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Optimization of reinforced concrete frames using integrated analysis and reliability

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Optimization of reinforced concrete frames using integrated analysis and reliability
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Optimization of reinforced concrete frames using integrated analysis and reliability
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Soeiro, Alfredo V.,
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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.




Full Text
172
C***************************************************************
ndof=iqh
do 20 j=l,ndof
P(j)=r(j)
20 continue
C** ****************** ******c***** ********************* ************
c ordering theta and r matrices
c***************************************************************
do 710 k=l,numec-l
jflag=0
do 720 i=l,ndof
if(abs(rb(i, k)).gt.0.0)j flag=i
720 continue
if (jflag.eq.0)then
do 730 l=k+l,numec
do 740 li=l,ndof
if(abs(rb(li,l)).gt.0.0)then
do 750 lj=l,ndof
temp(lj,1)=rb(lj,1)
rb(lj ,l)=rb(lj, jflag)
rb (1 j j flag) =temp (1 j 1)
750 continue
do 760 lj=l,2*nel
temp(1j,1)=theta(1j,1)
theta(lj,l)=theta(lj,jflag)
theta(1j,j flag)=temp(1j,1)
760 continue
go to 733
endif
740 continue
733 continue
730 continue
endif
710 continue
c***************************************************************
c normalizing theta and r vectors
c**************************************************************
do 810 i=l,numec
do 820 j=l,2*nel
if(abs(theta(j,i)).ne.1.O.and.
* theta(j,i).ne.0.0)then
fact=abs(1./theta(j,i))
do 830 jj=l,2*nel
theta(jj,i)=theta(jj,i)*fact
830 continue
do 840 jj=l,ndof
rb(jj ,i)=rb(j j ,i) *fact
840 continue
go to 734
endif
820 continue
734 continue
810 continue
c**************************************************************
c transpose theta and r matrices
c**************************************************************


144
substantially the size of the problem. However, the benefits
of this change could be significant.
Changing the nonlinear analysis method from secant
stiffness approach to a tangent stiffness approach could be
another solution to the lack of convergence. In this case a
two stage process would be adopted. The first would consist
of a linear optimization up to the formation of a hinge
followed by a phase with a sequence of incremental loading
and optimization procedures until convergence was obtained.


81
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.


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


99
both unitary costs by the cost of concrete and the result is
as follows:
where
f(x) = (x Xj + xk 10) Lp
Lp length of element p, p = l,m.
Equality constraints, one for each global degree of
freedom, are defined as
hq 2r (kqr x3n+r) Rq/ <3"rl,...,m
where
kqr global stiffness coefficient;
x3n+r global displacement r;
Rq external force q.
Inequality constraints that control the maximum global
displacements and impose a minimum element reliability are
as follows
where
9i x3n+r dr i1,...,m
gj = relj betaj, < 0 j=l,...n
dr maximum absolute value of global displacement r;
relj reliability index of element j;
betaj minimum reliability index prescribed for
element j.


164
c*************************************************************
x=47./60.*b*fc
y=0.004*es*aste-aste*fy
z=-0.004*es*co*aste
if((y*y).It.(4.*x*z))then
y=sqrt(4.*x*z)
end if
vkd=(-y+sqrt(y*y-4.*x*z))/(2.*x)
epcs=0.004*(vkd-co)/vkd
if(epcs.ge.epsy)then
epcs=epsy
endif
c***************************************************************
C CONCRETE FORCE IN REGION AB
c***************************************************************
alphal=2./3.
ccab=alphal*b*0.5*vkd*fc
c* ************************************************************ *
c CONCRETE FORCE IN REGION BC
c****************************************************** *******
alpha2=0.9
ccbc=alpha2*b*0.5*vkd*fc
(^* ************************************************************* *
C DISTANCE OF CENTROID TO TOP IN AB
C**************************************************************
gama1=0.875*vkd
c******** *************************************** ****************
C DISTANCE OF CENTROID TO TOP IN BC
C***************************************************** *********
gama2=0.259255*vkd
c**********************************************************
C COEFFICIENTS FOR FAILURE FUNCTION
c************************************************************
al=(ccab*(dd-gamal)+ccbc*(dd-gama2))/fc
a2=-l.
c***********************************************************
c COSINE DIRECTORS
c******************************************** **************
tetal=al*sigmal*fc
teta2=a2*sigma2*vm
c************************************************************
C INDEPENDENT TERM
c***********************************************************
fps=0.004*es*(vkd-co)/vkd
bi=aste*fps*(dd-co)
c***********************************************************
C RELIABILITY INDEX
Q***********************************************************
beta(kel)=(al*fc+a2*vm+bi)/sqrt(tetal*tetal+teta2*teta2)
c************************************************************
C ULTIMATE MOMENT AND ROTATION
c************************************************************
vmu(kel)=al*fc+bi
phiu=0.004/vkd
if((4.*phiy).It.phiu)then
vmu(kel)=(vmu(kel)-vmy)/(phiu-phiy)*3.*phiy+vmy


6
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:


73
where
^1
- mean
value
of
fc?
^2
- mean
value
of
Me;
<*1
- standard deviation of fc;
cr2 standard deviation of Me.
Replacing the standard normal variables in the flexural
failure function the expression assumes the following form:
z = alalul + a2a2u2 + alM-i + a2p.2 + 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:
6 = (axm + a2u2 + b) /(a 1o1 + a2o2)%


60
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


91
mi.
mi. mi.
mi. mi.
Mode 3
Mode 4
Mode 5
mi.
Mode 6
mi. mi. mi.
Mode 7 Mode 8
Figure 5.4. System failure modes.


