• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Structural optimization
 Integrated optimization of linear...
 Nonlinear reinforced concrete...
 Structural element reliability
 System reliability
 Procedure implementation
 Examples
 Conclusions and recommendation...
 Augmented lagrangian subroutin...
 Generalized reduced gradient...
 Generalized reduced gradient...
 Reference
 Biographical sketch
 Copyright














Title: Optimization of reinforced concrete frames using integrated analysis and reliability
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
    Abstract
        Page viii
        Page ix
    Structural optimization
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
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        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Integrated optimization of linear frames
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
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        Page 30
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    Nonlinear reinforced concrete element
        Page 37
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    Structural element reliability
        Page 54
        Page 55
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        Page 71
        Page 72
        Page 73
    System reliability
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
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        Page 96
    Procedure implementation
        Page 97
        Page 98
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        Page 118
    Examples
        Page 119
        Page 120
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        Page 137
        Page 138
    Conclusions and recommendations
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
    Augmented lagrangian subroutines
        Page 145
        Page 146
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    Generalized reduced gradient example
        Page 189
        Page 190
        Page 191
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        Page 193
        Page 194
    Generalized reduced gradient subroutines
        Page 195
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        Page 228
        Page 229
    Reference
        Page 230
        Page 231
        Page 232
        Page 233
        Page 234
        Page 235
    Biographical sketch
        Page 236
        Page 237
        Page 238
    Copyright
        Copyright 1
        Copyright 2
Full Text












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 elastic-plastic 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 one-component 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 so-called 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 cross-section. 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 P-e


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 Kuhn-Tucker

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 stress-strain diagram for concrete

and the elastic-plastic 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 318-83.

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 moment-rotation diagram corresponding to the

current global rotation. Values of the yielding and

ultimate moments were obtained by integrating the actual

stress-strain 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 analysis-optimization 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-

by-element 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 (23-26). 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,...,2n-1 --- 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 pseudo-objective

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

pseudo-objective 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 pseudo-objective 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 run-time.

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 pseudo-objective function. The gradient

vector represents the line along which there is the highest

variation of the pseudo-objective 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 pseudo-objective 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 pseudo-objective 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

pseudo-objective 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

ill-conditioning. 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


1-Initial Point
4-Final 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 stress-strain

diagrams. Consequently, when material nonlinearities are

included, modeling of the behavior of any composite element

is very difficult (29-30). 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 (31-33). 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 (34-35). 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 stress-strain 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 moment-rotation 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 one-component 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










One-Component 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 moment-curvature 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 moment-curvature diagram for reinforced

concrete elements is bilinear. It is obtained assuming

material stress-strain curves that are parabolic-linear 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 Stress-Strain Diagram



fs
'S


Steel Stress-Strain Diagram


Figure 3.2. Material behavior.









Cc + Cs = Ts
where

Cc compressive force in the concrete and is equal to

the area under stress-strain 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 moment-curvature diagram were the yielding and

ultimate values. These characteristic values were

determined considering the corresponding section strain

distribution, the stress-strain 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
yp-8


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


Moment-Rotation 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 10-4 and 10-7 over a 50 year expected design life.

In serviceability limit states the probability of failure

varies between 10-1 and 10-3.








A criterion proposed is as follows (41):



pf = 10-5 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 10-3 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 Lind-Hasofer 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 (49-50). 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 (52-53). 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 (54-55). 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 (56-57).

There is no doubt that the combination of optimum

design techniques and reliability-based 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 reliability-based 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 weakest-link 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 (62-63). 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 non-failure 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 T-1 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 (p-m).



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

Gauss-Jordan 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.




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