shortterm in duration and which
do not occur at fixed positions. Live loads on bridge structures are those loads that
result from either individual vehicles or from trains of closely spaced vehicles. Bridges
are typically designed and rated for large vehicles such as standard trucks, cranes, or
special overload vehicles. Two vehicle loading scenarios are generally considered when
modeling highway bridge structuresindividual moving vehicle loads and stationary
lane loads. Both of these conditions can be modeled using the preprocessor.
The first scenario represents normal highway traffic conditions in which
vehicles move across the bridge at usual traffic speeds. In this scheme the vehicles are
assumed to be moving with sufficient speed that, when they enter onto the bridge, there
is an impact effect that amplifies the magnitude of the loads exerted by the vehicle on
the bridge. There may be multiple vehicles simultaneously on the bridge in this
scenario depending on the number of spans, spans lengths, and number of traffic lanes.
To model individual vehicle loads using the preprocessor, the engineer simply
specifies the type, directionforward or reverse, and position of each of the vehicles
on the bridge. Vehicles may be placed at fixed locations, shifted in even increments, or
shifted relative to the finite element nodal locations. If the vehicles are moved using
either of the shifting methods, then the entire vehicle system is shifted as a single
entity. A vehicle system in this context refers to the collection of all vehicles
simultaneously on the bridge.
Vehicles may be positioned and moved on the bridge using any of the following
three methods.
1. Fixed positioning. A single position (location and direction) is specified for
each vehicle on the bridge.
2. Equal shifting. Each vehicle is placed at an initial position and subsequently
shifted a specified number of times in the lateral and longitudinal bridge
directions. The user specifies the incremental shift distances and has the
option of shifting only in the lateral direction, only in the longitudinal
direction, or in both directions.
3. Nodal shifting. Each vehicle is placed at an initial position after which it is
automatically shiftedby the preprocessorin the positive longitudinal
bridge direction such that each axle in the system is in turn placed at each
line of nodes running laterally across the bridge. This option is not available
in constant or variable skew bridge types.
Initial vehicle positions are specified by stating the coordinates of the centerline of the
vehicle's lead axle relative to the lateral and longitudinal directions of the bridge.
The second live loading scenario introduced at the beginning of this section
stationary lane loadingrepresents the case in which traffic is more or less stopped on
the bridge and vehicles are very closely spaced together. Lane loading is usually
thought of as a uniform load extending over specified spans in the longitudinal direction
and over a specified width in the lateral direction. AASHTO defines lane loads as being
ten feet wide. However, because lane loading is intended to represent a series of closely
spaced vehicles, the preprocessor instead models uniform lane loads as a series of
closely spaced axles with each having a width of six feetthe approximate width of a
vehicle axle. Lane loads are described by specifying which spans the lane load extends
over, and by specifying the lateral position of the centerline of the lane.
2.4.5 Line Loads and Overlay Loads
In addition to the live load modeling capabilities provided by the preprocessor,
an engineer may also specify the location and magnitude of long term dead loads such
as line loads and uniform overlays. Dead loads due to structural components such as
the deck slab, girders, and diaphragms are automatically accounted for in the bridge
models created by the preprocessor and therefore do not need to be specified as line or
overlay loads.
Dead loads due to nonstructural elements such as nonstructural parapets or
railings can be modeled by specifying the location and magnitude of line loads. For
example, the dead weight of a nonstructural parapet may be applied to the bridge by
specifying a line load having a magnitude equal to the dead weight of the stiffener per
unit length. Uniform dead loads, such as that which would result from the resurfacing
of a bridge, may be accounted for by specifying a uniform overlay load.
2.4.6 Prismatic and Nonprismatic Girders
Prismatic girderslab bridges, in which the cross sectional shape of the girders
remains the same along the entire length of the bridge, are the simplest type of girder
slab bridge. Prismatic girders are commonly used in prestressed concrete girder bridges
where standard precast cross sectional shapes are the norm. Most reinforced concrete
Tbeam bridges can also be classified as prismatic girderslab bridges.
Nonprismatic girders, in which the cross sectional shape of the girders varies
along the length of the bridge, are commonly used to minimize material and labor costs
in steel girder bridges. The cross sectional shape of a steel girder can be easily varied
by welding cover plates of various sizes to the top and bottom flanges of the girder,
thus optimizing the use of material. Nonprismatic girders are also used in post
tensioned prestressed concrete girder bridges in which thickened girder webs, called
end blocks, are often required at the anchor points of the posttensioning tendons.
Another class of nonprismatic girder occurs when the depth of a girder is varied
usually linearlyalong the length of a girder span. Linearly tapering girders occur in
both steel and prestressed concrete girder bridges.
Prismatic girders can be modeled for all of the bridge types treated by the
preprocessor, either for live load analysis and full load analysis. Nonprismatic girders
are also permitted for the following bridge types.
1. Steel girder. Constant skew steel girder bridges modeled for either live load
analysis or full load analysis and variable skew bridges modeled for full load
analysis.
2. Reinforced concrete Tbeam. Constant skew reinforced concrete Tbeam
bridges modeled for either live load analysis or full load analysis.
3. Prestressed girder. Prestressed girder bridges that are prismatic except for
the presence of end blocks can be modeled for full load analysis. Constant
skew nonprismatic prestressed girder bridges may be modeled for live load
analysis.
Using the preprocessor, the task of describing and modeling the cross sectional
variation of nonprismatic girders has been greatly simplified. Flexible generation
capabilities are provided that minimize the quantity of data that must be manually
prepared by the user. Refer to 2.3.3 for further details.
2.4.7 Composite Action
Composite action is developed when two structural elements are joined in such a
manner that they deform integrally and act as a single composite unit when loaded. In
the case of highway bridges, composite action may be developed between the concrete
deck slab and the supporting girders or between the deck slab and stiffening members
such as parapets. Designing a bridge based on composite action can result in lighter and
shallower girders, reduced construction costs, and increased span lengths.
The extent to which composite action is developed depends upon the strength of
bond that exists between the slab and the adjoining flexural members. In a fully
composite system, strains are continuous across the interface of the slab and the
flexural members and therefore no slip occurs between these elements. Vertical normal
stresses and interface shear stresses are developed at the boundary between the two
elements. Proper development of the interface shear stresses is necessary for composite
action to occur and is provided by a combination of friction and mechanical shear
connection schemes.
In steel girder bridges, as illustrated in Figure 2.5, shear studs are often welded
to the top flanges of the girders and embedded into the deck slab so that the two
elements deform jointly. Concrete girders and parapets may be mechanically connected
to the concrete deck slab by extending steel reinforcing bars from the concrete flexural
members into the deck slab during construction. In each of these shear connection
schemes the goal is to provide adequate mechanical bond between the members such
that they behave as a single composite unit.
In a noncomposite bridge system, there is a lack of bond between the top of the
girder and the bottom of the slab. As a result, the two elements are allowed to slide
relative to each other during deformation and do not act as a single composite unit.
Only vertical forces act between the two elements and there is a discontinuity of strain
at the boundary between the elements.
The preprocessor can represent the presence or absence of composite action in a
bridge by using one of three composite action models. The first model, called the
noncomposite model (NCM), represents situations in which composite action is not
present. The second and third models, termed the composite girder model (CGM) and
the eccentric girder model (EGM) respectively, simulate composite action using two
different finite element modeling techniques. Using the concept of an effective width,
GirderSlab
Interface
Shear Stud Intrface
Shear
I_Steel Girder Stresses
Figure 2.5 Composite Action Between a Girder and Slab
the composite girder model represents composite action by including an effective width
of slab into the calculation of girder cross sectional properties. A more accurate
approach using a pseudo three dimensional finite element model is used in the eccentric
girder model. Additional details of each of the composite action models are given in the
next chapter.
2.5 Modeling Features Specific to Prestressed Concrete Girder Bridges
Several of the modeling features available in the preprocessor relate specifically
to the modeling of prestressed concrete girder bridges (see Figure 2.6). This section
will provide an overview of those features.
2.5.1 Cross Sectional Property Databases
Databases containing cross sectional property data for standard prestressed
concrete girder sections have been embedded into the preprocessor to quicken the
modeling process and reduce errors. The databases contain cross sectional descriptions
of standard AASHTO girders and FDOT bulbT girder types. When modeling a bridge
based on one of these standard girder types, the engineer simply specifies a girder
identification symbol. The preprocessor then retrieves all of the cross sectional data
needed for finite element modeling from an internal database.
This technique saves the user time and eliminates the possibility that he or she
may accidentally enter erroneous data. Since the majority of prestressed concrete girder
bridges are constructed using standard girders, a typical user may never have to
manually enter cross sectional data. To cover cases in which nonstandard girders are
used, the preprocessor also allows the user to manually enter cross sectional data.
2.5.2 Pretensioning and PostTensioning
Prestressed concrete girder bridges modeled by the preprocessor may be either
of the pretensioned type or pre and posttensioned type. Pretensioning occurs during
the process of casting the concrete girders whereas posttensioning occurs after the
girders have been installed in a bridge. The preprocessor can model bridges having
either one or two phases of posttensioning, however, there are specificand distinct
construction sequences associated with each of these schemes.
Figure 2.6 Cross Section of a Typical Prestressed Concrete Girder Bridge
Each type of prestressing, whether it be pretensioning, phase1 posttensioning,
or phase2 posttensioning, is modeled by the preprocessor as a single tendon having a
single area, profile, and prestressing force. In reality, prestressing is usually made up
of many smaller strands located nearby one another so as to form a prestressing group.
This means that when specifying the profile of prestressing strands, the user needs to
specify the profile of the centroid of the prestressing group. Several methods of
describing tendon profiles are provided by the preprocessor including straight profiles,
single and dual point hold down profiles, and parabolically draped profiles.
2.5.3 Shielding of Pretensioning
Pretensioning is used to induce moments into an unloaded girder that will
eventually oppose moments produced by normal bridge loading. In many situations,
however, when normal loads are applied to bridge there is little or no moment at one or
both ends of the girders. In these situations, the pretensioning may be placed in a
profile that has zero eccentricity at the ends of the girder so that zero countermoment
is induced. An alternative is to use a straight pretensioning profile with selected
pretensioning strands being shielded near the ends of the girder.
Shieldingalso known as debondingis the process of preventing bond between
the pretensioning strand and the concrete so as to effectively eliminate a portion of the
pretensioning at a particular location. The preprocessor is capable of modeling
pretensioning shielding. The user must specify the percentages of pretensioning that are
shielded and the distances over which those percentages apply.
2.5.4 PostTensioning Termination
In certain situations, it can be advantageous to terminate posttensioning tendons
at locations other than at the ends of the girders. For example, selected tendons may be
brought out through the top of the girder near, but not at, the end of the girder. The
preprocessor can model early termination of posttensioning tendons in prestressed
concrete girder bridges provided that the user has specified the termination points.
Recall that all posttensioning in a bridge must be represented using either one
or two phases when using the preprocessor. Since each of these phases is represented
using a single tendon, all of the posttensioning for a particular phase must be
terminated at a common location. If multiple tendons are used for a particular phase of
posttensioning and those tendons do not terminate at the same location, then a single
approximate termination point for the entire phase must be determined by the user.
2.5.5 End Blocks
End blocks are regions of a girder in which the web has been significantly
thickened but the overall shape of the cross section remains unchanged. They are often
provided at the ends of prestressed concrete girders to increase the shear capacity of the
cross section and to accommodate the large quantity of reinforcing that is often
necessary at the anchorage points of posttensioning tendons.
The preprocessor models girders containing end blocks as specialcase
nonprismatic girders. End blocks are permitted at the end supports and permanent
interior supports of a bridge. The only information that the engineer must specify is the
thickness and length of each end block. End blocks are assumed to have the same
general shape as the normal girder cross section except for an increased web thickness
that extends some specified length along the end of the girder. A typical end block is
illustrated in Figure 2.7. Actual girders generally have a transition length in which the
web thickness varies from the thickness of the end block to the thickness of the normal
section. This transition length is not actually specified by the user, however, the
preprocessor will model the transition from the end block cross section to the normal
cross section using a single tapered girder element.
2.5.6 Temporary Shoring
There are practical limitations to the length of girders that can be fabricated and
transported. As a result, some bridges are built by employing a construction method in
which more than one prestressed girder is used to form each span. The preprocessor
can model bridges in which each main span is constructed from two individually
I End Block Length ni
Length
End Block I
WWb Widdh
+ End 'Block+
Cetroid Cenrooid
Cross Sectiono idupp To Cro Secon
Of Ed Block .. ,. Of Girder
Figure 2.7 End Block Region of a Prestressed Concrete Girder
prestressed girders that are temporarily supported, bonded together, and then post
tensioned for continuity. This feature of the preprocessor is only available for the
modeling of multiple span bridges and a maximum of one temporary shore per span is
permitted. Finally, all of the girders within a particular span must have the same
condition with respect to whether or not temporary shoring is present. Once the
preprocessor has determined which spans in the bridge contain temporary shoring, it
will create structural models for each stage of construction accounting for the presence
of shoring.
2.5.7 Stiffening of the Deck Slab Over the Girder Flanges
The top flanges of prestressed concrete girders are usually sufficiently thick that
they stiffen the portion of the deck slab lying directly above them. As a result the
lateral bending deformation in the portion of the slab that lies directly over the girder
flanges is markedly less than the deformation of the portion of the slab that spans
between the flanges of adjacent girders. In the models created by the preprocessor, this
stiffening effect is accounted for by attaching lateral beam elements to the slab
elements that lie directly above the girder flanges. The stiffnesses of the lateral beam
elements are computed in such a way that they reflect the approximate bending stiffness
of the girder flange. A more detailed discussion of this modeling procedure is presented
in the next chapter.
2.6 Modeling Features Specific to Steel Girder Bridges
Several of the modeling features available in the preprocessor relate specifically
to the modeling of steel girder bridges (see Figure 2.8). This section will provide an
overview of those features.
2.6.1 Diaphragms
In the steel girder bridge models created by the preprocessor diaphragms are
permitted to be either of the steel beam type or the steel cross brace type. Each of these
types is illustrated in Figure 2.9. The diaphragms connect adjacent girders together but
are not connected to the deck slab between the girders. Structurally, the diaphragms aid
in lateral load distribution, prevent movement of the girder ends relative to one
another, and are assumed to provide complete lateral bracing of the bottom flange in
negative moment regions. If a large number of diaphragms are used, as is often the
case for steel girder bridges, the diaphragms may have a significant effect on lateral
load distribution.
Cross brace diaphragms are constructed from relatively light steel members such
as angles and are often arranged in either an Xbrace configuration or a Kbrace
Conrete Concrete Steel Beam Steel
Parapet Deck Slab Diaphragm Girder
Figure 2.8 Cross Section of a Typical Steel Girder Bridge
configuration. The steel girder bridge type is the only bridge type for which the
preprocessor allows cross brace diaphragms to be modeled. A detailed study was
performed by Hays and Garcelon (see Hays et al. 1994, Appendix I) in which steel
girder bridges were studied using full three dimensional models. The studies indicated
that the behavior of bridges having Xbrace and Kbrace diaphragms were sufficiently
close that Kbrace diaphragms can adequately be modeled using the Xbrace
configuration. Thus, only the Xbrace configuration is modeled by the preprocessor.
The engineer must specify whether beam diaphragms or cross brace diaphragms
will be used and provide the section properties for either the steel beam or the elements
of the cross brace. These section properties are then used for all of the diaphragms in
the bridge. However, in the case of cross brace diaphragms, the depth of the
diaphragms will vary if the depth of the girders vary.
2.6.2 Hinges
Hinged girder connections are occasionally placed in steel girder bridges to
accommodate expansion joints or girder splices. The preprocessor is capable of
Deck Slab Steel Beam LightSteel Elements Light Steel Elements
Beam Diaphragm XBrace Diaphragm KBrace Diaphragm
Cross Brace Diaphragms
Figure 2.9 Diaphragm Types Available for Steel Girder Bridges
modeling hinge connections for constant skew steel girder bridges. Hinges are assumed
to run laterally across the entire width of the bridge, thus forming a lateral hinge line at
each hinge location. Along a hinge line, each of the girders contains a hinge connection
and the deck slab is assumed to be discontinuous. Modeling the slab as discontinuous
across the hinge line is consistent with the construction conditions of an expansion
joint.
2.6.3 Concrete Creep and Composite Action
Long term sustained dead loads on a bridge will cause the concrete deck slab to
creep. Concrete creep is time dependent nonrecoverable deformation that occurs as the
result of sustained loading on the concrete. Over time, the concrete will flow similar to
a plastic material and will incur permanent deformation.
In a steel girder bridge the deck slab and girders are constructed from materials
that have different elastic moduli and different sustained load characteristics. Steel has a
higher elastic modulus than concrete and does not creep under sustained loads as
concrete does. If long term dead loads are applied to the bridge after the concrete deck
slab and steel girders have begun to act compositelyt, the slab will be subjected to
sustained loading and creep will occur. As the deck slab undergoes creep deformation
but the steel girders do notmore and more of the load on the bridge will be carried by
the girders and girder stresses will consequently increase. Creeping of the deck slab
t Long term dead loads that are applied after composite action has begun include the
dead weight of structural parapets, line loadssuch as the weight of a railing or a
nonstructural parapet, and deck overlay loads.
essentially softens the composite girderslab load resisting system and therefore
increases the stresses in the girders.
When modeling steel girder bridges that fit the conditions described above, the
preprocessor will automatically account for the effects of concrete deck creep. Details
of this modeling procedure are presented in the next chapter.
2.7 Modeling Features Specific to Reinforced Concrete TBeam Bridges
Reinforced concrete Tbeam bridges (see Figure 2.10) are modeled very
similarly to prestressed concrete girder bridges except that there is no prestressing
present. A notable exception is that the cross sectional shape of a Tbeam girder is
completely defined by the depth and width of the girder web. A Tbeam girder consists
of a rectangular concrete web joined to the deck slaba portion of which acts as the
girder flange. Thus the engineer simply specifies the depth and width of the web of
each girder when modeling Tbeam bridges. Databases of standard cross sectional
shapes are not needed as was the case in prestressed concrete girder bridges.
Concrete Concrete
Parapet Deck Slab
Concrete
TBeam Web
Web Width
Figure 2.10 Cross Section of a Typical Reinforced Concrete TBeam Bridge
2.8 Modeling Features Specific to FlatSlab Bridges
Flatslab bridges (see Figure 2.11) consist of a thick reinforced concrete deck
slab and optional reinforced concrete edge stiffeners. Thus, unlike all of the other
bridge types modeled by the preprocessor, there are no girders in flatslab bridges.
However, there can still be composite action between the deck slab and edge stiffeners
such as parapets, if such stiffeners are present and considered structurally effective.
Support conditions for flatslab bridges are also unique among the bridge types
modeled by the preprocessor. Flatslab bridges are supported continuously across the
bridge in the lateral direction at each support location. This is in contrast to girderslab
bridges in which supports are only provided for girder elements and the remainder of
the bridge is assumed to be supported by the girders.
Conrete Concrete Flat Slab
Parapet Deck Slab Thickness
Figure 2.11 Cross Section of a Typical Flatslab Bridge
CHAPTER 3
MODELING BRIDGE COMPONENTS
3.1 Introduction
In creating finite element bridge models, the preprocessor utilizes modeling
procedures that have been devised specifically for the types of bridges considered.
Some of the procedures are used to model actual structural components such as girders
and diaphragms whereas others are used to model structural behavior such as composite
action and deck slab stiffening. This chapter will discuss the preprocessor modeling
procedures in detail.
3.2 Modeling the Common Structural Components
The common structural components that are modeled by the preprocessor
include the deck slab, girders, stiffenerssuch as parapets or railings, diaphragms, and
elastic supports. The modeling of these common structural components, which are the
components that are common to several or all of the bridge types considered, will be
discussed below.
3.2.1 Modeling the Deck Slab
Plate bending elements are used to model the bridge deck for the noncomposite
model (NCM) and the composite girder model (CGM). However, in cases where
composite action is modeled with the eccentric girder model (EGM), flat shell elements
are used. The shell element combines plate bending behavior and membrane behavior,
however the membrane response is not coupled with the plate bending response. The
thickness of the slab elements is specified by the engineer and is assumed to be constant
throughout the entire deck except over the girders, where a different thickness may be
specified.
The plate bending elements and the bending (flexural) portion of the flat shell
elements used in the present bridge modeling are based on the ReissnerMindlin thick
plate formulation (Hughes 1987, Mindlin 1951, Reissner 1945). In the Reissner
Mindlin formulation, transverse shear deformations, which can be significant in thick
plate situations such as in flatslab bridge modeling, are properly taken into account.
Consolazio (1990) studied the convergence characteristics of isoparametric elements
based on the thick plate formulation and found that these elements are appropriate for
bridge modeling applications.
While typical isoparametric plate and shell elements may generally have
between four and nine nodes, bilinear (four node) plate and shell elements are used for
all of the bridge models created by the preprocessor. This choice was made for a
number of reasons. Because vehicle axle loads occur at random locations on a bridge,
accurately describing these axle loads requires a substantial number of nodes in the
longitudinal direction. It is generally suggested that when using the preprocessor at
least twenty elements per span be used in the longitudinal direction. Use of biquadratic
(nine node) elements in models following this suggestion would require substantially
more solution time than would models using the simpler bilinear elements. This was
shown to be true by Consolazio (1990) for all but trivially small bridge models.
Another important reason for using bilinear elements instead of biquadratic
elements is related to the fact that both of these elements are known to be rank deficient
when their stiffness matrices are numerically underintegrated. Selective reduced
integration (Bathe 1982, Hughes 1987) is often used to alleviate the problem of shear
locking in plate elements. Shear locking, which results in greatly exaggerated structural
shear stiffness, occurs when elements based on the thick plate formulation are used in
thin plate situations. Selective reduced integration is used to soften the portion of the
element stiffness matrix that is associated with transverse shear.
When underintegrated elements are used in thick plate situations such as the
modeling of a flatslab bridge, zero energy modes may develop which can cause the
global stiffness matrix of the structure to be locally or globally singular (or nearly
singular). Both the bilinear and biquadratic elements suffer from zero energy modes.
However, it has been the author's experience that the mode associated with biquadratic
elements, illustrated in Figure 3.1, is excited far more frequently in bridge modeling
situations than the modes associated with bilinear elements. In fact the biquadratic
 Undefomed Element
SDfonmed Eleoment
In Zero Energy
Mode Configuration
(r.st) Local Element Directions
The tdiretion is the transverse
transnational dircion of the element
Figure 3.1 Zero Energy Mode in a Biquadratic Lagrangian Element
element zero energy mode occurs quite often in the modeling of flatslab bridges and
must be used with great caution in such situations.
One solution to this problem is to use the nine node heterosis element developed
by Hughes (1987) which inherits all of the advantages of using higher order shape
functions without the disadvantage of being rank deficient. Correct rank is
accomplished by utilizing standard lagrangian (ninenode) shape functions for all
element rotational degrees of freedom (DOFs) but serendipity (eightnode) shape
functions for the translational DOFs. Both a nine node standard lagrangian element and
a nine node heterosis element have been implemented by the author in a FEA program
that was developed as part of the present research. In tests on flatslab bridge models,
the heterosis element performed flawlessly in situations where lagrangian elements
suffer from zero energy modes.
However, because there is no translational degree of freedom associated with
the center nodes of heterosis elements, bridge meshing and distribution of wheel loads
is considerably more complex. Thus, a simpler solution is to simply use bilinear
elements and ensure that an adequate number of such elements is used both the lateral
and longitudinal directions of the bridge. This is the solution that has been adopted for
use in the preprocessor.
3.2.2 Modeling the Girders and Stiffeners
Girders and stiffeners are modeled using standard frame elements. Frame
elements consider flexural effects (pure beam bending), shear effects, axial effects, and
torsional effects. Each of these groups of effects are considered separately and are
therefore not coupled.
If the CGM is chosen, composite section properties are used for the elements
representing girders and stiffeners in the bridge. If the NCM is selected then the
noncomposite element properties are used. If the EGM is used, then the noncomposite
girder and stiffener properties are used and the composite action is modeled by locating
the frame elements eccentrically from the centroid of the slab.
In modeling steel girders using frame elements, the transverse shear
deformations in the elements are properly taking into account. Hays and Garcelon
(Hays et al. 1994, Appendix I) found that, when using the type of models created by
the preprocessor, shear deformations in the girders must be considered for the analysis
to be accurate. This conclusion was based on a study comparing the response of models
created by the preprocessor and the response of fully three dimensional models. Shear
deformations are not, and do not need to be, accounted for in concrete girders or
concrete parapets where such deformations are typically negligible.
The term stiffener, as used in this research, refers to structural elements such as
parapets, railings, and sidewalks that reside on the bridge deck. Stiffeners can improve
the load distribution characteristics of bridges by adding stiffness to the bridge deck,
usually near the lateral edges.
3.2.3 Modeling the Diaphragms
Diaphragms are bridge components that connect girders together so as to
provide a means of transferring deck loads laterally to adjacent girders. In prestressed
concrete girder bridges and R/C Tbeam bridges, the diaphragms are assumed to be
constructed as concrete beams and are thus modeled using frame elements. Beam
diaphragms are assumed to not act compositely with the deck slab. This is true whether
or not composite action is present between the girders, stiffeners, and deck slab.
Therefore the diaphragm elements in concrete girder bridges are located at the elevation
of the centroid of the slab, as illustrated in Figure 3.2. In this manner, the diaphragm
elements assist in distributing load laterally but do not act compositely with the deck
slab.
In steel girder bridges, diaphragms may be either steel beams or cross braces
constructed from relatively light steel members called struts. Steel beam diaphragms,
shown in Figure 3.2, are modeled in the same manner that concrete diaphragms are
modeled. Cross brace diaphragms, however, are modeled using axial truss elements
representing the strutsthat are located eccentrically from the centroid of the slab. The
struts are located eccentrically from the finite element nodes regardless of whether or
not composite action is present between the girders, stiffeners, and deck slab. Truss
eccentricities are computed as the distances from the centroid of the slab to the top and
Finite
Deck Concrete Concrete Deck Steel Stel Diaphragm Elemnt
Slab Diaphragm Girder Slab Diaphragm Girder Elements Ndes
Concrete Girder Bridge Steel Girder Bridge Elevation of Cntroid Of Deck Slab
Figure 3.2 Modeling Beam Diaphragms
bottom strut connection points, as is shown in Figure 3.3. Thus, the centroid of the
deck slab is used as a datum from which eccentricities are referenced. Of primary
importance in computing these eccentricities is correctly representing the slopes of the
cross struts, since these slopes determines how effective the diaphragm will be in
transferring loads laterally.
Finally, recall that the preprocessor models both Xbrace and Kbrace
diaphragms using the Xbrace configuration for the reasons that were discussed in the
previous chapter.
3.2.4 Modeling the Supports
Bridge models created by the preprocessor use axial truss elements to model
elastic spring supports rather than using rigid supports. In girderslab bridges, vertical
support is usually provided by elastomeric bearing pads located between the ends of the
girders and the abutments and at interior support piers. Bearing pads are modeled using
elastic axial truss elements of unit length, unit elastic modulus, and a cross sectional
area that results in the desired support stiffness. By default the preprocessor will
automatically provide reasonable values of bearing pad stiffness (see Hays et al. 1994
oi t 't Axial Tuss_
ut Elements
Figure 3.3 Modeling Cross Brace Diaphragms
for details) however the engineer may manually adjust these values if detailed bearing
stiffness data are available. In addition to vertical supports, horizontal supports must
also be provided to prevent rigid body instability of the models at each stage of
construction. Horizontal support is provided through finite element boundary condition
specification rather than by using elastic supports.
Flatslab bridges are supported continuously in the lateral direction at each
support in the bridge. Since bearing pads are not typically used in flatslab bridge
construction the support stiffnesses cannot not be easily determined. However, the
preprocessor assumes a reasonable value of bearing stiffness, which again can be
manually adjusted by the engineer if better data are available.
3.3 Modeling Composite Action
Composite action is developed when two structural elements are joined in such a
way that they deform together and act as a single unit when loaded. In the case of
bridges, composite action can occur between the concrete slab and the supporting
concrete or steel girders. The extent to which composite action can be developed
depends upon the strength of bond that exists between the slab and the girders.
Composite action may also occur between stiffeners and the deck slab. In a composite
system there is continuity of strain at the slabgirder interface and therefore no slip
occurs between these elements. Horizontal shear forces and vertical forces are
developed at the boundary between the two elements. The interaction necessary for the
development of composite action between the slab and the girder is provided by friction
and the use of mechanical shear connectors.
In a noncomposite girderslab system, there is a lack of bond between the top of
the girder and the bottom of the slab. As a result, the two elements are allowed to slide
relative to each other during deformation and do not act as a single composite unit.
Only vertical forces act between the two elements and there is a discontinuity of strain
at the boundary between the elements.
The preprocessor allows the engineer to model the girderslab interaction as
either noncomposite, or as composite using one of two composite modeling techniques.
The girderslab interaction models available in the preprocessor are illustrated in
Figure 3.4.
Noncomposite action is modeled using the noncomposite model (NCM) in
which the centroid of the girder is effectively at the same elevation as the centroid of
the slab. The section properties specified for the girders are those of the bare girders
alone. In this model the primary function of the slab elements is to distribute the wheel
loads laterally to the girders, therefore plate bending elements are used to model the
deck slab.
Composite action between the slab and the girder is modeled in one of two ways
using the preprocessor. One way involves the use of the composite girder model
(CGM) and the other the eccentric girder model (EGM). These composite action
models are also illustrated in Figure 3.4.
Effective Width
'I H i st Shell)
Centroid of
Girder Alone Eccentricty
Slab Elements Centmid of (Plate Bending) OfGirder
(Plate Bending) Composite Aloe
Section A
Noncomposite Composite Girder Eccentric Girder
Model (NCM) Model (CGM) Model (EGM)
Figure 3.4 Modeling Noncomposite Action and Composite Action
3.3.1 Modeling Composite Action with the Composite Girder Model
One method of modeling composite action is by utilizing composite properties
for the girder elements. The centroid of the composite girder section is at the same
elevation as the midsurface of the slab in the finite element model. A composite girder
section is a combination of the bare girder and an effective width of the slab that is
considered to participate in the flexural behavior.
Due to shear strain in the plane of the slab, the compressive stresses in the
girder flange and slab are not uniform, and typically decrease in magnitude with
increasing lateral distance away from the girder web. This effect is often termed shear
lag. In certain cases of laterally nonsymmetric bridge loading, the compressive stress in
the deck may vary such that the stress is higher at the edge of the bridge than over the
centerline of a girder. An effective slab width over which the compressive stress in the
deck is assumed to be uniform is used to model the effect of the slab acting compositely
with the girder. The effective width is computed in such a way that the total force
carried within the effective slab width due to the uniform stress is approximately equal
to the total force carried in the slab under the actual nonuniform stress condition.
In order to compute composite section properties, the effective width must be
determined. Standard AASHTO recommendations are used to compute the effective
width for the various bridge types that can be modeled using the preprocessor. In
computing composite girder properties, the width of effective concrete slab that is
assumed to act compositely with the girder must be transformed into equivalent girder
material. This transformation is accomplished by using the modular ratio, n, given by
=E (3.1)
where Ec is the modulus of elasticity of the concrete slab and Eg is the modulus of
elasticity of the girder. For steel girders the modulus of elasticity is taken as 29,000
ksi. For concrete, the modulus of elasticity is computed based on the concrete strength
using the AASHTO criteria for normal weight concrete.
When using the composite girder model, composite action is approximated by
using composite section properties for the girder members. The primary function of the
slab elements in the CGM finite element model is to distribute wheel loads laterally to
the composite girders, thus plate bending elements are used to model the deck slab.
3.3.2 Modeling Composite Action with the Eccentric Girder Model
The second method available for modeling composite action involves the use of
a pseudo three dimensional bridge model that is called the eccentric girder model
(EGM). In this model, the girders are represented as frame elements that have the
properties of the bare girder cross section but which are located eccentrically from the
slab centroid by using rigid links. By locating the girder elements eccentrically from the
slab, the girder and slab act together as a single composite flexural unit. In general,
each componentthe slab and the girdermay undergo flexure individually and may
therefore sustain moments. However, because the components are coupled together by
rigid links, the composite section resists loads through the development not only of
moments but also of axial forces in the elements.
Rigid links, also referred to as eccentric connections, are assumed to be
infinitely rigid and therefore can be represented exactly using a mathematical
transformation. Thus, by using the mathematical transformation, additional link
elements do not need be added to the finite element model to represent the coupling of
the slab and girder elements.
In the eccentric girder model, shear lag in the deck is properly taken into
account because the deck slab is modeled with flat shell elementselements created by
the superposition of plate bending elements and membrane elements. Because the slab
and girders are eccentric to one another and because they are coupled together in a
three dimensional sense, the EGM is referred to as a pseudo three dimensional model.
It is not a fully three dimensional model because the coupling is accomplished through
the use of infinitely rigid links. In an actual bridge the axial force in the slab must be
transferred to the girder centroid through a flexiblenot infinitely rigidgirder web. In
a fully three dimensional model, the girder webs would have to be modeled using shell
elements as was done by Hays and Garcelon (Hays et al. 1991). Therefore the models
created by the preprocessor are pseudo three dimensional models.
The main deficiency of using rigid links occurs in modeling weak axis girder
behavior. The use of rigid links causes the weak axis moment of inertia of the girders
to become coupled to the rotational degrees of freedom of the deck slab. This coupling
will generally result in an overestimation of the lateral stiffness of the girders. To avoid
such a problem the preprocessor sets the weak axis moment of inertia of the girders to a
negligibly small value. This procedure allows rigid links to be used in modeling
composite action under longitudinal bridge flexure but does not result in overestimation
of lateral stiffness. Since the preprocessor models bridges for gravity and vehicle
loading and not for lateral effects such as wind or seismic loading, this procedure is
reasonable.
Illustrated in Figure 3.5 is the eccentric girder model for a girderslab system
consisting of a concrete deck slab and a nonprismatic steel girder. The system is
assumed to consist of multiple spans of which the first span is shown in the figure. In
modeling the slab and girder, a total of six elements have been used in the longitudinal
direction in the span shown. A width of two elements in the lateral direction are shown
modeling the deck slab. Nodes in the finite element model are located at the elevation
of the slab centroid. The girder elements are located eccentrically from the nodes using
rigid links whose lengths are the eccentricities between the centroid of the slab and the
centroid of the girder. Because the girder is nonprismatic, the elevation of the girder
centroid varies as one moves along the girder in the longitudinal direction. For this
Deck Slab Centroid Of
(Shell) Element Shell Elements
; yi i I IEccntricity
Ar Girder
(Frame) Girder Rigid Link So AA
Element Centrid (Eccentricity) Set A
 Finite Element Node
/ Flat Shell Element
SEccentricity Betceen
Slan Ce atro And
Girder Certroid
Figure 3.5 Modeling Composite Action with the Eccentric Girder Model
reason the lengths of the rigid linksi.e. the eccentricities of the girder elementsalso
vary in the longitudinal direction. Displacements at the ends of the girder elements are
related to the nodal displacement DOF at the finite element nodes by rigid link
transformations.
Slab elements, modeled using flat shell elements, are connected directly to the
finite element nodes without eccentricity. Recall that in the EGM the slab elements are
allowed to deform axially as are the girder elements. In this manner the slab and girder
elements function jointly in resisting load applied to the structure. Since the slab
elements must be allowed to deform axially a translating roller support is provided at
the end of the first span. By using a roller support, the girder and slab are permitted to
deform axially as well as flexurally and can therefore act compositely as a single unit.
The EGM composite action modeling technique is generally considered to be
more accurate than the CGM modeling technique. This is because in the CGM an
approximate effective width must be used in the determination of the composite section
properties. However, while the EGM is more accurate, the analysis results must be
interpreted with greater care since the effect of the axial girder forces must be taken
into consideration when the total moment in the girder section is determined. Also,
when using the EGM, it is necessary to use a sufficient number of longitudinal
elements to ensure that compatibility of longitudinal strains between the slab and girder
elements is approached (Hays and Schultz 1988). It is therefore recommended that at
least twenty elements in the longitudinal direction be used in each span.
3.4 Modeling Prestressed Concrete Girder Bridge Components
The modeling of structural components and structural behavior that occur only
in prestressed concrete girder bridges will be described in the sections below.
3.4.1 Modeling Prestressing Tendons
Prestressed concrete girder bridges make use of pretensioning and post
tensioning tendons to precompress the concrete girders, thus reducing or eliminating
the development of tensile stresses. The tendons used for pretensioning and post
tensioning of concrete will be referred to collectively as prestressing tendons in this
context. Prestressed bridges are pretensioned and may optionally be posttensioned in
either one or two phases when using the preprocessor. Posttensioned continuous
concrete girder bridges are modeled assuming that the girders are pretensioned, placed
on their supports and then posttensioned together to provide structural continuity.
In order to model prestressing tendons using finite elements, both the tendon
geometry and the prestressing force must be represented. Tendons are modeled as axial
Girder Prestressing Rigid Prestressing Finit
(Fme) Tendon (Truss) Link Tendon Element
Element Element (Eccentricity) Centroid Node
Prestressing irder
Hold Don Points Cros Setion
Figure 3.6 Modeling the Profile of a Prestressing Tendon
truss elements that are eccentrically connected to the finite element nodes by rigid links
(see Figure 3.6). Since straight truss elements are used between each set of nodes in the
tendon, a piecewise linear approximation of the tendon profile results. The tendon is
divided into a number of segments that is equal to the number of longitudinal elements
per span specified by the user. As long as a reasonable number of elements per span is
specified, this method of representing the profile will yield results of ample accuracy.
The reference point from which tendon element eccentricities are specified in
the model varies depending on the particular type of composite action modeling being
used and on the particular construction stage being modeled. In the noncomposite
model (NCM) tendon element eccentricities are always referenced to the centroid of the
barei.e. noncompositegirder cross section. In the composite model (CGM), for
construction stages where the slab and girder are acting compositely, the eccentricities
are referenced to the centroid of the composite girder cross section which includes an
effective width of deck slab. Eccentricities in the eccentric girder model (EGM) for
construction stages where the slab and girder are acting compositely are referenced to
the midsurface of the slab. Prestressing element eccentricities for construction stages in
which the deck slab is structurally effective are illustrated in Figure 3.7. In construction
stages where the deck slab has not yet become structurally effective, the tendon
eccentricities are always referenced to the centroid of the bare girder cross section
regardless of which composite action model is being used.
When using the preprocessor, the engineer always specifies the location of the
prestressing tendons relative to the top of the girder. With this data, the preprocessor
computes the proper truss element eccentricities for each construction stage of the
bridge based on the composite action model in effect. The example girder that is shown
in Figure 3.6 has a piecewise linear tendon profile tendon created by dual hold down
points. Note however that the preprocessor is capable of approximating any tendon
profile, linear or not, as a series of linear segments.
In addition to modeling the profile of the prestressing tendons, the prestressing
force must also be represented. This is accomplished simply by specifying an initial
tensile force for each tendon (truss) element in the model. Since the tendons are
modeled using elastic truss elements, prestress losses due to elastic shortening of the
concrete girder are automatically accounted for in the analysis. However, nonelastic
Nonconposie Composite Girder Eccentric Girder
Model (NCM) Model (CGM) Model (EGM)
Figure 3.7 Modeling the Profile of a Prestressing Tendon
losses due to effects such as friction, anchorage slip, creep and shrinkage of concrete,
and relaxation of tendon steel are not incorporated into the model. (In the BRUFEM
system, these nonelastic losses are accounted for automatically by the postprocessor
based on the appropriate AASHTO specifications). Thus, the tendon forces specified by
the engineer must be the initial pretensioning or posttensioning forces in the tendons,
prior to any losses.
3.4.2 Increased Stiffening of the Slab Over the Concrete Girders
During lateral flexure in prestressed (and also reinforced concrete Tbeam)
girder bridges, the relatively thin slab between the girders deforms much more than the
portion of the slab over the flanges of the girders. This behavior is due to the fact that
the girder flange and web have a stiffening effect on the portion of the slab that lies
directly above the girder. Rather than using thicker plate elements over the girders,
lateral beam elements are used to model this stiffening effect. These lateral beam
elements are located at the midsurface of the slab and extend over the width of the
girder flanges, as is shown in Figure 3.8.
Slab Lateral Beam _
Elemen\ I / Elements Slab
SElements
L / Girder
ThickneOf / Elements
Tnicness Of sy
Slab Over Girder Lateral
Thcktness Of Bm
Slab Over Girder Elements
Plus Effective Range 4
Thickne L
Cross Sectional View Plan View
Figure 3.8 Modeling the Stiffening Effect of the Girder Flanges on the Deck Slab
The lateral bending stiffness of these elements is assumed to be that of a wide
beam of width SY (SY/2 for elements at the ends of the bridge). From plate theory
(Timoshenko and WoinowksyKrieger 1959) the flexural moment in a plate is given by
Et3
Mx= x + 4 ) (3.2)
12( 1 v2)
where M, is the moment per unit width of plate, E is the modulus of elasticity, t is
the plate thickness, v is Poisson's ratio, and x, and ,y are the curvatures in the x
and ydirections respectively. Since the value of Poisson's ratio for concrete is small
(v= 0.15), it can be assumed that the quantity (1 v) is approximately unity. Also,
since only bending in the lateral direction (xdirection) is of interest for the lateral beam
members, only the xcurvature x, is taken into consideration. From Equation (3.2)
and the simplifications stated above, the moment of inertia of a plate element having
thickness t is
I (SY) (3.3)
12
Since the moment of inertia of the slab is automatically accounted for through the
inclusion of the plate elements in the bridge model, the effective moment of inertia of
the lateral beam element is given by
I =t(+f) t (SY) (3.4)
12
where t(sg+ef) is the thickness of the slab over the girder plus the effective flange
thickness of the girder and t, is the thickness of the slab of the girder.
The torsional moment of inertia of the lateral beam members is obtained in a
similar manner. From plate theory the twisting moment in a plate of thickness t is
given by
SGt3 (3.5)
6y 6
where G is the shear modulus of elasticity, and 4y is the cross (or torsional)
curvature in the plate. Thus, the effective torsional moment of inertia of the lateral
beam elements is given by
J sg+ef) t (SY) (3.6)
6
where the parameters t(sg+ef) and tg are the same as those described earlier.
3.5 Modeling Steel Girder Bridge Components
The modeling of structural components and structural behavior that occur only
in steel girder bridges will be described in the sections below. One of the areas of
modeling that is specific to steel girder bridges is that of modeling cross brace steel
diaphragms. However, since this topic was already considered in 3.2.3 in a general
discussion of diaphragm modeling, it will not be repeated here.
3.5.1 Modeling Hinges
Hinged girder connections are occasionally placed in steel girder bridges to
accommodate expansion joints or girder splices. Hinges are assumed to run laterally
across the entire width of the bridge, thus forming a lateral hinge line at each hinge
location. Along a hinge line, each of the girders contains a hinge connection and the
deck slab is made discontinuous.
When using the preprocessor, the engineer inserts hinges into the bridge by
specifying the distances from the beginning of the bridge to the hinges. If the hinge
distances specified do not match the locations of finite element nodal lines, then the
hinge lines are moved to the location of the nearest nodal line. Also, note that the
insertion of hinges into a bridge must not cause the structure to become unstable. For
example, one may not insert a hinge into a single span bridge since this would result in
an unstable structure.
Hinge modeling is accomplished by placing a second set of finite element nodes
along the hinge line at the same locations as the original nodes. In Figure 3.9, this is
depicted by showing a small finite distance between the two set of nodes at the hinge
line. In the actual finite element bridge model the distance between the two lines of
nodes is zero. Girder, stiffener, and slab elements on each side of the hinge line are
then connected only to the set of nodes on their corresponding side of the hinge. At this
point the bridge is completely discontinuous across the hinge line.
Figure 3.9 Modeling a Hinge in a Steel Girder Bridge
Girders are the only structural components assumed to be even partially
continuous across hinges. The deck slab and stiffeners are assumed to be completely
discontinuous across hinges. Girder are continuous with respect to vertical translation
and, in some cases, axial translation but not flexural rotation. As a result, the end
rotations of the girder elements to either side of a hinge are uncoupledi.e. a hinge is
formed. Displacement constraints are used to provide continuity at the points where
girder elements cross a hinge line. In bridges modeled using the NCM or CGM
composite action models, the vertical translations of the nodes that connect girder
elements across a hinge line are constrained. When the EGM composite action model is
used, all three translations must be constrained due to the axial effects in the model.
Nodes to which girder elements are not connected are left unconstrained and therefore
are allowed displace independently.
3.5.2 Accounting for Creep in the Concrete Deck Slab
As was explained in the previous chapter, long term sustained dead loads on a
bridge will cause the concrete deck slab to creep. In steel girder bridges, the deck slab
and girders are constructed from materials that have different elastic moduli and
therefore different sustained load characteristics. As the concrete slab creeps over time,
increasingly more of the dead loads will be carried by the steel girders. Since the
models created by the preprocessor are not time dependent finite element models, the
creep effect must be accounted for in some approximate manner. Depending on the
composite action model being used, creep is accounted for in one of the ways discussed
below.
If the CGM is being used to represent composite action, then the creep effect is
accounted for when computing composite section properties for the girders. Normally,
when composite section properties are computed, the effective width of concrete slab
that acts compositely with the girders is transformed into an equivalent width of steel
by dividing by the modular ratio. This equivalent width of steel is then included in the
computation of composite section properties for the girder. The modular ratio is a
measure of the relative stiffnesses of steel and concrete and is given by
Ec
Eg (3.7)
where Ec is the modulus of elasticity of the concrete deck slab and Eg is the modulus
of elasticity of the steel girders. In order to account for the increased deformation that
will arise from creeping of the deck, the preprocessor uses a modified modular ratio
when computing composite girder properties. When transforming the effective width of
concrete slab into equivalent steel, a modular ratio of 3n is used instead of n. This
yields a smaller width of equivalent steel and therefore smaller section properties.
Because the section properties are reduced, the stiffness of the girders are reduced,
deformations increase, and stresses in the girders increase.
If the EGM is used to represent composite action, then the effect of creep is
accounted for by employing orthotropic material properties in the slab elements. In the
EGM, the deck slab is modeled using a mesh of shell elements. By using orthotropic
material properties for these shell elements, different elastic moduli may be specified
for the longitudinal and lateral directions. To account for creep, the elastic modulus of
the shell elements is divided by a factor of 3.0 in the longitudinal direction, but left
unmodified in the lateral direction. Using this technique, the effects of creep, which
relate primarily to stresses in the longitudinal direction, are accounted for without
disturbing the lateral load distribution behavior of the deck.
If the noncomposite model (NCM) is used, then the girders and deck slab do not
act compositely and the girders are assumed to carry all of the load on the bridge. Since
the girders carry all of the load, the effects of creep in the concrete deck will be
negligible and therefore no special modeling technique is necessary.
3.6 Modeling Reinforced Concrete TBeam Bridge Components
Reinforced concrete (R/C) Tbeam bridges are modeled by the preprocessor in
essentially the same manner that prestressed concrete girder bridges are modeled. The
obvious exception to this statement is that R/C Tbeam bridges lack the prestressing
(pretensioning and possibly posttensioning) tendons that are present in prestressed
concrete girder bridges. However, the girders in each of these bridge types are
constructed from the same materialconcreteand are therefore modeled in the same
manner. Diaphragms have precisely the same configuration in both of these bridge
types and are therefore modeled in identical fashion. Finally, the deck slab stiffening
effect that was discussed in 3.4.2 in the context of prestressed concrete girders also
occurs in R/C Tbeam bridges. In R/C Tbeam bridges, this stiffening effect is
represented using the same lateral beam elements that were discussed earlier for
prestressed concrete girders.
3.7 Modeling FlatSlab Bridge Components
A flatslab bridge consists of a thick reinforced concrete slab and optional edge
stiffeners such as parapets. If stiffeners are present and structurally effective, they are
modeled using frame elements as is the case for girderslab bridges. If stiffeners are not
present on the bridge or are not considered structurally effective, then the slab is
modeled using plate elements and the entire bridge is represented as a largepossibly
multiple spanplate structure. When stiffeners are present on the bridge but do not act
compositely with the slaband are therefore not considered structurally effectivethey
should be specified as line loads by the engineer.
If stiffeners are considered structurally effective, then the engineer can choose
either the CGM or EGM of composite action. If the CGM is chosen, then the slab is
modeled using plate elements and composite section properties are computed for the
stiffener elements using the effective width concept. If the EGM is chosen, the slab is
modeled using shell elements and the stiffeners are located eccentrically above the flat
slab using rigid links. The NCM is not available for flatslabs because if sufficient bond
does not exist between the stiffeners and slab to transfer horizontal shear forces, it
cannot be assumed that there is sufficient bond to transfer vertical forces either. In
order for stiffenerswhich are above the slabto assist in resisting loads, there must
be sufficient bond to transfer vertical forces to and from the slab.
3.8 Modeling the Construction Stages of Bridges
To properly analyze a bridge for evaluation purposes, such as in a design
verification or the rating of an existing bridge, each distinct structural stage of
construction must be represented in the model. Using the preprocessor, this can be
accomplished by creating a full load modela model in which the full construction
sequence is considered. Each of the individual construction stage models, which
collectively constitute the full load model, simulates a particular stage of construction
and the dead or live loads associated with that stage.
Modeling individual construction stages is very important in prestressed
concrete girder bridges, important in steel girder bridges, and of lesser importance in
R/C Tbeam and flatslab bridges. For each of these bridge types, the preprocessor
assumes a particular sequence of structural configurations and associated loadings.
These sequences will be briefly described below, however, for complete and highly
detailed descriptions of each sequence see Hays et al. (1994).
Several different types of prestressed concrete girder bridges may be modeled
using the preprocessor. These include bridges that have a single span, multiple spans,
pretensioning, one phase of posttensioning, two phases of posttensioning, and
temporary shoring. Each of these variations has its own associated sequence of
construction stages the highlights of which are described below.
All prestressed girder bridges begin with an initial construction stage in which
the girders are the only components that are structurally effective. In multiple span
prestressed concrete girder bridges, the girders are not continuous over interior
supports at this stage but rather consist of a number of simply supported spans. At this
stage, the bridge is subjected to pretensioning forces and to dead loads resulting from
the weight of the girders, diaphragms, andin most casesthe deck slab. In
subsequent construction stages the bridge components that caused dead loads in this
first stage, e.g. the diaphragms and deck slab, will become structurally effective and
will assist the girders in resisting loads due to posttensioning, temporary support
removal, stiffener dead weight, line loads, overlay loads, and ultimately vehicle loads.
Prestressed concrete girder bridges may go through a construction stage that
represents the effect of additional long term dead loads acting on the compound
structure. Loads that are classified as additional long term dead loads include the dead
weight of the stiffeners, line loads, and overlay loads. The compound structure is
defined to be the stage of construction at which the deck slab has hardened and all
structural components, except for stiffeners, are structurally effective. The term
compound is used to refer to the fact that the various structural components act as a
single compound unit but implies nothing with regard to the composite or noncomposite
nature of girderslab interaction. Since the deck slab has hardened at this construction
stage, the girders and deck slab may act compositely. Also, lateral beam elements are
included in the bridge model to represent the increased stiffening of the girder on the
deck slab.
As construction progresses from one stage to the next, the bridge components
become structurally effective in the following ordergirders, pretensioning,
diaphragms, deck slab, lateral beam elements, phase1 posttensioning (if present),
stiffeners, and phase2 posttensioning (if present). The final construction stage is
represented by a live load model, i.e. a model which represents the final structural
configuration of the bridge and its associated live loading. At this stage, each and every
structural component of the bridge is active in resisting loads and the loads applied to
the bridge are those resulting solely from vehicle loading and lane loading.
Steel girder bridges have simpler construction sequences than prestressed
concrete girder bridges due to the lack of prestressing and temporary shoring. Steel
girder construction sequences begin with a construction stage in which the girders and
diaphragms are assumed to be immediately structurally effective. The bridge model
consists only of girder elements and diaphragm elements which, acting together, resist
the dead load of the girders, diaphragms, and the deck slab.
It is assumed that the girders in multiple span steel girder bridges are
immediately continuous over interior supports. The assumption of immediate continuity
of the girders is reasonable since multiple span steel bridges are not typically
constructed as simple spans that are subsequently made continuous as is the case in
prestressed concrete girder bridges. It is also assumed that the diaphragms in steel
girder bridges are installed and structurally effective prior to the casting of the deck
slab.
Steel girder bridges may go through a construction stage that represents
additional long term dead loads acting on the compound structure just as was described
above for prestressed concrete girder bridges. Since the deck slab has hardened at this
construction stage, the girders and deck slab may act compositely. However, lateral
beam elements are not used in steel girder bridge models as they are in prestressed
concrete girder bridge models. Also, in steel girder bridges, the effects of concrete
deck creep under long term dead loading must be properly accounted for. This is
accomplished by using the techniques presented in 3.5.2.
In the final (live load) construction stage of steel girder bridges, each and every
structural component of the bridge is active in resisting loads. At this stage, cracking of
the concrete deck in negative moment regions of multiple span steel girder bridges may
be modeled by the preprocessor. This deck crackingthe extent of which is specified
by the engineeris assumed to occur only in the final live load configuration of the
bridge. In modeling deck cracking, the preprocessor assumes a region in which
negative moment is likely to be present. This assumption is necessary since only after
the finite element analysis has been completed will the actual region of negative
moment be known. Thus, the preprocessor assumes negative moment will be present in
regions of the bridge that are within a distance of twotenths of the span length to either
side of interior supports. See Hays et al. (1994) for further details of the cracked deck
modeling procedure used by the preprocessor.
In R/C Tbeam and flatslab bridges, the construction sequence is not of great
significance and has little effect on the analysis and rating of the bridge. During the
construction of these bridge types, the bridge often remains shored until all of the
bridge components have become structurally effective. In such situations, all of the
structural components become structurally effective and able to carry load before the
shoring is removed. The preprocessor therefore assumes shored construction for these
two bridge types.
Based on the shored construction assumption, all of the structural elements used
to model R/C Tbeam and flat bridges are assumed to be immediately structurally
effective in resisting dead load. Thus, there exists only a single dead load construction
stage in which all of the dead loadsincluding additional long term dead loadsare
applied to the fully effective structural bridge model. The final live load construction
stage consists of the same structural model as that used in the dead load stage, but
subjected to vehicle and lane loading instead of dead loading.
3.9 Modeling Vehicle Loads
Using the vehicle live loading features provided by the preprocessor, an
engineer can quickly describe complicated live loading conditions with relative ease. In
describing live loading conditions, the engineer only needs to specify the type, location,
and direction of each vehicle on the bridge and the location and extent of lane loads.
Once these data have been obtained, the preprocessor can generate a complete finite
element representation of the live loading conditions.
Recall from the previous chapter that the preprocessor models lane loads not as
uniform loads on the deck slab, but instead as trains of closely spaced axles. Thus, both
individual vehicles and lane loads are modeled using collections of axles. To model
these live loads using the finite element method, the preprocessor must convert each
axle, whether from a vehicle or a lane load, into an equivalent set of concentrated nodal
loads that are applied to the slab nodes in the model. In order to accomplish this
conversion to nodal loads the multiple step procedure illustrated in Figure 3.10 is
performed by the preprocessor.
81
Vehicle or
moidge Axle VWheel Lod
Vh ic I dnast Geometry Distribution
Axle Loads  Wheel Loads  Nodal Loads
Lane Loads
Figure 3.10 Conversion of Vehicle and Lane Loads to Nodal Loads
Given the location of a vehicle on the bridge, the preprocessor computes the
location of each axle within the vehicle, and then the location of each wheel within
each axle. In the case of a lane load, the preprocessor computes the location of each
axle in the axle train, and then the computes the location of each wheel within each
axle. Finally, after computing the magnitude of each wheel load, the preprocessor
distributes each wheel load to the finite element nodes that are closest to its location.
Each wheel load is idealized as a single concentrated load acting at the location of the
contact point of the wheel. Wheel loads are distributed to the finite element nodes using
the static distribution technique illustrated in Figure 3.11. This distribution technique is
used for zero, constant, and variable skew bridges.
X2
Equivalet Whel N4
Nodal Load Load Node Statically Equivalent
Pw Number Nodal Loads N3
P4
P1 : N1 PI Pw (1a(l) ( .
N2 P2 = Pw (a)(1p) N2 Y
N3 P3 Pw (1)(P) NI
N4 P4 Pw (a)()
a SIIl, emen t  I Slab
Sb E t Static Distribution Factors Element
Finite a = XI / X2
Prspetive Element 3 YI / Y2
Plan View
Figure 3.11 Static Distribution of Wheel Loads
Wheel loads that are off the bridge in the longitudinal direction are ignored.
Wheel loads that are off the bridge in the lateral direction are converted into statically
equivalent loads by assuming rigid arms connect the loads to the line of slab elements
at the extreme edge of the bridge. After making this assumption, the same static
distribution procedure described above is used to compute the equivalent nodal loads.
This procedure of assuming a rigid arm allows the engineer to model rare cases in
which the outside wheels of a vehicle are off the portion of the deck that is considered
to be structurally effective.
The ability of the preprocessor to perform vehicle load distribution is one of its
most time saving features since very often many load cases must be explored to
determine which ones are critical for design or rating. Each of these load cases consists
of many wheel loads that must be converted into even more numerous equivalent nodal
loads. The number of nodal loads required to represent a moderate number of load
cases can quickly reach into the thousands.
CHAPTER 4
DATA COMPRESSION IN FINITE ELEMENT ANALYSIS
4.1 Introduction
In the analysis of highway bridges, the amount of outofcore storage that is
available to hold data during the analysis can frequently constrain the size of models
that can be analyzed. It is not uncommon for a bridge analysis to require hundreds of
megabytes of outofcore storage for the duration of the analysis. Also, while the size
of the bridge model may be physically constrained by the availability of outofcore
storage, it may also be effectively constrained by the amount of execution time required
to perform the analysis. The use of computer assisted modeling software such as the
bridge modeling preprocessor presented in Chapters 2 and 3 further increases the
demand on computing resources. Using the preprocessor, an engineer can create
models that are substantially larger and more complex than anything that could have
been created manually.
To address the issues of the large storage requirements and lengthy execution
times arising from the analysis of bridge structures, a realtime data compression
strategy suitable for FEA software has been developed. This chapter will describe that
data compression strategy, its implementation, and parametric studies performed to
evaluate the effectiveness of the technique in the analysis of several bridge structures. It
will also be shown that by utilizing data compression techniques, the quantity of outof
core storage required to perform a bridge analysis can be vastly reduced. Also, in many
cases the execution time required for an analysis can be dramatically reduced by using
the same data compression techniques.
4.2 Background
During the past three decades a primary focal point of research efforts aimed at
improving the performance of FEA software has been the area of equation solving
techniques. This is due to the fact that the equation solving phase of a FEA program is
one of the most computationally intensive and time consuming portions of the program.
As a result of research efforts in this area, equation solvers that are optimized both for
speed and for minimization of required incore memory are now widely available. The
performance of FEA programs implementing such equation solvers is now often
constrained not by the quality of the equation solver but instead by the speed at which
data may be moved back and forth between the program core to outofcore storage and
by the availability of outofcore storage.
Although the coding details of FEA software vary greatly from program to
program, all FEA programs must perform certain common procedures. As the size of
the finite element model increases, three of these common procedures begin to
dominate a program in terms of the portion of total execution time that is spent in these
procedures. They are solving the global set of equilibrium equations, forming element
stiffness and force recovery matrices, and performing element force recovery.
In many FEA programs, the element stiffness and force recovery matrices are
formed incore and then written to disk sequentially as each element is formed. This
procedure requires that all of the element stiffness and force recovery matrices be
moved from the program core to disk. In all but the smallest of finite element models,
this is necessary because there is insufficient memory to hold all of the element
matrices for the duration of the analysis. Once the element matrices have been written
to disk, the global equilibrium equations are formed by assembling the element stiffness
and load matrices into the global stiffness and load matrices. This step requires that all
of the element matrices that have been written to disk be moved from disk storage back
to the program core. Finally, when the global equilibrium equations have been solved
and displacements are known, the element force recovery matrices must be transferred
from disk back to the program core to perform element force recovery. Under certain
circumstances, such as in analyses involving multiple load casesan extremely common
occurrence in bridge modelingor in nonlinear analyses where element state
determination and element matrix formation must be performed many times, some or
all of the steps discussed above may need to be performed more than once.
Consequently, element matrices must be transferred to and from disk many times
during the analysis.
Due to rapidly improving computational power and relatively low cost, the
personal computer (PC) has gained widespread use in the area of FEA rivaling more
expensive workstations in terms of raw computational power. However, PCs are
generally slower than workstations in the area of disk input/output (I/O) speed and
often have far less available disk space. To address the issue of slow I/O speed on PCs,
some FEA software developers write custom disk I/O routines in assembly language.
This results in an FEA code that is considerably faster than that which can be achieved
using only the disk I/O routines provided in standard high level languages.
However, while the use of assembly language yields increased disk I/O speed, it
does so at the cost of portability. This is because assembly language is intimately tied to
the architecture of central processing unit (CPU) on which the software is running and
is therefore not portable to machines having different CPU architectures. Furthermore,
the use of assembly language I/O routines achieves nothing with respect to the problem
of the large of outofcore storage demands made by FEA software.
Instead of using assembly language disk I/O routines, the author has chosen a
different approach in which the quantity of data written to the disk is reduced while
preserving the information content of that data. This is accomplished by using a data
compression technique to compress the data before writing it to disk, and decompress it
when reading the data back from disk. The result is faster data transfer and vastly
reduced outofcore storage requirements.
4.3 Data Compression in Finite Element Software
In using compressed disk I/O in finite element software the goals are twofold
to reduce the quantity of outofcore storage required during the analysis and to reduce
the execution time of the analysis. Data compression is used to accomplish these goals
by taking a block of data that the FEA software must transfer to or from disk storage
and compressing it to a smaller size before performing the transfer. Compression
preserves the information content of the data block while reducing its size, thus
reducing the amount of time that must be spent performing disk I/O. Of course the
process of compressing the data before writing it to disk requires an additional quantity
of time. Also the data must now be decompressed into a usable form when reading it
back from disk which also requires additional time. However, the end result is often
that the amount of additional time required to compress and decompress the data is
small in comparison to the amount of time saved by performing less disk I/O. This is
especially true on PC platforms where disk I/O is known to be particularly slow.
While one benefit of using compressed I/O can be a reduction in the execution
time required by FEA softwarewhich is often the critical measure of performancea
second benefit is that the quantity of disk space required to hold data files created by
the software is substantially reduced. A typical FEA program will create both
temporary data files, which exist only for the duration of the program, and data files
containing analysis results which may be read by post processing software. The data
compression strategy presented herein compresses only files that are binary data files,
i.e. raw data files. This is opposed to formatted readable output files that the user of a
FEA program might view using a text editor. Binary data files containing element
matrices, the global stiffness and load matrices, or analysis results such as
displacements, stresses, and forces are typically the largest files created by FEA
software and are the types of files which are therefore compressed. Formatted output
files can just as easily be compressed but the user of the FEA software would have to
decompress them before being able to view their contents.
For reasons of accuracy and minimization of roundoff error, virtually all FEA
programs perform floating point arithmetic using double precision data values. In
addition, much or all of the integer data in a program consists of long (4 byte) integers
as opposed to short (2 byte) integers either because the range of a short integer is not
sufficient or because long integers are the default in the language (as is the case in
Fortran 77). An underlying consequence of using double precision floating point and
long integer data types is that there is a tremendous amount of repetition in data files
created by FEA software. Consider as an example the element load vector for a nine
node plate bending element. A plate bending element has two rotational and one
translational degrees of freedom at each node in the local coordinate system, but when
rotated to the global coordinate system there are six degrees of freedom per node.
Thus, for a single load case the rotated element load vector which might be saved for
later assembly into the global load vector will have 9*6 = 54 entries. If the entries are
double precision floating point values then each of the 54 entries in the vector is made
up of 8 bytes resulting in a total of 54*8 = 432 bytes. Now consider an unloaded plate
element of this type where the load vector contains all zeros. Typical floating point
standards represent floating point values as a combination of a sign bit, a mantissa, and
an exponent. A value of zero can be represented by a zero sign bit, zero exponent, and
zero mantissa. Thus a double precision representation of the value zero may consist of
eight zero bytes. A zero byte is defined as containing eight unset (zero) bits.
Consequently, the load vector for an unloaded plate element will consist of 432
repeated zero bytes resulting, in a considerable amount of repetition within the data
file. This type of data repetition, in which there are sequences of repeated bytes, will
be referred to hereafter as small scale repetition.
In addition to the small scale repetition described above, data files created by
FEA software contain large scale repetitions of data as well. Consider the element
stiffness of the plate element described above. When rotated to the global coordinate
system, the element stiffness will be a 54 x 54 matrix of double precision values. Using
the symmetric property of stiffness matrices, assume that only the upper triangle of the
matrix is saved to disk for later assembly into the global stiffness matrix. Thus, a total
of 54*(54+1)/2 = 1,485 double precision values, or 1,485*8 = 11,800 bytes of
information must be saved to disk for a single element. Now consider a rectangular slab
model of constant thickness consisting of a 10 x 10 grid of elements where there is a
high degree of regularity in the finite element mesh. Assume that the rotated element
stiffness for each element in the model is identical to that of all the other elements. To
save the element stiffnesses for each of the elements in the model, a total of
10*10*11,800 = 1,188,000 bytes of data must be transferred from the program core to
disk, and then back to the program core at assembly time when there are actually only
11,448 unique bytes of information in that data.
Somewhere in between the small and large scale repetitions described above lies
what will be appropriately referred to as medium scale repetition. Medium scale
repetition refers to sequences of repeated byte patterns. If the length of the pattern is a
single byte, then medium scale repetition degenerates to small scale repetition, whereas
if the length of the pattern is the size of an entire stiffness matrix, then medium scale
repetition degenerates to large scale repetition. As used herein, the term medium scale
repetition will refer to patterns ranging from two bytes in length to a few hundred bytes
in length.
It should be noted the data compression strategy being presented herein is
intended to supplement rather than replace efficient programming practices. Take as an
example, the case of the repeated plate element stiffness matrices described above. A
well implemented FEA code might attempt to recognize such repetition and write only
a single representative stiffness matrix followed by repetition codes instead of writing
each complete element stiffness. In this case the compression library will supplement
the already efficient FEA coding by opaquely compressing the single representative
element stiffness that must be saved. The term opaquely is used to indicate that the
details of the data compression process, and the very fact that data compression is even
being performed, are not visible toi.e. are opaquely hidden fromthe FEA code.
Thus, the reduction of data volume is performed in a separate and self contained
manner which requires no special changes to the FEA software. If however, the FEA
code makes no such special provisions for detecting element based repetition, then the
compression library described herein will reduce the volume of data that must be saved
by compressing each element stiffness matrix as it is written.
In each case, compression is accomplished primarily by recognizing small and
medium scale repetition within the data being saved. In fact, due to the small size of
the hash key used in the hashing portion of the compression algorithmdescribed in
detail later in this chapterthe likelihood of the compression library identifying large
scale repetitions such as entire stiffness matrices is very remote. Instead, the vast
majority of the compression is accomplished by recognizing small and medium scale
repetition, both of which are abundant in the type of data produced by FEA software.
4.4 Compressed I/O Library Overview
The compressed disk I/O library being presented herein uses a buffered data
compression technique that performs compression on both small and large scale
repetitions like those described above. Written in the ANSI C language, the compressed
I/O library was originally intended for use in FEA software written in C. From this
point on ANSI C will be referred to simply as C, and the ANSI C I/O functions will be
referred to as standard C I/O functions to distinguish them from the compressed I/O
functions being presented. In its present form, the library can be used to manipulate
sequential binary 1/O files, although it could also be adapted to manipulate direct access
(as opposed to sequential), and formatted (as opposed to binary) data files.
Contained in the library are compressed I/O functions that are direct
replacements for the standard C I/O functions that a FEA program would normally
utilize. The compressed I/O library functions have identical prototypes to their standard
C counterparts. As a result, converting a FEA program written in C which utilizes
sequential binary I/O over to compressed I/O requires only that the names of the
standard C functions called by the program be replaced with the corresponding
compressed I/O function names. Table 4.1 lists the compressed I/O functions that are
provided in the library. The CioModify function has no counterpart in the standard C
library because it allows the FEA code to modify certain characteristics of the

130
been implemented in NetSim. The use of the hybrid backpropagation algorithm
typically results in accelerated convergence of the training process which in turn leads
to more robust networks. In this context the term robust is used to mean that the
network generalizes the desired inputoutput relationship well. NetSim also allows the
user to control virtually every aspect of the training algorithm so that training can be
customized to the training data set being learned.
5.6 Supervised Training Techniques
In supervised training the goal is to arrive at a set of network connection
weights which accurately embody the inputoutput relationship represented by the
training data. In order to determine if a particular set of connection weights accurately
models the training data, we will develop a scalar measure of goodness or
badness. This scalar measure indicates the degree of error that is present in the
networks representation of the data. A convenient measure of error for use in neural
network training is
(5.3)
N N
1Y train 1 v out
where NtraÂ¡ is the number of training data pairs, Nout is the number of neurons in
the output layer of the network, Ok is the output value of the k,h neuron in the output
layer of the network, and Tk is the target value (training data) of the klh neuron. Thus,
just as in regression analysis, we choose here to use a mean squared error statistic to
115
approximately 4 percent and 9 percent, respectively, of their original sizes. The storage
savings for these models is even more dramatic than the earlier results from the SEAS
runs.
It is interesting to note from the plot that the use of even very small buffer sizes
can produce a significant degrees of compression. Using a buffer size of only 256
bytesequivalent to the size of just 32 double precision valuesthe outofcore storage
requirements of the prestressed and steel bridge analyses were reduced to approximately
11 percent and 17 percent, respectively, of their original sizes. This is clear evidence
that there is a great deal of smallscale repetition in FEA data since this is the only type
of repetition that can be recognized when using such a small buffer size.
0.40 1 1 1 1 1 1 i i
0.35
Zero Skew Prestressed / 4096 Hai 
Variable Skew Steel / 4096 Hash
0.30
0.00
0
2000 4000 6000 8000 10000 12000 14000 16000 18000
Buffer Size (bytes)
Figure 4.8 SIMPAL File Compression Results
16
review were general purpose analysis programs such as SAP and STRUDL as well as
specialized bridge analysis programs such as GENDEK, CURVBRG, and MUPDI. In
addition, simplified analysis packages such as SALOD (Hays and Miller 1987) were
also reviewed. Each of the various programs were evaluated and compared primarily
on the basis of the analysis accuracy. However, the modeling capabilities of the
software were not of primary concern in the review.
At present, there are several commercial packages available for bridge modeling
and analysis, however, their modeling capabilities and analysis accuracy vary widely.
The commercial program BARS is widely used by state departments of transportation
(DOTs) throughout the United States. However BARS utilizes a simplified one
dimensional beam model to represent the behavior of the bridge and therefore cannot
accurately account for lateral load distribution between adjacent girders or skewed
bridge geometry.
Another commercial package is CBRIDGE. CBR1DGEand its design
counterpart CBDesignhave the ability to model and analyze straight and curved
girder bridges as well as generate bridge geometry and vehicular loading conditions.
Although CBRIDGE is now a commercial package, the original analysis methods were
developed under funded research programs at Syracuse University. A limitation of the
CBRIDGE package is that the bridge models created do not account for the individual
construction stages of the bridge.
Public domain (noncommercial) programs for finite element modeling and
analysis include the CSTRUCT and XBUILD packages developed by Austin.
190
If steepest descent directions are used, convergence to the minimum of the error
surface can be very slow in many cases. In CG, conjugate directions are used instead
of steepest descent directions. Conjugate directions are directions which very closely
approximate the steepest descent directions but which are also conjugate to all of the
previous directions followed on the surface thus far. The result of using conjugate
directions instead of steepest descent directions is quicker convergence to the minimum
of the error surface and therefore quicker solution to the problem Ax = b (Golub and
Van Loan 1989, Jennings and McKeown 1992, Press et al. 1991). The Conjugate
Gradient algorithm is given by the following steps.
form initial guess X(o)
(7.4)
o
II
o
II
o
1
Ji
(7.5)
a &0
a(0 T A
P(I)AP(I)
(7.6)
*0+1) = *0) +a(,P( 0
(7.7)
r0+l) = r0)a0)AP(0
(7.8)
n .. r0+1/0+1)
P(i+1) T
rvm
(7.9)
P(i+l) = r(/+l) +P(i+l)/(0
(7.10)
When an iterative method exhibits slow convergence, the matrix equation
Ax = b may be preconditioned to accelerate convergence. In theory, preconditioning
may be applied to any iterative method, however there are several practical
103
chosen as the FEA code to be used in the studies. SEAS was developed independently
by the author as part of the research being reported on in this dissertation. It is a
general purpose linear finite element package supporting threedimensional truss,
frame, and plate (ninenode lagrangian and heterosis) elements.
The compression library was implemented in SEAS in such a way that binary
I/O could be performed in either standard or compressed format. In addition, the
parameters of the compression algorithmnamely the buffer size and hash table size
may be specified by the user at runtime when using compressed I/O. The data files
that were compressed were those containing the element matrices, stress recovery
matrices, and stress results from the analysis. These files account for the bulk of outof
core storage required during a FEA analysis.
A pair of twospan flatslab bridge models subjected to moving vehicle loads
were created and analyzed by SEAS. The relevant parameters of each of the models are
listed in Table 4.3. The first model is a twospan concrete flatslab bridge having zero
skew geometry while the second is identical except that it has variable skew geometry.