212
do 305 k=l,m
c(k)=a(k,iflag)
a(k,iflag)=a(k,i)
a(k,i)=c(k)
305 continue
do 310 k=l,nt
c(k)=b(k,iflag)
b(k,iflag)=b(k,i)
b(k,i)=c(k)
310 continue
do 280 j=i+l,nt
if (abs(a(i,j)).gt.0.00001) then
fact=-a(i,j)/a(i,i)
do 290 kk=l,m
a(kk,j)=a(kk,j)+a(kk,i)*fact
290 continue
do 291 kk=l,nt
b(kk,j)=b(kk,j)+b(kk,i)*fact
291 continue
endif
280 continue
200 continue
numec=nt-3*n
c* ************************************************************* *
c Forming bl
c***************************************************** **********
lcount=nt-numec+l
ki=l
do 800 i=lcount,nt
do 810 j=l,nt
bl(j/ki)=b(j,i)
810 continue
ki=ki+l
800 continue
c**************************************************************
c Creating Theta matrix
c******************* ******************************************
do 156 j=l,numec
do 157 i=l,2*n
k=iqh+i
theta(i,j)=bl(k,j)
157 continue
156 continue
c***************************************************************
c Creating virtual displacements
c***************************************************** *********
do 169 j=l,numec
do 158 i=l,iqh
r(i,j)=bl(i,j)
158 continue
c************************************************************
c Adding joint mechanisms
c**************************************************************
do 161 i=3,iqh,3
if (abs(r(i,j)).gt.0.000001)then
r(i,j)=0.0


29
XI area of beam
X2 inertia of beam
X3 horizontal tip displacement
X4 vertical tip displacement
X5 tip rotation
VARIABLE
INITIAL
FINAL
XI (in2)
i.O
6.65
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.


52
Spring Moment-Rotation Diagram
Hu Ultimate moment
My Yielding moment
Kl 10e30
K2 (Hu My)/(Ou fly)
Ksec Spring stiffness for
M > My
Figure 3.6. Secant spring stiffness.


223
al=a2
a2=a
resl=res2
res2=res
go to 100
endif
arml=dd-a+2./3.*xl
arm2=dd-gama*(a-xl)
vmy=ccl*arail+epcs*es*(dd-co)*aste+cc2*arm2
phiy=epsy/(dd-a)
c************************************************ ********
c LINEAR APPROXIMATION
Q********************************************************
ro=aste/(b*d)
vn=es/ec
vk=sqrt(4.*ro*ro*vn*vn+2.*(ro+ro*co/d)*vn)-2.*ro*vn
vkd=vk*d
exc=epsy/(d-vkd)*vkd
cc=0.5*vkd*b*ec*exc
excs=exc/vkd*(vkd-co)
fps=excs*es
cs=fps*aste
vmy=cc*(d-vkd/3.)+cs*(d-co)
phiy=exc/vkd
write(*,*)'cc=1,cc,'cs=',cs,'exc=',exc,'excs=',exes
write(*,*)'linear approx','vmy=',vmy,'phiy=',phiy
stop
end


158
tvag=tvag+clag(k)*psi+psi*psi*rp
200 continue
c***************************************************************
C VALUE OF LAGRANGIAN FUNCTION
C***************************************************************
call valobf (n,ntot,vof,x,cl)
avof = cv*vof
vlag = avof+tvah+tvag
return
end
subroutine modsti(area,ec,vki,vkj,cl,tinert,cosl,cos2)
implicit double precision (a-h,o-z)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
C* ************************************************************* *
C FLEXIBILITY MATRIX (2x2)
C********************************************************* ******
do 10 i=l,6
do 20 j=l,6
ck(i,j)=0.0
20 continue
10 continue
x=cl/(3*ec*tinert)+l./vki
y=cl/(3*ec*tinert)+l./vkj
z=-cl/(6*ec*tinert)
C***************************************************************
C INVERSION OF MATRIX
C******* ********************* ******************** ******* * * *
det=x*y-z*z
a=y/det
b=-z/det
c=b
d=x/det
C**************************************************************
C EXPANDED MATRIX (6x6)
Q************************************************************ *
ckll=ec*area/cl
ckl4=-ckll
ck41=-ckll
ck44=ckll
ck22=(a+b+c+d)/(cl*cl)
ck25=-ck22
ck52=ck25
Ck55=ck22
ck23=(a+c)/cl
Ck53=-Ck23
ck26=(b+d)/cl
Ck56=-Ck26
ck33=a


