OPTIMIZATION OF REINFORCED
CONCRETE FRAMES USING
INTEGRATED ANALYSIS AND RELIABILITY
By
ALFREDO V. SOEIRO
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
1989
ACKNOWLEDGEMENTS
I want to express my sincerest gratitude to Dr. Marc
Hoit for monitoring my research, for the inventive ideas and
for his constant support. I am specially thankful to Dr.
Fernando Fagundo for the productive discussions, his
friendship and his vigorous encouragement. I owe to these
two my best recollections from the University of Florida.
My sincerest appreciation is extended to Dr. Prabhat Hajela
for the teachings and the careful reading of my
dissertation. My indebtment goes also to Dr. Clifford Hays
and Dr. John Lybas for the useful conversations and their
activity in my committee. I am also grateful to Dr. David
Bloomquist for his support to initiate my geotechnical
reliability research and his continuous disposition to help.
I would also like to acknowledge my appreciation to the
Fulbright Comission and to the Department of Civil
Engineering of the University of Florida for their financial
aid and the opportunity to study the fascinating area of
structural optimization. My thanks go also to the
Department of Civil Engineering of the University of Porto,
specially to Dr. Adao da Fonseca, for giving me the
possibility to research and study in the USA.
My sincere appreciation and best remembrances go to my
friends in the Gainesville Portuguese community and to my
colleagues Jose, Joon, Lin and Prasit that helped smooth
the life contours created by the research work. Finally,
my gratitude goes to my wife, Paula, for her work, her
patience and her support throughout the whole period during
which this dissertation was completed.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS. ................................... ii
LIST OF TABLES. .......... ... ..... . ...... ....... vi
LIST OF FIGURES.. .. .............. ............... vii
ABSTRACT ............. .. ........ ..... .. .. . .. Viii
CHAPTERS
1 STRUCTURAL OPTIMIZATION....................... 1
Introduction.......................... ....... 1
Historical Background....................... 3
Methods ...... ............................... 6
Typical Applications......................... 8
Study Objectives.............................. 15
Summary... ..................... ............. 17
2 INTEGRATED OPTIMIZATION OF LINEAR FRAMES...... 21
Original Research........................... 21
Augmented Lagrangian Function................ 22
Unconstrained Minimization Techniques......... 25
Final Results........................ ..... .... 28
Further Improvements.......................... 32
3 NONLINEAR REINFORCED CONCRETE ELEMENT......... 37
Introduction ........................ ....... 37
Element Modeling Survey...................... 38
Beam Element with Inelastic Hinges............ 40
Beam Element Stiffness........................ 49
4 STRUCTURAL ELEMENT RELIABILITY ............... 54
Introduction............... . .. .. .. .. 54
Two Dimensional Space Example................ 60
Reinforced Concrete Element Reliability....... 69
5 SYSTEM RELIABILITY.......................... 74
Introduction.... .............................. 74
System Reliability and Optimization........... 75
Methods................... ................ '77
Generation of Failure Modes................. 82
Beta Unzipping Method ....................... 90
Page
6 PROCEDURE IMPLEMENTATION..................... 97
Introduction .......................... ... 97
Augmented Lagrangian Formulation.............. 98
Generalized Reduced Gradient.................. 108
Reliability................................ 114
7 EXAMPLES....................................... 119
Introduction......................... ..... 119
Result Verification.......................... 120
Debug Frame............................ ....... 121
Compared Frame............. .................... 131
Building Frame................................ 136
8 CONCLUSIONS AND RECOMMENDATIONS............... 139
Linear Material Behavior...................... 139
Nonlinear Material Behavior................... 141
Future Work ................................ 142
APPENDICES
A AUGMENTED LAGRANGIAN SUBROUTINES.............. 145
B GENERALIZED REDUCED GRADIENT EXAMPLE.......... 189
C GENERALIZED REDUCED GRADIENT SUBROUTINES...... 195
REFERENCES......................... ............ 230
BIOGRAPHICAL SKETCH................................... 236
LIST OF TABLES
Table Page
7.1. Debug frame (GRG): linear version results......... 124
7.2. Debug frame: Augmented Lagrangian version.......... 126
7.3. Debug frame (GRG): yielding stiffness results..... 127
7.4. Debug frame (GRG): secant stiffness results....... 129
7.5. Debug frame: element moments...................... 130
7.6. Compared frame: initial steel
area reinforcement...................... 133
7.7. Compared frame results............................ 135
7.8. Building frame results............................ 138
LIST OF FIGURES
Figure Pae
1.1. Implicit optimization............................ 5
1.2. Element optimization........................... 10
1.3. Truss optimization ............................. 11
1.4. System optimization ............................. 13
1.5. Geometry optimization ...... .................... 14
2.1. Pattern Search......... ....... .................. 27
2.2. Cantilever beam .................................. 29
2.3. One bay frame ................................... 31
2.4. Gradient method.................................. 34
3.1. Element model................. .............. ... 41
3.2. Material behavior.............. .................. 43
3.3. Reinforced concrete section...................... 45
3.4. Element deformation diagrams..................... 48
3.5. Curvature integration............................ 50
3.6. Secant spring stiffness.......................... 52
4.1. Design safety region............................. 61
4.2. Probabilistic functions.......................... 64
4.3. Safety checks ...................................... 66
4.4. Reliability index................................. 68
5.1. System models .................... ............... 78
5.2. Failure graph ................ .............. ... 83
5.3. Element displacements definition................. 85
5.4. System failure modes ............................ 91
5.5. Combinatorial tree............................... 96
6.1. Augmented lagrangian function plot............... 104
6.2. Augmented Lagrangian version flowchart........... 106
6.3. Generalized Reduced Gradient version flowchart... 113
6.4. Bilinear elasticplastic diagram................. 117
7.1. Displacement verification ....................... 122
7.2. Debug frame ................................ .......... 123
7.3. Compared frame................................... 132
7.4. Building frame................................... 137
B.1. Integrated optimization example.................. 191
vii
Abstract of the Dissertation Presented to the Graduate
School of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
OPTIMIZATION OF REINFORCED
CONCRETE FRAMES USING
INTEGRATED ANALYSIS AND RELIABILITY
By
ALFREDO V. SOEIRO
August 1989
Chairman: Dr. Marc I. Hoit
Cochairman: Dr. Fernando E. Fagundo
Major Department: Civil Engineering
Simultaneous analysis and design were considered
in the optimization of reinforced concrete frames. Frame
elements had rectangular cross sections with double steel
reinforcement. Design variables were the section
dimensions, the area of steel reinforcement and the
structure global displacements. Equality constraints were
the equilibrium equations and inequality constraints were
generated by element reliability requirements, code
reinforcement ratios and section dimension bounds.
Optimization strategies were based on the Augmented
Lagrangian formulation and on the Generalized Reduced
Gradient method.
Reliability of the frames was considered at the element
and system level. An element failure function was defined
using moment forces and flexural strength. The random
viii
variables considered were flexural strength of concrete and
external loads. System reliability was evaluated at the
mechanism level using combinations of the elementary failure
mechanisms.
Optimization of the frames considering material
nonlinear behavior was also investigated. Inclusion of this
property was performed using a onecomponent model for the
reinforced concrete element. Inelastic rotational springs
were added to the ends of the linear elastic element. The
element matrix was obtained by condensation of element
elastic stiffness and secant spring stiffness.
Three frames were researched. Respective results using
linear material behavior were discussed. In these three
cases the optimal solutions were found. Element reliability
constraints were active and system reliability was
satisfied. The integrated formulation was validated in the
linear behavior range. The nonlinear material behavior
results were presented for the smaller frame.
CHAPTER 1
STRUCTURAL OPTIMIZATION
Introduction
Optimization is a state of mind that is always
implicitly present in the structural engineering process.
From experience engineers learn to recognize good initial
dimension ratios so that their preliminary designs demand
small changes through the iterative process and that
elements are not overdesigned. The motivation behind this
attitude is to create a structure that for given purposes is
simultaneously useful and economic.
Structural optimization theory tries to rationalize
this methodology for several reasons. The main one is to
reduce the design time, specially for repetitive projects.
It provides a systematized logical design procedure and
yields some design improvement over conventional methods.
It tries to avoid bias due to engineering intuition and
experience. It also increases the possibility of obtaining
improved designs and requires a minimal amount of human
machine interaction.
There are, however, some limitations and disadvantages
when using design optimization techniques. The first one is
the increase in computational time when the number of design
variables becomes large. Another disadvantage is that the
applicability of the specific analysis program that results
from the optimization formulation is generally limited to
the particular purpose to which it was developed. A common
inconvenience is that conceptual errors and incomplete
formulations are frequent. Another drawback is that most
optimization algorithms have difficulty in dealing with
nonlinear and discontinuous functions and, hence, caution
must be exercised when formulating the design problem.
Another factor of concern is that the optimization algorithm
does not guarantee convergence to the global optimum design;
yielding on most occasions local optimum points. These
facts lead to the conclusion that optimization results may
often be misleading and, therefore, should always be
examined.
Therefore, some authors suggest that the word
"optimization" in structural design should be replaced by
"design improvement" as a better expression to materialize
the root and outcome of this structural design activity (1).