The finite element meshes for each bridge are illustrated in Figure 4.3. Each of the
bridges was modeled using ninenode lagrangian plate elements supported on elastic
truss elements.
Table 4.3 Parameters of Bridge Models Used in SEAS Parametric Studies
Geometry
Nodes
Degrees of
Freedom
Truss
Elements
Frame
Elements
Plate
Elements
Load
Cases
Zero Skew
924
2583
63
0
200
110
Variable Skew
924
2583
63
0
200
110
107
fully exploiting largescale repetition in FEA data. This is because the hash key used in
the compression algorithm is too small to identify patterns as large as a complete
element stiffness matrices. The plot in Figure 4.4 confirms this presumption and
indicates that compression is actually achieved by recognizingand compressing
medium and smallscale repetitions in the data.
Finally, the plot also yields the very important result that the degree of
compression achieved does not vary greatly with respect to the size of the hash table.
Since each element of the hash table is four bytes in size, this result is very important
because it indicates that relatively small hash tables may be used to achieve
considerable compression. Ross (1992) states that the hashing algorithm in RDC is
optimized for a hash table of 4096 elements which would require 16384 bytes of
memory. However, Figure 4.4 suggests thatat least in FEA applicationsa hash table
as small as 512 elementstotaling only 2048 bytescan be used without sacrificing
compression performance.
Plotted in Figure 4.5 are the normalized execution times for the various flatslab
bridge models tested. The execution times plotted are for analysis runs which were
made on a UNIX workstation. Each compressed I/O execution time was normalized
with respect to the execution time required when the analysis was performed using
standard I/O. The plot indicates that the time taken for an analysis is not highly
dependent on the size of the hash table used. The difference in execution times between
cases where the minimum and maximum hash table sizes were used averages around
just 2 or 3 percent. As a result, we may again conclude that small hash tables can be
used without sacrificing a great deal of performance in terms of execution time.
85
In many FEA programs, the element stiffness and force recovery matrices are
formed incore and then written to disk sequentially as each element is formed. This
procedure requires that all of the element stiffness and force recovery matrices be
moved from the program core to disk. In all but the smallest of finite element models,
this is necessary because there is insufficient memory to hold all of the element
matrices for the duration of the analysis. Once the element matrices have been written
to disk, the global equilibrium equations are formed by assembling the element stiffness
and load matrices into the global stiffness and load matrices. This step requires that all
of the element matrices that have been written to disk be moved from disk storage back
to the program core. Finally, when the global equilibrium equations have been solved
and displacements are known, the element force recovery matrices must be transferred
from disk back to the program core to perform element force recovery. Under certain
circumstances, such as in analyses involving multiple load casesan extremely common
occurrence in bridge modelingor in nonlinear analyses where element state
determination and element matrix formation must be performed many times, some or
all of the steps discussed above may need to be performed more than once.
Consequently, element matrices must be transferred to and from disk many times
during the analysis.
Due to rapidly improving computational power and relatively low cost, the
personal computer (PC) has gained widespread use in the area of FEA rivaling more
expensive workstations in terms of raw computational power. However, PCs are
generally slower than workstations in the area of disk input/output (I/O) speed and
10
the overall process of compressing and subsequently decompressing a data set. It should
be clear from context which meaning is intended.
Data compression techniques may be broadly divided into two categories
lossless data compression and lossy data compression. In lossless data compression, the
data set may be translated from its original format into a compressed format and
subsequently back to the original format without any loss, corruption, or distortion of
the data. In contrast, lossy data compression techniques allow some distortion of the
data to occur during the translation process. This can result in greater compression than
that which can be achieved using lossless techniques. Lossy compression methods are
widely used in image compression where a modest amount of distortion of the data can
be tolerated.
In the compression of numeric FEA data such as finite element matrices it is
necessary to utilize lossless data compression methods since corruption of the data to
any extent would invalidate the analysis. Thus, in the present work, in order to
capitalize on the repetitive nature of FEA data, a realtime lossless data compression
strategy has been developed, implemented, and tested in bridge FEA software.
The term realtime is used to indicate that the FEA data is not created and then
subsequently compressed as a separate step but instead is compressed in realtime as the
data is being created. Thus the compression may be looked upon as a filter through
which a stream of numeric FEA data is passed in, and a stream of compressed data
emerges. This type of compression is also more loosely referred to as onthefly data
compression. Of course, the direction of the data stream must eventually be reversed so
71
location. Along a hinge line, each of the girders contains a hinge connection and the
deck slab is made discontinuous.
When using the preprocessor, the engineer inserts hinges into the bridge by
specifying the distances from the beginning of the bridge to the hinges. If the hinge
distances specified do not match the locations of finite element nodal lines, then the
hinge lines are moved to the location of the nearest nodal line. Also, note that the
insertion of hinges into a bridge must not cause the structure to become unstable. For
example, one may not insert a hinge into a single span bridge since this would result in
an unstable structure.
Hinge modeling is accomplished by placing a second set of finite element nodes
along the hinge line at the same locations as the original nodes. In Figure 3.9, this is
depicted by showing a small finite distance between the two set of nodes at the hinge
line. In the actual finite element bridge model the distance between the two lines of
nodes is zero. Girder, stiffener, and slab elements on each side of the hinge line are
then connected only to the set of nodes on their corresponding side of the hinge. At this
point the bridge is completely discontinuous across the hinge line.
Figure 3.9 Modeling a Hinge in a Steel Girder Bridge
246
Smith, M. (1993). Neural Networks for Statistical Modelling, Van Nostrand Reinhold,
New York.
Szewczyk, Z.P., Hajela, P. (1994). Damage Detection in Structures Based on Feature
Sensitive Neural Networks, Journal of Computing in Civil Engineering, ASCE,
Vol.8, No.2, pp. 163178.
Timoshenko, S., WoinowskyKrieger, S. (1959). Theory of Plates and Shells, 2nd
Edition, McGrawHill, New York.
Van der Vorst, H.A. (1982). High Performance Preconditioning, SIAM Journal on
Scientific and Statistical Computing, Vol.10, No.6, pp. 11741185.
Vanluchene, R.D., Sun R. (1990). Neural Networks in Structural Engineering, Micro
computers in Civil Engineering, Vol.5, pp.207215.
Warren S.S. (1994). Neural Network Implementation in SAS Software, Proceedings of
the Nineteenth Annual SAS Users Group International Conference, SAS Institute
Inc., Cary, NC, USA.
Wassermann, P.D. (1989). Neural Computing Theory and Practice, Van Nostrand
Reinhold, New York.
Wilson, E.L., Bathe, K.J., and Doherty, W.P. (1974). Direct Solution of Large
Systems of Linear Equations, Computers and Structures, Vol.4, pp. 363372.
Wilson, E.L., Dovey, H.H. (1978). Solution or Reduction of Equilibrium Equations
for Large Complex Structural Systems, Advances in Engineering Software,
Vol.l, No.l, pp.1925.
Wu, X., Ghaboussi, J., Garrett, J.H.Jr. (1992). Use of Neural Networks in Detection
of Structural Damage, Computers and Structures, Vol.42, No.4, pp.649659.
Ziv, J., Lempel, A. (1977). A Universal Algorithm for Sequential Data Compression,
IEEE Transactions on Information Theory, Vol. IT23, No.3. pp.337343.
Zokaie, T. (1992). Distribution of Wheel Loads on Highway Bridges, Research Results
Digest, NCHRP, Transporation Research Board, National Research Council,
Vol. 187.
186
be solved and it not the goal of this dissertation to survey them ail. However, some
general classifications will be useful for the discussions that follow.
1. Direct solvers. Solution strategies in which the matrix equation is solved in a
direct manner, without need for iteration.
2. Iterative solvers. Solution strategies in which the matrix equation is solved
by iteratively refining the estimates of the unknowns being determined. In
such cases, the number of iterations required for convergence is dependent
on a number of problem dependent parameters and cannot generally be
determined a priori.
3. Elementbyelement solvers. Solution strategies in which the full matrix
equation is never actually formed. Instead, the elements which would be
used to form the full matrix equation are processed individually. Element
byelement solvers can be further classified as either direct or iterative.
4. Sparse solvers. Solution strategies in which patterns of sparsity in the matrix
equation are exploited to reduce the amount of time and memory that will be
required to solve the system of equations. Many types of sparsity can be
accounted for (symmetric, banded, profile) and sparsity can be employed in
either direct or iterative solvers.
What all of these equation solving schemes have in common is that they are all
more or less general purpose in nature. These schemes may exploit a particular
structure of sparsity (e.g. bandedness), or a particular property of the matrices involved
(e.g. postivedefmiteness), but the actual physical problem being solved is not
considered in the solution scheme. In this chapter, a new solution strategy will be
presented which combines neural networks with an iterative equation solving scheme to
produce a hybrid method specific to the area of highway bridge FEA.
7,2 Exploiting Domain Knowledge
In the present research, a domain specific equation solver has been created for
the analysis of highway bridge structures. The term domain specific is used to designate
89
file. This type of data repetition, in which there are sequences of repeated bytes, will
be referred to hereafter as small scale repetition.
In addition to the small scale repetition described above, data files created by
FEA software contain large scale repetitions of data as well. Consider the element
stiffness of the plate element described above. When rotated to the global coordinate
system, the element stiffness will be a 54 x 54 matrix of double precision values. Using
the symmetric property of stiffness matrices, assume that only the upper triangle of the
matrix is saved to disk for later assembly into the global stiffness matrix. Thus, a total
of 54*(54+l)/2 = 1,485 double precision values, or 1,485*8 = 11,800 bytes of
information must be saved to disk for a single element. Now consider a rectangular slab
model of constant thickness consisting of a 10 x 10 grid of elements where there is a
high degree of regularity in the finite element mesh. Assume that the rotated element
stiffness for each element in the model is identical to that of all the other elements. To
save the element stiffnesses for each of the elements in the model, a total of
10*10*11,800 = 1,188,000 bytes of data must be transferred from the program core to
disk, and then back to the program core at assembly time when there are actually only
11,448 unique bytes of information in that data.
Somewhere in between the small and large scale repetitions described above lies
what will be appropriately referred to as medium scale repetition. Medium scale
repetition refers to sequences of repeated byte patterns. If the length of the pattern is a
single byte, then medium scale repetition degenerates to small scale repetition, whereas
if the length of the pattern is the size of an entire stiffness matrix, then medium scale
232
applications this may not be cause for concern because the overall magnitude of b
may always be small relative to the magnitude at a. Thus, the error at b, although
large relative itself, is small when put in the context of the overall problem. However,
in the neural networks used in this research, the normalized displacement surfaces are
scaled before arriving at a set of final displacements.
When a large scaling factor is applied to a value similar to point b of the
simplified example, a significant error can be introduced into the problem. This is
precisely the situation which occurs when large residual forces are generated during the
solution process. Since neural network seeding causes large initial residuals, a large
amount error is quickly introduced into the solution process and it essentially saturates.
A large residual causes a large error in a displacement calculation which in turn causes
a larger residual. As this cycle repeats, the residual forces grow rapidly.
Looking at the NNPCG algorithm (Equations (7.33)(7.40)) one can see that
the a term (Equation (7.36)) serves as a step size parameter. Displacement changes are
first multiplied by the a step size factor before using them to update the total
displacements. However, at each iteration, the denominator of the a calculation grows
faster than the numerator because of the growing residuals. As a result, a quickly
tends toward zero. Table 7.2 shows the values of a which occur during a full NN
PCG solution of the flatslab bridge with vehicular loading. It is clear from the table
that the growing residuals cause the denominator of a to grow and a to quickly
diminish. As a tends towards zero, refinement in the displacements essentially stalls.
224
Longitudinal Direction (indies) Longitudinal Direction (indies)
Figure 7.13 Zero Seeded Convergence of Displacements (TzTranslations) and
Residuals (FzForces) for a Flatslab Bridge Under Vehicular
Loading
113
Table 4.5 Parameters of Bridge Models Used in SIMPAL Parametric Studies
Name
Skew
Nodes
DOFs
Truss
Elements
Frame
Elements
Plate
Elements
Load
Cases
Prestressed
Zero
1271
4434
516
972
1176
123
Steel
Variable
963
2829
20
324
880
65
element meshes for both bridges are illustrated in Figure 4.7. The bridges are the same
as those of example problems PRE.5 and STL.5 presented in Hays et al. (1994). They
were chosen because they are reasonably large in size and represent realistic highway
bridge structures.
In the zero skew prestressed bridge model there is a great deal of regularity in
the geometry of the finite element mesh. As a result, there will be a high degree of
largescale repetitionas well as both medium and smallscale repetitionin the FEA
data created during the analysis. In contrast, the variable skew geometry of the steel
girder bridge model will prevent the formation of largescale repetition in the FEA
data. However, it will be shown later that medium and smallscale repetition still
abound.
Each of the models were analyzed using SIMPAL in compressed I/O mode with
buffer sizes of 64, 128, 256, 512, 1024, 2048, 4096, 8192, 12288, and 16384 bytes,
and a hash table size of 4096 elements. In addition, the models were also analyzed
using SIMPAL in a mode which uses the standard Fortran I/O functions instead of the
compressed I/O functions.
155
In the present study, the ability to handle and an arbitrary number of loads of
arbitrary magnitude was accomplished using superposition and separation.
Superposition was used to handle the variable number of loads. Each loading condition
is broken down into as many individual loads as are necessary to represent the overall
loading. The displacements due to each individual component are then computed and
accumulated with all of the other loads to form the total set of displacements for the
structure. This is a straight forward, standard technique of structural analysis.
Proper handling of variable magnitude loads was achieved by separating
displacement shape data from displacement magnitude data. The concept behind this
technique stems from the assumption of linear structural behavior. Place a single load
on a structurewhich is assumed to be linearand compute the displacements. Now
double the load and compute the displacements. The second set of displacements will
simply be the first set scaled by a factor of 2.0assuming linear behavior. This is
illustrated for a simple propped cantilever beam in Figure 6.2
There is really only one characteristic displacement shape for the structure for
each particular loading condition. We will term this characteristic shape the
normalized shape" of the structure and define it to be the displaced shape (set of
displacements) of the structure normalized with respect to the largest magnitude
Load P, Max. Displacement = A Load = 2P. Max. Displacement = 2A Normalized (Characteristic) Shape
Max. Displacement = 1
Figure 6.2 Linearly Proportional Displacements
142
the network may appear to learn more rapidly in the early stages of training, it may
slow significantly during the later stages because it is using incomplete gradient data.
A procedure called batching attempts to combine the benefits of exampieby
example training with the robustness of the pure backpropagation procedure. In
batching, a mean squared pseudo error E is computed not for a single training pair, as
was the case in examplebyexample training, but for a batch of training pairs. A
parameter called the batch size determines how many training pairs are processed
before the connection weights are updated. If the batch size is chosen to be one, then
this procedure reverts to examplebyexample training. On the other hand, if the batch
size is chosen to be equal to the number of training pairs in the training seta choice
referred to as full batchingthen the procedure reverts to pure backpropagation. In
practice, a batch size somewhere between these two extremes is usually chosen.
Batching shares the rapid initial learning characteristics of examplebyexample
learning but is also somewhat more robust than the examplebyexample procedure.
However, the gradient followed is still only an approximation of the gradient of the
true E and can therefore lead the learning process in incorrect directions, thereby
slowing the training process.
In the NetSim package, the user may specify the batch size to be used in
updating the connection weights. By default, full batching is performed which
corresponds to the pure backpropagation procedure, however the user may explore
other possibilities by specifying smaller batch sizes.
153
may be used in a separate process, e.g. a matrix equation solve, to produce a set of
structural displacements.
In contrast, although a neural network can also encode the loaddisplacement
relationship for a bridge structures, none of the steps employed in building that
encoding are directly related to the behavior of structures. Instead, a set of training data
is generated and used to train the network. It is the training data and the process used to
generate that training data that are related to the behavior bridge structuresnot the
network training process itself. The only reason the network learns the load
displacement relationship for a bridge is because it is this relationship that is
represented by the training data.
In the present research, the network training data was generated using analytical
FEA results from bridge analyses. However, this does not have to be the case. For
example, one might instead choose to apply loads to a structure and experimentally
measure the resulting displacements. In this manner, a set of training data could be
generated using only the experimental loaddisplacement data. Using this data, a
network could again be trained to implicitly encode the relationship without any need
for a structural behavior rule base from which to work.
6.3 Separation of Shane and Magnitude
In constructing neural networks for bridge analysis, it is essential to create
networks that are capable of handling arbitrary loading conditions. Bridge loads may
vary in magnitude, location, type (force, moment), and source (point load, uniform
14
PCG algorithm was specifically chosen for this application because one component of
that algorithm involves the use of an approximate stiffness matrix to precondition the
problem. Preconditioning reduces the effective condition number of the system and thus
increases the rate of convergence of the iterative process. A more detailed discussion of
this phenomenon will be presented later in this work.
Implicitly embodied in the connection weights of the neural networks is the
relationship between applied loads and resulting displacements in flatslab bridge
structures. This is precisely the same relationship that is captured in the more
traditional stiffness matrix of FEA. Since the PCG algorithm calls for an approximation
of the stiffness matrix to precondition the problem, what is actually needed is an
approximation of the relationship between loads and displacements. While that
approximate relationship is usually expressed explicitly in terms of an approximate
stiffness matrix, in the present research it is expressed implicitly within the neural
networks.
Thus, the current application of neural networks seeks to accelerate the equation
solving process by
1. Using the embedded domain knowledge to yield very accurate initial
estimates of the solution.
2. Using the implicit relationship between loads and displacements embodied in
the networks to precondition, and thus accelerate, the convergence of the
PCG solution process.
Detailed descriptions of neural network theory, the representation of bridge data,
network training, and implementation of the trained networks into a PCG solver will be
presented in later chapters.
144
0.0 < (a. < LO controls the degree of influence that previous weight changes have on the
current change. If a value of p = 0 is used, then previous changes have no influence
on the current change and momentum is disabled. As the value of p is increased,
previous changes have more and more influence on the current change.
One of the advantages of using momentum is that it accelerates the descent into
ravines in the error surface. Narrow ravineswhich appear to be fairly common in
neural network error surfaceshave steep sides and a relatively flat bottom. When the
steepest descent directions are computed, they generally point toward the bottom of the
ravine, and are perpendicular to the direction of the actual minimum in the ravine.
Thus, a steepest descent procedure will oscillate back and forth between the two sides
of the ravine and waste a large amount of computational effort before finally settling in
the bottom.
When momentum is applied to this situation, the bottom of the ravine is quickly
reached with very little wasted oscillation. To understand why this is the case, consider
the name of the methodmomentum. The method is called momentum because the
effect of the exponential averaging is very much like the effect of momentum on a
moving body. If, over several epochs of training, the changes to a particular network
connection weight are consistently of the same sign, then the weight is moving in a
particular direction and the use of the exponential averaging tends to give the motion in
that direction momentum.
If, however, over several epochs of training the changes to a connection weight
toggle back and forth between positive and negative, then the training process is
163
process. Validation data can be used for this purpose. (See Chapter 5 for a detailed
discussion on the use of validation data).
Although the concepts developed above were illustrated using the simple
propped beam example, they can be directly extended to the case of flatslab bridges.
Shape and magnitude networks are constructed for bridges in the same manner as that
just described. However, instead of using only unit force loads, one must now use unit
force loads (Fz), and unit moment loads (Mx, My). Likewise, instead of computing
only translations, now translations (Tz), and rotations (Rx, Ry) must be computed.
Finally, a load location now means a twodimensional location on a flatslab bridge
instead of a onedimensional location along a beam.
6.4 Generating Analytical Training Data
In order to construct neural networks for flatslab bridge analysis, training data
of the sort described in the previous section had to be generated. In this study,
analytical FEA displacement results were used as the basis for the neural network
training data. SEASthe FEA package developed for this researchwas used to
generate all network training data. Twospan flatslab bridges of varying sizes were
analyzed for a large number of unit load conditions. From the analysis results, the
maximum magnitude displacements were determined and recorded for each load case
and displacement type. Next, the displacements for each displacement type and load
case were normalized with respect to the appropriate maximum magnitude term.
RMS Residual Load Error (kips) RMS Residual Load Error (kips)
219
Figure 7.8 Convergence Behavior of an ICPCG Solver Applied to the Analysis
of a Flatslab Bridge Under Uniform Loading
Figure 7.9 Convergence Behavior of an ICPCG Solver Applied to the Analysis
of a Flatslab Bridge Under Vehicular Loading
86
often have far less available disk space. To address the issue of slow I/O speed on PCs,
some FEA software developers write custom disk I/O routines in assembly language.
This results in an FEA code that is considerably faster than that which can be achieved
using only the disk I/O routines provided in standard high level languages.
However, while the use of assembly language yields increased disk I/O speed, it
does so at the cost of portability. This is because assembly language is intimately tied to
the architecture of central processing unit (CPU) on which the software is running and
is therefore not portable to machines having different CPU architectures. Furthermore,
the use of assembly language I/O routines achieves nothing with respect to the problem
of the large of outofcore storage demands made by FEA software.
Instead of using assembly language disk I/O routines, the author has chosen a
different approach in which the quantity of data written to the disk is reduced while
preserving the information content of that data. This is accomplished by using a data
compression technique to compress the data before writing it to disk, and decompress it
when reading the data back from disk. The result is faster data transfer and vastly
reduced outofcore storage requirements.
4,3 Data Compression in Finite Element Software
In using compressed disk I/O in finite element software the goals are twofold
to reduce the quantity of outofcore storage required during the analysis and to reduce
the execution time of the analysis. Data compression is used to accomplish these goals
by taking a block of data that the FEA software must transfer to or from disk storage
and compressing it to a smaller size before performing the transfer. Compression
2
Whereas the development of such packages for the analysis and design of
buildingtype structures has roughly kept pace with the demands of industry, the same
is not true for the case of highway bridge analysis. This is probably attributable to the
fact that there are simply many more buildingtype structures constructed than there are
highway bridge structures, and therefore a greater demand exists. However, this is not
to say that there is not a demand for such software in bridge analysis. With an
inventory of more than half a million bridges in the United States alone, and roughly
20 percent of those bridges considered structurally deficient and in need of evaluation,
the demand for computer assisted bridge analysis packages exists.
Modeling highway bridges for FEA presents certain challenges that are not
present in the analysis of building structures. For example, in addition to being
subjected to the usual fixed location loads, bridges are also subjected moving vehicular
loads which are often complex and cumbersome to describe with the level of detail
needed for FEA. Also, because moving vehicle loads are typically represented using a
large number of discrete vehicle locations, bridge analyses often contain a large number
of load cases. As a direct result, the engineer is faced not only with the daunting task of
describing the loads, but also of interpreting the vast quantity of results that will be
generated by the analysis.
In order to properly analyze bridge systems for evaluation purposes, as in a
design verification or rating of an existing bridge, each distinct structural stage of
construction should be represented in the model. This is because the bridge has a
distinct structural configuration at each stage of construction, and it is that structural
156
displacement occurring under a particular loading condition. Therefore, a different
normalized shape exists for each loading condition. The ordinates of the normalized
shape will then lie in the range [1,1], which is particularly suitable for implementation
using neural networks.
Thus, we can separate the task of computing structural displacements using
neural networks into two distinct taskscomputing normalized shape values and
computing magnitude scaling values. Shape networks are used to accomplish the first
task while scaling networks are used to accomplish the second task. When an actual
displacement must be computed, a shape network is invoked to compute a normalized
shape ordinate, a magnitude network is invoked to compute a scaling factor, and the
two values are multiplied together.
In the present neural network implementation, separate networks have been
constructed for each combination of load (Fz, Mx, My), displacement (Tz, Rx, Ry),
and task (shape, magnitude). Thus, there are a total of 18 (3x3x2) neural networks
which, when operating collectively as a single unit, can compute true structural
displacements in flatslab bridges (see Figure 6.3).
In subdividing the overall task into several subtasks and assigning a separate
neural network to each subtask, the goal was to minimize the size of the neural
networks. Also, previous experience with training neural networks had suggested that
partitioning the overall task into smaller pieces would increase the likelihood of success
during the training phase.
239
seeding methods and it was found that neural network seeding is a very effective
method for accelerating the convergence of iterative methods. Additional tests revealed
that neural network preconditioning is not an effective methodat least not in the form
tested. Conceptually, neural network preconditioning is a sound idea, however it has
been found that a higher level of network accuracy is required to make the method
effective in practice.
During the course of training these networks, several important observations
were made which can serve as guidelines for other studies involving the use of neural
networks. These guidelines are summarized below.^
1. Start out small. When choosing topological parameters such as the number
of layers and the number of neurons in each layer, start out small. Err on
the low side when choosing the initial network size. Attempt to train the
network repeatedly using different seed values (see Guideline 6 below). If
training consistently fails to lower the network error to an acceptable level,
then increase the size of the network by a small amount and repeat the
process.
2. Avoid networks that are too large. If the network is too largei.e. there are
more degrees of freedom (DOFs) in the network than can be trained using
the available training datathen one of two things will occur. Either the
network will train to an acceptably low level or training will simply halt. In
the former case, the network will have memorized the training data and
will serve simply as a lookup table. (In most applications, this condition is
not desirable.) More oftenat least in the authors experiencethe latter
case will occur in which the network simply never trains.
3. Avoid extremely small convergence tolerances. When choosing the
convergence tolerance for network training, avoid choosing very small
^ In providing these guidelines, it is assumed throughout that obtaining neural networks
which generalize well is desirable. Although this is virtually always the case, one can
conceive applications in which a neural network that functions simply as a lookup
table would be useful. In such cases, these guidelines do not apply. Furthermore, the
guidelines are intended for applications involving continuous values parameters
although many of them also apply to networks involving discrete parameters.
145
oscillating. If exponential averaging (momentum) is applied to this situation, the size of
the weight changes will quickly decreasedue to the sign reversalsand the network
will settle in the bottom of the ravine with little oscillation. From that point it may then
start to build momentum moving along the ravine bottom in a direction perpendicular
to the sides. This process is illustrated graphically in Figure 5.6. For a more detailed
description of the phenomenon, see Smith (1993).
Another primary benefit of using momentum is that it allows the training
process to escape shallow local minima in the error surface. As a weight moves
consistently in one direction, it builds momentum in that direction. If a shallow
minimum is encountered, the momentum of the connection weight can carry it through
the shallow spot and on to a lowerand hopefullyglobal minimum. A pure steepest
descent approach will become trapped in such a situation because the gradient will point
toward the bottom of the local minima.
Therefore, using a gradient descent approach together with momentum can be a
very effective solution to the neural network training problem. NetSim allows the user
Training Without Momentum
Training With Momentum
Figure 5.6 Using Momentum to Dampen Training Oscillations
245
Lee, H., Hajela, P. (1994). Prediction of Turbine Performance Using Mulitlayer
Feedforward Networks by Reducing Mapping Nonlinearity, Artificial
Intelligence and Object Oriented Approaches for Structural Engineering,
pp.99105, CIVILCOMP Ltd., Edinburgh, Scotland.
Manteuffel, T.A. (1980). An Incomplete Factorization Technique for Positive Definite
Linear Systems, Mathematics of Computation, Vol.34, No. 150, pp.473497.
Meijerink, J.A., van der Vorst, H.A. (1977). An Iterative Solution Method for Linear
Systems of Which the Coefficient Matrix is a Symmetrix MMatrix,
Mathematics of Computation, Vol.31, No.137, pp.148162.
Mindlin, R.D. (1951). Influence of Rotatory Inertia and Shear on Flexural Motions of
Isotropic Elastic Plates, Journal of Applied Mechanics, Vol.73, pp.3138.
Optimization of FiniteElement Software Using DataCompression Techniques,
Microcomputers in Civil Engineering, Vol.9, No.3, pp. 161173.
Papadrakakis, M., Dracopouios, M.C. (1991). Improving the Efficiency of Pre
conditioning for Iterative Methods, Computers and Structures, Vol.41, No.6,
pp.12631272.
Press, W.H., Flannery B.P., Teukolsky, S.A., Vetterling, W.T. (1991), Numerical
Recipes in C. The Art of Scientific Computing, Cambridge University Press,
New York.
Rehak, D.R,, Thewalt, C.R., Doo, L.B. (1989). Neural Network Approaches in
Structural Mechanics Computations, Computer Utilization in Stuctural
Engineering, Proceedings of the Structures Congress 1989, ASCE, Edited by
Nelson, J.K.Jr., ASCE, New York, pp. 168176.
Reissner, E. (1945). The Effect of Transverse Shear Deformation on the Bending of
Elastic Plates, Journal of Applied Mechanics, Trans. ASME, Vol.67, pp.A67 
A77.
Rogers, J.L. (1994) Simulating Structural Analysis with Neural Networks, Journal of
Computing in Civil Engineering, ASCE, Vol.8, No.2, pp.252265.
Ross, E. (1992). A Simple DataCompression Technique, C Users Journal, Vol. 10,
No. 10, pp. 113120.
Rumelhart, D.E., Hinton, G.E., Williams, R.J. (1986), Learning Internal
Representations By Error Backpropogation, Parallel Distributed Processing,
Vol.l, pp.318362.
Sedgewick, R. (1990). Algorithms in C, AddisonWesley, Reading Massachusetts.
233
Table 7.2 Decreasing Step Size (a) During Neural Network Preconditioning
Iteration i
r(TnNN(r(i))
P0)Mn
ro)NN(r{i))
1 T A
P(i)AP<)
0
0.885778E+04
0.561218E02
0.561218E02
1
0.136434E+03
0.172476E03
0.172476E03
2
0.465492E+01
0.576414E05
0.576414E05
3
0.219950E+00
0.265231E06
0.26523 IE06
4
0.131720E01
0.154710E07
0.154710E07
5
0.950806E03
0.108810E08
0.108810E08
6
0.800233E04
0.892588E10
0.892588E10
7
0.766925E05
0.834038E11
0.834038E11
8
0.822289E06
0.872152E12
0.872152E12
9
0.972948E07
0.100675E12
0.100675E12
One solution to this problem is to use a more sophisticated method of
controlling error tolerance during network training. Another solution is to recombine
each shape and magnitude network pair into a single network. In this way terms which
are small, and which will remain small in the context of the overall problem, can
tolerate a larger relative amount of error because they will have less influence on the
overall problem. Both of these approaches would require the neural networks to be
retrained using new training data and new training control parameters.
18
1.3.2 Data Compression in FEA
During the past few decades a great deal of effort by FEA researchers has been
directed at both optimizing the use of available incore storage in FEA software and
optimizing the numerical efficiency of matrix equation solvers. However, relatively
little attention has been focused on the optimization of outofcore storage
requirements. It is true that researchers have developed various special purpose
bookkeeping strategies that can moderately reduce outofcore storage demands in
specific situations. However, aside from his own work (Consolazio and Hoit 1994), the
author has been unable to find any references in the literature regarding general
purpose strategies directly incorporating the use of data compression techniques in FEA
software.
In contrast, the development of advanced data compression techniques has been
an active area of research in the Computer and Information Science (CIS) field for at
least two decades. In recent years, system software developers have realized the many
fold benefits of using realtime data compression and have begun embedding data
compression directly into the computer operating systems they develop. However, no
such applications of data compression in FEA have appeared in the engineering
literature.
1.3.3 Neural Network Annlications in Structural Engineering
During the past five to ten years, there has been a steadily increasing interest in
applying the neural network solution paradigm to structural engineering problems.
133
Figure 5.4 Network Error Statistics During Training
One final note regarding the use of validation data should be made. If the error
statistics E and V are plotted together and the value of V never increases, this also
indicates an important training problem. If the value of V does not reach a minimum
and then begin to rise, the network does not have sufficient complexity (number of
neurons, number of hidden layers) to overfit. This situation is undesirable because it
indicates that better a representation of the problem could be achieved by adding
additional network complexity. Essentially, there are insufficient degrees of freedom in
the system to accurately model the problem.
5.7 Gradient Descent and Stochastic Training Techniques
Training a neural network essentially involves solving a nonlinear unconstrained
optimization problem. The error function E is the objective function to be minimized
and the connection weights are the design variables that may be varied to minimize E.
The problem is a nonlinear one because each of the computing neurons present in the
network contains a nonlinear transfer function that is used to process network signals.
229
7.6.2 Preconditioning Using Neural Networks
In addition to using neural networks to seed the solution vector, recall that the
NNPCG algorithm also uses neural networks to perform preconditioning. To test the
effectiveness of neural network preconditioning, the same bridge models analyzed in
the previous section were also analyzed using the full NNPCG algorithmi.e. neural
network seeding and preconditioning. In neural network preconditioning, the networks
are used within the PCG iteration loop as well as to initially seed the solution vector.
Figure 7.16 illustrates the vertical translations (Tz) and vertical residual forces
(Fz) resulting from a full NNPCG analysis of the vehicular loading case. It is evident
from the figure that there is a problem with the solution process. In fact, the iterative
refinement process stalls almost immediately. The displacements are changed by only a
very small fraction and the residuals do not diminish. Although only iterations 0, 1,
and 2 are shown in the figure, the plots for subsequent iterations are virtually identical.
Conceptually, the NNPCG solution algorithm is sound. Unfortunately, a
practical neural network training issue arises which is responsible for the stalling
phenomenon encountered. When neural networks are constructed, they are iteratively
trained to within an error tolerance that is deemed acceptable for the problem being
solved.
In the neural networks constructed for the NNPCG algorithm, displacements
calculations are split into two componentsshape and magnitude (refer to Chapter 6 for
a detailed discussion). Shape networks essentially encode normalized displacement
surfaces for the bridge while magnitude networks encode shape scaling information.
30
Second, if the user wishes to recreate the bridge model at a future date, all of the
necessary data must again be reentered exactly as before. Similarly, if the user wishes
to recreate the model but with a small variation in some parameter, all of the data must
again be reentered including the modified parameter.
To circumvent these problems, the preprocessor maintains a history file
containing each of the responses interactively entered by the user. Thus, there is a
permanentand commentedrecord of what data was in fact entered by the user
should this ever become a matter of dispute in the future. Since the history file contains
all of the data provided by the user, it may also be used to recreate an entire bridge
model. The user simply tells the preprocessor to read input data from the history file
instead of interactively from the user.
In addition to the uses mentioned above, the history file may also be used to
resume a suspended input session, revise selected bridge parameters, or revise the
vehicle loading conditions imposed on a bridge. Thus, the combination of an interactive
program interface and a reusableand editablehistory file results in a program that
exhibits the advantages of both the interactive and batch approaches without the
exhibiting the disadvantage of each.
2.4 Common Modeling Features and Concepts
Many of the modeling features available in the preprocessor are common to
several of the types of bridges that can modeled. Recall that the preprocessor is capable
of modeling prestressed girder, steel girder, reinforced concrete Tbeam, and flab slab
160
a displacementsampling location as input and return the normalized displacement at the
sampling location specified.
Using the training data described above, neural networks can be trained to
encode the shape and scaling data. With regard to Figure 6.4, this means that the
networks are trained to predict the curves shown in the figure, not just the data points.
To accomplish this successfully, a sufficient number of training pairs must be
generatedmore than the three points used in the example.
6.3.2 Using Trained Shape and Scaling Networks
Once shape and scaling neural networks have been trained for a specific type of
structure, they may be used to compute true displacements. By assuming linear
structural behavior, the principle of superposition can be used to handle an arbitrary
Table 6.1 Network Training Data Generated For Propped Cantilever Beam
Scaling Networks
Shape Networks
(3 Training Pairs)
(9 Training Pairs)
Load
Maximum
Load
Displacement
Normalized
Location
Magnitude
Location
Location
Displacement
(Input)
(Output)
(Input)
(Input)
(Output)
xa
\a
A max
xa
xa
.aa
A norm
h
xb
A max
xa
Xb
A norm
Xc
\c
A max
*a
xc
\ca
^ norm
*b
xa
A norm
hb
*b
Xb
A norm
*b
Xc
A norm
. ao
xc
Xa
A norm
xc
Xb
A norm
*c
Xc
\cc
A norm
6
system. This is especially true in situations where multiple analyses will need to be
performed, as in an iterative designevaluation cycle or within a nonlinear analysis
scheme.
Thus, it is evident that in order for a computer assisted bridge modeling system
to be practical and useful, the FEA analysis component must be as numerically efficient
as possible so as to minimize the required analysis time and minimize the use of outof
core storage.
1.2 Present Research
The research reported on in this dissertation focuses on achieving three primary
objectives with respect to FEA bridge modeling. They are :
1. Developing an interactive bridge modeling preprocessor capable of
generating FEA models that can account for bridge construction stages and
vehicular loading conditions.
2. Developing a realtime data compression strategy that, once installed into
the FEA engine of a bridge analysis package, will reduce the computational
demands of the analysis.
3. Developing a domain specific equation solver based on neural network
technology and the subsequent installation of that solver into the FEA engine
of a bridge analysis package.
Each of these objectives attempts to address and overcome a specific difficulty
encountered when applying FEA techniques to the analysis of highway bridge systems.
The following sections describein greater detaileach objective and the methods used
to attain those objectives.
73
composite action model being used, creep is accounted for in one of the ways discussed
below.
If the CGM is being used to represent composite action, then the creep effect is
accounted for when computing composite section properties for the girders. Normally,
when composite section properties are computed, the effective width of concrete slab
that acts compositely with the girders is transformed into an equivalent width of steel
by dividing by the modular ratio. This equivalent width of steel is then included in the
computation of composite section properties for the girder. The modular ratio is a
measure of the relative stiffnesses of steel and concrete and is given by
where Ec is the modulus of elasticity of the concrete deck slab and Eg is the modulus
of elasticity of the steel girders. In order to account for the increased deformation that
will arise from creeping of the deck, the preprocessor uses a modified modular ratio
when computing composite girder properties. When transforming the effective width of
concrete slab into equivalent steel, a modular ratio of 3 is used instead of n. This
yields a smaller width of equivalent steel and therefore smaller section properties.
Because the section properties are reduced, the stiffness of the girders are reduced,
deformations increase, and stresses in the girders increase.
If the EGM is used to represent composite action, then the effect of creep is
accounted for by employing orthotropic material properties in the slab elements. In the
EGM, the deck slab is modeled using a mesh of shell elements. By using orthotropic
128
output parameters. For example, in the present research the loaddisplacement data
used to train the networks consisted of a limited number of discrete choices of load
locations and displacement sampling locations. If the networks could only memorize
these training pairs and look them up during the network use stage, the networks would
not be very useful.
Instead, the networks are trained to learn the generalized relationship between
load and displacement. In this way, if a load or displacement location is specified
during the network use stage that does not correspond to one of the training pairs in the
training data, the network will generalize the relationship and still be able to predict the
correct displacementor at least a good approximation thereof.
When networks generalize in the manner just described, they are effectively
performing a type of high dimensional interpolation or extrapolation. (Networks may
also be trained to perform essentially as classification engines but this type of network
has not been used in this research.) To ensure that a network can properly generalize
the relationship for which has been trained, several checks and safeguards must be
employed. Strategies such as the use of validation data to avoid network overtraining
(or overfitting) will be discussed later in this chapter.
5.5 The NetSim Neural Network Package
To perform network training and explore the factors involved in robust network
training, the author has written a neural network training and simulation package called
NetSim as part of this research. The NetSim package is written in the C programming
227
From the figures one can see that in the ICPCG algorithm the displacements
seem to spread or ripple away from the initial load points as convergence
progresses. This is easiest to see in Figures 7.13 and 7.14 which correspond to the
vehicular loading condition analyzed with zero and diagonal seeding respectively. One
may observe from the residual force plots that the residual forces gradually spread
away from the initial load points. In response to the spreading residual forces,
displacements gradually change from zero values to nonzero values in the vicinity of
the residual forces. Therefore, immediate global displacement changes do not occur in
response to localized residuals forces. This fact explains the similar convergence
behavior of the zero and diagonal seeding cases.
Essentially the diagonal seeding case starts out approximately one iteration
ahead of the zero seeding case. At iteration 0, the diagonal seeding case uses the non
zero forces in the load vector and the diagonals of the stiffness matrix to compute
displacement changes at the locations of those forces. At iteration 0 of the zero seeding
case, the solution vector is simply filled with zeros. However, at the next iteration, the
residuals in the zero seeding case will be the same as the original loads in the diagonal
seeding case. Since the displacement changes are very local in nature, the displacement
changes computed for iteration 1 of the zero seeding case will be very similar to (but
not exactly the same as) those of the diagonal seeding case at iteration 0.
The residual spreading effect described above also explains why the residual
force error remains large for many iterations in the zero and diagonal seeding cases. A
sufficient number of iterationsand a sufficient degree of residual force spreading
38
1. Steel girder. Constant skew steel girder bridges modeled for either live load
analysis or full load analysis and variable skew bridges modeled for full load
analysis.
2. Reinforced concrete Tbeam. Constant skew reinforced concrete Tbeam
bridges modeled for either live load analysis or full load analysis.
3. Prestressed girder. Prestressed girder bridges that are prismatic except for
the presence of end blocks can be modeled for full load analysis. Constant
skew nonprismatic prestressed girder bridges may be modeled for live load
analysis.
Using the preprocessor, the task of describing and modeling the cross sectional
variation of nonprismatic girders has been greatly simplified. Flexible generation
capabilities are provided that minimize the quantity of data that must be manually
prepared by the user. Refer to Â§2.3.3 for further details.
2.4.7 Composite Action
Composite action is developed when two structural elements are joined in such a
manner that they deform integrally and act as a single composite unit when loaded. In
the case of highway bridges, composite action may be developed between the concrete
deck slab and the supporting girders or between the deck slab and stiffening members
such as parapets. Designing a bridge based on composite action can result in lighter and
shallower girders, reduced construction costs, and increased span lengths.
The extent to which composite action is developed depends upon the strength of
bond that exists between the slab and the adjoining flexural members. In a fully
composite system, strains are continuous across the interface of the slab and the
flexural members and therefore no slip occurs between these elements. Vertical normal
157
Loads
r: Fz = Force Along ZAxis
r:Mx Moment About XAxis
r:My = Moment About YAxis
Di placements
q:Tz = Translation Along ZAxis
^:Rx = Rotation About XAxis
<7:Ry = Rotation About YAxis
Figure 6.3 Organization of SubTask Neural Networks
6.3.1 Generating Network Training Data
To illustrate the process of separating shape data from magnitude (scaling)
dataand subsequently training networks using that dataconsider the propped
cantilever illustrated in Figure 6.4. For simplicity, only three points (a, b, and
c) on the beam will be used to generate shape and scaling data. In practice, many
more points would be used so that a large quantity of data for neural network training
would be available. The separation procedure is outlined below.
1. Apply loads. Apply unit loads (forces, moments) to the structure at the
loading points.
2. Compute displacements. For each applied unit load, compute the
displacements that arise in the structure. Displacements are computed at the
selected displacementsampling points.
CHAPTER 5
NEURAL NETWORKS
5.1 Introduction
Artificial neural networks^ are simple computational models which crudely
mimic the operation of biological neural network systems such as the human brain.
Many of the original concepts of neural network (NN) operation were developed to
mimic the brain and to produce artificially intelligent systems. Over time the
researchers in this field have gradually drifted into different corners of the NN field.
There are still many researchersworking primarily in the biological scienceswho
seek to model the behavior of biological systems as accurately as possible. Others are
less concerned with the relationship to biological systems and more concerned with
whether or not a mathematical NN model can solve a particular engineering problem.
The latter group can be further subdivided into researchers interested in using
NNs to produce artificial intelligence and researchers interested in using NNs simply as
a new type of statistical modeling tool. In each of these cases, the details of the NN
operationsuch as architectures and learning ruleshas more to do with mathematical
reasoning than biological mimicry. In the research being reported on herein, NNs are
used primarily as a trainable statistical modeling tool.
f From this point forward, artificial neural networks will be referred to simply as
neural networks with the understanding that the networks being discussed are
computationaland not biologicalin nature.
119
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS iii
ABSTRACT viii
CHAPTERS
1 INTRODUCTION 1
1.1 Background 1
1.1.1 Computer Assisted Bridge Modeling 1
1.1.2 Computational Aspects of Highway Bridge Analysis 3
1.2 Present Research 6
1.2.1 Computer Assisted Bridge Modeling 7
1.2.2 RealTime Data Compression 9
1.2.3 Neural Network Equation Solver 12
1.3 Literature Review 15
1.3.1 Computer Assisted Bridge Modeling 15
1.3.2 Data Compression in FEA 18
1.3.3 Neural Network Applications in Structural Engineering 18
2 A PREPROCESSOR FOR BRIDGE MODELING 21
2.1 Introduction 21
2.2 Overview of the Bridge Modeling Preprocessor 22
2.3 Design Philosophy of the Preprocessor 25
2.3.1 Internal Preprocessor Databases 25
2.3.2 The Basic Model and Extra Members 26
2.3.3 Generation 28
2.3.4 The Preprocessor History File 29
2.4 Common Modeling Features and Concepts 30
2.4.1 Bridge Directions 31
2.4.2 Zero Skew, Constant Skew, and Variable Skew Bridge Geometry ...32
2.4.3 Live Load Models and Full Load Models 33
2.4.4 Live Loads 34
2.4.5 Line Loads and Overlay Loads 36
2.4.6 Prismatic and Nonprismatic Girders 37
IV
169
[0,1] and [1,1] respectively, all output parameters must be encoded into one of these
two ranges. While network input parameters do not have to be encoded, there are
several benefits to doing so. Therefore, in the present research, input parameters as
well as output parameters are encoded.
As will be seen in the following sections, most of the input parameters of the
neural networks consist of encoded bridge coordinates. Load locations and
displacementsampling locations are encoded in a manner that is not only suitable, but
preferable for use with neural networks. Figure 6.8 illustrates the normalized bridge
coordinate system which was defined to accomplish this encoding. Normalized
Xcoordinates (lateral positions) were defined simply as varying linearly from a value
of X=0.0 at the left edge of the bridge to a value of X = 1.0 at the right edge.
Encoding the Ycoordinates could have been accomplished in the same
mannerlinearly varying the coordinates from a value of Y=0.0 at the beginning of
the bridge to a value of Y = 1.0 at the end. This, however, would have been a poor
choice of encoding. To understand why, consider a simple example. Assume we have a
60 ft. bridge with span lengths of 40 ft. and 20 ft. for the first and second spans
respectively. Assume further that a load is placed at a location 30 ft. from the
beginning of the bridge. In this case the normalized Ycoordinate of the load would be
Y =0.5=30/60. Using this encoding scheme, assume that a network is trained to learn
the loaddisplacement relationship for the bridge.
222
Figure 7.11 Diagonal Seeded Convergence of Displacements (TzTranslations)
and Residuals (FzForces) for a Flatslab Bridge Under Uniform
Loading
244
Hays, C., Hoit, M., Selvappalam, M., Vinayagamoorthy, M., Consolazio, G. (1990).
Bridge Rating of GirderSlab Bridges With Automated Finite Element Modeling
Techniques, Structures and Materials Research Report No. 903, Interim
Research Report, Engineering and Industrial Experiment Station, University of
Florida, Gainesville, Florida.
Hays, C., Miller, M. (1987). Develop Modifications and Enhancements to Existing
Bridge Rating Programs SALOD and FORCE, Structures and Materials
Research Report No. 871, Engineering and Industrial Experiment Station,
University of Florida, Gainesville, Florida.
Hays, C., Schultz, L. (1988). Verification of Improved Analytical Procedures for
Design and Rating of Highway Bridges, Structures and Materials Research
Report No. 881, Final Research Report, Engineering and Industrial Experiment
Station, University of Florida, Gainesville, Florida.
HechtNielsen, R. (1991). Neurocomputing, AddisonWesley Publishing Company,
Reading Massachusetts.
Hegazy, T., Fazio, P., Moselhi, O. (1994). Developing Practical Neural Network
Applications Using BackPropogation, Microcomputers in Civil Engineering,
Vol.9, No.2, pp. 145159.
Hoit, M.I., Stoker, D., Consolazio, G.R. (1994). Neural Networks for Equation
Renumbering, Computers and Structures, Vol.52, No.5, pp.10111021.
Hoit, M.I., and Wilson, E.L. (1983). An Equation Numbering Algorithm Based on a
Minimum Front Criteria, Computers and Structures, Vol.16, pp.225239.
Hughes, T.J.R. (1987). The Finite Element Method Linear Static and Dynamic Finite
Element Analysis, Prentice Hall, Englewood Cliffs, NJ.
Jacobs, R.A. (1988). Increased Rates of Convergence Through Learning Rate
Adaption, Neural Networks, Vol.l, pp.295307.
Jennings, A., McKeown, J.J. (1992). Matrix Computation, 2nd Ed., Chapter 11 :
Iterative Methods for Linear Equations, John Wiley & Sons, New York.
Kershaw, D.S. (1978). The Incomplete CholeskyConjugate Gradient Method for the
Iterative Solution of Systems of Linear Equations, Journal of Computational
Physics, Vol.26, pp.4365.
59
development of composite action between the slab and the girder is provided by friction
and the use of mechanical shear connectors.
In a noncomposite girderslab system, there is a lack of bond between the top of
the girder and the bottom of the slab. As a result, the two elements are allowed to slide
relative to each other during deformation and do not act as a single composite unit.
Only vertical forces act between the two elements and there is a discontinuity of strain
at the boundary between the elements.
The preprocessor allows the engineer to model the girderslab interaction as
either noncomposite, or as composite using one of two composite modeling techniques.
The girderslab interaction models available in the preprocessor are illustrated in
Figure 3.4.
Noncomposite action is modeled using the noncomposite model (NCM) in
which the centroid of the girder is effectively at the same elevation as the centroid of
the slab. The section properties specified for the girders are those of the bare girders
alone. In this model the primary function of the slab elements is to distribute the wheel
loads laterally to the girders, therefore plate bending elements are used to model the
deck slab.
Composite action between the slab and the girder is modeled in one of two ways
using the preprocessor. One way involves the use of the composite girder model
(CGM) and the other the eccentric girder model (EGM). These composite action
models are also illustrated in Figure 3.4.
135
Figure 5.5 Hypothetical Error Surface for a Network Having Two Weights
Then, from the starting point, a vector is constructed pointing in the direction of
the steepest descent, i.e. the direction in which the error surface drops off in elevation
most quickly. The next point (set of connection weights) to be examined will lie
somewhere along that vector. A stepsize parameter controls how far along the vector
the next point chosen will be. Once the move to the next point has been madei.e. the
new values of the design variables have been computedthe whole process repeats.
Thus, the minimum point ishopefullylocated by iteratively moving down the error
surface in finite size steps.
While the method of steepest descent has the appeal of being conceptually
simple, it suffers from many problems when applied to practical network training
situations. Therefore, several variations on the approach have been developed which
attempt to overcome the problems associated with steepest descent. While still using
gradient information, these hybrid methods also bring in additional information and
31
bridges. The features and concepts discussed below are common to manyor allof
these bridges types.
2.4.1 Bridge Directions
In discussing the preprocessor, the meaning of certain terminology regarding
bridge directions must be established. In this context, the longitudinal direction of a
bridge is the direction along which traffic moves. The lateral direction of the bridge is
the direction perpendicular to and ninety degrees clockwise from the longitudinal
direction. Finally, the transverse direction is taken as the direction perpendicular to the
bridge deck and positive upward from the bridge. These directions are illustrated in
Figure 2.3. The lateral, longitudinal, and transverse bridge directions correspond to the
global X, y, and zdirections respectively in the global coordinate system of the finite
element model.
Longitudinal
Direction
o
0
Plan
View
Of
Truck
Plan View
Of Bridge
Lateral
Direction
(YDirection)
Figure 2.3 Lateral, Longitudinal, and Transverse Bridge Directions
48
modeling hinge connections for constant skew steel girder bridges. Hinges are assumed
to run laterally across the entire width of the bridge, thus forming a lateral hinge line at
each hinge location. Along a hinge line, each of the girders contains a hinge connection
and the deck slab is assumed to be discontinuous. Modeling the slab as discontinuous
across the hinge line is consistent with the construction conditions of an expansion
joint.
2.6.3 Concrete Creep and Composite Action
Long term sustained dead loads on a bridge will cause the concrete deck slab to
creep. Concrete creep is time dependent nonrecoverable deformation that occurs as the
result of sustained loading on the concrete. Over time, the concrete will flow similar to
a plastic material and will incur permanent deformation.
In a steel girder bridge the deck slab and girders are constructed from materials
that have different elastic moduli and different sustained load characteristics. Steel has a
higher elastic modulus than concrete and does not creep under sustained loads as
concrete does. If long term dead loads are applied to the bridge after the concrete deck
slab and steel girders have begun to act composite^, the slab will be subjected to
sustained loading and creep will occur. As the deck slab undergoes creep deformation
but the steel girders do notmore and more of the load on the bridge will be carried by
the girders and girder stresses will consequently increase. Creeping of the deck slab
1 Long term dead loads that are applied after composite action has begun include the
dead weight of structural parapets, line loadssuch as the weight of a railing or a
nonstructural parapet, and deck overlay loads.
196
In banded preconditioning, a banded slice of the matrix A is extracted and
inserted into an otherwise empty matrix to form M (see Figure 7.2). In the author's
implementation of this preconditioning scheme, the width of the band to be extracted
from A is specified as part of the FEA control parameters. The width may be specified
either as an absolute width or as a fraction of the total number of equations in the
system. Thus, different band widths can be examined to determine appropriate
parameters for a particular type of bridge structure. Also, note that although the portion
extracted from the matrix A is called a banded slice, this data is inserted into M in
profile (skyline) format.
Here again, theoretically, one would need to form the preconditioning matrix as
P = M 1 and premultiply each side of the matrix equation by P to precondition it.
However, unlike the case of diagonal preconditioning, the matrix inverse M l will not
be easy to compute in general. In practice, and in the authors implementation,
33
2.4.3 Live Load Models and Full Load Models
Broadly speaking, there are two basic classes of bridge models that can be
created by the preprocessorlive load models and full load models. Live load models
are used primarily to compute lateral load distribution factors (LLDFs) for girderslab
bridges (see Hays et al. 1994 for more information regarding LLDFs). A live load
model represents only the final structural configuration of a bridgethat is the bridge
configuration that is subjected to live vehicle loads.
By contrast, a full load model is actually not a model at all but rather a series of
models that represent the different stages of construction of a single bridge. Full load
models are analyzed so that a bridge rating can subsequently be performed using the
analysis results. Each of the individual construction stage models, which collectively
constitute a full load model, simulates a particular stage of construction and the dead or
live loads associated with that stage. After all of the construction stage models have
been analyzed a rating may be performed by superimposing the force^ results from
each of the analyses. This is a very important pointeach analysis considers only
incremental loads, not accumulated loads. In fact, this procedure must be used in order
to account for locked in forces, i.e. forces that are developed at a particular stage of
construction and locked into the structure from that point forward.
The last construction stage model in any series of full load models is always a
live load model, i.e. a model representing the final structural configuration of the
^ In this context, the term force is used in a general sense to mean either a shear force,
axial force, bending moment, shear stress, axial stress, or bending stress.
201
the large magnitude negative terms will be included in M. As a result, the matrix M
does not really approximate A and M~l is a poor preconditioner for A .
In theory, one could remedy this situation by increasing the size of the band so
as to include the large negative terms. However, the bandwidth chosen would then have
to include the vast majority of the matrix A Recall that we must be able to efficiently
solve subproblems of the form Mq = r within the PCG iteration process. If M is
virtually the entire matrix A, then the cost of solving Mq = r will be roughly
equivalent to solving the original system Ax = b and there is no point to using an
iterative process at all.
7.4.2 Incomplete Choleskv Decomposition Preconditioning
It is apparent from the discussion above that the diagonal and band
preconditioning schemes will only be effective for cases in which the matrix A is
diagonally dominant or nearly so. An alternative preconditioning scheme must be
developed for cases such as FEA of bridge structures. The incomplete cholesky
decomposition (ICD) preconditioning scheme offers just such an alternative. When this
preconditioning scheme is combined with the standard CG algorithm, the resulting
algorithm is called an ICCG (Incomplete Cholesky Conjugate Gradient) algorithm.
Therefore ICCG solvers are just a specific type of PCG solver. From this point
forward, the ICCG algorithm will be referred to as the ICPCG (Incomplete Cholesky
Preconditioned Conjugate Gradient) algorithm to be consistent with terminology that
174
computed neuron output values were scaled up by a factor of 1.2 to produce data again
in the range [1,1]. Automatic scaling of network input and output parameters is one of
the features built into the NetSim software (see Chapter 5). As a result, the neural
network code generated automatically by NetSim handles this scaling internally.
Application code calling the neural network modules never needs to be concerned with
the scaling.
Training the normalized shape networks was accomplished using the NetSim
software. The 2250 loaddisplacement pairs described earlier in this chapter were used
as training data. Table 6.2 lists the relevant parameters of the final trained networks.
Two types of error statistics are reported in the tablemaximum error and average
error. The maximum error statistic is a worst case measure of network error. Each of
the 2250 training pairs used for training will have a different error associated with it.
One of these 2250 pairs will have an associated error that is larger than that of all the
other pairs. It is the error for this worst case training pair that is reported as the
maximum error in the table. The average error, which is the sum of the errors over all
of the training pairs divided by the number of training pairs, is also reported in the
table.
The table indicates, that while the worst case errors encountered during network
training were significant, the networks performed very well on average. Whereas the
largest of the maximum errors encountered was approximately 33%, the largest of the
average errors was only around 3% indicating that the network errors were very small
in the vast majority of cases.
217
Rotation Rx (radians)
Rotation Ry (radians)
Figure 7.7 Neural Network Predicted Displacements and Analytical (FEA)
Displacements for a Flatslab Bridge Under Vehicular Loading
237
running on. However, whether or not the savings in I/O time will completely or only
partially offset the time spent compressing and decompressing data will depend on
characteristics of the problem and the computer platform. In particular, systems in
which disk I/O can become a performance bottleneck will usually benefit greatly from
data compression. Personal computers (PCs) running the DOS operating system tend to
fall into this category.
The data compression strategy presented in this study was implemented using
the C programming language. It was installed in the SEAS FEA package (which is also
written in C) for testing. An interface layer was developed (also in the C language) to
allow the data compression library to be called directly from software written in
Fortran. Using this interface, data compression was implemented and tested in a
Fortran coded FEA package that is used in the analysis of highway bridges. Tests
results indicated that Fortran I/O is often less efficient than C I/O and as a result FEA
software written in Fortran can greatly benefit from data compression. In tests
performed using moderate size bridge models, analysis execution times often decreased
by a factor of 2 to 3. Simultaneously, there was a (roughly) order of magnitude
decrease in outofcore storage requirements.
Due to the nature of the buffering scheme used in the data compression library,
only sequential data files can be compressed. An area of future research which should
be pursued is that of adapting the buffering algorithm to also handle direct access files.
Once this is done, data compression will be able to be applied in more applications than
is currently possible. Also, the use of data compression for incore memory
compression is currently being examined. Some preliminary work has already been
154
pressure, selfweight). The networks must be constructed considering all of these
possibilities, otherwise they will have only limited applicability.
In the bridges studied in this research only gravity loads are considered,
therefore externally applied vertical forces are present but externally applied moments
are not. Moment loading still needs to be considered, however, if one deals with
iterative processes involving outofbalance (residual) forces. Such forces arise when
using iterative equation solving algorithms and also when dealing with nonlinear
analysis. Since the neural networks presented in this chapter were created with the
intention of installing them in an iterative equation solver (see Chapter 7), moment
loading had to be considered.
In the flatslab bridge models studied, plate bending elements were used to
model the slab. Therefore, at each slab node of the model, there were three active
degrees of freedom (DOF)one outofplane translation (Tz) and two inplane rotations
(Rx, Ry). The coordinate system used in the bridge models is such that the Xdirection
is the lateral direction, the Ydirection is the longitudinal direction (the direction of
vehicular travel), and the Zdirection is the transverse direction (perpendicular to the
slab and positive as one moves vertically upward). Thus, three types of loads can occur
and must be consideredvertical forces (Fz), moments about the xaxis (Mx), and
moments about the yaxis (My). Gravity loads will virtually always be represented as
vertical forces (Fz), but residual forces may cause any of the three load types (Fz, Mx,
or My).
BIOGRAPHICAL SKETCH
The author was born in Washington, D.C., in 1966 and grew up in Derwood,
Maryland, until 1978, at which time his family moved to Brooksville, Florida. After
attending junior high and high school in Brooksville, he began attending the University
of Florida in 1984 as a freshman. In 1988 he and his wife Lorraine were married in
Micanopy, Florida. During the following year, 1989, he received the degree of
Bachelor of Science in civil engineering from the University of Florida.
After working briefly for a civil engineering consulting firm, he entered the
graduate school of the University of Florida from which he received the degree of
Master of Engineering in 1990. He then began work on a doctorate, also at the
University of Florida. During the course of his doctoral work, he and his wife had two
childrena son, Michael, born in 1992, and a daughter, Molly, born in 1995. In 1995,
he received the degree of Doctor of Philosophy from the University of Florida after
which he joined the faculty at Rutgers University, in New Jersey, as an Assistant
Professor in the Department of Civil and Environmental Engineering.
247
108
The fact that the normalized execution times in the plot are all greater than 1.0
indicates that the use of data compression increased the quantity of time required to
perform the analyses. This is not always the case (see Consolazio 1994) and it will be
shown in the next section that in many cases the use of data compression can
substantially reduce the required execution time. However, as a general rule, when data
compression is used in FEA software coded in C, there can be a modest penalty in the
form of increased execution time.
The increase in execution time arises because, when using compressed I/O,
additional computational work must be performed to compress and decompress the data
during the analysis. Under favorable conditions, the added amount of time required for
i
H
Z
1.35
1.30
1.25
1.20
1.15
1.10
1.05
1.00
8000 10000
Buffer Size (bytes)
12000
14000 16000
Zero Skew / 512 Hash / Workstation (UNIX)
Zero Skew / 2048 Hash / Workstation (UNIX)
Zero Skew / 4096 Hash / Workstation (UNIX)
Variable Skew / 512 Hash / Workstation (UNIX)
Variable Skew / 2048 Hash / Workstation (UNIX)
Variable Skew / 4096 Hash / Workstation (UNIX)
Figure 4.5 SEAS Execution Time Results for Workstation Analyses
93
operations will be copied to the end of the previous blocks of data so that data in the
I/O buffer is accumulated. Once the I/O buffer is full, its contents are compressed into
the compression buffer and then the compressed block is written to disk as is illustrated
in Figure 4.1.
Buffering is necessary to achieve a high degree of compression because if the
library were to perform compression each time a write operation were performed, only
repetitions within the small block of data being written could be compressed. In the
example of a plate element stiffness matrix, which might be written using one write for
each row in the matrix, only repetitions within a row could be compressed. If however,
buffered I/O is used and the buffer size is large enough to hold larger portions of the
matrix, then higher degrees of compression can be achieved because the larger blocks
containing repeated information are simultaneously in the I/O buffer when it is
compressed. During read operations, the process is reversed so that compressed data
I/O Buffer Size
1. Copy data block to empty
I/O buffer.
2. Append data block to I/O
buffer.
3. Append partial data block
to I/O buffer.
4. Compress the I/O buffer
into die compression
buffer.
5. Write the compressed
block to disk
6. Copy the remaining portion
of the data block to the
emptied I/O buffer.
7. Append data block to I/O
buffer. Repeat.
Figure 4.1 Buffered and Compressed Writing Procedure
218
In neural network seeding, of course, the solution vector is estimated by having
the networks compute their prediction of the displaced shape of the bridge under the
specified loading condition.
In order to isolate the effect of the seeding method from other factors in the
solution processsuch as the choice of preconditioning schemean ICPCG equation
solver was used for all of the seeding tests performed. Thus, precisely the same
incomplete cholesky preconditioning scheme was used in each case.* Each of the two
loading conditions described earlier were analyzed using each of the three seeding
methods for a total of six tests. The RMS (root mean square) of the residual load vector
was chosen as a scalar measure of the solution error at each iteration.* The results of
these tests are plotted in Figures 7.8 and 7.9.
In both cases, it is clear that neural network solution seeding is a highly
effective technique for accelerating convergence. Compared to the other two seeding
methods, the RMS load error for the neural network seeded case drops off quickly and
more or less monotonically. In addition, there are a few interesting and important
observations to be made regarding these plots.
* To prevent the formation of negative diagonal terms during the incomplete
decomposition process, diagonal terms in the approximation matrix M were scaled
by a factor of 1000 prior to decomposition. Offdiagonal terms in M were left
unaltered.
* In practice, both an RMS load error and an RMS displacementchange error should
be used in monitoring convergence. Use of an RMS load error is critical because it
provides a quantitative measure of how far away from equilibrium the structure is at a
particular iteration. In the author's implementation of the NNPCG solver, both of
the error statistics are computed and reported. However, for purposes of comparing
different seeding methods, examining only the RMS load error is sufficient.
52
composite action is modeled with the eccentric girder model (EGM), flat shell elements
are used. The shell element combines plate bending behavior and membrane behavior,
however the membrane response is not coupled with the plate bending response. The
thickness of the slab elements is specified by the engineer and is assumed to be constant
throughout the entire deck except over the girders, where a different thickness may be
specified.
The plate bending elements and the bending (flexural) portion of the flat shell
elements used in the present bridge modeling are based on the ReissnerMindlin thick
plate formulation (Hughes 1987, Mindlin 1951, Reissner 1945). In the Reissner
Mindlin formulation, transverse shear deformations, which can be significant in thick
plate situations such as in flatslab bridge modeling, are properly taken into account.
Consolazio (1990) studied the convergence characteristics of isoparametric elements
based on the thick plate formulation and found that these elements are appropriate for
bridge modeling applications.
While typical isoparametric plate and shell elements may generally have
between four and nine nodes, bilinear (four node) plate and shell elements are used for
all of the bridge models created by the preprocessor. This choice was made for a
number of reasons. Because vehicle axle loads occur at random locations on a bridge,
accurately describing these axle loads requires a substantial number of nodes in the
longitudinal direction. It is generally suggested that when using the preprocessor at
least twenty elements per span be used in the longitudinal direction. Use of biquadratic
(nine node) elements in models following this suggestion would require substantially
8
3.Computing potentially hundreds of discrete vehicle positions and
subsequently computing and specifying the load data required for FEA.
All of these barriers are overcome through the use of the preprocessor because it
handles these tasks in a semiautomated fashion. The term semiautomated, which is
used synonymously with computer assisted in this dissertation, alludes to the fact that
there is an interaction between the engineer and the modeling software. Neither has
complete responsibility for controlling the creation of the bridge model. General
characteristics of bridge structures and bridge loading are built into the preprocessor so
as to allow rapid modeling of such structures. However, the engineer retains the right
to introduce engineering judgmentwhere appropriateinto the creation of the model
by interacting with the software. Thus, the engineer is freed from the tedium of
manually preparing all of the data needed for FEA and allowed to focus on more
important aspects of the rating or design process.
In addition to handling the primary modeling tasks discussed above, the
preprocessor handles numerous other tasks which are required in bridge modeling. The
most important of these are listed here.
1. Modeling composite action between the girders and slab, in some cases
including the calculation of composite girder section properties based on the
recommended AASHTO (AASHTO 1992) procedure.
2. Modeling pretensioning and posttensioning tendons, including the
specification of finite element end eccentricities.
3. Modeling variable cross section girders, including the generation and
calculation of all necessary cross sectional properties and eccentricities.
4. Modeling complex bridge geometry such as variable skew.
5. Modeling live loading conditions considering not only a single standard
vehicle but often several different standard vehicles.
199
In addition, the diagonal terms of the matrix are guaranteed to be positive and are
generally large (in magnitude) relative to the off diagonal terms.
Recall from earlier discussion that the matrix M must be a good
approximation of A if P = M~l is to be an effective preconditioner for the system
Ax = b. Therefore the question becomes, Is a diagonal slice M or banded slice M
of the matrix A sufficiently representative of the information content of AT. The
answer to this questionat least for the bridge structures studiedis No.. A slice of
the stiffness matrix does not contain sufficient information to approximately represent
the information content of the overall stiffness matrix. This fact is directly attributable
to the structure of sparsity in the matrix.
In the bridge models studied, the stiffness matrices have a structure similar to
that illustrated in Figure 7.3. One can see that there are essentially two distinct bands
of nonzero terms in the matrix separated by a void of zero terms. There are two
distinct bands of data (not three) because the matrix is known to be symmetric and
therefore the lower band is known to be the mirror image of the upper band. As a
result, the lower band introduces no new information. Although the sparsity pattern
illustrated is for a small bridge modelapproximately 150 degrees of freedomthe
sparsity patterns for larger bridge models is very similar. The primary difference is that
the bands of nonzero terms become increasingly narrowerrelative to the overall size
of the matrixas the bridge models grow in size.
Figure 7.3 illustrates all of the nonzero terms in the matrix including terms
having large magnitude as well those having small magnitude. The largest magnitude
4
as secondary storage, low speed storage, and backing store. Incore storage generally
refers to the amount of physical random access memory (RAM) available on a
computer, although on computers running virtual memory operating systems there can
also be virtual incore memory. Outofcore storage generally refers to available space
on hard disks, also called fixed disks.
Optimizing the use of available incore storage has been an area of considerable
research during the past few decades. In contrast, little research has been performed
that addresses the large outofcore storage requirements often imposed by FEA. Out
ofcore storage is used for three primary purposes in FEA :
1. To hold temporary data such as element stiffness, load, and stress recovery
matrices (collectively referred to as element matrices) that exist only for the
duration of the analysis.
2. To hold analysis results such as global displacements and element stresses
that will later be read by postprocessing software.
3. To perform blocked, outofcore equation solutions in cases where the
global stiffness or global load matrices are too large to be contained incore
as a single contiguous unit.
In cases 1 and 3, once the analysis is complete the storage is no longer needed, i.e. the
storage is temporary in nature. In case 2, the storage will be required at least until the
analysis results have been read by postprocessing software.
In the analysis of highway bridges, the amount of outofcore storage that is
available to hold element matrices can frequently become a constraint on the size of
model that can be analyzed. It is not uncommon for a bridge analysis to require
hundreds of megabytes of outofcore storage during an analysis. Also, as a result of
the proliferation of low cost personal computers (PCs), there has been a migration of
184
ability of the neural networks to compute accurate displacements in flatslab bridges is
a topic covered in detail in that chapter, presentation of neural network test results will
be delayed until that time.
112
described aboveare shown in Figure 4.6. Also shown in the figure are the results
from cases in which the stress files were compressed. One can see that there is a
reduction of approximately 10 percent in execution time when the stress files are not
compressed. This reduction in execution time results from the fact that no effort is
being put forth to compress the stress fileseffort which equates to added execution
time. Since RDC does not greatly reduce size of the stress data, it is recommended that
this type of data should not be compressed.
4.8.2 Data Compression in FEA Software Coded in Fortran
To evaluate the effectiveness of data compression in FEA software coded in
Fortran, the author modified a version of the SIMPAL programwritten by Dr. Marc
Hoitto use compressed C I/O instead of the normal Fortran I/O. Recall that SIMPAL
is the Fortran FEA module of the BRUFEM system that was described in earlier
chapters. The program was converted from standard Fortran I/O to compressed I/O by
implementing both the compressed I/O and Fortran interface libraries described earlier
in this chapter.
Since SIMPAL serves as the FEA engine of the BRUFEM system, the test
models used in this portion of the data compression study were full size bridge models
created by the preprocessor described in Chapters 2 and 3. The relevant parameters of
each of the models are listed in Table 4.5. The first bridge is a twospan prestressed
concrete girder bridge having zero skew geometry, pretensioning tendons, post
tensioning tendons, and temporary supports. The second bridge is a threespan steel
girder bridge having variable skew geometry and nonprismatic girders. The finite
138
modified in multiple layer neural network was not solved for a number of years. The
development of the backpropagation algorithm by Rumelhart et. al (1986) solved this
problem by using an backward error propagation procedure based on the chain rule of
derivatives. In backpropagation, the error data computed at the output layer of the
network is propagated backward through the network and combined with computed
transfer function derivatives. When combined, these two pieces of data allow for the
computation of the partial derivative of E with respect to connection weights in hidden
layers of the network. Once these partial derivatives are computed, a steepest descent
approach may again be used to modify the corresponding connection weights.
For a detailed description of the mathematics behind backpropagation, the
reader is referred to any one of the numerous texts written on the subject. The main
objective here is to outline and discuss the relative merits of the method and its many
variations. It is important to understand that in the pure version of backpropagation just
described, the connection weights in the network are only updated once per epoch. That
is, a complete epoch must be completed before any network learning takes place. This
fact derives from the definition of E. Since E was defined as the error over the entire
training set, for backpropagation to be a true steepest descent approach, it can only be
applied to the errors that are accumulated over an entire epoch.
When there are a large number of training pairs in the training setas is the
case in the present researchpure backpropagation can be a very slow process. Some
of the variant backpropagation methods employed in the NetSim software package are
described in the following sections. While these variant methods are not true steepest
100
function with the correct argument list to perform the requested operation. The
interface library simply serves as a pathway between Fortran and C, with the actual
compression and disk I/O operations still being performed by the compressed I/O
library as described earlier.
Unlike the compressed C I/O library functions, which attempt to closely parallel
the behavior of the standard ANSI C I/O functions they replace, the Fortran interface
functions do not attempt to provide all of the functionality of standard Fortran I/O
statements. For example, the ERR and END parameters provided in the Fortran READ
and WRITE statements are not provided in the interface. Also, the Fortran file modes
NEW, OLD, and UNKNOWN must be replaced with C language file modes in the
calls to the interface library. Minor changes must also be made to Fortran I/O
statements to accommodate the fact that C I/O functions write and read blocks of
contiguous memory only. Therefore Fortran write and read operations which contain
implicit loops must be modified to perform block write and read operations in which all
of the data within a single block is contiguous in memory.