APPENDIX A
AUGMENTED LAGRANGIAN SUBROUTINES


214
do 121 ikl=l,iqh
vahk(ikl)=0.0
121 continue
n=nel
n3=n+n+n
do 100 k = l,n
sigma2=0.0
do 111 ipo=l,6
u(ipo)=0.0
111 continue
do 200 i = 1,6
m=lm(i,k)
if (m.eq.O) go to 200
if(cvload(m).gt.sigma2)sigma2=cvload(m)
1 = n3 + m
u(i) = x(l)
200 continue
cl=cosl(k)
c2=cos2(k)
d2=-c2*u(1)+cl*u(2)
d3=u(3)
d5=-c2*U(4)+Cl*U(5)
d6=U(6)
c***************************************************************
c element forces
c***************************************************************
c = cl(k)
base = x(3*k-2)
height = x(3*k-l)
aste = x(3*k)
area = base height
tinert = area*height*height/12.0
al = ec*tinert/(c*c*c)
call eley (ec,tinert,c,vksi(k),vksj(k),d2,d3,d5,d6,
* fo3,fo6)
sigmal=cvmu(k)
c***************************************************************
c ultimate and yield moments
c***************************************************************
call newmum(k,x,sigmal,sigma2,fo3,fo6,vksi(k),vksj(k),
* d2,d5,d3,d6)
c***************************************************************
c global modified stiffness
c***************************************************************
call modsti(k,tinert,area,vksi(k),vksj(k))
do 300 1=1,6
j=lm(l,k)
if (j.eq.0) go to 300
do 400 11=1,6
m = lm(ll,k)
if (m.eq.O) go to 400
jj = n3+m
vahk(j)=vahk(j)+ck(l,11)*x(jj)
400 continue
300 continue
100 continue


122
i j
O- plastic
hinge
location
i.j element
node
location
A
Element AB
Ks = GJ/L
where
Ks Secant spring stiffness;
G Shear modulus;
J Torsional moment of inertia;
L Element length.
Figure 7.1. Displacement verification.


71
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*CJ = 3500 + 1.34 550 = 4237 psi
or
f'cr = f'c + 2.33*cr 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.


210
implicit double precision (a-h,o-z)
dimension a(100,100),b(100,100),c(100),cm(100,100),
* cost(n),qa(100,100),sint(n),cl(n),q(100,100),
* lm(6,n),am(100,100),bl(100,100),theta(200,100),
* r(iqh,100)
£* ************************************************************ *
c Constraint matrix for the structure
q**************************************************** **********
do 300 i=l,3*n
do 400 j=l,6*n
cm(i,j)=0.0
q(i,j)=0.0
400 continue
300 continue
do 60 k=l,n
i=3*k
j=6*k
at=l.0/cl(k)
im2=i-2
iml=i-l
jml=j-1
jm2=j-2
jm3=j-3
jm4=j-4
jm5=j-5
cm(im2,j m5)=-l.0
cm(im2,jm2)=1.0
cm(iml,jm3)=l.0
cm(i,j)=1.0
cm(iml,jm4)=-at
cm(iml,jml)=at
cm(i,jm4)=-at
cm(i,jml)=at
60 continue
a*************************************************************it
c Coordinate trnsformation matrix
c**************************************************************
do 70 k=l,n
co=cost(k)
si=sint(k)
j=6*k
do 80 i=l, 2
jum=j-3*i+l
jdois=j-3*i+2
jtres=j-3*i+3
q( jum,jum)=co
q(jum, jdois)=si
q(jdois,jum)=-si
q(jdois,jdois)=co
q(jtres,jtres)=1.0
80 continue
70 continue
Q* ************************************************************ *
c Compatibilibity matrix from LM matrix
c**************************************************************
n6=6*n


31
106Kin lOSKin
ELEMENT
INITIAL
FINAL
1
Area (in2)
1.0
25.4
Inertia (in4)
1.0
120224
2
Area (in2)
1.0
179
Inertia (in4)
1.0
5912
3
Area Cin2)
1.0
35.1
Inertia (in4)
1.0
17058
Figure 2.3. One bay frame.


226
Parameter indicating alteration of the limit of number
of iterations.
Line 24
Maximum number of consecutive iterations without
objective function improvement, maximum number of
consecutive Newton iterations, maximum number of
completed one dimensional searches.
Line 25
Parameter that controls the quantity of information in
the output file.
Line 26
Number indicating minimum printed information.
Line 27
Indication that tangent vector extrapolation should be
used for estimating initial values of basic variables.
Line 28
Number of design variables iniatially included in the
basis.
Line 29
Numbers of design variables of the initial basis.
Line 30
Parameter that indicates if new data should be read.


179
*
8003
8004
if(invmec(lmi).eq.m)
go to 8004
continue
end if
nk=nk+l
numele=numele+l
invmec(n)=m
go to 8000
continue
end if
endif
8002 continue
8001 continue
8000 continue
c*************************************************************
c control of system reliability
c*************************************************************
write(8,*)
write(8,*)BETA MINIMAL FOR THE SYSTEM = ';betmin
write(8,*)
jflag=0
if(betmin.It.relind)then
delta=(relind-betmin)/relind
do 3891 i=l,nel
do 3892 j=l,numele
if(invmec(j).eq.i)then
elerel(i)=(1.+delta)*elerel(i)
endif
3892 continue
3891 continue
jflag=l
endif
return
end
subroutine mecsys(n,iqh,cl,cost,sint,lm,numec,r,theta)
implicit double precision (a-h,o-z)
dimension a(100,100),b(100,100),c(100),cm(100,100),
* cost(n),qa(100,100),sint(n),cl(n),q(100,100),
* lm(6,n),am(100,100),bl(100,100),theta(200,100),
* r(iqh,100)
c**************************************************************
c Constraint matrix for the structure
c**************************************************************
do 300 i=l,3*n
do 400 j=l,6*n
cm(i,j)=0.0
q(i,j)=0.0
400
continue


38
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


79
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 *i=in('l P(ei=0))
where
7T product;
ej_ = 0 if element is in a failure state,


123
5.000 lb
Coefficients
of
variation
fc 0-15
loads -0.15
Materials
fc = 3,000 psi
fy = 40.000 psi
Hinge location
Figure 7.2. Debug frame.