Nevertheless, there is an increasing recognition that it is
a convenient and valuable tool to improve structural designs
has been increasing among the designers community.
Historical Background
Throughout time there have been various attempts to
address structural optimization. The earliest ideas of
optimum design can be found in Galileo's works concerning
the bending strength of beams. Other eminent scientists
like Bernouilli, Lagrange, Young, worked on structural
optimum design based on applied mechanics concepts (2).
These pioneering attempts were based on a close relation to
the thoughts and accomplishments of structural mechanics.
They started with hypotheses of stress distribution in
flexural elements and ended with material fatigue laws.
The accepted first work in structural optimization
discusses layout theory, or structural topology. The paper
focused on the grouping of truss bars that creates the
minimum weight structure for a given set of loads and
materials. The author of this primary achievement was
Maxwell, in 1854, and Michell developed and publicized these
concepts in 1904 (3). The practical application of these
theorems was never accomplished since significant
constraints were not included in the original works.
Some procedures widely used by structural designers are
nothing more than techniques of structural optimization. A
well known example is the socalled Magnel's diagram (4).
It is used to find the optimal eccentricity of the cable
that leads to the smallest prestressing force without
exceeding the limits imposed on the stresses in prestressed
concrete beams with excess capacity. This is a typical
maximization problem in a linear design space, where the
design variables are the eccentricity and the inverse of the
cable prestressing force. The objective function is the
value of the inverse of the cable prestressing force, and is
to be maximized. The constraints represent the allowable
stresses in tension and compression at the top and bottom of
the crosssection. The problem is solved using a graphic
representation of the problem, as shown in Figure 1.1, but
could be solved numerically using the Simplex method.
Numerical optimization methods and techniques have been
widely researched and used in the operations research area,
commonly known as Mathematical Programming. The practical
application of these theories has been carried out in
several areas for some decades like management, economic
analysis, warfare, and industrial production. Lucien Schmit
was the first to use nonlinear programming techniques in
structural engineering design (5). The main purpose of
structural optimization methods was to supply an automated
tool to help the designer distribute scanty resources.
Presently, anyone who wants to consider optimum structural
design must become familiar with recent synthesis approaches
as well as with accepted analysis procedures.
5
Magnel' s Diagram
Optimum Pair Pe
optimum
Feasible region
P Initial prestressing force:
e Eccentricity of cable;
e* maximum cable eccentricity;
a),b) minimum 1/P;
c).d) maximum 1/P.
Figure 1.1 Implicit optimization.
1/P
Methods
In the last twenty years researchers have made
considerable advances in developing techniques of optimum
design. Research and exploration of these methods were
mainly developed in the aeronautical and mechanical
industries, where the need for more economical and efficient
final products was extremely important. More recently, with
the availability of increasing computer capabilities, civil
engineering researchers and designers have increased their
participation in structural optimization following the lines
defined by the other engineering disciplines. Optimization
methods are, nevertheless, common to these different
engineering design areas and are mainly divided in two
groups. These are commonly known by the names Optimality
Criteria and Mathematical Programming (6). Another area in
structural optimization researched by a few scientists is
based on duality theory concepts, and is an attempt to unify
the two basic methodologies (7).
Optimality Criteria methods are based on an iterative
approach where the conditions for an optimum solution are
previously defined. The concept can be used as the basis
for the selection of a structure with minimum volume. This
methodology derives from the extreme principles of
structural mechanics and has been limited to simple
structural forms and loading conditions. The formulation
can be mathematically expressed as follows:
xk+l = (p (xk,uk+l)
where x is the vector of design variables, uk+ is an
estimative of lagrangian multipliers and p is an adequate
recurrence relation. Estimation of the lagrangian
multipliers is made using the active constraints, those
inequality or equality constraints with value close to zero.
Recurrence relation p and lagrangian multipliers represent
the necessary conditions for optimality known as KuhnTucker
conditions.
On the other hand, the Mathematical Programming
approach establishes an iterative method that updates the
search direction. It seeks the maximum or minimum of
multivariable function subject to limitations expressed by
constraint functions. The iterative procedure may be
defined as follows:
Xk+l = xk + ak dk
where ak is the step size and dk is the search direction.
The search direction is obtained through an analysis of the
optimization problem and the step size depends on the one
dimensional search along that direction. Methods of the
second class may be divided in two areas. These areas are
transformation methods, like penalty functions, barrier
functions and method of multipliers, and primal methods,
such as sequential linear and quadratic programming,
gradient projection method, generalized reduced gradient and
method of feasible directions.
Typical Applications
In structural optimal design applications there are
several types of problems. They address different targets in
structural design such as the best configuration for a truss
or the cross sections of a prestressed concrete beam. There
are four main properties of any structure that may be
focused by structural optimization. These are mechanical or
physical properties of the material, topology of the
structure, geometric layout of the structure and cross
sectional dimensions. Main types of applications are
optimization of elements, truss bars, flexural systems,
continuum systems, geometry and topology (8).
In the case dealing with element optimization, the
search is done with a reduced number of variables and the
use of code provisions transformed adequately to the
optimization formulation. Element forces are found, element
cross section is optimized, updated element forces are
computed and the process is repeated until there is
convergence. For instance, the optimal design of steel wide
flange sections may have as design variables the width and
thickness of the web and flanges. Constraints may be
obtained in an explicit form, as the evaluation of the
objective and constraint functions does not require matrix
structural analysis. The minimization technique may be
chosen as any one of the available direct search methods
(9). Examples of design variables in element optimization
are presented in Figure 1.2.
Optimization of truss bar sections has been thoroughly
studied due to the simplicity of truss structural
optimization problems. There is a decline of interest since
they are now rarely used in present structural engineering.
Each bar is represented by one variable and the global
stiffness matrix terms are linear functions of these
variables. Of the various improvement techniques one is
based on variable linking, consequently reducing the size of
the problem. Another technique to decrease the size is
based on constraint deletion, where inactive constraints are
temporarily kept out of the optimization process. There are
various formulations for the analysis model based on plastic
analysis, force or displacement method (10). An example of
the formulation used for truss optimization is presented in
Figure 1.3.
The problem of system optimization is commonly
addressed using design sensitivity analysis and explicit
approximations of constraint functions. The intent is to
improve the performance of the chosen algorithm. Design
sensitivity analysis is the calculation of the analytical
derivatives of the objective and constraint functions with
respect to the design variables. This information about the
change in the value of a constraint related to the changes
STEEL SECTION
DESIGN VARIABLES
Flange width
Flange thickness
Web height
Web thickness
CONCRETE SECTION
DESIGN VARIABLES
Width
Height
Top reinforcement
Bottom reinforcement
Figure 1.2. Element optimization.
M ///~/////
15 ft
5 ft 12 ft 10 ft 10 ft 12 ft 5 t
Minimize Z LiAi
subject to
Fi < Fc
Fi < Ft
where
Li length of truss bar i
Ai area of truss bar i
Fi stress in truss bar i
Fc allowable compressive stress
Ft allowable tensile stress
Figure 1.3. Truss optimization.
in the design variables, contributes to the reduction of the
exact analyses required during the optimization process.
Explicit approximations of the constraint functions using a
first order Taylor series expansion are widely used in
Optimality Criteria and Mathematical Programming methods.
In large and continuum systems some other techniques are
used. For example, the sequential optimization of
substructures or decomposition using model coordination
techniques are used to improve the performance (11). An
example of a type of system optimization is illustrated in
Figure 1.4.
Geometric and topologic optimization creates geometric
design variables that are, for instance, the coordinates of
nodes in a finite element mesh or the pier location for a
continuous bridge. In certain cases where the areas of the
elements have zero as lower bounds, the unnecessary elements
can be eliminated by the optimization algorithm. Sometimes
the concept of separate design spaces, one for joint
coordinates and the other for cross sectional element sizes,
is used when trying to reduce the size of the design space
considered at any stage (12). An example of optimal
configuration is presented in Figure 1.5.
In large optimization problems it is usual to use
multilevel optimization techniques where the structural
designer has to coordinate and optimize at several levels of
the design process. This technique is also useful when the
main goal is to find the optimum geometry besides optimizing
Optimization of a Two span prestressed beam
XI
X2
XI to X6 Section geometry
X7 to X9 Eccentricities of draped cable
X1O Prestressing force
Figure 1.4. System optimization.
OPTIMIZATION
OF
TRUSS GEOMETRY
Initial Configuration
S.
S.
S.
S.
S.
S.
S.
S.
S.
Optimal Configuration
Figure 1.5. Geometry optimization.
Load
Load
the structural elements. Design variables that control the
geometry are often handled better when considered separately
from the set of sizing variables (13).
Study Objectives
The main objectives of the present work are to combine
adequately optimization and reliability concepts and to test
the performance of the integrated approach to reinforced
concrete frames. Reliability requirements are imposed at
the element and the system level. At element level a
maximum probability of failure is imposed for each element
and at the system level a minimum reliability index is
imposed for the failure mechanisms.
The material behavior of the reinforced concrete
elements is separated in two phases. The first considers
linear material behavior and the second includes the
concrete and steel nonlinear behavior.