Table 4.2 FortranCallable Interface Library Functions
F77Callable
Interface
Function
Compressed
I/O Library
Function
Differences between Compressed and Fortran I/O
copen
CioOpen
File modes are specified as C modes, not Fortran
cwrite
CioWrite
Must write contiguous blocks of memory
cflush
CioFlush
Not provided in standard Fortran
crewind
CioRewind
Same basic functionality as standard Fortran
cread
CioRead
Must read contiguous blocks of data
celse
CioClose
Same basic functionality as standard Fortran
emode
CioModify
Can only modify default characteristics
179
Tz (inches) Geometry 0.
0.020
Normalized^'
Lateral 0.75
Direction 1.00
Tz (inches) Geometry 1.2
0.020,
Lateral 0.75
Direction 1.00'
Normalized Longitudinal Direction
Normalized Longitudinal Direction
Figure 6.11 Maximum Magnitude Translations (Tz) Caused By Unit Forces (Fz)
(Training Data for Scaling Neural Networks)
Rx (radians) Geometry 0.6
1.00*04
8.00e05
Rx (radians) Geometry 1.0
1.00*041
Rx (radians) Geometry 1.2
1.00*04
8.00eO5
0.25'
Normalized ^^
Lateral 0.
Direction 1.00'
Normalized Longitudinal Direction
Normalized Longitudinal Direction
Figure 6.12 Maximum Magnitude Rotations (Rx) Caused By Unit Forces (Fz)
(Training Data for Scaling Neural Networks)
106
compressed I/O library was able to reduce the outofcore storage requirements of the
zero skew and variable skew bridge analyses to approximately 7 percent and 12
percent, respectively, of their original sizes.
Also, note that although there is really only a single unique plate element in the
zero skew modeli.e. all of the plate elements in the model are identical to each
otherthe compression algorithm was not able to fully capitalize on this fact. If the
compression algorithm had recognized this largescale repetition, only one of the two
hundred plate elements making up the model would have needed to be stored. This
would have resulted in a compression ratio on the order of 1/200 = 0.005, or about 0.5
percentconsiderably less than the 7 percent achieved. Earlier in this chapter (see
Â§4.3), it was stated that the compression strategy presented herein is not capable of
69
The lateral bending stiffness of these elements is assumed to be that of a wide
beam of width SY (SY/2 for elements at the ends of the bridge). From plate theory
(Timoshenko and WoinowksyKrieger 1959) the flexural moment in a plate is given by
(3.2)
where Mx is the moment per unit width of plate, E is the modulus of elasticity, t is
the plate thickness, v is Poissons ratio, and <(>* and (> v are the curvatures in the x
and ydirections respectively. Since the value of Poissons ratio for concrete is small
be assumed that the quantity (1 v21 is approximately unity. Also,
(v = 0.15), it can
since only bending in the lateral direction (xdirection) is of interest for the lateral beam
members, only the xcurvature Â§x is taken into consideration. From Equation (3.2)
and the simplifications stated above, the moment of inertia of a plate element having
thickness t is
iJ(SY) <33)
12
Since the moment of inertia of the slab is automatically accounted for through the
inclusion of the plate elements in the bridge model, the effective moment of inertia of
the lateral beam element is given by
(3.4)
where t(sg+ef) 's the thickness of the slab over the girder plus the effective flange
thickness of the girder and t^ is the thickness of the slab of the girder.
236
8.2 Data Compression in FEA
A realtime data compression strategy appropriate for inclusion in FEA software
has been developed, implemented, and extensively tested. In most analysis situations
the data generated by FEA software is rich in repetition. By integrating data
compression techniques directly into FEA software, it has been shown that the amount
of outofcore storage required by finite element analyses can be vastly reduced. This is
especially true in cases where the FEA model is highly regular geometrically or
otherwise. Just such a situation arises when computer assisted modeling software is
used because this type of software generally creates very regular finite element meshes.
Thus, models created by software such as the preprocessor developed in this
study are prime candidates for the application of data compression. Parametric studies
revealed that the outofcore storage requirements for such analyses could be reduced
by an order or magnitude in many cases. As an additional benefit of the use of data
compression, the execution time required for many of the analysis situations examined
was also reduced. This benefit is attributable to the fact that, when using data
compression, a substantially reduced amount of disk input/output (I/O) must be
performed. In many cases, the amount time spent compressing and decompressing data
during the analysis is more than offset by the time saved by performing less disk
activity.
Whereas a reduction in outofcore storage is virtually guaranteed by the use of
data compression, a reduction in the required execution time is not. The degree of
compression achieved is independent of the computer platform that the software is
159
c). This does not have to be the case and is in fact not the case in the flatslab
bridges studied in this research. Finally, note that in this example, the shape ordinates
will lie in the range [0,1] since there are no negative displacements (assuming positive
is vertical downward). Clearly, this will not always be the case and in genera! the shape
ordinates will lie in the range [1,1].
The steps listed above are illustrated in Figure 6.4. Unit loads are applied at
each of the loading points and the vertical displacements computed. The maximum
magnitude displacements Aamax, and Acnlax are determined and used to build
the maximum displacement curve shown. For a given location on the beam, the height
of this curve is equal to the magnitude of the maximum displacement that occurs when
a unit load is applied at that location. Each unit load application generates a single point
on this curve. The set of all such generated points constitutes a training data set for
constructing scaling neural networks. In this simple example, three training pairs are
generated (see Table 6.1). Scaling networks take load locations as input and return
corresponding magnitude scaling factors (maximum magnitude) as output.
The formation of normalized displacement shapes is also illustrated in
Figure 6.4 where three shape are generated by the application of three units loads. In
each of these cases, the displacements are scaled so that the largest magnitude term in
each shape is unity. In the example problem, the displacementsampling locations are at
the same locations as the load points. Therefore, there are a total of nine training pairs
generated for network training (see Table 6.1). Shape networks take a load location and
105
medium and smallscale repetition in the data produced by both of the models. Thus,
these models were appropriate for examining the effects of model complexity on the
ability of the compression library to reduce outofcore storage requirements.
Also of interest was determining what effect the buffer size and hash table size
had on the compression of the data. Thus, each of the flatslab models were analyzed
using compressed I/O with buffer sizes of 64, 128, 256, 512, 1024, 2048, 4096, 8192,
12288, and 16384 bytes, and hash table sizes of 512, 2048, and 4096 elements (where
each element consisted of a four byte pointer).
In addition, the models were also analyzed using SEAS in a mode which uses
the standard C I/O functions instead of the compressed I/O functions. The file size and
execution time data from these two standard C I/O runs served as reference data against
which the compressed runs were evaluated.
A normalized compression ratio was computed for each compressed I/O run by
dividing the outofcore storage requirements of the compressed analysisdefined as
the total size of the element stiffness, load, force recovery, and stress filesby the out
ofcore usage required when analyzed by SEAS in the standard C I/O mode. Thus, a
compression ratio of 0.25 would indicate that the compressed outofcore storage
requirements were only 25 percent of the normal outofcore storage requirements of
the software.
Compression ratio results for the SEAS runs are plotted in Figure 4.4. As one
might anticipate, the plot confirms that a greater degree of compression can be
achieved in regular, highly repetitive models than in more irregular models. The
50
2.8 Modeling Features Specific to FlatSlab Bridges
Flatslab bridges (see Figure 2.11) consist of a thick reinforced concrete deck
slab and optional reinforced concrete edge stiffeners. Thus, unlike all of the other
bridge types modeled by the preprocessor, there are no girders in flatslab bridges.
However, there can still be composite action between the deck slab and edge stiffeners
such as parapets, if such stiffeners are present and considered structurally effective.
Support conditions for flatslab bridges are also unique among the bridge types
modeled by the preprocessor. Flatslab bridges are supported continuously across the
bridge in the lateral direction at each support location. This is in contrast to girderslab
bridges in which supports are only provided for girder elements and the remainder of
the bridge is assumed to be supported by the girders.
Concrete Concrete Flat Slab
Parapet Deck Slab Thickness
I I I I I I I I I I I I I I I i i i i
Figure 2.11 Cross Section of a Typical Flatslab Bridge
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Loc VuQuoc
Associate Professor of Aerospace Engineering,
Mechanics, and Engineering Science
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
William J. Sullivan
Assooiflte Professor of Germanic and Slavic
Languages and Literatures
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August 1995
Â£
Winfred M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School
36
The second live loading scenario introduced at the beginning of this section
stationary lane loadingrepresents the case in which traffic is more or less stopped on
the bridge and vehicles are very closely spaced together. Lane loading is usually
thought of as a uniform load extending over specified spans in the longitudinal direction
and over a specified width in the lateral direction. AASHTO defines lane loads as being
ten feet wide. However, because lane loading is intended to represent a series of closely
spaced vehicles, the preprocessor instead models uniform lane loads as a series of
closely spaced axles with each having a width of six feetthe approximate width of a
vehicle axle. Lane loads are described by specifying which spans the lane load extends
over, and by specifying the lateral position of the centerline of the lane.
2,4,5 Line Loads and Overlay Loads
In addition to the live load modeling capabilities provided by the preprocessor,
an engineer may also specify the location and magnitude of long term dead loads such
as line loads and uniform overlays. Dead loads due to structural components such as
the deck slab, girders, and diaphragms are automatically accounted for in the bridge
models created by the preprocessor and therefore do not need to be specified as line or
overlay loads.
Dead loads due to nonstructural elements such as nonstructural parapets or
railings can be modeled by specifying the location and magnitude of line loads. For
example, the dead weight of a nonstructural parapet may be applied to the bridge by
specifying a line load having a magnitude equal to the dead weight of the stiffener per
207
networks serve two functions, each of which is aimed at accelerating the overall
iteration process.
1. Seeding the solution vector. The initial solution vector used to start the
iterative process is generated using the neural networks.
2. Preconditioning. The neural networks are used in place of the matrix M to
precondition the matrix equation being solved.
Every iterative equation solving method begins with a common stepcomputing an
initial guess at the solution vector. The closer the initial guess is to the actual solution,
the faster convergence to that solution will be. Neural networks provide a mechanism
for computing the initial guess with little computational effort.
Since the neural networks have been trained to learn the loaddisplacement
relationship for flatslab bridges, they can be directly called upon to compute an initial
approximation of the displaced shape of a bridge. An initial preprocessing stage must
be added to the FEA engine so that for each degree of freedom (DOF) in the structural
model, the relevant neural network input parameters can be easily accessed.' After the
global load vector has been formed by the usual finite element procedures, an initial
estimate of the solution vector is computed. This is accomplished by using the principle
of superposition.
Initially, the global displacement vectori.e. the solution vectoris filled with
zeros. Then, the global load vector is scanned for nonzero terms. Each time a non
zero load term is encountered, the displacements due to that load are computed for
^ The relevant parameters include the normalized bridge coordinates and displacement
type of each DOF in the bridge. (Refer to Chapter 6 for more information.)
28
unsymmetrical bridges can be modeled. A limit of three extra left members and three
extra right members is enforced by the preprocessor.
2.3.3 Generation
In order to further reduce the amount of time that an engineer must spend in
describing a bridge, the preprocessor performs many types of generation automatically.
Generation in this context means that the user needs only to specify a small set of data
that the preprocessor will use to generate, or create, a much larger set of data needed
for complete bridge modeling. To illustrate the types of generation that the
preprocessor performs, consider the following example.
Bridges containing nonprismatic girders, i.e. girders that have varying cross
sectional shape, can be easily modeled using the preprocessor. To describe a non
prismatic girder, the user only needs to define the shape of the girder cross section at
key definition points. Definition points are the unique locations along the girder that
completely describe the cross sectional variation of the girder. In the example steel
girder illustrated in Figure 2.2, the user only needs to specify the cross sectional shapes
A1 through A6 at the six definition points. Using this data, the preprocessor will auto
matically generate cross sectional descriptions of the girder at each of the finite element
nodal location in the model. Also, the preprocessor will generate cross sectional
properties at each of these nodes and assign those properties to the correct elements in
the final model. Thus, the amount of data that must be manually prepared by the
engineer is kept to a minimum.
180
Ry (radians) Geometry 0.6
8.00e05i
6.00e05
Ry (radians) Geometry 0.
8.00e05
6.00e05
Normalized
Lateral 0.75
Direction 100
Normalized Laigitudiial Direction
Ry (radians) Geometry 1.0
8.00e05
6.00&05
Ry (radians) Geometry 1.2
8.00e05
6.00e05
Normalized
Lateral
Direction LOO'
Normalized Longitudinal Direction
Figure 6.13 Maximum Magnitude Rotations (Ry) Caused By Unit Forces (Fz)
(Training Data for Scaling Neural Networks)
T z (inches) Geometry 0.6 Tz (inches) Geometry 1.0
Tz (inches) Geometry 1.2
1.00e04
8.00e05
Normalized
Lateral 75
Direction 100
Nonnalisd Longitudinal Direction
Figure 6.14 Maximum Magnitude Translations (Tz) Caused By Unit Moments (Mx)
(Training Data for Scaling Neural Networks)
202
will be introduced later in this chapter. The prefix, IC in this case, designates the type
of preconditioning scheme used.
When a direct solution algorithm is applied to a sparse matrix such as the one
shown in Figure 7.3, a great deal of fillin occurs during the decomposition process.
Fillin refers to terms inside the matrix profile which are initially zero but which
become nonzero (i.e. fillin) as the decomposition is performed. As a result of fill
in, the sparsity of the original matrix is not preserved in the decomposed matrix. Fillin
terms often have values which are small in magnitude relative to the other terms in the
matrix. This fact forms the basis of the Incomplete Cholesky Decomposition concept
(Jennings 1992, Manteuffel 1980, Meijerink and van der Vorst 1977, Papadrakakis and
Dracopoulos 1991, Radicati di Brozolo and Vitaletti 1989). For a sparse matrix such as
the one shown in Figure 7.3, calculations involving fillin terms account for a large
proportion of the total set of calculations that must be performed during the
decomposition.
Since these terms are often small in magnitude, the ICD concept says that we
can ignore their effect during the decomposition and perhaps still get an approximate
decomposition of the original matrix. In fact, in an ICD the fillin in terms are ignored
all together. As a result, we have constrained the decomposed matrix to have the same
sparsity pattern as the original matrixalthough the values in the matrix have changed
considerably during decomposition. Because fillin is not allowed, the number of
calculations involved in the decomposition is vastly reduced from that of a true
decomposition.
149
When the method of adaptive learning rates is applied to this situation (see Figure 5.7),
the learning rate is allowed to grow progressively larger as long as the slope is not
identically zero. Thus, after several epochs of training, the step size can grow large
enough to finally make some progress in moving off the plateau.
In implementing the method of adaptive learning rates, an exponential averaging
scheme is used to determine which direction each weight has been moving recently
where recently means the past few epochs. A control parameter, directly analogous to
the p parameter in the momentum method, is used to control how much influence the
previous training history has on the current calculations. Next, the direction in which
the weight must be moved to reduce E is computed and compared to the direction the
weight has been moving recently. If these two directions have the different signs, then
the learning rate for this weight is reduced by multiplying it by a fractional valued less
than one. If the two directions have the same sign, then the learning rate is increased by
adding a constant value. Thus, reductions in learning rates can occur very quickly, e.g.
to dampen oscillations, while increases in learning rates are attained more gradually.
The method of adaptive learning rates has been implemented in the NetSim
neural network training software. The user can specify the values of the four
parameters which control the learning rate adaptation(1) the initial learning rate,
(2) the recentness factor, (3) the learning rate reduction factor, and (4) the learning
rate growth factor. In addition, NetSim allows the user to mix the batching method, the
momentum method, and the method of adaptive learning rates together to produce the
most robust training combination for the particular problem being solved. While the
40
I i
TTT''TTTT
7^
Figure 2.5 Composite Action Between a Girder and Slab
the composite girder model represents composite action by including an effective width
of slab into the calculation of girder cross sectional properties. A more accurate
approach using a pseudo three dimensional finite element model is used in the eccentric
girder model. Additional details of each of the composite action models are given in the
next chapter.
2.5 Modeling Features Specific to Prestressed Concrete Girder Bridges
Several of the modeling features available in the preprocessor relate specifically
to the modeling of prestressed concrete girder bridges (see Figure 2.6). This section
will provide an overview of those features.
2.5.1 Cross Sectional Property Databases
Databases containing cross sectional property data for standard prestressed
concrete girder sections have been embedded into the preprocessor to quicken the
modeling process and reduce errors. The databases contain cross sectional descriptions
63
elements as was done by Hays and Garcelon (Hays et al. 1991). Therefore the models
created by the preprocessor are pseudo three dimensional models.
The main deficiency of using rigid links occurs in modeling weak axis girder
behavior. The use of rigid links causes the weak axis moment of inertia of the girders
to become coupled to the rotational degrees of freedom of the deck slab. This coupling
will generally result in an overestimation of the lateral stiffness of the girders. To avoid
such a problem the preprocessor sets the weak axis moment of inertia of the girders to a
negligibly small value. This procedure allows rigid links to be used in modeling
composite action under longitudinal bridge flexure but does not result in overestimation
of lateral stiffness. Since the preprocessor models bridges for gravity and vehicle
loading and not for lateral effects such as wind or seismic loading, this procedure is
reasonable.
Illustrated in Figure 3.5 is the eccentric girder model for a girderslab system
consisting of a concrete deck slab and a nonprismatic steel girder. The system is
assumed to consist of multiple spans of which the first span is shown in the figure. In
modeling the slab and girder, a total of six elements have been used in the longitudinal
direction in the span shown. A width of two elements in the lateral direction are shown
modeling the deck slab. Nodes in the finite element model are located at the elevation
of the slab centroid. The girder elements are located eccentrically from the nodes using
rigid links whose lengths are the eccentricities between the centroid of the slab and the
centroid of the girder. Because the girder is nonprismatic, the elevation of the girder
centroid varies as one moves along the girder in the longitudinal direction. For this
81
Figure 3.10 Conversion of Vehicle and Lane Loads to Nodal Loads
Given the location of a vehicle on the bridge, the preprocessor computes the
location of each axle within the vehicle, and then the location of each wheel within
each axle. In the case of a lane load, the preprocessor computes the location of each
axle in the axle train, and then the computes the location of each wheel within each
axle. Finally, after computing the magnitude of each wheel load, the preprocessor
distributes each wheel load to the finite element nodes that are closest to its location.
Each wheel load is idealized as a single concentrated load acting at the location of the
contact point of the wheel. Wheel loads are distributed to the finite element nodes using
the static distribution technique illustrated in Figure 3.11. This distribution technique is
used for zero, constant, and variable skew bridges.
Node
Number
Statically Equivalent
Nodal Loads
N1
PI = Pw (la)(lP)
N2
P2 = Pw (Â£X)(lp)
N3
P3 = Pw (la)(P)
N4
P4 = Pw (a)(p)
Static Distribution Factors
a XI / X2
p = Y1 / Y2
Figure 3.11 Static Distribution of Wheel Loads
181
Rx (radians) Geometry 0.8
4.00e06
3.00e06
Rx (radians) Geometry 1.2
4.00e06,
Lateral 0.75
Direction 1.00
Normalized Longitudinal Direction
Normalized Longitudinal Direction
Figure 6.15 Maximum Magnitude Rotations (Rx) Caused By Unit Moments (Mx)
(Training Data for Scaling Neural Networks)
Ry (radians) Geometry 1.0
8.00e07
6.00e07
Ry (radians) Geometry 0.8
8.00e07r
6.00e07
4.00e07
2.00e07
O.OOe+OO
0.00
0.2^
Normalized 050N
Lateral
Direction
SksSS
0.75N
l.OO^
i.00
U25
Ry (radians) Geometry 1.2
8.00e07i
Normalized Longitudinal Direction
Normalized Longitudinal Direction
Figure 6.16 Maximum Magnitude Rotations (Ry) Caused By Unit Moments (Mx)
(Training Data for Scaling Neural Networks)
Copyright 1995
by
Gary Raph Consolazio
161
number of loads. Further, by utilizing shape and scaling neural networks, loads of
arbitrary magnitude can be properly handled.
Continuing with the propped cantilever example from the previous section,
Figure 6.5 illustrates the process of computing true displacements using superposition,
shape networks, and scaling networks. Displacements from the two loads, P and Q, are
computed separately for several points along the beam. Then, using the principle of
superposition, they are summed together to form the true, total displaced shape.
To compute the displacements corresponding to each load, shape and scaling
neural networks are used in conjunction with the load magnitudes P and Q. For
illustrative purposes only, we will compute the displacements at the three points (a,
b, c) only and assume that these can be used to crudely represent the displaced
shape of the structure. In realistic applications, many more points would need to be
used.
Figure 6.5 Using Shape and Scaling Networks To Compute True Displacements
58
for details) however the engineer may manually adjust these values if detailed bearing
stiffness data are available. In addition to vertical supports, horizontal supports must
also be provided to prevent rigid body instability of the models at each stage of
construction. Horizontal support is provided through finite element boundary condition
specification rather than by using elastic supports.
Flatslab bridges are supported continuously in the lateral direction at each
support in the bridge. Since bearing pads are not typically used in flatslab bridge
construction the support stiffnesses cannot not be easily determined. However, the
preprocessor assumes a reasonable value of bearing stiffness, which again can be
manually adjusted by the engineer if better data are available.
3.3 Modeling Composite Action
Composite action is developed when two structural elements are joined in such a
way that they deform together and act as a single unit when loaded. In the case of
bridges, composite action can occur between the concrete slab and the supporting
concrete or steel girders. The extent to which composite action can be developed
depends upon the strength of bond that exists between the slab and the girders.
Composite action may also occur between stiffeners and the deck slab. In a composite
system there is continuity of strain at the slabgirder interface and therefore no slip
occurs between these elements. Horizontal shear forces and vertical forces are
developed at the boundary between the two elements. The interaction necessary for the
25
2.3 Design Philosophy of the Preprocessor
In the design of the preprocessor, the basic philosophy has been to exploit
regularity and repetition whenever and wherever possible in the creation of the bridge
model. This idea applies to bridge layout, bridge geometry, girder cross sectional
shape, vehicle configuration, and vehicle movement as well as several other bridge
variables.
2.3.1 Internal Preprocessor Databases
Regularity in the form of standardized bridge components and loading has been
accounted for by using databases. Standard girder cross sectional shapes, such as the
AASHTO girder cross sections, and standard vehicle descriptions are included in
internal databases that make up part of the preprocessor. Thus, instead of having to
completely describe the configuration of, say for example, an HS20 truck, the user
simply specifies an identification symbol, in this case HS20, and the preprocessor
retrieves all of the relevant information from an internal database.
The vehicle database embedded in the preprocessor contains all of the
information necessary for modeling loads due to the standard vehicles H20, HS20,
HS25, ST5, SFT, SU2, SU3, SU4, C3, C4, C5 TC95, T112, T137, T150, and T174.
In addition to these standard vehicle specifications, the user may create specifications
for customi.e. nonstandardvehicles by specifying all of the relevant information in
a text data file.
165
MyRx, MyRy. Recall from previous discussion that these are loaddisplacement
pairingsthat is particular components of displacement caused by particular
components of load. For example, FzTz represents the displacements Tz (translations
in the Zdirection) caused by loads Fz (forces in the Zdirection).
The load and displacementsampling points used to generate shape network
training data are illustrated in Figure 6.6. There are 50 load points and 45
displacementsampling points, the result of which is the generation of 2250=50x45
combinations, and consequently 2250 neural network training pairs. Due to the large
number of training pairs, only the single bridge geometry shown in Figure 6.6 was
considered for the shape networks in this study. While this choice limits the flexibility
of the networks, it was felt that if the technique could be shown to be successful for
this limited case, it could then be expanded to encompass more general bridge
geometries.
The load and displacementsampling points used to generate scaling network
training data are illustrated in Figure 6.7. The 231 points served both as load points
and displacementsampling points. A denser grid than that used in the shape network
case was used here for two reasons. First, it was found that the grid used in the shape
network was too coarse to locate the true maximum magnitude displacements in the
bridge. That is, the true mximums often occurred at locations other than the
displacementsampling points and were therefore not captured.
109
compression and decompression of the data is fully compensated by the fact that a
reduced quantity of I/O must be performed. Whether or not this compensation is
complete or partial depends on the programming language and computer platform being
used. It has been the authors experience that in the case of the C language, the binary
I/O functions native to the language are often implemented in a very efficient manner.
The time required for data compression and decompression is only partially
compensated for by the reduced quantity of I/O that must be performedthis results in
increased execution time. However, if the availability of outofcore storage is a
constraining factor in determining whether or not an analysis can be performed, the use
of data compression can eliminate the constraint if a mild penalty in execution time can
be tolerated.
The initial steepness of the plots in Figures 4.4 and 4.5 also reveals the
important fact that the maximum degree of compression that can be achieved is quickly
approached for fairly small buffer sizes. The use of large buffer sizes produces only
marginal increases in the performance of the compression library and is not justified on
systems where memory usage must be minimized.
Using a fixed hash table size of 4096 elements, the flatslab bridge models were
analyzed again to evaluate the effectiveness of data compression on the PC platform.
Both of the flatslab bridges were analyzedusing buffer sizes of 64, 128, 256, 512,
1024, 2048, 4096, 8192, 12288, and 16384 bytes as beforeon a personal computer
running Microsoft Windows NT as an operating system. An additional set of analyses
were also performed on the PC in which the stress files produced by the analyses were
24
will be discussed in detail later. Flatslab bridges are constructed as thick slabs lacking
girders and resisting loads directly through longitudinal flexure of the slab.
The preprocessor has been developed so as to allow maximum flexibility with
respect to the types of bridges that can be modeled. Each of the following bridge types
can be modeled using the preprocessor.
1. Prestressed concrete girder. Bridges consisting of precast prestressed
concrete girders, optional reinforced concrete edge stiffeners, a reinforced
concrete deck slab, and reinforced concrete diaphragms.
2. Steel girder. Bridges consisting of steel girders, optional reinforced concrete
edge stiffeners, a reinforced concrete deck slab, and steel diaphragms.
3. Reinforced concrete Tbeam. Bridges consisting of reinforced concrete T
beam girders, optional reinforced concrete edge stiffeners, a reinforced
concrete deck slab, and reinforced concrete diaphragms.
4. Reinforced concrete flatslab. Bridges consisting of a thick reinforced
concrete deck slab and optional reinforced concrete edge stiffeners.
The general characteristics of each these bridge types are built into the preprocessor so
as to allow rapid modeling. Information regarding the construction sequence of each
type of bridge is also embedded in the preprocessor. This information includes not only
the structural configuration of the bridge at each stage of construction but also the
sequence in which dead loads of various types are applied to the bridge.
Finally, the preprocessor allows the engineer to rapidly and easily model live
loading conditions consisting of combinations of vehicle loads and lane loads. Vehicle
data, such as wheel loads and axle spacing, for a wide range of standard vehiclesfor
example the HS20are embedded in the preprocessor. In addition, there are a variety
of methods available to the user for specifying vehicle locations and shifting.
101
Combining the compressed I/O library with Fortran code using mixed language
programming on workstation platforms is fairly straightforward, requiring only minor
naming convention changes. Mixed language programming using PC compilers is more
complicated due to differences in naming conventions, calling conventions, and link
libraries.
4,8 Data Compression Parameter Study and Testing
Efficient use of system resources is always of importance and especially so in
the FEA of bridge structures where resource demand can quickly exceed resource
availability. On virtually all computer platforms, memory is the most precious resource
used by the software and must therefore be used very efficiently. Although the goal of
using data compression is to increase performance of FEA software with respect to
total execution time and disk usage, improvements in these areas must not be made at
great expense to the overall functionality of the software. Since data compression
requires that memory be allocated for use as I/O buffers, hash tables, and compression
buffers, it is evident that using data compression reduces the memory available to the
software for other uses. Using data compression could adversely affect the functionality
of the software if memory is used inefficiently or excessively.
Thus, parametric studies were performed with the goal of determining values
for key compression parameters which will produce considerable performance
improvements without sacrificing any more memory than is necessary. The three
parameters chosen to be examined due to their influence on the performance of the
compression algorithm are listed below.
99
control word was shortened to 8bits (1byte) which may be safely written to any valid
memory address on a workstation platforms.
4.7 Fortran Interface to the Compressed I/O Library
Although the compressed I/O library is written in C and was originally intended
for use in FEA software also written in C, there exists a vast body of engineering
software in use today which is written in Fortran 77referred to simply as Fortran
from this point onand which can also benefit from data compression. Most of this
software performs quite adequately and should not have to be completely overhauled
and rewritten in C simply to take advantage of the compressed I/O library. In addition,
much of this software has been specifically optimized for the Fortran language and
might suffer loss in performance if the rewritten code was not properly optimized for
the C language. It should be noted however, that the compressed I/O library itself
cannot be implemented in an efficient manner in Fortran 77 because the compression
functions make extensive use of the bitlevel operators which are part of the C language
but not a part of the Fortran 77 language.
Since C and Fortran I/O functions differ greatly in their method of controlling
files and writing and reading data, an interface library was developed in C to allow
Fortran programs to indirectly call the compressed I/O library functions. Table 4.2 lists
the Fortrancallable Clanguage function available in the interface library. The interface
library maintains a simple linkedlist datastructure which matches Fortran unit
numbers with C file handles (identifiers). When a Fortran program needs to perform an
I/O operation, the interface library calls the appropriate compressed I/O library
64
Figure 3.5 Modeling Composite Action with the Eccentric Girder Model
reason the lengths of the rigid linksi.e. the eccentricities of the girder elementsalso
vary in the longitudinal direction. Displacements at the ends of the girder elements are
related to the nodal displacement DOF at the finite element nodes by rigid link
transformations.
Slab elements, modeled using flat shell elements, are connected directly to the
finite element nodes without eccentricity. Recall that in the EGM the slab elements are
allowed to deform axially as are the girder elements. In this manner the slab and girder
elements function jointly in resisting load applied to the structure. Since the slab
elements must be allowed to deform axially a translating roller support is provided at
the end of the first span. By using a roller support, the girder and slab are permitted to
deform axially as well as flexurally and can therefore act compositely as a single unit.
The EGM composite action modeling technique is generally considered to be
more accurate than the CGM modeling technique. This is because in the CGM an
approximate effective width must be used in the determination of the composite section
132
Nvalid \Nout
\ I
(=1
I {okTky
U=i
^ valid ^out
(5.4)
where NvaÂ¡id is the number of validation data pairs, Noul is the number of neurons in
the output layer of the network, O is the output value of the k,h neuron in the output
layer of the network, and 7\ is the target value (validation data) of the klh neuron.
During network training, we use the data pairs in the validation group to evaluate how
well the network is learning the relationship represented by the data in the training set.
Howeverand this is a crucial pointthe training process is never allowed to use the
information in the validation data set to modify the network connection weights. The
validation data is used only to evaluate the training process, but never to guide the
process.
If both E and V are plotted as a function of the progression of training (see
Figure 5.4), one will find that there is a point at which V minimizes and then begins to
grow. The point at which V minimizes represents the ideal stopping point for training.
At the minimum point, the network is able to generalize the inputoutput relationship as
well as it ever will for that particular training run. If training continues, the network
will begin to memorize the data in the training set and lose its ability to generalize.
Thus, the validation data error V will begin to rise even though the training data error E
will continue to decrease. This phenomenon is similar to overfitting data using high
order polynomialsthe fitting function begins to fit the data points rather than the
overall function represented by the data points.
39
stresses and interface shear stresses are developed at the boundary between the two
elements. Proper development of the interface shear stresses is necessary for composite
action to occur and is provided by a combination of friction and mechanical shear
connection schemes.
In steel girder bridges, as illustrated in Figure 2.5, shear studs are often welded
to the top flanges of the girders and embedded into the deck slab so that the two
elements deform jointly. Concrete girders and parapets may be mechanically connected
to the concrete deck slab by extending steel reinforcing bars from the concrete flexural
members into the deck slab during construction. In each of these shear connection
schemes the goal is to provide adequate mechanical bond between the members such
that they behave as a single composite unit.
In a noncomposite bridge system, there is a lack of bond between the top of the
girder and the bottom of the slab. As a result, the two elements are allowed to slide
relative to each other during deformation and do not act as a single composite unit.
Only vertical forces act between the two elements and there is a discontinuity of strain
at the boundary between the elements.
The preprocessor can represent the presence or absence of composite action in a
bridge by using one of three composite action models. The first model, called the
noncomposite model (NCM), represents situations in which composite action is not
present. The second and third models, termed the composite girder model (CGM) and
the eccentric girder model (EGM) respectively, simulate composite action using two
different finite element modeling techniques. Using the concept of an effective width,
65
properties. However, while the EGM is more accurate, the analysis results must be
interpreted with greater care since the effect of the axial girder forces must be taken
into consideration when the total moment in the girder section is determined. Also,
when using the EGM, it is necessary to use a sufficient number of longitudinal
elements to ensure that compatibility of longitudinal strains between the slab and girder
elements is approached (Hays and Schultz 1988). It is therefore recommended that at
least twenty elements in the longitudinal direction be used in each span.
3.4 Modeling Prestressed Concrete Girder Bridge Components
The modeling of structural components and structural behavior that occur only
in prestressed concrete girder bridges will be described in the sections below.
3.4,1 Modeling Prestressing Tendons
Prestressed concrete girder bridges make use of pretensioning and post
tensioning tendons to precompress the concrete girders, thus reducing or eliminating
the development of tensile stresses. The tendons used for pretensioning and post
tensioning of concrete will be referred to collectively as prestressing tendons in this
context. Prestressed bridges are pretensioned and may optionally be posttensioned in
either one or two phases when using the preprocessor. Posttensioned continuous
concrete girder bridges are modeled assuming that the girders are pretensioned, placed
on their supports and then posttensioned together to provide structural continuity.
In order to model prestressing tendons using finite elements, both the tendon
geometry and the prestressing force must be represented. Tendons are modeled as axial
32
2.4.2 Zero Skew. Constant Skew, and Variable Skew Bridge Geometry
Bridges modeled using the preprocessor may be broadly divided into two
categories based on the bridge geometryconstant skew and variable skew. A constant
skew bridge is one in which all of the support lines form the same angle with the lateral
direction of the bridge. The constant skew category includes right bridges as a
particular case since the skew angle in a right bridge is a constant value of zero. Right
bridges are those bridges in which the support lines are at right angles to the direction
of vehicle movement. Constant skew geometry, including zero skew, can be modeled
for all of the bridge types treated by the preprocessor.
Variable skew geometry may also be modeled using the preprocessor but only
for steel girder bridges and single span prestressed girder bridges. In a variable skew
bridge, each support line may form a different angle with the lateral direction of the
bridge. Each of the bridge skew cases considered is illustrated in Figure 2.4.
End Support Lines
Interior Support Line
oo
dood
DO00
DO00
DO00
Right Bridge
(Zero Skew)
Variable Skew Bridge
Constant Skew Geometry
Figure 2.4 Zero Skew, Constant Skew, and Variable Skew Bridge Geometry
205
original matrix A and therefore its ICD does not function well as a preconditioner
for A.
The ICD procedure used in this research is called a rejection by position scheme
because terms are rejected based on their position in the matrix. If a term is located at a
position not in the sparsity pattern of the original matrix, then it is rejected for use
during the decomposition. Papadrakakis and Dracopoulos (1991) state that rejection by
position methods often do not perform satisfactorily in structural mechanics problems.
An alternative procedure is to perform rejection by magnitude in which terms smaller
than a particular threshold magnitude are rejected from the decomposition process.
While this rejection scheme generally results in better preconditioners, it is more
difficult to implement because the sparsity pattern of the decomposed matrix is not
known until the decomposition is complete. This can be undesirable since efficient
storage strategies are essential to the solution of very large problems.
7.5 A Domain Specific Equation Solver
In the present research, a domain specific equation solver has been developed by
combining neural networks (NN) with the preconditioned conjugate gradient (PCG)
algorithm to create a hybrid NNPCG algorithm. The hybrid algorithm is domain
specific because the neural networks that comprise a portion of the algorithm were
trained specifically for one class of structurestwo span flatslab highway bridges. In
developing this solver, the goal was to accelerate the equation solving stage of highway
bridge FEA by encoding domain knowledge directly into the solver.
CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS
8.1 Computer Assisted Modeling
A modeling preprocessor has been developed that can be used by engineers to
model highway bridges for finite element analysis (FEA). The preprocessor considers
the types of bridge construction most commonly encountered in practiceprestressed
concrete girder, steel girder, reinforced concrete Tbeam, and reinforced concrete flat
slab bridges. All of the basic structural components of these bridge types are modeled
by the preprocessor. Basic structural components include girders, parapets,
diaphragms, pretensioning tendons, posttensioning tendons, hinges, and the deck slab.
Three separate finite element modeling methods of representing composite
action between the girders (parapets) and slab are provided by the preprocessor. The
first method represents cases in which composite action is not present. The second and
third methods represent the presence of full composite action between the girders
(parapets) and slab by using two and threedimensional modeling techniques
respectively. Shear lag in the deck slab is properly accounted for in the three
dimensional model.
Several features provided in the preprocessor are aimed specifically at
facilitating rapid data generation. Some of these features include embedded databases
234
141
5.8.1 ExampleBvExample Training and Batching
In pure backpropagation, the connection weights in the network are only
updated once per epoch. This is because the error E is defined as the error over the
entire training set. According to this definition, the gradient of the error surface cannot
be determined until the errors over all of the training pairs have been summed. When
there are a large number of training pairsas is often the case in practical applications
of neural networksthis scheme of only updating the connection weights once per
epoch can result in very slow network learning.
One method of accelerating the learning process is to use examplebyexample
training. In examplebyexample training, the connection weights in the network are
updated after each presentation of a training pair. For example, a training pair (an input
vector and the corresponding output vector) are presented to the network. The network
computes an output vector, compares it to the desired output vector in the training pair,
and computes an error vector. The mean squared error of this error vector is then
computed and is used as a pseudo E. After computing the partial derivatives of this
pseudo E with respect to the weights, a pseudo steepest descent direction is determined
and the connection weights in the network are immediately updated in that direction.
One advantage of examplebyexample training is that the network appears to
learn more rapidly than in pure backpropagation because it is allowed to update the
connection weights very frequently. On the other hand, examplebyexample training
follows the gradient of a pseudo E not the gradient of the true E. As a result, although
212
iteration. However, in an actual implementation of this algorithm, efficient coding may
be employed so that the AW() operator is invoked only once per iteration. Also note
that the neural networks are used both to compute the initial estimate of the
displacements and to solve the Mq = r subproblems (i.e. perform preconditioning).
7.6 Implementation and Results
Implementation of the NNPCG algorithm was accomplished by merging the
neural networks presented in Chapter 6 with a PCG solver.^ In the merged NNPCG
equation solver, neural networks are called upon to perform the tasks described in the
previous section, namely solution seeding and preconditioning. The hybrid NNPCG
solver was coded using both the Fortran and C programming languages and was
installed in the SIMPAL FEA program written by Hoit (Hays et al. 1994). To begin
construction of the NNPCG solver, an ICPCG solver was written in Fortran and
installed in the FEA software.
Next, each of the direct Mq = r subproblem solutions performed in the
ICPCG solver was replaced with a call to a neural network control module. Thus, for
a given load vector q, the network control module is called to compute the
corresponding displacement vector r which is in turn passed back to the PCG
iteration module. The neural network control modulewhich is written in Fortran
^ During the course of this research, several iterative equation solving algorithms were
coded and tested by the author to evaluate their effectiveness in highway bridge
analysis. In particular, equation solvers were written and tested for the standard CG
algorithm, the diagonal PCG algorithm, the band PCG algorithm, the ICPCG
algorithm, and the NNPCG algorithm.
27
Parapet Deck Slab Diaphragm Girder
Extra Extra I , I Extra Extra
Left Left \ Basic Model \ Right Right
Parapet Girder Girder Parapet
Figure 2.1 Cross Section of a Girderslab Bridge Illustrating the Basic
Model and Extra Left and Right Edge Members
only need to be specified once by the user. For example, in the bridge shown in
Figure 2.1 notice that the girder spacing SbasÂ¡c is constant within the basic model and
that the cross sectional shape of each of the girders in the basic model is the same. In
this case, the user would only need to specify SbasÂ¡c once and describe the girder cross
sectional shape once for all four of the girders in the basic model of this bridge.
While the technique of using a basic model to describe a bridge can greatly
speed the process of gathering input from the user, most bridges possess additional
members near the edges that do not fit into the basic model scheme. In the preprocessor
these edge members are termed extra members and are appended to either side of the
basic model to complete the description of the bridge. For example, on each side of the
bridge in Figure 2.1 there is an extra girder and an extra stiffener. In this case the extra
girders have different cross sectional shapes and spacings than the girders of the basic
model. In addition, edge stiffening parapets are present which clearly are different from
the girders of the basic model. In the example shown, the bridge is symmetric but this
need not be the case. By specifying some of the girders in a bridge as extra girders,
189
The Jacobi and GaussSeidel iterative algorithms are the simplest and easiest to
implement, however, they have slow converge characteristics in most cases. To
accelerate the convergence rate a scale factor la having a value in the range 1 < a < 2,
may be added to the GaussSeidel method to produce the SOR (Successive Over
Relaxation) method. SOR generally converges more rapidly than GaussSeidel,
however choosing the optimum scale factor aopt is a difficult process. One may use an
iterative approach in which many different scale factors are examined and then the one
producing the fastest convergence is selected for use. This approach is only useful in
cases where many problems of similar type will be solved and will hopefully have
similar values of (Â¡>op[. Alternatively, one may perform an eigen analysis to determine
an approximate value of opt (Golub and Van Loan 1989), however this will require a
substantial amount of computational effort and may offset any savings derived from
using an iterative solution scheme.
The Conjugate Gradient (CG) method and its variants constitute another class of
iterative solution method which may be used in FEA. In the CG method, the matrix
equation Ax = b is solved by minimizing the residual
r = bAx (7.3)
where r is the residual vector and x is an approximation of the exact solution x. The
residual is actually minimized indirectly by formulating an error functionwhich is
quadratic in r and minimizing the error function. The method is called the Conjugate
Gradient method because the optimization process traverses the error function using
conjugate directions instead of steepest descent directions.
178
Finally, in addition to the scaling just described, a final scaling is also used to
compact the range of values which must be operated on by the networks. In the scaling
networks, all output values initially in the range [0,1] are compacted into the range
[0.1,0.9] to improve network training. The reasons for performing this type of scaling
were given in the previous section with regard to the shape networks. The reader is
therefore referred to that section for more information.
Training the scaling neural networks was accomplished using the NetSim
software. The 924 loaddisplacement pairs described earlier in this chapter were used as
training data. Figures 6.116.19 graphically illustrate the data that was used to train
each of the nine scaling networks.
In addition to the 924 training pairs used to train the networks, 38 validation
pairs were used to monitor the generalization capabilities of the networks. Validation
data consisted of selected maximum displacement magnitudes from a GSF=0.9
geometry case that was generated separately from the training data.
Table 6.3 lists the relevant parameters of the trained scaling networks. The error
statistics shown the table indicate that the neural networks were able to model the
training data and match the validation data to within a reasonable level of error. The
maximum error statistics for both the training and validation data sets were less than
10% while the average error statistics were less than 2%.
231
Normalized Exact
Displacement Shape
Neural Network
Displacement Shape
Upper & Lower
Tolerance Bounds
Figure 7.17 OneDimensional Illustration of Convergence Criteria
cause the developement of large residuals which in turn cause the NNPCG algorithm
to stop converging toward the correct solution (see discussion below).
The second problem arises in choosing a training tolerance for the shape
networks. Each displacement surface is normalized with respect to the largest
magnitude displacement occurring in that particular shape. Thus, the output values of
the shape networks are always in the normalized range [1,1].^ When a convergence
tolerance is chosen, it is chosen in the range [0,1]. A simplified onedimensional
example is shown in Figure 7.17. All of the ordinates on the normalized shape have
values in the range [0,1]. Assume that a convergence tolerance of 0.1 is chosen. This
choice says that up to a 10% error at the maximum magnitude location a can be
tolerated for this problem.
Thus the oscillatingand hypothetically neural network computedshape
shown in the figure is within the specified tolerance. Note, however, that by choosing a
tolerance of 0.1, we allow a 50% error to occur at b relative to itself. In some
t Actually, the output range of the shape networks is confined to a range narrower than
[1,1]. This is done so that only the nearly linear portion of the sigmoid output
function is used (see Chapter 6). However, the constricted range is always expanded
back to [1,1] when data is sent from the neural network back to the equation solver.
134
To minimize the network error E with respect to the connections weights, two basic
optimization strategies are often usedgradient descent methods and stochastic
methods.
In discussing neural network training algorithms, it is useful to visualize the
network error Â£ as a high dimensional surface. The height of the surface corresponds
to the value of the error function E produced by choosing a particular set of connection
weights. For example, consider a very small network having only two connection
weights. We could compute the error E produced by choosing various sets of the
weights W( and W2 and then plot the errors as a function of the weights. Such an error
surface might look like the surface in Figure 5.5. The goal of network training is to
locate point of minimum elevation on that surface, i.e. determine the values of the
connection weights that minimize E.
Gradient descent methods attempt to find the minimum point by following the
slope of the error surface in a downward direction, i.e. in a direction of decreasing E.
There are many types of gradient descent optimization, each having advantages and
disadvantages. The simplest gradient descent method is called the method of steepest
descent. This process begins by picking a set of weights which constitutes a starting
point on the surface. The function E and its derivative (gradient) are then computed at
the starting point. Based on the computed gradient, a direction of steepest descent is
determined.
12
time required for the bridge analysis was shown to decrease to as little as approximately
one third of the execution time needed when compression was not utilized.
1.2.3 Neural Network Equation Solver
In order for a computer assisted bridge modeling system to be effective, the
time required to perform each FEA analysis must be minimized. To address this issue,
an application of Artificial Neural Networks (ANNs) has been used to create a domain
specific equation solver. Since the equation solving stage of a FEA accounts for a large
portion of the total time required to perform an analysis, increasing the speed of this
stage will have a significant effect on the speed of the overall analysis.
In the present work, the approach taken to minimize the analysis execution time
is to implicitly embed, using ANNs, domain knowledge related to bridge analysis into
the equation solver itself. In this way a domain specific equation solver, i.e. an
equation solver constructed to solve problems within the specific problem domain of
bridge analysis, is created. The concept behind such an equation solver is that by
exploiting knowledge of the problem, e.g. knowing displacement characteristics of
bridge structures, the solver will be able to more rapidly arrive at the solution.
In the present application ANNs have been trained to learn displacement
characteristics of twospan flatslab bridges under generalized loading conditions. Using
analytical displacement data generated by numerous finite element analyses, a set of
network training data was created with which to train the ANNs. Next, using ANN
training software that was developed as part of the present research, several neural
41
of standard AASHTO girders and FDOT bulbT girder types. When modeling a bridge
based on one of these standard girder types, the engineer simply specifies a girder
identification symbol. The preprocessor then retrieves all of the cross sectional data
needed for finite element modeling from an internal database.
This technique saves the user time and eliminates the possibility that he or she
may accidentally enter erroneous data. Since the majority of prestressed concrete girder
bridges are constructed using standard girders, a typical user may never have to
manually enter cross sectional data. To cover cases in which nonstandard girders are
used, the preprocessor also allows the user to manually enter cross sectional data.
2.5.2 Pretensionine and PostTensioning
Prestressed concrete girder bridges modeled by the preprocessor may be either
of the pretensioned type or pre and posttensioned type. Pretensioning occurs during
the process of casting the concrete girders whereas posttensioning occurs after the
girders have been installed in a bridge. The preprocessor can model bridges having
either one or two phases of posttensioning, however, there are specificand distinct
construction sequences associated with each of these schemes.
Concrete Concrete Concrete Concrete
Parapet Deck Slab Diaphragm Girder
Figure 2.6 Cross Section of a Typical Prestressed Concrete Girder Bridge
29
Figure 2.2 Nonprismatic Steel Girder Bridge With User Specified Definition
Points and Finite Element Nodes
The methods by which a user positions vehicles on a bridge provides another
illustration of the types of generation performed by the preprocessor. As will be seen
later, the user needs only to provide a minimal amount of information in order to
generate potentially hundreds of discrete vehicle positions.
2.3.4 The Preprocessor History File
When using a primarily interactive program such as the preprocessor, the
majority of required data is gathered directly from the user, as opposed to being
gathered from an input data file as in a batch program. An interactive approach to data
collection generally results in easier program operation from the viewpoint of the user.
However, one disadvantage of this approach is that because the user has not prepared
an input data file in advance, as is the case in batch programs, there is no record of the
data given by the user. This is undesirable for two reasons. First, there is no permanent
record of what data was specified by the user and therefore there is no paper trail that
can be used to trace the source of an error should one be detected at some later date.
2.4.7Composite Action 38
2.5 Modeling Features Specific to Prestressed Concrete Girder Bridges 40
2.5.1 Cross Sectional Property Databases 40
2.5.2 Pretensioning and PostTensioning 41
2.5.3 Shielding of Pretensioning 42
2.5.4 PostTensioning Termination 43
2.5.5 End Blocks 43
2.5.6 Temporary Shoring .44
2.5.7 Stiffening of the Deck Slab Over the Girder Flanges 45
2.6 Modeling Features Specific to Steel Girder Bridges 46
2.6.1 Diaphragms 46
2.6.2 Hinges 47
2.6.3 Concrete Creep and Composite Action 48
2.7 Modeling Features Specific to Reinforced Concrete TBeam Bridges 49
2.8 Modeling Features Specific to FlatSlab Bridges 50
3 MODELING BRIDGE COMPONENTS 51
3.1 Introduction 51
3.2 Modeling the Common Structural Components 51
3.2.1 Modeling the Deck Slab 51
3.2.2 Modeling the Girders and Stiffeners 54
3.2.3 Modeling the Diaphragms 55
3.2.4 Modeling the Supports 57
3.3 Modeling Composite Action 58
3.3.1 Modeling Composite Action with the Composite Girder Model 60
3.3.2 Modeling Composite Action with the Eccentric Girder Model 61
3.4 Modeling Prestressed Concrete Girder Bridge Components 65
3.4.1 Modeling Prestressing Tendons 65
3.4.2 Increased Stiffening of the Slab Over the Concrete Girders 68
3.5 Modeling Steel Girder Bridge Components 70
3.5.1 Modeling Hinges 70
3.5.2 Accounting for Creep in the Concrete Deck Slab 72
3.6 Modeling Reinforced Concrete TBeam Bridge Components 74
3.7 Modeling FlatSlab Bridge Components 75
3.8 Modeling the Construction Stages of Bridges 76
3.9 Modeling Vehicle Loads 80
4 DATA COMPRESSION IN FINITE ELEMENT ANALYSIS 83
4.1 Introduction 83
4.2 Background 84
4.3 Data Compression in Finite Element Software 86
4.4 Compressed I/O Library Overview 91
4.5 Compressed I/O Library Operation 92
4.6 Data Compression Algorithm 95
v
62
properties of the bare girder cross section but which are located eccentrically from the
slab centroid by using rigid links. By locating the girder elements eccentrically from the
slab, the girder and slab act together as a single composite flexural unit. In general,
each componentthe slab and the girdermay undergo flexure individually and may
therefore sustain moments. However, because the components are coupled together by
rigid links, the composite section resists loads through the development not only of
moments but also of axial forces in the elements.
Rigid links, also referred to as eccentric connections, are assumed to be
infinitely rigid and therefore can be represented exactly using a mathematical
transformation. Thus, by using the mathematical transformation, additional link
elements do not need be added to the finite element model to represent the coupling of
the slab and girder elements.
In the eccentric girder model, shear lag in the deck is properly taken into
account because the deck slab is modeled with flat shell elementselements created by
the superposition of plate bending elements and membrane elements. Because the slab
and girders are eccentric to one another and because they are coupled together in a
three dimensional sense, the EGM is referred to as a pseudo three dimensional model.
It is not a fully three dimensional model because the coupling is accomplished through
the use of infinitely rigid links. In an actual bridge the axial force in the slab must be
transferred to the girder centroid through a flexiblenot infinitely rigidgirder web. In
a fully three dimensional model, the girder webs would have to be modeled using shell
76
3.8 Modeling the Construction Stases of Bridges
To properly analyze a bridge for evaluation purposes, such as in a design
verification or the rating of an existing bridge, each distinct structural stage of
construction must be represented in the model. Using the preprocessor, this can be
accomplished by creating a full load modela model in which the full construction
sequence is considered. Each of the individual construction stage models, which
collectively constitute the full load model, simulates a particular stage of construction
and the dead or live loads associated with that stage.
Modeling individual construction stages is very important in prestressed
concrete girder bridges, important in steel girder bridges, and of lesser importance in
R/C Tbeam and flatslab bridges. For each of these bridge types, the preprocessor
assumes a particular sequence of structural configurations and associated loadings.
These sequences will be briefly described below, however, for complete and highly
detailed descriptions of each sequence see Hays et al. (1994).
Several different types of prestressed concrete girder bridges may be modeled
using the preprocessor. These include bridges that have a single span, multiple spans,
pretensioning, one phase of posttensioning, two phases of posttensioning, and
temporary shoring. Each of these variations has its own associated sequence of
construction stages the highlights of which are described below.
All prestressed girder bridges begin with an initial construction stage in which
the girders are the only components that are structurally effective. In multiple span
prestressed concrete girder bridges, the girders are not continuous over interior
75
represented using the same lateral beam elements that were discussed earlier for
prestressed concrete girders.
5.7 Modeling FlatSlab Bridge Components
A flatslab bridge consists of a thick reinforced concrete slab and optional edge
stiffeners such as parapets. If stiffeners are present and structurally effective, they are
modeled using frame elements as is the case for girderslab bridges. If stiffeners are not
present on the bridge or are not considered structurally effective, then the slab is
modeled using plate elements and the entire bridge is represented as a largepossibly
multiple spanplate structure. When stiffeners are present on the bridge but do not act
compositely with the slaband are therefore not considered structurally effectivethey
should be specified as line loads by the engineer.
If stiffeners are considered structurally effective, then the engineer can choose
either the CGM or EGM of composite action. If the CGM is chosen, then the slab is
modeled using plate elements and composite section properties are computed for the
stiffener elements using the effective width concept. If the EGM is chosen, the slab is
modeled using shell elements and the stiffeners are located eccentrically above the flat
slab using rigid links. The NCM is not available for flatslabs because if sufficient bond
does not exist between the stiffeners and slab to transfer horizontal shear forces, it
cannot be assumed that there is sufficient bond to transfer vertical forces either. In
order for stiffenerswhich are above the slabto assist in resisting loads, there must
be sufficient bond to transfer vertical forces to and from the slab.
143
5.8.2 Momentum
In examplebyexample training, the network connection weights are modified
using a gradient that is computed based on a single training pair. Batching, on the other
hand, uses a gradient that is computed based on a number of training pairs. A gradient
computed using the batching scenario can be thought of as being an average gradient
over all the training pairs in the batch. Thus, the batching procedure essentially makes
use of averaged derivative data to stabilize the oscillations which occur frequently in
examplebyexample training.
Momentum is a different approach to solving the same problem that batching
attempts to solve. In momentum, we average the connection weight changes instead of
averaging the computed derivatives. The steepest descent direction is determined and
the weight changes which would normally be made are computed. However, instead of
using these changes directly, they are averaged with the last set of changes that were
made to the weights. Then, the averaged set of weight changes are used to actually
update the connection weights. Note that the previous weight changes made were
themselves averaged changes, therefore the effect of previous gradient data has an
influence on all subsequent weight updates made. An exponential averaging of the form
Awaveraged (5.5)
is used, where Aw, is the computed weight change for the current epoch (or batch),
Aw,_! is the weight change which occurred during the previous epoch (or batch), and
Awfveraged is the exponentially averaged weight change which will actually be used to
update the connection weight. The parameter p, which must be in the range
204
fillin terms are ignored, the decomposition process can become unstable and produce
unreasonable results. For example, the procedure of ignoring fillin often results in the
formation of negative terms on the diagonal of the decomposed matrixan
unreasonable condition in FEA. Jennings (1992) presents a stabilization process that
can be used to avoid this pitfall, however, it requires additional processing of the
coefficient matrix A and may result in a modification of the sparsity pattern. Kershaw
(1977) handled the development of negative diagonal termswhich he stated occurred
infrequently in the types of problems he was studyingby assigning the offending
diagonal the value given by
z1 n
SN (728)
7=1 7=+1
and then proceeding with the decomposition. This procedure was implemented and
tested by the author in FEA bridge analysis applications, however, it generally
performed poorly. Whereas the problems studied by Kershaw seldom required this
fix to be made, the bridge analysis problems studied by the author required the fix
very frequently and the resulting decomposition performed poorly as a preconditioner.
Diagonal scaling was also implemented and tested as a method for countering the
formation of negative diagonal terms during the decomposition. It was found that by
scaling the diagonal of M by a factor in the range [I03,106] prior to decomposition,
negative diagonal terms were not formed during decomposition. However, this
procedure essentially converts the matrix M into a nearly diagonal matrix of large
positive values. As a result, the scaled matrix M no longer reflects the character of the
92
Table 4.1 Compressed and Standard C Binary I/O Functions
Compressed
I/O function
Standard C
I/O function
Use of Function
CioOpen
fopen
Open a binary file for I/O
CioWrite
fwrite
Write a block of data to binary file
CioFlush
fflush
Force a flush of internal I/O buffers to disk
CioRewind
rewind
Rewind to top of a binary file
CioRead
fread
Read a block of data from a binary file
CioClose
fclose
Close a binary file and flush I/O buffers
CioModify
n/a
Modify characteristics of compression
compression algorithm at run time. However, the FEA code is not required to call
CioModify as default compression parameters are built into the library.
4,5 Compressed I/O Library Operation
One or more files may simultaneously be under the control of the compressed
I/O library at any given time. Each time a file is opened, a new entry in a linked list
that is maintained by the library is created to hold information pertaining to that
particular file. For each file under the control of the library, memory is allocated for an
I/O buffer that will be used for writing and reading operations and for a hash table that
will be used during compression. In addition, if there is at least one file under the
control of the library then a single compression buffer is also allocated and shared by
all files controlled by the library. To maximize the degree of compression that can be
achieved during binary I/O, the compressed I/O library performs buffered I/O.
Buffering simply means that when the calling program requests that a block of data be
written to disk, instead of immediately compressing and writing the data to disk the
library will copy the data into an I/O buffer in memory. Data from subsequent write
183
1.00
/ O Q Dll /mtilarto\ ^0nmi4ni 1 ^
1.00
Normalized Lcngitudiial Direction
Figure 6.19 Maximum Magnitude Rotations (Ry) Caused By Unit Moments (My)
(Training Data for Scaling Neural Networks)
6.8 Implementation and Testing
In order to perform loaddisplacement calculations for flatslab bridges using the
neural networks developed herein, the networks were integrated together through a
common control module. Given a particular loading condition on a bridge, it is the
responsibility of the control module to determine which neural networks must be
invoked to compute the structural displacements for the bridge. It is also the
responsibility of the control module to perform load superposition and to combine
shape and scaling data from the networks in the correct manner.
The eighteen component neural networks and the control module were
integrated into an iterative equation solver as a separate part of this research. The
development and testing of that equation solver are the topics of Chapter 7. Since the
110
not compressed. The results for the workstation runs and two sets of PC runs are
plotted in Figure 4.6. Note that the each of the curves are normalized with respect to
the platform on which the analysis was run. Thus the curves represent a comparison not
of the relative speeds of the different platforms, but instead a comparison how data
compression affects the performance on each platform.
The plots indicate that under each set of conditions, the use of data compression
increased the required analysis time. The increase in execution time appears to be more
severe for cases in which data compression is used on the PC platform. However, it
will be shown in the next section that this is not a general rule and in many cases just
the opposite behavior occurs.
In performing the compression studies, the observation was made that the stress
files produced by the analyses did not compress as well as the element files. Whereas
Figure 4.6 SEAS Execution Time Results for Workstation and PC Analyses
(All Analyses Run With a Hash Table Size of 4096 Elements)
136
mechanisms which are used in traversing the error surface. Most of the variations
attempt to accelerate the process of minimizing E, however some address issues such as
avoiding local minima and plateaus in the error surface. A few of these methods, which
have been implemented in the NetSim package, will be discussed later in this chapter.
Whereas gradient descent methods use gradient information to search the error
surface for a minimum, stochastic methods use probability to guide the search. Given a
particular point on the error surface, each connection weight in the network is given a
random perturbation in some direction after which the error function E is reevaluated.
If the error function decreases as a result of the random motion, then the motion is
accepted. If the error function increases as a result of the motion, then there is still a
small probability that the motion will be accepted. A pseudo temperature parameter
controls the likelihood (probability) that movement in a direction of an increasing E
will be accepted. During the training (optimization) process, the pseudo temperature
parameter is gradually reduced according to an annealing schedule so that the
probability that bad movements are accepted is gradually reduced zero.
Stochastic methods have the advantage that they virtually always find the global
minimum of E without getting trapped in local minimaa situation which can occur
with gradient descent techniques. However, stochastic methods also tend to be much
slower than gradient descent techniques. Also, an appropriate annealing schedule must
be developed in order for stochastic methods to be effective.
226
Longitudinal Direction (inches) Longitudnal Direction (inches)
Figure 7.15 Neural Network Seeded Convergence of Displacements
(TzTranslations) and Residuals (FzForces) for a Flatslab Bridge
Under Vehicular Loading
208
every DOF in the model and are added to the previous values in the solution vector.
Thus, the effect of each load term is superimposed on the effects of all previously
encountered load terms until all of the load terms have been processed. When this
process finishes, the solution vector will hold a neural network generated
approximation of the true displaced shape of the structure.
In using the superposition principle, note that the networks will be called upon
numerous times before the final displaced shape approximation is completely generated.
However, in parallelcomputing or networkdistributedcomputing environments, each
of these separate neural network invocations can be performed independently and
simultaneously (in parallel). Therefore, the concept of using superposition in the
manner just described can lead to very high performance of the software on parallel
computing platforms.
Consider also the fact that for vehicular loading conditions, only a small
percentage of the terms in the global load vector will be nonzero. Consider one of the
wheels of a vehicle that is sitting on the bridge deck. The wheel load will be distributed
only to a very few translational DOF that are in the immediate vicinity of the wheel.
As a result, only a few terms in the global load vector will be affected by the load.
Even in the case of uniform pressure loading, the number of nonzero load terms will
be small relative to the total number of DOF in the model. This is because the uniform
pressure loads acting on the deck elementswhether they are plate or shell elements
are converted into loads which correspond only to the transverse translational DOF in
the model. Terms in the global load vector which correspond to inplane rotational
129
language and therefore runs on most computer architectures. NetSim is set up in a
general manner so that the user can construct neural networks to solve virtually any
type of problem for which sufficient training data is available. The user can freely
specify the topology of the network, training data, convergence tolerances, and
numerous options related to the actual process of training the connection weights in the
network.
NetSim can be used to construct, train, test, and implement neural networks for
any situation described by the user. Since network training is an iterative process, it is
important to be able to watch the progress of the training process. Therefore, numerous
error statistics are displayed during the training process so that the progression of
training can be easily monitored. NetSim can also process training validation data to
ensure that the network is capable of generalizing the relationship being learned and is
not simply memorizing the training data.
Once a network has been completely trained, it may also be tested using the
NetSim package. If the testing process indicates an acceptable network, NetSim can be
used to automatically generate C code modules that will emulate the trained network.
The code thus generated will properly account for the trained connection weights,
biases, transfer functions, and scaling of input and output parameters. Also, the code
forms a self contained module that can be easily called from any C or Fortran program.
Since the most difficult part of constructing a neural network is usually the
training stage, this is the task for which NetSim has been primarily designed. A hybrid
version of the backpropagation training algorithmdiscussed in the next sectionhas
CHAPTER 2
A PREPROCESSOR FOR BRIDGE MODELING
2.1 Introduction
This chapter will describe the development and capabilities of an interactive
bridge modeling preprocessor that has been created to facilitate rapid computer assisted
modeling of highway bridges. This preprocessor is one component of a larger system of
programs collectively called the BRUFEM system (Hays et al. 1994). BRUFEM,
which is an acronym for Bridge Rating Using the Finite Element Method, is a software
package consisting of a series of Fortran 77 programs capable of rapidly and accurately
modeling, analyzing, and rating most typical highway bridge structures.
The development of the BRUFEM system was funded by the Florida
Department of Transportation (FDOT) with the goal of creating a computer assisted
system for rating highway bridges in the state of Florida. Bridge rating is the process
of evaluating the structural fitness of a bridge under routine and overload vehicle
loading conditions. With a significant portion of existing highway bridges in the United
States nearing or exceeding their design life, the need for engineers to be able to
accurately and efficiently evaluate the health of such bridges is evident.
Development of the complete BRUFEM system was accomplished in
incremental stages of progress spanning several years and involving the efforts of
21
26
The cross sectional shape databases embedded in the preprocessor contain
complete descriptions of the following standard cross sections used for girders and
parapets.
1. AASHTO prestressed concrete girder types I, II, III, IV, V, and VI
2. Florida DOT prestressed concrete girder bulbT types I, II, III, and IV
3. Standard parapetsold and new standards
In addition to these standard cross sectional shapes, the user may describe nonstandard
cross sectional shapes interactively to the preprocessor.
2.3.2 The Basic Model and Extra Members
Girderslab bridges typically contain a central core of equally spaced girders
that is referred to as the basic model when discussing the preprocessor. In addition to
this central core the bridge may have extra girders at unequal spacings, parapets, and
railings near the bridge edges. The basic model and extra edge members are depicted in
Figure 2.1. Equal girder spacing arises because it simplifies the design, analysis, and
construction of the bridge. Flatslab bridges also contain a central core, or basic model,
in which the deck slab has a uniform reinforcing pattern. While there are no girders in
flatslab bridges, these bridges may have edge elements such as parapets or railings just
as girderslab bridges may.
Almost all of the bridge types considered by the preprocessor utilize the concept
of a basic model to simplify the specification of bridge data. Exceptions to this rule are
the variable skew bridge types in which the concept of a basic model is not applicable.
Within the basic model all bridge parameters are assumed to be constant and therefore
42
Each type of prestressing, whether it be pretensioning, phase1 posttensioning,
or phase2 posttensioning, is modeled by the preprocessor as a single tendon having a
single area, profile, and prestressing force. In reality, prestressing is usually made up
of many smaller strands located nearby one another so as to form a prestressing group.
This means that when specifying the profile of prestressing strands, the user needs to
specify the profile of the centroid of the prestressing group. Several methods of
describing tendon profiles are provided by the preprocessor including straight profiles,
single and dual point hold down profiles, and parabolically draped profiles.
2.5.3 Shielding of Pretensioning
Pretensioning is used to induce moments into an unloaded girder that will
eventually oppose moments produced by normal bridge loading. In many situations,
however, when normal loads are applied to bridge there is little or no moment at one or
both ends of the girders. In these situations, the pretensioning may be placed in a
profile that has zero eccentricity at the ends of the girder so that zero countermoment
is induced. An alternative is to use a straight pretensioning profile with selected
pretensioning strands being shielded near the ends of the girder.
Shieldingalso known as debondingis the process of preventing bond between
the pretensioning strand and the concrete so as to effectively eliminate a portion of the
pretensioning at a particular location. The preprocessor is capable of modeling
pretensioning shielding. The user must specify the percentages of pretensioning that are
shielded and the distances over which those percentages apply.
162
Consider the load P first. This load occurs precisely at the point a which was
earlier used to generate network training data. To compute the displacements at a,
b, and c due to P, we begin by computing the normalized displacements Arm,
A1'arm > an
normalized displacements are correct, their absolute magnitudes are not. If we scale the
normalized displacements by A1^computed using the scaling networkwe obtain
the displacements that would result from a unit load acting instead of a load of
magnitude P acting. Therefore, we must scale once more, by a factor P, to obtain the
total displacements at each of the three points.
Load Q is handled in precisely the same manner as load P. The sole difference
is that, in the case of load Q, the shape and scaling networks are called upon to predict
values that were not part of the training data. Since load Q is located at point q that
was not a point in the training data, the neural networks must interpolate to obtain the
values of A^^, Aafiorm, Ab^orm, and Acflorm. Networks are said to generalize well if
they can perform this sort of interpolationor sometimes extrapolationin a
meaningful, reliable, and accurate manner. In such cases, the networks have been able
to take a discrete set of training data, and generalize to the continuous relationship
represented by the discrete points.
Therefore, the ability to train robust networks that are capable of generalizing
well is very important in neural network applications. It also becomes evident that
monitoring the networks ability to generalize should be part of the overall training
241
method can be proven to be usefuleven very effective. If this could be accomplished,
then efficient equation solvers could be developed for specific bridge types of interest.
In particular, future efforts should be directed at training neural networks to learn the
loaddisplacement relationship of girderslab bridges.
53
more solution time than would models using the simpler bilinear elements. This was
shown to be true by Consolazio (1990) for all but trivially small bridge models.
Another important reason for using bilinear elements instead of biquadratic
elements is related to the fact that both of these elements are known to be rank deficient
when their stiffness matrices are numerically underintegrated. Selective reduced
integration (Bathe 1982, Hughes 1987) is often used to alleviate the problem of shear
locking in plate elements. Shear locking, which results in greatly exaggerated structural
shear stiffness, occurs when elements based on the thick plate formulation are used in
thin plate situations. Selective reduced integration is used to soften the portion of the
element stiffness matrix that is associated with transverse shear.
When underintegrated elements are used in thick plate situations such as the
modeling of a flatslab bridge, zero energy modes may develop which can cause the
global stiffness matrix of the structure to be locally or globally singular (or nearly
singular). Both the bilinear and biquadratic elements suffer from zero energy modes.
However, it has been the authors experience that the mode associated with biquadratic
elements, illustrated in Figure 3.1, is excited far more frequently in bridge modeling
situations than the modes associated with bilinear elements. In fact the biquadratic
.... Undeformed Element
Deformed Element
In Zero Energy
Mode Configuration
(r.s.t) Local Element Directions
The tdirection is the transverse
translational direction of the element.
Figure 3.1 Zero Energy Mode in a Biquadratic Lagrangian Element
66
Girder Prestressing Rigid Prestressing Finite
(Frame) Tendon (Truss) Link Tendon Element
Element Element (Eccentricity) Centroid Node
Girder
Cross Section
Figure 3.6 Modeling the Profile of a Prestressing Tendon
truss elements that are eccentrically connected to the finite element nodes by rigid links
(see Figure 3.6). Since straight truss elements are used between each set of nodes in the
tendon, a piecewise linear approximation of the tendon profile results. The tendon is
divided into a number of segments that is equal to the number of longitudinal elements
per span specified by the user. As long as a reasonable number of elements per span is
specified, this method of representing the profile will yield results of ample accuracy.
The reference point from which tendon element eccentricities are specified in
the model varies depending on the particular type of composite action modeling being
used and on the particular construction stage being modeled. In the noncomposite
model (NCM) tendon element eccentricities are always referenced to the centroid of the
barei.e. noncompositegirder cross section. In the composite model (CGM), for
construction stages where the slab and girder are acting compositely, the eccentricities
are referenced to the centroid of the composite girder cross section which includes an
effective width of deck slab. Eccentricities in the eccentric girder model (EGM) for
construction stages where the slab and girder are acting compositely are referenced to
the midsurface of the slab. Prestressing element eccentricities for construction stages in
90
repetition degenerates to large scale repetition. As used herein, the term medium scale
repetition will refer to patterns ranging from two bytes in length to a few hundred bytes
in length.
It should be noted the data compression strategy being presented herein is
intended to supplement rather than replace efficient programming practices. Take as an
example, the case of the repeated plate element stiffness matrices described above. A
well implemented FEA code might attempt to recognize such repetition and write only
a single representative stiffness matrix followed by repetition codes instead of writing
each complete element stiffness. In this case the compression library will supplement
the already efficient FEA coding by opaquely compressing the single representative
element stiffness that must be saved. The term opaquely is used to indicate that the
details of the data compression process, and the very fact that data compression is even
being performed, are not visible toi.e. are opaquely hidden fromthe FEA code.
Thus, the reduction of data volume is performed in a separate and self contained
manner which requires no special changes to the FEA software. If however, the FEA
code makes no such special provisions for detecting element based repetition, then the
compression library described herein will reduce the volume of data that must be saved
by compressing each element stiffness matrix as it is written.
In each case, compression is accomplished primarily by recognizing small and
medium scale repetition within the data being saved. In fact, due to the small size of
the hash key used in the hashing portion of the compression algorithmdescribed in
detail later in this chapterthe likelihood of the compression library identifying large
70
The torsional moment of inertia of the lateral beam members is obtained in a
similar manner. From plate theory the twisting moment in a plate of thickness t is
given by
GtJ
(3.5)
xy ~ 6 vxy
where G is the shear modulus of elasticity, and <>,y is the cross (or torsional)
curvature in the plate. Thus, the effective torsional moment of inertia of the lateral
beam elements is given by
J =
Ksg+ef) fsg
6
(SY)
(3.6)
where the parameters sg+ef) an(* lsg are the same as those described earlier.
3.5 Modeling Steel Girder Bridge Components
The modeling of structural components and structural behavior that occur only
in steel girder bridges will be described in the sections below. One of the areas of
modeling that is specific to steel girder bridges is that of modeling cross brace steel
diaphragms. However, since this topic was already considered in Â§3.2.3 in a general
discussion of diaphragm modeling, it will not be repeated here.