83
Truss Bars
^ Load
FAILURE GRAPH
2F
F Bar Failure
Figure 5.2. Failure graph.


18
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.


84
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, to Sg,
may be represented as in Figure 5.3. Element deformation
parameters may be defined by three independent quantities
S'i displacement about node i;
S'2 ~ rotation of node i;
S'3 rotation of node j.


118
avoid possible precocious eliminations the fundamental
mechanisms are ordered such that those that involve external
work are placed before joint mechanisms. To facilitate the
combination of the mechanisms all virtual displacements are
scaled so that all hinge rotations are unitary. At the end,
if the mechanism with lower reliability index is not
acceptable, the elements with hinges that belong to this
failure mechanism have their required element reliability
indices increased. The optimization process is restarted
with these indices modified by the same percentage of the
system reliability violation.


Table 7.2. Debug frame: Augmented Lagrangian version
Element
Section
1
2
3
4
Total
Total
Initial
Final
Initial
Final
Reliability
Base
Height
Area
Base
Height
Area
Index
(in)
(in)
(in2)
(in)
(in)
(in2)
5.0
10.0
1.0
2.0*
6.0*
0.06*
4.4
5.0
10.0
1.0
2.0*
6.8
0.07*
4.3
5.0
10.0
1.0
2.1*
8.4
0.09*
5.2
5.0
10.0
1.0
2.0*
9.4
0.09*
5.7
Initial
Cost..
* -
lower
bounds.
Final Cost....
Global Displacements
1
2
3
4
5
6
7
8
9
(in)
(in)
(rad)
(in)
(in)
(rad)
(in)
(in)
(rad)
.23
0.0
-.002
.23
-.041'
0.0
.23
-.003
0.0
1.1*
0.0
-.009
1.1*
-.126
.004
1.1*
-.008
-.006
Secant Spring Stiffness
(lb.in/rad)
1, 3, 4, 5, 6, 7, 8
30
Hinge Number
Spring Stiffness
10x10
2
6.7X108


225
User's Manual
Generalized Reduced Gradient Method
Example: Debug Frame
Input File: DATA
Line 1
Problem title.
Line 2
Number of variables, number of constraints, number of
equality constraints.
Line 3
Number of variables with lower bounds.
Line 4 to line 11
Variable number and respective lower bound.
Line 12
Number of constraints with upper bounds.
Line 13 to line 16
Constraint number and respective upper bound.
Line 17 to line 19
Initial values of design variables.
Line 20
Number of prescribed optimization parameters.
Line 21
Constraint tolerance.
Line 22
Convergence tolerance.
Line 23


157
call jacequ(x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,
* fy,ntot, iqh,vah,r,vahk,es,ecm,beta,cvmu,cvload,k,
* vmu,vjac)
call sol(iqh,vjac,r,vinv)
do 5890 jgo=l,iqh
x(jgo+n3)=vinv(jgo)
5890 continue
200 continue
C*********************************************************
C REINITIALIZE VALUES
C********************************************************
k=0
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vo f,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
return
end
subroutine lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,cg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,kl,vmu,rph)
implicit double precision (a-h,o-z)
dimension x(ntot),cl(n),cosl(n),cos2(n),lm(6,n),d(iqg),
* clah(iqh), clag(iqgn), vag(iqgn), r(iqh), vah(iqh),
* ch(iqh), cg(iqgn), vahk(iqh), beta(n),
* cvmu(n), cvload(iqh),vmu(n)
common /parr/ decfc,fcinc,cv,alpl,ec,rp,fc,es,ecm,relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
tvah =0.0
tvag = 0.0
rp2 = 2.0*rp
C**************************************************************
C EQUALITY CONSTRAINTS
C**************************************************************
call equeon (x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,
* fy,ntot,iqh,vah,r,vahk,es,ecm,beta,cvmu,cvload,kl,vmu)
do 100 k=l,iqh
vc=vah(k)*ch(k)
tvah=tvah+clah(k)*vc+rph*vc*vc
100 continue
C***************************************************************
C DISPLACEMENT CONSTRAINTS
C**************************************************************
call inecon (iqg,n,ntot,vag,x,d,relind,beta)
do 200 k=l,iqgn
v=vag(k)*cg(k)
z=-clag(k)/rp2
psi=max(v,z)


46
The force in the compressed steel is given by
where
cs ~ As fcs
As steel area;
fcs stress in compressed steel.
The force in the steel under tension is determined by
Ts As
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 Cl, that
defines the centroid of the concrete compressive stress
diagram, is introduced as
Cl = 1 -
ca
ec fc dec /(£ca
eca
fc dec)
0
These parameters, a and il, 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, Mji, may be defined as