Structural frame optimization problems have usually
been formulated based on the cycling between two distinct
phases, analysis and optimal design. This methodology is
the classical approach in structural optimization. The
first phase consists in an initial sizing or structure
definition. In the second phase, a structural analysis is
performed and in the third phase, the structure is resized
or redefined using Mathematical Programming or Optimality
Criteria methods. The cycling between phases two and three
is interrupted when the termination criteria are met.
The research option summarized here combines phases two
and three into one only stage. This is accomplished by the
addition of the global displacements to the set of design
variables. This addition implies that the equilibrium
equations, solved explicitly in the cycling approach, are
added to the set of constraints as equality constraints.
These new equality constraints will be solved iteratively
while in the cycling approach the solution is obtained using
a Gauss type decomposition. The main objective behind the
adoption of this strategy was to experiment this formulation
where the variables related with element stiffness
definition and the displacement variables are in the same
design space. For that reason the simultaneous
optimization and iterative solution of equilibrium equations
could be more efficient than the classical nested approach.
The application of this formulation was initially
performed with elastic linear frames subjected to static
loading. The constraints consisted of limiting the global
displacements and the element stresses, besides the
additional set of qualities representing the equilibrium
constraints. The optimization method used consisted of
unconstrained minimization of an augmented lagrangian
function of the initial objective function and the equality
and inequality constraints (14).
Summary
Results obtained with the integrated approach were
encouraging and proved that the method was acceptable for
elastic design purposes with displacement and stress
constraints. Despite the fact that optimum values were
obtained there was however an increase in the size of the
problem. This modification of the problem size was due to
the fact that the number of variables and the set of
constraints augmented.
The final type of optimization problem considered in
this work was the minimization of the total cost of a
reinforced concrete plane frame submitted to static loading
considering the actual stressstrain diagram for concrete
and the elasticplastic behavior of the reinforcing steel.
A typical element had a constant rectangular section and
doubly reinforced with equal amount of flexural steel on
both sides. Width and height of the cross sections had
prescribed lower bounds, representing practical requirements
and an adequate ratio between the height and the width. The
amount of steel was limited by lower and upper bounds
dictated by the minimum and maximum reinforcing steel ratios
requested by the Building Code Requirements for Reinforced
Concrete, commonly known as ACI 31883.
Inequality constraints considered included maximum
values for the global displacements and a minimum
reliability index for the element flexural failure function.
Displacements allowed were based on serviceability
requirements like cracking and relative story drift. The
reliability indices were based on usual values of
probability of failure used in design codes. Only the
flexural behavior of the frames was analyzed since it is the
most important for usual structures and the members were
modeled as beam elements.
Inelastic behavior of the structure due to the material
nonlinearities imposes a change of the global stiffness
terms independently of those dictated by the alterations of
the dimensions during the optimization search. For that
reason, the reinforced concrete element was modeled as a
linear elastic beam with nonlinear rotational springs at
each end. Rotational spring stiffness was considered
infinite when the moment was below the yielding moment.
Above that value the element stiffness was inverted to its
flexibility and the inverse of the secant spring stiffness
value was added to the corresponding diagonal terms. Spring
stiffness was calculated using the secant value of the
bilinear momentrotation diagram corresponding to the
current global rotation. Values of the yielding and
ultimate moments were obtained by integrating the actual
stressstrain diagram for the compressive force in the
concrete. The corresponding rotation at a hinge was
calculated by integrating the curvature diagram along the
element.
Element reliability was evaluated using a Level 2
method, i.e., an approximation to the evaluation of the
exact probability of failure. The statistical variables
considered were those assumed to have greater influence on
the final result. These were the compressive strength of
concrete and the external loads, assumed as normal
distributed variables. The corresponding reliability index
was calculated for constraint evaluation using the ultimate
moment obtained from the integration of the respective
strain diagram.
Optimization techniques tested were based on the
Augmented Lagrangian and the Generalized Reduced Gradient
methods. The optimization problem was run, and after
termination, the structure probability of failure was
compared with the assigned value. If the result was not
satisfactory, the process was restarted with updated values
of the element reliability indices for the members involved
in the most probable collapse mechanism.
Evaluation of the system reliability was divided in two
phases. First phase consisted of the identification of the
elementary collapse mechanisms. In the second phase these
elementary mechanisms were linearly combined to generate all
significant mechanisms. System reliability was calculated
considering the frame as a series system where each element
is one of these mechanisms with higher probability of
failure.
Generation of the fundamental collapse mechanisms was
made using Watwood's method (15). The automatic procedure
consisted of using the geometric configuration of the frame
and external loading to find all the.one degree of freedom
failure mechanisms. The reliabilities of these mechanisms
was calculated using the corresponding failure functions
System reliability was evaluated using the Beta
unzipping method (16). The elementary mechanisms were
linearly combined to obtain other failure mechanisms. The
corresponding failure functions were created and the
associated reliability indices calculated. In each set of
combinations only those in the closeness of the minimum were
considered for the next combination (17).
CHAPTER 2
INTEGRATED OPTIMIZATION OF LINEAR FRAMES
Original Research
Integrated formulations for structural optimization
problems has received little attention in the published
literature. The works of L. Schmit and R. L. Fox are
considered the pioneering work as applied to integrated
structural optimization (18). The concept of this
structural synthesis problem is to combine the design
variables with the behavioral variables.
The immediate consequences of this concept are that the
problem has a larger number of design variables and the
traditional nested analysisoptimization process is avoided.
This approach has not been popular since past performance
was not comparable to the iterative techniques based on
Optimality Criteria and Mathematical Programming concepts.
In the integrated formulation the equilibrium constraints
generate a large additional number of equality constraints.
Several researchers have recently adopted the
integrated approach with encouraging results. These recent
attempts have been motivated by new solution procedures
21
considered more adequate for this type of formulation and by
computer hardware development. An example is the
optimization of elements with stiffness and strength
properties proportional to the transverse size of the
elements with linearization of the displacement constraints
(19). Another algorithm uses the incremental load approach
and conjugate gradient methods to optimize a structure
subjected to nonlinear collapse constraints (20). In this
case the stiffness matrix is approximated using the element
byelement technique (21). A more recent work uses a new
solution technique based on Geometric Programming theory
(22). In this formulation the equilibrium constraints are
the sum of geometric terms that are function of the design
variables.
This chapter describes research that was conducted to
study the integrated analysis approach for portal frames
with linear behavior and static loading (2326). The
initial phase addressed only constraints on the
displacements. Stress constraints were added on a second
phase. Throughout this part of the study the frame elements
had continuous prismatic rectangular cross section.
Augmented Lagrangian Function
The optimization technique of cycling unconstrained
minimizations of a penalty function, based on an pseudo
objective augmented lagrangian function, was chosen as the
solution scheme (27). The design variables were the areas
and inertias of each element and the global displacements.
Since it is a planar frame there are three degrees of
freedom for each joint in the structure.
The merit function used was the volume of the
structure. In frames made with one material, volume is
generally considered to be proportional to the structure
cost. This value was calculated as the sum, for all
elements, of the product of the element area times the
respective length. The set of inequality constraints was
generated by the structure physical behavior and material
properties. Limits were imposed on the global displacements
and, in the final stage, the element stresses were also
bounded.
The compatibility and equilibrium requirements were
guaranteed by the additional group of equality constraints.
This set was given by the product of the stiffness matrix
and the vector of global displacements from which the vector
of external global loads was subtracted.
A brief description of the problem variables and
respective formulation for a typical planar frame is the
following:
Structural parameters
n structural elements;
m number of global degrees of freedom;
R vector of static external loads;
D vector of bounds of m;
Design variables
Xk, k=l,3,...,2n1  area of element (k+l)/2;
xj, j=2,4,...,2n  inertia of element j/2;
xi, i=2n+l,2n+2,...,2n+m  global displacements;
Objective Function
f(K) = Z lpxkk
p=l,n
where
lp length of element p;
Equality Constraints
H(x) = K x* R
where
K global stiffness matrix;
x* displacement vector;
Inequality Constraints
G(x) = x* D < 0
Augmented Lagrangian Function
L(x,u,v) = f(x) + u H + P H H + v G' + P G'G'
where
u, V lagrangian multipliers;
P penalty factor;
G' maximum of (G, y/2P}.
The optimization procedure consists of several cycles
of unconstrained minimization of the pseudoobjective
function. The values of the lagrangian multipliers are kept
constant during each cycle of the unconstrained
minimization. At the end of an unconstrained minimization
cycle, the multipliers are updated using an appropriate rule
(12). The procedure is repeated for successive cycles until
there is no significant change of the objective function.
At this point the primal and dual optima have been found and
the algorithm stops.
Unconstrained Minimization Techniques
Initially the technique used for the unconstrained
minimization of the augmented lagrangian function was a zero
order method referred to as the Hooke and Jeeves method or
Pattern Search. The classification as a zero order method
means that it does not utilize any information about the
form or shape of the function. After the phase when stress
constraints were added, a first order method, Steepest
Descent, was tested as an improvement in the algorithm's
performance (27). The technique is based on the gradient of
the function that indicates the direction with the highest
slope at a given point. Second order methods were
determined inappropriate because the pseudo inequality
constraints, g', have discontinuous second derivatives.