3.5.1 Modeling Hinges
Hinged girder connections are occasionally placed in steel girder bridges to
accommodate expansion joints or girder splices. Hinges are assumed to run laterally
across the entire width of the bridge, thus forming a lateral hinge line at each hinge
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fullv^/adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philoso
Hoit, Chair
ciate Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Clifford O. Hays, Cochair
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Ronald A. Cook
Associate Professor of Civil Engineering
I certify that 1 have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Q/r
)
L

John M. Lybas
Associate Professor of CV
V
1 Engineering
77
supports at this stage but rather consist of a number of simply supported spans. At this
stage, the bridge is subjected to pretensioning forces and to dead loads resulting from
the weight of the girders, diaphragms, andin most casesthe deck slab. In
subsequent construction stages the bridge components that caused dead loads in this
first stage, e.g. the diaphragms and deck slab, will become structurally effective and
will assist the girders in resisting loads due to posttensioning, temporary support
removal, stiffener dead weight, line loads, overlay loads, and ultimately vehicle loads.
Prestressed concrete girder bridges may go through a construction stage that
represents the effect of additional long term dead loads acting on the compound
structure. Loads that are classified as additional long term dead loads include the dead
weight of the stiffeners, line loads, and overlay loads. The compound structure is
defined to be the stage of construction at which the deck slab has hardened and all
structural components, except for stiffeners, are structurally effective. The term
compound is used to refer to the fact that the various structural components act as a
single compound unit but implies nothing with regard to the composite or noncomposite
nature of girderslab interaction. Since the deck slab has hardened at this construction
stage, the girders and deck slab may act compositely. Also, lateral beam elements are
included in the bridge model to represent the increased stiffening of the girder on the
deck slab.
As construction progresses from one stage to the next, the bridge components
become structurally effective in the following ordergirders, pretensioning,
diaphragms, deck slab, lateral beam elements, phase1 posttensioning (if present),
60
Effective Width
Slab Elements
Noncomposite
Model (NCM)
Composite Girder
Model (CGM)
Figure 3.4 Modeling Noncomposite Action and Composite Action
3.3.1 Modeling Composite Action with the Composite Girder Model
One method of modeling composite action is by utilizing composite properties
for the girder elements. The centroid of the composite girder section is at the same
elevation as the midsurface of the slab in the finite element model. A composite girder
section is a combination of the bare girder and an effective width of the slab that is
considered to participate in the flexural behavior.
Due to shear strain in the plane of the slab, the compressive stresses in the
girder flange and slab are not uniform, and typically decrease in magnitude with
increasing lateral distance away from the girder web. This effect is often termed shear
lag. In certain cases of laterally nonsymmetric bridge loading, the compressive stress in
the deck may vary such that the stress is higher at the edge of the bridge than over the
centerline of a girder. An effective slab width over which the compressive stress in the
deck is assumed to be uniform is used to model the effect of the slab acting compositely
with the girder. The effective width is computed in such a way that the total force
240
valuese.g. 0.0001 is generally too small. The neural network will either
never train to the desired tolerance or it will train to that level but will likely
function simply as a lookup table without the ability to generalize.
4. Use validation data. When training a neural network, always use validation
data to monitor the ability of the network to generalize. Methods of selecting
validation data are available even in cases where very limited training data it
available.
5. Use compacted normalization. When normalizing dataespecially output
parametersnormalize the data into a compacted portion of the output range
of the transfer function being used. For example, if the output range of the
transfer function is [0,1], then normalize the data into the range [0.1,0.9],
This is important because most transfer functionse.g. the sigmoid family
of functionsonly approach the limiting values (asymptotically). Therefore
it will be difficult for the network to ever predict these values to within a
small tolerance.
6. Try many different random seed values. For a given network topology, the
training process should be repeated several times (e.g. 510) using
differentrandomly selectedstarting points on the error surface. This
reduces the likelihood that the training process will fail due to local minima
or plateaus in the error surfacehopefully at least a few of the training runs
will succeed.
7. Avoid applications requiring a high degree of numeric accuracy. Avoid
choosing applications in which a high degree of numeric accuracy is needed
for continuous valued parameterssuch accuracy is difficult to achieve. The
output values of networks involving continuous valued parameters will
always contain a certain level of error. Consider this fact carefully when
determining if an application is suitable for solution by neural networks.
All of these guidelines were followed in training the neural networks used in this
research and were proven to be useful. Many of them were not available at the outset,
but were formulated instead through a process of trialanderror during the work.
Future work in applying neural networks to highway bridge analysis should
concentrate on solving the problems encountered in neural network preconditioning. By
using modified convergence criteria for the network training process it is felt that this
72
Girders are the only structural components assumed to be even partially
continuous across hinges. The deck slab and stiffeners are assumed to be completely
discontinuous across hinges. Girder are continuous with respect to vertical translation
and, in some cases, axial translation but not flexural rotation. As a result, the end
rotations of the girder elements to either side of a hinge are uncoupledi.e. a hinge is
formed. Displacement constraints are used to provide continuity at the points where
girder elements cross a hinge line. In bridges modeled using the NCM or CGM
composite action models, the vertical translations of the nodes that connect girder
elements across a hinge line are constrained. When the EGM composite action model is
used, all three translations must be constrained due to the axial effects in the model.
Nodes to which girder elements are not connected are left unconstrained and therefore
are allowed displace independently.
3.5.2 Accounting for Creep in the Concrete Deck Slab
As was explained in the previous chapter, long term sustained dead loads on a
bridge will cause the concrete deck slab to creep. In steel girder bridges, the deck slab
and girders are constructed from materials that have different elastic moduli and
therefore different sustained load characteristics. As the concrete slab creeps over time,
increasingly more of the dead loads will be carried by the steel girders. Since the
models created by the preprocessor are not time dependent finite element models, the
creep effect must be accounted for in some approximate manner. Depending on the
68
losses due to effects such as friction, anchorage slip, creep and shrinkage of concrete,
and relaxation of tendon steel are not incorporated into the model. (In the BRUFEM
system, these nonelastic losses are accounted for automatically by the postprocessor
based on the appropriate AASHTO specifications). Thus, the tendon forces specified by
the engineer must be the initial pretensioning or posttensioning forces in the tendons,
prior to any losses.
3.4.2 Increased Stiffening of the Slab Over the Concrete Girders
During lateral flexure in prestressed (and also reinforced concrete Tbeam)
girder bridges, the relatively thin slab between the girders deforms much more than the
portion of the slab over the flanges of the girders. This behavior is due to the fact that
the girder flange and web have a stiffening effect on the portion of the slab that lies
directly above the girder. Rather than using thicker plate elements over the girders,
lateral beam elements are used to model this stiffening effect. These lateral beam
elements are located at the midsurface of the slab and extend over the width of the
girder flanges, as is shown in Figure 3.8.
Figure 3.8 Modeling the Stiffening Effect of the Girder Flanges on the Deck Slab
195
3. Incomplete Cholesky Decomposition Preconditioning. The approximation
matrix M is formed by extracting only the nonzero terms of A and
copying them into M. Subsequently, an incomplete decomposition
(ignoring fillin) is performed on the matrix M.
Each of these preconditioning schemes were implemented by the author in the
S1MPAL finite element analysis programwritten by Hoitthat is part of the
BRUFEM system (Hays et al. 1994). In implementing the various preconditioners in
SIMPAL, the profile, blocked, outofcore capabilities of the original direct solver
were also provided so that preconditioning for large FEA bridge models could be
studied.
7.4.1 Diagonal and Band Preconditioning
In diagonal preconditioning^ the main diagonal of A is extracted and inserted
into an otherwise empty matrix to form M (see Figure 7.1). The preconditioning
matrix P is then formed as P=M~l which is just a diagonal matrix in which
Pa = . To precondition the system, each side of the matrix equation Ax = b is
Mu
premultiplied by P. In practice, the matrix P which is extremely sparsedoes not
actually need to be formed since its effect on the system PAx = Pb can be computed
easily and directly.
^ In the discussion that follows, the matrix A is assumed to be symmetric and positive
definiteas is often the case in FEA.
158
3. Record the maximum magnitude displacements. For the each applied unit
load, determine the maximum magnitude displacement and record it for later
use in normalization and scaling.
4. Normalize the displacements. Normalize the displacements with respect to
the recorded maximum magnitude displacements. For each load case, the
displacements are normalized using the maximum magnitude value that
occurs in that load case. After normalization, the ordinates of each
displacement shape will lie in the range [1,1].
Several simplifications have been made in the propped cantilever beam example to keep
the discussion clear. First, only vertical loads (forces) and vertical displacements
(translations) have been considered. In practice, moment loads and rotational
displacements would also need to be considered. Next, the loading points and
displacementsampling points have been chosen to be at the same locations (a, b,
A ax = Maximum Magnitude Displacement AP At "cf Due To A Unit Load Applied
Caused By Unit Load Applied At "a" nt>rm At "P" And Normalized With Respect To
1
L
Normalized Shape Data That Shape
Networks Must Encode
o Neural Network Training Data Point
Figure 6.4 Separation of Shape and Scaling (Magnitude) Data
46
2.6 Modeling Features Specific to Steel Girder Bridges
Several of the modeling features available in the preprocessor relate specifically
to the modeling of steel girder bridges (see Figure 2.8). This section will provide an
overview of those features.
2.6.1 Diaphragms
In the steel girder bridge models created by the preprocessor diaphragms are
permitted to be either of the steel beam type or the steel cross brace type. Each of these
types is illustrated in Figure 2.9. The diaphragms connect adjacent girders together but
are not connected to the deck slab between the girders. Structurally, the diaphragms aid
in lateral load distribution, prevent movement of the girder ends relative to one
another, and are assumed to provide complete lateral bracing of the bottom flange in
negative moment regions. If a large number of diaphragms are used, as is often the
case for steel girder bridges, the diaphragms may have a significant effect on lateral
load distribution.
Cross brace diaphragms are constructed from relatively light steel members such
as angles and are often arranged in either an Xbrace configuration or a Kbrace
Concrete Concrete Steel Beam Steel
Figure 2.8 Cross Section of a Typical Steel Girder Bridge
13
networks were trained to predict displacement patterns in flatslab bridge under
generalized loading conditions. Once the networks were trained, a preconditioned
conjugate gradient (PCG) equation solver was implemented using the neural networks
both to seed the solution vector and to act as an implicit preconditioner.
In the case of seeding the solution vector, the networks attempt to predict the
actual set of displacements that would occur in the bridge under the given loading
condition. These displacements are then used as the initial estimate of the solution
vector in the equation solving process. Conceptually, the idea here is to make use of
the domain knowledge embedded in the ANNs to allow for the computation of a very
good initial guess at the solution vector. Clearly, for any iterative method, the ideal
initial solution estimate would be the exact solution since in that case no iteration would
be required.
Since the exact solution is obviously not known, it is typically necessary to use
a simplified scheme to estimate the solution vector. Such schemes include seeding the
solution vector with random numbers, zeros, or values based on the assumption of
diagonal dominance. None of these methods works particularly well for bridge
structures. In the present research, these simplistic methods are replaced by a
sophisticated set of neural networks that can predict very good initial estimates by
exploiting their knowledge of the problem.
In general, the neural networks are not be able to predict the exact set of
displacements that occur in the bridge. Therefore it will be necessary to perform
iterations within the PCG algorithm In order to converge on the exact solution. The
126
Networks store the relationship between input and output implicitly in
connection weights instead of explicitly as a set of rules as is the case in deterministic
systems. In other words, if a given set of input parameters for a particular problem are
sent to a network, there does not exist an explicit set of deterministic rules which can
be followed to arrive at what should be the output of the network. Instead, the input
signal is sent through the network and processed by an implicit set of rules which are
represented by the various connection strengths in the network. In a feed forward
network, this processing of the input data is performed using the weight summations
and transfer functions described earlier.
Thus, in order to construct a network that can solve a particular engineering
problem, the connection weights of the network must be trained to learn the
relationship between the input and output. Again, this is in contrast to the traditional
deterministic approach in which one formulates an explicit set of rules describing the
solution of the problem and then encodes those rules into an algorithm using a
computer language.
There are two basic approaches used in training networkssupervised training
and unsupervised training. In supervised training, network connection weights are
trained using example training pairs. A training pair consists of an input vector and a
desired output vector. Numerous training pairs are presented to the network and the
connection weights of the network are gradually adjusted so that it will eventually be
able to independently compute the correct output and also to generalize the relationship
represented by the training data.
Ill
the element files contain a great deal of repetition, the stress files contain more or less
random data from the viewpoint of the compression algorithm. To determine whether
the reduced compression of stress files was specific to the RDC algorithm or was
common to data compression algorithms in general, the stress files were independently
compressed using the RDC and LZ77 algorithms.
The public domain program ZIPa compressed file archiveruses LZ77 (Ziv
and Lempel 1977) as its compression algorithm and was used to compress the stress
files for comparison with RDC. Results from the compression of the stress files
produced by zero skew flatslab bridge analyses are summarized in Table 4.4. It can be
seen that while LZ77 achieved a greater compression than RDC, it also took
approximately twice as much time to do soa situation that is unacceptable in FEA
applications.
Since RDC does not produce dramatic outofcore storage savings when
compressing stress files, a series of parametric runs were performed to determine if
significant savings in execution time could be achieved by not compressing the stress
files. The results from these runswhich were performed on the same PC platform
Table 4.4 Comparison of RDC and LZ77 Compression Algorithms for Stress
Files Produced by Zero Skew Flatslab Bridge Analyses
Compression
Algorithm
Uncompressed
File Size
(bytes)
Compressed
File Size
(bytes)
Compression
Ratio
Compression
Time
(seconds)
RDCt
4,066,120
3,346,120
0.823
21.7
LZ77
4,066,120
2,821,704
0.694
40.5
t Buffer size of 16384 bytes and hash table size of 4096 elements.
172
lesson regarding what happens when the same load is applied in the second span.
Neuron2 provides a crossover between the two categorizations which can be useful in
preserving the ability of the network to generalize.
Thus, in this study, a singleneuron linear encoding was used to represent
Xcoordinates and a threeneuron piecewise linear encoding was used to represent
Ycoordinates. The threeneuron encoding was found to be especially important in
creating the normalized shape networks. These networks are very sensitive to being
able to distinguish between effects (loads, displacements) occurring in different spans.
6.6 Shape Neural Networks
Illustrated in Figure 6.9 is the basic layout of the shape neural networks used in
this research. As the figure indicates, there are eight input parameters and a single
output parameter for each of these networks. Recall that nine of these networks were
constructed so as to consider each combination of load type (Fz, Mx, My) and
displacement type (Tz, Rx, Ry). Also, note that there is no loadmagnitude input
parameterthese networks predict normalized shapes only.
The input parameters consist of the location (lateral and longitudinal
coordinates) of the applied load and the location at which the displacement is to be
sampled. These coordinates are encoded as was described in the previous sectionone
neuron for the Xcoordinate and three neurons for the Ycoordinate.
Up to this point, it has been stated that normalized displacements are always in
the range [1,1]. While this is essentially true (and is the preferable way of discussing
the concepts involved in encoding displacement data) there is one additional scaling that
87
preserves the information content of the data block while reducing its size, thus
reducing the amount of time that must be spent performing disk I/O. Of course the
process of compressing the data before writing it to disk requires an additional quantity
of time. Also the data must now be decompressed into a usable form when reading it
back from disk which also requires additional time. However, the end result is often
that the amount of additional time required to compress and decompress the data is
small in comparison to the amount of time saved by performing less disk I/O. This is
especially true on PC platforms where disk I/O is known to be particularly slow.
While one benefit of using compressed I/O can be a reduction in the execution
time required by FEA softwarewhich is often the critical measure of performancea
second benefit is that the quantity of disk space required to hold data files created by
the software is substantially reduced. A typical FEA program will create both
temporary data files, which exist only for the duration of the program, and data files
containing analysis results which may be read by post processing software. The data
compression strategy presented herein compresses only files that are binary data files,
i.e. raw data files. This is opposed to formatted readable output files that the user of a
FEA program might view using a text editor. Binary data files containing element
matrices, the global stiffness and load matrices, or analysis results such as
displacements, stresses, and forces are typically the largest files created by FEA
software and are the types of files which are therefore compressed. Formatted output
files can just as easily be compressed but the user of the FEA software would have to
decompress them before being able to view their contents.
23
1. Discretizing each and every structural component of the bridge into discrete
finite elements and subsequently specifying the characteristicsgeometry,
material properties, connectivities, eccentricities, etc.of each of those
elements.
2. Modeling the structural configuration and the appropriate dead loads at each
distinct stage of construction.
3. Computing potentially hundreds of discrete vehicle positions and
subsequently computing and specifying the load data required for FEA.
Each of these obstacles is overcome through the use of the preprocessor because
it handles these tasks in a semiautomated fashion in which the engineer and the
software both contribute to the creation of the model.
Bridge modeling is accomplished using the preprocessor in an interactive
manner in which the user is asked a series of questions regarding the characteristics of
the bridge being modeled. Each response given by the user determines which questions
will be asked subsequently. For example, assume that the user is asked to specify the
number of the spans in the bridge and a response of 2 is given. Then the user may
later be askeddepending on the type of bridge being modeledto specify the amount
of deck steel effective in negative moment regionsi.e. a parameter that is only
applicable to bridges having more than one span.
Both girderslab bridges and flatslab bridges may be modeled using the
preprocessor. Girderslab bridges are those characterized as having large flexural
elements, called girders, that run in the longitudinal direction of the bridge and which
are the primary means of applied bridge loads. In a girderslab bridge, the girders and
deck slab are often joined together in such a way that they act compositely in resisting
loads through flexure. This type of structural behavior is called composite action and
37
unit length. Uniform dead loads, such as that which would result from the resurfacing
of a bridge, may be accounted for by specifying a uniform overlay load.
2.4.6 Prismatic and Nonorismatic Girders
Prismatic girderslab bridges, in which the cross sectional shape of the girders
remains the same along the entire length of the bridge, are the simplest type of girder
slab bridge. Prismatic girders are commonly used in prestressed concrete girder bridges
where standard precast cross sectional shapes are the norm. Most reinforced concrete
Tbeam bridges can also be classified as prismatic girderslab bridges.
Nonprismatic girders, in which the cross sectional shape of the girders varies
along the length of the bridge, are commonly used to minimize material and labor costs
in steel girder bridges. The cross sectional shape of a steel girder can be easily varied
by welding cover plates of various sizes to the top and bottom flanges of the girder,
thus optimizing the use of material. Nonprismatic girders are also used in post
tensioned prestressed concrete girder bridges in which thickened girder webs, called
end blocks, are often required at the anchor points of the posttensioning tendons.
Another class of nonprismatic girder occurs when the depth of a girder is varied
usually linearlyalong the length of a girder span. Linearly tapering girders occur in
both steel and prestressed concrete girder bridges.
Prismatic girders can be modeled for all of the bridge types treated by the
preprocessor, either for live load analysis and full load analysis. Nonprismatic girders
are also permitted for the following bridge types.
116
It is important to note that in implementing the compression library in SIMPAL,
it was decided that the stress files should not be compressed. This decision was based
on two factors. First, results from the SEAS parameter studies described the previous
section indicated that FEA stress files do not compress well enough to warrant the
computational effort associated with compressing them. Second, in the BRUFEM
system, the stress files written by SIMPAL must be read by a bridge rating post
processor that is coded in Fortran and which utilizes Fortran I/O. By having SIMPAL
write the stress files in Fortran format instead of compressed format, I/O modification
to the postprocessor were avoided.
Thus, in interpreting Figure 4.8, one must keep in mind that the reduction in
outofcore storage indicated in the plots does not reflect the storage required by the
stress files. The plots indicate the degree of compression which was achieved for the
element data flies which were compressed. While these files often constitute the major
portion of storage required by an analysis, stress files can also require substantial
storage as wellespecially in cases involving a large number of load cases.
Plotted in Figure 4.9 are the normalized execution times for the prestressed and
steel bridge models tested. The execution times plotted are for analysis runs performed
on a workstation running the UNIX operating system and on a PC running the DOS
operating system. Since it was of interest to compare the speed of standard Fortran I/O
to that of compressed I/O, the execution times for each of these tests were normalized
with respect execution time of the standard Fortran I/O runs.
49
essentially softens the composite girderslab load resisting system and therefore
increases the stresses in the girders.
When modeling steel girder bridges that fit the conditions described above, the
preprocessor will automatically account for the effects of concrete deck creep. Details
of this modeling procedure are presented in the next chapter.
2.7 Modeling Features Specific to Reinforced Concrete TBeam Bridges
Reinforced concrete Tbeam bridges (see Figure 2.10) are modeled very
similarly to prestressed concrete girder bridges except that there is no prestressing
present. A notable exception is that the cross sectional shape of a Tbeam girder is
completely defined by the depth and width of the girder web. A Tbeam girder consists
of a rectangular concrete web joined to the deck slaba portion of which acts as the
girder flange. Thus the engineer simply specifies the depth and width of the web of
each girder when modeling Tbeam bridges. Databases of standard cross sectional
shapes are not needed as was the case in prestressed concrete girder bridges.
Concrete
Parapet
Concrete
Deck Slab
Concrete Concrete
Diaphragm TBeam Web
KH
h 1
Figure 2.10 Cross Section of a Typical Reinforced Concrete TBeam Bridge
67
which the deck slab is structurally effective are illustrated in Figure 3.7. In construction
stages where the deck slab has not yet become structurally effective, the tendon
eccentricities are always referenced to the centroid of the bare girder cross section
regardless of which composite action model is being used.
When using the preprocessor, the engineer always specifies the location of the
prestressing tendons relative to the top of the girder. With this data, the preprocessor
computes the proper truss element eccentricities for each construction stage of the
bridge based on the composite action model in effect. The example girder that is shown
in Figure 3.6 has a piecewise linear tendon profile tendon created by dual hold down
points. Note however that the preprocessor is capable of approximating any tendon
profile, linear or not, as a series of linear segments.
In addition to modeling the profile of the prestressing tendons, the prestressing
force must also be represented. This is accomplished simply by specifying an initial
tensile force for each tendon (truss) element in the model. Since the tendons are
modeled using elastic truss elements, prestress losses due to elastic shortening of the
concrete girder are automatically accounted for in the analysis. However, nonelastic
Noncomposite
Model (NCM)
Effective Width
' 'I
f
M
Prestressing
Centroid
Cl
Tendon
Eccentricity
1
Composite Girder
Model (CGM)
Eccentric Girder
Model (EGM)
Figure 3.7 Modeling the Profile of a Prestressing Tendon
4.7 Fortran Interface to the Compressed I/O Library 99
4.8 Data Compression Parameter Study and Testing 101
4.8.1 Data Compression in FEA Software Coded in C 102
4.8.2 Data Compression in FEA Software Coded in Fortran 112
5 NEURAL NETWORKS 119
5.1 Introduction 119
5.2 Network Architecture and Operation 120
5.3 Problem Solving Using Neural Networks 124
5.4 Network Learning 125
5.5 The NetSim Neural Network Package 128
5.6 Supervised Training Techniques 130
5.7 Gradient Descent and Stochastic Training Techniques 133
5.8 Backpropagation Neural Network Training 137
5.8.1 ExampleByExample Training and Batching 141
5.8.2 Momentum 143
5.8.3 Adaptive Learning Rates 146
6 NEURAL NETWORKS FOR HIGHWAY BRIDGE ANALYSIS 151
6.1 Introduction 151
6.2 Encoding Structural Behavior 151
6.3 Separation of Shape and Magnitude 153
6.3.1 Generating Network Training Data 157
6.3.2 Using Trained Shape and Scaling Networks 160
6.4 Generating Analytical Training Data 163
6.5 Encoding Bridge Coordinates 168
6.6 Shape Neural Networks 172
6.7 Scaling Neural Networks 175
6.8 Implementation and Testing 183
7 ITERATIVE EQUATION SOLVERS FOR HIGHWAY BRIDGE
ANALYSIS 185
7.1 Introduction 185
7.2 Exploiting Domain Knowledge 186
7.3 Iterative FEA Equation Solving Schemes 188
7.4 Preconditioning in Highway Bridge Analysis 194
7.4.1 Diagonal and Band Preconditioning 195
7.4.2 Incomplete Cholesky Decomposition Preconditioning 201
7.5 A Domain Specific Equation Solver 205
7.6 Implementation and Results 212
7.6.1 Seeding the Solution Vector Using Neural Networks 213
7.6.2 Preconditioning Using Neural Networks 229
vi
54
element zero energy mode occurs quite often in the modeling of flatslab bridges and
must be used with great caution in such situations.
One solution to this problem is to use the nine node heterosis element developed
by Hughes (1987) which inherits all of the advantages of using higher order shape
functions without the disadvantage of being rank deficient. Correct rank is
accomplished by utilizing standard lagrangian (ninenode) shape functions for all
element rotational degrees of freedom (DOFs) but serendipity (eightnode) shape
functions for the translational DOFs. Both a nine node standard lagrangian element and
a nine node heterosis element have been implemented by the author in a FEA program
that was developed as part of the present research. In tests on flatslab bridge models,
the heterosis element performed flawlessly in situations where lagrangian elements
suffer from zero energy modes.
However, because there is no translational degree of freedom associated with
the center nodes of heterosis elements, bridge meshing and distribution of wheel loads
is considerably more complex. Thus, a simpler solution is to simply use bilinear
elements and ensure that an adequate number of such elements is used both the lateral
and longitudinal directions of the bridge. This is the solution that has been adopted for
use in the preprocessor.
3.2.2 Modeling the Girders and Stiffeners
Girders and stiffeners are modeled using standard frame elements. Frame
elements consider flexural effects (pure beam bending), shear effects, axial effects, and
96
format without any loss, corruption, or distortion of the data. In contrast, lossy data
compression techniques permit some distortion of the data to occur during the
translation process. Lossy techniques are used in applications such as image and sound
compression in which a limited degree of distortion can be tolerated. However, in
applying data compression to FEA, it is the floating point representations of numeric
values which are compressed and therefore distortion of the data to any extent would
invalidate the analysis. Thus, in FEA it is necessary to use a lossless compression
technique such as RDC.
In RDC, the fundamental unit of data is the byte, so that all data is examined on
a byte by byte basis to determine if there are runlengths of repeated bytes or larger
matching patterns consisting of blocks of bytes. All data, including single and double
precision floating point values, short and long integers, characters and strings are
compressed on a byte basis. In runlength encoding (Ross 1992, Sedgewick 1990), a
repeated sequence of a byte is compressed by replacing the sequence by a single
instance of the byte and an additional runlength code indicating the number of times
the byte is to be repeated. RLE compression can be used to compress small scale
repetitions as in the example of the plate element load vector discussed earlier which
contained a sequence of repeated zero bytes.
Large scale pattern matching, as in the example of the plate element stiffness
matrix presented earlier, is accomplished in RDC by using a sliding dictionary to match
a multiple byte pattern at the current location in the buffer with an earlier occurrence of
the same pattern. A hashing algorithm (Ross 1992, Sedgewick 1990) is used to search
213
NNPCG Iteration Neural Network
Module Control Module
Figure 7.4 Relationship Between NNPCG Solver Modules
calls individual neural network modules to calculate specific components of the
displacement vector. Using the principle of superposition, the neural network modules
are called repeatedly to form the complete displacement vector for the structure under
the specified loading conditions.
Each of the neural network modules constitutes a selfcontained neural network
that has been trained to solve one aspect of the overall loaddisplacement calculation
problem. Since each of the networks were trained using the NetSim package (see
discussion in Chapters 5 and 6), the automatic code generation capability that software
was used to generate the individual C code modules corresponding to each network.
Figure 7.4 illustrates the relationship between the various components of the NNPCG
solver.
7,6.1 Seeding the Solution Vector Using Neural Networks
In order to test the effectiveness of the NNPCG algorithm, numerous tests were
performed using FEA models of a typical twospan flatslab bridge. The bridge, which
206
In this research, neural networks serve as the domain knowledge encoding
mechanism. The networks have been trainedto within an acceptable level of errorto
learn the loaddisplacement relationship for twospan flatslab concrete bridges. Once
trained, the networks are used to predict displacement patterns for these bridge
structures under specified loading conditions. Finally, the neural network displacement
predictions are used within the larger framework of the PCG algorithm to create a
hybrid equation solver.
Recall from Chapter 5 that network training is an iterative process which is
halted only when the chosen measure of error has decreased below an acceptable
tolerance level. Therefore, the loaddisplacement relationship encoded by the networks
is not exact but rather an approximation. Since it is not an exact encoding, one cannot
use the networks to directly solve for the exact displacements in bridge structures.
However, the network can be used to rapidly solve for approximate displacements in
these structures and it is this fact which is exploited by the hybrid NNPCG algorithm.
Conceptually, the idea of the hybrid solver is to exploit the fact that the neural
networks can rapidly predict approximate displacements with little required
computation. However, the use of the networks must be placed into an overall
algorithm in which exact displacements can ultimately be computed. Iterative equation
solving methods are the obvious choice for such a framework. Neural networks can be
used to compute approximate displacements at each iteration while the overall iterative
process guides convergence to the exact solution. In the NNPCG algorithm, the neural
47
configuration. The steel girder bridge type is the only bridge type for which the
preprocessor allows cross brace diaphragms to be modeled. A detailed study was
performed by Hays and Garcelon (see Hays et al. 1994, Appendix I) in which steel
girder bridges were studied using full three dimensional models. The studies indicated
that the behavior of bridges having Xbrace and Kbrace diaphragms were sufficiently
close that Kbrace diaphragms can adequately be modeled using the Xbrace
configuration. Thus, only the Xbrace configuration is modeled by the preprocessor.
The engineer must specify whether beam diaphragms or cross brace diaphragms
will be used and provide the section properties for either the steel beam or the elements
of the cross brace. These section properties are then used for all of the diaphragms in
the bridge. However, in the case of cross brace diaphragms, the depth of the
diaphragms will vary if the depth of the girders vary.
2,6.2 Hinges
Hinged girder connections are occasionally placed in steel girder bridges to
accommodate expansion joints or girder splices. The preprocessor is capable of
Figure 2.9 Diaphragm Types Available for Steel Girder Bridges
168
Second, each load point in this case generates only a single neural network
training pair. For each load, the displacements at all the grid points are computed and
scanned, but then only the largest magnitude term is retainedall other values are
discarded. Therefore, the volume of training data that generated for the scaling
networks was considerably less than that of the shape networks. As a result, more
bridge geometries could be treated in this case than was possible for the shape network
case. A base geometry was chosen that matched the geometry used for the shape
networks. Then three additional scaled variations of this base geometry were analyzed.
In all, Geometric Scale Factors (GSFs) of 0.6, 0.8, 1.0, and 1.2 were treated.
Strictly speaking, a shape network trained for a GSF of 1.0 (the only case
considered for shape networks) should not be able to be used with a scaling network
trained using a GSF of, say, 0.8. However, by separating displacement shape data from
magnitude data it was reasoned that the normalized shape might remain roughly
unchanged for various GSFs. By using scaling networks trained on multiple GSFs, the
normalized displacement shape could then be scaled by appropriate scaling factors.
Thus, using four GSFs and 231 grid points, a total of 924=4*231 neural network
training pairs were generated for each loaddisplacement combination (e.g. Fz loads
causing Tz translations).
6.5 Encoding Bridge Coordinates
When using neural networks, all network output parameters must be encoded in
a form that is consistent with the transfer functions being used. Since the output ranges
of the sigmoid transfer functions g and h (see Chapter 5) used in this research are
193
Preconditioned Conjugate Gradient (PCG) algorithm results. PCG equation solvers can
be further classified based on the choice of preconditioner used. (Some preconditioners
that can be used in the FEA of highway bridge structures are discussed in the next
section.) One particularly appealing aspect of combining preconditioning with the CG
algorithm is that the preconditioned coefficient matrix A can be formed implicitly.
Therefore when implementing this algorithm, the transformed matrix A = PA
never needs to be explicitly computed (and therefore also never needs to be stored).
The Preconditioned Conjugate Gradient algorithm is given by the following steps.
form initial guess
(7.15)
'(0) ~b~ ^x(0)
(7.16)
O
II
5;
1
o
(7.17)
rfi)M V(i)
(0 T .
Pa)Mi)
(7.18)
x0'+l) = x(i) +a(i)P( 0
(7.19)
r0+D = r(0 ad)Mo
(7.20)
T l
r0+V)M r(/+1)
P(+1) J uK
r(i)M r(i)
(7.21)
P(i+1) = M 1''(i+l) + P(i+l)P(i
j (7.22)
where M is a matrix that closely approximates
the coefficient matrix A. The
preconditioning matrix in this algorithm is P = M 1 but if M is chosen carefully, then
M 1 should be fairly easy to compute. Actually, M
1 never needs to be computed at
7
1.2.1 Computer Assisted Bridge Modeling
Widespread use of FEA techniques in highway bridge analysis has been
curtailed by a lack of requisite pre and postprocessing tools. Routine use of FEA in
bridge analysis can only occur when computer assisted modeling software has been
developed specifically with the highway bridge engineer in mind. To address this issue,
an interactive bridge modeling program has been developed as part of the research
reported on herein. The resulting bridge modeling preprocessor, called BRUFEM1, is
one component of the overall BRUFEM system (Hays et al. 1990, 1991, 1994).
BRUFEM, which is an acronym for Bridge Rating Using the Finite Element Method, is
a software package consisting of a series of Fortran 77 programs which, when working
collectively as a system, is capable of modeling, analyzing, and rating many types of
highway bridge structures.
The BRUFEM preprocessor, which will hereafter be referred to simply as the
preprocessor, allows an engineer to create detailed FEA bridge models by specifying
interactivelya minimal amount of bridge data. Information needed specifically for the
modeling of bridge structures and bridge loading is embedded directly into the
software. Thus the usual barriers that would prevent an engineer from manually
constructing the FEA bridge model are overcome. The primary barriers are :
1. Discretizing each and every structural component of the bridge into discrete
finite elements and subsequently specifying the characteristicsgeometry,
material properties, connectivities, eccentricities, etc.of each of those
elements.
2. Modeling the structural configuration and the appropriate dead loads at each
distinct stage of construction.
177
displacement magnitudes all fall in the range [0,1]. Note that when using the
networksas opposed to when training themthe normalized maximum displacement
magnitudes produced by the networks must be scaled from [0,1] to a true structural
range. The overall scaling factors (the maximum displacement magnitudes from the
GSF = 1.2 case) used for purpose this are listed in Table 6.3 .
Table 6.3 Trained Shape Networks
Load
Type
Disp.
Type
Network
Topology
Num.
Conn.
Num.
Epoch
Overall
Max.
Max.
Training
(Validation)
Errorf
Avg.
Training
(Validation)
Errort
Fz
Tz
5x15x15x1
315
10000
1.6502e02
0.07265
(0.02705)
0.00745
(0.00705)
Fz
Rx
5x15x15x1
315
15000
9.2530e05
0.06189
(0.04727)
0.01123
(0.01131)
Fz
Ry
5x15x15x1
315
20000
6.9303e05
0.06310
(0.03980)
0.00869
(0.00939)
Mx
Tz
5x15x15x1
315
10000
9.2771e05
0.06586
(0.04303)
0.01194
(0.01255)
Mx
Rx
5x15x15x1
315
40000
3.6506e06
0.07944
(0.03670)
0.00829
(0.00705)
Mx
Ry
5x15x15x1
315
40000
6.5784e07
0.05909
(0.03823)
0.00769
(0.00868)
My
Tz
5x15x15x1
315
20000
6.7015e05
0.06947
(0.03839)
0.01188
(0.01029)
My
Rx
5x15x15x1
315
30000
6.5784e07
0.05544
(0.01020)
0.02640
(0.00855)
My
Ry
5x18x18x1
432
20000
3.5500e06
0.09878
(0.07304)
0.01774
(0.01666)
f The errors statistics reported here have already been made relative to the range [0,1]
instead of the compacted range [0.1,0.9].
88
For reasons of accuracy and minimization of roundoff error, virtually all FEA
programs perform floating point arithmetic using double precision data values. In
addition, much or all of the integer data in a program consists of long (4 byte) integers
as opposed to short (2 byte) integers either because the range of a short integer is not
sufficient or because long integers are the default in the language (as is the case in
Fortran 77). An underlying consequence of using double precision floating point and
long integer data types is that there is a tremendous amount of repetition in data files
created by FEA software. Consider as an example the element load vector for a nine
node plate bending element. A plate bending element has two rotational and one
translational degrees of freedom at each node in the local coordinate system, but when
rotated to the global coordinate system there are six degrees of freedom per node.
Thus, for a single load case the rotated element load vector which might be saved for
later assembly into the global load vector will have 9*6 = 54 entries. If the entries are
double precision floating point values then each of the 54 entries in the vector is made
up of 8 bytes resulting in a total of 54*8 = 432 bytes. Now consider an unloaded plate
element of this type where the load vector contains all zeros. Typical floating point
standards represent floating point values as a combination of a sign bit, a mantissa, and
an exponent. A value of zero can be represented by a zero sign bit, zero exponent, and
zero mantissa. Thus a double precision representation of the value zero may consist of
eight zero bytes. A zero byte is defined as containing eight unset (zero) bits.
Consequently, the load vector for an unloaded plate element will consist of 432
repeated zero bytes resulting, in a considerable amount of repetition within the data
230
Figure 7.16 Displacements (TzTranslations) and Residuals (FzForces) for a
Flatslab Bridge Under Vehicular Loading Using the Full NNPCG
Algorithm (Network Seeding and Network Preconditioning)
Thus, to compute a displacement, a shape network is called to compute a normalized
displacement Aorm and a magnitude network is called to compute a scaling factor
AScaie The actual structural displacement is then given by their product,
A = A norm^ scale
The problems here are twofold. First, recall from Chapter 6 that, while the
networks were trained to a low average error, there were a few training cases in which
the maximum errors were quite significant. It is evident from these tests that large
network errors can not be tolerated, even if they are limited in number. Such errors
203
Thus, in the ICPCG method, the matrix M is chosen to be exactly equal to the
matrix A As a result, we start off with a matrix M that exactly approximates the
coefficient matrix A However, we then use an approximate method to form the
decomposition of M. This can be roughly thought of as forming an approximate
inverse since the approximate decomposition will be used to solve the Mq = r sub
problems during the PCG iterationsinstead of using q = M~lr. (Refer to the
previous section for more details on the Mq = r subproblem). Since fillin is not
allowed, the incomplete decomposition is computationally inexpensivea situation
which is necessary for an effective preconditioning scheme.
Also, since fillin is never allowed, the sparsity of the matrices A and M can
be fully exploited. In a direct solver, the profile storage scheme is the most efficient
storage scheme available since fillin must be considered. Since the ICD scheme
preserves the sparsity of the matrix, the terms which would normally be filledin do not
need not be stored since they will be ignored. Thus, fully compact storage schemes in
which only the nonzero terms of the matrix are stored can be used in an ICPCG
solver. In fact, it is because iterative methods can fully exploit the sparsity of the
matrices that they can outperform direct methods for very large problems. In such
problems, calculations involving fillin require a great deal of storage and
computational effort which can be eliminated using iterative methods and the ICD
preconditioning scheme.
Unfortunately, despite all of its advantages, the ICD preconditioning scheme
still performs poorly in FEA bridge analysis. The primary problem is that when the
ANALYSIS OF HIGHWAY BRIDGES USING
COMPUTER ASSISTED MODELING, NEURAL NETWORKS,
AND DATA COMPRESSION TECHNIQUES
By
GARY RAPH CONSOLAZIO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995
164
Several modifications were made to SEAS to facilitate rapid and accurate
generation of displacement training data. A feature was implemented that causes SEAS
to automatically normalize all displacement data and to report the maximum magnitude
displacements which occurred. Thus, the displacement data reported by SEAS was
already in the form needed for network training.
An additional plate bending elementthe heterosis element (Hughes 1987)was
also added to SEAS and was used in all of the analyses performed. Originally, a
biquadratic (9node) lagrangian element was used to model the flatslab bridges.
However, it was found early on that this element was prone to zero energy modes when
used to model thick flatslabs. Therefore, a heterosis element, which does not suffer
from the same zero energy modes+. was implemented and tested in SEAS. Tests
showed that the heterosis element performed well in situations where zero energy
modes arose when using lagrangian elements.
To generate shape network training data, a grid of load locations and a grid of
displacement sampling locations were chosen. For a given loaddisplacement
combination (e.g. Fz loads causing Tz translations), unit loads were applied at each
point in the load grid. Each of these unit loads generated one load case in the analysis.
The displacements at each of the displacementsampling points were then extracted
from the FEA output. This process was repeated for all nine of the loaddisplacement
combinations that were consideredFzTz, FzRx, FzRy, MxTz, MxRx, MxRy, MyTz,
The advantages of heterosis plate elements over lagrangian plate elements in thick
plate applications were discussed in detail in Chapter 3.
182
Tz (indies) Geometry 0.6
8.00e05
6.00&05
Tz (inches) Geometry 1.0
8.00e05,
Tz (inches) Geometry 0.8
8.00e05
6.00e05
4.00e05
2.00e05
0.00e+00p
0.00
0.25'
Normalized1^
Lateral 0.75
Direction 1.00
Normalized Longitudinal Direction
Normalized Longitudinal Direction
Figure 6.17 Maximum Magnitude Translations (Tz) Caused By Unit Moments (My)
(Training Data for Scaling Neural Networks)
Rx (radians) Geometry 1.0
8.00e07
6.00e07
Rx (radians) Geometry 0.8
8.00e07,
Normalized
Lateral 0.75
Direction 1.00
Normalized Longitudinal Direction
Rx (radians) Geometry 1.2
8.00e07i
kt i 050
Normalized
100 Lateral 0.75
Normalized Longitudinal Direction
Figure 6.18 Maximum Magnitude Rotations (Rx) Caused By Unit Moments (My)
(Training Data for Scaling Neural Networks)
198
Therefore, the matrix inverse M 1 really never needs to be formed as long as one can
solve the matrix equation Mq = r. In practice, this is generally accomplished by
performing a direct equation solution on the subproblem Mq = r of the overall
T
problem Ax = b. In the present research the symmetric decomposition LDL is used,
however, any reasonable technique can be employed. Observe then that one component
of the iterative PCG solution process involves the use of a direct solver on a sub
problem, therefore a PCG solver will generally contain iterative solution code as well
as direct solution code.
To determine the effectiveness of diagonal and band preconditioning in the FEA
of highway bridge structures, several bridge models were constructed using the
preprocessor described in Chapters 2 and 3. These models were then analyzed by the
author's PCG solver using both diagonal and band preconditioning. Diagonal
preconditioning was studied by simply using a bandwidth of 1 for the band
preconditioning casei.e. special coding to compute the diagonal inverse was not
written.
Results of these studies indicated that neither diagonal preconditioning nor band
preconditioning work well in the analysis of highway bridge structures. To understand
why this is the case, one must examine the structure (sparsity pattern) of the coefficient
matrices that arise in bridge FEA. In bridge FEA, the generic coefficient matrix A
becomes the global finite element stiffness matrix of the structure. It is well known that
such stiffness matrices are often sparse and exhibit either banded or skyline structure.
175
The various network topologies reported in Table 6.2 were arrived at by trial
and error. Each of the final network configurations shown in the table was the result of
training several different size networks and selecting the one which produced the least
error. Also, note that for each topology trained, several (510) separate training runs
were performed starting from different, randomly selected points on the error surface.
6.7 Scaling Neural Networks
Figure 6.10 illustrates the basic layout of the scaling (magnitude) neural
networks used in this research. As the figure indicates, there are five input parameters
and a single output parameter for each of these networks. Nine networks were
constructed so that each combination of load type (Fz, Mx, My) and displacement type
(Tz, Rx, Ry) was covered.
Table 6.2 Trained Shape Networks
Load
Type
Disp.
Type
Network
Topology
Number of
Connections
Number of
Epochs
Maximum
Error1
Average
Error1
Fz
Tz
8x16x26x1
570
15000
0.27310
0.02298
Fz
Rx
8x16x26x1
570
20000
0.21745
0.02135
Fz
Ry
8x18x28x1
676
25000
0.21791
0.02605
Mx
Tz
8x18x20x1
524
10000
0.26698
0.02870
Mx
Rx
8x18x20x1
524
15000
0.10630
0.01361
Mx
Ry
8x20x24x1
664
10000
0.32554
0.03590
My
Tz
8x20x24x1
664
25000
0.20696
0.02442
My
Rx
8x18x20x1
524
40000
0.24641
0.02887
My
Ry
8x18x20x1
524
10000
0.11923
0.01420
^ The errors statistics reported here have already been made relative to the range [1,1]
instead of the compacted range [1/1.2,1/1.2].
192
will have a more compact eigenvalue spectrumnamely the eigenvalue spectrum of A
instead of A.
For a symmetric positivedefinite matrix, the smallest value which k can take
on is 1.0. Therefore an ideal preconditioning matrix P will transform A into the
identity matrix / for which the condition number is 1.0. It then becomes apparent that
the ideal preconditioner for any system Ax = b is P= A~l, the inverse of the original
coefficient matrix. Clearly, however, the amount of work involved in obtaining A~
cannot be justified simply to accelerate the convergence of an iterative equation solver.
If A~l was available, then the system Ax = b could be trivially solved and there would
be no need for an iterative solver at all.
However, one can see from this discussion that the closer the preconditioned
coefficient matrix A = PA is to the identity matrix /, the faster the iterative solution
process will converge to x. This fact can serve as a guide for evaluating different
preconditioning strategies. Preconditioning is only economical if the computational
savings resulting from accelerated convergence more than offset the additional work
involved in obtaining P and transforming Ax = b into Axb. Thus, a good
preconditioner must not be excessively expensive to compute and must substantially
reduce the condition number of the matrix A .
While preconditioning canin theorybe applied to any iterative method, it is
particularly well suited to the Conjugate Gradient method (Jennings and McKeown
Â§11.13, 1992). When preconditioning is combined with the CG algorithm, a
214
is illustrated in Figure 7.5, was subjected to both uniform loading and vehicular
loading conditionstypical situations arising in highway bridge analysis. Specifically,
the loading conditions examined are those listed below.
1. Uniform Loading. A uniform load (surface pressure load) extending over the
entire length and width of the bridge.
2. Vehicular Loading. Two HS20 trucks positioned endtoend near the right
edge of the bridge.
These loading conditions were used throughout the testing phase and will be referred to
repeatedly in the discussion that follows. The FEA bridge models were prepared using
the preprocessor described in Chapters 2 and 3. Table 7.1 lists the important
parameters of the FEA models.
One of the goals in developing the NNPCG algorithm was to accelerate the
convergence of the solution process by utilizing an intelligent initial guess. By
seeding the solution vector with displacements calculated using neural networks, the
goal is to start the iterative process at a point very near the correct solution. In the
bridge models studied, the slab is modeled using flat plate elements that have three
30 ft. (360 in.)
10 Elements
* 40 ft. (480 in.)
* 25 ft. (300 in.) *
10 Elements
10 Elements
Figure 7.5 Configuration of Flatslab Bridge Test Models
Abstract of Dissertation Presented to the Graduate
School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANALYSIS OF HIGHWAY BRIDGES USING
COMPUTER ASSISTED MODELING, NEURAL NETWORKS,
AND DATA COMPRESSION TECHNIQUES
By
GARY RAPH CONSOLAZIO
August 1995
Chairman: Marc I. Hoit
Major Department: Civil Engineering
By making use of modern computing facilities, it is now possible to routinely
apply finite element analysis (FEA) techniques to the analysis of complex structural
systems. While these techniques may be successfully applied to the area of highway
bridge analysis, there arise certain considerations specific to bridge analysis that must
be addressed.
To properly analyze bridge systems for rating purposes, it is necessary to model
each distinct structural stage of construction. Also, due to the nature of moving
vehicular loading, the modeling of such loads is complex and cumbersome. To address
these issues, computer assisted modeling software has been developed that allows an
engineer to easily model both the construction stages of a bridge and complex vehicular
loading conditions.
Using the modeling software an engineer can create large, refined FEA models
that otherwise would have required prohibitively large quantities of time to prepare
manually. However, as the size of these models increases so does the demand on the
viii
215
DOF at each nodevertical translations Tz, rotations about the Xaxis (lateral
direction) Rx, and rotations about the Yaxis (longitudinal direction) Ry. Thus, to form
the initial solution vector, three displacements (Tz, Rx, Ry) must be computed at each
node in the model, i.e. at each active DOF.
Using the bridge models described above, neural networks were employed to
compute displacements for use in solution seeding. Figures 7.6 and 7.7 compare the
neural network predicted displacements and analytical (FEA) displacements for the
uniform and vehicular loading conditions respectively. One can clearly see from the
figures that the neural networks do a respectable job of matching the analytical results.
The most notable differences occur in the prediction of the Rx rotations where
noticeable differences can be seen. However, overall the figures suggest that neural
networks may be very effective in seeding the solution vector since their predictions
closely match analytical results.
To test the presumption, three different seeding methods were examinedzero
seeding, diagonal seeding, and neural network seeding. Zero seeding starts the iterative
process by simply filling the solution vector with zero values. Diagonal seeding
computes each term in the solution vector as x, = A^/b, which implicitly assumes that
the matrix is nearly diagonally dominant.
Table 7.1 Structural Parameters of Flatslab Bridge Test Models
Geometry
Span1 Length 40 ft.
Span2 Length 25 ft.
Bridge Width 30 ft.
Slab Thickness 14 in.
Skew Angle 0 deg.
Material Properties
Concrete 4 ksi
Poisson's 0.15
Elastic Mod. 3605 ksi
Bearing 50 ksi
Modeling
Span1 Elements 10
Span2 Elements 10
Width Elements 10
Number of DOFs 693
Number of Nodes 264
123
neuron to fire otherwise there is no output signal produced. Others produce binary
output signals which are tied to the value of the weighted sum.
In this research, two types of continuous valued transfer functions, called
sigmoid functions, are used to produce the output signal of computing neurons. Those
sigmoid functions, which have the form
and
g(x)
1
l + e~x
(5.1)
h(x) =
\e
l+e~
(5.2)
serve to modifyor squashthe weighted sum of the input signals into a confined
range of output values. This squashing is performed in a nonlinear manner as is
illustrated in the plots of Figure 5.3. Sigmoid function g has output range of [0,1] and
is used in applications where the output of the NN will always be positive. If the NN
output values must be able to take on positive or negative values, then the sigmoid
function h is used which has an output range of [1,1].
Once the weighted sum of input signals has been passed through the transfer
function, and the value of the transfer function computed, this value becomes the
output signal of the neuron. If the network consists of only an input layer and an output
layer, then the output signals of the computing neurons are also the output signals of
the overall network. This type of network is commonly referred to as a single layer
network since there is only one layer of computing neurons. In a multiple layer network
there is more than one layer of computing neurons. Every network layer that is made
20
vibrational signature of a structure. Szewczyk and Hajela (1994) extended the concept
once again by utilizing counterpropagation neural networks instead of the more often
used backpropagation neural networks. Counterpropagation networks can be trained
much more rapidly than traditional plain vanilla backpropagation networks and are
therefore well suited for damage detection applications where a large number of
training sets need to be learned.
Several other diverse applications of neural networks in structural engineering
have also appeared in the literature. Garcelon and Nevill (1990) explored the use of
neural networks in the qualitative assessment of preliminary structural designs. Hoit
etal. (1994) investigated the use of neural networks in renumbering the equilibrium
equations that must be solved during a structural analysis. Gagarin et al. (1994) used
neural networks to determine truck attributes (velocity, axle spacings, axle loads) of in
motion vehicles on highway bridges using only girder strain data. Rogers (1994)
illustrated how neural network based structural analyses can be combined with
optimization software to produce efficient structural optimization systems.
55
torsional effects. Each of these groups of effects are considered separately and are
therefore not coupled.
If the CGM is chosen, composite section properties are used for the elements
representing girders and stiffeners in the bridge. If the NCM is selected then the
noncomposite element properties are used. If the EGM is used, then the noncomposite
girder and stiffener properties are used and the composite action is modeled by locating
the frame elements eccentrically from the centroid of the slab.
In modeling steel girders using frame elements, the transverse shear
deformations in the elements are properly taking into account. Hays and Garcelon
(Hays et al. 1994, Appendix I) found that, when using the type of models created by
the preprocessor, shear deformations in the girders must be considered for the analysis
to be accurate. This conclusion was based on a study comparing the response of models
created by the preprocessor and the response of fully three dimensional models. Shear
deformations are not, and do not need to be, accounted for in concrete girders or
concrete parapets where such deformations are typically negligible.
The term stiffener, as used in this research, refers to structural elements such as
parapets, railings, and sidewalks that reside on the bridge deck. Stiffeners can improve
the load distribution characteristics of bridges by adding stiffness to the bridge deck,
usually near the lateral edges.
3.2.3 Modeling the Diaphragms
Diaphragms are bridge components that connect girders together so as to
provide a means of transferring deck loads laterally to adjacent girders. In prestressed
45
prestressed girders that are temporarily supported, bonded together, and then post
tensioned for continuity. This feature of the preprocessor is only available for the
modeling of multiple span bridges and a maximum of one temporary shore per span is
permitted. Finally, all of the girders within a particular span must have the same
condition with respect to whether or not temporary shoring is present. Once the
preprocessor has determined which spans in the bridge contain temporary shoring, it
will create structural models for each stage of construction accounting for the presence
of shoring.
2,5.7 Stiffening of the Deck Slab Over the Girder Flanges
The top flanges of prestressed concrete girders are usually sufficiently thick that
they stiffen the portion of the deck slab lying directly above them. As a result the
lateral bending deformation in the portion of the slab that lies directly over the girder
flanges is markedly less than the deformation of the portion of the slab that spans
between the flanges of adjacent girders. In the models created by the preprocessor, this
stiffening effect is accounted for by attaching lateral beam elements to the slab
elements that lie directly above the girder flanges. The stiffnesses of the lateral beam
elements are computed in such a way that they reflect the approximate bending stiffness
of the girder flange. A more detailed discussion of this modeling procedure is presented
in the next chapter.
117
Figure 4.9 SEAS Execution Time Results for Workstation and PC Analyses
The plots indicate that in all cases, a substantial reduction in execution time was
achieved through the use of data compression. Table 4.6 summarizes the average
executiontime results that were obtained when buffer sizes in excess of 2000 bytes
were used in the data compression algorithm. Based on these results two important
observations may be made. First, observe that by using compressed I/O in Fortran
Table 4.6 Summary of SIMPAL Execution Time Results
Model
Name
Computer
Platform
Normalized
Execution Time'
Savings in
Execution Time
Prestressed
Workstation (UNIX)
0.35
65%
Steel
Workstation (UNIX)
0.49
51%
Prestressed
PC (DOS)
0.30
70%
Steel
PC (DOS)
0.36
64%
t Average normalized execution times when data compression was used and the buffer
size exceeded 2000 bytes.
238
done to evaluate the feasibility of constructing a compressed core equation solver using
data compression techniques.
8.3 Neural Networks and Iterative Equation Solvers
The use of artificial neural networks (hereafter simply neural networks) in
highway bridge analysis has been explored in depth in this research. A complete
software package for constructing, training, and testing neural networks has been
developed. That neural network packagecalled NetSimhas been used construct a
series of networks which encode the loaddisplacement relationship of twospan flat
slab concrete bridges. Eighteen networks were used to encode the overall load
displacement relationship of the bridgesnine shape networks and nine magnitude
networks.
Once constructed and trained, the neural networks were integrated into an
iterative equation solver for use in highway bridge analysis. A hybrid neural network
preconditioned conjugate gradient (NNPCG) equation solver was developed by
merging neural networks with a PCG solver. Within the NNPCG solver, neural
networks are used to seed the solution vector and to perform neural network
preconditioning. Prior to constructing the NNPCG solver, several traditional
preconditioning strategies were tested in the analysis of highway bridges and were
found to yield unsatisfactory performance.
Several tests were performed to evaluate the effectiveness of the new NNPCG
solver in the analysis of highway bridges. Studies were performed using various
35
assumed to be moving with sufficient speed that, when they enter onto the bridge, there
is an impact effect that amplifies the magnitude of the loads exerted by the vehicle on
the bridge. There may be multiple vehicles simultaneously on the bridge in this
scenario depending on the number of spans, spans lengths, and number of traffic lanes.
To model individual vehicle loads using the preprocessor, the engineer simply
specifies the type, directionforward or reverse, and position of each of the vehicles
on the bridge. Vehicles may be placed at fixed locations, shifted in even increments, or
shifted relative to the finite element nodal locations. If the vehicles are moved using
either of the shifting methods, then the entire vehicle system is shifted as a single
entity. A vehicle system in this context refers to the collection of all vehicles
simultaneously on the bridge.
Vehicles may be positioned and moved on the bridge using any of the following
three methods.
1. Fixed positioning. A single position (location and direction) is specified for
each vehicle on the bridge.
2. Equal shifting. Each vehicle is placed at an initial position and subsequently
shifted a specified number of times in the lateral and longitudinal bridge
directions. The user specifies the incremental shift distances and has the
option of shifting only in the lateral direction, only in the longitudinal
direction, or in both directions.
3. Nodal shifting. Each vehicle is placed at an initial position after which it is
automatically shiftedby the preprocessorin the positive longitudinal
bridge direction such that each axle in the system is in turn placed at each
line of nodes running laterally across the bridge. This option is not available
in constant or variable skew bridge types.
Initial vehicle positions are specified by stating the coordinates of the centerline of the
vehicles lead axle relative to the lateral and longitudinal directions of the bridge.
REFERENCES
American Association of State Highway and Transportation Officials (1992). Standard
Specifications for Highway Bridges, American Association of State Highway
and Transportation Officials, Fifteenth Edition, Washington, D.C.
Austin, M.A., Creighton, S., Albrecht, P. (1993). XBUILD : Preprocessor for Finite
Element Analysis of Steel Bridge Structures, Journal of Computing in Civil
Engineering, ASCE, Vol.7, No.l, pp.5470.
Austin, M.A., Mahin, S.A., Pister, K.S. (1989). CSTRUCT : Computer Environment
for the Design of Steel Structures, Journal of Computing in Civil Engineering,
ASCE, Vol.3, No.3, pp.209227.
Bathe, K.J. (1982). Finite Element Procedures in Engineering Analysis, Prentice Hall,
Englewood Cliffs, NJ.
Bathe, K.J., and Wilson, E.L. (1976). Numerical Methods in Finite Element Analysis,
Prentice Hall, Englewood Cliffs, NJ.
Brown, D.A., Murthy, P.L.N., Berke, L. (1991). Computational Simulation of
Composite Ply Micromechanics Using Artificial Neural Networks, Micro
computers in Civil Engineering, Vol.6, pp.8797.
Carpenter, William C., Barthelemy, JeanFrancois (1994). Common Misconceptions
About Neural Networks as Approximators, Journal of Computing in Civil
Engineering, ASCE, Vol.8, No.3, pp.345358, July.
Consolazio, G.R. (1990) A Review and Evaluation of Bilinear and Biquadratic
Isoparametric Plate Bending Elements, Masters Report, Department of Civil
Engineering, University of Florida.
Consolazio, G.R., Hoit, M.I. (1994). Optimization of FiniteElement Software Using
DataCompression Techniques, Microcomputers in Civil Engineering, Vol.9,
No.3, pp. 161173.
Creighton, S., Austin, M.A., Albrecht, P. (1990). PreProcessor for Finite Element
Analysis of Highway Bridges, Technical Research Report TR 9054, Systems
Research Center, University of Maryland, College Park, Md.
242
187
the fact that the equation solver is setup specifically for one domain (class) of
structures. In the present research, the class of structures chosen to be studied was that
of twospan reinforced concrete flatslab bridges. Although this was the only domain
considered in this study, the concepts and methods described herein can be easily
extended to other types of bridge structures.
Central to the idea of creating a domain specific equation solver is the goal of
accelerating the equation solving process by embedding knowledge of the problem
domain directly into the solver. Thus, a custom purpose equation solver can be created
that is very fast for one particular type of bridge structure. This approach is especially
applicable to situations in which similar types of structures are frequently analyzed
using similar types of structural modeling. Just such a situation exists when computer
assisted structural modeling software, such as the bridge modeling preprocessor
described in Chapters 2 and 3, is used.
The bridge modeling preprocessor is capable of modeling four basic types of
highway bridgesprestressed concrete girder, steel girder, teebeam, and flatslab
bridges. While the geometry, properties, and loading of these structures can vary, the
basic structural configuration and modeling of the bridges is similar within each class
of bridge. Therefore, this type of modeling and analysis environment is a prime
candidate for using a domain specific equation solver. Flatslab bridges were chosen for
this research because they are the simplest of the four classes of bridges modeled by the
preprocessor. Once the concepts and methods have been developed for the flatslab
bridge type, they can be extended to the other bridge types.
124
up of computing neuronsexcept the output layeris referred to as a hidden layer. The
term hidden layer is used to denote the fact that these layers have no direct connection
to the input or output leads of the network and are therefore hidden from everything
outside of the network. Thus, the NN shown in Figure 5.1 has one input layer (non
computing), two hidden layers (computing), and one output layer (computing).
5.3 Problem Solving Using Neural Networks
Problem solving using neural networks consists of two primary stagesnetwork
training and network use. During the training stage, the network is taught how to solve
a particular problem of interest. During the network use stage, the network is told to
solve the problem for a particular set of input parameters.
102
1. Size of the I/O and compression buffers.
2. Size of the hash table.
3. Complexity and repetitiveness inherent in the structure being analyzed.
If the I/O buffers for the files under the control of the compressed I/O library are all of
the same sizewhich will usually be the casethen the compression buffer used by
these files will be the same size as the I/O buffers. For this reason, the parameter
examined will be referred to simply as buffer size from this point on, with the
understanding that this parameter actually corresponds to the I/O buffer size and the
compression buffer size.
Although the complexity and repetitiveness of the structure are not directly
related to the memory requirements of the compression algorithm, they have
considerable effect on the reductions in file size and execution time that can be
achieved. In addition, they indirectly affect the memory requirements of the
compression library in that complex, nonrepetitive structures will require larger buffer
and hash table sizes to achieve the same level of performance that can be achieved
using smaller buffer and hash table sizes for simpler, more repetitive structures.
4,8.1 Data Compression in FEA Software Coded in C
Parametric studies were performed to investigate the influences of buffer size
and hash table size in the compression of FEA data as well as to evaluate the
effectiveness of data compression in Ccoded FEA software. The studies were
performed by implementing the data compression library into a FEA code written in
the C language. SEAS, an acronym for Structural Engineering Analysis Software, was
220
1. Similarity of zero and diagonal seeding. The convergence characteristics of
the zero and diagonal seeding methods are very similar.
2. Slowly diminishing error in zero and diagonal seeding. In both the zero and
diagonal seeding cases, the RMS load error diminishes slowly and
sporadically, remaining large for many iterations.
3. Large initial error in neural network seeding. For each loading condition the
RMS load error of the neural network seeding case is very large during the
early iterations.
All three of these convergence characteristics can be understood by considering the
manner in which the ICPCG algorithm iteratively modifies its estimates of the
displacement unknowns.
Figures 7.107.15 illustrate, in storyboard sequences, the convergence behavior
of the six seeding tests performed. In each figure, the vertical translations (Tz) of the
slab are plotted sidebyside with the corresponding vertical residual (outofbalance)
forces (Fz). Story frames are shown for iterations 0, 5, 25, 50, and 100. The vertical
translation plots roughly correspond to the displaced shapes of the model at the various
stages of convergence. Similarly, the vertical outofbalance force plots illustrate, in a
qualitative manner, how far away from equilibrium the structure is at different stages of
convergence. Plots of the rotations (Rx and Ry) and outofbalance moments (Mx and
My) have been omitted in the interest of space, however, many of the trends exhibited
in the plots shown carry over to those not shown.
173
Disp. YCoordinate(
Disp. YCoordinate(
Disp. YCoordinate(
Load YCoordinate(
Load YCoordinatef
Load YCoordinate(:
Disp. XCoonfinate
Load XCoordinate
All Input Parameter
Are Normalized To
The Range [0,1].
Sizes Vary
Hidden Layer
Figure 6.9 Configuration of Normalized Shape Neural Networks
will now be introduced to improve the trainability of networks. In the shape networks,
the sigmoid transfer function h is used so that the networks are capable of predicting
negative as well as positive normalized displacements. By using only a compacted
portion of the neuron output range, specifically the range [1/1.2,1/1.2], the networks
become easier to train and therefore the risk of overtraining is reduced.
While the theoretical output range of the sigmoid function h is [1,1], this
range is only approached asymptotically. If output parameters in the training data take
on values of 1 or 1, the network will never be able to exactly match these values. Of
course, it will be able to approximate these bounding values to within a small tolerance.
However, by simply scaling the output data into a compacted range, the problem is
alleviated.
Therefore, prior to training, the normalized displacement data generated by the
FEAwhich was in the range [1,1]was scaled down by a factor 1/1.2 and then
passed to the network. Later when the trained network was used in an application, the
188
7.3 Iterative FEA Equation Solving Schemes
T
Direct matrix solution schemessuch as the LDL decompositionare often
used in FEA when there are a moderate number of equations to be solved. However, as
the number of equations becomes very large, iterative methods become more efficient
if full advantage is taken of matrix sparsity (Jennings and McKeown 1992). Also, in
cases where an approximation of the solution is known, iterative methods can
outperform direct methods.
Several types of iterative methods may be used to solve the matrix equations
arising from FEA. The general objective of these solution methods is to solve a matrix
equation of the form
Ax = b (7.1)
where A is a coefficient matrix, b is the right hand side (RHS) vector, and x is the
solution vector. In a typical FEA situation, A is the global stiffness matrix, b is the
global load vector, and x is the vector of structural displacements. The matrix equation
will be solved iteratively, meaning that there will be a set of approximate solution
vectors
*(0).*(1)>*(2)* (72)
whichunder favorable conditionswill converge to the exact solution x. The
differences between the various iterative methods lie primarily in how the estimates
are updated at the end of each iteration.
152
Explicit Encoding of LoadDisplacement Relationship
Implicit Encoding of LoadDisplacement Relationship
Figure 6.1 Encoding the LoadDisplacement Relationship
formed, the global stiffness matrix may be used in conjunction with a global load
vector to solve for the displacements in the structure.
Just as a global stiffness matrix can be used to explicitly encode the load
displacement relationship, neural networks can be used to implicitly encode the same
relationship. These encoding styles are illustrated in Figure 6.1. The neural network
representation of the loaddisplacement relationship is said to be implicit because the
relationship is encoded through a network training process that is unrelated to the
structural behavior of bridgesexcept that the training data was generated by either
analytical or experimental tests of bridges.
Therefore, the fundamental difference between an explicit and an implicit
encoding is the process by which the encoding is generated. In each of these methods,
the end results is a set of numeric values that are used to compute displacements within
a structure. Flowever, it is the process by which those numeric values were generated
that is different between explicit and implicit encodings. In the explicit encoding, a set
of rules that relate to structural behavior (e.g. constitutive laws, solid mechanics) are
used to generate the numeric values in the stiffness matrix. Later, these numeric values
97
for previous instances of byte patterns. As the compressor scans through the data buffer
being compressed, it uses the three bytes at the current location as a hash key. A
hashing function is then applied to this key to produce an index into a hash table. A
hash table entry is a pointer to the location in the buffer of the last occurrence of the
hash key. If the three byte key has occurred earlier in the buffer, then the entry in the
hash tablewhich the key has hashed towill hold a pointer to the previous instance
of the key. This establishes that at least three bytes are repeated.
Next, the algorithm determines how many additional bytes (following the hash
key) are repeated at the previous location in the buffer. Once the end of a matching
byte pattern is found, a code is written that references the previous location in the
buffer where the same data can be found and the number of bytes that are in the
pattern. Thus, instead of writing the actual byte pattern again, a compact repetition
code is written.
Figure 4.2 illustrates the process of pattern matching using a hashing algorithm
and a sliding dictionary. Hashing collisions, where two different keys hash to the same
entry in the hash table, are not resolved in RDC. Instead, the most recent key that has
hashed to a particular entry in the hash table replaces the previous occupant of that
entry, thus giving rise to the term sliding dictionary. The reader is referred to
Sedgewick (1990) for a more thorough treatment of hashing algorithms and pattern
matching.
3
configuration that resists loads at that point in the construction sequence. Stresses and
forces developed at each of these stages will be locked into the structure in subsequent
stages of construction. Such conditions cannot simply be modeled by applying all dead
loads to the final structural configuration of the bridge. Modeling of the distinct
construction stages is important in the analysis of steel girder bridges and is very
important in prestressed concrete girder bridges.
Therefore, in addition to describing complex vehicular loading conditions the
engineer is also faced with preparing multiple FEA models to represent each distinct
stage of the bridge construction. Thus, the need for the development of computer
assisted bridge modeling software is clear.
1.1.2 Computational Aspects of Highway Bridge Analysis
Assuming that modeling software for highway bridges exists, an actual analysis
must still be performed. As a result of advances in computational hardware and decades
of refinement of FEA code, is it now possible to perform analyses of complex
structural systems on a more or less routine basis. However, there arise certain
considerations specific to bridge analysis that must still be addressed if the full potential
of computer assisted modeling is to be realized.
In the FEA of bridge structures, the computational demands imposed by the
analysis generally fall into one of two categoriesrequired storage and required
execution time. Required storage can be subdivided into incore storage, also referred
to as primary or high speed storage, and outofcore storage, also referred to variously
CHAPTER 6
NEURAL NETWORKS FOR HIGHWAY BRIDGE ANALYSIS
6.1 Introduction
In the present research, the use of neural networks in the area of highway bridge
analysis has been studied. This chapter will discuss the construction of a group of
neural networks that are able to approximately encode the loaddisplacement
relationship for twospan flatslab bridges.^ Given a set of loads, the networks can be
used to compute displacements that would result from those loads. Chapter 7 will then
discuss the installation of the networks into an iterative equation solver, the product of
which is a hybrid solver customized specifically for highway bridge analysis.
6.2 Encoding Structural Behavior
In traditional FEA, the loaddisplacement relationship for a structure is encoded
explicitly within the global stiffness matrix. The relationship is said to be encoded
explicitly because there is an explicit, mathematically based set of rulesrelated to the
behavior of structureswhich are followed to form the global stiffness matrix. Once
^ It was the goal of this research to develop neural network analysis techniques
specifically for twospan flatslab bridges but which could subsequently be extended
to encompass other bridge types.
151
139
descent optimization methods, they are still gradient descent methods and generally
result in reduced training times and more robust training when used appropriately.
Understanding that backpropagation is a steepest descent approach to locating
the minimum on the error surface E, it should also be clear that this is an iterative
process. The process begins by choosingusually randomlya set of connection
weights for the network. This choice constitutes a starting point on the multiple
dimensional error surface. At this point the steepest gradient of the error surface is
determined and the connection weights are modified in that direction. This connection
weight modification represents a jump from the starting point to a new point on the
error surface. In an ideal situation this new point would be the minimum of the error
surface, however this is never the case in practice. Therefore, the process of computing
the steepest descent directions and jumping in those directions must be repeated
iteratively.
Gradually, this iterative process will move to a minimum point on the error
surface. A minimum point is simply a location of zero gradient. Thus the minimum
point located may not be the global minimum of the error surface at all. Instead it may
be a plateau (flat spot) in the surface or a local minimum. Whereas pure
backpropagation will essentially stall at such anomalies in the error surface, some of the
variants described in the following sections can cope with these conditions. However all
of the backpropagation methods share one factthey move down the error surface in
finite size jumps.
216
Rotation Ry (radians)
Neural Network 
Analytical (FEA) 
Longitudinal Direction (inches)
Figure 7.6 Neural Network Predicted Displacements and Analytical (FEA)
Displacements for a Flatslab Bridge Under Uniform Loading
44
thickness and length of each end block. End blocks are assumed to have the same
general shape as the normal girder cross section except for an increased web thickness
that extends some specified length along the end of the girder. A typical end block is
illustrated in Figure 2.7. Actual girders generally have a transition length in which the
web thickness varies from the thickness of the end block to the thickness of the normal
section. This transition length is not actually specified by the user, however, the
preprocessor will model the transition from the end block cross section to the normal
cross section using a single tapered girder element.
2.5.6 Temporary Shoring
There are practical limitations to the length of girders that can be fabricated and
transported. As a result, some bridges are built by employing a construction method in
which more than one prestressed girder is used to form each span. The preprocessor
can model bridges in which each main span is constructed from two individually
Web Widlh N
1
Girder
Centroid
Cross Section
Of Girder
Figure 2.7 End Block Region of a Prestressed Concrete Girder
166
Figure 6.6 Load and Displacement Sampling Points Used To Generate
Training Data For Normalized Shape Networks
X 50 Load Application Points T T Nine Node Heterosis
45 Displacement Sampling Points I l I Plate Bending Element
34
bridge and live loading. When analyzed, the force results from this analysis do not
represent the true forces in the structure but rather the increment of forces due only to
applied live loading. These force results must be combined with the force results from
the other construction stage modelsi.e. the stages that contain dead loadsin order to
determine the actual forces present in the structure.
In the BRUFEM bridge rating system, the superposition of analysis results is
performed automatically by the postprocessor. The analysis results are also factored
according to the type of loading that produced thembefore they are superimposed.
Thus, the preprocessor always creates bridge models that are subjected to unfactored
loads. Load factoring is then performed later in the rating process when the post
processor reads the analysis results.
2.4.4 Live Loads
The term live load is applied to loads that are shortterm in duration and which
do not occur at fixed positions. Live loads on bridge structures are those loads that
result from either individual vehicles or from trains of closely spaced vehicles. Bridges
are typically designed and rated for large vehicles such as standard trucks, cranes, or
special overload vehicles. Two vehicle loading scenarios are generally considered when
modeling highway bridge structuresindividual moving vehicle loads and stationary
lane loads. Both of these conditions can be modeled using the preprocessor.