125
The displacements satisfied the global equilibrium equations
and all the constraints were satisfied.
The optimization problem with material nonlinear
behavior was solved using two minimization techniques
described in Chapter 6. Results of the version using Hooke
and Jeeves method are listed in Table 7.2. However,
evaluation of element moments did not correspond to the
location of the hinges, i.e., the element moment values in
some hinges were below the yielding moment value. There was
no force equilibrium in some of the nodes. An extensive set
of initial design points and optimization parameters were
tested with negative results. For that reason the
optimization technique was tentatively replaced by the
Generalized Reduced Gradient.
First attempt to optimize with the Generalized Reduced
Gradient assuming nonlinear material behavior, was with a
secant spring stiffness equal to the ratio between the
yielding moment and the yielding rotation. The option of
using this spring stiffness had the advantage of avoiding
the oscillation of the spring stiffness between the rigid
and lower values. The implementation resulting from this
choice was named yielding stiffness. Although providing
incorrect displacements as the spring stiffness values did
not represent the true material behavior, the yielding
stiffness would model a situation somewhere between the
linear and the nonlinear material behavior. The results are
presented in Table 7.3. After adequate analysis it was


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.
ii


175
if(lj.It.-0.1)then
kmu=-l
lj=abs(lj)
endif
do 400 kk=l,nel
kkj=2*kk-l
thesum(kk)=thesum(kk)+theta(lj ,kkj) *kmu
thesul(kk)=thesul(kk)+theta(lj,kkj+l)*kmu
400 continue
kmu=l
300 continue
c**************************************************************
c reliability of combined mechanisms
c (external work)
c**************************************************************
do 372 l=l,nucome
lcl=lc(j+l-l)
if(lcl.It.-0.1)then
kmu=-l
lcl=abs(lcl)
endif
do 472 kk=l,ndof
if(abs(p(kk)).lt.0.001)go to 472
dispsu(kk)=dispsu(kk)+rb(lcl,kk)*kmu
472 continue
kmu=l
372 continue
5564 continue
c*************************************************************
c combination with fundamental mechanisms
c (internal work)
c*************************************************************
do 100 k=l,numec
vmeanr=0.0
vmeanl=0.0
stdevr=0.0
stdevl=0.0
vmanrm=0.0
vmanlm=0.0
stdvrm=0.0
stdvlm=0.0
do 499 lll=j,j+nucome-1
if(abs(lc(lll)).ge.k)go to 100
499 continue
if(nucome.gt.1)then
if (icontr.eq.1)then
becomi=400.
becopl=500.
go to 5574
endif
endif
thesu=0.0
thesu2=0.0
thesui=0.0
theslm=0.0
do 600 kk=l,nel


98
during the implementation and testing are presented and
discussed.
Augmented Lagrangian Formulation
The set of design variables is divided in two main
groups. These are the dimensions and steel area, defining
each element cross section, and the global displacements.
The objective function is the cost of the structure as a
function of the the volume of concrete and steel. Equality
constraints are defined by the global equilibrium equations.
Inequality constraints include the bounds on global
displacements and the minimum element reliability.
In a reinforced concrete portal frame the set of design
variables x having n elements and m global degrees of
freedom is partitioned as follows:
x, i=l,4,...,3n-2 base of rectangular element section;
Xj, j=2,5,...,3n-l height of rectangular element section;
xjr, k=3,6,..., 3n area of steel on one side of section;
xi, l=3n+l, ..., 3n+m global displacements.
The objective function used in this formulation was
defined using the average costs of cast in place concrete
for reinforced concrete frames and main reinforcing steel
(71). Combined relative cost function was obtained dividing


33
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


168
end
subroutine inecon (iqg,n,ntot,vag,x,d,relind,beta)
implicit double precision (a-h,o-z)
dimension vag(iqg+n),x(ntot),d(iqg),beta(n)
nel3=3*n
c* ******************************************************** *
c displacements
c**********************************************************
do 100 k l,iqg
j = nel3 + k
vag(k) = abs(x(j)) / d(k) 1.
100 continue
c************************************************* *********
c reliability
c**********************************************************
iqgpl=iqg+l
iqgn=iqg+n
do 200 k=iqgpl,iqgn
vag(k) = relind/beta(k-iqg)-1.
200 continue
return
end
subroutine inputd (cl,cosl,cos2,iqh,jdir,jm,lm,n,nj,
* nol,no2,r,xc,yc)
implicit double precision (a-h,o-z)
dimension nol(n),no2(n),jdir(3),xc(nj),yc(nj),
* jm(6,nj),lm(6,n),cl(n),cosl(n),cos2(n),r(iqh)
c**********************************************************
c filling lm matrix
c**********************************************************
do 600 i = l,n
j = nol(i)
k = no2(i)
do 600 1 = 1,3
lm(l,i) = jm (1, j)
lm(l+3,i) = jm(l,k)
600 continue
c**********************************************************
c geometric characteristics
Q* ***************** ************************************** *
do 800 ii = l,n


76
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:
subject to
Minimize CD
Gi(X) < 0, i=l,m
Pu ^ puo
ps ^ pso