Hooke and Jeeves method is an iterative procedure where
each step may involve two kinds of moves. The first type of
moves explores the local configuration of the pseudo
objective function along the directions of the design
variables. The investigation is done within a prescribed
step size from the current temporary design point. Each
variable is investigated one at a time. The value of the
step size is increased or decreased with success or failure
in the exploration. This search along the coordinate
directions will eventually lead to a smaller value of the
pseudoobjective function. Otherwise the optimum has been
reached and the exploration stops.
Once all variables have been searched, a pattern move
is attempted. The pattern direction is defined by the
starting and final points of the variable search and a move
is made along that direction. The process of exploration
and pattern moves is repeated until there is no significant
improvement of the pseudoobjective function. A graphic
example is presented in Figure 2.1. The initial point of
the variable search, 1, and the final point of that cycle,
4, define a pattern direction that yields a better design
point, 5.
HOOKE and JEEVES
I Initial Point
6 Final Point
X24
4/5 Pattern Move
Function Contours
Figure 2.1. Pattern Search.
A computer program was written in accordance with the
previous statements and discussions. The structure of the
program was conceived by taking into account future
inclusions of other types of constraints, changes in the
minimization techniques, element replacements and extension
to nonlinear and dynamic problems. Hence the program was
divided into separate subprograms for the independent tasks
(26).
Final Results
The performance and accuracy of the formulation
described above was evaluated. Test examples for that
purpose were structures with an explicit optimal
configuration or simple frames. In the isostatic examples
the optimal explicit solutions could be obtained and
compared to the computer results. For the other structures,
several runs were made with different initial design points
and the optimal configuration was determined.
Minimum values were imposed for the dimensions of the
cross sections, represented by lower bounds of the areas and
moments of inertia. The optimization results show the final
values of the displacement variables as the exact solutions
for the equilibrium equations. The final area and moment of
inertia are also the expected optimal values. Results of a
cantilever beam are presented in Figure 2.2.
XI.X2 X
:1 area of beam
:2 inertia of beam
3 horizontal tip displacement
4 vertical tip displacement
:5 tip rotation
VARIABLE INITIAL FINAL
Xl (in2) 1.0 6.55
X2 (in4) 1.0 78625
X3 (in) 0.4 0.500
X4 (in) 0.4 0.353
X5 (rad) 0.4 0.006
Figure 2.2. Cantilever beam.
/
Penalty factors used in these runs were of an order of
magnitude greater than that of the objective function and
constraints. They were kept constant during each
optimization cycle. Scaling was also mandatory since the
various terms of augmented lagrangian function have
different orders of magnitude. The adopted scaling method
consisted of using the inverse of the initial value of the
expressions concerned. Initial guesses for the design
variables were also important for the algorithm performance.
The closer these initial designs were to the optimum, the
faster the convergence rate.
An updated version of this algorithm was created with
the addition of stress constraints. The results of the
structures used to test this addition illustrated the
adequacy of the method for this type of problems. Again,
for the cantilever beam with the explicit solution, the
optimum results were obtained. For the frame, the final
answer corresponded to what was expected and convergence was
obtained. Final mass distribution resembles that previously
attained just with displacement constraints. The geometry
and related values are presented in Figure 2.3.
A tapered cantilever loaded at the tip was compared
with the results obtained using a recursive relation between
the dimensions and displacements (12). The two sets of
results, those from the reference and those from the program
run, are very close. The maximum absolute difference
10 Kin
1OO00K
10 Kin
^1
ELEMENT INITIAL FINAL
Area (in2) 1.0 25.4
Inertia (in4) 1.0 120224
Area (in2) 1.0 179
2
Inerti. (in4) 1.0 5912
Area (in2) 1.0 35.1
Inertia (in4) 1.0 17058
Figure 2.3. One bay frame.
between the correspondent section dimensions is less than
five percent.
Further Improvements
In subsequent developments, some other improvements
were added to the algorithm that used the augmented
lagrangian formulation. The first consisted of eliminating
from the search those constraints that were inactive. Those
constraints whose value did not show a change when the line
search was along one of the design variable, were skipped
from recalculation. This savings in computational effort
allowed a reduction of forty per cent of the total runtime.
This feature was discarded when the gradient search method
was implemented. With this technique the changes in the
design variables were done simultaneously, all constraints
were altered and selective recalculation was no longer
possible.
Another significant improvement was achieved by
starting the solution with feasible displacements. The
displacement variables were calculated at the beginning of
the program corresponding to the initial loading and frame
dimensions. This led to the situation where the equality
constraints were exactly satisfied at the start of the
iteration procedure. This addition was kept in the version
using the gradient search. Work was also done on selecting
the initial cross section dimensions. Rules of thumb were
found to expedite calculations to obtain acceptable initial
values.
The method of steepest descent makes use of the
gradient of the pseudoobjective function. The gradient
vector represents the line along which there is the highest
variation of the pseudoobjective function at the actual
design point. Moving in the direction defined by the
negative of the gradient vector is expected to decrease the
value of the pseudoobjective function. This direction is
called the steepest descent. A graphical representation of
the method is displayed in Figure 2.4. Since the explicit
formulation of the gradient of the pseudoobjective function
was not practical to obtain, the gradient vector was
obtained using a finite difference technique. To obtain the
minimum point along the gradient direction another design
point along that line is found such that it has a higher
pseudoobjective function value. Then, the optimum value
should lie in this interval and a line search is performed
using the golden section method.
The gradient vector was normalized to avoid numerical
illconditioning. For the same reason, constraints and the
design variables were also scaled. Numerical difficulties
are predictable if just one of the constraint function, or
the objective function, is of different magnitude than the
rest of the terms or its rate of change is considerably
different from the others. Scaling factors for each
constraint were evaluated as the ratio between the gradient
STEEPEST DESCENT
1Initial Point
4Final Point
X2 Function Contours
Figure 2.4. Gradient method.
of the objective function and the gradient of that
constraint. Scaling of the design variables was also tried.
The normalization of the design variables consisted of
applying scaling factors that reduced them to a single order
of magnitude.
The results obtained with this unconstrained
minimization technique were inferior to those using the
Hooke and Jeeves method. The apparent reason was the shape
of the surface generated by the augmented lagrangian
function. Around a relative local optimal point, where the
equality constraints are satisfied, the variation of the
augmented lagrangian function was very abrupt.
Consequently, any line search performed starting at a
relative optimal point would invariably return to the same
initial point.
When using a set of design variables that was not a
relative local optimum, the gradient search would still not
converge. The reason for this lack of convergence was the
numerical error created by the steep slope of the function.
This fact could not be avoided despite the several
combinations of the constraint and variable scalings aimed
at smoothing the shape of the augmented lagrangian function.
Another phase of research consisted in using a mixed
method for the search. In a first phase, Hooke and Jeeves
was used to obtain a better second point than the starting
design point. This second point was then used to apply the
gradient search. The procedure was repeated with the
36
consequent updates of the lagrangian multipliers. This
mixed method did not present any improvement over the Hooke
and Jeeves method. The important conclusion from the
results of this mixed strategy was that convergence could
only be obtained when enough iterations of the Hooke and
Jeeves phase were completed. Consequently, the adopted
unconstrained minimization method for the optimization of
the augmented lagrangian function in the linear static
formulation was the Hooke and Jeeves method.
CHAPTER 3
NONLINEAR REINFORCED CONCRETE ELEMENT
Introduction
Reinforced concrete elements are made of two different
materials, concrete and steel. Concrete is the massive
component, has a high compressive strength and fails easily
when submitted to tension. Steel is embedded whenever
tensile strength is required. For that reason the
additional steel bars are commonly designated as reinforcing
steel.
Adequate combination of these two materials originates
a symbiotic composite material that has been widely used
(28). These elements are designed with bending, compression
and torsion requirements for code and safety compliance. In
some cases tension is also allowed.
Concrete and steel have nonlinear stressstrain
diagrams. Consequently, when material nonlinearities are
included, modeling of the behavior of any composite element
is very difficult (2930). When loads produce a tensile
stress greater than the maximum allowable value for the
concrete cracking results. When reinforcing steel stress
37
reaches the yielding value there is a large strain and
section curvature increase. Geometric nonlinearities are
then created by extra rotations of flexural elements from
the cracking and steel yielding.
A basic assumption in nonlinear analysis of reinforced
concrete frames is that the element rotations with relation
to the line defined by the nodes, chord rotations, are small
and the theory for straight elements may be applied with
some adaptations. The most popular analysis techniques are
based on incremental loadings of the structure and are known
by the initial stiffness and tangent stiffness methods. A
technique based on the application of the entire load at a
single step is known by the secant stiffness method. This
last technique was chosen for the analysis of the structure
since it is more adequate to the optimization formulation.
Element Modeling Survey
In the last three decades there have been many attempts
to create a simplified beam model of the inelastic
reinforced concrete element (3133). The main objective for
this research has been to advance a solution providing
precise results within reasonable computational and memory
storage limits. The study has a significant importance for
the analysis of reinforced concrete structures submitted to
dynamic loads (3435). In these examples the moments at the
ends are close to the ultimate allowable values. This
closeness implies that the concrete and steel stresses are
in the nonlinear intervals of the stressstrain diagrams.