The first scenario represents normal highway traffic conditions in which
vehicles move across the bridge at usual traffic speeds. In this scheme the vehicles are
225
Longitudinal Direction (indies) Longitudinal Direction (indies)
Figure 7.14 Diagonal Seeded Convergence of Displacements (TzTranslations)
and Residuals (FzForces) for a Flatslab Bridge Under Vehicular
Loading
223
Longitudinal Direction (inches)
Residual (kips) Iteration 100
Longitudinal Direction (inches)
Figure 7.12 Neural Network Seeded Convergence of Displacements
(TzTranslations) and Residuals (FzForces) for a Flatslab Bridge
Under Uniform Loading
125
The training stage can be time very consuming and computationally expensive,
however in many cases training only needs to be performed once. Once the network
has been trained, it may repeatedly called upon to solve numerous problems of the type
it was trained to solve. Also, while network training is often a computationally
expensive process, network use is usually quite cheap. In fact, network use simply
refers to the process of performing a forward pass through the networkas was
described in the previous sectionfor a specified set of input parameters. The output
parameters predicted by the network are essentially the components of the solution to
the problem (although some postprocessing of the solution data may be necessary).
Therefore the training stage can be looked upon as an investment, the product of
which is an easy to use, computationally efficient tool for solving a particular type of
problem. One may think of the training process as being analogous to the development
and coding of a rule based algorithm in traditional deterministic problem solving. Once
the algorithm has been encoded, it often will never need to be modified again but
instead simply used repeatedly to solve problems of the appropriate type.
5.4 Network Learning
In order to use a neural network to solve a particular engineering problem, the
network must first be trained to learn the problem. The relationship between the input
and output parameters is stored in the connection weights of the network. Therefore, it
is these connection weights that must be trained in order for the network to learn a
particular problem.
CHAPTER 4
DATA COMPRESSION IN FINITE ELEMENT ANALYSIS
4.1 Introduction
In the analysis of highway bridges, the amount of outofcore storage that is
available to hold data during the analysis can frequently constrain the size of models
that can be analyzed. It is not uncommon for a bridge analysis to require hundreds of
megabytes of outofcore storage for the duration of the analysis. Also, while the size
of the bridge model may be physically constrained by the availability of outofcore
storage, it may also be effectively constrained by the amount of execution time required
to perform the analysis. The use of computer assisted modeling software such as the
bridge modeling preprocessor presented in Chapters 2 and 3 further increases the
demand on computing resources. Using the preprocessor, an engineer can create
models that are substantially larger and more complex than anything that could have
been created manually.
To address the issues of the large storage requirements and lengthy execution
times arising from the analysis of bridge structures, a realtime data compression
strategy suitable for FEA software has been developed. This chapter will describe that
data compression strategy, its implementation, and parametric studies performed to
evaluate the effectiveness of the technique in the analysis of several bridge structures. It
83
121
FanOut Direction Of Nonlinear
Neurons Signal Travel Neurons
Figure 5.1 Layout of a Feed Forward Neural Network
Output
Vectors
Input vectors are shown being applied to the input layer of the network. Each element
of each input vector contains a single piece of information that has been encoded in a
manner which is appropriate for network use. Encoding data for neural network
applications will be discussed in greater detail later.
Collectively, all of the elements of an input vector constitute a single input
pattern which is applied to the input layer of the network. The input layer is different
than all of the other layers in the network because it is the only layer which is made up
of fanout neurons. A fanout neuron (see Figure 5.2) serves simply as a signal
distribution point or fanout point. The signal applied to the input side is copied and
sent out on every connection at the output side. Thus it serves simply to distribute a
single input signal to each of the neurons in the next layer.
The output signals from the input layer neurons are sent along weighted
connections to the nonlinear computing neurons in the next layer. As the signals travel
along these connections, they are either inhibited or excited depending on the strength
of the connections. The strengths of the interlayer connections are key to a networks
ability to represent the relationship between the input space and the output space. The
15
1.3 Literature Review
The research being reported on herein focuses on three distinct yet strongly
linked topics related to FEA of highway bridge structures. In the following sections the
work of previous researchers in each of these three areas will be surveyed.
1.3.1 Computer Assisted Bridge Modeling
The widespread proliferation of FEA as the tool of choice for solid mechanics
analysis has resulted in the demand for and creation of numerous computer assisted
modeling packages during the past few decades. In the area of structural analysis, these
modeling packages generally fall into one of three general classificationsgeneral
purpose, building oriented, or bridge oriented. Computer assisted preprocessors that are
intended for use in the modeling and analysis of highway bridge structures can be
further classified as commercial packages or research packages.
Software packages for the modeling, analysis, and postprocessing of bridge
structures are often bundled together and distributed or sold as a single system. For this
reason, bundled packages falling into the general category of bridge analysis will be
considered here along with packages which belong to the narrower category of bridge
modeling. Also, because the determination of wheel load distributions on highway
bridges is often needed during both design and evaluation phases, packages that are
aimed at determining such distribution characteristics are also considered here.
Zokaie (1992) performed an extensive review and evaluation of software
capable of predicting wheel load distributions on highway bridges. Included in the
56
concrete girder bridges and R/C Tbeam bridges, the diaphragms are assumed to be
constructed as concrete beams and are thus modeled using frame elements. Beam
diaphragms are assumed to not act compositely with the deck slab. This is true whether
or not composite action is present between the girders, stiffeners, and deck slab.
Therefore the diaphragm elements in concrete girder bridges are located at the elevation
of the centroid of the slab, as illustrated in Figure 3.2. In this manner, the diaphragm
elements assist in distributing load laterally but do not act compositely with the deck
slab.
In steel girder bridges, diaphragms may be either steel beams or cross braces
constructed from relatively light steel members called struts. Steel beam diaphragms,
shown in Figure 3.2, are modeled in the same manner that concrete diaphragms are
modeled. Cross brace diaphragms, however, are modeled using axial truss elements
representing the strutsthat are located eccentrically from the centroid of the slab. The
struts are located eccentrically from the finite element nodes regardless of whether or
not composite action is present between the girders, stiffeners, and deck slab. Truss
eccentricities are computed as the distances from the centroid of the slab to the top and
Deck Concrete Concrete Deck Steel Steel
Slab Diaphragm Girder Slab Diaphragm Girder
=?
i
N
CT {Jh
Diaphragm
Elements
Finite
Element
Nodes
Generic
Girders
Concrete Girder Bridge
Steel Girder Bridge
Elevation of Centroid Of Deck Slab
Figure 3.2 Modeling Beam Diaphragms
61
carried within the effective slab width due to the uniform stress is approximately equal
to the total force carried in the slab under the actual nonuniform stress condition.
In order to compute composite section properties, the effective width must be
determined. Standard AASHTO recommendations are used to compute the effective
width for the various bridge types that can be modeled using the preprocessor. In
computing composite girder properties, the width of effective concrete slab that is
assumed to act compositely with the girder must be transformed into equivalent girder
material. This transformation is accomplished by using the modular ratio, n, given by
where Ec is the modulus of elasticity of the concrete slab and Eg is the modulus of
elasticity of the girder. For steel girders the modulus of elasticity is taken as 29,000
ksi. For concrete, the modulus of elasticity is computed based on the concrete strength
using the AASHTO criteria for normal weight concrete.
When using the composite girder model, composite action is approximated by
using composite section properties for the girder members. The primary function of the
slab elements in the CGM finite element model is to distribute wheel loads laterally to
the composite girders, thus plate bending elements are used to model the deck slab.
3.3.2 Modeling Composite Action with the Eccentric Girder Model
The second method available for modeling composite action involves the use of
a pseudo three dimensional bridge model that is called the eccentric girder model
(EGM). In this model, the girders are represented as frame elements that have the
5
analysis software away from the large mainframe computers of the past toward the
smaller PC and workstation platforms of today. This migration has resulted in greater
demands being placed on smaller computerscomputers that often have only moderate
amounts of incore and outofcore storage.
Although the development of preprocessing tools is necessary to make routine
use of FEA in bridge analysis a reality, it also introduces a new problem. Using
computer assisted modeling software, it becomes quite simple for an engineer to create
very large FEA bridge modelsmodels that would otherwise would be too tedious to
prepare manually. While this is generally regarded as desirable from the standpoint of
analysis accuracy it also has the adverse effect of greatly increasing the demand for out
ofcore storage. It is clear then that the issue of outofcore storage optimization must
addressed in conjunction with the development of computer assisted modeling software
if the full potential of the latter is to be realized.
While the size of FEA bridge models may be physically constrained by the
availability of outofcore storage, these same models may also be practically
constrained by the amount of execution time required to perform the analysis. When
moving vehicle loads are modeled using a large number of discrete vehicle positions,
the number of load cases that must be analyzed can quickly reach into the hundreds.
Combine this fact with the aforementioned ease with which complex FEA models can
be createdusing preprocessing softwareand the result is the need to analyze large
bridge models for potentially hundreds of load cases. In such situations, the execution
time required to perform the analysis may diminish the usefulness of the overall
CHAPTER 3
MODELING BRIDGE COMPONENTS
3.1 Introduction
In creating finite element bridge models, the preprocessor utilizes modeling
procedures that have been devised specifically for the types of bridges considered.
Some of the procedures are used to model actual structural components such as girders
and diaphragms whereas others are used to model structural behavior such as composite
action and deck slab stiffening. This chapter will discuss the preprocessor modeling
procedures in detail.
3.2 Modeling the Common Structural Components
The common structural components that are modeled by the preprocessor
include the deck slab, girders, stiffenerssuch as parapets or railings, diaphragms, and
elastic supports. The modeling of these common structural components, which are the
components that are common to several or all of the bridge types considered, will be
discussed below.
3.2.1 Modeling the Deck Slab
Plate bending elements are used to model the bridge deck for the noncomposite
model (NCM) and the composite girder model (CGM). However, in cases where
51
78
stiffeners, and phase2 posttensioning (if present). The final construction stage is
represented by a live load model, i.e. a model which represents the final structural
configuration of the bridge and its associated live loading. At this stage, each and every
structural component of the bridge is active in resisting loads and the loads applied to
the bridge are those resulting solely from vehicle loading and lane loading.
Steel girder bridges have simpler construction sequences than prestressed
concrete girder bridges due to the lack of prestressing and temporary shoring. Steel
girder construction sequences begin with a construction stage in which the girders and
diaphragms are assumed to be immediately structurally effective. The bridge model
consists only of girder elements and diaphragm elements which, acting together, resist
the dead load of the girders, diaphragms, and the deck slab.
It is assumed that the girders in multiple span steel girder bridges are
immediately continuous over interior supports. The assumption of immediate continuity
of the girders is reasonable since multiple span steel bridges are not typically
constructed as simple spans that are subsequently made continuous as is the case in
prestressed concrete girder bridges. It is also assumed that the diaphragms in steel
girder bridges are installed and structurally effective prior to the casting of the deck
slab.
Steel girder bridges may go through a construction stage that represents
additional long term dead loads acting on the compound structure just as was described
above for prestressed concrete girder bridges. Since the deck slab has hardened at this
construction stage, the girders and deck slab may act compositely. However, lateral
beam elements are not used in steel girder bridge models as they are in prestressed
209
DOF and inplane translational DOF are not affected by such pressure loads. Thus, the
computation of the initial displaced shape of a bridge using neural networks is a
computationally inexpensive procedure.
Thus far, the use of neural networks in predicting initial solution estimates could
be applied to any iterative equation solution algorithmnot just the PCG algorithm.
However, the PCG algorithm is the most suitable choice because it allows for the use
of the neural networks not just in predicting the initial solution estimate, but also in
preconditioning the system.
In solving the equation Axb, the PCG algorithm requires an approximation
matrix M which closely approximates the character of A. The better the
approximation is, the faster the process will converge. Recall that the approximation
matrix M is used to solve subproblems of the form Mq = r within each PCG
iteration. In the hybrid NNPCG algorithm, neural networkswhich approximate the
loaddisplacement relationship for the structuretake the place of the matrix M. In
other words, the neural networks are used to approximate the loaddisplacement
relationship that is formally encoded in the structural stiffness matrix. This novel
approach to preconditioning is termed neural network preconditioning.
Each of the steps in the PCG algorithm which requires the solution of an
Mq = r subproblem is now solved using the neural networks instead of using a direct
equation solver. In the PCG algorithm, these subproblems are theoretically solved as
q = M V
(7.29)
170
Now consider another bridge in which the load remains at the same longitudinal
location but the span lengths are reversed. In this case, the normalized coordinate of the
load would still be Y=0.5=30/60. If the data obtained from analyses of these two
bridges were combined to train a single network, the training process would fail.
Training would fail because the two sets of data contradict each other. In one case, a
longitudinal load coordinate of Y=0.5 corresponds to a load in the first span whereas
in the second case, the same coordinate corresponds to a load in the second span.
Clearly, the structural displacements for these two cases will be opposite to each other,
but the network has no mechanism with which to distinguish why they are opposite.
The scenario described above can be remedied by using a piecewise linear
normalization such as the one shown in Figure 6.8. In this encoding scheme the interior
support of the bridge always lies at a normalized coordinate of Y=0.5. The
normalization is then linear within each spani.e. piecewise linear over the entire
Lateral XDirection
1.0
*1.0
1.0
Y=0.0 Y=0.2 Y=0.5 Y=1.0
Neuron3 Value = 0.0
r*
 Neuron2 Value = 0.4
Neuron1 Value = 0.6
1 Tl.O
ho
Normalized Bridge Coordinate System
Multiple Neuron Encoding Of YCoordinates
Figure 6.8 Encoding Bridge Coordinates
104
In the zero skew bridge, the model is extremely regular and there is a high
degree of element repetition. Each span is square in aspect ratio and is divided into a
ten by ten grid of plate elements. As a result, the element matriceselement stiffness,
load, rotation, and stress recovery matricesare identical for all of the plate elements
in the model.
In contrast, due to the variation in nodal geometry, each plate element in the
variable skew model has a slightly different shape and therefore slightly different
element matrices. The first bridge represents a case in which there is a great deal of
largescale repetition in the data while the second bridge represents a case in which
there is no largescale repetition. As will be shown below, there is considerable
I 1 Longitudinal Direction
Figure 4.3 Bridge Models Used in SEAS Parametric Studies
150
bookkeeping required to implement the combination of all three of these methodsin
addition to pure backpropagationis complex and will not be discussed here, the use of
the software is straightforward.
On a final note, the backpropagation variants described herein are by far not the
only variants currently available. The variants described here are only the ones which
the author has implemented in the NetSim software. Another important class of variants
are the second order methods, which use not only first order derivatives but also second
order derivatives to guide the training process. The interested reader is referred to the
relevant literature on the subject.
f
t
IE
t
r4
i)
"<1 
ii
GSF*30 ft. (GSF*360 in.)
 Lateral Direction
a
si
la
iÂ£
a
i 0
i f
j a
a a
Figure 6.7 Load and Displacement Sampling Points Used To Generate
Training Data For Scaling (Magnitude) Networks
8 CONCLUSIONS AND RECOMMENDATIONS 234
8.1 Computer Assisted Modeling 234
8.2 Data Compression in FEA 236
8.3 Neural Networks and Iterative Equation Solvers 238
REFERENCES 242
BIOGRAPHICAL SKETCH 247
vii
79
concrete girder bridge models. Also, in steel girder bridges, the effects of concrete
deck creep under long term dead loading must be properly accounted for. This is
accomplished by using the techniques presented in Â§3.5.2.
In the final (live load) construction stage of steel girder bridges, each and every
structural component of the bridge is active in resisting loads. At this stage, cracking of
the concrete deck in negative moment regions of multiple span steel girder bridges may
be modeled by the preprocessor. This deck crackingthe extent of which is specified
by the engineeris assumed to occur only in the final live load configuration of the
bridge. In modeling deck cracking, the preprocessor assumes a region in which
negative moment is likely to be present. This assumption is necessary since only after
the finite element analysis has been completed will the actual region of negative
moment be known. Thus, the preprocessor assumes negative moment will be present in
regions of the bridge that are within a distance of twotenths of the span length to either
side of interior supports. See Hays et al. (1994) for further details of the cracked deck
modeling procedure used by the preprocessor.
In R/C Tbeam and flatslab bridges, the construction sequence is not of great
significance and has little effect on the analysis and rating of the bridge. During the
construction of these bridge types, the bridge often remains shored until all of the
bridge components have become structurally effective. In such situations, all of the
structural components become structurally effective and able to carry load before the
shoring is removed. The preprocessor therefore assumes shored construction for these
two bridge types.
17
CSTRUCT (Austin et al. 1989) is an interactive program developed for the design,
modeling, and analysis of planar steel frames under both static and seismic loading
conditions. Although CSTRUCT is not capable of modeling highway bridges, the
general approach to usersoftware interaction developed in that package was later
extended in the development of the XBU1LD bridge modeling system (Austin et al.
1993, Creighton et al. 1990).
The XBUILD package allows a user to interactively build, and simultaneously
view via a graphical interface, finite element models of steel girder highway bridge
structures. XBUILD also allows the user to interactively specify the location and type
of vehicle loading present on the bridge. However XBUILD also has several important
limitations.
1. It can only model steel girder bridges. Thus, the modeling of other types of
bridges such as prestressed concrete girder, reinforced concrete Tbeam, and
flatslab bridges cannot be accomplished.
2. It can only model right (90 degree) bridges having a rectangular finite
element mesh. Thus, neither constant skew nor variable skew bridges can be
modeled.
3. It cannot model the construction stages of the bridge.
In summary, although the XBUILD package provides a user friendly environment for
bridge modeling as well as some powerful graphical features, it is still limited in scope.
While additional bridge modeling packages do exist which have not been
mentioned here, the vast majority of these packages never appear in the literature. This
is due to the fact that such modeling systems are often informal projects developed by
engineering firms strictly for inhouse use.
LD
1780
1995
cm
UNIVERSITY OF FLORIDA
3 1262 08554 8757
118
FEA software, not only can the outofcore storage requirements be reduced byin
many casesan order of magnitude, but the required analysis time can also be
simultaneously reduced by a substantial fraction.
Next, observe that the savings in execution time produced by the use of
compressed I/O is more pronounced on the PC computer platform than on the
workstation platform. This has important implications given the fact that PCs are
increasingly being used to perform inexpensive desktop FEA.
221
Residual (kips) Iteration 50
Longitudinal Direction (inches)
Longitudinal Direction (inches)
Figure 7.10 Zero Seeded Convergence of Displacements (TzTranslations) and
Residuals (FzForces) for a Flatslab Bridge Under Uniform Loading
ANALYSIS OF HIGHWAY BRIDGES USING
COMPUTER ASSISTED MODELING, NEURAL NETWORKS,
AND DATA COMPRESSION TECHNIQUES
By
GARY RAPH CONSOLAZIO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995
Copyright 1995
by
Gary Raph Consolazio
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Professor Marc I. Hoit for his
guidance in both research and professional issues, for his generous support, and for his
enthusiastic encouragement throughout the duration of my doctoral program. In
addition I would like to express my gratitude to Professor Clifford O. Hays, Jr., for his
guidance and support, especially during the initial part of my doctoral program. I
would also like to thank Professors Ron A. Cook, John M. Lybas, W. J. Sullivan, and
Loc VuQuoc for serving on my committee.
I would especially like to thank my wife, Lori, for the enduring patience and
support she has shown during my graduate educationâ€”one could not possibly hope for
a more supportive spouse. I would like to thank my parents, Lynne and Bruce, for
instilling in me the importance of an education and my grandfather, William V.
Consolazio, for encouraging my interest in science and making it possible for me to
pursue an advanced degree.
Finally, I would like to thank my friend and colleague Petros Christou and all
my other fellow graduate studentsâ€”especially Wilson Moy and Prashant Andradeâ€”for
their friendship and encouragement.
The work presented in this dissertation was partially sponsored by the Florida
iii
Department of Transportation.
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS in
ABSTRACT viii
CHAPTERS
1 INTRODUCTION 1
1.1 Background 1
1.1.1 Computer Assisted Bridge Modeling 1
1.1.2 Computational Aspects of Highway Bridge Analysis 3
1.2 Present Research 6
1.2.1 Computer Assisted Bridge Modeling 7
1.2.2 RealTime Data Compression 9
1.2.3 Neural Network Equation Solver 12
1.3 Literature Review 15
1.3.1 Computer Assisted Bridge Modeling 15
1.3.2 Data Compression in FEA 18
1.3.3 Neural Network Applications in Structural Engineering 18
2 A PREPROCESSOR FOR BRIDGE MODELING 21
2.1 Introduction 21
2.2 Overview of the Bridge Modeling Preprocessor 22
2.3 Design Philosophy of the Preprocessor 25
2.3.1 Internal Preprocessor Databases 25
2.3.2 The Basic Model and Extra Members 26
2.3.3 Generation 28
2.3.4 The Preprocessor History File 29
2.4 Common Modeling Features and Concepts 30
2.4.1 Bridge Directions 31
2.4.2 Zero Skew, Constant Skew, and Variable Skew Bridge Geometry ...32
2.4.3 Live Load Models and Full Load Models 33
2.4.4 Live Loads 34
2.4.5 Line Loads and Overlay Loads 36
2.4.6 Prismatic and Nonprismatic Girders 37
IV
2.4.7Composite Action 38
2.5 Modeling Features Specific to Prestressed Concrete Girder Bridges 40
2.5.1 Cross Sectional Property Databases 40
2.5.2 Pretensioning and PostTensioning 41
2.5.3 Shielding of Pretensioning 42
2.5.4 PostTensioning Termination . 43
2.5.5 End Blocks 43
2.5.6 Temporary Shoring .44
2.5.7 Stiffening of the Deck Slab Over the Girder Flanges 45
2.6 Modeling Features Specific to Steel Girder Bridges 46
2.6.1 Diaphragms 46
2.6.2 Hinges 47
2.6.3 Concrete Creep and Composite Action 48
2.7 Modeling Features Specific to Reinforced Concrete TBeam Bridges 49
2.8 Modeling Features Specific to FlatSlab Bridges 50
3 MODELING BRIDGE COMPONENTS 51
3.1 Introduction 51
3.2 Modeling the Common Structural Components 51
3.2.1 Modeling the Deck Slab 51
3.2.2 Modeling the Girders and Stiffeners 54
3.2.3 Modeling the Diaphragms 55
3.2.4 Modeling the Supports 57
3.3 Modeling Composite Action 58
3.3.1 Modeling Composite Action with the Composite Girder Model 60
3.3.2 Modeling Composite Action with the Eccentric Girder Model 61
3.4 Modeling Prestressed Concrete Girder Bridge Components 65
3.4.1 Modeling Prestressing Tendons 65
3.4.2 Increased Stiffening of the Slab Over the Concrete Girders 68
3.5 Modeling Steel Girder Bridge Components 70
3.5.1 Modeling Hinges 70
3.5.2 Accounting for Creep in the Concrete Deck Slab 72
3.6 Modeling Reinforced Concrete TBeam Bridge Components 74
3.7 Modeling FlatSlab Bridge Components 75
3.8 Modeling the Construction Stages of Bridges 76
3.9 Modeling Vehicle Loads 80
4 DATA COMPRESSION IN FINITE ELEMENT ANALYSIS 83
4.1 Introduction 83
4.2 Background 84
4.3 Data Compression in Finite Element Software 86
4.4 Compressed I/O Library Overview 91
4.5 Compressed I/O Library Operation 92
4.6 Data Compression Algorithm 95
v
4.7 Fortran Interface to the Compressed I/O Library 99
4.8 Data Compression Parameter Study and Testing 101
4.8.1 Data Compression in FEA Software Coded in C 102
4.8.2 Data Compression in FEA Software Coded in Fortran 112
5 NEURAL NETWORKS 119
5.1 Introduction 119
5.2 Network Architecture and Operation 120
5.3 Problem Solving Using Neural Networks 124
5.4 Network Learning 125
5.5 The NetSim Neural Network Package 128
5.6 Supervised Training Techniques 130
5.7 Gradient Descent and Stochastic Training Techniques 133
5.8 Backpropagation Neural Network Training 137
5.8.1 ExampleByExample Training and Batching 141
5.8.2 Momentum 143
5.8.3 Adaptive Learning Rates 146
6 NEURAL NETWORKS FOR HIGHWAY BRIDGE ANALYSIS 151
6.1 Introduction 151
6.2 Encoding Structural Behavior 151
6.3 Separation of Shape and Magnitude 153
6.3.1 Generating Network Training Data 157
6.3.2 Using Trained Shape and Scaling Networks 160
6.4 Generating Analytical Training Data 163
6.5 Encoding Bridge Coordinates 168
6.6 Shape Neural Networks 172
6.7 Scaling Neural Networks 175
6.8 Implementation and Testing 183
7 ITERATIVE EQUATION SOLVERS FOR HIGHWAY BRIDGE
ANALYSIS 185
7.1 Introduction 185
7.2 Exploiting Domain Knowledge 186
7.3 Iterative FEA Equation Solving Schemes 188
7.4 Preconditioning in Highway Bridge Analysis 194
7.4.1 Diagonal and Band Preconditioning 195
7.4.2 Incomplete Cholesky Decomposition Preconditioning 201
7.5 A Domain Specific Equation Solver 205
7.6 Implementation and Results 212
7.6.1 Seeding the Solution Vector Using Neural Networks 213
7.6.2 Preconditioning Using Neural Networks 229
vi
8 CONCLUSIONS AND RECOMMENDATIONS 234
8.1 Computer Assisted Modeling 234
8.2 Data Compression in FEA 236
8.3 Neural Networks and Iterative Equation Solvers 238
REFERENCES 242
BIOGRAPHICAL SKETCH 247
vii
Abstract of Dissertation Presented to the Graduate
School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANALYSIS OF HIGHWAY BRIDGES USING
COMPUTER ASSISTED MODELING, NEURAL NETWORKS,
AND DATA COMPRESSION TECHNIQUES
By
GARY RAPH CONSOLAZIO
August 1995
Chairman: Marc I. Hoit
Major Department: Civil Engineering
By making use of modern computing facilities, it is now possible to routinely
apply finite element analysis (FEA) techniques to the analysis of complex structural
systems. While these techniques may be successfully applied to the area of highway
bridge analysis, there arise certain considerations specific to bridge analysis that must
be addressed.
To properly analyze bridge systems for rating purposes, it is necessary to model
each distinct structural stage of construction. Also, due to the nature of moving
vehicular loading, the modeling of such loads is complex and cumbersome. To address
these issues, computer assisted modeling software has been developed that allows an
engineer to easily model both the construction stages of a bridge and complex vehicular
loading conditions.
Using the modeling software an engineer can create large, refined FEA models
that otherwise would have required prohibitively large quantities of time to prepare
manually. However, as the size of these models increases so does the demand on the
viii
computing facilities used to perform the analysis. This is especially true in regard to
temporary storage requirements and required execution time.
To address these issues a real time lossless data compression strategy suitable
for FEA software has been developed, implemented, and tested. The use of this data
compression strategy has resulted in dramatically reduced storage requirements and, in
many cases, also a significant reduction in the analysis execution time. The latter result
can be attributed to the reduced quantity of physical data transfer which must be
performed during the analysis.
In a further attempt to reduce the analysis execution time, a neural network has
been employed to create a domain specific equation solver. The chosen domain is that
of twospan flatslab bridges. A neural network has been trained to predict
displacement patterns for these bridges under various loading conditions. Subsequently,
a preconditioned conjugate gradient equation solver was constructed using the neural
network both to seed the solution vector and to act as a preconditioner. Results are
promising but further network training is needed to fully realize the potential of the
application.
ix
CHAPTER 1
INTRODUCTION
1.1 Background
In spite of the widespread success with which finite element analysis (FEA)
techniques have been applied to problems in solid mechanics and structural analysis,
the use of FEA in highway bridge analysis has suffered from a lack of requisite pre
and postprocessing tools. Without question, the finite element method (FEM) affords
engineers a powerful and flexible tool with which to solve problems ranging in
complexity from static linear elastic analyses to dynamic nonlinear analyses. During the
past few decades, numerous high quality FEA software packages have been developed
both in the form of commercial products and research codes.
1.1.1 Computer Assisted Bridge Modeling
In addition to containing a core FEA engine many of these packagesâ€”especially
the commercial onesâ€”include or can be linked to separate pre and postprocessing
modules to aid the engineer in preparing and interpreting FEA data. Modeling
preprocessors for building structures that allow the engineer to accurately and
efficiently prepare FEA models are common. Once the core FEA engine has been
executed, postprocessing packages facilitate interpretation of the often voluminous
quantities of analysis results generated by such software.
1
2
Whereas the development of such packages for the analysis and design of
buildingtype structures has roughly kept pace with the demands of industry, the same
is not true for the case of highway bridge analysis. This is probably attributable to the
fact that there are simply many more buildingtype structures constructed than there are
highway bridge structures, and therefore a greater demand exists. However, this is not
to say that there is not a demand for such software in bridge analysis. With an
inventory of more than half a million bridges in the United States alone, and roughly
20 percent of those bridges considered structurally deficient and in need of evaluation,
the demand for computer assisted bridge analysis packages exists.
Modeling highway bridges for FEA presents certain challenges that are not
present in the analysis of building structures. For example, in addition to being
subjected to the usual fixed location loads, bridges are also subjected moving vehicular
loads which are often complex and cumbersome to describe with the level of detail
needed for FEA. Also, because moving vehicle loads are typically represented using a
large number of discrete vehicle locations, bridge analyses often contain a large number
of load cases. As a direct result, the engineer is faced not only with the daunting task of
describing the loads, but also of interpreting the vast quantity of results that will be
generated by the analysis.
In order to properly analyze bridge systems for evaluation purposes, as in a
design verification or rating of an existing bridge, each distinct structural stage of
construction should be represented in the model. This is because the bridge has a
distinct structural configuration at each stage of construction, and it is that structural
3
configuration that resists loads at that point in the construction sequence. Stresses and
forces developed at each of these stages will be locked into the structure in subsequent
stages of construction. Such conditions cannot simply be modeled by applying all dead
loads to the final structural configuration of the bridge. Modeling of the distinct
construction stages is important in the analysis of steel girder bridges and is very
important in prestressed concrete girder bridges.
Therefore, in addition to describing complex vehicular loading conditions the
engineer is also faced with preparing multiple FEA models to represent each distinct
stage of the bridge construction. Thus, the need for the development of computer
assisted bridge modeling software is clear.
1.1.2 Computational Aspects of Highway Bridge Analysis
Assuming that modeling software for highway bridges exists, an actual analysis
must still be performed. As a result of advances in computational hardware and decades
of refinement of FEA code, is it now possible to perform analyses of complex
structural systems on a more or less routine basis. However, there arise certain
considerations specific to bridge analysis that must still be addressed if the full potential
of computer assisted modeling is to be realized.
In the FEA of bridge structures, the computational demands imposed by the
analysis generally fall into one of two categoriesâ€”required storage and required
execution time. Required storage can be subdivided into incore storage, also referred
to as primary or high speed storage, and outofcore storage, also referred to variously
4
as secondary storage, low speed storage, and backing store. Incore storage generally
refers to the amount of physical random access memory (RAM) available on a
computer, although on computers running virtual memory operating systems there can
also be virtual Incore memory. Outofcore storage generally refers to available space
on hard disks, also called fixed disks.
Optimizing the use of available incore storage has been an area of considerable
research during the past few decades. In contrast, little research has been performed
that addresses the large outofcore storage requirements often imposed by FEA. Out
ofcore storage is used for three primary purposes in FEA :
1. To hold temporary data such as element stiffness, load, and stress recovery
matrices (collectively referred to as element matrices) that exist only for the
duration of the analysis.
2. To hold analysis results such as global displacements and element stresses
that will later be read by postprocessing software.
3. To perform blocked, outofcore equation solutions in cases where the
global stiffness or global load matrices are too large to be contained incore
as a single contiguous unit.
In cases 1 and 3, once the analysis is complete the storage is no longer needed, i.e. the
storage is temporary in nature. In case 2, the storage will be required at least until the
analysis results have been read by postprocessing software.
In the analysis of highway bridges, the amount of outofcore storage that is
available to hold element matrices can frequently become a constraint on the size of
model that can be analyzed. It is not uncommon for a bridge analysis to require
hundreds of megabytes of outofcore storage during an analysis. Also, as a result of
the proliferation of low cost personal computers (PCs), there has been a migration of
5
analysis software away from the large mainframe computers of the past toward the
smaller PC and workstation platforms of today. This migration has resulted in greater
demands being placed on smaller computersâ€”computers that often have only moderate
amounts of incore and outofcore storage.
Although the development of preprocessing tools is necessary to make routine
use of FEA in bridge analysis a reality, it also introduces a new problem. Using
computer assisted modeling software, it becomes quite simple for an engineer to create
very large FEA bridge modelsâ€”models that would otherwise would be too tedious to
prepare manually. While this is generally regarded as desirable from the standpoint of
analysis accuracy it also has the adverse effect of greatly increasing the demand for out
ofcore storage. It is clear then that the issue of outofcore storage optimization must
addressed in conjunction with the development of computer assisted modeling software
if the full potential of the latter is to be realized.
While the size of FEA bridge models may be physically constrained by the
availability of outofcore storage, these same models may also be practically
constrained by the amount of execution time required to perform the analysis. When
moving vehicle loads are modeled using a large number of discrete vehicle positions,
the number of load cases that must be analyzed can quickly reach into the hundreds.
Combine this fact with the aforementioned ease with which complex FEA models can
be createdâ€”using preprocessing softwareâ€”and the result is the need to analyze large
bridge models for potentially hundreds of load cases. In such situations, the execution
time required to perform the analysis may diminish the usefulness of the overall
6
system. This is especially true in situations where multiple analyses will need to be
performed, as in an iterative designevaluation cycle or within a nonlinear analysis
scheme.
Thus, it is evident that in order for a computer assisted bridge modeling system
to be practical and useful, the FEA analysis component must be as numerically efficient
as possible so as to minimize the required analysis time and minimize the use of outof
core storage.
1.2 Present Research
The research reported on in this dissertation focuses on achieving three primary
objectives with respect to FEA bridge modeling. They are :
1. Developing an interactive bridge modeling preprocessor capable of
generating FEA models that can account for bridge construction stages and
vehicular loading conditions.
2. Developing a realtime data compression strategy that, once installed into
the FEA engine of a bridge analysis package, will reduce the computational
demands of the analysis.
3. Developing a domain specific equation solver based on neural network
technology and the subsequent installation of that solver into the FEA engine
of a bridge analysis package.
Each of these objectives attempts to address and overcome a specific difficulty
encountered when applying FEA techniques to the analysis of highway bridge systems.
The following sections describeâ€”in greater detailâ€”each objective and the methods used
to attain those objectives.
7
1.2.1 Computer Assisted Bridge Modeling
Widespread use of FEA techniques in highway bridge analysis has been
curtailed by a lack of requisite pre and postprocessing tools. Routine use of FEA in
bridge analysis can only occur when computer assisted modeling software has been
developed specifically with the highway bridge engineer in mind. To address this issue,
an interactive bridge modeling program has been developed as part of the research
reported on herein. The resulting bridge modeling preprocessor, called BRUFEM1, is
one component of the overall BRUFEM system (Hays et al. 1990, 1991, 1994).
BRUFEM, which is an acronym for Bridge Rating Using the Finite Element Method, is
a software package consisting of a series of Fortran 77 programs which, when working
collectively as a system, is capable of modeling, analyzing, and rating many types of
highway bridge structures.
The BRUFEM preprocessor, which will hereafter be referred to simply as the
preprocessor, allows an engineer to create detailed FEA bridge models by specifyingâ€”
interactivelyâ€”a minimal amount of bridge data. Information needed specifically for the
modeling of bridge structures and bridge loading is embedded directly into the
software. Thus the usual barriers that would prevent an engineer from manually
constructing the FEA bridge model are overcome. The primary barriers are :
1. Discretizing each and every structural component of the bridge into discrete
finite elements and subsequently specifying the characteristicsâ€”geometry,
material properties, connectivities, eccentricities, etc.â€”of each of those
elements.
2. Modeling the structural configuration and the appropriate dead loads at each
distinct stage of construction.
8
3.Computing potentially hundreds of discrete vehicle positions and
subsequently computing and specifying the load data required for FEA.
All of these barriers are overcome through the use of the preprocessor because it
handles these tasks in a semiautomated fashion. The term semiautomated, which is
used synonymously with computer assisted in this dissertation, alludes to the fact that
there is an interaction between the engineer and the modeling software. Neither has
complete responsibility for controlling the creation of the bridge model. General
characteristics of bridge structures and bridge loading are built into the preprocessor so
as to allow rapid modeling of such structures. However, the engineer retains the right
to introduce engineering judgmentâ€”where appropriateâ€”into the creation of the model
by interacting with the software. Thus, the engineer is freed from the tedium of
manually preparing all of the data needed for FEA and allowed to focus on more
important aspects of the rating or design process.
In addition to handling the primary modeling tasks discussed above, the
preprocessor handles numerous other tasks which are required in bridge modeling. The
most important of these are listed here.
1. Modeling composite action between the girders and slab, in some cases
including the calculation of composite girder section properties based on the
recommended AASHTO (AASHTO 1992) procedure.
2. Modeling pretensioning and posttensioning tendons, including the
specification of finite element end eccentricities.
3. Modeling variable cross section girders, including the generation and
calculation of all necessary cross sectional properties and eccentricities.
4. Modeling complex bridge geometry such as variable skew.
5. Modeling live loading conditions considering not only a single standard
vehicle but often several different standard vehicles.
9
These features facilitate the rapid development of FEA bridge models by alleviating
the user of manually performing these tasks. Detailed descriptions of the capabilities of
the preprocessor will be given in subsequent chapters.
1.2.2 RealTime Data Compression
To address the issue of the large storage and execution time requirements arising
from the analysis of bridge structures, a realtime data compression strategy suitable for
FEA software has been developed and implemented. In the of discretization stage of
FEA modeling, any repetition or regularity in either structural geometry or
configuration is usually exploited to the fullest possible extent. This exploitation of
regularity has the advantage of not only minimizing the effort needed to prepare the
model but also of generally leading to a model that is desirable from the standpoint of
accuracy. An additional yet largely unexploited benefit of this regularity is that because
the model itself is highly repetitive, the data generated by the analysis software will
also be highly repetitive. Such conditions are ideal for the use of data compression.
Data compression is the process of taking one representation of set of data and
translating it into a different representation that requires less space to store while
preserving the information content of the original data set. Since compressed data
cannot be directly used in its compressed format, it must be decompressed at some later
stage in the life cycle of the data. This process is called either decompression or
uncompression of the data. However, the term data compression is also used to refer to
10
the overall process of compressing and subsequently decompressing a data set. It should
be clear from context which meaning is intended.
Data compression techniques may be broadly divided into two categoriesâ€”
lossless data compression and lossy data compression. In lossless data compression, the
data set may be translated from its original format into a compressed format and
subsequently back to the original format without any loss, corruption, or distortion of
the data. In contrast, lossy data compression techniques allow some distortion of the
data to occur during the translation process. This can result in greater compression than
that which can be achieved using lossless techniques. Lossy compression methods are
widely used in image compression where a modest amount of distortion of the data can
be tolerated.
In the compression of numeric FEA data such as finite element matrices it is
necessary to utilize lossless data compression methods since corruption of the data to
any extent would invalidate the analysis. Thus, in the present work, in order to
capitalize on the repetitive nature of FEA data, a realtime lossless data compression
strategy has been developed, implemented, and tested in bridge FEA software.
The term realtime is used to indicate that the FEA data is not created and then
subsequently compressed as a separate step but instead is compressed in realtime as the
data is being created. Thus the compression may be looked upon as a filter through
which a stream of numeric FEA data is passed in, and a stream of compressed data
emerges. This type of compression is also more loosely referred to as onthefly data
compression. Of course, the direction of the data stream must eventually be reversed so
11
that the numeric FEA data can be obtained by decompressing the compressed data. This
reversed process is also performed in realtime with the data being decompressed and
retrieved on demand as required by the FEA software.
The compression strategy developed in the present work consists of the
combination of a file input/output (i/o) library and a buffering algorithm both wrapped
around a core data compression algorithm called Ross Data Compression (RDC). RDC
is a sequential data compression algorithm that utilizes run length encoding (RLE) and
pattern matching to compress sequential data streams. Once developed, the technique
was implemented into two FEA programs used in the analysis of highway bridge
structures and tested using several realistic FEA bridge models.
Due to the repetitive nature of FEA bridge models, the data compression
strategy of the present work has been shown to greatly reduce the storage requirements
of FEA software. In the bridge models tested, the storage requirements for FEA
software equipped with data compression were roughly an order of magnitude smaller
than the storage requirements of the same FEA software lacking data compression.
Also, the use of data compression was shown to substantially decrease the
analysis execution time in many cases. This is due to the fact that when using data
compression, the quantity of disk i/o that must be performed by the FEA software is
greatly decreased often resulting in decreased execution time. This benefit has been
shown to be especially advantageous on workstation and personal computer platforms
running FEA software written in Fortran 77. Under such circumstances, the execution
12
time required for the bridge analysis was shown to decrease to as little as approximately
one third of the execution time needed when compression was not utilized.
1.2.3 Neural Network Equation Solver
In order for a computer assisted bridge modeling system to be effective, the
time required to perform each FEA analysis must be minimized. To address this issue,
an application of Artificial Neural Networks (ANNs) has been used to create a domain
specific equation solver. Since the equation solving stage of a FEA accounts for a large
portion of the total time required to perform an analysis, increasing the speed of this
stage will have a significant effect on the speed of the overall analysis.
In the present work, the approach taken to minimize the analysis execution time
is to implicitly embed, using ANNs, domain knowledge related to bridge analysis into
the equation solver itself. In this way a domain specific equation solver, i.e. an
equation solver constructed to solve problems within the specific problem domain of
bridge analysis, is created. The concept behind such an equation solver is that by
exploiting knowledge of the problem, e.g. knowing displacement characteristics of
bridge structures, the solver will be able to more rapidly arrive at the solution.
In the present application ANNs have been trained to learn displacement
characteristics of twospan flatslab bridges under generalized loading conditions. Using
analytical displacement data generated by numerous finite element analyses, a set of
network training data was created with which to train the ANNs. Next, using ANN
training software that was developed as part of the present research, several neural
13
networks were trained to predict displacement patterns in flatslab bridge under
generalized loading conditions. Once the networks were trained, a preconditioned
conjugate gradient (PCG) equation solver was implemented using the neural networks
both to seed the solution vector and to act as an implicit preconditioner.
In the case of seeding the solution vector, the networks attempt to predict the
actual set of displacements that would occur in the bridge under the given loading
condition. These displacements are then used as the initial estimate of the solution
vector in the equation solving process. Conceptually, the idea here is to make use of
the domain knowledge embedded in the ANNs to allow for the computation of a very
good initial guess at the solution vector. Clearly, for any iterative method, the ideal
initial solution estimate would be the exact solution since in that case no iteration would
be required.
Since the exact solution is obviously not known, it is typically necessary to use
a simplified scheme to estimate the solution vector. Such schemes include seeding the
solution vector with random numbers, zeros, or values based on the assumption of
diagonal dominance. None of these methods works particularly well for bridge
structures. In the present research, these simplistic methods are replaced by a
sophisticated set of neural networks that can predict very good initial estimates by
exploiting their â€˜knowledgeâ€™ of the problem.
In general, the neural networks are not be able to predict the exact set of
displacements that occur in the bridge. Therefore it will be necessary to perform
iterations within the PCG algorithm In order to converge on the exact solution. The
14
PCG algorithm was specifically chosen for this application because one component of
that algorithm involves the use of an approximate stiffness matrix to precondition the
problem. Preconditioning reduces the effective condition number of the system and thus
increases the rate of convergence of the iterative process. A more detailed discussion of
this phenomenon will be presented later in this work.
Implicitly embodied in the connection weights of the neural networks is the
relationship between applied loads and resulting displacements in flatslab bridge
structures. This is precisely the same relationship that is captured in the more
traditional stiffness matrix of FEA. Since the PCG algorithm calls for an approximation
of the stiffness matrix to precondition the problem, what is actually needed is an
approximation of the relationship between loads and displacements. While that
approximate relationship is usually expressed explicitly in terms of an approximate
stiffness matrix, in the present research it is expressed implicitly within the neural
networks.
Thus, the current application of neural networks seeks to accelerate the equation
solving process by
1. Using the embedded domain knowledge to yield very accurate initial
estimates of the solution.
2. Using the implicit relationship between loads and displacements embodied in
the networks to precondition, and thus accelerate, the convergence of the
PCG solution process.
Detailed descriptions of neural network theory, the representation of bridge data,
network training, and implementation of the trained networks into a PCG solver will be
presented in later chapters.
15
1.3 Literature Review
The research being reported on herein focuses on three distinct yet strongly
linked topics related to FEA of highway bridge structures. In the following sections the
work of previous researchers in each of these three areas will be surveyed.
1.3.1 Computer Assisted Bridge Modeling
The widespread proliferation of FEA as the tool of choice for solid mechanics
analysis has resulted in the demand for and creation of numerous computer assisted
modeling packages during the past few decades. In the area of structural analysis, these
modeling packages generally fall into one of three general classificationsâ€”general
purpose, building oriented, or bridge oriented. Computer assisted preprocessors that are
intended for use in the modeling and analysis of highway bridge structures can be
further classified as commercial packages or research packages.
Software packages for the modeling, analysis, and postprocessing of bridge
structures are often bundled together and distributed or sold as a single system. For this
reason, bundled packages falling into the general category of bridge analysis will be
considered here along with packages which belong to the narrower category of bridge
modeling. Also, because the determination of wheel load distributions on highway
bridges is often needed during both design and evaluation phases, packages that are
aimed at determining such distribution characteristics are also considered here.
Zokaie (1992) performed an extensive review and evaluation of software
capable of predicting wheel load distributions on highway bridges. Included in the
16
review were general purpose analysis programs such as SAP and STRUDL as well as
specialized bridge analysis programs such as GENDEK, CURVBRG, and MUPDI. In
addition, simplified analysis packages such as SALOD (Hays and Miller 1987) were
also reviewed. Each of the various programs were evaluated and compared primarily
on the basis of the analysis accuracy. However, the modeling capabilities of the
software were not of primary concern in the review.
At present, there are several commercial packages available for bridge modeling
and analysis, however, their modeling capabilities and analysis accuracy vary widely.
The commercial program BARS is widely used by state departments of transportation
(DOTs) throughout the United States. However BARS utilizes a simplified one
dimensional beam model to represent the behavior of the bridge and therefore cannot
accurately account for lateral load distribution between adjacent girders or skewed
bridge geometry.
Another commercial package is CBRIDGE. CBR1DGEâ€”and its design
counterpart CBDesignâ€”have the ability to model and analyze straight and curved
girder bridges as well as generate bridge geometry and vehicular loading conditions.
Although CBRIDGE is now a commercial package, the original analysis methods were
developed under funded research programs at Syracuse University. A limitation of the
CBRIDGE package is that the bridge models created do not account for the individual
construction stages of the bridge.
Public domain (noncommercial) programs for finite element modeling and
analysis include the CSTRUCT and XBUILD packages developed by Austin.
17
CSTRUCT (Austin et al. 1989) is an interactive program developed for the design,
modeling, and analysis of planar steel frames under both static and seismic loading
conditions. Although CSTRUCT is not capable of modeling highway bridges, the
general approach to usersoftware interaction developed in that package was later
extended in the development of the XBU1LD bridge modeling system (Austin et al.
1993, Creighton et al. 1990).
The XBUILD package allows a user to interactively build, and simultaneously
view via a graphical interface, finite element models of steel girder highway bridge
structures. XBUILD also allows the user to interactively specify the location and type
of vehicle loading present on the bridge. However XBUILD also has several important
limitations.
1. It can only model steel girder bridges. Thus, the modeling of other types of
bridges such as prestressed concrete girder, reinforced concrete Tbeam, and
flatslab bridges cannot be accomplished.
2. It can only model right (90 degree) bridges having a rectangular finite
element mesh. Thus, neither constant skew nor variable skew bridges can be
modeled.
3. It cannot model the construction stages of the bridge.
In summary, although the XBUILD package provides a user friendly environment for
bridge modeling as well as some powerful graphical features, it is still limited in scope.
While additional bridge modeling packages do exist which have not been
mentioned here, the vast majority of these packages never appear in the literature. This
is due to the fact that such modeling systems are often informal projects developed by
engineering firms strictly for inhouse use.
18
1.3.2 Data Compression in FEA
During the past few decades a great deal of effort by FEA researchers has been
directed at both optimizing the use of available incore storage in FEA software and
optimizing the numerical efficiency of matrix equation solvers. However, relatively
little attention has been focused on the optimization of outofcore storage
requirements. It is true that researchers have developed various special purpose
bookkeeping strategies that can moderately reduce outofcore storage demands in
specific situations. However, aside from his own work (Consolazio and Hoit 1994), the
author has been unable to find any references in the literature regarding general
purpose strategies directly incorporating the use of data compression techniques in FEA
software.
In contrast, the development of advanced data compression techniques has been
an active area of research in the Computer and Information Science (CIS) field for at
least two decades. In recent years, system software developers have realized the many
fold benefits of using realtime data compression and have begun embedding data
compression directly into the computer operating systems they develop. However, no
such applications of data compression in FEA have appeared in the engineering
literature.
1.3.3 Neural Network Annlications in Structural Engineering
During the past five to ten years, there has been a steadily increasing interest in
applying the neural network solution paradigm to structural engineering problems.
19
VanLuchene and Sun (1990) illustrated some potential uses of neural networks in
structural engineering applications by training networks to perform simple concrete
beam selection and concrete slab analysis. Since that time, researchers have begun
using neural networks in many areas of structural engineering.
Ghaboussi et al. (1991) utilized neural networks for material modeling of
concrete under static and cyclic loading conditions. Neural networks were employed to
capture the materiallevel behavior characteristics (constitutive laws) of concrete using
experimentally collected results as network training data. In this way, the constitutive
laws of the material were derived directly from experimental data and implicitly
embedded in the networks. This is in contrast to the traditional method of formulating a
set of explicit mathematical rules that collectively form the constitutive laws of a
material.
Wu et al. (1992) explored the use of neural networks in carrying out damage
detection in structures subjected to seismic loading. This was accomplished by training
a network to recognize the displacement behavior of a frame structure under various
damage states each of which represented the damage of a single structural component.
Elkordy and Chang (1993) refined this concept by using analytically generated training
data to train networks to detect changes in the dynamic properties of structures.
Accurate prediction of the absolute dynamic properties of a structure by analytical
techniques such as FEA can be very difficult. Instead, Elkordy and Chang used
analytical models to identify changes in dynamic properties. In this way they were able
to train neural networks to detect structural damage by recognizing shifts in the
20
vibrational signature of a structure. Szewczyk and Hajela (1994) extended the concept
once again by utilizing counterpropagation neural networks instead of the more often
used backpropagation neural networks. Counterpropagation networks can be trained
much more rapidly than traditional â€œplain vanillaâ€ backpropagation networks and are
therefore well suited for damage detection applications where a large number of
training sets need to be learned.
Several other diverse applications of neural networks in structural engineering
have also appeared in the literature. Garcelon and Nevill (1990) explored the use of
neural networks in the qualitative assessment of preliminary structural designs. Hoit
etal. (1994) investigated the use of neural networks in renumbering the equilibrium
equations that must be solved during a structural analysis. Gagarin et al. (1994) used
neural networks to determine truck attributes (velocity, axle spacings, axle loads) of inÂ¬
motion vehicles on highway bridges using only girder strain data. Rogers (1994)
illustrated how neural network based structural analyses can be combined with
optimization software to produce efficient structural optimization systems.
CHAPTER 2
A PREPROCESSOR FOR BRIDGE MODELING
2.1 Introduction
This chapter will describe the development and capabilities of an interactive
bridge modeling preprocessor that has been created to facilitate rapid computer assisted
modeling of highway bridges. This preprocessor is one component of a larger system of
programs collectively called the BRUFEM system (Hays et al. 1994). BRUFEM,
which is an acronym for Bridge Rating Using the Finite Element Method, is a software
package consisting of a series of Fortran 77 programs capable of rapidly and accurately
modeling, analyzing, and rating most typical highway bridge structures.
The development of the BRUFEM system was funded by the Florida
Department of Transportation (FDOT) with the goal of creating a computer assisted
system for rating highway bridges in the state of Florida. Bridge rating is the process
of evaluating the structural fitness of a bridge under routine and overload vehicle
loading conditions. With a significant portion of existing highway bridges in the United
States nearing or exceeding their design life, the need for engineers to be able to
accurately and efficiently evaluate the health of such bridges is evident.
Development of the complete BRUFEM system was accomplished in
incremental stages of progress spanning several years and involving the efforts of
21
22
several researchers. Early work on the BRUFEM preprocessor was performed by
Selvappalam and Hays (Hays et al. 1990). Subsequently, the author took over
responsibility for the development of the preprocessor and took this portion of the
BRUFEM project to completion.
The four primary component programs that make up the BRUFEM system are
the following.
1. BRUFEM1. An interactive bridge modeling preprocessor.
2. SIMPAL. A core finite element analysis engine.
3. BRUFEM3. An interactive bridge rating postprocessor.
4. SIMPLOT. A graphical postprocessor for displaying analysis results.
The modeling capabilities of the preprocessor, BRUFEM1, will be the focus of this
chapter and also Chapter 3. Enhancements to the FEA program, SIMPAL, using data
compression techniques will be discussed later in Chapter 4. For complete descriptions
of the other component programs, see Hays et al. (1994).
2.2 Overview of the Bridge Modeling Preprocessor
The primary design goal in developing the preprocessor has been to create an
easy to use, interactive tool with which engineers can model complete bridge systems
for later finite element analysis. By using a computer assisted modeling preprocessor,
the usual barriers that would prevent an engineer from manually constructing an FEA
bridge model are overcome. These barriers, which were first introduced in Chapter 1,
are listed below.
23
1. Discretizing each and every structural component of the bridge into discrete
finite elements and subsequently specifying the characteristicsâ€”geometry,
material properties, connectivities, eccentricities, etc.â€”of each of those
elements.
2. Modeling the structural configuration and the appropriate dead loads at each
distinct stage of construction.
3. Computing potentially hundreds of discrete vehicle positions and
subsequently computing and specifying the load data required for FEA.
Each of these obstacles is overcome through the use of the preprocessor because
it handles these tasks in a semiautomated fashion in which the engineer and the
software both contribute to the creation of the model.
Bridge modeling is accomplished using the preprocessor in an interactive
manner in which the user is asked a series of questions regarding the characteristics of
the bridge being modeled. Each response given by the user determines which questions
will be asked subsequently. For example, assume that the user is asked to specify the
number of the spans in the bridge and a response of â€˜2â€™ is given. Then the user may
later be askedâ€”depending on the type of bridge being modeledâ€”to specify the amount
of deck steel effective in negative moment regionsâ€”i.e. a parameter that is only
applicable to bridges having more than one span.
Both girderslab bridges and flatslab bridges may be modeled using the
preprocessor. Girderslab bridges are those characterized as having large flexural
elements, called girders, that run in the longitudinal direction of the bridge and which
are the primary means of applied bridge loads. In a girderslab bridge, the girders and
deck slab are often joined together in such a way that they act compositely in resisting
loads through flexure. This type of structural behavior is called composite action and
24
will be discussed in detail later. Flatslab bridges are constructed as thick slabs lacking
girders and resisting loads directly through longitudinal flexure of the slab.
The preprocessor has been developed so as to allow maximum flexibility with
respect to the types of bridges that can be modeled. Each of the following bridge types
can be modeled using the preprocessor.
1. Prestressed concrete girder. Bridges consisting of precast prestressed
concrete girders, optional reinforced concrete edge stiffeners, a reinforced
concrete deck slab, and reinforced concrete diaphragms.
2. Steel girder. Bridges consisting of steel girders, optional reinforced concrete
edge stiffeners, a reinforced concrete deck slab, and steel diaphragms.
3. Reinforced concrete Tbeam. Bridges consisting of reinforced concrete T
beam girders, optional reinforced concrete edge stiffeners, a reinforced
concrete deck slab, and reinforced concrete diaphragms.
4. Reinforced concrete flatslab. Bridges consisting of a thick reinforced
concrete deck slab and optional reinforced concrete edge stiffeners.
The general characteristics of each these bridge types are built into the preprocessor so
as to allow rapid modeling. Information regarding the construction sequence of each
type of bridge is also embedded in the preprocessor. This information includes not only
the structural configuration of the bridge at each stage of construction but also the
sequence in which dead loads of various types are applied to the bridge.
Finally, the preprocessor allows the engineer to rapidly and easily model live
loading conditions consisting of combinations of vehicle loads and lane loads. Vehicle
data, such as wheel loads and axle spacing, for a wide range of standard vehiclesâ€”for
example the HS20â€”are embedded in the preprocessor. In addition, there are a variety
of methods available to the user for specifying vehicle locations and shifting.
25
2.3 Design Philosophy of the Preprocessor
In the design of the preprocessor, the basic philosophy has been to exploit
regularity and repetition whenever and wherever possible in the creation of the bridge
model. This idea applies to bridge layout, bridge geometry, girder cross sectional
shape, vehicle configuration, and vehicle movement as well as several other bridge
variables.
2.3.1 Internal Preprocessor Databases
Regularity in the form of standardized bridge components and loading has been
accounted for by using databases. Standard girder cross sectional shapes, such as the
AASHTO girder cross sections, and standard vehicle descriptions are included in
internal databases that make up part of the preprocessor. Thus, instead of having to
completely describe the configuration of, say for example, an HS20 truck, the user
simply specifies an identification symbol, in this case â€˜HS20â€™, and the preprocessor
retrieves all of the relevant information from an internal database.
The vehicle database embedded in the preprocessor contains all of the
information necessary for modeling loads due to the standard vehicles H20, HS20,
HS25, ST5, SFT, SU2, SU3, SU4, C3, C4, C5 TC95, T112, T137, T150, and T174.
In addition to these standard vehicle specifications, the user may create specifications
for customâ€”i.e. nonstandardâ€”vehicles by specifying all of the relevant information in
a text data file.
26
The cross sectional shape databases embedded in the preprocessor contain
complete descriptions of the following standard cross sections used for girders and
parapets.
1. AASHTO prestressed concrete girder types I, II, III, IV, V, and VI
2. Florida DOT prestressed concrete girder bulbT types I, II, III, and IV
3. Standard parapetsâ€”old and new standards
In addition to these standard cross sectional shapes, the user may describe nonstandard
cross sectional shapes interactively to the preprocessor.
2.3.2 The Basic Model and Extra Members
Girderslab bridges typically contain a central core of equally spaced girders
that is referred to as the basic model when discussing the preprocessor. In addition to
this central core the bridge may have extra girders at unequal spacings, parapets, and
railings near the bridge edges. The basic model and extra edge members are depicted in
Figure 2.1. Equal girder spacing arises because it simplifies the design, analysis, and
construction of the bridge. Flatslab bridges also contain a central core, or basic model,
in which the deck slab has a uniform reinforcing pattern. While there are no girders in
flatslab bridges, these bridges may have edge elements such as parapets or railings just
as girderslab bridges may.
Almost all of the bridge types considered by the preprocessor utilize the concept
of a basic model to simplify the specification of bridge data. Exceptions to this rule are
the variable skew bridge types in which the concept of a basic model is not applicable.
Within the basic model all bridge parameters are assumed to be constant and therefore
27
Parapet Deck Slab Diaphragm Girder
Extra Extra I m m I Extra Extra
Left Left \ Basic Model \ Right Right
Parapet Girder Girder Parapet
Figure 2.1 Cross Section of a Girderslab Bridge Illustrating the Basic
Model and Extra Left and Right Edge Members
only need to be specified once by the user. For example, in the bridge shown in
Figure 2.1 notice that the girder spacing SbasÂ¡c is constant within the basic model and
that the cross sectional shape of each of the girders in the basic model is the same. In
this case, the user would only need to specify SbasÂ¡c once and describe the girder cross
sectional shape once for all four of the girders in the basic model of this bridge.
While the technique of using a basic model to describe a bridge can greatly
speed the process of gathering input from the user, most bridges possess additional
members near the edges that do not fit into the basic model scheme. In the preprocessor
these edge members are termed extra members and are appended to either side of the
basic model to complete the description of the bridge. For example, on each side of the
bridge in Figure 2.1 there is an extra girder and an extra stiffener. In this case the extra
girders have different cross sectional shapes and spacings than the girders of the basic
model. In addition, edge stiffening parapets are present which clearly are different from
the girders of the basic model. In the example shown, the bridge is symmetric but this
need not be the case. By specifying some of the girders in a bridge as extra girders,
28
unsymmetrical bridges can be modeled. A limit of three extra left members and three
extra right members is enforced by the preprocessor.
2.3.3 Generation
In order to further reduce the amount of time that an engineer must spend in
describing a bridge, the preprocessor performs many types of generation automatically.
Generation in this context means that the user needs only to specify a small set of data
that the preprocessor will use to generate, or create, a much larger set of data needed
for complete bridge modeling. To illustrate the types of generation that the
preprocessor performs, consider the following example.
Bridges containing nonprismatic girders, i.e. girders that have varying cross
sectional shape, can be easily modeled using the preprocessor. To describe a nonÂ¬
prismatic girder, the user only needs to define the shape of the girder cross section at
key definition points. Definition points are the unique locations along the girder that
completely describe the cross sectional variation of the girder. In the example steel
girder illustrated in Figure 2.2, the user only needs to specify the cross sectional shapes
A1 through A6 at the six definition points. Using this data, the preprocessor will autoÂ¬
matically generate cross sectional descriptions of the girder at each of the finite element
nodal location in the model. Also, the preprocessor will generate cross sectional
properties at each of these nodes and assign those properties to the correct elements in
the final model. Thus, the amount of data that must be manually prepared by the
engineer is kept to a minimum.
29
Figure 2.2 Nonprismatic Steel Girder Bridge With User Specified Definition
Points and Finite Element Nodes
The methods by which a user positions vehicles on a bridge provides another
illustration of the types of generation performed by the preprocessor. As will be seen
later, the user needs only to provide a minimal amount of information in order to
generate potentially hundreds of discrete vehicle positions.
2.3.4 The Preprocessor History File
When using a primarily interactive program such as the preprocessor, the
majority of required data is gathered directly from the user, as opposed to being
gathered from an input data file as in a batch program. An interactive approach to data
collection generally results in easier program operation from the viewpoint of the user.
However, one disadvantage of this approach is that because the user has not prepared
an input data file in advance, as is the case in batch programs, there is no record of the
data given by the user. This is undesirable for two reasons. First, there is no permanent
record of what data was specified by the user and therefore there is no â€˜paper trailâ€™ that
can be used to trace the source of an error should one be detected at some later date.
30
Second, if the user wishes to recreate the bridge model at a future date, all of the
necessary data must again be reentered exactly as before. Similarly, if the user wishes
to recreate the model but with a small variation in some parameter, all of the data must
again be reentered including the modified parameter.
To circumvent these problems, the preprocessor maintains a history file
containing each of the responses interactively entered by the user. Thus, there is a
permanentâ€”and commentedâ€”record of what data was in fact entered by the user
should this ever become a matter of dispute in the future. Since the history file contains
all of the data provided by the user, it may also be used to recreate an entire bridge
model. The user simply tells the preprocessor to read input data from the history file
instead of interactively from the user.
In addition to the uses mentioned above, the history file may also be used to
resume a suspended input session, revise selected bridge parameters, or revise the
vehicle loading conditions imposed on a bridge. Thus, the combination of an interactive
program interface and a reusableâ€”and editableâ€”history file results in a program that
exhibits the advantages of both the interactive and batch approaches without the
exhibiting the disadvantage of each.
2.4 Common Modeling Features and Concepts
Many of the modeling features available in the preprocessor are common to
several of the types of bridges that can modeled. Recall that the preprocessor is capable
of modeling prestressed girder, steel girder, reinforced concrete Tbeam, and flab slab
31
bridges. The features and concepts discussed below are common to manyâ€”or allâ€”of
these bridges types.
2.4.1 Bridge Directions
In discussing the preprocessor, the meaning of certain terminology regarding
bridge directions must be established. In this context, the longitudinal direction of a
bridge is the direction along which traffic moves. The lateral direction of the bridge is
the direction perpendicular to and ninety degrees clockwise from the longitudinal
direction. Finally, the transverse direction is taken as the direction perpendicular to the
bridge deck and positive upward from the bridge. These directions are illustrated in
Figure 2.3. The lateral, longitudinal, and transverse bridge directions correspond to the
global X, y, and zdirections respectively in the global coordinate system of the finite
element model.
Longitudinal
Direction
o
0
Plan
View
Of
Truck
Plan View
Of Bridge
Lateral
Direction
(YDirection)
Figure 2.3 Lateral, Longitudinal, and Transverse Bridge Directions
32
2.4.2 Zero Skew. Constant Skew, and Variable Skew Bridge Geometry
Bridges modeled using the preprocessor may be broadly divided into two
categories based on the bridge geometryâ€”constant skew and variable skew. A constant
skew bridge is one in which all of the support lines form the same angle with the lateral
direction of the bridge. The constant skew category includes right bridges as a
particular case since the skew angle in a right bridge is a constant value of zero. Right
bridges are those bridges in which the support lines are at right angles to the direction
of vehicle movement. Constant skew geometry, including zero skew, can be modeled
for all of the bridge types treated by the preprocessor.
Variable skew geometry may also be modeled using the preprocessor but only
for steel girder bridges and single span prestressed girder bridges. In a variable skew
bridge, each support line may form a different angle with the lateral direction of the
bridge. Each of the bridge skew cases considered is illustrated in Figure 2.4.
â€” End Support Lines
â€” Interior Support Line
oâ€”o
DOâ€”DC
DOâ€”00
DOâ€”00
DOâ€”00
Right Bridge
(Zero Skew)
Variable Skew Bridge
Constant Skew Geometry
Figure 2.4 Zero Skew, Constant Skew, and Variable Skew Bridge Geometry
33
2.4.3 Live Load Models and Full Load Models
Broadly speaking, there are two basic classes of bridge models that can be
created by the preprocessorâ€”live load models and full load models. Live load models
are used primarily to compute lateral load distribution factors (LLDFs) for girderslab
bridges (see Hays et al. 1994 for more information regarding LLDFs). A live load
model represents only the final structural configuration of a bridgeâ€”that is the bridge
configuration that is subjected to live vehicle loads.
By contrast, a full load model is actually not a model at all but rather a series of
models that represent the different stages of construction of a single bridge. Full load
models are analyzed so that a bridge rating can subsequently be performed using the
analysis results. Each of the individual construction stage models, which collectively
constitute a full load model, simulates a particular stage of construction and the dead or
live loads associated with that stage. After all of the construction stage models have
been analyzed a rating may be performed by superimposing the force^ results from
each of the analyses. This is a very important pointâ€”each analysis considers only
incremental loads, not accumulated loads. In fact, this procedure must be used in order
to account for locked in forces, i.e. forces that are developed at a particular stage of
construction and locked into the structure from that point forward.
The last construction stage model in any series of full load models is always a
live load model, i.e. a model representing the final structural configuration of the
^ In this context, the term force is used in a general sense to mean either a shear force,
axial force, bending moment, shear stress, axial stress, or bending stress.
34
bridge and live loading. When analyzed, the force results from this analysis do not
represent the true forces in the structure but rather the increment of forces due only to
applied live loading. These force results must be combined with the force results from
the other construction stage modelsâ€”i.e. the stages that contain dead loadsâ€”in order to
determine the actual forces present in the structure.
In the BRUFEM bridge rating system, the superposition of analysis results is
performed automatically by the postprocessor. The analysis results are also factoredâ€”
according to the type of loading that produced themâ€”before they are superimposed.
Thus, the preprocessor always creates bridge models that are subjected to unfactored
loads. Load factoring is then performed later in the rating process when the postÂ¬
processor reads the analysis results.
2.4.4 Live Loads
The term live load is applied to loads that are shortterm in duration and which
do not occur at fixed positions. Live loads on bridge structures are those loads that
result from either individual vehicles or from trains of closely spaced vehicles. Bridges
are typically designed and rated for large vehicles such as standard trucks, cranes, or
special overload vehicles. Two vehicle loading scenarios are generally considered when
modeling highway bridge structuresâ€”individual moving vehicle loads and stationary
lane loads. Both of these conditions can be modeled using the preprocessor.
The first scenario represents normal highway traffic conditions in which
vehicles move across the bridge at usual traffic speeds. In this scheme the vehicles are
35
assumed to be moving with sufficient speed that, when they enter onto the bridge, there
is an impact effect that amplifies the magnitude of the loads exerted by the vehicle on
the bridge. There may be multiple vehicles simultaneously on the bridge in this
scenario depending on the number of spans, spans lengths, and number of traffic lanes.
To model individual vehicle loads using the preprocessor, the engineer simply
specifies the type, directionâ€”forward or reverse, and position of each of the vehicles
on the bridge. Vehicles may be placed at fixed locations, shifted in even increments, or
shifted relative to the finite element nodal locations. If the vehicles are moved using
either of the shifting methods, then the entire vehicle system is shifted as a single
entity. A vehicle system in this context refers to the collection of all vehicles
simultaneously on the bridge.
Vehicles may be positioned and moved on the bridge using any of the following
three methods.
1. Fixed positioning. A single position (location and direction) is specified for
each vehicle on the bridge.
2. Equal shifting. Each vehicle is placed at an initial position and subsequently
shifted a specified number of times in the lateral and longitudinal bridge
directions. The user specifies the incremental shift distances and has the
option of shifting only in the lateral direction, only in the longitudinal
direction, or in both directions.
3. Nodal shifting. Each vehicle is placed at an initial position after which it is
automatically shiftedâ€”by the preprocessorâ€”in the positive longitudinal
bridge direction such that each axle in the system is in turn placed at each
line of nodes running laterally across the bridge. This option is not available
in constant or variable skew bridge types.
Initial vehicle positions are specified by stating the coordinates of the centerline of the
vehicleâ€™s lead axle relative to the lateral and longitudinal directions of the bridge.
36
The second live loading scenario introduced at the beginning of this section
stationary lane loadingâ€”represents the case in which traffic is more or less stopped on
the bridge and vehicles are very closely spaced together. Lane loading is usually
thought of as a uniform load extending over specified spans in the longitudinal direction
and over a specified width in the lateral direction. AASHTO defines lane loads as being
ten feet wide. However, because lane loading is intended to represent a series of closely
spaced vehicles, the preprocessor instead models uniform lane loads as a series of
closely spaced axles with each having a width of six feetâ€”the approximate width of a
vehicle axle. Lane loads are described by specifying which spans the lane load extends
over, and by specifying the lateral position of the centerline of the lane.
2,4,5 Line Loads and Overlay Loads
In addition to the live load modeling capabilities provided by the preprocessor,
an engineer may also specify the location and magnitude of long term dead loads such
as line loads and uniform overlays. Dead loads due to structural components such as
the deck slab, girders, and diaphragms are automatically accounted for in the bridge
models created by the preprocessor and therefore do not need to be specified as line or
overlay loads.
Dead loads due to nonstructural elements such as nonstructural parapets or
railings can be modeled by specifying the location and magnitude of line loads. For
example, the dead weight of a nonstructural parapet may be applied to the bridge by
specifying a line load having a magnitude equal to the dead weight of the stiffener per
37
unit length. Uniform dead loads, such as that which would result from the resurfacing
of a bridge, may be accounted for by specifying a uniform overlay load.
2.4.6 Prismatic and Nonorismatic Girders
Prismatic girderslab bridges, in which the cross sectional shape of the girders
remains the same along the entire length of the bridge, are the simplest type of girder
slab bridge. Prismatic girders are commonly used in prestressed concrete girder bridges
where standard precast cross sectional shapes are the norm. Most reinforced concrete
Tbeam bridges can also be classified as prismatic girderslab bridges.
Nonprismatic girders, in which the cross sectional shape of the girders varies
along the length of the bridge, are commonly used to minimize material and labor costs
in steel girder bridges. The cross sectional shape of a steel girder can be easily varied
by welding cover plates of various sizes to the top and bottom flanges of the girder,
thus optimizing the use of material. Nonprismatic girders are also used in post
tensioned prestressed concrete girder bridges in which thickened girder webs, called
end blocks, are often required at the anchor points of the posttensioning tendons.
Another class of nonprismatic girder occurs when the depth of a girder is variedâ€”
usually linearlyâ€”along the length of a girder span. Linearly tapering girders occur in
both steel and prestressed concrete girder bridges.
Prismatic girders can be modeled for all of the bridge types treated by the
preprocessor, either for live load analysis and full load analysis. Nonprismatic girders
are also permitted for the following bridge types.
38
1. Steel girder. Constant skew steel girder bridges modeled for either live load
analysis or full load analysis and variable skew bridges modeled for full load
analysis.
2. Reinforced concrete Tbeam. Constant skew reinforced concrete Tbeam
bridges modeled for either live load analysis or full load analysis.
3. Prestressed girder. Prestressed girder bridges that are prismatic except for
the presence of end blocks can be modeled for full load analysis. Constant
skew nonprismatic prestressed girder bridges may be modeled for live load
analysis.
Using the preprocessor, the task of describing and modeling the cross sectional
variation of nonprismatic girders has been greatly simplified. Flexible generation
capabilities are provided that minimize the quantity of data that must be manually
prepared by the user. Refer to Â§2.3.3 for further details.
2.4.7 Composite Action
Composite action is developed when two structural elements are joined in such a
manner that they deform integrally and act as a single composite unit when loaded. In
the case of highway bridges, composite action may be developed between the concrete
deck slab and the supporting girders or between the deck slab and stiffening members
such as parapets. Designing a bridge based on composite action can result in lighter and
shallower girders, reduced construction costs, and increased span lengths.
The extent to which composite action is developed depends upon the strength of
bond that exists between the slab and the adjoining flexural members. In a fully
composite system, strains are continuous across the interface of the slab and the
flexural members and therefore no slip occurs between these elements. Vertical normal
39
stresses and interface shear stresses are developed at the boundary between the two
elements. Proper development of the interface shear stresses is necessary for composite
action to occur and is provided by a combination of friction and mechanical shear
connection schemes.