182
do 290 kk=l,m
a(kk,j)=a(kk,j)+a(kk,i)*fact
290 continue
do 291 kk=l,nt
b(kk,j)=b(kk,j)+b(kk,i)*fact
291 continue
endif
280 continue
200 continue
numec=nt-3*n
c***************************************************** **********
c Forming bl
c***************************************************************
lcount=nt-numec+l
ki=l
do 800 i=lcount,nt
do 810 j=l,nt
bl(j ,ki)=b(j ,i)
810 continue
ki=ki+l
800 continue
q**************************************************************
c Creating Theta matrix
c*************************************************************
do 156 j=l,numec
do 157 i=l,2*n
k=iqh+i
theta(i,j)=bl(k,j)
157 continue
156 continue
c******** ********************************************** *********
c Creating virtual displacements
c****************************************************** ********
do 169 j=l,numec
do 158 i=l,iqh
r(i,j)=bl(i,j)
158 continue
c************************************************************
c Adding joint mechanisms
c******************************************************* *******
do 161 i=3,iqh/3
if (abs(r(i,j)).gt.0.000001)then
r(i,j)=0.0
do 162 k=l,n
if(lm(3,k).eq.i)then
lpo=2*k-l
theta(lpo,j)=l
endif
if(lm(6,k).eq.i)then
lpo=2*k
theta(lpo,j)=1
endif
162 continue
endif
161 continue
169 continue


SERIES SYSTEM
PARALLEL SYSTEM
Y
Figure 5.1. System models.


42
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:


206
499 continue
if(nucome.gt.1)then
if (icontr.eq.l)then
becomi=400.
becopl=500.
go to 5574
endif
endif
thesu=0.0
thesu2=0.0
thesui=0.0
theslm=0.0
do 600 kk=l,nel
kkj=2*kk-l
thesu=thesum(kk)+theta(k,kkj)
thesu2=thesul(kk)+theta(k,kkj+1)
term=(abs(thesu)+abs(thesu2))*cvmu(kk)
* *vmu(kk)
vmeanr=term/cvmu (kk) +vmeanr
stdevr=stdevr+term*term
thesui=thesum(kk)-theta(k,kkj)
theslm=thesul(kk)-theta(k,kkj+1)
termm=(abs(thesui)tabs(theslm))*cvmu(kk)
* *vmu(kk)
vmannu=tennm/cvmu (kk) +vmannn
stdvrm=stdvrm+tennin*termm
600 continue
c*************************************************************
c combination with fundamental mechanisms
c (external work)
Q*************************************************************
do 672 kk=l,ndof
if(abs(p(kk)).lt.0.001)go to 672
dispkk=(dispsu(kk)+rb(k,kk))*p(kk)
vmeanl=vmeanl+dispkk
stdevl=stdevl+dispkk*cvload(kk)*dispkk
* *cvload(kk)
dispkm=(dispsu(kk)-rb(k,kk))*p(kk)
vmanlm=vmanlm+dispkm
stdvlm=stdvlm+dispkm*cvload(kk)*dispkm
* *cvload(kk)
672
5574
138
continue
becopl=(vmeanr-vmeanl)/sqrt(stdevr+stdevl)
becomi=(vmanrm-vmanlm)/sqrt(stdvrm+stdvlm)
continue
if(becomi.lt.becopl) then
do 138 lk=l,nucome
ltemp=ltemp+l
lc(ltemp)=lc(j+lk-1)
continue
1temp=ltemp+1
lc(ltemp)=-k
lctlp=-k
becote=becomi
else
do 139 lk=l,nucome


154
c = cl(kel)
45 continue
base = x(3*kel-2)
height = x(3*kel-l)
aste = x(3*kel)
area = base height
tinert = area*height*height/12.0
al = ec*tinert/(c*c*c)
fo3 = al*(6.0*c*(d2-d5) + 2.0*c*c*(2.0*d3+d6))
fo6 = al*(6.0*c*(d2-d5) + 2.0*c*c*(d3+2.0*d6))
C****************************************************************
C ULTIMATE AND YIELD MOMENTS
C****************************************************************
call mumy(base,height,aste,vn,co,epsy,ec,fc,fy,
* vksi(kel),vksj(kel),c,fo3,fo6,es,ecm,betak,
* sigmal,sigma2,tinert,kl,vmuk,vmy,ijflag)
if(vksi(kel).It.10e20.or.vksj(kel).It.l0e20)then
call eley(ec,tinert,c,vksi(kel),vksj(kel),
* d2,d3,d5,d6,fo3,fo6)
call mumy(base,height,aste,vn,co,epsy,ec,fc,fy,
* vksi(kel),vksj(kel),c,fo3,fo6,es,ecm,betak,
* sigmal,sigma2,tinert,kl,vmuk,vmy,ijflag)
endif
beta(kel)=betak
vmu(kel)=vmuk
C****************************************************************
C GLOBAL MODIFIED STIFFNESS
Q******************************************************* *********
call modsti(area,ec,vksi(kel),vksj(kel),c,tinert,cl,c2)
do 300 1 = 1,6
j = lm (l,kel)
if (j.eq.O) go to 300
do 400 11 = 1,6
m = lm (11,kel)
if (m.eq.0) go to 400
jj = n3+m
vahk(j)=vahk(j)+ck(l,ll)*x(jj)
400 continue
300 continue
100 continue
Q**********************************************************
C SUBTRACTION OF EXTERNAL GLOBAL FORCES
C***********************************************************
rmax=0.01
do 510 ilj=l,iqh
if(abs(r(ilj)).gt.rmax)rmax=abs(r(ilj))
510 continue
do 500 kpj = l,iqh
if(abs(r(kpj)).It.0.0001)then
vah(kpj)=vahk(kpj)/rmax
go to 500
endif
vah(kpj) = (vahk(kpj) r(kpj))/rmax
500 continue
return
end