The frame behaves as if inelastic plastic hinges have formed
due to concrete cracking and steel yielding.
Initial studies in this area addressed simple
structures with momentrotation relationship conditioned by
the moments at the beam extremities. This produced the one
component model with nonlinear rotational springs at the
ends. Later, another theory assumed a bilinear moment
resistance with two parallel elements, one to simulate
yielding and the other to represent strain hardening.
Several variations of these two theories have been developed
and experimentally tested (36).
Recent improvements in computer software led to
sophisticated modeling of reinforced concrete elements using
nontraditional finite element techniques. A simple approach
to this type of problem is based on the theory of damage
mechanics (37). The beam element is modeled as a
macroelement divided in models with explicit and accurate
behavior. The behavior of the whole structure is then
extrapolated from the small elements.
These types of models have been tested thoroughly to
ascertain its reliability and accuracy (38). These
evaluations, made mostly by comparison of computer program
results with experimental test data, provided a great deal
of information for further enhancements and refinements.
The option for this study had to fall on a element model
that is a compromise between the accuracy required and the
cyclic nature of the optimization process (39). Repeated
evaluation of the element stiffness due to the changes of
the physical properties of the elements is required. For
this reason it is highly desirable to choose a model with
low computational requirements.
Beam Element with Inelastic Hinges
Given the available solutions for the model of the
reinforced concrete element, the onecomponent model was
chosen as shown in Figure 3.1. It is a simple idealization
that doesn't increase the total number of elements of the
structure. This model has shown to accurately model the
nonlinear behavior of reinforced concrete, even for dynamic
loadings (40). Some basic assumptions and simplifications
were made for the definition of the model. For example, the
fact that concrete cracks under tensile loading, causing
local nonlinear behavior, was not accounted for. Time
dependent properties of the concrete were not considered.
Shear effects were not included in this formulation. The
loads were considered applied at the nodes and elements with
loads in the span can be approximated by a discrete number
of elements with nodal loads.
The unique element internal action considered was
flexure. Yielding of the reinforcing steel may only take
place in the hinges at the element ends. Strain hardening
OneComponent Model
Reinforced Concrete Element
Linear Elastic Element
i/G)~v
I
:_Q1y
with Secant Stiffness
Figure 3.1. Element model.
Sprin
and related altered element stiffness are simulated by the
linear element with nonlinear rotational springs at the
extremities. Inelastic rotations of reinforced concrete
hinges at the element ends are determined as a function of
the respective momentcurvature relationship for each
element. These curves are redefined every time any element
sectional properties changes during the optimization process
since the ultimate and yielding moments also change.
A typical momentcurvature diagram for reinforced
concrete elements is bilinear. It is obtained assuming
material stressstrain curves that are paraboliclinear for
the concrete and bilinear for the reinforcing steel as shown
in Figure 3.2 (28). The stress in the concrete is
designated by fc and the stress in the steel reinforcement
is represented by fs. The algorithm used to compute the
moment corresponding to a certain strain diagram is an
iterative Newton based iteration that determines the depth
of the neutral axis guaranteeing equilibrium of the internal
forces. Then, after determining the internal coupled forces
the related moment is computed.
All reinforced concrete elements are doubly reinforced
with equal areas of steel on both sides. This assumption is
valid for columns and acceptable for beams since in
continuous frames there are moments of different sign along
the beams. Evaluation of the moments for each reinforced
concrete section was based on the exact internal equilibrium
equations as follows:
B
f C
c C
A _
0.002 0.004
Concrete StressStrain Diagram
fs
'S
Steel StressStrain Diagram
Figure 3.2. Material behavior.
Cc + Cs = Ts
where
Cc compressive force in the concrete and is equal to
the area under stressstrain curve corresponding
to concrete strain Ec;
Cs compressive force in the steel area As
corresponding to steel strain Ecs;
Ts tensile force in the steel area As corresponding
to steel strain Es (ES y Ey).
Typical element moments necessary to define the
bilinear momentcurvature diagram were the yielding and
ultimate values. These characteristic values were
determined considering the corresponding section strain
distribution, the stressstrain diagrams for concrete and
steel, the location of neutral axis and the moment of the
internal forces as shown in Figure 3.3. The compressive
force of the concrete is given at any time by
Cc = a fcm b kd
where
Ieca
a = fc/(fcmEca)dec;
0
fcm maximum flexural concrete stress;
Eca concrete strain at the top compression fiber;
b element cross section base;
kd distance of neutral axis from top compressed fiber.
SECTION CHARACTERISTICS
AS
v_1^
h'
As
EC
yCC
yp8
Geometry
Strain
Diagram
Forces
Figure 3.3. Reinforced concrete section.
The force in the compressed steel is given by
Cs = As fcs
where
As steel area;
fcs stress in compressed steel.
The force in the steel under tension is determined by
Ts = As fy
where
fy yielding steel stress.
For instance, the internal ultimate moment is given by
the moments of these three internal forces about the top
compressed fiber. For that reason a parameter 0, that
defines the centroid of the concrete compressive stress
diagram, is introduced as
*Eca Ieca
S=1 ec fc dec /(Eca fc dEc)
0 0
These parameters, a and n, when the ultimate concrete
strain is defined as ec = 0.004, become
a = 2/3 (region AB) n = 3/8 (region AB)
a = 0.9 (region BC) n = 0.51851 (region BC)
where the regions AB and BC are defined in Figure 3.2. The
section flexural strength, Mi, may be defined as
Mi = Cs d'+ Cc 0 kd Ts d
where
d' distance of Cs to top compressed fiber;
d distance of Ts to top compressed fiber.
Element curvatures corresponding to these yielding and
ultimate moments are obtained assuming that plane sections
remain plane after deformation and there is no strain
hardening of the reinforcing steel. These formulas are as
follows:
#y = (Ey + Eca)/d
u = (Esa + Ecu)/d
where
y yielding curvature;
Ey Es / fy;
Eg 29x106 psi;
fy yielding stress of reinforcing steel;
Eca maximum concrete compressive strain;
u ultimate curvature;
Esa actual tensile strain of steel;
Ecu ultimate compressive strain of concrete.
These section characteristics define section diagrams
as shown on Figure 3.4. The value of the ultimate rotation
was given by the integration of the curvature along the
element. Two types of curvature diagrams were considered
Moment Curvature Diagram
Linearized
Diagram
Moy u 0D
Moment Rotation Diagram
Figure 3.4. Element deformation diagrams.
1 u ,
for integration. The first one was when moments at element
ends had the same rotational direction and the second when
the rotational directions were opposite. In both cases a
simplified method was used to integrate the curvature along
the element to find the corresponding rotation since the
moments at the other end were kept constant. Yielding
rotation for any node of the element was calculated assuming
the yielding moment at that node and keeping the other
moment unchanged. The same method was applied for the
calculation of the ultimate rotation where a modified
curvature diagram was used as schematically exemplified in
Figure 3.5.
Beam Element Stiffness
The elastic element chosen has a stiffness derived in
classical terms. End rotational springs had variable
stiffness depending on element moments at the nodes. A
large value was assigned to the secant spring stiffness when
moments were below the yielding value assured a linear
behavior. The secant stiffness value obtained from the
moment rotation diagram was used for moment values above
yielding. The strain hardening ratio of the linearized
moment rotation diagram was computed as the difference
between ultimate and yielding moments divided by the
difference between the ultimate and yielding rotations. A
MOMENT DIAGRAM
Mi
M Mi Mj
j Mi Moment at node i
Mj Moment at node j
My Yielding moment
CURVATURE DIAGRAM
] (u Ultimate curvature
y Yielding curvature
(i Curvature at node j
Figure 3.5. Curvature integration.
graphical description of these definitions is presented in
Figure 3.6.
The element modified stiffness was derived from the
condensation of elastic stiffness matrices of the linear
elastic element and the rotational spring elements. To
condense the two matrices the first step consisted of
inverting the sum of the corresponding flexibility matrices
concerning the independent element rotational degrees of
freedom. The next step was the expansion of this element
stiffness to include the axial displacements, uncoupled from
the spring rotations, and the other dependent element
degrees of freedom. The main steps of this step are the
following:
1/Ksi
0
1
1
0 1/3 1/6
+ 3EI/L
1/Ksj 1/6 1/3
1 0 0 1 0 0
[ a ] = 0 1/L 1 0 1/L 0
0 1/L 0 0 1/L 1
[ Kmod ] = [ a ]t [ Ks* ] [ a ]
Ks secant stiffness matrix with element rotations;
Ks* expanded secant stiffness matrix with
uncoupled axial stiffness;
Ksi stiffness of spring at node i;
Ksj stiffness of spring at node j;
[ K ]
where
Moment
Mu
My
Rotation
Spring
MomentRotation Diagram
Mu Ultimate moment
Yielding moment
10e30
(Mu My)/(u
 Oy)
Ksec Spring stiffness for
M > My
Figure 3.6. Secant spring stiffness.
My 
Kl 
K2 
E element modulus of elasticity;
I element moment of inertia;
L element length;
a expansion matrix;
Kmod modified element matrix.