In steel girder bridges, as illustrated in Figure 2.5, shear studs are often welded
to the top flanges of the girders and embedded into the deck slab so that the two
elements deform jointly. Concrete girders and parapets may be mechanically connected
to the concrete deck slab by extending steel reinforcing bars from the concrete flexural
members into the deck slab during construction. In each of these shear connection
schemes the goal is to provide adequate mechanical bond between the members such
that they behave as a single composite unit.
In a noncomposite bridge system, there is a lack of bond between the top of the
girder and the bottom of the slab. As a result, the two elements are allowed to slide
relative to each other during deformation and do not act as a single composite unit.
Only vertical forces act between the two elements and there is a discontinuity of strain
at the boundary between the elements.
The preprocessor can represent the presence or absence of composite action in a
bridge by using one of three composite action models. The first model, called the
noncomposite model (NCM), represents situations in which composite action is not
present. The second and third models, termed the composite girder model (CGM) and
the eccentric girder model (EGM) respectively, simulate composite action using two
different finite element modeling techniques. Using the concept of an effective width,
40
H Ã ? T f Ã
r
Interface
Shear
Stresses
Figure 2.5 Composite Action Between a Girder and Slab
the composite girder model represents composite action by including an effective width
of slab into the calculation of girder cross sectional properties. A more accurate
approach using a pseudo three dimensional finite element model is used in the eccentric
girder model. Additional details of each of the composite action models are given in the
next chapter.
2,5 Modeling Features Specific to Prestressed Concrete Girder Bridges
Several of the modeling features available in the preprocessor relate specifically
to the modeling of prestressed concrete girder bridges (see Figure 2.6). This section
will provide an overview of those features.
2.5.1 Cross Sectional Property Databases
Databases containing cross sectional property data for standard prestressed
concrete girder sections have been embedded into the preprocessor to quicken the
modeling process and reduce errors. The databases contain cross sectional descriptions
41
of standard AASHTO girders and FDOT bulbT girder types. When modeling a bridge
based on one of these standard girder types, the engineer simply specifies a girder
identification symbol. The preprocessor then retrieves all of the cross sectional data
needed for finite element modeling from an internal database.
This technique saves the user time and eliminates the possibility that he or she
may accidentally enter erroneous data. Since the majority of prestressed concrete girder
bridges are constructed using standard girders, a typical user may never have to
manually enter cross sectional data. To cover cases in which nonstandard girders are
used, the preprocessor also allows the user to manually enter cross sectional data.
2.5.2 Pretensionine and PostTensioning
Prestressed concrete girder bridges modeled by the preprocessor may be either
of the pretensioned type or pre and posttensioned type. Pretensioning occurs during
the process of casting the concrete girders whereas posttensioning occurs after the
girders have been installed in a bridge. The preprocessor can model bridges having
either one or two phases of posttensioning, however, there are specificâ€”and distinct
construction sequences associated with each of these schemes.
Figure 2.6 Cross Section of a Typical Prestressed Concrete Girder Bridge
42
Each type of prestressing, whether it be pretensioning, phase1 posttensioning,
or phase2 posttensioning, is modeled by the preprocessor as a single tendon having a
single area, profile, and prestressing force. In reality, prestressing is usually made up
of many smaller strands located nearby one another so as to form a prestressing group.
This means that when specifying the profile of prestressing strands, the user needs to
specify the profile of the centroid of the prestressing group. Several methods of
describing tendon profiles are provided by the preprocessor including straight profiles,
single and dual point hold down profiles, and parabolically draped profiles.
2.5.3 Shielding of Pretensioning
Pretensioning is used to induce moments into an unloaded girder that will
eventually oppose moments produced by normal bridge loading. In many situations,
however, when normal loads are applied to bridge there is little or no moment at one or
both ends of the girders. In these situations, the pretensioning may be placed in a
profile that has zero eccentricity at the ends of the girder so that zero countermoment
is induced. An alternative is to use a straight pretensioning profile with selected
pretensioning strands being shielded near the ends of the girder.
Shieldingâ€”also known as debondingâ€”is the process of preventing bond between
the pretensioning strand and the concrete so as to effectively eliminate a portion of the
pretensioning at a particular location. The preprocessor is capable of modeling
pretensioning shielding. The user must specify the percentages of pretensioning that are
shielded and the distances over which those percentages apply.
43
2.5.4 PostTensionine Termination
In certain situations, it can be advantageous to terminate posttensioning tendons
at locations other than at the ends of the girders. For example, selected tendons may be
brought out through the top of the girder near, but not at, the end of the girder. The
preprocessor can model early termination of posttensioning tendons in prestressed
concrete girder bridges provided that the user has specified the termination points.
Recall that all posttensioning in a bridge must be represented using either one
or two phases when using the preprocessor. Since each of these phases is represented
using a single tendon, all of the posttensioning for a particular phase must be
terminated at a common location. If multiple tendons are used for a particular phase of
posttensioning and those tendons do not terminate at the same location, then a single
approximate termination point for the entire phase must be determined by the user.
2.5.5 End Blocks
End blocks are regions of a girder in which the web has been significantly
thickened but the overall shape of the cross section remains unchanged. They are often
provided at the ends of prestressed concrete girders to increase the shear capacity of the
cross section and to accommodate the large quantity of reinforcing that is often
necessary at the anchorage points of posttensioning tendons.
The preprocessor models girders containing end blocks as specialcase
nonprismatic girders. End blocks are permitted at the end supports and permanent
interior supports of a bridge. The only information that the engineer must specify is the
44
thickness and length of each end block. End blocks are assumed to have the same
general shape as the normal girder cross section except for an increased web thickness
that extends some specified length along the end of the girder. A typical end block is
illustrated in Figure 2.7. Actual girders generally have a transition length in which the
web thickness varies from the thickness of the end block to the thickness of the normal
section. This transition length is not actually specified by the user, however, the
preprocessor will model the transition from the end block cross section to the normal
cross section using a single tapered girder element.
2.5.6 Temporary Shoring
There are practical limitations to the length of girders that can be fabricated and
transported. As a result, some bridges are built by employing a construction method in
which more than one prestressed girder is used to form each span. The preprocessor
can model bridges in which each main span is constructed from two individually
Web Width N
1
Girder
Centroid
Cross Section
Of Girder
Figure 2.7 End Block Region of a Prestressed Concrete Girder
45
prestressed girders that are temporarily supported, bonded together, and then post
tensioned for continuity. This feature of the preprocessor is only available for the
modeling of multiple span bridges and a maximum of one temporary shore per span is
permitted. Finally, all of the girders within a particular span must have the same
condition with respect to whether or not temporary shoring is present. Once the
preprocessor has determined which spans in the bridge contain temporary shoring, it
will create structural models for each stage of construction accounting for the presence
of shoring.
2,5.7 Stiffening of the Deck Slab Over the Girder Flanges
The top flanges of prestressed concrete girders are usually sufficiently thick that
they stiffen the portion of the deck slab lying directly above them. As a result the
lateral bending deformation in the portion of the slab that lies directly over the girder
flanges is markedly less than the deformation of the portion of the slab that spans
between the flanges of adjacent girders. In the models created by the preprocessor, this
stiffening effect is accounted for by attaching lateral beam elements to the slab
elements that lie directly above the girder flanges. The stiffnesses of the lateral beam
elements are computed in such a way that they reflect the approximate bending stiffness
of the girder flange. A more detailed discussion of this modeling procedure is presented
in the next chapter.
46
2.6 Modeling Features Specific to Steel Girder Bridges
Several of the modeling features available in the preprocessor relate specifically
to the modeling of steel girder bridges (see Figure 2.8). This section will provide an
overview of those features.
2.6.1 Diaphragms
In the steel girder bridge models created by the preprocessor diaphragms are
permitted to be either of the steel beam type or the steel cross brace type. Each of these
types is illustrated in Figure 2.9. The diaphragms connect adjacent girders together but
are not connected to the deck slab between the girders. Structurally, the diaphragms aid
in lateral load distribution, prevent movement of the girder ends relative to one
another, and are assumed to provide complete lateral bracing of the bottom flange in
negative moment regions. If a large number of diaphragms are used, as is often the
case for steel girder bridges, the diaphragms may have a significant effect on lateral
load distribution.
Cross brace diaphragms are constructed from relatively light steel members such
as angles and are often arranged in either an Xbrace configuration or a Kbrace
Concrete Concrete Steel Beam Steel
Figure 2.8 Cross Section of a Typical Steel Girder Bridge
47
configuration. The steel girder bridge type is the only bridge type for which the
preprocessor allows cross brace diaphragms to be modeled. A detailed study was
performed by Hays and Garcelon (see Hays et al. 1994, Appendix I) in which steel
girder bridges were studied using full three dimensional models. The studies indicated
that the behavior of bridges having Xbrace and Kbrace diaphragms were sufficiently
close that Kbrace diaphragms can adequately be modeled using the Xbrace
configuration. Thus, only the Xbrace configuration is modeled by the preprocessor.
The engineer must specify whether beam diaphragms or cross brace diaphragms
will be used and provide the section properties for either the steel beam or the elements
of the cross brace. These section properties are then used for all of the diaphragms in
the bridge. However, in the case of cross brace diaphragms, the depth of the
diaphragms will vary if the depth of the girders vary.
2,6.2 Hinges
Hinged girder connections are occasionally placed in steel girder bridges to
accommodate expansion joints or girder splices. The preprocessor is capable of
Figure 2.9 Diaphragm Types Available for Steel Girder Bridges
48
modeling hinge connections for constant skew steel girder bridges. Hinges are assumed
to run laterally across the entire width of the bridge, thus forming a lateral hinge line at
each hinge location. Along a hinge line, each of the girders contains a hinge connection
and the deck slab is assumed to be discontinuous. Modeling the slab as discontinuous
across the hinge line is consistent with the construction conditions of an expansion
joint.
2.6.3 Concrete Creep and Composite Action
Long term sustained dead loads on a bridge will cause the concrete deck slab to
creep. Concrete creep is time dependent nonrecoverable deformation that occurs as the
result of sustained loading on the concrete. Over time, the concrete will flow similar to
a plastic material and will incur permanent deformation.
In a steel girder bridge the deck slab and girders are constructed from materials
that have different elastic moduli and different sustained load characteristics. Steel has a
higher elastic modulus than concrete and does not creep under sustained loads as
concrete does. If long term dead loads are applied to the bridge after the concrete deck
slab and steel girders have begun to act compositelyL the slab will be subjected to
sustained loading and creep will occur. As the deck slab undergoes creep deformationâ€”
but the steel girders do notâ€”more and more of the load on the bridge will be carried by
the girders and girder stresses will consequently increase. Creeping of the deck slab
1 Long term dead loads that are applied after composite action has begun include the
dead weight of structural parapets, line loadsâ€”such as the weight of a railing or a
nonstructural parapet, and deck overlay loads.
49
essentially softens the composite girderslab load resisting system and therefore
increases the stresses in the girders.
When modeling steel girder bridges that fit the conditions described above, the
preprocessor will automatically account for the effects of concrete deck creep. Details
of this modeling procedure are presented in the next chapter.
2.7 Modeling Features Specific to Reinforced Concrete TBeam Bridges
Reinforced concrete Tbeam bridges (see Figure 2.10) are modeled very
similarly to prestressed concrete girder bridges except that there is no prestressing
present. A notable exception is that the cross sectional shape of a Tbeam girder is
completely defined by the depth and width of the girder web. A Tbeam girder consists
of a rectangular concrete web joined to the deck slabâ€”a portion of which acts as the
girder flange. Thus the engineer simply specifies the depth and width of the web of
each girder when modeling Tbeam bridges. Databases of standard cross sectional
shapes are not needed as was the case in prestressed concrete girder bridges.
Concrete
Parapet
Concrete
Deck Slab
Concrete Concrete
Diaphragm TBeam Web
KH
h 1
Figure 2.10 Cross Section of a Typical Reinforced Concrete TBeam Bridge
50
2.8 Modeling Features Specific to FlatSlab Bridges
Flatslab bridges (see Figure 2.11) consist of a thick reinforced concrete deck
slab and optional reinforced concrete edge stiffeners. Thus, unlike all of the other
bridge types modeled by the preprocessor, there are no girders in flatslab bridges.
However, there can still be composite action between the deck slab and edge stiffeners
such as parapets, if such stiffeners are present and considered structurally effective.
Support conditions for flatslab bridges are also unique among the bridge types
modeled by the preprocessor. Flatslab bridges are supported continuously across the
bridge in the lateral direction at each support location. This is in contrast to girderslab
bridges in which supports are only provided for girder elements and the remainder of
the bridge is assumed to be supported by the girders.
Concrete Concrete Flat Slab
Parapet Deck Slab Thickness
I I I I I I I I I I I I I I I I I I I
Figure 2.11 Cross Section of a Typical Flatslab Bridge
CHAPTER 3
MODELING BRIDGE COMPONENTS
3.1 Introduction
In creating finite element bridge models, the preprocessor utilizes modeling
procedures that have been devised specifically for the types of bridges considered.
Some of the procedures are used to model actual structural components such as girders
and diaphragms whereas others are used to model structural behavior such as composite
action and deck slab stiffening. This chapter will discuss the preprocessor modeling
procedures in detail.
3.2 Modeling the Common Structural Components
The common structural components that are modeled by the preprocessor
include the deck slab, girders, stiffenersâ€”such as parapets or railings, diaphragms, and
elastic supports. The modeling of these common structural components, which are the
components that are common to several or all of the bridge types considered, will be
discussed below.
3.2.1 Modeling the Deck Slab
Plate bending elements are used to model the bridge deck for the noncomposite
model (NCM) and the composite girder model (CGM). However, in cases where
51
52
composite action is modeled with the eccentric girder model (EGM), flat shell elements
are used. The shell element combines plate bending behavior and membrane behavior,
however the membrane response is not coupled with the plate bending response. The
thickness of the slab elements is specified by the engineer and is assumed to be constant
throughout the entire deck except over the girders, where a different thickness may be
specified.
The plate bending elements and the bending (flexural) portion of the flat shell
elements used in the present bridge modeling are based on the ReissnerMindlin thick
plate formulation (Hughes 1987, Mindlin 1951, Reissner 1945). In the Reissner
Mindlin formulation, transverse shear deformations, which can be significant in thick
plate situations such as in flatslab bridge modeling, are properly taken into account.
Consolazio (1990) studied the convergence characteristics of isoparametric elements
based on the thick plate formulation and found that these elements are appropriate for
bridge modeling applications.
While typical isoparametric plate and shell elements may generally have
between four and nine nodes, bilinear (four node) plate and shell elements are used for
all of the bridge models created by the preprocessor. This choice was made for a
number of reasons. Because vehicle axle loads occur at random locations on a bridge,
accurately describing these axle loads requires a substantial number of nodes in the
longitudinal direction. It is generally suggested that when using the preprocessor at
least twenty elements per span be used in the longitudinal direction. Use of biquadratic
(nine node) elements in models following this suggestion would require substantially
53
more solution time than would models using the simpler bilinear elements. This was
shown to be true by Consolazio (1990) for all but trivially small bridge models.
Another important reason for using bilinear elements instead of biquadratic
elements is related to the fact that both of these elements are known to be rank deficient
when their stiffness matrices are numerically underintegrated. Selective reduced
integration (Bathe 1982, Hughes 1987) is often used to alleviate the problem of shear
locking in plate elements. Shear locking, which results in greatly exaggerated structural
shear stiffness, occurs when elements based on the thick plate formulation are used in
thin plate situations. Selective reduced integration is used to soften the portion of the
element stiffness matrix that is associated with transverse shear.
When underintegrated elements are used in thick plate situations such as the
modeling of a flatslab bridge, zero energy modes may develop which can cause the
global stiffness matrix of the structure to be locally or globally singular (or nearly
singular). Both the bilinear and biquadratic elements suffer from zero energy modes.
However, it has been the authorâ€™s experience that the mode associated with biquadratic
elements, illustrated in Figure 3.1, is excited far more frequently in bridge modeling
situations than the modes associated with bilinear elements. In fact the biquadratic
....â€” Undeformed Element
Deformed Element
In Zero Energy
Mode Configuration
(r.s.t) Local Element Directions
The tdirection is the transverse
translational direction of the element.
Figure 3.1 Zero Energy Mode in a Biquadratic Lagrangian Element
54
element zero energy mode occurs quite often in the modeling of flatslab bridges and
must be used with great caution in such situations.
One solution to this problem is to use the nine node heterosis element developed
by Hughes (1987) which inherits all of the advantages of using higher order shape
functions without the disadvantage of being rank deficient. Correct rank is
accomplished by utilizing standard lagrangian (ninenode) shape functions for all
element rotational degrees of freedom (DOFs) but serendipity (eightnode) shape
functions for the translational DOFs. Both a nine node standard lagrangian element and
a nine node heterosis element have been implemented by the author in a FEA program
that was developed as part of the present research. In tests on flatslab bridge models,
the heterosis element performed flawlessly in situations where lagrangian elements
suffer from zero energy modes.
However, because there is no translational degree of freedom associated with
the center nodes of heterosis elements, bridge meshing and distribution of wheel loads
is considerably more complex. Thus, a simpler solution is to simply use bilinear
elements and ensure that an adequate number of such elements is used both the lateral
and longitudinal directions of the bridge. This is the solution that has been adopted for
use in the preprocessor.
3.2.2 Modeling the Girders and Stiffeners
Girders and stiffeners are modeled using standard frame elements. Frame
elements consider flexural effects (pure beam bending), shear effects, axial effects, and
55
torsional effects. Each of these groups of effects are considered separately and are
therefore not coupled.
If the CGM is chosen, composite section properties are used for the elements
representing girders and stiffeners in the bridge. If the NCM is selected then the
noncomposite element properties are used. If the EGM is used, then the noncomposite
girder and stiffener properties are used and the composite action is modeled by locating
the frame elements eccentrically from the centroid of the slab.
In modeling steel girders using frame elements, the transverse shear
deformations in the elements are properly taking into account. Hays and Garcelon
(Hays et al. 1994, Appendix I) found that, when using the type of models created by
the preprocessor, shear deformations in the girders must be considered for the analysis
to be accurate. This conclusion was based on a study comparing the response of models
created by the preprocessor and the response of fully three dimensional models. Shear
deformations are not, and do not need to be, accounted for in concrete girders or
concrete parapets where such deformations are typically negligible.
The term stiffener, as used in this research, refers to structural elements such as
parapets, railings, and sidewalks that reside on the bridge deck. Stiffeners can improve
the load distribution characteristics of bridges by adding stiffness to the bridge deck,
usually near the lateral edges.
3.2.3 Modeling the Diaphragms
Diaphragms are bridge components that connect girders together so as to
provide a means of transferring deck loads laterally to adjacent girders. In prestressed
56
concrete girder bridges and R/C Tbeam bridges, the diaphragms are assumed to be
constructed as concrete beams and are thus modeled using frame elements. Beam
diaphragms are assumed to not act compositely with the deck slab. This is true whether
or not composite action is present between the girders, stiffeners, and deck slab.
Therefore the diaphragm elements in concrete girder bridges are located at the elevation
of the centroid of the slab, as illustrated in Figure 3.2. In this manner, the diaphragm
elements assist in distributing load laterally but do not act compositely with the deck
slab.
In steel girder bridges, diaphragms may be either steel beams or cross braces
constructed from relatively light steel members called struts. Steel beam diaphragms,
shown in Figure 3.2, are modeled in the same manner that concrete diaphragms are
modeled. Cross brace diaphragms, however, are modeled using axial truss elementsâ€”
representing the strutsâ€”that are located eccentrically from the centroid of the slab. The
struts are located eccentrically from the finite element nodes regardless of whether or
not composite action is present between the girders, stiffeners, and deck slab. Truss
eccentricities are computed as the distances from the centroid of the slab to the top and
Deck Concrete Concrete Deck Steel Steel
Slab Diaphragm Girder Slab Diaphragm Girder
=? â€”
i
â€”â€”â€œN
Cr~â€”{Jh
Diaphragm
Elements
Finite
Element
Nodes
Generic
Girders
Concrete Girder Bridge
Steel Girder Bridge
â€” Elevation of Centroid Of Deck Slab
Figure 3.2 Modeling Beam Diaphragms
57
bottom strut connection points, as is shown in Figure 3.3. Thus, the centroid of the
deck slab is used as a datum from which eccentricities are referenced. Of primary
importance in computing these eccentricities is correctly representing the slopes of the
cross struts, since these slopes determines how effective the diaphragm will be in
transferring loads laterally.
Finally, recall that the preprocessor models both Xbrace and Kbrace
diaphragms using the Xbrace configuration for the reasons that were discussed in the
previous chapter.
3.2.4 Modeling the Supports
Bridge models created by the preprocessor use axial truss elements to model
elastic spring supports rather than using rigid supports. In girderslab bridges, vertical
support is usually provided by elastomeric bearing pads located between the ends of the
girders and the abutments and at interior support piers. Bearing pads are modeled using
elastic axial truss elements of unit length, unit elastic modulus, and a cross sectional
area that results in the desired support stiffness. By default the preprocessor will
automatically provide reasonable values of bearing pad stiffness (see Hays et al. 1994
58
for details) however the engineer may manually adjust these values if detailed bearing
stiffness data are available. In addition to vertical supports, horizontal supports must
also be provided to prevent rigid body instability of the models at each stage of
construction. Horizontal support is provided through finite element boundary condition
specification rather than by using elastic supports.
Flatslab bridges are supported continuously in the lateral direction at each
support in the bridge. Since bearing pads are not typically used in flatslab bridge
construction the support stiffnesses cannot not be easily determined. However, the
preprocessor assumes a reasonable value of bearing stiffness, which again can be
manually adjusted by the engineer if better data are available.
3.3 Modeling Composite Action
Composite action is developed when two structural elements are joined in such a
way that they deform together and act as a single unit when loaded. In the case of
bridges, composite action can occur between the concrete slab and the supporting
concrete or steel girders. The extent to which composite action can be developed
depends upon the strength of bond that exists between the slab and the girders.
Composite action may also occur between stiffeners and the deck slab. In a composite
system there is continuity of strain at the slabgirder interface and therefore no slip
occurs between these elements. Horizontal shear forces and vertical forces are
developed at the boundary between the two elements. The interaction necessary for the
59
development of composite action between the slab and the girder is provided by friction
and the use of mechanical shear connectors.
In a noncomposite girderslab system, there is a lack of bond between the top of
the girder and the bottom of the slab. As a result, the two elements are allowed to slide
relative to each other during deformation and do not act as a single composite unit.
Only vertical forces act between the two elements and there is a discontinuity of strain
at the boundary between the elements.
The preprocessor allows the engineer to model the girderslab interaction as
either noncomposite, or as composite using one of two composite modeling techniques.
The girderslab interaction models available in the preprocessor are illustrated in
Figure 3.4.
Noncomposite action is modeled using the noncomposite model (NCM) in
which the centroid of the girder is effectively at the same elevation as the centroid of
the slab. The section properties specified for the girders are those of the bare girders
alone. In this model the primary function of the slab elements is to distribute the wheel
loads laterally to the girders, therefore plate bending elements are used to model the
deck slab.
Composite action between the slab and the girder is modeled in one of two ways
using the preprocessor. One way involves the use of the composite girder model
(CGM) and the other the eccentric girder model (EGM). These composite action
models are also illustrated in Figure 3.4.
60
Effective Width
Slab Elements
Noncomposite
Model (NCM)
Composite Girder
Model (CGM)
Figure 3.4 Modeling Noncomposite Action and Composite Action
3.3.1 Modeling Composite Action with the Composite Girder Model
One method of modeling composite action is by utilizing composite properties
for the girder elements. The centroid of the composite girder section is at the same
elevation as the midsurface of the slab in the finite element model. A composite girder
section is a combination of the bare girder and an effective width of the slab that is
considered to participate in the flexural behavior.
Due to shear strain in the plane of the slab, the compressive stresses in the
girder flange and slab are not uniform, and typically decrease in magnitude with
increasing lateral distance away from the girder web. This effect is often termed shear
lag. In certain cases of laterally nonsymmetric bridge loading, the compressive stress in
the deck may vary such that the stress is higher at the edge of the bridge than over the
centerline of a girder. An effective slab width over which the compressive stress in the
deck is assumed to be uniform is used to model the effect of the slab acting compositely
with the girder. The effective width is computed in such a way that the total force
61
carried within the effective slab width due to the uniform stress is approximately equal
to the total force carried in the slab under the actual nonuniform stress condition.
In order to compute composite section properties, the effective width must be
determined. Standard AASHTO recommendations are used to compute the effective
width for the various bridge types that can be modeled using the preprocessor. In
computing composite girder properties, the width of effective concrete slab that is
assumed to act compositely with the girder must be transformed into equivalent girder
material. This transformation is accomplished by using the modular ratio, n, given by
where Ec is the modulus of elasticity of the concrete slab and Eg is the modulus of
elasticity of the girder. For steel girders the modulus of elasticity is taken as 29,000
ksi. For concrete, the modulus of elasticity is computed based on the concrete strength
using the AASHTO criteria for normal weight concrete.
When using the composite girder model, composite action is approximated by
using composite section properties for the girder members. The primary function of the
slab elements in the CGM finite element model is to distribute wheel loads laterally to
the composite girders, thus plate bending elements are used to model the deck slab.
3.3.2 Modeling Composite Action with the Eccentric Girder Model
The second method available for modeling composite action involves the use of
a pseudo three dimensional bridge model that is called the eccentric girder model
(EGM). In this model, the girders are represented as frame elements that have the
62
properties of the bare girder cross section but which are located eccentrically from the
slab centroid by using rigid links. By locating the girder elements eccentrically from the
slab, the girder and slab act together as a single composite flexural unit. In general,
each componentâ€”the slab and the girderâ€”may undergo flexure individually and may
therefore sustain moments. However, because the components are coupled together by
rigid links, the composite section resists loads through the development not only of
moments but also of axial forces in the elements.
Rigid links, also referred to as eccentric connections, are assumed to be
infinitely rigid and therefore can be represented exactly using a mathematical
transformation. Thus, by using the mathematical transformation, additional link
elements do not need be added to the finite element model to represent the coupling of
the slab and girder elements.
In the eccentric girder model, shear lag in the deck is properly taken into
account because the deck slab is modeled with flat shell elementsâ€”elements created by
the superposition of plate bending elements and membrane elements. Because the slab
and girders are eccentric to one another and because they are coupled together in a
three dimensional sense, the EGM is referred to as a pseudo three dimensional model.
It is not a fully three dimensional model because the coupling is accomplished through
the use of infinitely rigid links. In an actual bridge the axial force in the slab must be
transferred to the girder centroid through a flexibleâ€”not infinitely rigidâ€”girder web. In
a fully three dimensional model, the girder webs would have to be modeled using shell
63
elements as was done by Hays and Garcelon (Hays et al. 1991). Therefore the models
created by the preprocessor are pseudo three dimensional models.
The main deficiency of using rigid links occurs in modeling weak axis girder
behavior. The use of rigid links causes the weak axis moment of inertia of the girders
to become coupled to the rotational degrees of freedom of the deck slab. This coupling
will generally result in an overestimation of the lateral stiffness of the girders. To avoid
such a problem the preprocessor sets the weak axis moment of inertia of the girders to a
negligibly small value. This procedure allows rigid links to be used in modeling
composite action under longitudinal bridge flexure but does not result in overestimation
of lateral stiffness. Since the preprocessor models bridges for gravity and vehicle
loading and not for lateral effects such as wind or seismic loading, this procedure is
reasonable.
Illustrated in Figure 3.5 is the eccentric girder model for a girderslab system
consisting of a concrete deck slab and a nonprismatic steel girder. The system is
assumed to consist of multiple spans of which the first span is shown in the figure. In
modeling the slab and girder, a total of six elements have been used in the longitudinal
direction in the span shown. A width of two elements in the lateral direction are shown
modeling the deck slab. Nodes in the finite element model are located at the elevation
of the slab centroid. The girder elements are located eccentrically from the nodes using
rigid links whose lengths are the eccentricities between the centroid of the slab and the
centroid of the girder. Because the girder is nonprismatic, the elevation of the girder
centroid varies as one moves along the girder in the longitudinal direction. For this
64
Figure 3.5 Modeling Composite Action with the Eccentric Girder Model
reason the lengths of the rigid linksâ€”i.e. the eccentricities of the girder elementsâ€”also
vary in the longitudinal direction. Displacements at the ends of the girder elements are
related to the nodal displacement DOF at the finite element nodes by rigid link
transformations.
Slab elements, modeled using flat shell elements, are connected directly to the
finite element nodes without eccentricity. Recall that in the EGM the slab elements are
allowed to deform axially as are the girder elements. In this manner the slab and girder
elements function jointly in resisting load applied to the structure. Since the slab
elements must be allowed to deform axially a translating roller support is provided at
the end of the first span. By using a roller support, the girder and slab are permitted to
deform axially as well as flexurally and can therefore act compositely as a single unit.
The EGM composite action modeling technique is generally considered to be
more accurate than the CGM modeling technique. This is because in the CGM an
approximate effective width must be used in the determination of the composite section
65
properties. However, while the EGM is more accurate, the analysis results must be
interpreted with greater care since the effect of the axial girder forces must be taken
into consideration when the total moment in the girder section is determined. Also,
when using the EGM, it is necessary to use a sufficient number of longitudinal
elements to ensure that compatibility of longitudinal strains between the slab and girder
elements is approached (Hays and Schultz 1988). It is therefore recommended that at
least twenty elements in the longitudinal direction be used in each span.
3.4 Modeling Prestressed Concrete Girder Bridge Components
The modeling of structural components and structural behavior that occur only
in prestressed concrete girder bridges will be described in the sections below.
3.4,1 Modeling Prestressing Tendons
Prestressed concrete girder bridges make use of pretensioning and postÂ¬
tensioning tendons to precompress the concrete girders, thus reducing or eliminating
the development of tensile stresses. The tendons used for pretensioning and postÂ¬
tensioning of concrete will be referred to collectively as prestressing tendons in this
context. Prestressed bridges are pretensioned and may optionally be posttensioned in
either one or two phases when using the preprocessor. Posttensioned continuous
concrete girder bridges are modeled assuming that the girders are pretensioned, placed
on their supports and then posttensioned together to provide structural continuity.
In order to model prestressing tendons using finite elements, both the tendon
geometry and the prestressing force must be represented. Tendons are modeled as axial
66
Girder Prestressing Rigid Prestressing Finite
(Frame) Tendon (Truss) Link Tendon Element
Element Element (Eccentricity) Centroid Node
Girder
Cross Section
Figure 3.6 Modeling the Profile of a Prestressing Tendon
truss elements that are eccentrically connected to the finite element nodes by rigid links
(see Figure 3.6). Since straight truss elements are used between each set of nodes in the
tendon, a piecewise linear approximation of the tendon profile results. The tendon is
divided into a number of segments that is equal to the number of longitudinal elements
per span specified by the user. As long as a reasonable number of elements per span is
specified, this method of representing the profile will yield results of ample accuracy.
The reference point from which tendon element eccentricities are specified in
the model varies depending on the particular type of composite action modeling being
used and on the particular construction stage being modeled. In the noncomposite
model (NCM) tendon element eccentricities are always referenced to the centroid of the
bareâ€”i.e. noncompositeâ€”girder cross section. In the composite model (CGM), for
construction stages where the slab and girder are acting compositely, the eccentricities
are referenced to the centroid of the composite girder cross section which includes an
effective width of deck slab. Eccentricities in the eccentric girder model (EGM) for
construction stages where the slab and girder are acting compositely are referenced to
the midsurface of the slab. Prestressing element eccentricities for construction stages in
67
which the deck slab is structurally effective are illustrated in Figure 3.7. In construction
stages where the deck slab has not yet become structurally effective, the tendon
eccentricities are always referenced to the centroid of the bare girder cross section
regardless of which composite action model is being used.
When using the preprocessor, the engineer always specifies the location of the
prestressing tendons relative to the top of the girder. With this data, the preprocessor
computes the proper truss element eccentricities for each construction stage of the
bridge based on the composite action model in effect. The example girder that is shown
in Figure 3.6 has a piecewise linear tendon profile tendon created by dual hold down
points. Note however that the preprocessor is capable of approximating any tendon
profile, linear or not, as a series of linear segments.
In addition to modeling the profile of the prestressing tendons, the prestressing
force must also be represented. This is accomplished simply by specifying an initial
tensile force for each tendon (truss) element in the model. Since the tendons are
modeled using elastic truss elements, prestress losses due to elastic shortening of the
concrete girder are automatically accounted for in the analysis. However, nonelastic
Noncomposite
Model (NCM)
Effective Width
' 'I
f
M
â™¦ *T
Prestressing
Centroid
Cl
Tendon
Eccentricity
1
Composite Girder
Model (CGM)
Eccentric Girder
Model (EGM)
Figure 3.7 Modeling the Profile of a Prestressing Tendon
68
losses due to effects such as friction, anchorage slip, creep and shrinkage of concrete,
and relaxation of tendon steel are not incorporated into the model. (In the BRUFEM
system, these nonelastic losses are accounted for automatically by the postprocessor
based on the appropriate AASHTO specifications). Thus, the tendon forces specified by
the engineer must be the initial pretensioning or posttensioning forces in the tendons,
prior to any losses.
3.4.2 Increased Stiffening of the Slab Over the Concrete Girders
During lateral flexure in prestressed (and also reinforced concrete Tbeam)
girder bridges, the relatively thin slab between the girders deforms much more than the
portion of the slab over the flanges of the girders. This behavior is due to the fact that
the girder flange and web have a stiffening effect on the portion of the slab that lies
directly above the girder. Rather than using thicker plate elements over the girders,
lateral beam elements are used to model this stiffening effect. These lateral beam
elements are located at the midsurface of the slab and extend over the width of the
girder flanges, as is shown in Figure 3.8.
Figure 3.8 Modeling the Stiffening Effect of the Girder Flanges on the Deck Slab
69
The lateral bending stiffness of these elements is assumed to be that of a wide
beam of width SY (SY/2 for elements at the ends of the bridge). From plate theory
(Timoshenko and WoinowksyKrieger 1959) the flexural moment in a plate is given by
(3.2)
where Mx is the moment per unit width of plate, E is the modulus of elasticity, t is
the plate thickness, v is Poissonâ€™s ratio, and <(>* and are the curvatures in the x
and ydirections respectively. Since the value of Poissonâ€™s ratio for concrete is small
be assumed that the quantity (l â€” v21 is approximately unity. Also,
(v = 0.15), it can
since only bending in the lateral direction (xdirection) is of interest for the lateral beam
members, only the xcurvature Â§x is taken into consideration. From Equation (3.2)
and the simplifications stated above, the moment of inertia of a plate element having
thickness t is
/=ÃV) (33)
12
Since the moment of inertia of the slab is automatically accounted for through the
inclusion of the plate elements in the bridge model, the effective moment of inertia of
the lateral beam element is given by
(3.4)
where t(sg+ef) Â¡s the thickness of the slab over the girder plus the effective flange
thickness of the girder and is the thickness of the slab of the girder.
70
The torsional moment of inertia of the lateral beam members is obtained in a
similar manner. From plate theory the twisting moment in a plate of thickness t is
given by
GtJ
(3.5)
â€˜xy ~ 6 vxy
where G is the shear modulus of elasticity, and <>,y is the cross (or torsional)
curvature in the plate. Thus, the effective torsional moment of inertia of the lateral
beam elements is given by
J =
Ksg+ef) fsg
6
(SY)
(3.6)
where the parameters f(sg+ef) an(* lsg are the same as those described earlier.
3.5 Modeling Steel Girder Bridge Components
The modeling of structural components and structural behavior that occur only
in steel girder bridges will be described in the sections below. One of the areas of
modeling that is specific to steel girder bridges is that of modeling cross brace steel
diaphragms. However, since this topic was already considered in Â§3.2.3 in a general
discussion of diaphragm modeling, it will not be repeated here.
3.5.1 Modeling Hinges
Hinged girder connections are occasionally placed in steel girder bridges to
accommodate expansion joints or girder splices. Hinges are assumed to run laterally
across the entire width of the bridge, thus forming a lateral hinge line at each hinge
71
location. Along a hinge line, each of the girders contains a hinge connection and the
deck slab is made discontinuous.
When using the preprocessor, the engineer inserts hinges into the bridge by
specifying the distances from the beginning of the bridge to the hinges. If the hinge
distances specified do not match the locations of finite element nodal lines, then the
hinge lines are moved to the location of the nearest nodal line. Also, note that the
insertion of hinges into a bridge must not cause the structure to become unstable. For
example, one may not insert a hinge into a single span bridge since this would result in
an unstable structure.
Hinge modeling is accomplished by placing a second set of finite element nodes
along the hinge line at the same locations as the original nodes. In Figure 3.9, this is
depicted by showing a small finite distance between the two set of nodes at the hinge
line. In the actual finite element bridge model the distance between the two lines of
nodes is zero. Girder, stiffener, and slab elements on each side of the hinge line are
then connected only to the set of nodes on their corresponding side of the hinge. At this
point the bridge is completely discontinuous across the hinge line.
Figure 3.9 Modeling a Hinge in a Steel Girder Bridge
72
Girders are the only structural components assumed to be even partially
continuous across hinges. The deck slab and stiffeners are assumed to be completely
discontinuous across hinges. Girder are continuous with respect to vertical translation
and, in some cases, axial translation but not flexural rotation. As a result, the end
rotations of the girder elements to either side of a hinge are uncoupledâ€”i.e. a hinge is
formed. Displacement constraints are used to provide continuity at the points where
girder elements cross a hinge line. In bridges modeled using the NCM or CGM
composite action models, the vertical translations of the nodes that connect girder
elements across a hinge line are constrained. When the EGM composite action model is
used, all three translations must be constrained due to the axial effects in the model.
Nodes to which girder elements are not connected are left unconstrained and therefore
are allowed displace independently.
3.5.2 Accounting for Creep in the Concrete Deck Slab
As was explained in the previous chapter, long term sustained dead loads on a
bridge will cause the concrete deck slab to creep. In steel girder bridges, the deck slab
and girders are constructed from materials that have different elastic moduli and
therefore different sustained load characteristics. As the concrete slab creeps over time,
increasingly more of the dead loads will be carried by the steel girders. Since the
models created by the preprocessor are not time dependent finite element models, the
creep effect must be accounted for in some approximate manner. Depending on the
73
composite action model being used, creep is accounted for in one of the ways discussed
below.
If the CGM is being used to represent composite action, then the creep effect is
accounted for when computing composite section properties for the girders. Normally,
when composite section properties are computed, the effective width of concrete slab
that acts compositely with the girders is transformed into an equivalent width of steel
by dividing by the modular ratio. This equivalent width of steel is then included in the
computation of composite section properties for the girder. The modular ratio is a
measure of the relative stiffnesses of steel and concrete and is given by
where Ec is the modulus of elasticity of the concrete deck slab and Eg is the modulus
of elasticity of the steel girders. In order to account for the increased deformation that
will arise from creeping of the deck, the preprocessor uses a modified modular ratio
when computing composite girder properties. When transforming the effective width of
concrete slab into equivalent steel, a modular ratio of 3Â« is used instead of n. This
yields a smaller width of equivalent steel and therefore smaller section properties.
Because the section properties are reduced, the stiffness of the girders are reduced,
deformations increase, and stresses in the girders increase.
If the EGM is used to represent composite action, then the effect of creep is
accounted for by employing orthotropic material properties in the slab elements. In the
EGM, the deck slab is modeled using a mesh of shell elements. By using orthotropic
74
material properties for these shell elements, different elastic moduli may be specified
for the longitudinal and lateral directions. To account for creep, the elastic modulus of
the shell elements is divided by a factor of 3.0 in the longitudinal direction, but left
unmodified in the lateral direction. Using this technique, the effects of creep, which
relate primarily to stresses in the longitudinal direction, are accounted for without
disturbing the lateral load distribution behavior of the deck.
If the noncomposite model (NCM) is used, then the girders and deck slab do not
act compositely and the girders are assumed to carry all of the load on the bridge. Since
the girders carry all of the load, the effects of creep in the concrete deck will be
negligible and therefore no special modeling technique is necessary.
3.6 Modeling Reinforced Concrete TBeam Bridge Components
Reinforced concrete (R/C) Tbeam bridges are modeled by the preprocessor in
essentially the same manner that prestressed concrete girder bridges are modeled. The
obvious exception to this statement is that R/C Tbeam bridges lack the prestressing
(pretensioning and possibly posttensioning) tendons that are present in prestressed
concrete girder bridges. However, the girders in each of these bridge types are
constructed from the same materialâ€”concreteâ€”and are therefore modeled in the same
manner. Diaphragms have precisely the same configuration in both of these bridge
types and are therefore modeled in identical fashion. Finally, the deck slab stiffening
effect that was discussed in Â§3.4.2 in the context of prestressed concrete girders also
occurs in R/C Tbeam bridges. In R/C Tbeam bridges, this stiffening effect is
75
represented using the same lateral beam elements that were discussed earlier for
prestressed concrete girders.
5.7 Modeling FlatSlab Bridge Components
A flatslab bridge consists of a thick reinforced concrete slab and optional edge
stiffeners such as parapets. If stiffeners are present and structurally effective, they are
modeled using frame elements as is the case for girderslab bridges. If stiffeners are not
present on the bridge or are not considered structurally effective, then the slab is
modeled using plate elements and the entire bridge is represented as a largeâ€”possibly
multiple spanâ€”plate structure. When stiffeners are present on the bridge but do not act
compositely with the slabâ€”and are therefore not considered structurally effectiveâ€”they
should be specified as line loads by the engineer.
If stiffeners are considered structurally effective, then the engineer can choose
either the CGM or EGM of composite action. If the CGM is chosen, then the slab is
modeled using plate elements and composite section properties are computed for the
stiffener elements using the effective width concept. If the EGM is chosen, the slab is
modeled using shell elements and the stiffeners are located eccentrically above the flat
slab using rigid links. The NCM is not available for flatslabs because if sufficient bond
does not exist between the stiffeners and slab to transfer horizontal shear forces, it
cannot be assumed that there is sufficient bond to transfer vertical forces either. In
order for stiffenersâ€”which are above the slabâ€”to assist in resisting loads, there must
be sufficient bond to transfer vertical forces to and from the slab.
76
3.8 Modeling the Construction Stases of Bridges
To properly analyze a bridge for evaluation purposes, such as in a design
verification or the rating of an existing bridge, each distinct structural stage of
construction must be represented in the model. Using the preprocessor, this can be
accomplished by creating a full load modelâ€”a model in which the full construction
sequence is considered. Each of the individual construction stage models, which
collectively constitute the full load model, simulates a particular stage of construction
and the dead or live loads associated with that stage.
Modeling individual construction stages is very important in prestressed
concrete girder bridges, important in steel girder bridges, and of lesser importance in
R/C Tbeam and flatslab bridges. For each of these bridge types, the preprocessor
assumes a particular sequence of structural configurations and associated loadings.
These sequences will be briefly described below, however, for complete and highly
detailed descriptions of each sequence see Hays et al. (1994).
Several different types of prestressed concrete girder bridges may be modeled
using the preprocessor. These include bridges that have a single span, multiple spans,
pretensioning, one phase of posttensioning, two phases of posttensioning, and
temporary shoring. Each of these variations has its own associated sequence of
construction stages the highlights of which are described below.
All prestressed girder bridges begin with an initial construction stage in which
the girders are the only components that are structurally effective. In multiple span
prestressed concrete girder bridges, the girders are not continuous over interior
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supports at this stage but rather consist of a number of simply supported spans. At this
stage, the bridge is subjected to pretensioning forces and to dead loads resulting from
the weight of the girders, diaphragms, andâ€”in most casesâ€”the deck slab. In
subsequent construction stages the bridge components that caused dead loads in this
first stage, e.g. the diaphragms and deck slab, will become structurally effective and
will assist the girders in resisting loads due to posttensioning, temporary support
removal, stiffener dead weight, line loads, overlay loads, and ultimately vehicle loads.
Prestressed concrete girder bridges may go through a construction stage that
represents the effect of additional long term dead loads acting on the compound
structure. Loads that are classified as additional long term dead loads include the dead
weight of the stiffeners, line loads, and overlay loads. The compound structure is
defined to be the stage of construction at which the deck slab has hardened and all
structural components, except for stiffeners, are structurally effective. The term
compound is used to refer to the fact that the various structural components act as a
single compound unit but implies nothing with regard to the composite or noncomposite
nature of girderslab interaction. Since the deck slab has hardened at this construction
stage, the girders and deck slab may act compositely. Also, lateral beam elements are
included in the bridge model to represent the increased stiffening of the girder on the
deck slab.
As construction progresses from one stage to the next, the bridge components
become structurally effective in the following orderâ€”girders, pretensioning,
diaphragms, deck slab, lateral beam elements, phase1 posttensioning (if present),
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stiffeners, and phase2 posttensioning (if present). The final construction stage is
represented by a live load model, i.e. a model which represents the final structural
configuration of the bridge and its associated live loading. At this stage, each and every
structural component of the bridge is active in resisting loads and the loads applied to
the bridge are those resulting solely from vehicle loading and lane loading.
Steel girder bridges have simpler construction sequences than prestressed
concrete girder bridges due to the lack of prestressing and temporary shoring. Steel
girder construction sequences begin with a construction stage in which the girders and
diaphragms are assumed to be immediately structurally effective. The bridge model
consists only of girder elements and diaphragm elements which, acting together, resist
the dead load of the girders, diaphragms, and the deck slab.
It is assumed that the girders in multiple span steel girder bridges are
immediately continuous over interior supports. The assumption of immediate continuity
of the girders is reasonable since multiple span steel bridges are not typically
constructed as simple spans that are subsequently made continuous as is the case in
prestressed concrete girder bridges. It is also assumed that the diaphragms in steel
girder bridges are installed and structurally effective prior to the casting of the deck
slab.
Steel girder bridges may go through a construction stage that represents
additional long term dead loads acting on the compound structure just as was described
above for prestressed concrete girder bridges. Since the deck slab has hardened at this
construction stage, the girders and deck slab may act compositely. However, lateral
beam elements are not used in steel girder bridge models as they are in prestressed
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concrete girder bridge models. Also, in steel girder bridges, the effects of concrete
deck creep under long term dead loading must be properly accounted for. This is
accomplished by using the techniques presented in Â§3.5.2.
In the final (live load) construction stage of steel girder bridges, each and every
structural component of the bridge is active in resisting loads. At this stage, cracking of
the concrete deck in negative moment regions of multiple span steel girder bridges may
be modeled by the preprocessor. This deck crackingâ€”the extent of which is specified
by the engineerâ€”is assumed to occur only in the final live load configuration of the
bridge. In modeling deck cracking, the preprocessor assumes a region in which
negative moment is likely to be present. This assumption is necessary since only after
the finite element analysis has been completed will the actual region of negative
moment be known. Thus, the preprocessor assumes negative moment will be present in
regions of the bridge that are within a distance of twotenths of the span length to either
side of interior supports. See Hays et al. (1994) for further details of the cracked deck
modeling procedure used by the preprocessor.
In R/C Tbeam and flatslab bridges, the construction sequence is not of great
significance and has little effect on the analysis and rating of the bridge. During the
construction of these bridge types, the bridge often remains shored until all of the
bridge components have become structurally effective. In such situations, all of the
structural components become structurally effective and able to carry load before the
shoring is removed. The preprocessor therefore assumes shored construction for these
two bridge types.
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Based on the shored construction assumption, all of the structural elements used
to model R/C Tbeam and flat bridges are assumed to be immediately structurally
effective in resisting dead load. Thus, there exists only a single dead load construction
stage in which all of the dead loadsâ€”including additional long term dead loadsâ€”are
applied to the fully effective structural bridge model. The final live load construction
stage consists of the same structural model as that used in the dead load stage, but
subjected to vehicle and lane loading instead of dead loading.
3.9 Modeling Vehicle Loads
Using the vehicle live loading features provided by the preprocessor, an
engineer can quickly describe complicated live loading conditions with relative ease. In
describing live loading conditions, the engineer only needs to specify the type, location,
and direction of each vehicle on the bridge and the location and extent of lane loads.
Once these data have been obtained, the preprocessor can generate a complete finite
element representation of the live loading conditions.
Recall from the previous chapter that the preprocessor models lane loads not as
uniform loads on the deck slab, but instead as trains of closely spaced axles. Thus, both
individual vehicles and lane loads are modeled using collections of axles. To model
these live loads using the finite element method, the preprocessor must convert each
axle, whether from a vehicle or a lane load, into an equivalent set of concentrated nodal
loads that are applied to the slab nodes in the model. In order to accomplish this
conversion to nodal loads the multiple step procedure illustrated in Figure 3.10 is
performed by the preprocessor.
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Figure 3.10 Conversion of Vehicle and Lane Loads to Nodal Loads
Given the location of a vehicle on the bridge, the preprocessor computes the
location of each axle within the vehicle, and then the location of each wheel within
each axle. In the case of a lane load, the preprocessor computes the location of each
axle in the axle train, and then the computes the location of each wheel within each
axle. Finally, after computing the magnitude of each wheel load, the preprocessor
distributes each wheel load to the finite element nodes that are closest to its location.
Each wheel load is idealized as a single concentrated load acting at the location of the
contact point of the wheel. Wheel loads are distributed to the finite element nodes using
the static distribution technique illustrated in Figure 3.11. This distribution technique is
used for zero, constant, and variable skew bridges.
Node
Number
Statically Equivalent
Nodal Loads
N1
PI = Pw (la)(lP)
N2
P2 = Pw (ot)(lP)
N3
P3 = Pw (la)(P)
N4
P4 = Pw (a)(p)
Static Distribution Factors
a  XI / X2
p = Y1 / Y2
Figure 3.11 Static Distribution of Wheel Loads
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Wheel loads that are off the bridge in the longitudinal direction are ignored.
Wheel loads that are off the bridge in the lateral direction are converted into statically
equivalent loads by assuming rigid arms connect the loads to the line of slab elements
at the extreme edge of the bridge. After making this assumption, the same static
distribution procedure described above is used to compute the equivalent nodal loads.
This procedure of assuming a rigid arm allows the engineer to model rare cases in
which the outside wheels of a vehicle are off the portion of the deck that is considered
to be structurally effective.
The ability of the preprocessor to perform vehicle load distribution is one of its
most time saving features since very often many load cases must be explored to
determine which ones are critical for design or rating. Each of these load cases consists
of many wheel loads that must be converted into even more numerous equivalent nodal
loads. The number of nodal loads required to represent a moderate number of load
cases can quickly reach into the thousands.
CHAPTER 4
DATA COMPRESSION IN FINITE ELEMENT ANALYSIS
4.1 Introduction
In the analysis of highway bridges, the amount of outofcore storage that is
available to hold data during the analysis can frequently constrain the size of models
that can be analyzed. It is not uncommon for a bridge analysis to require hundreds of
megabytes of outofcore storage for the duration of the analysis. Also, while the size
of the bridge model may be physically constrained by the availability of outofcore
storage, it may also be effectively constrained by the amount of execution time required
to perform the analysis. The use of computer assisted modeling software such as the
bridge modeling preprocessor presented in Chapters 2 and 3 further increases the
demand on computing resources. Using the preprocessor, an engineer can create
models that are substantially larger and more complex than anything that could have
been created manually.
To address the issues of the large storage requirements and lengthy execution
times arising from the analysis of bridge structures, a realtime data compression
strategy suitable for FEA software has been developed. This chapter will describe that
data compression strategy, its implementation, and parametric studies performed to
evaluate the effectiveness of the technique in the analysis of several bridge structures. It
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will also be shown that by utilizing data compression techniques, the quantity of outof
core storage required to perform a bridge analysis can be vastly reduced. Also, in many
cases the execution time required for an analysis can be dramatically reduced by using
the same data compression techniques.
4.2 Background
During the past three decades a primary focal point of research efforts aimed at
improving the performance of FEA software has been the area of equation solving
techniques. This is due to the fact that the equation solving phase of a FEA program is
one of the most computationally intensive and time consuming portions of the program.
As a result of research efforts in this area, equation solvers that are optimized both for
speed and for minimization of required incore memory are now widely available. The
performance of FEA programs implementing such equation solvers is now often
constrained not by the quality of the equation solver but instead by the speed at which
data may be moved back and forth between the program core to outofcore storage and
by the availability of outofcore storage.
Although the coding details of FEA software vary greatly from program to
program, all FEA programs must perform certain common procedures. As the size of
the finite element model increases, three of these common procedures begin to
dominate a program in terms of the portion of total execution time that is spent in these
procedures. They are solving the global set of equilibrium equations, forming element
stiffness and force recovery matrices, and performing element force recovery.
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In many FEA programs, the element stiffness and force recovery matrices are
formed incore and then written to disk sequentially as each element is formed. This
procedure requires that all of the element stiffness and force recovery matrices be
moved from the program core to disk. In all but the smallest of finite element models,
this is necessary because there is insufficient memory to hold all of the element
matrices for the duration of the analysis. Once the element matrices have been written
to disk, the global equilibrium equations are formed by assembling the element stiffness
and load matrices into the global stiffness and load matrices. This step requires that all
of the element matrices that have been written to disk be moved from disk storage back
to the program core. Finally, when the global equilibrium equations have been solved
and displacements are known, the element force recovery matrices must be transferred
from disk back to the program core to perform element force recovery. Under certain
circumstances, such as in analyses involving multiple load casesâ€”an extremely common
occurrence in bridge modelingâ€”or in nonlinear analyses where element state
determination and element matrix formation must be performed many times, some or
all of the steps discussed above may need to be performed more than once.
Consequently, element matrices must be transferred to and from disk many times
during the analysis.
Due to rapidly improving computational power and relatively low cost, the
personal computer (PC) has gained widespread use in the area of FEA rivaling more
expensive workstations in terms of raw computational power. However, PCs are
generally slower than workstations in the area of disk input/output (I/O) speed and
86
often have far less available disk space. To address the issue of slow I/O speed on PCs,
some FEA software developers write custom disk I/O routines in assembly language.
This results in an FEA code that is considerably faster than that which can be achieved
using only the disk I/O routines provided in standard high level languages.
However, while the use of assembly language yields increased disk I/O speed, it
does so at the cost of portability. This is because assembly language is intimately tied to
the architecture of central processing unit (CPU) on which the software is running and
is therefore not portable to machines having different CPU architectures. Furthermore,
the use of assembly language I/O routines achieves nothing with respect to the problem
of the large of outofcore storage demands made by FEA software.
Instead of using assembly language disk I/O routines, the author has chosen a
different approach in which the quantity of data written to the disk is reduced while
preserving the information content of that data. This is accomplished by using a data
compression technique to compress the data before writing it to disk, and decompress it
when reading the data back from disk. The result is faster data transfer and vastly
reduced outofcore storage requirements.
4,3 Data Compression in Finite Element Software
In using compressed disk I/O in finite element software the goals are twofoldâ€”
to reduce the quantity of outofcore storage required during the analysis and to reduce
the execution time of the analysis. Data compression is used to accomplish these goals
by taking a block of data that the FEA software must transfer to or from disk storage
and compressing it to a smaller size before performing the transfer. Compression
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preserves the information content of the data block while reducing its size, thus
reducing the amount of time that must be spent performing disk I/O. Of course the
process of compressing the data before writing it to disk requires an additional quantity
of time. Also the data must now be decompressed into a usable form when reading it
back from disk which also requires additional time. However, the end result is often
that the amount of additional time required to compress and decompress the data is
small in comparison to the amount of time saved by performing less disk I/O. This is
especially true on PC platforms where disk I/O is known to be particularly slow.
While one benefit of using compressed I/O can be a reduction in the execution
time required by FEA softwareâ€”which is often the critical measure of performanceâ€”a
second benefit is that the quantity of disk space required to hold data files created by
the software is substantially reduced. A typical FEA program will create both
temporary data files, which exist only for the duration of the program, and data files
containing analysis results which may be read by post processing software. The data
compression strategy presented herein compresses only files that are binary data files,
i.e. raw data files. This is opposed to formatted readable output files that the user of a
FEA program might view using a text editor. Binary data files containing element
matrices, the global stiffness and load matrices, or analysis results such as
displacements, stresses, and forces are typically the largest files created by FEA
software and are the types of files which are therefore compressed. Formatted output
files can just as easily be compressed but the user of the FEA software would have to
decompress them before being able to view their contents.
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For reasons of accuracy and minimization of roundoff error, virtually all FEA
programs perform floating point arithmetic using double precision data values. In
addition, much or all of the integer data in a program consists of long (4 byte) integers
as opposed to short (2 byte) integers either because the range of a short integer is not
sufficient or because long integers are the default in the language (as is the case in
Fortran 77). An underlying consequence of using double precision floating point and
long integer data types is that there is a tremendous amount of repetition in data files
created by FEA software. Consider as an example the element load vector for a nine
node plate bending element. A plate bending element has two rotational and one
translational degrees of freedom at each node in the local coordinate system, but when
rotated to the global coordinate system there are six degrees of freedom per node.
Thus, for a single load case the rotated element load vector which might be saved for
later assembly into the global load vector will have 9*6 = 54 entries. If the entries are
double precision floating point values then each of the 54 entries in the vector is made
up of 8 bytes resulting in a total of 54*8 = 432 bytes. Now consider an unloaded plate
element of this type where the load vector contains all zeros. Typical floating point
standards represent floating point values as a combination of a sign bit, a mantissa, and
an exponent. A value of zero can be represented by a zero sign bit, zero exponent, and
zero mantissa. Thus a double precision representation of the value zero may consist of
eight zero bytes. A zero byte is defined as containing eight unset (zero) bits.
Consequently, the load vector for an unloaded plate element will consist of 432
repeated zero bytes resulting, in a considerable amount of repetition within the data
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file. This type of data repetition, in which there are sequences of repeated bytes, will
be referred to hereafter as small scale repetition.
In addition to the small scale repetition described above, data files created by
FEA software contain large scale repetitions of data as well. Consider the element
stiffness of the plate element described above. When rotated to the global coordinate
system, the element stiffness will be a 54 x 54 matrix of double precision values. Using
the symmetric property of stiffness matrices, assume that only the upper triangle of the
matrix is saved to disk for later assembly into the global stiffness matrix. Thus, a total
of 54*(54+l)/2 = 1,485 double precision values, or 1,485*8 = 11,800 bytes of
information must be saved to disk for a single element. Now consider a rectangular slab
model of constant thickness consisting of a 10 x 10 grid of elements where there is a
high degree of regularity in the finite element mesh. Assume that the rotated element
stiffness for each element in the model is identical to that of all the other elements. To
save the element stiffnesses for each of the elements in the model, a total of
10*10*11,800 = 1,188,000 bytes of data must be transferred from the program core to
disk, and then back to the program core at assembly time when there are actually only
11,448 unique bytes of information in that data.
Somewhere in between the small and large scale repetitions described above lies
what will be appropriately referred to as medium scale repetition. Medium scale
repetition refers to sequences of repeated byte patterns. If the length of the pattern is a
single byte, then medium scale repetition degenerates to small scale repetition, whereas
if the length of the pattern is the size of an entire stiffness matrix, then medium scale
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repetition degenerates to large scale repetition. As used herein, the term medium scale
repetition will refer to patterns ranging from two bytes in length to a few hundred bytes
in length.
It should be noted the data compression strategy being presented herein is
intended to supplement rather than replace efficient programming practices. Take as an
example, the case of the repeated plate element stiffness matrices described above. A
well implemented FEA code might attempt to recognize such repetition and write only
a single representative stiffness matrix followed by repetition codes instead of writing
each complete element stiffness. In this case the compression library will supplement
the already efficient FEA coding by opaquely compressing the single representative
element stiffness that must be saved. The term opaquely is used to indicate that the
details of the data compression process, and the very fact that data compression is even
being performed, are not visible toâ€”i.e. are opaquely hidden fromâ€”the FEA code.
Thus, the reduction of data volume is performed in a separate and self contained
manner which requires no special changes to the FEA software. If however, the FEA
code makes no such special provisions for detecting element based repetition, then the
compression library described herein will reduce the volume of data that must be saved
by compressing each element stiffness matrix as it is written.
In each case, compression is accomplished primarily by recognizing small and
medium scale repetition within the data being saved. In fact, due to the small size of
the hash key used in the hashing portion of the compression algorithmâ€”described in
detail later in this chapterâ€”the likelihood of the compression library identifying large
91
scale repetitions such as entire stiffness matrices is very remote. Instead, the vast
majority of the compression is accomplished by recognizing small and medium scale
repetition, both of which are abundant in the type of data produced by FEA software.
4.4 Compressed I/O Library Overview
The compressed disk I/O library being presented herein uses a buffered data
compression technique that performs compression on both small and large scale
repetitions like those described above. Written in the ANSI C language, the compressed
I/O library was originally intended for use in FEA software written in C. From this
point on ANSI C will be referred to simply as C, and the ANSI C I/O functions will be
referred to as standard C I/O functions to distinguish them from the compressed I/O
functions being presented. In its present form, the library can be used to manipulate
sequential binary I/O files, although it could also be adapted to manipulate direct access
(as opposed to sequential), and formatted (as opposed to binary) data files.
Contained in the library are compressed I/O functions that are direct
replacements for the standard C I/O functions that a FEA program would normally
utilize. The compressed I/O library functions have identical prototypes to their standard
C counterparts. As a result, converting a FEA program written in C which utilizes
sequential binary I/O over to compressed I/O requires only that the names of the
standard C functions called by the program be replaced with the corresponding
compressed I/O function names. Table 4.1 lists the compressed I/O functions that are
provided in the library. The CioModify function has no counterpart in the standard C
library because it allows the FEA code to modify certain characteristics of the
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Table 4.1 Compressed and Standard C Binary I/O Functions
Compressed
I/O function
Standard C
I/O function
Use of Function
CioOpen
fopen
Open a binary file for I/O
CioWrite
fwrite
Write a block of data to binary file
CioFlush
fflush
Force a flush of internal I/O buffers to disk
CioRewind
rewind
Rewind to top of a binary file
CioRead
fread
Read a block of data from a binary file
CioCIose
fclose
Close a binary file and flush I/O buffers
CioModify
n/a
Modify characteristics of compression
compression algorithm at run time. However, the FEA code is not required to call
CioModify as default compression parameters are built into the library.
4,5 Compressed I/O Library Operation
One or more files may simultaneously be under the control of the compressed
I/O library at any given time. Each time a file is opened, a new entry in a linked list
that is maintained by the library is created to hold information pertaining to that
particular file. For each file under the control of the library, memory is allocated for an
I/O buffer that will be used for writing and reading operations and for a hash table that
will be used during compression. In addition, if there is at least one file under the
control of the library then a single compression buffer is also allocated and shared by
all files controlled by the library. To maximize the degree of compression that can be
achieved during binary I/O, the compressed I/O library performs buffered I/O.
Buffering simply means that when the calling program requests that a block of data be
written to disk, instead of immediately compressing and writing the data to disk the
library will copy the data into an I/O buffer in memory. Data from subsequent write
93
operations will be copied to the end of the previous blocks of data so that data in the
I/O buffer is accumulated. Once the I/O buffer is full, its contents are compressed into
the compression buffer and then the compressed block is written to disk as is illustrated
in Figure 4.1.
Buffering is necessary to achieve a high degree of compression because if the
library were to perform compression each time a write operation were performed, only
repetitions within the small block of data being written could be compressed. In the
example of a plate element stiffness matrix, which might be written using one write for
each row in the matrix, only repetitions within a row could be compressed. If however,
buffered I/O is used and the buffer size is large enough to hold larger portions of the
matrix, then higher degrees of compression can be achieved because the larger blocks
containing repeated information are simultaneously in the I/O buffer when it is
compressed. During read operations, the process is reversed so that compressed data
I/O Buffer Size
1. Copy data block to empty
I/O buffer.
2. Append data block to I/O
buffer.
3. Append partial data block
to I/O buffer.
4. Compress the I/O buffer
into die compression
buffer.
5. Write the compressed
block to disk
6. Copy the remaining portion
of the data block to the
emptied I/O buffer.
7. Append data block to I/O
buffer. Repeat.
Figure 4.1 Buffered and Compressed Writing Procedure
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blocks are read from the disk into the decompression buffer and then decompressed into
the I/O buffer. Read operations extract data from the I/O buffer until it is empty at
which time another compressed block is brought in from disk and the cycle repeats.
Although the sequence in which data blocks are written and read is important,
the size of the data blocks written and read is not important. Clearly the total quantity
of data read must equal that of the data written, however, the individual block sizes
used to read the data need not be the same block sizes as those used to write the data.
Take for example the case of saving double precision coordinate triplets containing the
x, y, and z coordinates of each node in a structure which contains a total of 100 nodes.
If each triplet of double precision values is written using a single write operation, then
the size of the data blocks written is 3*8 = 24 bytes. The 100 required write operation
would results in a total of 100*24 = 2400 bytes of data being written into the I/O
buffer. These 2400 bytes would be compressed before writing them to disk and later
decompressed after reading them from disk. If during the reading of the coordinates it
is for some reason more convenient or desirable to read each of the individual x, y, and
z components separately, then we may read each component with a single read
operation and the size of the data blocks read is 1*8 = 8 bytes. The required 100*3
read operations will result in a total of 100*3*8 = 2400 bytes of data being read.
Since a compression or decompression operation only uses the compression
buffer for the duration of the operation, a single buffer can be shared by all of the files
controlled by the library. However, the size of the shared compression buffer must be
at least as big as the largest of all the I/O buffers associated with the files that are
95
currently open. Therefore, if a new file is opened with an I/O buffer size larger than
that of any of the other currently open files, then the compression buffer must be
expanded.
All of the memory associated with buffers, hash tables, and linked list entries
used by the library are dynamically allocated at runtime as files are opened, and later
released when those same files are closed. The CioModify function may be used to
change the size of a buffer or hash table associated with a particular file or may be used
to modify the default buffer or hash table size so that all files subsequently opened will
use the modified default sizes.
All of the information pertaining to each file controlled by the library such as
the size, location, and status of I/O buffers, hash tables, and the compression buffer,
the state of the file (write, read, idle), the amount of free space remaining in the I/O
buffers, and file handles for each file are automatically maintained by the library.
During a compression or decompression operation, a buffer of data is passed to a
separate set of functions which actually perform the compression or decompression
using the algorithm discussed below.
4.6 Data Compression Algorithm
Data compression in the compressed I/O library is accomplished using a
technique called Ross Data Compression (RDC) which performs runlength encoding
(RLE) and pattern matching types of compression (Ross 1992). RDC is a lossless data
compression technique. In lossless data compression, a data set may be translated from
its original format into a compressed format and subsequently back to the original
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format without any loss, corruption, or distortion of the data. In contrast, lossy data
compression techniques permit some distortion of the data to occur during the
translation process. Lossy techniques are used in applications such as image and sound
compression in which a limited degree of distortion can be tolerated. However, in
applying data compression to FEA, it is the floating point representations of numeric
values which are compressed and therefore distortion of the data to any extent would
invalidate the analysis. Thus, in FEA it is necessary to use a lossless compression
technique such as RDC.
In RDC, the fundamental unit of data is the byte, so that all data is examined on
a byte by byte basis to determine if there are runlengths of repeated bytes or larger
matching patterns consisting of blocks of bytes. All data, including single and double
precision floating point values, short and long integers, characters and strings are
compressed on a byte basis. In runlength encoding (Ross 1992, Sedgewick 1990), a
repeated sequence of a byte is compressed by replacing the sequence by a single
instance of the byte and an additional runlength code indicating the number of times
the byte is to be repeated. RLE compression can be used to compress small scale
repetitions as in the example of the plate element load vector discussed earlier which
contained a sequence of repeated zero bytes.
Large scale pattern matching, as in the example of the plate element stiffness
matrix presented earlier, is accomplished in RDC by using a sliding dictionary to match
a multiple byte pattern at the current location in the buffer with an earlier occurrence of
the same pattern. A hashing algorithm (Ross 1992, Sedgewick 1990) is used to search
97
for previous instances of byte patterns. As the compressor scans through the data buffer
being compressed, it uses the three bytes at the current location as a hash key. A
hashing function is then applied to this key to produce an index into a hash table. A
hash table entry is a pointer to the location in the buffer of the last occurrence of the
hash key. If the three byte key has occurred earlier in the buffer, then the entry in the
hash tableâ€”which the key has â€œhashed toâ€â€”will hold a pointer to the previous instance
of the key. This establishes that at least three bytes are repeated.
Next, the algorithm determines how many additional bytes (following the hash
key) are repeated at the previous location in the buffer. Once the end of a matching
byte pattern is found, a code is written that references the previous location in the
buffer where the same data can be found and the number of bytes that are in the
pattern. Thus, instead of writing the actual byte pattern again, a compact repetition
code is written.
Figure 4.2 illustrates the process of pattern matching using a hashing algorithm
and a sliding dictionary. Hashing collisions, where two different keys hash to the same
entry in the hash table, are not resolved in RDC. Instead, the most recent key that has
hashed to a particular entry in the hash table replaces the previous occupant of that
entry, thus giving rise to the term sliding dictionary. The reader is referred to
Sedgewick (1990) for a more thorough treatment of hashing algorithms and pattern
matching.
98
Since the compression algorithm presented by Ross (1992) was designed
primarily for use on machines having a 16bit architecture, modifications were
necessary to enable it to be successfully used on 32bit workstation platforms. As
originally designed, the compressor and decompressor write and read words of control
bits which indicate whether subsequent data in the file consists of raw data, RLE
control codes, or pattern control codes. The 16bit (2byte) control words originally
used in the algorithm cause pointer alignment faults on many workstation platforms.
When a 2byte control word must be written to a memory address that is not aligned
with the word size of the machine, a fault can occur. To overcome this problem, the
12345 6789 10
1. Initialize all of the entries in the hash table.
2. Apply the hashing algorithm to the hash key to
produce an index into the hash table.
3. Place the location of the key in the I/O buffer
into the hash table. No previous entry exists
therefore there is no matching pattern. Since
there is no match, the byte 'Z' is copied to the
output verbatim. Advance to next byte.
4. Apply the hashing algorithm to the hash key to
produce an index into the hash table.
5. Place the location of the key in the I/O buffer
into the hash table. No previous entry exists
therefore there is no matching pattern. Since
there is no match, the byte 'Y' is copied to the
output verbatim. Advance to next byte.
6. Scanning of the I/O buffer continues in the same
manner until a key is matched. Now apply the
hashing algorithm to the hash key to produce an
index into the hash table.
7. The entry in the hash table is already occupied
indicating that a matching pattern exists earlier in
the I/O buffer. Replace the previous hash table
value with the new location in the I/O buffer of
the repeated hash key.
8. Check for additional repeated bytes following the
hash key and determine the total length of
repeated data. Instead of copying the repeated
pattern 'ZYXD' to the output verbatim, a
compact compression code is written. Scanning
resumes at the byte 'F' following the compressed
pattern.
Figure 4.2 Pattern Matching Using a Hashing Algorithm
99
control word was shortened to 8bits (1byte) which may be safely written to any valid
memory address on a workstation platforms.
4.7 Fortran Interface to the Compressed I/O Library
Although the compressed I/O library is written in C and was originally intended
for use in FEA software also written in C, there exists a vast body of engineering
software in use today which is written in Fortran 77â€”referred to simply as Fortran
from this point onâ€”and which can also benefit from data compression. Most of this
software performs quite adequately and should not have to be completely overhauled
and rewritten in C simply to take advantage of the compressed I/O library. In addition,
much of this software has been specifically optimized for the Fortran language and
might suffer loss in performance if the rewritten code was not properly optimized for
the C language. It should be noted however, that the compressed I/O library itself
cannot be implemented in an efficient manner in Fortran 77 because the compression
functions make extensive use of the bitlevel operators which are part of the C language
but not a part of the Fortran 77 language.
Since C and Fortran I/O functions differ greatly in their method of controlling
files and writing and reading data, an interface library was developed in C to allow
Fortran programs to indirectly call the compressed I/O library functions. Table 4.2 lists
the Fortrancallable Clanguage function available in the interface library. The interface
library maintains a simple linkedlist datastructure which matches Fortran unit
numbers with C file handles (identifiers). When a Fortran program needs to perform an
I/O operation, the interface library calls the appropriate compressed I/O library
100
function with the correct argument list to perform the requested operation. The
interface library simply serves as a pathway between Fortran and C, with the actual
compression and disk I/O operations still being performed by the compressed I/O
library as described earlier.
Unlike the compressed C I/O library functions, which attempt to closely parallel
the behavior of the standard ANSI C I/O functions they replace, the Fortran interface
functions do not attempt to provide all of the functionality of standard Fortran I/O
statements. For example, the ERR and END parameters provided in the Fortran READ
and WRITE statements are not provided in the interface. Also, the Fortran file modes
NEW, OLD, and UNKNOWN must be replaced with C language file modes in the
calls to the interface library. Minor changes must also be made to Fortran I/O
statements to accommodate the fact that C I/O functions write and read blocks of
contiguous memory only. Therefore Fortran write and read operations which contain
implicit loops must be modified to perform block write and read operations in which all
of the data within a single block is contiguous in memory.
Table 4.2 FortranCallable Interface Library Functions
F77Callable
Interface
Function
Compressed
I/O Library
Function
Differences between Compressed and Fortran I/O
copen
CioOpen
File modes are specified as C modes, not Fortran
cwrite
CioWrite
Must write contiguous blocks of memory
cflush
CioFlush
Not provided in standard Fortran
crewind
CioRewind
Same basic functionality as standard Fortran
cread
CioRead
Must read contiguous blocks of data
celÃ³se
CioClose
Same basic functionality as standard Fortran
emode
CioModify
Can only modify default characteristics
101
Combining the compressed I/O library with Fortran code using mixed language
programming on workstation platforms is fairly straightforward, requiring only minor
naming convention changes. Mixed language programming using PC compilers is more
complicated due to differences in naming conventions, calling conventions, and link
libraries.