27
HOOKE and JEEVES
l Initial Point 4/5 Pattern Move
6 Final Point
Figure 2.1. Pattern Search.


82
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


227
Example: Debug Frame
Input File: DATA1
4,5
1,2
2.3
3.4
4.5
1,1,1,0,0
0,0,0,0,100
0,0,0,50,100
0,0,0,100,100
1,1,1,100,0
2,1,5000
3,2,-5000
'0,0,0
3000,40000,1
29e6,0.004
2
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15


149
c**************************************************************
call datini (clah,clag,ch,eg)
c***************************************************************
c subroutine optimization
c***************************************************************
call optimi (vlag,r,x,cl,cosl,cos2,lm,d,clah,xol,
* vag,toll,clag,vah,ch,eg,xo,vahk,grad,xu,xl,xl,x2,
* delta,alpha,alp,vaho,vago,vn,co,epsy,fy,ast,beta,theta,
* numec,vmu,cvmu,rv,cvload,become,1c,thesum,thesuml,
* dispsum,ni,nucomb,lct,xll,xl2,romin,rph,grad,vahold,
* vjac,vinv,bfal,epsilo)
c***************************************************************
c write final data
q*****************************************************************
write (8,880)
write (8,890) iter
write (8,910) vlag
write (8,840)
do 1000 k = l,n
na = 3 k
ne = na 1
no = ne 1
write (8,851) k, x(no), x(ne), x(na)
1000 continue
write (8,960)
do 1151 k = n21, ntot
i = k 3*n
write (8,970) i, x(k)
1151 continue
write (8,920)
do 1101 k = l,iqh
write (8,930) k, vah(k)
1101 continue
write (8,940)
do 1200 k = l,iqg
write (8,950) k, vag(k)
1200 continue
write (8,944)
do 1211 k=iqg+l,iqgn
write(8,946)k-iqh,beta(k-iqh)
1211 continue
write(8,945)
do 1241 k=l,n
write(8,947)k,vksi(k),vksj(k)
124 continue
write(8,*)
write(8,*)' LAGRANGIAN MULTIPLIERS EQUALITIES'
write(8,*)
do 1277 i=l,iqh
write(8,966)clah(i)
1277 continue
write(8,*)
write(8,*)' LAGRANGIAN MULTIPLIERS INEQUALITIES'
write(8,*)
do 1278 i=l,iqh+n
write(8,966)clag(i)


142
and using a secant approach relies upon the fact the exact
secant spring stiffness value is obtained. There are
certain approximations in the determination of the yielding
and ultimate rotations, that define the moment rotation
diagram from which the secant stiffness is evaluated. All
these instabilities and approximations may create the lack
of convergence that the results have shown.
Future Work
A good approach to improve the adequacy of the
formulation assuming linear material behavior would be the
determination of the proper values for the mean and standard
deviation values of the external loads and concrete
strength. Presently, there is a lack of information to
allow a practical choice of these parameters for each
particular design situation. More research should be done
to examine the influence of including other statistical
parameters such as the cross section dimensions, position of
reinforcing steel, steel strength and load characteristics.
Addition of other element effects will transform this
formulation into a more complete optimization package. The
main element force to be considered is the axial force that
is decisive for column design. This will transform the
system reliability evaluation and the element reliability
constraints. Fundamental failure mechanisms will include
axial failures coupled with flexural failures and there will


120
Hooke and Jeeves and the Generalized Reduced Gradient
methods. The relevant results of these examples are
presented in Tables 7.1 through 7.8.
Result Verification
Validation of results from the three types of frames
described above was accomplished with a common strategy
implemented at three levels. These were element
reliability, compatibility with element moment rotation
diagram, and global structure equilibrium and compatibility.
Control of results was extensively performed for the debug
frame and carefully administered in the other two cases.
To evaluate the element reliability and the
compatibility of the moment rotation diagram at the end or
during the optimization process, a group of two programs was
used. These computer programs called YIEL and ELTES are
listed in Appendix C. The input data is composed of the
dimensions of the cross section, the steel area
reinforcement, the values of the secant stiffness of the
springs, the length of the element and the global
displacements of the element nodes. The output includes the
element moments at the ends, the yielding and ultimate
moments, the yielding and ultimate rotations, and the
element reliability. These values are compared with those
reported by the program results.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
LIST OF TABLES vi
LIST OF FIGURES V
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
iv


192
dh2/dX!
dh2/dx2
dh2/dx3
dh2/dx4
dh2/dx5
dh3/dxi
dh3/dx3
dh3/dx5
x23 (-0.15x3 + x4);
XiX22(-0.45x3 + 3x4);
-0.15x^x23;
xix23;
0;
dh3/dx2 = dh3/dx4 = 0;
-0.45x3;
1;
Step c Initial Design Point and Initial Values
Dependent variables d^t = {xi,x2,x3};
Independent variables dj^ = {X4,xs};
ddfc = {1,10,-0.1333} difc = {-0.02,0.4}
grad = {100,10,0} grad fit = {0,0}
where grad f is gradient of f;
H = [ J I C ]
where H is Hessian matrix of the equalities
J
-l -0.3 30
0 0 -150
-0.6-0.12 0
C
-150 0
1000 0
0 1
Step D Recurrence Formulas
dkt = {ddk I dik};