After evaluating the modified element stiffness matrix
it was transformed from the local coordinates to the global
coordinates by the use of the corresponding rotation matrix.
The values of the terms of this element stiffness matrix
were then used to compute the corresponding updated equality
constraint values. The process was similar to assembling a
structure global matrix using a location matrix relating the
element degrees of freedom with the structure global degrees
of freedom.
CHAPTER 4
STRUCTURAL ELEMENT RELIABILITY
Introduction
Design and checking of structures in the field of Civil
Engineering has been traditionally based on deterministic
analysis. Adequate dimensions, material properties and
loads are assumed and an analysis is carried out to obtain
the required evaluation. Nevertheless, variations of all
these parameters and questions related to the structural
model may impose a different behavior than expected (41).
It must be emphasized that if there were no uncertainties
related to the prediction of loads, materials and structure
modeling, then the respective safety would be more easily
guaranteed.
For these reasons the use of probabilistic principles
and methodologies in structural design has been increasing.
Design for safety and performance should consider the
conflict between safety and risk. The objective of
probability concepts and methods is to develop a framework
where the effects of these uncertainties are considered.
Structural reliability has received the attention of several
54
researchers and, consequently, it is introduced into almost
all recent structural codes worldwide.
It is a relatively young structural science that
evolved in the same way as other new areas where theoretical
studies dictate the general principles for systematic
treatment of problems. There are however practical
difficulties in obtaining enough statistical data and
handling the sophistication of the probabilistic methods.
For these reasons the analytical processes involved in the
determination of structural reliability were grouped in
different working levels (42). These working levels depend
on the problem considered and the desired accuracy for the
reliability evaluation. There are three basic levels and
the classification increases with the sophistication of the
method used and the amount of statistical data that is
manipulated.
Level 1 uses a methodology that provides a structural
member with an adequate structural reliability by the
specification of partial safety factors and characteristic
values of design variables. This is the method currently
used in structural design codes (43). Level 2 includes all
methods that control the probability of failure at certain
points on the failure boundary defined by a limit state
equation (44). Level 3 groups all techniques that perform a
complete and exact analysis of the structure taking into
account the joint probability function of all the variables
involved (45).
In this chapter, the technique used to analyze the
structural reliability of each reinforced concrete beam
element is described. Due to the nature of the problem,
where optimization and reliability evaluation are performed
simultaneously at the element level, a Level 2 method was
chosen. Since the concepts of limit state design and
probability of failure are intimately connected with
structural reliability, a brief description is also
included.
Concept of limit state may be described as that state
beyond which a structure, or part of it, can no longer
fulfill the functions or satisfy the conditions for which it
was designed. Namely, the structure is said to reach a
limit state when a specific response parameter attains a
threshold value. Examples of ultimate limit states are the
loss of equilibrium of a part or the whole of the structure
considered as a rigid body, failure or excessive plasticity
of critical sections due to static actions, transformation
of the structure into a mechanism, buckling due to elastic
or plastic instability, fatigue, excessive deflections and
abundant cracking.
Modern codes divide limit states into two main groups.
Ultimate limit states, corresponding to the maximum load
carrying capacity, and serviceability limit states, related
to the criteria governing normal use and durability (46).
For each of these groups the importance of damage is
different and is represented by the adopted respective
probability of failure.
For instance, in reinforced and prestressed concrete,
code checks for the ultimate limit states are based on
element forces, except in the plastic analysis where the
design variables are the loads. In cases where fatigue is
involved, stresses are also the control variables. The
service limit states are the cracking limit state and the
deformation limit state. In this work only the ultimate
flexural limit state and the global deformation limit state
are addressed since they are the more relevant for the
optimization study.
Acceptable risks of failure for any structure are
affected by the nature of the structure itself and its
expected application. These are dependent on social and
local variations. It is common for structural engineers to
balance the contradiction between the economy and safety of
the structure. This particular aspect is the main reason
why it is so appealing to combine reliability and
optimization in structural design.
Probabilities of failure used in limit state designs
vary with the risk of loss of human lives, the number of
lives affected and economic consequences. In ultimate limit
states the range of probability of failure adopted is
between 104 and 107 over a 50 year expected design life.
In serviceability limit states the probability of failure
varies between 101 and 103.
A criterion proposed is as follows (41):
pf = 105 U T / L
where
U 0.005 ........ Places of public assembly, dams;
0.05 ......... Domestic, office, industry, travel;
0.5 .......... Bridges;
5 ............ Towers, masts, offshore structures;
T life period of the structure(years);
L number of people involved.
These values must be interpreted carefully. For
example, the value of 103 means theoretically that, on the
average, out of 1000 nominally identical buildings, one will
crack or deform excessively. It is evident that in civil
engineering 1000 identical buildings rarely occur, even
neglecting the fact that a statistically significant number
require samples at least 10 to 20 times larger.
Moreover, the determination of these low probabilities
requires extrapolations of statistical properties that are
experimentally known only around the mean values of the
random quantities. For these reasons, the probabilities of
failure in civil engineering have no real statistical
significance and they must be considered not as
deterministic quantities but just as conventional
comparative values.
In consequence of the above considerations, the
differences between the methods used in each of the three
levels are rather operational than conceptual. There are no
rigid boundaries between them. They are used in accordance
with the required accuracy and the nature of the problem to
be studied.
Level 3 methods require a complete analysis of the
problem and also the integration of the joint distribution
density of the random variables extended over the safety
domain. They remain in the field of research and are used
to check the validity of approximations, idealizations and
simplifications performed in the other two levels.
Level 2 methods use random variables characterized by
their known or assumed distribution functions, defined in
terms of important parameters as means and variances. This
avoids the multidimensional integration of the previous
method. These methods may be used by engineers to solve
problems of special technical and economical importance.
Code committees engaged in drafting and revising standard
codes of practice use them to evaluate the partial safety
factors. It is possible that computational developments in
the near future will allow for such methods to be more
commonly used by the practicing engineer. The probabilistic
aspect of the problem in the Level 1 methods is represented
by characteristic values of the random variables involved.
With these characteristic values partial safety factors are
derived using Level 2 methods. They are used by most
engineers where reliability theory and probabilistic methods
are the basis of their code provisions.
These Level 1 methods could be replaced by the Level 2
methods if an agreement was obtained in the following
issues: selection of basic random variables for each
specific problem, their distribution types and relative
statistical parameters; form of the various limit state
equations and choice of models; operational reliability
levels to be adopted in different design situations.
It must be emphasized that the advantage of Level 1
schemes over Level 2 are their great operational simplicity
due to the use of fixed and constant partial safety factors
for a given class of design situations. The main
disadvantage of Level 1 is the selection of partial safety
factors for a given structural class in such a way that the
efficiency of the method proposed is satisfactory. It must
assure that the deviation of the reliability of a design
made on the basis of the adopted coefficients from the
desired reliability level laid down in the code is
acceptable.
Two Dimensional Space Example
Let R and S be two random variables, where R defines
strength and S the load. Then the limit state function z
shown in Figure 4.1 is defined as
r z r 0O
( z>0 ) ijil
SAFE
D' (z
UNSAFE
Safe and Unsafe Design Regions
Figure 4.1. Design safety region.
z = r s
where
r resistance function;
s load function.
The domain D (z>0) is the safe domain and D'(zO0) is
the failure domain. The probability of failure, pf, is the
probability that a point (R,S) belongs to D'. Once the
statistical distributions of the random variables R and S
are known, the numerical solution of the corresponding
equation will determine pf. Assuming that both variables R
and S have a Gaussian distribution, and further defining rm
and sm as the mean values, and aR and aS as standard
deviations of R and S, respectively, the random variable Z
will also be normal and its statistical parameters are
defined as
zm = rm sm
az = (a2R + a2S)
where
zm mean value function;
az standard deviation function.
Defining Fz as the cumulative normal distribution
function, the probability of failure may be calculated as
pf = P(Z
A graphic representation of these functions is
presented in Figure 4.2. Introducing the standardized
variable u and the reliability index as
u = (z zm) / aZ
S= zm / Cz = (rm sm) / (a2R + a2S)
then the probability of failure may be expressed as
pf = Fu(zm / az) = Fu(B)
An important concept widely used in structural safety
when considering random variables is the Central Safety
Factor. It relates the mean values and coefficients of
variation of R and S to determine a probabilistic safety
factor (44). It is a simplistic way of establishing some
influence on the design variables of the respective random
characteristics.
To consider a more detailed study a Level 2 method is
applied in the element reliability evaluation. In this
method safety checks are made at a finite number of points
of the failure boundary. A graphic representation in a two
dimensional space is presented in Figure 4.3. In the case
where this check is made at only one point, the parameter to
be determined is the minimum distance between the origin of
the system of the standardized variables to the boundary of
the safety domain.
Z mean
Probability Density Function
Cumulative Density Function
Figure 4.2. Probabilistic functions.
It is possible to associate this distance with a precise
meaning in terms of reliability. A technique derived from
this concept is the LindHasofer Minimum Distance method
illustrated in Figure 4.3 (47).