4,8 Data Compression Parameter Study and Testing
Efficient use of system resources is always of importance and especially so in
the FEA of bridge structures where resource demand can quickly exceed resource
availability. On virtually all computer platforms, memory is the most precious resource
used by the software and must therefore be used very efficiently. Although the goal of
using data compression is to increase performance of FEA software with respect to
total execution time and disk usage, improvements in these areas must not be made at
great expense to the overall functionality of the software. Since data compression
requires that memory be allocated for use as I/O buffers, hash tables, and compression
buffers, it is evident that using data compression reduces the memory available to the
software for other uses. Using data compression could adversely affect the functionality
of the software if memory is used inefficiently or excessively.
Thus, parametric studies were performed with the goal of determining values
for key compression parameters which will produce considerable performance
improvements without sacrificing any more memory than is necessary. The three
parameters chosen to be examined due to their influence on the performance of the
compression algorithm are listed below.
102
1. Size of the I/O and compression buffers.
2. Size of the hash table.
3. Complexity and repetitiveness inherent in the structure being analyzed.
If the I/O buffers for the files under the control of the compressed I/O library are all of
the same sizeâ€”which will usually be the caseâ€”then the compression buffer used by
these files will be the same size as the I/O buffers. For this reason, the parameter
examined will be referred to simply as buffer size from this point on, with the
understanding that this parameter actually corresponds to the I/O buffer size and the
compression buffer size.
Although the complexity and repetitiveness of the structure are not directly
related to the memory requirements of the compression algorithm, they have
considerable effect on the reductions in file size and execution time that can be
achieved. In addition, they indirectly affect the memory requirements of the
compression library in that complex, nonrepetitive structures will require larger buffer
and hash table sizes to achieve the same level of performance that can be achieved
using smaller buffer and hash table sizes for simpler, more repetitive structures.
4,8.1 Data Compression in FEA Software Coded in C
Parametric studies were performed to investigate the influences of buffer size
and hash table size in the compression of FEA data as well as to evaluate the
effectiveness of data compression in Ccoded FEA software. The studies were
performed by implementing the data compression library into a FEA code written in
the C language. SEAS, an acronym for Structural Engineering Analysis Software, was
103
chosen as the FEA code to be used in the studies. SEAS was developed independently
by the author as part of the research being reported on in this dissertation. It is a
general purpose linear finite element package supporting threedimensional truss,
frame, and plate (ninenode lagrangian and heterosis) elements.
The compression library was implemented in SEAS in such a way that binary
I/O could be performed in either standard or compressed format. In addition, the
parameters of the compression algorithmâ€”namely the buffer size and hash table size
may be specified by the user at runtime when using compressed I/O. The data files
that were compressed were those containing the element matrices, stress recovery
matrices, and stress results from the analysis. These files account for the bulk of outof
core storage required during a FEA analysis.
A pair of twospan flatslab bridge models subjected to moving vehicle loads
were created and analyzed by SEAS. The relevant parameters of each of the models are
listed in Table 4.3. The first model is a twospan concrete flatslab bridge having zero
skew geometry while the second is identical except that it has variable skew geometry.
The finite element meshes for each bridge are illustrated in Figure 4.3. Each of the
bridges was modeled using ninenode lagrangian plate elements supported on elastic
truss elements.
Table 4.3 Parameters of Bridge Models Used in SEAS Parametric Studies
Geometry
Nodes
Degrees of
Freedom
Truss
Elements
Frame
Elements
Plate
Elements
Load
Cases
Zero Skew
924
2583
63
0
200
110
Variable Skew
924
2583
63
0
200
110
104
In the zero skew bridge, the model is extremely regular and there is a high
degree of element repetition. Each span is square in aspect ratio and is divided into a
ten by ten grid of plate elements. As a result, the element matricesâ€”element stiffness,
load, rotation, and stress recovery matricesâ€”are identical for all of the plate elements
in the model.
In contrast, due to the variation in nodal geometry, each plate element in the
variable skew model has a slightly different shape and therefore slightly different
element matrices. The first bridge represents a case in which there is a great deal of
largescale repetition in the data while the second bridge represents a case in which
there is no largescale repetition. As will be shown below, there is considerable
I â€” Longitudinal Direction
Figure 4.3 Bridge Models Used in SEAS Parametric Studies
105
medium and smallscale repetition in the data produced by both of the models. Thus,
these models were appropriate for examining the effects of model complexity on the
ability of the compression library to reduce outofcore storage requirements.
Also of interest was determining what effect the buffer size and hash table size
had on the compression of the data. Thus, each of the flatslab models were analyzed
using compressed I/O with buffer sizes of 64, 128, 256, 512, 1024, 2048, 4096, 8192,
12288, and 16384 bytes, and hash table sizes of 512, 2048, and 4096 elements (where
each element consisted of a four byte pointer).
In addition, the models were also analyzed using SEAS in a mode which uses
the standard C I/O functions instead of the compressed I/O functions. The file size and
execution time data from these two standard C I/O runs served as reference data against
which the compressed runs were evaluated.
A normalized compression ratio was computed for each compressed I/O run by
dividing the outofcore storage requirements of the compressed analysisâ€”defined as
the total size of the element stiffness, load, force recovery, and stress filesâ€”by the out
ofcore usage required when analyzed by SEAS in the standard C I/O mode. Thus, a
compression ratio of 0.25 would indicate that the compressed outofcore storage
requirements were only 25 percent of the normal outofcore storage requirements of
the software.
Compression ratio results for the SEAS runs are plotted in Figure 4.4. As one
might anticipate, the plot confirms that a greater degree of compression can be
achieved in regular, highly repetitive models than in more irregular models. The
106
compressed I/O library was able to reduce the outofcore storage requirements of the
zero skew and variable skew bridge analyses to approximately 7 percent and 12
percent, respectively, of their original sizes.
Also, note that although there is really only a single unique plate element in the
zero skew modelâ€”i.e. all of the plate elements in the model are identical to each
otherâ€”the compression algorithm was not able to fully capitalize on this fact. If the
compression algorithm had recognized this largescale repetition, only one of the two
hundred plate elements making up the model would have needed to be stored. This
would have resulted in a compression ratio on the order of 1/200 = 0.005, or about 0.5
percentâ€”considerably less than the 7 percent achieved. Earlier in this chapter (see
Â§4.3), it was stated that the compression strategy presented herein is not capable of
107
fully exploiting largescale repetition in FEA data. This is because the hash key used in
the compression algorithm is too small to identify patterns as large as a complete
element stiffness matrices. The plot in Figure 4.4 confirms this presumption and
indicates that compression is actually achieved by recognizingâ€”and compressing
medium and smallscale repetitions in the data.
Finally, the plot also yields the very important result that the degree of
compression achieved does not vary greatly with respect to the size of the hash table.
Since each element of the hash table is four bytes in size, this result is very important
because it indicates that relatively small hash tables may be used to achieve
considerable compression. Ross (1992) states that the hashing algorithm in RDC is
optimized for a hash table of 4096 elements which would require 16384 bytes of
memory. However, Figure 4.4 suggests thatâ€”at least in FEA applicationsâ€”a hash table
as small as 512 elementsâ€”totaling only 2048 bytesâ€”can be used without sacrificing
compression performance.
Plotted in Figure 4.5 are the normalized execution times for the various flatslab
bridge models tested. The execution times plotted are for analysis runs which were
made on a UNIX workstation. Each compressed I/O execution time was normalized
with respect to the execution time required when the analysis was performed using
standard I/O. The plot indicates that the time taken for an analysis is not highly
dependent on the size of the hash table used. The difference in execution times between
cases where the minimum and maximum hash table sizes were used averages around
just 2 or 3 percent. As a result, we may again conclude that small hash tables can be
used without sacrificing a great deal of performance in terms of execution time.
108
The fact that the normalized execution times in the plot are all greater than 1.0
indicates that the use of data compression increased the quantity of time required to
perform the analyses. This is not always the case (see Consolazio 1994) and it will be
shown in the next section that in many cases the use of data compression can
substantially reduce the required execution time. However, as a general rule, when data
compression is used in FEA software coded in C, there can be a modest penalty in the
form of increased execution time.
The increase in execution time arises because, when using compressed I/O,
additional computational work must be performed to compress and decompress the data
during the analysis. Under favorable conditions, the added amount of time required for
i
H
Z
1.25
1.20
1.40
1.35
1.30
1.15
1.10
1.05
1.00
8000 10000
Buffer Size (bytes)
12000
14000 16000
Zero Skew / 512 Hash / Workstation (UNIX)
Zero Skew / 2048 Hash / Workstation (UNIX)
Zero Skew / 4096 Hash / Workstation (UNIX)
Variable Skew / 512 Hash / Workstation (UNIX)
Variable Skew / 2048 Hash / Workstation (UNIX)
Variable Skew / 4096 Hash / Workstation (UNIX)
Figure 4.5 SEAS Execution Time Results for Workstation Analyses
109
compression and decompression of the data is fully compensated by the fact that a
reduced quantity of I/O must be performed. Whether or not this compensation is
complete or partial depends on the programming language and computer platform being
used. It has been the authorâ€™s experience that in the case of the C language, the binary
I/O functions native to the language are often implemented in a very efficient manner.
The time required for data compression and decompression is only partially
compensated for by the reduced quantity of I/O that must be performedâ€”this results in
increased execution time. However, if the availability of outofcore storage is a
constraining factor in determining whether or not an analysis can be performed, the use
of data compression can eliminate the constraint if a mild penalty in execution time can
be tolerated.
The initial steepness of the plots in Figures 4.4 and 4.5 also reveals the
important fact that the maximum degree of compression that can be achieved is quickly
approached for fairly small buffer sizes. The use of large buffer sizes produces only
marginal increases in the performance of the compression library and is not justified on
systems where memory usage must be minimized.
Using a fixed hash table size of 4096 elements, the flatslab bridge models were
analyzed again to evaluate the effectiveness of data compression on the PC platform.
Both of the flatslab bridges were analyzedâ€”using buffer sizes of 64, 128, 256, 512,
1024, 2048, 4096, 8192, 12288, and 16384 bytes as beforeâ€”on a personal computer
running Microsoft Windows NT as an operating system. An additional set of analyses
were also performed on the PC in which the stress files produced by the analyses were
110
not compressed. The results for the workstation runs and two sets of PC runs are
plotted in Figure 4.6. Note that the each of the curves are normalized with respect to
the platform on which the analysis was run. Thus the curves represent a comparison not
of the relative speeds of the different platforms, but instead a comparison how data
compression affects the performance on each platform.
The plots indicate that under each set of conditions, the use of data compression
increased the required analysis time. The increase in execution time appears to be more
severe for cases in which data compression is used on the PC platform. However, it
will be shown in the next section that this is not a general rule and in many cases just
the opposite behavior occurs.
In performing the compression studies, the observation was made that the stress
files produced by the analyses did not compress as well as the element files. Whereas
Figure 4.6 SEAS Execution Time Results for Workstation and PC Analyses
(All Analyses Run With a Hash Table Size of 4096 Elements)
Ill
the element files contain a great deal of repetition, the stress files contain more or less
random data from the viewpoint of the compression algorithm. To determine whether
the reduced compression of stress files was specific to the RDC algorithm or was
common to data compression algorithms in general, the stress files were independently
compressed using the RDC and LZ77 algorithms.
The public domain program ZIPâ€”a compressed file archiverâ€”uses LZ77 (Ziv
and Lempel 1977) as its compression algorithm and was used to compress the stress
files for comparison with RDC. Results from the compression of the stress files
produced by zero skew flatslab bridge analyses are summarized in Table 4.4. It can be
seen that while LZ77 achieved a greater compression than RDC, it also took
approximately twice as much time to do soâ€”a situation that is unacceptable in FEA
applications.
Since RDC does not produce dramatic outofcore storage savings when
compressing stress files, a series of parametric runs were performed to determine if
significant savings in execution time could be achieved by not compressing the stress
files. The results from these runsâ€”which were performed on the same PC platform
Table 4.4 Comparison of RDC and LZ77 Compression Algorithms for Stress
Files Produced by Zero Skew Flatslab Bridge Analyses
Compression
Algorithm
Uncompressed
File Size
(bytes)
Compressed
File Size
(bytes)
Compression
Ratio
Compression
Time
(seconds)
RDCt
4,066,120
3,346,120
0.823
21.7
LZ77
4,066,120
2,821,704
0.694
40.5
t Buffer size of 16384 bytes and hash table size of 4096 elements.
112
described aboveâ€”are shown in Figure 4.6. Also shown in the figure are the results
from cases in which the stress files were compressed. One can see that there is a
reduction of approximately 10 percent in execution time when the stress files are not
compressed. This reduction in execution time results from the fact that no effort is
being put forth to compress the stress filesâ€”effort which equates to added execution
time. Since RDC does not greatly reduce size of the stress data, it is recommended that
this type of data should not be compressed.
4.8.2 Data Compression in FEA Software Coded in Fortran
To evaluate the effectiveness of data compression in FEA software coded in
Fortran, the author modified a version of the SIMPAL programâ€”written by Dr. Marc
Hoitâ€”to use compressed C I/O instead of the normal Fortran I/O. Recall that SIMPAL
is the Fortran FEA module of the BRUFEM system that was described in earlier
chapters. The program was converted from standard Fortran I/O to compressed I/O by
implementing both the compressed I/O and Fortran interface libraries described earlier
in this chapter.
Since SIMPAL serves as the FEA engine of the BRUFEM system, the test
models used in this portion of the data compression study were full size bridge models
created by the preprocessor described in Chapters 2 and 3. The relevant parameters of
each of the models are listed in Table 4.5. The first bridge is a twospan prestressed
concrete girder bridge having zero skew geometry, pretensioning tendons, postÂ¬
tensioning tendons, and temporary supports. The second bridge is a threespan steel
girder bridge having variable skew geometry and nonprismatic girders. The finite
113
Table 4.5 Parameters of Bridge Models Used in SIMPAL Parametric Studies
Name
Skew
Nodes
DOFs
Truss
Elements
Frame
Elements
Plate
Elements
Load
Cases
Prestressed
Zero
1271
4434
516
972
1176
123
Steel
Variable
963
2829
20
324
880
65
element meshes for both bridges are illustrated in Figure 4.7. The bridges are the same
as those of example problems PRE.5 and STL.5 presented in Hays et al. (1994). They
were chosen because they are reasonably large in size and represent realistic highway
bridge structures.
In the zero skew prestressed bridge model there is a great deal of regularity in
the geometry of the finite element mesh. As a result, there will be a high degree of
largescale repetitionâ€”as well as both medium and smallscale repetitionâ€”in the FEA
data created during the analysis. In contrast, the variable skew geometry of the steel
girder bridge model will prevent the formation of largescale repetition in the FEA
data. However, it will be shown later that medium and smallscale repetition still
abound.
Each of the models were analyzed using SIMPAL in compressed I/O mode with
buffer sizes of 64, 128, 256, 512, 1024, 2048, 4096, 8192, 12288, and 16384 bytes,
and a hash table size of 4096 elements. In addition, the models were also analyzed
using SIMPAL in a mode which uses the standard Fortran I/O functions instead of the
compressed I/O functions.
114
The filesize and executiontime data from these two standard Fortran I/O runs
served as reference data against which the compressed runs were evaluated. For each
compressed run, a normalized compression ratio and normalized execution time was
computed by normalizing the compressed I/O data with respect to data from the pure
Fortran I/O runs. Compression ratio results for the SIMPAL runs are plotted in
Figure 4.8. The SIMPAL models confirm what was already found to be true in the
SEAS runsâ€”a greater degree of compression can be achieved in regular, highly
repetitive models such as the zero skew prestressed bridge than in irregular models like
the variable skew bridge. The compressed I/O library was able to reduce the outof
core storage requirements of the zero skew and variable skew bridge analyses to
Longitudinal Direction
Prestressed Concrete Girder Bridge
T
Temporary Support
Zi
T~
â– Temporary Support
(Lateral
Direction
142'0"
(20 Elements)
'1Â¡ VSâ€™ 142'0
(2 Elements) (20 Elements)
 Longitudinal Direction
Steel Girder Bridge
Figure 4.7 Bridge Models Used in SIMPAL Parametric Studies
115
approximately 4 percent and 9 percent, respectively, of their original sizes. The storage
savings for these models is even more dramatic than the earlier results from the SEAS
runs.
It is interesting to note from the plot that the use of even very small buffer sizes
can produce a significant degrees of compression. Using a buffer size of only 256
bytesâ€”equivalent to the size of just 32 double precision valuesâ€”the outofcore storage
requirements of the prestressed and steel bridge analyses were reduced to approximately
11 percent and 17 percent, respectively, of their original sizes. This is clear evidence
that there is a great deal of smallscale repetition in FEA data since this is the only type
of repetition that can be recognized when using such a small buffer size.
0.40 1 1 1 1 1 1 i i
0.35
Zero Skew Prestressed / 4096 Hadi â€¢â€”
Variable Skew Steel / 4096 Hash
0.30
0.00
0
2000 4000 6000 8000 10000 12000 14000 16000 18000
Buffer Size (bytes)
Figure 4.8 SIMPAL File Compression Results
116
It is important to note that in implementing the compression library in SIMPAL,
it was decided that the stress files should not be compressed. This decision was based
on two factors. First, results from the SEAS parameter studies described the previous
section indicated that FEA stress files do not compress well enough to warrant the
computational effort associated with compressing them. Second, in the BRUFEM
system, the stress files written by SIMPAL must be read by a bridge rating postÂ¬
processor that is coded in Fortran and which utilizes Fortran I/O. By having SIMPAL
write the stress files in Fortran format instead of compressed format, I/O modification
to the postprocessor were avoided.
Thus, in interpreting Figure 4.8, one must keep in mind that the reduction in
outofcore storage indicated in the plots does not reflect the storage required by the
stress files. The plots indicate the degree of compression which was achieved for the
element data flies which were compressed. While these files often constitute the major
portion of storage required by an analysis, stress files can also require substantial
storage as wellâ€”especially in cases involving a large number of load cases.
Plotted in Figure 4.9 are the normalized execution times for the prestressed and
steel bridge models tested. The execution times plotted are for analysis runs performed
on a workstation running the UNIX operating system and on a PC running the DOS
operating system. Since it was of interest to compare the speed of standard Fortran I/O
to that of compressed I/O, the execution times for each of these tests were normalized
with respect execution time of the standard Fortran I/O runs.
117
Figure 4.9 SEAS Execution Time Results for Workstation and PC Analyses
The plots indicate that in all cases, a substantial reduction in execution time was
achieved through the use of data compression. Table 4.6 summarizes the average
executiontime results that were obtained when buffer sizes in excess of 2000 bytes
were used in the data compression algorithm. Based on these results two important
observations may be made. First, observe that by using compressed I/O in Fortran
Table 4.6 Summary of SIMPAL Execution Time Results
Model
Name
Computer
Platform
Normalized
Execution Time'
Savings in
Execution Time
Prestressed
Workstation (UNIX)
0.35
65%
Steel
Workstation (UNIX)
0.49
51%
Prestressed
PC (DOS)
0.30
70%
Steel
PC (DOS)
0.36
64%
t Average normalized execution times when data compression was used and the buffer
size exceeded 2000 bytes.
118
FEA software, not only can the outofcore storage requirements be reduced byâ€”in
many casesâ€”an order of magnitude, but the required analysis time can also be
simultaneously reduced by a substantial fraction.
Next, observe that the savings in execution time produced by the use of
compressed I/O is more pronounced on the PC computer platform than on the
workstation platform. This has important implications given the fact that PCs are
increasingly being used to perform inexpensive desktop FEA.
CHAPTER 5
NEURAL NETWORKS
5.1 Introduction
Artificial neural networks^ are simple computational models which crudely
mimic the operation of biological neural network systems such as the human brain.
Many of the original concepts of neural network (NN) operation were developed to
mimic the brain and to produce artificially intelligent systems. Over time the
researchers in this field have gradually drifted into different corners of the NN field.
There are still many researchersâ€”working primarily in the biological sciencesâ€”who
seek to model the behavior of biological systems as accurately as possible. Others are
less concerned with the relationship to biological systems and more concerned with
whether or not a mathematical NN model can solve a particular engineering problem.
The latter group can be further subdivided into researchers interested in using
NNs to produce artificial intelligence and researchers interested in using NNs simply as
a new type of statistical modeling tool. In each of these cases, the details of the NN
operationâ€”such as architectures and learning rulesâ€”has more to do with mathematical
reasoning than biological mimicry. In the research being reported on herein, NNs are
used primarily as a trainable statistical modeling tool.
f From this point forward, artificial neural networks will be referred to simply as
neural networks with the understanding that the networks being discussed are
computationalâ€”and not biologicalâ€”in nature.
119
120
Readers interested in the history and development of NN theories and
applications are referred to the numerous textbooks and journal articles that have been
written on the subject. In particular, HechtNielsen(1991) presents a thorough treatment
of the NN subject matter including a history of the field, discussion of the various NN
architectures, the current state of the art, and future directions of NN theory and
applications. Smith (1993) approaches the use of NNs from a more narrowâ€”but also
more focusedâ€”point of view in which NNs are treated solely as a tool for the statistical
modeling of data. Finally, Wassermann (1989) provides a good introductory level
textbook discussing the various NN architectures and learning rules.
5.2 Network Architecture and Operation
The architecture of a neural network refers to the layout and connection of the
various elements which collectively make up the network. There are numerous types of
neural network architectures and equally numerous ways of categorizing those
architectures. One way of classifying NNs is based on the path of a signal traveling
through the network. If a signal travels from the input layer to the output layer in a
single forward direction, then the neural network is called a feedforward network. In
contrast, if the output signal from a neuron is not only sent to a subsequent layer but is
also sent to a previous layer, i.e. is fed back into the system, then the system is called a
recurrent network.
A neural network is an organized collection of layers of computational units
called neurons that are connected to adjacent layers through modifiable connection
weights. Figure 5.1 illustrates the basic layout of a simple feedforward neural network.
121
FanOut
Neurons
Direction Of
Signal Travel
Nonlinear
Neurons
Input
Vectors
Output
Vectors
Figure 5.1 Layout of a Feed Forward Neural Network
Input vectors are shown being applied to the input layer of the network. Each element
of each input vector contains a single piece of information that has been encoded in a
manner which is appropriate for network use. Encoding data for neural network
applications will be discussed in greater detail later.
Collectively, all of the elements of an input vector constitute a single input
pattern which is applied to the input layer of the network. The input layer is different
than all of the other layers in the network because it is the only layer which is made up
of fanout neurons. A fanout neuron (see Figure 5.2) serves simply as a signal
distribution point or fanout point. The signal applied to the input side is copied and
sent out on every connection at the output side. Thus it serves simply to distribute a
single input signal to each of the neurons in the next layer.
The output signals from the input layer neurons are sent along weighted
connections to the nonlinear computing neurons in the next layer. As the signals travel
along these connections, they are either inhibited or excited depending on the strength
of the connections. The strengths of the interlayer connections are key to a networkâ€™s
ability to represent the relationship between the input space and the output space. The
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FanOut Neuron
Figure 5.2 Neurons Types Used in Artificial Neural Networks
inputoutput relationship modeled by the network is implicitly stored in the connection
strengths. When constructing a neural network, the connection strengths are the
parameters which are trained to learn the problem to be solved. This will be discussed
in detail in the section on network learning later in this chapter.
Each layer in a networkâ€”except the input layerâ€”is made up of nonlinear
computing neurons of the type shown in Figure 5.2. Input signals sent from the neurons
in the previous layer arrive at the input side of a computing neuron for processing.
Each of these signals was sent by a single neuron in the previous layer, whether that
previous layer was the input layer of the network or a hidden layer of computing
neurons. The signals are multiplied by the appropriate connection strengths and
summed, thus forming a weighted sum to which an additional bias term is also added.
This weighted sum is then passed through a transfer function to generate the final
output signal which will leave the output side of the neuron. Several types of transfer
functionsâ€”which are also called activation functionsâ€”ase available for use in the
construction of neural networks. Some transfer functions act as hard limiting threshold
functions in which the weighted sum must exceed a certain threshold in order for the
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neuron to fire otherwise there is no output signal produced. Others produce binary
output signals which are tied to the value of the weighted sum.
In this research, two types of continuous valued transfer functions, called
sigmoid functions, are used to produce the output signal of computing neurons. Those
sigmoid functions, which have the form
and
g(x)
1
l + e~x
(5.1)
h(x) =
\e
l+e~
(5.2)
serve to modifyâ€”or squashâ€”the weighted sum of the input signals into a confined
range of output values. This squashing is performed in a nonlinear manner as is
illustrated in the plots of Figure 5.3. Sigmoid function g has output range of [0,1] and
is used in applications where the output of the NN will always be positive. If the NN
output values must be able to take on positive or negative values, then the sigmoid
function h is used which has an output range of [1,1].
Once the weighted sum of input signals has been passed through the transfer
function, and the value of the transfer function computed, this value becomes the
output signal of the neuron. If the network consists of only an input layer and an output
layer, then the output signals of the computing neurons are also the output signals of
the overall network. This type of network is commonly referred to as a single layer
network since there is only one layer of computing neurons. In a multiple layer network
there is more than one layer of computing neurons. Every network layer that is made
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up of computing neuronsâ€”except the output layerâ€”is referred to as a hidden layer. The
term hidden layer is used to denote the fact that these layers have no direct connection
to the input or output leads of the network and are therefore hidden from everything
outside of the network. Thus, the NN shown in Figure 5.1 has one input layer (nonÂ¬
computing), two hidden layers (computing), and one output layer (computing).
5.3 Problem Solving Using Neural Networks
Problem solving using neural networks consists of two primary stagesâ€”network
training and network use. During the training stage, the network is taught how to solve
a particular problem of interest. During the network use stage, the network is told to
solve the problem for a particular set of input parameters.
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The training stage can be time very consuming and computationally expensive,
however in many cases training only needs to be performed once. Once the network
has been trained, it may repeatedly called upon to solve numerous problems of the type
it was trained to solve. Also, while network training is often a computationally
expensive process, network use is usually quite cheap. In fact, network use simply
refers to the process of performing a forward pass through the networkâ€”as was
described in the previous sectionâ€”for a specified set of input parameters. The output
parameters predicted by the network are essentially the components of the solution to
the problem (although some postprocessing of the solution data may be necessary).
Therefore the training stage can be looked upon as an investment, the product of
which is an easy to use, computationally efficient tool for solving a particular type of
problem. One may think of the training process as being analogous to the development
and coding of a rule based algorithm in traditional deterministic problem solving. Once
the algorithm has been encoded, it often will never need to be modified again but
instead simply used repeatedly to solve problems of the appropriate type.
5.4 Network Learning
In order to use a neural network to solve a particular engineering problem, the
network must first be trained to learn the problem. The relationship between the input
and output parameters is stored in the connection weights of the network. Therefore, it
is these connection weights that must be trained in order for the network to learn a
particular problem.
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Networks store the relationship between input and output implicitly in
connection weights instead of explicitly as a set of rules as is the case in deterministic
systems. In other words, if a given set of input parameters for a particular problem are
sent to a network, there does not exist an explicit set of deterministic rules which can
be followed to arrive at what should be the output of the network. Instead, the input
signal is sent through the network and processed by an implicit set of rules which are
represented by the various connection strengths in the network. In a feed forward
network, this processing of the input data is performed using the weight summations
and transfer functions described earlier.
Thus, in order to construct a network that can solve a particular engineering
problem, the connection weights of the network must be trained to learn the
relationship between the input and output. Again, this is in contrast to the traditional
deterministic approach in which one formulates an explicit set of rules describing the
solution of the problem and then encodes those rules into an algorithm using a
computer language.
There are two basic approaches used in training networksâ€”supervised training
and unsupervised training. In supervised training, network connection weights are
trained using example training pairs. A training pair consists of an input vector and a
desired output vector. Numerous training pairs are presented to the network and the
connection weights of the network are gradually adjusted so that it will eventually be
able to independently compute the correct output and also to generalize the relationship
represented by the training data.
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In unsupervised training, input data is applied to the network but desired output
data is not. These types of networks are called self organizing networks. However, they
are less widely used and will not be mentioned here again. The interested reader is
referred to the literature for more information on unsupervised training scenarios.
In the present research, supervised network training is used. As an example, in
this research neural networks have been trained to learn the loaddisplacement
relationship for highway bridge structures. This relationship is usually explicitly
encoded in the process of forming the global stiffness and load matrices for the
structure. Instead, in this research, the loaddisplacement relationship is implicitly
encoded in the connection weights of neural networks. Input vectors applied to the
networks contain parameters which describe bridge characteristics, the location and
magnitude of loads, and the location at which displacements are desired. The output
value is the displacement at the specified locations as predicted by the neural network.
To train these networks, example training pairs were generated that consisted of
input parameters and the correct or desired output values, i.e. the correct displacements
at the specified points of interest. Through a supervised training process, the connection
weights of the networks were gradually modified until the displacements predicted by
the networks matched the known displacements to within a small tolerance of error.
It must be emphasized here, though, that the goal of network training is not
simply to construct a network that memorizes a set of training pairs. Indeed, a network
of this type would have very limited usefulness. Instead, the goal is to construct a
network that uses the training pairs to generalize the relationship between the input and
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output parameters. For example, in the present research the loaddisplacement data
used to train the networks consisted of a limited number of discrete choices of load
locations and displacement sampling locations. If the networks could only memorize
these training pairs and look them up during the network use stage, the networks would
not be very useful.
Instead, the networks are trained to learn the generalized relationship between
load and displacement. In this way, if a load or displacement location is specified
during the network use stage that does not correspond to one of the training pairs in the
training data, the network will generalize the relationship and still be able to predict the
correct displacementâ€”or at least a good approximation thereof.
When networks generalize in the manner just described, they are effectively
performing a type of high dimensional interpolation or extrapolation. (Networks may
also be trained to perform essentially as classification engines but this type of network
has not been used in this research.) To ensure that a network can properly generalize
the relationship for which has been trained, several checks and safeguards must be
employed. Strategies such as the use of validation data to avoid network overtraining
(or overfitting) will be discussed later in this chapter.
5.5 The NetSim Neural Network Package
To perform network training and explore the factors involved in robust network
training, the author has written a neural network training and simulation package called
NetSim as part of this research. The NetSim package is written in the C programming
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language and therefore runs on most computer architectures. NetSim is set up in a
general manner so that the user can construct neural networks to solve virtually any
type of problem for which sufficient training data is available. The user can freely
specify the topology of the network, training data, convergence tolerances, and
numerous options related to the actual process of training the connection weights in the
network.
NetSim can be used to construct, train, test, and implement neural networks for
any situation described by the user. Since network training is an iterative process, it is
important to be able to watch the progress of the training process. Therefore, numerous
error statistics are displayed during the training process so that the progression of
training can be easily monitored. NetSim can also process training validation data to
ensure that the network is capable of generalizing the relationship being learned and is
not simply memorizing the training data.
Once a network has been completely trained, it may also be tested using the
NetSim package. If the testing process indicates an acceptable network, NetSim can be
used to automatically generate C code modules that will emulate the trained network.
The code thus generated will properly account for the trained connection weights,
biases, transfer functions, and scaling of input and output parameters. Also, the code
forms a self contained module that can be easily called from any C or Fortran program.
Since the most difficult part of constructing a neural network is usually the
training stage, this is the task for which NetSim has been primarily designed. A hybrid
version of the backpropagation training algorithmâ€”discussed in the next sectionâ€”has
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been implemented in NetSim. The use of the hybrid backpropagation algorithm
typically results in accelerated convergence of the training process which in turn leads
to more robust networks. In this context the term robust is used to mean that the
network generalizes the desired inputoutput relationship well. NetSim also allows the
user to control virtually every aspect of the training algorithm so that training can be
customized to the training data set being learned.
5.6 Supervised Training Techniques
In supervised training the goal is to arrive at a set of network connection
weights which accurately embody the inputoutput relationship represented by the
training data. In order to determine if a particular set of connection weights accurately
models the training data, we will develop a scalar measure of â€œgoodnessâ€ or
â€œbadnessâ€. This scalar measure indicates the degree of error that is present in the
networkâ€™s representation of the data. A convenient measure of error for use in neural
network training is
(5.3)
N . N
1Y train 1 v out
where NtraÂ¡â€ž is the number of training data pairs, Nout is the number of neurons in
the output layer of the network, Ok is the output value of the k'h neuron in the output
layer of the network, and Tk is the target value (training data) of the k,h neuron. Thus,
just as in regression analysis, we choose here to use a mean squared error statistic to
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measure the goodness or badness of the networkâ€™s fit to the training data. A small value
of E represents a network that accurately matches the training data while a large value
of E indicates a network that differs significantly from the training data.
While a small value of Â£ is a necessary condition for a network to accurately
embody a problem, it is not a sufficient condition. This fact is readily apparent.
Consider a case in which the network has been trained to such an extent that it has
essentially memorized the problem. That is, the network has done a very good job of
learning the data points in the training set. In such a case, the value of E would
presumably be very small. However, the more general inputoutput relationship
represented by the discrete data points of the training set may not still not be modeled
well by the network. The training process may have passed the point at which the
network was still able to generalize the inputoutput relationship. Since in the present
researchâ€”and indeed in most network applicationsâ€”we are generally interested in
developing networks that generalize well, we must also develop a measure of how well
the network generalizes. This may be accomplished by partitioning the available
problem data into two groupsâ€”a training group and a validation group.
As an example, assume that for a particular problem there are 1000 inputoutput
pairs available for use in training. Then we might partition this group into a training set
of 800 pairs and a validation set of 200 pairs. The 200 validation pairs should be
chosen randomly from the overall set of 1000 pairs. Then we define a validation data
error statisticâ€”similar to the one defined earlier for the training dataâ€”as
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Nvalid \Nout
\ Z
Â¿ (=1
Z (okTky
[ *=i
^ valid N0ut
(5.4)
where NvaÂ¡id is the number of validation data pairs, Noul is the number of neurons in
the output layer of the network, OÂ¿ is the output value of the k,h neuron in the output
layer of the network, and is the target value (validation data) of the klh neuron.
During network training, we use the data pairs in the validation group to evaluate how
well the network is learning the relationship represented by the data in the training set.
Howeverâ€”and this is a crucial pointâ€”the training process is never allowed to use the
information in the validation data set to modify the network connection weights. The
validation data is used only to evaluate the training process, but never to guide the
process.
If both E and V are plotted as a function of the progression of training (see
Figure 5.4), one will find that there is a point at which V minimizes and then begins to
grow. The point at which V minimizes represents the ideal stopping point for training.
At the minimum point, the network is able to generalize the inputoutput relationship as
well as it ever will for that particular training run. If training continues, the network
will begin to memorize the data in the training set and lose its ability to generalize.
Thus, the validation data error V will begin to rise even though the training data error E
will continue to decrease. This phenomenon is similar to overfitting data using high
order polynomialsâ€”the fitting function begins to fit the data points rather than the
overall function represented by the data points.
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Figure 5.4 Network Error Statistics During Training
One final note regarding the use of validation data should be made. If the error
statistics E and V are plotted together and the value of V never increases, this also
indicates an important training problem. If the value of V does not reach a minimum
and then begin to rise, the network does not have sufficient complexity (number of
neurons, number of hidden layers) to overfit. This situation is undesirable because it
indicates that better a representation of the problem could be achieved by adding
additional network complexity. Essentially, there are insufficient degrees of freedom in
the system to accurately model the problem.
5.7 Gradient Descent and Stochastic Training Techniques
Training a neural network essentially involves solving a nonlinear unconstrained
optimization problem. The error function E is the objective function to be minimized
and the connection weights are the design variables that may be varied to minimize E.
The problem is a nonlinear one because each of the computing neurons present in the
network contains a nonlinear transfer function that is used to process network signals.
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To minimize the network error E with respect to the connections weights, two basic
optimization strategies are often usedâ€”gradient descent methods and stochastic
methods.
In discussing neural network training algorithms, it is useful to visualize the
network error Â£ as a high dimensional surface. The â€œheightâ€ of the surface corresponds
to the value of the error function E produced by choosing a particular set of connection
weights. For example, consider a very small network having only two connection
weights. We could compute the error E produced by choosing various sets of the
weights W( and W2 and then plot the errors as a function of the weights. Such an error
surface might look like the surface in Figure 5.5. The goal of network training is to
locate point of minimum elevation on that surface, i.e. determine the values of the
connection weights that minimize E.
Gradient descent methods attempt to find the minimum point by following the
slope of the error surface in a downward direction, i.e. in a direction of decreasing E.
There are many types of gradient descent optimization, each having advantages and
disadvantages. The simplest gradient descent method is called the method of steepest
descent. This process begins by picking a set of weights which constitutes a starting
point on the surface. The function E and its derivative (gradient) are then computed at
the starting point. Based on the computed gradient, a direction of steepest descent is
determined.
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Figure 5.5 Hypothetical Error Surface for a Network Having Two Weights
Then, from the starting point, a vector is constructed pointing in the direction of
the steepest descent, i.e. the direction in which the error surface drops off in elevation
most quickly. The next point (set of connection weights) to be examined will lie
somewhere along that vector. A stepsize parameter controls how far along the vector
the next point chosen will be. Once the move to the next point has been madeâ€”i.e. the
new values of the design variables have been computedâ€”the whole process repeats.
Thus, the minimum point isâ€”hopefullyâ€”located by iteratively moving down the error
surface in finite size steps.
While the method of steepest descent has the appeal of being conceptually
simple, it suffers from many problems when applied to practical network training
situations. Therefore, several variations on the approach have been developed which
attempt to overcome the problems associated with steepest descent. While still using
gradient information, these hybrid methods also bring in additional information and
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mechanisms which are used in traversing the error surface. Most of the variations
attempt to accelerate the process of minimizing E, however some address issues such as
avoiding local minima and plateaus in the error surface. A few of these methods, which
have been implemented in the NetSim package, will be discussed later in this chapter.
Whereas gradient descent methods use gradient information to search the error
surface for a minimum, stochastic methods use probability to guide the search. Given a
particular point on the error surface, each connection weight in the network is given a
random perturbation in some direction after which the error function E is reevaluated.
If the error function decreases as a result of the random motion, then the motion is
accepted. If the error function increases as a result of the motion, then there is still a
small probability that the motion will be accepted. A pseudo temperature parameter
controls the likelihood (probability) that movement in a direction of an increasing E
will be accepted. During the training (optimization) process, the pseudo temperature
parameter is gradually reduced according to an annealing schedule so that the
probability that bad movements are accepted is gradually reduced zero.
Stochastic methods have the advantage that they virtually always find the global
minimum of E without getting trapped in local minimaâ€”a situation which can occur
with gradient descent techniques. However, stochastic methods also tend to be much
slower than gradient descent techniques. Also, an appropriate annealing schedule must
be developed in order for stochastic methods to be effective.
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5.8 Backpronaeation Neural Network Training
The backpropagation network training procedure is currently the most popular
training procedure used in neural network applications. Backpropagation was the first
network learning law developed that could train multiple layer neural networks in a
systematic manner. In its simplest form, backpropagation is a steepest descent
optimization method with an additional provision for propagating error information
backward through the network. In training a network, recall that the error E is the error
over all of the training pairs in the training set. Therefore, the errors computed at the
output layer of the network must be summed over all of the training pairs in order to
correctly form the error E.
When each of the example pairs in the training data set has been presented to
the network exactly once, an epoch of training is said to have occurred. The amount of
effort required to train a network is usually measured by the number of epochs that
were required for convergence. In simple backpropagation, the network errors are
accumulated over the duration of one epoch of training. Then, at the end of each
epoch, the connection weights at the output layer of the network are modified using a
steepest descent optimization approach. This is possible because the partial derivatives
of the error E with respect to each of these output layer connection weights are easily
computed.
However, modifying the connection weights in hidden layers is not as simple
because the partial derivatives of E with respect to these weights are not as easily
computed. In fact, the problem of how hidden layer connection weights should be
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modified in multiple layer neural network was not solved for a number of years. The
development of the backpropagation algorithm by Rumelhart et. al (1986) solved this
problem by using an backward error propagation procedure based on the chain rule of
derivatives. In backpropagation, the error data computed at the output layer of the
network is propagated backward through the network and combined with computed
transfer function derivatives. When combined, these two pieces of data allow for the
computation of the partial derivative of E with respect to connection weights in hidden
layers of the network. Once these partial derivatives are computed, a steepest descent
approach may again be used to modify the corresponding connection weights.
For a detailed description of the mathematics behind backpropagation, the
reader is referred to any one of the numerous texts written on the subject. The main
objective here is to outline and discuss the relative merits of the method and its many
variations. It is important to understand that in the pure version of backpropagation just
described, the connection weights in the network are only updated once per epoch. That
is, a complete epoch must be completed before any network learning takes place. This
fact derives from the definition of E. Since E was defined as the error over the entire
training set, for backpropagation to be a true steepest descent approach, it can only be
applied to the errors that are accumulated over an entire epoch.
When there are a large number of training pairs in the training setâ€”as is the
case in the present researchâ€”pure backpropagation can be a very slow process. Some
of the variant backpropagation methods employed in the NetSim software package are
described in the following sections. While these variant methods are not true steepest
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descent optimization methods, they are still gradient descent methods and generally
result in reduced training times and more robust training when used appropriately.
Understanding that backpropagation is a steepest descent approach to locating
the minimum on the error surface E, it should also be clear that this is an iterative
process. The process begins by choosingâ€”usually randomlyâ€”a set of connection
weights for the network. This choice constitutes a starting point on the multiple
dimensional error surface. At this point the steepest gradient of the error surface is
determined and the connection weights are modified in that direction. This connection
weight modification represents a jump from the starting point to a new point on the
error surface. In an ideal situation this new point would be the minimum of the error
surface, however this is never the case in practice. Therefore, the process of computing
the steepest descent directions and jumping in those directions must be repeated
iteratively.
Gradually, this iterative process will move to a minimum point on the error
surface. A minimum point is simply a location of zero gradient. Thus the minimum
point located may not be the global minimum of the error surface at all. Instead it may
be a plateau (flat spot) in the surface or a local minimum. Whereas pure
backpropagation will essentially stall at such anomalies in the error surface, some of the
variants described in the following sections can cope with these conditions. However all
of the backpropagation methods share one factâ€”they move down the error surface in
finite size jumps.
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In pure backpropagation, the size of each jump on the error surface is controlled
by a stepsize parameter that is called the learning rate. A single learning rate is used
to compute the size of the jump in each of the dimensions of the problem. Recall that
each of the connection weights in the network represents a dimension of the space that
we are conceptually calling a surface. For each network training situation, there exists
an optimal learning rate that will result in the quickest convergence of the training
process.
Unfortunately, the optimal learning rate is dependent on the parameters of both
the network and the training data and is therefore unique to each problem and cannot be
predetermined. If the learning rate is chosen to be too small, network training can
progress painfully slowly and an acceptably low level of error may never be attained.
On the other hand, if the choice of learning rate is too large, the training process can
become unstable in which case the connection weights may oscillate badly between
epochs or simply diverge.
While simple heuristics for choosing the learning rate are available, none are
especially reliable. Therefore the process of choosing the learning rate is usually an
iterative one in which several different learning rates are tried and the one which
produces the best results is selected. Again, some of the backpropagation variants
described in the next section address this problem. Specifically, the method of adaptive
learning rates works quite well.
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5.8.1 ExampleBvExample Training and Batching
In pure backpropagation, the connection weights in the network are only
updated once per epoch. This is because the error E is defined as the error over the
entire training set. According to this definition, the gradient of the error surface cannot
be determined until the errors over all of the training pairs have been summed. When
there are a large number of training pairsâ€”as is often the case in practical applications
of neural networksâ€”this scheme of only updating the connection weights once per
epoch can result in very slow network learning.
One method of accelerating the learning process is to use examplebyexample
training. In examplebyexample training, the connection weights in the network are
updated after each presentation of a training pair. For example, a training pair (an input
vector and the corresponding output vector) are presented to the network. The network
computes an output vector, compares it to the desired output vector in the training pair,
and computes an error vector. The mean squared error of this error vector is then
computed and is used as a pseudo E. After computing the partial derivatives of this
pseudo E with respect to the weights, a pseudo steepest descent direction is determined
and the connection weights in the network are immediately updated in that direction.
One advantage of examplebyexample training is that the network appears to
learn more rapidly than in pure backpropagation because it is allowed to update the
connection weights very frequently. On the other hand, examplebyexample training
follows the gradient of a pseudo E not the gradient of the true E. As a result, although
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the network may appear to learn more rapidly in the early stages of training, it may
slow significantly during the later stages because it is using incomplete gradient data.
A procedure called batching attempts to combine the benefits of exampieby
example training with the robustness of the pure backpropagation procedure. In
batching, a mean squared pseudo error E is computed not for a single training pair, as
was the case in examplebyexample training, but for a batch of training pairs. A
parameter called the batch size determines how many training pairs are processed
before the connection weights are updated. If the batch size is chosen to be one, then
this procedure reverts to examplebyexample training. On the other hand, if the batch
size is chosen to be equal to the number of training pairs in the training setâ€”a choice
referred to as full batchingâ€”then the procedure reverts to pure backpropagation. In
practice, a batch size somewhere between these two extremes is usually chosen.
Batching shares the rapid initial learning characteristics of examplebyexample
learning but is also somewhat more robust than the examplebyexample procedure.
However, the gradient followed is still only an approximation of the gradient of the
true E and can therefore lead the learning process in incorrect directions, thereby
slowing the training process.
In the NetSim package, the user may specify the batch size to be used in
updating the connection weights. By default, full batching is performed which
corresponds to the pure backpropagation procedure, however the user may explore
other possibilities by specifying smaller batch sizes.
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5.8.2 Momentum
In examplebyexample training, the network connection weights are modified
using a gradient that is computed based on a single training pair. Batching, on the other
hand, uses a gradient that is computed based on a number of training pairs. A gradient
computed using the batching scenario can be thought of as being an average gradient
over all the training pairs in the batch. Thus, the batching procedure essentially makes
use of averaged derivative data to stabilize the oscillations which occur frequently in
examplebyexample training.
Momentum is a different approach to solving the same problem that batching
attempts to solve. In momentum, we average the connection weight changes instead of
averaging the computed derivatives. The steepest descent direction is determined and
the weight changes which would normally be made are computed. However, instead of
using these changes directly, they are averaged with the last set of changes that were
made to the weights. Then, the averaged set of weight changes are used to actually
update the connection weights. Note that the previous weight changes made were
themselves averaged changes, therefore the effect of previous gradient data has an
influence on all subsequent weight updates made. An exponential averaging of the form
^averaged + (5.5)
is used, where Aw, is the computed weight change for the current epoch (or batch),
Aw,_ is the weight change which occurred during the previous epoch (or batch), and
Awfveraged is the exponentially averaged weight change which will actually be used to
update the connection weight. The parameter p, which must be in the range
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0.0 < (a. < LO , controls the degree of influence that previous weight changes have on the
current change. If a value of p = 0 is used, then previous changes have no influence
on the current change and momentum is disabled. As the value of p is increased,
previous changes have more and more influence on the current change.
One of the advantages of using momentum is that it accelerates the descent into
ravines in the error surface. Narrow ravinesâ€”which appear to be fairly common in
neural network error surfacesâ€”have steep sides and a relatively flat bottom. When the
steepest descent directions are computed, they generally point toward the bottom of the
ravine, and are perpendicular to the direction of the actual minimum in the ravine.
Thus, a steepest descent procedure will oscillate back and forth between the two sides
of the ravine and waste a large amount of computational effort before finally settling in
the bottom.
When momentum is applied to this situation, the bottom of the ravine is quickly
reached with very little wasted oscillation. To understand why this is the case, consider
the name of the methodâ€”momentum. The method is called momentum because the
effect of the exponential averaging is very much like the effect of momentum on a
moving body. If, over several epochs of training, the changes to a particular network
connection weight are consistently of the same sign, then the weight is moving in a
particular direction and the use of the exponential averaging tends to give the motion in
that direction momentum.
If, however, over several epochs of training the changes to a connection weight
toggle back and forth between positive and negative, then the training process is
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oscillating. If exponential averaging (momentum) is applied to this situation, the size of
the weight changes will quickly decreaseâ€”due to the sign reversalsâ€”and the network
will settle in the bottom of the ravine with little oscillation. From that point it may then
start to build momentum moving along the ravine bottom in a direction perpendicular
to the sides. This process is illustrated graphically in Figure 5.6. For a more detailed
description of the phenomenon, see Smith (1993).
Another primary benefit of using momentum is that it allows the training
process to escape shallow local minima in the error surface. As a weight moves
consistently in one direction, it builds momentum in that direction. If a shallow
minimum is encountered, the momentum of the connection weight can carry it through
the shallow spot and on to a lowerâ€”and hopefullyâ€”global minimum. A pure steepest
descent approach will become trapped in such a situation because the gradient will point
toward the bottom of the local minima.
Therefore, using a gradient descent approach together with momentum can be a
very effective solution to the neural network training problem. NetSim allows the user
Training Without Momentum
Training With Momentum
Figure 5.6 Using Momentum to Dampen Training Oscillations
146
to control whether or not momentum is used during the training process and if it is, to
specify the momentum factor p.
5.8.3 Adaptive Learning Rates
The final variant of backpropagation that will be discussed here is the method of
adaptive learning rates. Originally developed by Jacobs (1988) under the name the
deltabardelta method, this method is discussed in some detail in Smith (1993). Recall
from earlier discussion that in network training the learning rate is a stepsize parameter
which controls the size of each jump taken when moving along the error surface. It was
stated earlier that the learning rate must be chosen very carefully, otherwise the training
process may never converge. Also, determining the optimal learning rate for a network
is a difficult, iterative process.
In the method of adaptive learning rates, the problems associated with correctly
choosing a network learning rate are eliminated. In addition, network training using
this method is generally much faster than pure backpropagation. (An order of
magnitude reduction in training time is not an uncommon situation). In this method,
each connection weight in the network is assigned its own learning rate instead of
having a single global learning rate. Initially all of the learning rates are set to some
specified value, however, unlike pure backpropagation the choice of this initial value is
not critical. As the training process progresses, each learning rate is adapted according
to the history of the training for its associated connection weight.
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Adapting the learning rates for each connection weight individually instead of
using a single global learning rate has a number of significant advantages.
1. Initial choice of learning rate is not critical
2. Minimizes oscillation during training
3. Rapid convergence
It should be clear that because the learning rates are adapted during the training
process, the initial choice has little effect on the overall training process. Any
reasonable value can be used to start the process. To understand why this method
minimizes oscillation and accelerates convergence, we must examine the actual
adaptation procedure.
In one way, the adaptation procedure is conceptually similar to the idea behind
momentum. Consider a single connection weight in the network. If over several
epochs, the changes which must be applied to this connection weightâ€”to reduce the
value of Eâ€”toggle back and forth between positive and negative, then the training
process is oscillating and consequently wasting computational effort. Such oscillation
can be damped by reducing the size of the learning rate associated with the weight (see
Figure 5.7). As the learning rate is reduced, the size of the jumps made on the error
surface are reduced, and the training process will quickly settle into crevasses in the
error surface. Thus, in the scenario of oscillation, the adaptive learning rates method
achieves much the same goal as momentum.
148
Now consider the opposite case in which changes of the same sign are
consistently made to a particular connection weight in the network. In such a situation,
the weight is consistently being moved in the same direction to reduce the overall
network error E. Thus, it seems reasonable to assume that this pattern of behavior
might continue. One way then of accelerating the training process (see Figure 5.7) is to
increase the size of the jumps made on the error surfaceâ€”i.e. increase the learning rate
for the connection weight. Since the size of each jump made is proportional to the
learning rate, increasing the learning rate will increase the jump size. As the jumps
grow in size, the training process will accelerate in the dimension associated with the
connection weight.
The use of increased learning rates can be critically important when the training
process must traverse an area of nearzero slope on the error surface. Recall that the
size of the updates applied to a connection weight are proportional not only to the
learning rate, but are also proportional to the partial derivative of E with respect to the
weight. Therefore, an area of nearzero slope can cause weight updates to become very
small, essentially stalling training along the dimension associated with the weight.
Damping Oscillation By Reducir^
The Learning Rate
Accelerating Training By Increasing
The Learning Rate
Coping With NearZero Slope By
Increasing The Learning Rate
Figure 5.7 The Effects of Using Adaptive Learning Rates
149
When the method of adaptive learning rates is applied to this situation (see Figure 5.7),
the learning rate is allowed to grow progressively larger as long as the slope is not
identically zero. Thus, after several epochs of training, the step size can grow large
enough to finally make some progress in moving off the plateau.
In implementing the method of adaptive learning rates, an exponential averaging
scheme is used to determine which direction each weight has been moving recentlyâ€”
where recently means the past few epochs. A control parameter, directly analogous to
the p parameter in the momentum method, is used to control how much influence the
previous training history has on the current calculations. Next, the direction in which
the weight must be moved to reduce E is computed and compared to the direction the
weight has been moving recently. If these two directions have the different signs, then
the learning rate for this weight is reduced by multiplying it by a fractional valued less
than one. If the two directions have the same sign, then the learning rate is increased by
adding a constant value. Thus, reductions in learning rates can occur very quickly, e.g.
to dampen oscillations, while increases in learning rates are attained more gradually.
The method of adaptive learning rates has been implemented in the NetSim
neural network training software. The user can specify the values of the four
parameters which control the learning rate adaptationâ€”(1) the initial learning rate,
(2) the â€œrecentnessâ€ factor, (3) the learning rate reduction factor, and (4) the learning
rate growth factor. In addition, NetSim allows the user to mix the batching method, the
momentum method, and the method of adaptive learning rates together to produce the
most robust training combination for the particular problem being solved. While the
150
bookkeeping required to implement the combination of all three of these methodsâ€”in
addition to pure backpropagationâ€”is complex and will not be discussed here, the use of
the software is straightforward.
On a final note, the backpropagation variants described herein are by far not the
only variants currently available. The variants described here are only the ones which
the author has implemented in the NetSim software. Another important class of variants
are the second order methods, which use not only first order derivatives but also second
order derivatives to guide the training process. The interested reader is referred to the
relevant literature on the subject.
CHAPTER 6
NEURAL NETWORKS FOR HIGHWAY BRIDGE ANALYSIS
6.1 Introduction
In the present research, the use of neural networks in the area of highway bridge
analysis has been studied. This chapter will discuss the construction of a group of
neural networks that are able to approximately encode the loaddisplacement
relationship for twospan flatslab bridges.^ Given a set of loads, the networks can be
used to compute displacements that would result from those loads. Chapter 7 will then
discuss the installation of the networks into an iterative equation solver, the product of
which is a hybrid solver customized specifically for highway bridge analysis.
6.2 Encoding Structural Behavior
In traditional FEA, the loaddisplacement relationship for a structure is encoded
explicitly within the global stiffness matrix. The relationship is said to be encoded
explicitly because there is an explicit, mathematically based set of rulesâ€”related to the
behavior of structuresâ€”which are followed to form the global stiffness matrix. Once
^ It was the goal of this research to develop neural network analysis techniques
specifically for twospan flatslab bridges but which could subsequently be extended
to encompass other bridge types.
151
152
Explicit Encoding of LoadDisplacement Relationship
Implicit Encoding of LoadDisplacement Relationship
Figure 6.1 Encoding the LoadDisplacement Relationship
formed, the global stiffness matrix may be used in conjunction with a global load
vector to solve for the displacements in the structure.
Just as a global stiffness matrix can be used to explicitly encode the load
displacement relationship, neural networks can be used to implicitly encode the same
relationship. These encoding styles are illustrated in Figure 6.1. The neural network
representation of the loaddisplacement relationship is said to be implicit because the
relationship is encoded through a network training process that is unrelated to the
structural behavior of bridgesâ€”except that the training data was generated by either
analytical or experimental tests of bridges.
Therefore, the fundamental difference between an explicit and an implicit
encoding is the process by which the encoding is generated. In each of these methods,
the end results is a set of numeric values that are used to compute displacements within
a structure. Flowever, it is the process by which those numeric values were generated
that is different between explicit and implicit encodings. In the explicit encoding, a set
of â€œrulesâ€ that relate to structural behavior (e.g. constitutive laws, solid mechanics) are
used to generate the numeric values in the stiffness matrix. Later, these numeric values
153
may be used in a separate process, e.g. a matrix equation solve, to produce a set of
structural displacements.
In contrast, although a neural network can also encode the loaddisplacement
relationship for a bridge structures, none of the steps employed in building that
encoding are directly related to the behavior of structures. Instead, a set of training data
is generated and used to train the network. It is the training data and the process used to
generate that training data that are related to the behavior bridge structuresâ€”not the
network training process itself. The only reason the network learns the load
displacement relationship for a bridge is because it is this relationship that is
represented by the training data.
In the present research, the network training data was generated using analytical
FEA results from bridge analyses. However, this does not have to be the case. For
example, one might instead choose to apply loads to a structure and experimentally
measure the resulting displacements. In this manner, a set of training data could be
generated using only the experimental loaddisplacement data. Using this data, a
network could again be trained to implicitly encode the relationship without any need
for a structural behavior â€œrule baseâ€ from which to work.
6.3 Separation of Shane and Magnitude
In constructing neural networks for bridge analysis, it is essential to create
networks that are capable of handling arbitrary loading conditions. Bridge loads may
vary in magnitude, location, type (force, moment), and source (point load, uniform
154
pressure, selfweight). The networks must be constructed considering all of these
possibilities, otherwise they will have only limited applicability.
In the bridges studied in this research only gravity loads are considered,
therefore externally applied vertical forces are present but externally applied moments
are not. Moment loading still needs to be considered, however, if one deals with
iterative processes involving outofbalance (residual) forces. Such forces arise when
using iterative equation solving algorithms and also when dealing with nonlinear
analysis. Since the neural networks presented in this chapter were created with the
intention of installing them in an iterative equation solver (see Chapter 7), moment
loading had to be considered.
In the flatslab bridge models studied, plate bending elements were used to
model the slab. Therefore, at each slab node of the model, there were three active
degrees of freedom (DOF)â€”one outofplane translation (Tz) and two inplane rotations
(Rx, Ry). The coordinate system used in the bridge models is such that the Xdirection
is the lateral direction, the Ydirection is the longitudinal direction (the direction of
vehicular travel), and the Zdirection is the transverse direction (perpendicular to the
slab and positive as one moves vertically upward). Thus, three types of loads can occur
and must be consideredâ€”vertical forces (Fz), moments about the xaxis (Mx), and
moments about the yaxis (My). Gravity loads will virtually always be represented as
vertical forces (Fz), but residual forces may cause any of the three load types (Fz, Mx,
or My).
155
In the present study, the ability to handle and an arbitrary number of loads of
arbitrary magnitude was accomplished using superposition and separation.
Superposition was used to handle the variable number of loads. Each loading condition
is broken down into as many individual loads as are necessary to represent the overall
loading. The displacements due to each individual component are then computed and
accumulated with all of the other loads to form the total set of displacements for the
structure. This is a straight forward, standard technique of structural analysis.
Proper handling of variable magnitude loads was achieved by separating
displacement shape data from displacement magnitude data. The concept behind this
technique stems from the assumption of linear structural behavior. Place a single load
on a structureâ€”which is assumed to be linearâ€”and compute the displacements. Now
double the load and compute the displacements. The second set of displacements will
simply be the first set scaled by a factor of 2.0â€”assuming linear behavior. This is
illustrated for a simple propped cantilever beam in Figure 6.2
There is really only one characteristic displacement shape for the structure for
each particular loading condition. We will term this characteristic shape the
â€œnormalized shape" of the structure and define it to be the displaced shape (set of
displacements) of the structure normalized with respect to the largest magnitude
Load â€” P, Max. Displacement = A Load = 2P. Max. Displacement = 2A Normalized (Characteristic) Shape
Max. Displacement = 1
Figure 6.2 Linearly Proportional Displacements
156
displacement occurring under a particular loading condition. Therefore, a different
normalized shape exists for each loading condition. The ordinates of the normalized
shape will then lie in the range [1,1], which is particularly suitable for implementation
using neural networks.
Thus, we can separate the task of computing structural displacements using
neural networks into two distinct tasksâ€”computing normalized shape values and
computing magnitude scaling values. Shape networks are used to accomplish the first
task while scaling networks are used to accomplish the second task. When an actual
displacement must be computed, a shape network is invoked to compute a normalized
shape ordinate, a magnitude network is invoked to compute a scaling factor, and the
two values are multiplied together.
In the present neural network implementation, separate networks have been
constructed for each combination of load (Fz, Mx, My), displacement (Tz, Rx, Ry),
and task (shape, magnitude). Thus, there are a total of 18 (3x3x2) neural networks
which, when operating collectively as a single unit, can compute true structural
displacements in flatslab bridges (see Figure 6.3).
In subdividing the overall task into several subtasks and assigning a separate
neural network to each subtask, the goal was to minimize the size of the neural
networks. Also, previous experience with training neural networks had suggested that
partitioning the overall task into smaller pieces would increase the likelihood of success
during the training phase.
157
Loads
r: Fz = Force Along ZAxis
r:Mx  Moment About XAxis
r:My = Moment About YAxis
Displacements
q:Tz = Translation Along ZAxis
<7:Rx = Rotation About XAxis
<7:Ry = Rotation About YAxis
Figure 6.3 Organization of SubTask Neural Networks
6.3.1 Generating Network Training Data
To illustrate the process of separating shape data from magnitude (scaling)
dataâ€”and subsequently training networks using that dataâ€”consider the propped
cantilever illustrated in Figure 6.4. For simplicity, only three points (â€œaâ€, â€œbâ€, and
â€œc") on the beam will be used to generate shape and scaling data. In practice, many
more points would be used so that a large quantity of data for neural network training
would be available. The separation procedure is outlined below.
1. Apply loads. Apply unit loads (forces, moments) to the structure at the
loading points.
2. Compute displacements. For each applied unit load, compute the
displacements that arise in the structure. Displacements are computed at the
selected displacementsampling points.
158
3. Record the maximum magnitude displacements. For the each applied unit
load, determine the maximum magnitude displacement and record it for later
use in normalization and scaling.
4. Normalize the displacements. Normalize the displacements with respect to
the recorded maximum magnitude displacements. For each load case, the
displacements are normalized using the maximum magnitude value that
occurs in that load case. After normalization, the ordinates of each
displacement shape will lie in the range [1,1].
Several simplifications have been made in the propped cantilever beam example to keep
the discussion clear. First, only vertical loads (forces) and vertical displacements
(translations) have been considered. In practice, moment loads and rotational
displacements would also need to be considered. Next, the loading points and
displacementsampling points have been chosen to be at the same locations (â€œaâ€, â€œbâ€,
Aâ€œ ax = Maximum Magnitude Displacement AÂ«P _ At "cf Due To A Unit Load Applied
Caused By Unit Load Applied At "a" nt>rm At "P" And Normalized With Respect To
1
L
Normalized Shape Data That Shape
Networks Must Encode
o Neural Network Training Data Point
Figure 6.4 Separation of Shape and Scaling (Magnitude) Data
159
â€œcâ€). This does not have to be the case and is in fact not the case in the flatslab
bridges studied in this research. Finally, note that in this example, the shape ordinates
will lie in the range [0,1] since there are no negative displacements (assuming positive
is vertical downward). Clearly, this will not always be the case and in genera! the shape
ordinates will lie in the range [1,1].
The steps listed above are illustrated in Figure 6.4. Unit loads are applied at
each of the loading points and the vertical displacements computed. The maximum
magnitude displacements A1^, , and Aâ€˜Â¡nax are determined and used to build
the maximum displacement curve shown. For a given location on the beam, the height
of this curve is equal to the magnitude of the maximum displacement that occurs when
a unit load is applied at that location. Each unit load application generates a single point
on this curve. The set of all such generated points constitutes a training data set for
constructing scaling neural networks. In this simple example, three training pairs are
generated (see Table 6.1). Scaling networks take load locations as input and return
corresponding magnitude scaling factors (maximum magnitude) as output.
The formation of normalized displacement shapes is also illustrated in
Figure 6.4 where three shape are generated by the application of three units loads. In
each of these cases, the displacements are scaled so that the largest magnitude term in
each shape is unity. In the example problem, the displacementsampling locations are at
the same locations as the load points. Therefore, there are a total of nine training pairs
generated for network training (see Table 6.1). Shape networks take a load location and
160
a displacementsampling location as input and return the normalized displacement at the
sampling location specified.
Using the training data described above, neural networks can be trained to
encode the shape and scaling data. With regard to Figure 6.4, this means that the
networks are trained to predict the curves shown in the figure, not just the data points.
To accomplish this successfully, a sufficient number of training pairs must be
generatedâ€”more than the three points used in the example.
6.3.2 Using Trained Shape and Scaling Networks
Once shape and scaling neural networks have been trained for a specific type of
structure, they may be used to compute true displacements. By assuming linear
structural behavior, the principle of superposition can be used to handle an arbitrary
Table 6.1 Network Training Data Generated For Propped Cantilever Beam
Scaling Networks
Shape Networks
(3 Training Pairs)
(9 Training Pairs)
Load
Maximum
Load
Displacement
Normalized
Location
Magnitude
Location
Location
Displacement
(Input)
(Output)
(Input)
(Input)
(Output)
xa
\a
A max
xa
xa
\Â°a
A norm
b
xb
A max
xa
Xb
A norm
Xc
\c
A max
*a
xc
\ca
^ norm
*b
xa
A norm
bb
*b
Xb
A norm
*b
Xc
A norm
. ac
xc
Xa
A norm
xc
Xb
A norm
*c
Xc
\cc
A norm
161
number of loads. Further, by utilizing shape and scaling neural networks, loads of
arbitrary magnitude can be properly handled.
Continuing with the propped cantilever example from the previous section,
Figure 6.5 illustrates the process of computing true displacements using superposition,
shape networks, and scaling networks. Displacements from the two loads, P and Q, are
computed separately for several points along the beam. Then, using the principle of
superposition, they are summed together to form the true, total displaced shape.
To compute the displacements corresponding to each load, shape and scaling
neural networks are used in conjunction with the load magnitudes P and Q. For
illustrative purposes only, we will compute the displacements at the three points (â€œaâ€,
â€œbâ€, â€œcâ€) only and assume that these can be used to crudely represent the displaced
shape of the structure. In realistic applications, many more points would need to be
used.
Figure 6.5 Using Shape and Scaling Networks To Compute True Displacements
162
Consider the load P first. This load occurs precisely at the point â€œaâ€ which was
earlier used to generate network training data. To compute the displacements at â€œaâ€,
â€œbâ€, and â€œcâ€ due to P, we begin by computing the normalized displacements Aâ€œâ€œrm,
A1'â€œarm . an
normalized displacements are correct, their absolute magnitudes are not. If we scale the
normalized displacements by A1^â€”computed using the scaling networkâ€”we obtain
the displacements that would result from a unit load acting instead of a load of
magnitude P acting. Therefore, we must scale once more, by a factor P, to obtain the
total displacements at each of the three points.
Load Q is handled in precisely the same manner as load P. The sole difference
is that, in the case of load Q, the shape and scaling networks are called upon to predict
values that were not part of the training data. Since load Q is located at point â€œqâ€ that
was not a point in the training data, the neural networks must interpolate to obtain the
values of A^^, Aafiorm, Ah^orm, and AcÂ£orm. Networks are said to generalize well if
they can perform this sort of interpolationâ€”or sometimes extrapolationâ€”in a
meaningful, reliable, and accurate manner. In such cases, the networks have been able
to take a discrete set of training data, and generalize to the continuous relationship
represented by the discrete points.
Therefore, the ability to train robust networks that are capable of generalizing
well is very important in neural network applications. It also becomes evident that
monitoring the networkâ€™s ability to generalize should be part of the overall training
163
process. Validation data can be used for this purpose. (See Chapter 5 for a detailed
discussion on the use of validation data).
Although the concepts developed above were illustrated using the simple
propped beam example, they can be directly extended to the case of flatslab bridges.
Shape and magnitude networks are constructed for bridges in the same manner as that
just described. However, instead of using only unit force loads, one must now use unit
force loads (Fz), and unit moment loads (Mx, My). Likewise, instead of computing
only translations, now translations (Tz), and rotations (Rx, Ry) must be computed.
Finally, a load â€œlocationâ€ now means a twodimensional location on a flatslab bridge
instead of a onedimensional location along a beam.
6.4 Generating Analytical Training Data
In order to construct neural networks for flatslab bridge analysis, training data
of the sort described in the previous section had to be generated. In this study,
analytical FEA displacement results were used as the basis for the neural network
training data. SEASâ€”the FEA package developed for this researchâ€”was used to
generate all network training data. Twospan flatslab bridges of varying sizes were
analyzed for a large number of unit load conditions. From the analysis results, the
maximum magnitude displacements were determined and recorded for each load case
and displacement type. Next, the displacements for each displacement type and load
case were normalized with respect to the appropriate maximum magnitude term.
164
Several modifications were made to SEAS to facilitate rapid and accurate
generation of displacement training data. A feature was implemented that causes SEAS
to automatically normalize all displacement data and to report the maximum magnitude
displacements which occurred. Thus, the displacement data reported by SEAS was
already in the form needed for network training.
An additional plate bending elementâ€”the heterosis element (Hughes 1987)â€”was
also added to SEAS and was used in all of the analyses performed. Originally, a
biquadratic (9node) lagrangian element was used to model the flatslab bridges.
However, it was found early on that this element was prone to zero energy modes when
used to model thick flatslabs. Therefore, a heterosis element, which does not suffer
from the same zero energy modesf, was implemented and tested in SEAS. Tests
showed that the heterosis element performed well in situations where zero energy
modes arose when using lagrangian elements.
To generate shape network training data, a grid of load locations and a grid of
displacement sampling locations were chosen. For a given loaddisplacement
combination (e.g. Fz loads causing Tz translations), unit loads were applied at each
point in the load grid. Each of these unit loads generated one load case in the analysis.
The displacements at each of the displacementsampling points were then extracted
from the FEA output. This process was repeated for all nine of the loaddisplacement
combinations that were consideredâ€”FzTz, FzRx, FzRy, MxTz, MxRx, MxRy, MyTz,
The advantages of heterosis plate elements over lagrangian plate elements in thick
plate applications were discussed in detail in Chapter 3.
165
MyRx, MyRy. Recall from previous discussion that these are loaddisplacement
pairingsâ€”that is particular components of displacement caused by particular
components of load. For example, FzTz represents the displacements Tz (translations
in the Zdirection) caused by loads Fz (forces in the Zdirection).
The load and displacementsampling points used to generate shape network
training data are illustrated in Figure 6.6. There are 50 load points and 45
displacementsampling points, the result of which is the generation of 2250=50x45
combinations, and consequently 2250 neural network training pairs. Due to the large
number of training pairs, only the single bridge geometry shown in Figure 6.6 was
considered for the shape networks in this study. While this choice limits the flexibility
of the networks, it was felt that if the technique could be shown to be successful for
this limited case, it could then be expanded to encompass more general bridge
geometries.
The load and displacementsampling points used to generate scaling network
training data are illustrated in Figure 6.7. The 231 points served both as load points
and displacementsampling points. A denser grid than that used in the shape network
case was used here for two reasons. First, it was found that the grid used in the shape
network was too coarse to locate the true maximum magnitude displacements in the
bridge. That is, the true mÃ¡ximums often occurred at locations other than the
displacementsampling points and were therefore not captured.
166
Figure 6.6 Load and Displacement Sampling Points Used To Generate
Training Data For Normalized Shape Networks
X 50 Load Application Points T T Nine Node Heterosis
â–¡ 45 Displacement Sampling Points I i I Plate Bending Element
167
f
aâ€”Â»
t
IE
t
Â«â€”aâ€”aâ€”
f4
i)
Kl 
ii
GSF*30 ft. (GSF*360 in.)
 Lateral Direction
1â€œ:
H
1!
la
9 2
iÂ£
a
i O
i f
j a
a a
Figure 6.7 Load and Displacement Sampling Points Used To Generate
Training Data For Scaling (Magnitude) Networks
168
Second, each load point in this case generates only a single neural network
training pair. For each load, the displacements at all the grid points are computed and
scanned, but then only the largest magnitude term is retainedâ€”all other values are
discarded. Therefore, the volume of training data that generated for the scaling
networks was considerably less than that of the shape networks. As a result, more
bridge geometries could be treated in this case than was possible for the shape network
case. A base geometry was chosen that matched the geometry used for the shape
networks. Then three additional scaled variations of this base geometry were analyzed.
In all, Geometric Scale Factors (GSFs) of 0.6, 0.8, 1.0, and 1.2 were treated.
Strictly speaking, a shape network trained for a GSF of 1.0 (the only case
considered for shape networks) should not be able to be used with a scaling network
trained using a GSF of, say, 0.8. However, by separating displacement shape data from
magnitude data it was reasoned that the normalized shape might remain roughly
unchanged for various GSFs. By using scaling networks trained on multiple GSFs, the
normalized displacement shape could then be scaled by appropriate scaling factors.
Thus, using four GSFs and 231 grid points, a total of 924=4*231 neural network
training pairs were generated for each loaddisplacement combination (e.g. Fz loads
causing Tz translations).
6.5 Encoding Bridge Coordinates
When using neural networks, all network output parameters must be encoded in
a form that is consistent with the transfer functions being used. Since the output ranges
of the sigmoid transfer functions g and h (see Chapter 5) used in this research are
169
[0,1] and [1,1] respectively, all output parameters must be encoded into one of these
two ranges. While network input parameters do not have to be encoded, there are
several benefits to doing so. Therefore, in the present research, input parameters as
well as output parameters are encoded.