Let X (X1, X2,..., Xn) be the vector of the basic
random variables of a given structural problem that may be
assumed to be statistically uncorrelated, involved in a
given structural problem. Let z = g(xl, X2,...., Xn)= 0 be
the boundary of the safety domain. The values of X
belonging to the failure domain will satisfy the inequality
z = g(x) < 0
The method consists in projecting the function z in the
space of standardized variables defined as
ui = (Xi xmi) / axi
Measuring, in this space, the minimum distance B of the
transformed surface g (ul, u2,....., un) from the origin of
the axes. A design is regarded reliable if B > B*, where B*
is prescribed by an appropriate code provision.
In geometrical terms, the hypersphere having radius B*
and with center at the origin of the axes ui is required to
lie within the transformed safety domain. The justification
for such a method is that most of the joint probability
density of the variables involved will be concentrated in
Ul Z= 0
Z (UI,U2) > 0
Figure 4.3. Safety checks.
the hypersphere having radius 8*, and that consequently it
will be associated with values of vector X belonging to the
safety domain. Mathematically, the problem to be solved is
to find
B = min (E u2i)
In a great number of cases the safety boundary domain
is linear, and one can write an expression for z as follows:
z = g (xl, x2,...., Xn) = b + Z aixi
Then, B can be immediately determined as follows
g (ul, u2,..., un) = b + Z aixmi + Z aiaxiui = 0
and the distance of this hyperplane to the origin is
B = Z (ai.xmi + b) / (Z a2iax2i)
Expressing in terms of the standardized variables is
equivalent to replacing the hypersurface by the hyperplane
passing through P*, the point of minimum distance between
the two geometric elements. A graphical illustration of
this approximation in a two dimensional space is presented
in Figure 4.4. Finally, the probability of failure, pf, and
Z = 0
Z (U1,U2) < 0
>0
Figure 4.4. Reliability index.
the reliability index, B, are within certain approximations
related by
Pf = 1 p(B)
where p is the function of standardized cumulative normal
distribution.
Reinforced Concrete Element Reliability
The element actions considered in analysis are only the
moments at the member ends. These are the points of maximum
value since only concentrated nodal loading is considered.
The failure function z is then defined as
z = r s = Mi Me
where
Mi ultimate internal resisting moment;
Me maximum external element moment.
The external moment at the section is obtained from the
element displacements using the condensed element stiffness
matrix defined in the previous chapter. The expressions to
obtain the value of Mi were defined in the previous chapter.
The random values chosen in this study were the
characteristic strength of the concrete, f'c, and the
maximum external moment in the element, Me. All other
variables of the expression defining Mi could be taken as
random but concrete strength was chosen due to the high
coefficient of variation. Thus, the flexural failure
function is linear and the respective reliability of failure
can be easily calculated.
Compressive strength of concrete is influenced by a
large number of factors grouped in three main categories,
namely materials, production and testing. Material
variability depend on the cement quality, moisture content,
mineral composition, physical properties and particle shape
of aggregates. The production factors involve the type of
watching, transportation procedure and workmanship. Testing
includes sampling techniques, test methodology, specimen
preparation and curing (48).
It is difficult to evaluate correctly the importance of
these three groups of factors. Their importance is certain
to vary for different regions and construction projects. It
has been found that the distribution of concrete compressive
strengths can be approximated by the normal (Gaussian)
distribution (4950). Characteristic concrete compressive
strengths obtained from a sampling of test data leads to a
conclusion that for strength levels between 3,000 and 4,000
psi, the coefficient of variation is constant. For
strengths beyond that range the standard deviation is
constant (51). Since the values in reinforced concrete
frames used are generally within the first interval the
statistical value considered was the variance of f'c. The
average standard variation for 68 a good quality control
testing at the construction site is 550 psi. Using a 3500
psi specified compressive strength of concrete, f'c, the
required average compressive strength of concrete is the
larger of the following (51):
f'cr = f'c + 1.34*0 = 3500 + 1.34 550 = 4237 psi
or
f'cr = f'c + 2.33*0 500 = 3500 + 2.33*550 500 = 4282 psi
The coefficient of variation of f'c for this range of
characteristic compressive strength is then given by
V = a/f'c = 550/4282 = 0.128
and consequently the coefficient of variation of the
concrete compressive flexural strength was adopted to be 0.15.
External loads have different coefficients of variation
for the different types of loads (5253). For most design
and construction in the United States a good estimate for
the coefficient of variation of dead loads is 0.10. For the
live loads the coefficient of variations are very high and
range from 0.39 to 1.04. For that reason and since the
building codes prescribe large values for live loads that
exceed the mean value a single coefficient of variation of
0.15 was adopted for the combination of dead loads plus live
loads.
Basic variables considered, fc and Me, are assumed to
have a probability function with normal distribution. This
assumption is correct for the characteristic compressive
strength of concrete but it does not hold for all external
loads that create Me. In the case where a statistical
refinement of the basic variable Me is required, there are
techniques available to address the problem (47).
Since flexural failure function, z, is linear the
reliability index B of each element can be calculated for
any given external moment, section and material properties.
Denoting the basic variables fc as xl, and Me as x2, and
eliminating the other parameters involved in the equation,
the flexural failure function takes the form
z = al xI + a2 x2 + b
where
al = a 0 b (kd)2;
a2 = 1;
b = As fcs d' As fy d.
Standardizing the variables xl and x2 leads to a
replacement of the basic variables
ul = (xI l)/a1
U2 = (x2 P2)/a2
where
M1 mean value of fc;
p2 mean value of Me;
a1 standard deviation of fc;
a2 standard deviation of Me.
Replacing the standard normal variables in the flexural
failure function the expression assumes the following form:
z = alalul + a2a2u2 + alj1 + a292 + b
Then the reliability index for each element is given by
the distance from the standardized failure function to the
origin of the standardized basic variables as follows:
B = (al,1 + a242 + b) /(alal + a2a2)
CHAPTER 5
SYSTEM RELIABILITY
Introduction
Optimum structural design techniques are mainly based
on deterministic assumptions. There is no doubt that some
of the design variables should be considered including their
random nature (5455). Of course system reliability
problems are more complicated than element reliability
problems. This is evident since it must consider all
multiple element failure functions, the several failure
modes and, in some cases, the correspondent statistical
correlation.
Another reason for including reliability considerations
in structural optimization procedures is that, in some
instances, the optimal solutions found have less redundancy
and smaller ultimate load reserve than those solutions
obtained with traditional design techniques (5657).
There is no doubt that the combination of optimum
design techniques and reliabilitybased design procedures
creates a powerful tool to obtain a practical optimized
solution. The objective is to find a balanced design
74
between all those that satisfy the optimization constraints
and at the same time will have the lowest allowable
probability of failure (58).
The strategy employed to evaluate the system
reliability is described in the rest of the chapter. The
elementary failure mechanisms of the structure are
determined using Watwood's method. Then the system
reliability is approximated using the Beta unzipping method,
which consists of determining the relevant collapse
mechanisms through linear combinations with fundamental
mechanisms. The theory related with these techniques is
tentatively described.
System Reliability and Optimization
A possible inclusion of the system probability of
failure is to attribute a cost to system failure. This
option originated a formulation based on the minimization of
the total cost with the traditional optimization constraints
(59). The objective function is as follows:
Minimize Ct = Co + Cf Pf
where
Ct cost of the structure;
Co initial cost of the structure;
Cf cost of failure;
Pf probability of structure failure.
This option is not commonly used for inhabited
structures since it is difficult to evaluate the economic
value of a structural failure where human life losses are
expected. A more popular alternative is to include an
additional constraint representing the maximum probability
of failure allowed for the structure (60). The constraint
for the system reliability will be of the type
Pf(X) < Pm
where
Pf probability of system failure;
X vector of design variables;
Pm allowable probability of system failure.
When performing structural optimization one may
consider serviceability and ultimate limit states. This
possibility leads to another type of formulation where the
objective function and constraints for these limit states
are considered simultaneously (61). This type of problems
are called reliabilitybased optimization and can be
summarized as follows:
Minimize Co
subject to
Gi(X) < 0, i=l,m
Pu Puo
Ps p Pso
where
Gi optimization constraints;
m number of behavior constraints;
Pu probability of ultimate system failure;
Puo maximum probability of ultimate system failure;
Ps probability of serviceability failure;
Pso maximum probability of serviceability failure.
The option adopted consisted of adding a constraint on
the system failure. The value of the system failure at the
end of the optimization cycle is compared with the target
value. If it is not satisfactory the element requirements
are modified and the optimization is restarted.
Methods
In determinate structures the collapse of any member
will lead to system failure. The probability of system
failure can be obtained as the probability of the union of
member probability failures (16). These types of structural
systems are called series systems or weakestlink systems.
Redundant structures will fail only if all redundant members
collapse. If this condition does not arise, whenever a
member fails there will be a redistribution of loads in the
system. These types of structures are called parallel
systems. Graphic examples are presented in Figure 5.1.
PARALLEL SYSTEM
Load
Truss
Bars
Load
Figure 5.1. System models.
Truss
Bars
SERIES SYSTEM
~1
Series systems with n elements have n failure modes.