As will be seen in the following sections, most of the input parameters of the
neural networks consist of encoded bridge coordinates. Load locations and
displacementsampling locations are encoded in a manner that is not only suitable, but
preferable for use with neural networks. Figure 6.8 illustrates the normalized bridge
coordinate system which was defined to accomplish this encoding. Normalized
Xcoordinates (lateral positions) were defined simply as varying linearly from a value
of X=0.0 at the left edge of the bridge to a value of X = 1.0 at the right edge.
Encoding the Ycoordinates could have been accomplished in the same
mannerâ€”linearly varying the coordinates from a value of Y=0.0 at the beginning of
the bridge to a value of Y = 1.0 at the end. This, however, would have been a poor
choice of encoding. To understand why, consider a simple example. Assume we have a
60 ft. bridge with span lengths of 40 ft. and 20 ft. for the first and second spans
respectively. Assume further that a load is placed at a location 30 ft. from the
beginning of the bridge. In this case the normalized Ycoordinate of the load would be
Y =0.5=30/60. Using this encoding scheme, assume that a network is trained to learn
the loaddisplacement relationship for the bridge.
170
Now consider another bridge in which the load remains at the same longitudinal
location but the span lengths are reversed. In this case, the normalized coordinate of the
load would still be Y=0.5=30/60. If the data obtained from analyses of these two
bridges were combined to train a single network, the training process would fail.
Training would fail because the two sets of data contradict each other. In one case, a
longitudinal load coordinate of Y=0.5 corresponds to a load in the first span whereas
in the second case, the same coordinate corresponds to a load in the second span.
Clearly, the structural displacements for these two cases will be opposite to each other,
but the network has no mechanism with which to distinguish why they are opposite.
The scenario described above can be remedied by using a piecewise linear
normalization such as the one shown in Figure 6.8. In this encoding scheme the interior
support of the bridge always lies at a normalized coordinate of Y=0.5. The
normalization is then linear within each spanâ€”i.e. piecewise linear over the entire
Lateral XDirection
1.0
1.0
1.0
Y=0.0 Y=0.2 Y=0.5 Y=1.0
Neuron3 Value = 0.0
r**
 Neuron2 Value = 0.4
Neuron1 Value = 0.6
1 Tl.O
JlO
Normalized Bridge Coordinate System
Multiple Neuron Encoding Of YCoordinates
Figure 6.8 Encoding Bridge Coordinates
171
length of the bridge. If a load is applied at a normalized longitudinal coordinate of
Y=0.25, then this always corresponds to a load at midspan of the first span
regardless of the actual lengths of the spans. Neural networks trained using this
encoding scheme would then have a method of distinguishing between the behavior of
bridges loaded in the first span and bridges loaded in the second span.
This encoding scheme can be further improved by using more than one neuron
to represent the Ycoordinate. In the present work, three neurons were used (see
Figure 6.8) to represent each Ycoordinate parameter (e.g. load locations,
displacementsampling locations). The objective of such a scheme is primarily to
accelerate network learning and therefore to reduce the risk of overtraining. The
training process progresses more rapidly in the multiple neuron case because network
connections to the input layer are stratified into three groupsâ€”connections associated
solely with the first span, connections associated with both spans, and connections
associated with the second span.
In the threeneuron encoding, each of the neurons (neuron1, neuron2, and
neuron3) corresponds to one of the groups just described. More importantly, the
connection weights attached to neuron1 will have no effect on the connection weights
attached to neuron3. When neuron1 is nonzero, neuron3 will by definition be
identically zero and therefore will have no influence on the connection weights
associated with neuron1. The reverse case is, of course, also true. Thus when the
network is â€œlearningâ€ a lesson with regard to something happening in the first spanâ€”
for example the application of a unit loadâ€”it will not â€œunlearnâ€ a previously acquired
172
lesson regarding what happens when the same load is applied in the second span.
Neuron2 provides a crossover between the two categorizations which can be useful in
preserving the ability of the network to generalize.
Thus, in this study, a singleneuron linear encoding was used to represent
Xcoordinates and a threeneuron piecewise linear encoding was used to represent
Ycoordinates. The threeneuron encoding was found to be especially important in
creating the normalized shape networks. These networks are very sensitive to being
able to distinguish between effects (loads, displacements) occurring in different spans.
6.6 Shape Neural Networks
Illustrated in Figure 6.9 is the basic layout of the shape neural networks used in
this research. As the figure indicates, there are eight input parameters and a single
output parameter for each of these networks. Recall that nine of these networks were
constructed so as to consider each combination of load type (Fz, Mx, My) and
displacement type (Tz, Rx, Ry). Also, note that there is no loadmagnitude input
parameterâ€”these networks predict normalized shapes only.
The input parameters consist of the location (lateral and longitudinal
coordinates) of the applied load and the location at which the displacement is to be
sampled. These coordinates are encoded as was described in the previous sectionâ€”one
neuron for the Xcoordinate and three neurons for the Ycoordinate.
Up to this point, it has been stated that normalized displacements are always in
the range [1,1]. While this is essentially true (and is the preferable way of discussing
the concepts involved in encoding displacement data) there is one additional scaling that
173
Disp. YCoordinate(
Disp. YCoordinate(
Disp. YCoordinate(
Load YCoordinate(
Load YCoordinatef
Load YCoordinate(:
Disp. XCoonfinate
Load XCoordinate
All Input Parameter
Are Normalized To
The Range [0,1].
Sizes Vary
Hidden Layer
Figure 6.9 Configuration of Normalized Shape Neural Networks
will now be introduced to improve the trainability of networks. In the shape networks,
the sigmoid transfer function h is used so that the networks are capable of predicting
negative as well as positive normalized displacements. By using only a compacted
portion of the neuron output range, specifically the range [1/1.2,1/1.2], the networks
become easier to train and therefore the risk of overtraining is reduced.
While the theoretical output range of the sigmoid function h is [1,1], this
range is only approached asymptotically. If output parameters in the training data take
on values of 1 or 1, the network will never be able to exactly match these values. Of
course, it will be able to approximate these bounding values to within a small tolerance.
However, by simply scaling the output data into a compacted range, the problem is
alleviated.
Therefore, prior to training, the normalized displacement data generated by the
FEAâ€”which was in the range [1,1]â€”was scaled down by a factor 1/1.2 and then
passed to the network. Later when the trained network was used in an application, the
174
computed neuron output values were scaled up by a factor of 1.2 to produce data again
in the range [1,1]. Automatic scaling of network input and output parameters is one of
the features built into the NetSim software (see Chapter 5). As a result, the neural
network code generated automatically by NetSim handles this scaling internally.
Application code calling the neural network modules never needs to be concerned with
the scaling.
Training the normalized shape networks was accomplished using the NetSim
software. The 2250 loaddisplacement pairs described earlier in this chapter were used
as training data. Table 6.2 lists the relevant parameters of the final trained networks.
Two types of error statistics are reported in the tableâ€”maximum error and average
error. The maximum error statistic is a worst case measure of network error. Each of
the 2250 training pairs used for training will have a different error associated with it.
One of these 2250 pairs will have an associated error that is larger than that of all the
other pairs. It is the error for this worst case training pair that is reported as the
maximum error in the table. The average error, which is the sum of the errors over all
of the training pairs divided by the number of training pairs, is also reported in the
table.
The table indicates, that while the worst case errors encountered during network
training were significant, the networks performed very well on average. Whereas the
largest of the maximum errors encountered was approximately 33%, the largest of the
average errors was only around 3% indicating that the network errors were very small
in the vast majority of cases.
175
The various network topologies reported in Table 6.2 were arrived at by trial
and error. Each of the final network configurations shown in the table was the result of
training several different size networks and selecting the one which produced the least
error. Also, note that for each topology trained, several (510) separate training â€œrunsâ€
were performed starting from different, randomly selected points on the error surface.
6.7 Scaling Neural Networks
Figure 6.10 illustrates the basic layout of the scaling (magnitude) neural
networks used in this research. As the figure indicates, there are five input parameters
and a single output parameter for each of these networks. Nine networks were
constructed so that each combination of load type (Fz, Mx, My) and displacement type
(Tz, Rx, Ry) was covered.
Table 6.2 Trained Shape Networks
Load
Type
Disp.
Type
Network
Topology
Number of
Connections
Number of
Epochs
Maximum
Error1
Average
Error1
Fz
Tz
8x16x26x1
570
15000
0.27310
0.02298
Fz
Rx
8x16x26x1
570
20000
0.21745
0.02135
Fz
Ry
8x18x28x1
676
25000
0.21791
0.02605
Mx
Tz
8x18x20x1
524
10000
0.26698
0.02870
Mx
Rx
8x18x20x1
524
15000
0.10630
0.01361
Mx
Ry
8x20x24x1
664
10000
0.32554
0.03590
My
Tz
8x20x24x1
664
25000
0.20696
0.02442
My
Rx
8x18x20x1
524
40000
0.24641
0.02887
My
Ry
8x18x20x1
524
10000
0.11923
0.01420
^ The errors statistics reported here have already been made relative to the range [1,1]
instead of the compacted range [1/1.2,1/1.2].
176
Sizes Vary
Hidden Layer
FanOut Neurons
I  Computing Neurons
Normalized Maximum
Displacements (Tz.Rx.
or Ry) In Range [0.1 ]
Figure 6.10 Configuration of Magnitude Scaling Neural Networks
The input parameters consist of the location (lateral and longitudinal
coordinates) of the applied load and the Geometric Scale Factor (GSF) of the bridge.
The GSF input parameter is not normalized into the range [0,1] since there is no
particular advantage in doing so. The load coordinates are encoded using one neuron
for the Xcoordinate and three neurons for the Ycoordinate.
It is the function of the scaling networks to predict displacement (translation,
rotation) scaling factors which can be combined with shape network data so as to
compute true structural displacements. However, the output ranges of the scaling
networks are limited to the output ranges of the transfer functions used. In this study,
the sigmoid function g, having an output range [0,1], was used for all scaling
networks.
Therefore, the maximum magnitude displacements also had to be normalized
into the range [0,1] for neural network use. This was accomplished by normalizing all
of the maximum displacement magnitudes with respect to the maximum values that
occurred in the GSF=1.2 geometry case. After normalization then, the maximum
177
displacement magnitudes all fall in the range [0,1]. Note that when using the
networksâ€”as opposed to when training themâ€”the normalized maximum displacement
magnitudes produced by the networks must be scaled from [0,1] to a true structural
range. The â€œoverallâ€ scaling factors (the maximum displacement magnitudes from the
GSF = 1.2 case) used for purpose this are listed in Table 6.3 .
Table 6.3 Trained Shape Networks
Load
Type
Disp.
Type
Network
Topology
Num.
Conn.
Num.
Epoch
Overall
Max.
Max.
Training
(Validation)
Errorf
Avg.
Training
(Validation)
Errort
Fz
Tz
5x15x15x1
315
10000
1.6502e02
0.07265
(0.02705)
0.00745
(0.00705)
Fz
Rx
5x15x15x1
315
15000
9.2530e05
0.06189
(0.04727)
0.01123
(0.01131)
Fz
Ry
5x15x15x1
315
20000
6.9303e05
0.06310
(0.03980)
0.00869
(0.00939)
Mx
Tz
5x15x15x1
315
10000
9.2771e05
0.06586
(0.04303)
0.01194
(0.01255)
Mx
Rx
5x15x15x1
315
40000
3.6506e06
0.07944
(0.03670)
0.00829
(0.00705)
Mx
Ry
5x15x15x1
315
40000
6.5784e07
0.05909
(0.03823)
0.00769
(0.00868)
My
Tz
5x15x15x1
315
20000
6.7015e05
0.06947
(0.03839)
0.01188
(0.01029)
My
Rx
5x15x15x1
315
30000
6.5784e07
0.05544
(0.01020)
0.02640
(0.00855)
My
Ry
5x18x18x1
432
20000
3.5500e06
0.09878
(0.07304)
0.01774
(0.01666)
f The errors statistics reported here have already been made relative to the range [0,1]
instead of the compacted range [0.1,0.9].
178
Finally, in addition to the scaling just described, a final scaling is also used to
compact the range of values which must be operated on by the networks. In the scaling
networks, all output values initially in the range [0,1] are compacted into the range
[0.1,0.9] to improve network training. The reasons for performing this type of scaling
were given in the previous section with regard to the shape networks. The reader is
therefore referred to that section for more information.
Training the scaling neural networks was accomplished using the NetSim
software. The 924 loaddisplacement pairs described earlier in this chapter were used as
training data. Figures 6.116.19 graphically illustrate the data that was used to train
each of the nine scaling networks.
In addition to the 924 training pairs used to train the networks, 38 validation
pairs were used to monitor the generalization capabilities of the networks. Validation
data consisted of selected maximum displacement magnitudes from a GSF=0.9
geometry case that was generated separately from the training data.
Table 6.3 lists the relevant parameters of the trained scaling networks. The error
statistics shown the table indicate that the neural networks were able to model the
training data and match the validation data to within a reasonable level of error. The
maximum error statistics for both the training and validation data sets were less than
10% while the average error statistics were less than 2%.
179
Tz (inches) Geometry 0.6
0.020,
Tz (inches) Geometry 1.0
0.020
Tz (inches) Geometry 0.8
Tz (inches) Geometry 1.2
0.020
0.015
0.010
0.005
0.000
0.00
Lateral 0.75
Direction 1.00'
Normalized Longitudinal Direction
Figure 6.11 Maximum Magnitude Translations (Tz) Caused By Unit Forces (Fz)
(Training Data for Scaling Neural Networks)
Rx (radians) Geometry 0.6
1.00e04
8.00e05
Rx (radians) Geometry 1.0
1 .OOe04,
Rx (radians) Geometry 0.8
1.00*04,
0.25
Normalized 0.50
Lateral 0.75
Direction 1.00'
Normalized Longitudinal Direction
Rx (radians) Geometry 1.2
1.00e04
8.00e05
0.25
Normalized
Lateral 0.
Direction 1.00'
Normalized Longitudinal Direction
Figure 6.12 Maximum Magnitude Rotations (Rx) Caused By Unit Forces (Fz)
(Training Data for Scaling Neural Networks)
180
Ry (radians) Geometry 0.6
8.00e05i
6.00e05
Ry (radians) Geometry 0.
8.00e05
6.00e05
Normalized
Lateral 0.75
Direction 1 00
Normalized Lcngitudiial Direction
Ry (radians) Geometry 1.0
8.00e05
6.00&05
Ry (radians) Geometry 1.2
8.00e05
6.00e05
Normalized
Lateral 0.75
Direction LOO'
Normalized Longitudinal Direction
Figure 6.13 Maximum Magnitude Rotations (Ry) Caused By Unit Forces (Fz)
(Training Data for Scaling Neural Networks)
T z (inches) Geometry 0.6 Tz (inches) Geometry 1.0
Tz (inches) Geometry 1.2
1.00e04
8.00e05
Normalized ^
Lateral 0/75
Direction 100
Nonnalisd Longitudinal Direction
Figure 6.14 Maximum Magnitude Translations (Tz) Caused By Unit Moments (Mx)
(Training Data for Scaling Neural Networks)
181
Rx (radians) Geometry 0.8
4.00e06
3.00e06
Rx (radians) Geometry 1.2
4.00e06i
Lateral 0.75
Direction 1.00'
Normalized Longitudinal Direction
Normalized Lcngitudinal Direction
Figure 6.15 Maximum Magnitude Rotations (Rx) Caused By Unit Moments (Mx)
(Training Data for Scaling Neural Networks)
Ry (radians) Geometry 0.6
8.00e07
6.00e07
Ry (radians) Geometry 0.8
8.00e07r
6.00e07
4.00e07
2.00e07
O.OOe+OO
0.00
0.2^
Normalized 0^0N
i\ Â»Â«*
Lateral
Directum
0.75N
l.OO^
i.OO
U25
Ry (radians) Geometry 1.0
8.00&07
6.00e07
Normalized Lcngitudinal Direction
Normalized Longitudinal Direction
Figure 6.16 Maximum Magnitude Rotations (Ry) Caused By Unit Moments (Mx)
(Training Data for Scaling Neural Networks)
182
Tz (inches) Geometry 0.6
8.00e05
6.00&05
Tz (inches) Geometry 1.0
8.00e05,
Tz (inches) Geometry 0.8
8.00e05
6.00e05
4.00e05
2.00e05
0.00e+00p
0.00
0.25â€™
NormalizedÂ®'
Lateral 0.75
Direction 1.00
Normalized Longitudinal Direction
Normalized Longitudinal Direction
Figure 6.17 Maximum Magnitude Translations (Tz) Caused By Unit Moments (My)
(Training Data for Scaling Neural Networks)
Rx (radians) Geometry 1.0
8.00e07
6.00e07
Rx (radians) Geometry 0.8
8.00e07,
Normalized
Lateral 0.75
Direction 1.00
Normalized Longitudinal Direction
Rx (radians) Geometry 1.2
8.00e07i
kt i 050
Normalized
100 Lateral 0.73
Normaliaxi Longitudinal Direction
Figure 6.18 Maximum Magnitude Rotations (Rx) Caused By Unit Moments (My)
(Training Data for Scaling Neural Networks)
183
1.00
/rnf1*nne\ i O Q Dll /ludlqno\ Oonmi4m 1 ^
1.00
Normalized Lcngitudiial Direction
Figure 6.19 Maximum Magnitude Rotations (Ry) Caused By Unit Moments (My)
(Training Data for Scaling Neural Networks)
6.8 Implementation and Testing
In order to perform loaddisplacement calculations for flatslab bridges using the
neural networks developed herein, the networks were integrated together through a
common control module. Given a particular loading condition on a bridge, it is the
responsibility of the control module to determine which neural networks must be
invoked to compute the structural displacements for the bridge. It is also the
responsibility of the control module to perform load superposition and to combine
shape and scaling data from the networks in the correct manner.
The eighteen component neural networks and the control module were
integrated into an iterative equation solver as a separate part of this research. The
development and testing of that equation solver are the topics of Chapter 7. Since the
184
ability of the neural networks to compute accurate displacements in flatslab bridges is
a topic covered in detail in that chapter, presentation of neural network test results will
be delayed until that time.
CHAPTER 7
ITERATIVE EQUATION SOLVERS FOR HIGHWAY BRIDGE ANALYSIS
7.1 Introduction
One of the primary focal points of FEA research during the past few decades
has been the development of fast and efficient equation solvers. This is because every
finite element analysis requires that at least one set of simultaneous equations be solved
as part of the analysis. In cases such as nonlinear analysis or dynamic analysis, the
equation solving step may have to be performed many times during the analysis. Since
the number of equations that must be solved will grow as the FEA model becomes
more refined, the equation solving portion of an analysis can account for a significant
portion of the total analysis time. It is therefore desirable to create equation solvers
which are as numerically efficient as possible.
In the FEA of highway bridges, the number of equations to be solvedâ€”which
will be equal to the number of degrees of freedom on the modelâ€”generally will
number between one thousand and ten thousand. To solve matrix equations involving
thousands of degrees of freedom (DOFs), many general purpose solution techniques
have been developed which are efficient with respect to both speed and memory usage.
There are probably as many different types of solvers as there are types of problems to
185
186
be solved and it not the goal of this dissertation to survey them ail. However, some
general classifications will be useful for the discussions that follow.
1. Direct sol\ers. Solution strategies in which the matrix equation is solved in a
direct manner, without need for iteration.
2. Iterative solvers. Solution strategies in which the matrix equation is solved
by iteratively refining the estimates of the unknowns being determined. In
such cases, the number of iterations required for convergence is dependent
on a number of problem dependent parameters and cannot generally be
determined a priori.
3. Elementbyelement solvers. Solution strategies in which the full matrix
equation is never actually formed. Instead, the elements which would be
used to form the full matrix equation are processed individually. Element
byelement solvers can be further classified as either direct or iterative.
4. Sparse solvers. Solution strategies in which patterns of sparsity in the matrix
equation are exploited to reduce the amount of time and memory that will be
required to solve the system of equations. Many types of sparsity can be
accounted for (symmetric, banded, profile) and sparsity can be employed in
either direct or iterative solvers.
What all of these equation solving schemes have in common is that they are all
more or less general purpose in nature. These schemes may exploit a particular
structure of sparsity (e.g. bandedness), or a particular property of the matrices involved
(e.g. postivedefmiteness), but the actual physical problem being solved is not
considered in the solution scheme. In this chapter, a new solution strategy will be
presented which combines neural networks with an iterative equation solving scheme to
produce a hybrid method specific to the area of highway bridge FEA.
7,2 Exploiting Domain Knowledge
In the present research, a domain specific equation solver has been created for
the analysis of highway bridge structures. The term domain specific is used to designate
187
the fact that the equation solver is setup specifically for one domain (class) of
structures. In the present research, the class of structures chosen to be studied was that
of twospan reinforced concrete flatslab bridges. Although this was the only domain
considered in this study, the concepts and methods described herein can be easily
extended to other types of bridge structures.
Central to the idea of creating a domain specific equation solver is the goal of
accelerating the equation solving process by embedding knowledge of the problem
domain directly into the solver. Thus, a custom purpose equation solver can be created
that is very fast for one particular type of bridge structure. This approach is especially
applicable to situations in which similar types of structures are frequently analyzed
using similar types of structural modeling. Just such a situation exists when computer
assisted structural modeling software, such as the bridge modeling preprocessor
described in Chapters 2 and 3, is used.
The bridge modeling preprocessor is capable of modeling four basic types of
highway bridgesâ€”prestressed concrete girder, steel girder, teebeam, and flatslab
bridges. While the geometry, properties, and loading of these structures can vary, the
basic structural configuration and modeling of the bridges is similar within each class
of bridge. Therefore, this type of modeling and analysis environment is a prime
candidate for using a domain specific equation solver. Flatslab bridges were chosen for
this research because they are the simplest of the four classes of bridges modeled by the
preprocessor. Once the concepts and methods have been developed for the flatslab
bridge type, they can be extended to the other bridge types.
188
7.3 Iterative FEA Equation Solving Schemes
T
Direct matrix solution schemesâ€”such as the LDL decompositionâ€”are often
used in FEA when there are a moderate number of equations to be solved. However, as
the number of equations becomes very large, iterative methods become more efficient
if full advantage is taken of matrix sparsity (Jennings and McKeown 1992). Also, in
cases where an approximation of the solution is known, iterative methods can
outperform direct methods.
Several types of iterative methods may be used to solve the matrix equations
arising from FEA. The general objective of these solution methods is to solve a matrix
equation of the form
Ax = b (7.1)
where A is a coefficient matrix, b is the right hand side (RHS) vector, and x is the
solution vector. In a typical FEA situation, A is the global stiffness matrix, b is the
global load vector, and x is the vector of structural displacements. The matrix equation
will be solved iteratively, meaning that there will be a set of approximate solution
vectors
%Â».*(1)>*(2)* (72)
whichâ€”under favorable conditionsâ€”will converge to the exact solution x. The
differences between the various iterative methods lie primarily in how the estimates
X(i) are updated at the end of each iteration.
189
The Jacobi and GaussSeidel iterative algorithms are the simplest and easiest to
implement, however, they have slow converge characteristics in most cases. To
accelerate the convergence rate a scale factor la , having a value in the range 1 < a < 2,
may be added to the GaussSeidel method to produce the SOR (Successive Over
Relaxation) method. SOR generally converges more rapidly than GaussSeidel,
however choosing the optimum scale factor aopt is a difficult process. One may use an
iterative approach in which many different scale factors are examined and then the one
producing the fastest convergence is selected for use. This approach is only useful in
cases where many problems of similar type will be solved and will hopefully have
similar values of (Â¡>opt. Alternatively, one may perform an eigen analysis to determine
an approximate value of opt (Golub and Van Loan 1989), however this will require a
substantial amount of computational effort and may offset any savings derived from
using an iterative solution scheme.
The Conjugate Gradient (CG) method and its variants constitute another class of
iterative solution method which may be used in FEA. In the CG method, the matrix
equation Ax = b is solved by minimizing the residual
r = bAx (7.3)
where r is the residual vector and x is an approximation of the exact solution x. The
residual is actually minimized indirectly by formulating an error functionâ€”which is
quadratic in r â€”and minimizing the error function. The method is called the Conjugate
Gradient method because the optimization process traverses the error function using
conjugate directions instead of steepest descent directions.
190
If steepest descent directions are used, convergence to the minimum of the error
surface can be very slow in many cases. In CG, conjugate directions are used instead
of steepest descent directions. Conjugate directions are directions which very closely
approximate the steepest descent directions but which are also conjugate to all of the
previous directions followed on the surface thus far. The result of using conjugate
directions instead of steepest descent directions is quicker convergence to the minimum
of the error surface and therefore quicker solution to the problem Ax = b (Golub and
Van Loan 1989, Jennings and McKeown 1992, Press et al. 1991). The Conjugate
Gradient algorithm is given by the following steps.
form initial guess X(o)
(7.4)
;$>
Â©
II
o
II
o
1
Â©
(7.5)
II
S'
(7.6)
*(Â¡+1)  *(.) +a(Â»P(o
(7.7)
r(i+l) = r0)a0)AP(0
(7.8)
n .. r0+1/0+1)
P(i+1)  T
'â€¢(I/O)
(7.9)
P(i+l) = 'â– (/+!) +P(i+l)/â€™(0
(7.10)
When an iterative method exhibits slow convergence, the matrix equation
Ax = b may be preconditioned to accelerate convergence. In theory, preconditioning
may be applied to any iterative method, however there are several practical
191
considerations which preclude its use with some of the iterative methods. The essential
idea behind preconditioning is that the rate of convergence is governed by the ratio
_ ^max (711)
^â– rnin ^1
where k is the condition number of the matrix A, Anlflx is the largest eigenvalue of
the matrix A, and is the smallest eigenvalue of the matrix A. Note that the
discussion here deals only with matrices A that are symmetric and positivedefinite
(which is typically the case in FEA). Therefore, the eigenvalues of A are all real and
positive (A > 0) and the condition number will always be a real, positive value.
The convergence rate of an iterative method can be accelerated by reducing the
condition number, i.e. by compressing the eigenvalue spectrum. The smaller the value
of k , the faster the iterative process will converge. The acceleration can be achieved
by transforming the matrix equation
Ax = b (7.12)
into an new matrix equation
PAx = Pb
or
(7.13)
Ax = b (7.14)
where A = PA and b = Pb are the transformed coefficient matrix and RFIS vector
respectively, and P is a preconditioning matrix. Thus, by premultiplying each side of
the equation by a preconditioning matrix P, we obtain a new matrix equation which
192
will have a more compact eigenvalue spectrumâ€”namely the eigenvalue spectrum of A
instead of A.
For a symmetric positivedefinite matrix, the smallest value which k can take
on is 1.0. Therefore an ideal preconditioning matrix P will transform A into the
identity matrix / for which the condition number is 1.0. It then becomes apparent that
the ideal preconditioner for any system Ax = b is P= A~l, the inverse of the original
coefficient matrix. Clearly, however, the amount of work involved in obtaining A~
cannot be justified simply to accelerate the convergence of an iterative equation solver.
If A~l was available, then the system Ax = b could be trivially solved and there would
be no need for an iterative solver at all.
However, one can see from this discussion that the closer the preconditioned
coefficient matrix A = PA is to the identity matrix /, the faster the iterative solution
process will converge to x. This fact can serve as a guide for evaluating different
preconditioning strategies. Preconditioning is only economical if the computational
savings resulting from accelerated convergence more than offset the additional work
involved in obtaining P and transforming Ax = b into Axb. Thus, a â€œgoodâ€
preconditioner must not be excessively expensive to compute and must substantially
reduce the condition number of the matrix A .
While preconditioning canâ€”in theoryâ€”be applied to any iterative method, it is
particularly well suited to the Conjugate Gradient method (Jennings and McKeown
Â§11.13, 1992). When preconditioning is combined with the CG algorithm, a
193
Preconditioned Conjugate Gradient (PCG) algorithm results. PCG equation solvers can
be further classified based on the choice of preconditioner used. (Some preconditioners
that can be used in the FEA of highway bridge structures are discussed in the next
section.) One particularly appealing aspect of combining preconditioning with the CG
algorithm is that the preconditioned coefficient matrix A can be formed implicitly.
Therefore when implementing this algorithm, the transformed matrix A = PA
never needs to be explicitly computed (and therefore also never needs to be stored).
The Preconditioned Conjugate Gradient algorithm is given by the following steps.
form initial guess x^q)
(7.15)
'â– (0) ~b~ ^x(0)
(7.16)
o
II
5;
1
o
(7.17)
rfi)M V(i)
â€œ(0 T .
PioMn
(7.18)
X(l+1) = X(l) + Ct(l)/'(i)
(7.19)
'â€¢(1+1) ='â€¢(/)â€œ(oMo
(7.20)
T â€” l
â€ž 'â– (1+1)^ 'â– (i+i)
r(i)M r(l)
(7.21)
P(i+\)~M '''(i+i) + P(i+l)P(i
j (7.22)
where M is a matrix that closely approximates
the coefficient matrix A. The
preconditioning matrix in this algorithm is P = M 1 but if M is chosen carefully, then
M 1 should be fairly easy to compute. Actually, M
1 never needs to be computed at
194
all since it is only used to solve the matrix equations of the form Mq = r (more on this
later) which can just as easily be solved by any appropriate decomposition technique.
7,4 Preconditioning in Highway Bridge Analysis
The degree to which accelerated convergence will be obtained by using
preconditioning depends heavily on the choice of preconditioner. Unfortunately, there
are no truly â€œgeneral purposeâ€ preconditioning schemes which work well in all
situations. An effective preconditioner for one type of problem may be a very poor
choice for a different type of problem. However, if a preconditioning scheme works
well for one problem, often times it will also work well for other problems of the same
classâ€”i.e. those having similar matrix structure, matrix properties, or matrix origin.
Therefore, it is apparent that searching for effective preconditioners is a
worthwhile effort if one will often be solving problems of similar type. Highway bridge
analysis using automated FEA modeling software is one such situation in which
problems of similar type will often need to be solved. With this in mindâ€”and within
the context of the research being reported on hereinâ€”the author investigated the
effectiveness the following preconditioning schemes for FEA of highway bridge
structures.
1. Diagonal Preconditioning. The diagonal of the matrix A is extracted and
used to form a diagonal preconditioning matrix M.
2. Band Preconditioning. A banded â€œsliceâ€ of the matrix A is extracted and
used to form a banded preconditioning matrix M.
195
3. Incomplete Cholesky Decomposition Preconditioning. The approximation
matrix M is formed by extracting only the nonzero terms of A and
copying them into M. Subsequently, an â€œincompleteâ€ decomposition
(ignoring fillin) is performed on the matrix M.
Each of these preconditioning schemes were implemented by the author in the
S1MPAL finite element analysis programâ€”written by Hoitâ€”that is part of the
BRUFEM system (Hays et al. 1994). In implementing the various preconditioners in
SIMPAL, the profile, blocked, outofcore capabilities of the original direct solver
were also provided so that preconditioning for large FEA bridge models could be
studied.
7.4.1 Diagonal and Band Preconditioning
In diagonal preconditioning^ , the main diagonal of A is extracted and inserted
into an otherwise empty matrix to form M (see Figure 7.1). The preconditioning
matrix P is then formed as P=M~l which is just a diagonal matrix in which
Pa = â€”. To precondition the system, each side of the matrix equation Ax = b is
Mu
premultiplied by P. In practice, the matrix P â€”which is extremely sparseâ€”does not
actually need to be formed since its effect on the system PAx = Pb can be computed
easily and directly.
^ In the discussion that follows, the matrix A is assumed to be symmetric and positive
definiteâ€”as is often the case in FEA.
196
In banded preconditioning, a banded â€œsliceâ€ of the matrix A is extracted and
inserted into an otherwise empty matrix to form M (see Figure 7.2). In the author's
implementation of this preconditioning scheme, the width of the band to be extracted
from A is specified as part of the FEA control parameters. The width may be specified
either as an absolute width or as a fraction of the total number of equations in the
system. Thus, different band widths can be examined to determine appropriate
parameters for a particular type of bridge structure. Also, note that although the portion
extracted from the matrix A is called a banded slice, this data is inserted into M in
profile (skyline) format.
Here again, theoretically, one would need to form the preconditioning matrix as
P = M 1 and premultiply each side of the matrix equation by P to precondition it.
However, unlike the case of diagonal preconditioning, the matrix inverse M~l will not
be easy to compute in general. In practice, and in the authorâ€™s implementation,
197
P = A/1 is actually never computed for band preconditioning. Examining the PCG
algorithm. Equations (7.15)(7.22), we see that the preconditioning matrix P = M~l is
used in a number of steps involving terms of the form M~lr which is simply a vector
that we will call q for convenience. Thus, we have a number of steps involving
computation of the form
(1.23)
However, this equation is simply the formal notation for the solution to the matrix
equation
Mq = r (7.24)
since
M~xMq=M~Xr (725>
(7.26)
(7.27)
/ q = M V
q = M lr .
198
Therefore, the matrix inverse M 1 really never needs to be formed as long as one can
solve the matrix equation Mq = r. In practice, this is generally accomplished by
performing a direct equation solution on the subproblem Mq = r of the overall
T
problem Ax = b. In the present research the symmetric decomposition LDL is used,
however, any reasonable technique can be employed. Observe then that one component
of the iterative PCG solution process involves the use of a direct solver on a subÂ¬
problem, therefore a PCG solver will generally contain iterative solution code as well
as direct solution code.
To determine the effectiveness of diagonal and band preconditioning in the FEA
of highway bridge structures, several bridge models were constructed using the
preprocessor described in Chapters 2 and 3. These models were then analyzed by the
author's PCG solver using both diagonal and band preconditioning. Diagonal
preconditioning was studied by simply using a bandwidth of 1 for the band
preconditioning caseâ€”i.e. special coding to compute the diagonal inverse was not
written.
Results of these studies indicated that neither diagonal preconditioning nor band
preconditioning work well in the analysis of highway bridge structures. To understand
why this is the case, one must examine the structure (sparsity pattern) of the coefficient
matrices that arise in bridge FEA. In bridge FEA, the generic coefficient matrix A
becomes the global finite element stiffness matrix of the structure. It is well known that
such stiffness matrices are often sparse and exhibit either banded or skyline structure.
199
In addition, the diagonal terms of the matrix are guaranteed to be positive and are
generally large (in magnitude) relative to the off diagonal terms.
Recall from earlier discussion that the matrix M must be a â€œgoodâ€
approximation of A if P = M~l is to be an effective preconditioner for the system
Ax = b. Therefore the question becomes, â€œIs a diagonal slice M or banded slice M
of the matrix A sufficiently representative of the information content of AT. The
answer to this questionâ€”at least for the bridge structures studiedâ€”is â€œNo.â€. A slice of
the stiffness matrix does not contain sufficient information to approximately represent
the information content of the overall stiffness matrix. This fact is directly attributable
to the structure of sparsity in the matrix.
In the bridge models studied, the stiffness matrices have a structure similar to
that illustrated in Figure 7.3. One can see that there are essentially two distinct bands
of nonzero terms in the matrix separated by a void of zero terms. There are two
distinct bands of data (not three) because the matrix is known to be symmetric and
therefore the lower band is known to be the mirror image of the upper band. As a
result, the lower band introduces no new information. Although the sparsity pattern
illustrated is for a small bridge modelâ€”approximately 150 degrees of freedomâ€”the
sparsity patterns for larger bridge models is very similar. The primary difference is that
the bands of nonzero terms become increasingly narrowerâ€”relative to the overall size
of the matrixâ€”as the bridge models grow in size.
Figure 7.3 illustrates all of the nonzero terms in the matrix including terms
having large magnitude as well those having small magnitude. The largest magnitude
200
3M*sâ€” :us*Ã¼
â€”K+fcÂ» â€œIOÃÃm?
is;:;
â– 'jÃs^sajs
sOfiW
Large Negative Values Are
 Located Along The Center
Of The Outer Band
 Region of Zero Values
Large Positive Values Are
Located Along The Center
Of The Inner Band
Â«JHÃ JÂ«4Â«
Figure 7.3 Typical Matrix Sparsity for a FlatSlab Bridge
positive terms are located along the main diagonal of the matrix as would be expected
for a finite element stiffness matrix. However, there are also large magnitude negative
terms located along the centerline of the outer band.
These large negative terms are the primary reason that diagonal and band
preconditioning schemes do not work well for bridge structures. When the matrix M is
constructed, it must approximately represent the information content of A if P = A/1
is to be an effective preconditioner. If the diagonal or a small band of terms are
extracted from A to form M, and A has the structure shown in the figure, none of
201
the large magnitude negative terms will be included in M. As a result, the matrix M
does not really approximate A and M~y is a poor preconditioner for A .
In theory, one could remedy this situation by increasing the size of the band so
as to include the large negative terms. However, the bandwidth chosen would then have
to include the vast majority of the matrix A . Recall that we must be able to efficiently
solve subproblems of the form Mq = r within the PCG iteration process. If M is
virtually the entire matrix A, then the cost of solving Mq = r will be roughly
equivalent to solving the original system Ax = b and there is no point to using an
iterative process at all.
7.4.2 Incomplete Choleskv Decomposition Preconditioning
It is apparent from the discussion above that the diagonal and band
preconditioning schemes will only be effective for cases in which the matrix A is
diagonally dominant or nearly so. An alternative preconditioning scheme must be
developed for cases such as FEA of bridge structures. The incomplete cholesky
decomposition (ICD) preconditioning scheme offers just such an alternative. When this
preconditioning scheme is combined with the standard CG algorithm, the resulting
algorithm is called an ICCG (Incomplete Cholesky Conjugate Gradient) algorithm.
Therefore ICCG solvers are just a specific type of PCG solver. From this point
forward, the ICCG algorithm will be referred to as the ICPCG (Incomplete Cholesky
Preconditioned Conjugate Gradient) algorithm to be consistent with terminology that
202
will be introduced later in this chapter. The prefix, IC in this case, designates the type
of preconditioning scheme used.
When a direct solution algorithm is applied to a sparse matrix such as the one
shown in Figure 7.3, a great deal of fillin occurs during the decomposition process.
Fillin refers to terms inside the matrix profile which are initially zero but which
become nonzero (i.e. â€œfillinâ€) as the decomposition is performed. As a result of fill
in, the sparsity of the original matrix is not preserved in the decomposed matrix. Fillin
terms often have values which are small in magnitude relative to the other terms in the
matrix. This fact forms the basis of the Incomplete Cholesky Decomposition concept
(Jennings 1992, Manteuffel 1980, Meijerink and van der Vorst 1977, Papadrakakis and
Dracopoulos 1991, Radicati di Brozolo and Vitaletti 1989). For a sparse matrix such as
the one shown in Figure 7.3, calculations involving fillin terms account for a large
proportion of the total set of calculations that must be performed during the
decomposition.
Since these terms are often small in magnitude, the ICD concept says that we
can ignore their effect during the decomposition and perhaps still get an approximate
decomposition of the original matrix. In fact, in an ICD the fillin in terms are ignored
all together. As a result, we have constrained the decomposed matrix to have the same
sparsity pattern as the original matrixâ€”although the values in the matrix have changed
considerably during decomposition. Because fillin is not allowed, the number of
calculations involved in the decomposition is vastly reduced from that of a true
decomposition.
203
Thus, in the ICPCG method, the matrix M is chosen to be exactly equal to the
matrix A . As a result, we start off with a matrix M that exactly â€œapproximatesâ€ the
coefficient matrix A . However, we then use an approximate method to form the
decomposition of M. This can be roughly thought of as forming an approximate
inverse since the approximate decomposition will be used to solve the Mq = r subÂ¬
problems during the PCG iterationsâ€”instead of using q = M~lr. (Refer to the
previous section for more details on the Mq = r subproblem). Since fillin is not
allowed, the incomplete decomposition is computationally inexpensiveâ€”a situation
which is necessary for an effective preconditioning scheme.
Also, since fillin is never allowed, the sparsity of the matrices A and M can
be fully exploited. In a direct solver, the profile storage scheme is the most efficient
storage scheme available since fillin must be considered. Since the ICD scheme
preserves the sparsity of the matrix, the terms which would normally be filledin do not
need not be stored since they will be ignored. Thus, fully compact storage schemes in
which only the nonzero terms of the matrix are stored can be used in an ICPCG
solver. In fact, it is because iterative methods can fully exploit the sparsity of the
matrices that they can outperform direct methods for very large problems. In such
problems, calculations involving fillin require a great deal of storage and
computational effort which can be eliminated using iterative methods and the ICD
preconditioning scheme.
Unfortunately, despite all of its advantages, the ICD preconditioning scheme
still performs poorly in FEA bridge analysis. The primary problem is that when the
204
fillin terms are ignored, the decomposition process can become unstable and produce
unreasonable results. For example, the procedure of ignoring fillin often results in the
formation of negative terms on the diagonal of the decomposed matrixâ€”an
unreasonable condition in FEA. Jennings (1992) presents a stabilization process that
can be used to avoid this pitfall, however, it requires additional processing of the
coefficient matrix A and may result in a modification of the sparsity pattern. Kershaw
(1977) handled the development of negative diagonal termsâ€”which he stated occurred
infrequently in the types of problems he was studyingâ€”by assigning the offending
diagonal the value given by
z1 n
SM (728)
7=1 71+1
and then proceeding with the decomposition. This procedure was implemented and
tested by the author in FEA bridge analysis applications, however, it generally
performed poorly. Whereas the problems studied by Kershaw seldom required this
â€œfixâ€ to be made, the bridge analysis problems studied by the author required the fix
very frequently and the resulting decomposition performed poorly as a preconditioner.
Diagonal scaling was also implemented and tested as a method for countering the
formation of negative diagonal terms during the decomposition. It was found that by
scaling the diagonal of M by a factor in the range [I03,106] prior to decomposition,
negative diagonal terms were not formed during decomposition. However, this
procedure essentially converts the matrix M into a nearly diagonal matrix of large
positive values. As a result, the scaled matrix M no longer reflects the character of the
205
original matrix A and therefore its ICD does not function well as a preconditioner
for A.
The ICD procedure used in this research is called a rejection by position scheme
because terms are rejected based on their position in the matrix. If a term is located at a
position not in the sparsity pattern of the original matrix, then it is rejected for use
during the decomposition. Papadrakakis and Dracopoulos (1991) state that rejection by
position methods often do not perform satisfactorily in structural mechanics problems.
An alternative procedure is to perform rejection by magnitude in which terms smaller
than a particular threshold magnitude are rejected from the decomposition process.
While this rejection scheme generally results in better preconditioners, it is more
difficult to implement because the sparsity pattern of the decomposed matrix is not
known until the decomposition is complete. This can be undesirable since efficient
storage strategies are essential to the solution of very large problems.
7.5 A Domain Specific Equation Solver
In the present research, a domain specific equation solver has been developed by
combining neural networks (NN) with the preconditioned conjugate gradient (PCG)
algorithm to create a hybrid NNPCG algorithm. The hybrid algorithm is domain
specific because the neural networks that comprise a portion of the algorithm were
trained specifically for one class of structuresâ€”two span flatslab highway bridges. In
developing this solver, the goal was to accelerate the equation solving stage of highway
bridge FEA by encoding domain knowledge directly into the solver.
206
In this research, neural networks serve as the domain knowledge encoding
mechanism. The networks have been trainedâ€”to within an acceptable level of errorâ€”to
learn the loaddisplacement relationship for twospan flatslab concrete bridges. Once
trained, the networks are used to predict displacement patterns for these bridge
structures under specified loading conditions. Finally, the neural network displacement
predictions are used within the larger framework of the PCG algorithm to create a
hybrid equation solver.
Recall from Chapter 5 that network training is an iterative process which is
halted only when the chosen measure of error has decreased below an acceptable
tolerance level. Therefore, the loaddisplacement relationship encoded by the networks
is not exact but rather an approximation. Since it is not an exact encoding, one cannot
use the networks to directly solve for the exact displacements in bridge structures.
However, the network can be used to rapidly solve for approximate displacements in
these structures and it is this fact which is exploited by the hybrid NNPCG algorithm.
Conceptually, the idea of the hybrid solver is to exploit the fact that the neural
networks can rapidly predict approximate displacements with little required
computation. However, the use of the networks must be placed into an overall
algorithm in which exact displacements can ultimately be computed. Iterative equation
solving methods are the obvious choice for such a framework. Neural networks can be
used to compute approximate displacements at each iteration while the overall iterative
process guides convergence to the exact solution. In the NNPCG algorithm, the neural
207
networks serve two functions, each of which is aimed at accelerating the overall
iteration process.
1. Seeding the solution vector. The initial solution vector used to start the
iterative process is generated using the neural networks.
2. Preconditioning. The neural networks are used in place of the matrix M to
precondition the matrix equation being solved.
Every iterative equation solving method begins with a common stepâ€”computing an
initial guess at the solution vector. The closer the initial guess is to the actual solution,
the faster convergence to that solution will be. Neural networks provide a mechanism
for computing the initial guess with little computational effort.
Since the neural networks have been trained to learn the loaddisplacement
relationship for flatslab bridges, they can be directly called upon to compute an initial
approximation of the displaced shape of a bridge. An initial preprocessing stage must
be added to the FEA engine so that for each degree of freedom (DOF) in the structural
model, the relevant neural network input parameters can be easily accessed.' After the
global load vector has been formed by the usual finite element procedures, an initial
estimate of the solution vector is computed. This is accomplished by using the principle
of superposition.
Initially, the global displacement vectorâ€”i.e. the solution vectorâ€”is filled with
zeros. Then, the global load vector is scanned for nonzero terms. Each time a nonÂ¬
zero load term is encountered, the displacements due to that load are computed for
^ The relevant parameters include the normalized bridge coordinates and displacement
type of each DOF in the bridge. (Refer to Chapter 6 for more information.)
208
every DOF in the model and are added to the previous values in the solution vector.
Thus, the effect of each load term is superimposed on the effects of all previously
encountered load terms until all of the load terms have been processed. When this
process finishes, the solution vector will hold a neural network generated
approximation of the true displaced shape of the structure.
In using the superposition principle, note that the networks will be called upon
numerous times before the final displaced shape approximation is completely generated.
However, in parallelcomputing or networkdistributedcomputing environments, each
of these separate neural network invocations can be performed independently and
simultaneously (in parallel). Therefore, the concept of using superposition in the
manner just described can lead to very high performance of the software on parallel
computing platforms.
Consider also the fact that for vehicular loading conditions, only a small
percentage of the terms in the global load vector will be nonzero. Consider one of the
wheels of a vehicle that is sitting on the bridge deck. The wheel load will be distributed
only to a very few translational DOF that are in the immediate vicinity of the wheel.
As a result, only a few terms in the global load vector will be affected by the load.
Even in the case of uniform pressure loading, the number of nonzero load terms will
be small relative to the total number of DOF in the model. This is because the uniform
pressure loads acting on the deck elementsâ€”whether they are plate or shell elementsâ€”
are converted into loads which correspond only to the transverse translational DOF in
the model. Terms in the global load vector which correspond to inplane rotational
209
DOF and inplane translational DOF are not affected by such pressure loads. Thus, the
computation of the initial displaced shape of a bridge using neural networks is a
computationally inexpensive procedure.
Thus far, the use of neural networks in predicting initial solution estimates could
be applied to any iterative equation solution algorithmâ€”not just the PCG algorithm.
However, the PCG algorithm is the most suitable choice because it allows for the use
of the neural networks not just in predicting the initial solution estimate, but also in
preconditioning the system.
In solving the equation Axb, the PCG algorithm requires an approximation
matrix M which closely approximates the character of A. The better the
approximation is, the faster the process will converge. Recall that the approximation
matrix M is used to solve subproblems of the form Mq = r within each PCG
iteration. In the hybrid NNPCG algorithm, neural networksâ€”which approximate the
loaddisplacement relationship for the structureâ€”take the place of the matrix M. In
other words, the neural networks are used to approximate the loaddisplacement
relationship that is formally encoded in the structural stiffness matrix. This novel
approach to preconditioning is termed neural network preconditioning.
Each of the steps in the PCG algorithm which requires the solution of an
Mq = r subproblem is now solved using the neural networks instead of using a direct
equation solver. In the PCG algorithm, these subproblems are theoretically solved as
q = M V
(7.29)
210
In practical implementation of the PCG algorithm, this step would be performed not by
forming the matrix inverse M~l but instead by using a direction solution process.
M â€”> (decomposition) â€”> LDlJ
(forward reduction \
diagonal scaling ) â€”V r
backward substitution
(7.30)
(7.31)
Note that r can be thought of as a pseudo load vector for which a set of pseudo
displacements q must be computed. In the NNPCG algorithm, neural networks are
used to solve the same subproblems.
r â€”> (neural networks) â€”> q (7.32)
From the perspective of the networks, the vector r represents a set of pseudo loads on
the bridge structure for which pseudo displacements q must be computed. Thus, in the
NNPCG algorithm, each Mq = r ^ subproblem appearing in the PCG algorithm is
solved by the neural networks. Using the same superposition principle described earlier
in this section, each term of the pseudo load vector is examined and its contribution to
the overall displacement pattern is computed.
In essence, the neural networks are being used to precondition the system
Ax = b just as the matrix M was used for the same purpose in the preconditioning
schemes described earlier in this chapter. This type of preconditioning is therefore
f Recall that the generic symbols q and r in the Mq = r equation were arbitrarily
chosen and were introduced simply to facilitate discussion of the preconditioned
conjugate gradient algorithm. For example, in the PCG step of Equation (7.17) the
generic q term becomes P(Q) and the generic r becomes /(o) .
211
termed neural network preconditioning and has the advantage that it can be customized
for particular classes of structures which are of interest to the engineer. By training
neural networks to learn the loaddisplacement relationship for a particular type (class)
of structure, one can create a custom preconditioner that is appropriate for that
particular class of structure. This is particularly appealing in cases such a highway
bridge analysis where the more general purpose preconditioning schemes do not work
well.
The following steps constitute the combined Neural NetworkPreconditioned
Conjugate Gradient (NNPCG) algorithm used in this research.
Â©
II
1
o
(7.33)
r(0) = b ~ ^*(0)
(7.34)
;$
o
II
o
(7.35)
r&NNir^)
a0) T A
P(i)AP(i)
(7.36)
*(t+l) = x(i) +a(i)P(i)
(7.37)
+
II
1
Q
1
(7.38)
â€ž r0+l)NN(r(i+l))
('+1)â€œ r(^iW(r(0)
(7.39)
P(i+1) = AW(,'(i+l)) + P(Â¿+l)f,(i)
(7.40)
Here NN() is a neural network â€œoperatorâ€ which takes a column vector of loads as an
argument and returns a column vector of displacements.
As written above, the
algorithm seems to suggest that the NN() operator must be invoked several times per
212
iteration. However, in an actual implementation of this algorithm, efficient coding may
be employed so that the NN() operator is invoked only once per iteration. Also note
that the neural networks are used both to compute the initial estimate of the
displacements and to solve the Mq = r subproblems (i.e. perform preconditioning).
7.6 Implementation and Results
Implementation of the NNPCG algorithm was accomplished by merging the
neural networks presented in Chapter 6 with a PCG solver.^ In the merged NNPCG
equation solver, neural networks are called upon to perform the tasks described in the
previous section, namely solution seeding and preconditioning. The hybrid NNPCG
solver was coded using both the Fortran and C programming languages and was
installed in the SIMPAL FEA program written by Hoit (Hays et al. 1994). To begin
construction of the NNPCG solver, an ICPCG solver was written in Fortran and
installed in the FEA software.
Next, each of the direct Mq = r subproblem solutions performed in the
ICPCG solver was replaced with a call to a neural network control module. Thus, for
a given â€œloadâ€ vector q, the network control module is called to compute the
corresponding â€œdisplacementâ€ vector r which is in turn passed back to the PCG
iteration module. The neural network control moduleâ€”which is written in Fortranâ€”
^ During the course of this research, several iterative equation solving algorithms were
coded and tested by the author to evaluate their effectiveness in highway bridge
analysis. In particular, equation solvers were written and tested for the standard CG
algorithm, the diagonal PCG algorithm, the band PCG algorithm, the ICPCG
algorithm, and the NNPCG algorithm.
213
NNPCG Iteration Neural Network
Module Control Module
Figure 7.4 Relationship Between NNPCG Solver Modules
calls individual neural network modules to calculate specific components of the
displacement vector. Using the principle of superposition, the neural network modules
are called repeatedly to form the complete displacement vector for the structure under
the specified loading conditions.
Each of the neural network modules constitutes a selfcontained neural network
that has been trained to solve one aspect of the overall loaddisplacement calculation
problem. Since each of the networks were trained using the NetSim package (see
discussion in Chapters 5 and 6), the automatic code generation capability that software
was used to generate the individual C code modules corresponding to each network.
Figure 7.4 illustrates the relationship between the various components of the NNPCG
solver.
7,6.1 Seeding the Solution Vector Using Neural Networks
In order to test the effectiveness of the NNPCG algorithm, numerous tests were
performed using FEA models of a typical twospan flatslab bridge. The bridge, which
214
is illustrated in Figure 7.5, was subjected to both uniform loading and vehicular
loading conditionsâ€”typical situations arising in highway bridge analysis. Specifically,
the loading conditions examined are those listed below.
1. Uniform Loading. A uniform load (surface pressure load) extending over the
entire length and width of the bridge.
2. Vehicular Loading. Two HS20 trucks positioned endtoend near the right
edge of the bridge.
These loading conditions were used throughout the testing phase and will be referred to
repeatedly in the discussion that follows. The FEA bridge models were prepared using
the preprocessor described in Chapters 2 and 3. Table 7.1 lists the important
parameters of the FEA models.
One of the goals in developing the NNPCG algorithm was to accelerate the
convergence of the solution process by utilizing an intelligent initial guess. By
â€œseedingâ€ the solution vector with displacements calculated using neural networks, the
goal is to start the iterative process at a point very near the correct solution. In the
bridge models studied, the slab is modeled using flat plate elements that have three
30 ft. (360 in.)
10 Elements
* 40 ft. (480 in.)
* 25 ft. (300 in.) *
10 Elements
10 Elements
Figure 7.5 Configuration of Flatslab Bridge Test Models
215
DOF at each nodeâ€”vertical translations Tz, rotations about the Xaxis (lateral
direction) Rx, and rotations about the Yaxis (longitudinal direction) Ry. Thus, to form
the initial solution vector, three displacements (Tz, Rx, Ry) must be computed at each
node in the model, i.e. at each active DOF.
Using the bridge models described above, neural networks were employed to
compute displacements for use in solution seeding. Figures 7.6 and 7.7 compare the
neural network predicted displacements and analytical (FEA) displacements for the
uniform and vehicular loading conditions respectively. One can clearly see from the
figures that the neural networks do a respectable job of matching the analytical results.
The most notable differences occur in the prediction of the Rx rotations where
noticeable differences can be seen. However, overall the figures suggest that neural
networks may be very effective in seeding the solution vector since their predictions
closely match analytical results.
To test the presumption, three different seeding methods were examinedâ€”zero
seeding, diagonal seeding, and neural network seeding. Zero seeding starts the iterative
process by simply filling the solution vector with zero values. Diagonal seeding
computes each term in the solution vector as x, = A^/b, which implicitly assumes that
the matrix is nearly diagonally dominant.
Table 7.1 Structural Parameters of Flatslab Bridge Test Models
Geometry
Span1 Length 40 ft.
Span2 Length 25 ft.
Bridge Width 30 ft.
Slab Thickness 14 in.
Skew Angle 0 deg.
Material Properties
Concrete 4 ksi
Poisson's 0.15
Elastic Mod. 3605 ksi
Bearing 50 ksi
Modeling
Span1 Elements 10
Span2 Elements 10
Width Elements 10
Number of DOFs 693
Number of Nodes 264
216
Rotation Ry (radians)
Neural Network 
Analytical (FEA) 
Longitudinal Direction (inches)
Figure 7.6 Neural Network Predicted Displacements and Analytical (FEA)
Displacements for a Flatslab Bridge Under Uniform Loading
217
Rotation Rx (radians)
Rotation Ry (radians)
Figure 7.7 Neural Network Predicted Displacements and Analytical (FEA)
Displacements for a Flatslab Bridge Under Vehicular Loading
218
In neural network seeding, of course, the solution vector is estimated by having
the networks compute their prediction of the displaced shape of the bridge under the
specified loading condition.
In order to isolate the effect of the seeding method from other factors in the
solution processâ€”such as the choice of preconditioning schemeâ€”an ICPCG equation
solver was used for all of the seeding tests performed. Thus, precisely the same
incomplete cholesky preconditioning scheme was used in each case.* Each of the two
loading conditions described earlier were analyzed using each of the three seeding
methods for a total of six tests. The RMS (root mean square) of the residual load vector
was chosen as a scalar measure of the solution error at each iteration.* The results of
these tests are plotted in Figures 7.8 and 7.9.
In both cases, it is clear that neural network solution seeding is a highly
effective technique for accelerating convergence. Compared to the other two seeding
methods, the RMS load error for the neural network seeded case drops off quickly and
more or less monotonically. In addition, there are a few interesting and important
observations to be made regarding these plots.
* To prevent the formation of negative diagonal terms during the incomplete
decomposition process, diagonal terms in the approximation matrix M were scaled
by a factor of 1000 prior to decomposition. Offdiagonal terms in M were left
unaltered.
* In practice, both an RMS load error and an RMS displacementchange error should
be used in monitoring convergence. Use of an RMS load error is critical because it
provides a quantitative measure of how far away from equilibrium the structure is at a
particular iteration. In the author's implementation of the NNPCG solver, both of
the error statistics are computed and reported. However, for purposes of comparing
different seeding methods, examining only the RMS load error is sufficient.
RMS Residual Load Error (kips) RMS Residual Load Error (kips)
219
Figure 7.8 Convergence Behavior of an ICPCG Solver Applied to the Analysis
of a Flatslab Bridge Under Uniform Loading
Figure 7.9 Convergence Behavior of an ICPCG Solver Applied to the Analysis
of a Flatslab Bridge Under Vehicular Loading
220
1. Similarity of zero and diagonal seeding. The convergence characteristics of
the zero and diagonal seeding methods are very similar.
2. Slowly diminishing error in zero and diagonal seeding. In both the zero and
diagonal seeding cases, the RMS load error diminishes slowly and
sporadically, remaining large for many iterations.
3. Large initial error in neural network seeding. For each loading condition the
RMS load error of the neural network seeding case is very large during the
early iterations.
All three of these convergence characteristics can be understood by considering the
manner in which the ICPCG algorithm iteratively modifies its estimates of the
displacement unknowns.
Figures 7.107.15 illustrate, in storyboard sequences, the convergence behavior
of the six seeding tests performed. In each figure, the vertical translations (Tz) of the
slab are plotted sidebyside with the corresponding vertical residual (outofbalance)
forces (Fz). Story frames are shown for iterations 0, 5, 25, 50, and 100. The vertical
translation plots roughly correspond to the displaced shapes of the model at the various
stages of convergence. Similarly, the vertical outofbalance force plots illustrate, in a
qualitative manner, how far away from equilibrium the structure is at different stages of
convergence. Plots of the rotations (Rx and Ry) and outofbalance moments (Mx and
My) have been omitted in the interest of space, however, many of the trends exhibited
in the plots shown carry over to those not shown.
221
Residual (kips) Iteration 50
Longitudinal Direction (indies)
Longitudinal Direction (indies)
Figure 7.10 Zero Seeded Convergence of Displacements (TzTranslations) and
Residuals (FzForces) for a Flatslab Bridge Under Uniform Loading
222
Figure 7.11 Diagonal Seeded Convergence of Displacements (TzTranslations)
and Residuals (FzForces) for a Flatslab Bridge Under Uniform
Loading
223
Figure 7.12 Neural Network Seeded Convergence of Displacements
(TzTranslations) and Residuals (FzForces) for a Flatslab Bridge
Under Uniform Loading
224
Longitudinal Direction (indies) Longitudinal Direction (indies)
Figure 7.13 Zero Seeded Convergence of Displacements (TzTranslations) and
Residuals (FzForces) for a Flatslab Bridge Under Vehicular
Loading
225
Longitudinal Direction (indies) Longitudinal Direction (indies)
Figure 7.14 Diagonal Seeded Convergence of Displacements (TzTranslations)
and Residuals (FzForces) for a Flatslab Bridge Under Vehicular
Loading
226
Lcngitudinal Direction (inches) Lon gitudia al Direction (inches)
Figure 7.15 Neural Network Seeded Convergence of Displacements
(TzTranslations) and Residuals (FzForces) for a Flatslab Bridge
Under Vehicular Loading
227
From the figures one can see that in the ICPCG algorithm the displacements
seem to â€œspreadâ€ or â€œrippleâ€ away from the initial load points as convergence
progresses. This is easiest to see in Figures 7.13 and 7.14 which correspond to the
vehicular loading condition analyzed with zero and diagonal seeding respectively. One
may observe from the residual force plots that the residual forces gradually spread
away from the initial load points. In response to the spreading residual forces,
displacements gradually change from zero values to nonzero values in the vicinity of
the residual forces. Therefore, immediate global displacement changes do not occur in
response to localized residuals forces. This fact explains the similar convergence
behavior of the zero and diagonal seeding cases.
Essentially the diagonal seeding case starts out approximately one iteration
ahead of the zero seeding case. At iteration 0, the diagonal seeding case uses the nonÂ¬
zero forces in the load vector and the diagonals of the stiffness matrix to compute
displacement changes at the locations of those forces. At iteration 0 of the zero seeding
case, the solution vector is simply filled with zeros. However, at the next iteration, the
residuals in the zero seeding case will be the same as the original loads in the diagonal
seeding case. Since the displacement changes are very local in nature, the displacement
changes computed for iteration 1 of the zero seeding case will be very similar to (but
not exactly the same as) those of the diagonal seeding case at iteration 0.
The residual spreading effect described above also explains why the residual
force error remains large for many iterations in the zero and diagonal seeding cases. A
sufficient number of iterationsâ€”and a sufficient degree of residual force spreadingâ€”
228
must occur before the structure even begins to approximate the final configuration.
Thus, the residual load error remains large while the spreading behavior continues.
The neural network seeding case is exactly the opposite of the zero and diagonal
seeding casesâ€”instead of gradually migrating toward the final configuration, the neural
network seeded solution attempts to jump directly to the final solution. As a result, the
convergence rate is accelerated. However, as a side effect, the residual load error is
very large during the early stages of convergence. This is because the displacements
encountered in early iterations are immediately large, instead of starting out at
essentially zero as is the case in the other seeding methods.
Consider Figures 7.12 and 7.15 as an example. To the naked eye, there is very
little difference between any of the displaced shapes shown. However, looking at the
residual force plots reveals that there are indeed significant differences. The point is
thisâ€”when an algorithm tries to jump directly to the final solution, any error present in
the computed displacements will cause large residuals to be generated. Since the neural
networks are only approximate, the displacements they predict will oscillate around the
exact solution to within a tolerance. Because the magnitudes of these displacements will
be fairly largeâ€”on the order of magnitude of the exact displacementsâ€”the resulting
residuals will also be large. However, because the initial displaced shape is close to the
exact solution, convergence will still advance rapidly.
229
7.6.2 Preconditioning Using Neural Networks
In addition to using neural networks to seed the solution vector, recall that the
NNPCG algorithm also uses neural networks to perform preconditioning. To test the
effectiveness of neural network preconditioning, the same bridge models analyzed in
the previous section were also analyzed using the full NNPCG algorithmâ€”i.e. neural
network seeding and preconditioning. In neural network preconditioning, the networks
are used within the PCG iteration loop as well as to initially seed the solution vector.
Figure 7.16 illustrates the vertical translations (Tz) and vertical residual forces
(Fz) resulting from a full NNPCG analysis of the vehicular loading case. It is evident
from the figure that there is a problem with the solution process. In fact, the iterative
refinement process stalls almost immediately. The displacements are changed by only a
very small fraction and the residuals do not diminish. Although only iterations 0, 1,
and 2 are shown in the figure, the plots for subsequent iterations are virtually identical.
Conceptually, the NNPCG solution algorithm is sound. Unfortunately, a
practical neural network training issue arises which is responsible for the stalling
phenomenon encountered. When neural networks are constructed, they are iteratively
trained to within an error tolerance that is deemed acceptable for the problem being
solved.
In the neural networks constructed for the NNPCG algorithm, displacements
calculations are split into two componentsâ€”shape and magnitude (refer to Chapter 6 for
a detailed discussion). Shape networks essentially encode normalized displacement
surfaces for the bridge while magnitude networks encode shape scaling information.
230
Figure 7.16 Displacements (TzTranslations) and Residuals (FzForces) for a
Flatslab Bridge Under Vehicular Loading Using the Full NNPCG
Algorithm (Network Seeding and Network Preconditioning)
Thus, to compute a displacement, a shape network is called to compute a normalized
displacement Aâ€žorm and a magnitude network is called to compute a scaling factor
^scale The actual structural displacement is then given by their product,
A = ^norm^ scale 
The problems here are twofold. First, recall from Chapter 6 that, while the
networks were trained to a low average error, there were a few training cases in which
the maximum errors were quite significant. It is evident from these tests that large
network errors can not be tolerated, even if they are limited in number. Such errors
231
Normalized Exact
Displacement Shape
Neural Network
Displacement Shape
Upper & Lower
Tolerance Bounds
Figure 7.17 OneDimensional Illustration of Convergence Criteria
cause the developement of large residuals which in turn cause the NNPCG algorithm
to stop converging toward the correct solution (see discussion below).
The second problem arises in choosing a training tolerance for the shape
networks. Each displacement surface is normalized with respect to the largest
magnitude displacement occurring in that particular shape. Thus, the output values of
the shape networks are always in the normalized range [1,1].^ When a convergence
tolerance is chosen, it is chosen in the range [0,1]. A simplified onedimensional
example is shown in Figure 7.17. All of the ordinates on the normalized shape have
values in the range [0,1]. Assume that a convergence tolerance of 0.1 is chosen. This
choice says that up to a 10% error at the maximum magnitude location â€œaâ€ can be
tolerated for this problem.
Thus the oscillatingâ€”and hypothetically neural network computedâ€”shape
shown in the figure is within the specified tolerance. Note, however, that by choosing a
tolerance of 0.1, we allow a 50% error to occur at â€œbâ€ relative to itself. In some
t Actually, the output range of the shape networks is confined to a range narrower than
[1,1]. This is done so that only the â€œnearly linearâ€ portion of the sigmoid output
function is used (see Chapter 6). However, the constricted range is always expanded
back to [1,1] when data is sent from the neural network back to the equation solver.
232
applications this may not be cause for concern because the overall magnitude of â€œbâ€
may always be small relative to the magnitude at â€œaâ€. Thus, the error at â€œbâ€, although
large relative itself, is small when put in the context of the overall problem. However,
in the neural networks used in this research, the normalized displacement surfaces are
scaled before arriving at a set of final displacements.
When a large scaling factor is applied to a value similar to point â€œbâ€ of the
simplified example, a significant error can be introduced into the problem. This is
precisely the situation which occurs when large residual forces are generated during the
solution process. Since neural network seeding causes large initial residuals, a large
amount error is quickly introduced into the solution process and it essentially saturates.
A large residual causes a large error in a displacement calculation which in turn causes
a larger residual. As this cycle repeats, the residual forces grow rapidly.
Looking at the NNPCG algorithm (Equations (7.33)(7.40)) one can see that
the a term (Equation (7.36)) serves as a step size parameter. Displacement changes are
first multiplied by the a step size factor before using them to update the total
displacements. However, at each iteration, the denominator of the a calculation grows
faster than the numerator because of the growing residuals. As a result, a quickly
tends toward zero. Table 7.2 shows the values of a which occur during a full NN
PCG solution of the flatslab bridge with vehicular loading. It is clear from the table
that the growing residuals cause the denominator of a to grow and a to quickly
diminish. As a tends towards zero, refinement in the displacements essentially stalls.
233
Table 7.2 Decreasing Step Size (a) During Neural Network Preconditioning
Iteration i
r(TnNN(r(i))
P(oMn
ro)NN(r{i))
1 T A
pcoMo
0
0.885778E+04
0.561218E02
0.561218E02
1
0.136434E+03
0.172476E03
0.172476E03
2
0.465492E+01
0.576414E05
0.576414E05
3
0.219950E+00
0.265231E06
0.26523 IE06
4
0.131720E01
0.154710E07
0.154710E07
5
0.950806E03
0.108810E08
0.108810E08
6
0.800233E04
0.892588E10
0.892588E10
7
0.766925E05
0.834038E11
0.834038E11
8
0.822289E06
0.872152E12
0.872152E12
9
0.972948E07
0.100675E12
0.100675E12
One solution to this problem is to use a more sophisticated method of
controlling error tolerance during network training. Another solution is to recombine
each shape and magnitude network pair into a single network. In this way terms which
are small, and which will remain small in the context of the overall problem, can
tolerate a larger relative amount of error because they will have less influence on the
overall problem. Both of these approaches would require the neural networks to be
retrained using new training data and new training control parameters.
CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS
8.1 Computer Assisted Modeling
A modeling preprocessor has been developed that can be used by engineers to
model highway bridges for finite element analysis (FEA). The preprocessor considers
the types of bridge construction most commonly encountered in practiceâ€”prestressed
concrete girder, steel girder, reinforced concrete Tbeam, and reinforced concrete flat
slab bridges. All of the basic structural components of these bridge types are modeled
by the preprocessor. Basic structural components include girders, parapets,
diaphragms, pretensioning tendons, posttensioning tendons, hinges, and the deck slab.
Three separate finite element modeling methods of representing composite
action between the girders (parapets) and slab are provided by the preprocessor. The
first method represents cases in which composite action is not present. The second and
third methods represent the presence of full composite action between the girders
(parapets) and slab by using two and threedimensional modeling techniques
respectively. Shear lag in the deck slab is properly accounted for in the three
dimensional model.
Several features provided in the preprocessor are aimed specifically at
facilitating rapid data generation. Some of these features include embedded databases
234
235
of commonly used structural objects (such as standard girder crosssections and
standard vehicles); rapid generation of bridge geometry, vehicle positions, girder cross
sectional data, and tendon profiles; and a history mechanism for reusing or editing
previously created bridge models. Using the provided vehicle positioning features, an
engineer can quickly describe all of the vehicle positions that need to be investigated.
Once vehicle positions are specified, statically equivalent finite element nodal
loads are automatically generated by the preprocessor. This feature relieves the
engineer of the task of having to manually calculate equivalent nodal loads for each of
theâ€”quite possiblyâ€”thousands of wheel positions being investigated. Also, dead loads
are automatically generated for each of the structural components present in the bridge.
Dead load calculations are based on the specific weights of the construction materials
and the geometric bridge data. Construction stages are correctly accounted for in the
FEA models by computing the appropriate loading conditions and structural
configuration of the bridge at each distinct structural stage.
While the preprocessor provides a powerful and flexible tool for modeling
highway bridges, in a primarily interactive manner, there is room for improvement. In
particular, the interface between the user and the software would be more user friendly
and efficient if it were converted from the questionresponse input model currently used
to graphical input model. Future work on computer assisted modeling software will
concentrate in this area. Also, the ability to handle more general geometrical layoutsâ€”
for example curved bridge geometryâ€”would be a desirable addition to the
preprocessor.
236
8.2 Data Compression in FEA
A realtime data compression strategy appropriate for inclusion in FEA software
has been developed, implemented, and extensively tested. In most analysis situations
the data generated by FEA software is rich in repetition. By integrating data
compression techniques directly Into FEA software, it has been shown that the amount
of outofcore storage required by finite element analyses can be vastly reduced. This is
especially true in cases where the FEA model is highly regular geometrically or
otherwise. Just such a situation arises when computer assisted modeling software is
used because this type of software generally creates very regular finite element meshes.
Thus, models created by software such as the preprocessor developed in this
study are prime candidates for the application of data compression. Parametric studies
revealed that the outofcore storage requirements for such analyses could be reduced
by an order or magnitude in many cases. As an additional benefit of the use of data
compression, the execution time required for many of the analysis situations examined
was also reduced. This benefit is attributable to the fact that, when using data
compression, a substantially reduced amount of disk input/output (I/O) must be
performed. In many cases, the amount time spent compressing and decompressing data
during the analysis is more than offset by the time saved by performing less disk
activity.
Whereas a reduction in outofcore storage is virtually guaranteed by the use of
data compression, a reduction in the required execution time is not. The degree of
compression achieved is independent of the computer platform that the software is
237
running on. However, whether or not the savings in I/O time will completely or only
partially offset the time spent compressing and decompressing data will depend on
characteristics of the problem and the computer platform. In particular, systems in
which disk I/O can become a performance bottleneck will usually benefit greatly from
data compression. Personal computers (PCs) running the DOS operating system tend to
fall into this category.
The data compression strategy presented in this study was implemented using
the C programming language. It was installed in the SEAS FEA package (which is also
written in C) for testing. An interface layer was developed (also in the C language) to
allow the data compression library to be called directly from software written in
Fortran. Using this interface, data compression was implemented and tested in a
Fortran coded FEA package that is used in the analysis of highway bridges. Tests
results indicated that Fortran I/O is often less efficient than C I/O and as a result FEA
software written in Fortran can greatly benefit from data compression. In tests
performed using moderate size bridge models, analysis execution times often decreased
by a factor of 2 to 3. Simultaneously, there was a (roughly) order of magnitude
decrease in outofcore storage requirements.
Due to the nature of the buffering scheme used in the data compression library,
only sequential data files can be compressed. An area of future research which should
be pursued is that of adapting the buffering algorithm to also handle direct access files.
Once this is done, data compression will be able to be applied in more applications than
is currently possible. Also, the use of data compression for incore memory
compression is currently being examined. Some preliminary work has already been
238
done to evaluate the feasibility of constructing a compressed core equation solver using
data compression techniques.
8.3 Neu 