Parallel systems with n elements have more than n failure
modes. These failure modes in parallel systems are
dependent on whether the failure type of the elements is
brittle or ductile (6263). For redundant brittle systems
the failure of an element and consequent redistribution of
the loads will provoke the system failure. In these cases
the system behavior may be considered to be generally
identical to that of as a series system.
Probability of failure of a series system can be
considered as the union of the elements probability of
failure
Pfs = P(Ui(Zi: 0)i=l,n)
where
U union of events;
Pfs probability of system failure;
Zi failure function of element i.
If the element failure functions are not correlated
then the evaluation of Pfs is relatively easy and may be
assumed as
Pfs = 1 Vi=1n(l P(ei=0))
where
r product;
ei = 0 if element is in a failure state,
ei = 1 if element i is in a nonfailure state;
P(ei=0) probability of failure of element i.
When there is correlation between element failure
functions then the calculations become more complicated and
time consuming. To avoid the exact evaluation,
approximation and bound techniques have been developed (64
65). The best known is the simple bounds. In this approach
the upper bound for the probability of system failure
assumes that all element failure functions are uncorrelated
and the lower bound is obtained assuming full dependence
between the element failure functions. If a more
sophisticated bounding technique is necessary the Ditlevesen
bounds may be used (17). The drawback is that this
sophistication implies the calculation of event joint
probabilities. A similar simplified approach to that used
in series systems may be adopted to find the simple bounds
for the failure of a parallel system.
In the case of parallel systems the lower bound
corresponds to the case where there is no dependence between
the elements failure and the upper bound corresponds to full
dependence between all elements failure (66). Exact
evaluation of the probability of system failure is very
difficult to obtain if the system has more than three
elements. To solve a general problem, approximation or
bounding techniques are used. For instance, for redundant
ductile systems there is a large list of procedures, most of
them with limited application (67).
Some methods for redundant systems involve the
determination of all collapse modes and their respective
probability of failure. To obtain all collapse modes the
fundamental mechanisms are determined and a Monte Carlo
simulation is performed to generate all others. Afterwards
the respective probabilities of failure are determined.
This approach, although accurate, is very demanding in
computational effort if the system is complex, and
consequently is used mostly to validate the performance of
other methods.
In redundant ductile systems a variation of the Monte
Carlo approach PNET or Point Estimate of System Collapse
Probability is used. This consists in linearly combining
the fundamental failure modes with the coefficients as
variables. An objective function representing the
reliability index of that combination is minimized and the
most probable failure mechanisms are defined.
Concerning redundant structures with brittle or ductile
elements, other approximation and bounding techniques have
been developed and studied based on graph theory. Two of
those approaches for obtaining the probability of failure
are the failure mode approach and the stable configuration
approach (68). Both methods require the determination of
all possible failure modes and the use of algorithms based
on graph theory.
To exemplify the determination of all possible failure
modes the initial step is to build a directed network, or
directed graph, with all possible events involving element
failures that will lead to a collapse. Each node represents
a stable configuration and each branch corresponds to a
element failure. Each path is a set of branches connecting
the initial and final nodes. A cut of the graph is a set of
branches containing only one branch from every path. A
simple example is presented in Figure 5.2.
Methods based on the determination of fundamental
failure mechanisms using practical simplifications from
graph theory have been implemented (69). The Beta unzipping
method and the branch and bound method are two examples.
The principal advantages are that they are precise and easy
to use. The Beta unzipping method finds the significant
collapse mechanisms using combinations of fundamental
mechanisms and rejecting those combinations that are outside
a prescribed interval. The branch and bound method selects
all failure paths that have high probabilities of
occurrence. Although less exact, the Beta unzipping method
was chosen due to its simplicity and performance.
Generation of Failure Modes
To define all failure mechanisms, the first step
consists of determining the set through manipulation of
elementary failure mechanisms. To obtain these, the method
2 3
1Tru
Load
ss Bars
FAILURE GRAPH
2F
F Bar Failure
Figure 5.2. Failure graph.
adopted was conceived by Watwood (15). It is an automatic
tool to generate all failure mechanisms with one degree of
freedom, or elementary failure mechanisms, of a given frame.
The set of these mechanisms and all their linear
combinations constitute all possible collapse configurations
(70). The technique is relatively simple to use since the
input data for this method is the same for traditional
elastic analysis like joint and element information.
Elementary failure mechanisms are dictated by the
geometry of the structure and potential hinge locations
created by the external load configuration. Hinge locations
are considered at the end of each member. In the case where
there are loads in the middle of the element, they are also
considered at the points of concentrated or discretized
loads. The element axial collapse is not considered in this
formulation although it was included in the original
version.
Element global displacements of a planar frame form a
vector with six variables, {S}. Using a cartesian
referential set of axes x and y the displacements, S1 to Sg,
may be represented as in Figure 5.3. Element deformation
parameters may be defined by three independent quantities
S'1 displacement about node i;
S'2 rotation of node i;
S'3 rotation of node j.
S2
1 SI
i
c^\s
p
4)
SsI
Element
Displacements
Independent
Element
Displacements
Rigid Body
Displacements
Figure 5.3. Element displacements definition.
a@  0 i * ^ *' 
s5t
SS4'
When a mechanism is formed each element moves as a
rigid body. The rigid body motion of an element of a planar
frame can be defined by three parameters. They can be
expressed in terms of the global coordinates x,y as
S'4 translation in the x direction;
S'5 translation in the y direction;
S'6 rotation about node i.
Two sets of three independent displacements, rigid body
parameters and element deformations, create the transformed
coordinate vector, (S'}. A relation can be established
between local global coordinates and transformed coordinate
vector represented by a linear transformation [T].
(S) = [T] {S'}
where
0 0 0 1 0 0
0 0 0 0 1 0
0 1 0 0 0 1
[T] = 1 0 0 1 0 0
0 0 0 0 1 L
0 0 1 0 0 1
L element length.
For any elementary failure mechanism the element
deformations, S'1, S'2, S'3, must be zero. This is only for
elements that do not have plastic hinges. To materialize
this condition, a matrix Ck is introduced for each element
87
k. This matrix is created with the first three rows of
matrix T1 for the kth element. The global condition
matrix, C, is a block diagonal matrix consisting of the Ck
matrices as follows:
1
Ck = 0
0
C1
C 
0
O
0
0
1/L
1/L
1 0
0 1/L
0 1/L
Using the previous matrices and vectors
relation now holds
the following
C (S) = (S'd)
where
(s) =
 first element
 second element
 nth element
and
(S'd) =
r
S'1
S'2
S'3
S'1
S'2
S'3
 first element
 nth element
Compatibility between the element displacements, {S}
and the structure global degrees of freedom {r} can be
established
{S) = [Q] [A] {r}
where
[Q] rotation matrix;
[A] compatibility matrix.
From previous equations the following expression holds
[C] [Q] [A] {r} = {S'd}
or
[B] {r) = {S'd)
An elementary mechanism of the structure is a solution
of the homogeneous system
[B] {r} = 0
If the structure configuration is not a mechanism there
is no solution for the system except the trivial solution.
To obtain a mechanism, releases of the global degrees of
freedom must be introduced. Two releases per element are
added corresponding to the hinges at the ends or points of
application of concentrated or discretized loads. Each
release corresponds to an addition of an external global
degree of freedom.
Addition of external degrees of freedom is done by
replacing a row in matrix [Q] [A] with zeros. The changed
rows correspond to the element degrees of freedom S3 and S6,
the node rotations. For each row that is replaced, a unit
column vector is added to the matrix [Q] [A] with a 1 in the
row that has been replaced. The dimensionality of (r} is
increased by the number of rows replaced in [Q] [A]. The
total is a set of extra columns with a dimension that is
twice the number of elements. The homogeneous system
becomes
[C] ([Q] [A])* {ra) = [B'] {ra) = 0
where
([Q] [A])* matrix with extra 2n columns;
({a) vector of increased global degrees of freedom.
Matrix [B'] is not square and has a greater number of
columns than the number of rows. The solution of the system
of homogeneous equations above may be obtained using a
technique similar to that when solving for redundant
unknowns in the force method. Difference between number of
rows and number of columns is the number of independent
solutions, that coincides with the number of elementary
mechanisms. Suppose the rank of [B'] is the number of
columns, m, and that the number of columns is p. In this
case one can find a matrix [D], nonsingular with dimensions
p by p such that
[B'] [D] = [I I 0]
where
[I] identity matrix, m by m;
[0] null matrix, m by (pm).
Last columns of [D] are independent solutions of the
homogeneous system of equations since they are orthogonal to
the rows of [B']. To obtain [D], a reduction is performed
on the columns of [B'] that is conceptually identical to a
GaussJordan reduction (15). The solution of such a system
of equations is illustrated in Figure 5.4, where all
elementary failure mechanisms for a two story frame are
presented.
Beta Unzipping Method
Advantages of the Beta unzipping method, as stated
before, are important. It can be used for reliability
estimation of planar and spatial trusses and frames made
with ductile or brittle elements. The probability of
failure can be evaluated with different levels of accuracy.
It is also a method that can be easily implemented for
automated calculations.
Mode 1 Mode 2
Mode 3 Mode 4 Mode 5
Mode 5 Mode 7 Mode 8
Figure 5.4. System failure modes.