219
endif
return
end
subroutine coxticon (aste, dd, b, vmy, phiy)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100)/cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cV/ec^pjfC/es^cm^elind^o^y^psy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),
* ck(6,6)
common /equal/ vah(100),vmu(100)
node=0
epso=0.002
c************************************************ **************
c
c exc concrete strain
c epcs compressive steel strain
c epsy yield strain
c
c**************************************************************
c first value for a
c**************************************************************
al=dd/2.
exc=al*epsy/(dd-al)
epcs=exc*(al-co)/al
t=fy*aste
cs=epcs*es*aste
eces=exc/epso
alpha=eces-eces*eces/3.
cc=alpha*fc*b*al
resl=cc+cs-t
c******************************************************** ******
c second value for a
c* ************************************************************ *
a2=0.25*dd
exc=a2*epsy/(dd-a2)
epcs=exc*(a2-co)/a2
cc=fc*a2*alpha*b
eces=exc/epso
cs=epcs*es*aste
res2=cc+cs-t
c***************************************************************
c newton iteration
c***************************************************************
100 a=a2-res2*(a2-al)/(res2-resl)


Figure 6.2. Augmented Lagrangian version flowchart.


25
L(X,,V) = f(X) +UH+PHH+VG' +P G'G'
where
u, v lagrangian multipliers;
P penalty factor;
G' maximum of (G, -v/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


209
C*************************************************************
nk=l
numele=0
do 8000 m=l,nel
do 8001 k=l,numec
do 8002 l=l,nummec
if(abs(locmec(l)).eq.k)then
*
*
8003
8004
endif
8002 continue
8001 continue
8000 continue
ml=2*m-l
m2=2*m
if(abs(theta(ml,k)).gt.0.0.
or.abs(theta(m2,k)).gt.0.0)then
if(nk.gt.1)then
do 8003 lmi=l,nk
if(invmec(lmi).eq.m)
go to 8004
continue
endif
nk=nk+l
nume1e=nume1e+1
invmec(n)=m
go to 8000
continue
endif
c*************************************************************
c control of system reliability
c*************************************************************
write(3,*)
write(3,*)'BETA MINIMAL FOR THE SYSTEM = ',betmin
write(3,*)
jflag=0
if(betmin.It.relind)then
delta=(relind-betmin)/relind
do 3891 i=l,nel
do 3892 j=l,numele
if(invmec(j).eq.i)then
elerel(i)=(1.+delta)*elerel(i)
endif
3892 continue
3891 continue
jflag=l
endif
return
end
subroutine mecsys(n,iqh,cl,cost,sint,lm,numec,r,theta)


89
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 Sg,
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} = [B1] {ra} = 0
where
([Q] [A])* matrix with extra 2n columns;
{ra} 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


102
order method because it does not rely on information about
the shape of the function obtained from derivatives.
Methods based on the second derivatives were discarded since
the problem was highly nonlinear and inequality constraints
had discontinuous second derivatives.
Conjugate Gradient is based on obtaining consecutive
directions that are linearly independent, thus accelerating
the search. The algorithm for the method is summarized as
follows
Step 1: Calculate grad f(xk);
Step 2: dk = -grad f(x);
Step 3: Find ak so that f(xk + ak.dk) = rain;
Step 4: xk+1 = xk + ak.xk
Step 5: Check convergence. If converged, stop.
Step 6: dk+1 = dk + [grad f(xk+1)2/grad f(xk)2].dk
Go to 3.
Conjugate Gradient method proved to be unsuitable for
the type of function presented. Progress in the
minimization was minimal due to the ridge-type shape of the
function. Whenever the process started at any point where


112
As specified before, some subroutines were directly
used from the Augmented Lagrangian formulation while others
had to be adapted or created. Since the data transfer
between subroutines in the Generalized Reduced Gradient
program was made through common data blocks, the same
methodology was used for most of the added subroutines. The
flowchart of this package is presented in Figure 6.3. An
example of the input data files and the listing of the new
or modified subroutines is presented in Appendix C. The
unmodified subroutines perform the same tasks as described
before.
Essential structure of this program is the same as
presented by the authors of the optimization package. There
is a program, OPTIMI, that calls the main subroutines
PRINCI, DATAIN, GRG and OUTRES. PRINCI reads the initial
data from file DATA1 that is not abridged by the typical
input data of the optimization package, which is read in
subroutine DATAIN. The subroutine GRG performs the problem
optimization calling other subroutines. The only subroutine
written for this implementation was GCOMP that computes the
values of the equality constraints, the inequality
constraints, and the objective function. The system
reliability was evaluated and the process was restarted if
the results were unsatisfactory. Subroutine OUTRES writes
the final results of each optimization run to a file RESULT.
This subroutine was also modified to include the relevant


APPENDIX C
GENERALIZED REDUCED GRADIENT SUBROUTINES


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
vi


178
do 6981 klp=l,ndof
dispsu(kip)=0.0
6981 continue
200 continue
c*************************************************************
c control of maximum number of tree rows
c* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
lpti=lpti+nucomb(nucome+1)
nucome=nucome+1
ni(nucome)=ni(nucome-1)+(nucome-1)*nucomb(nucome-1)
if(nucome.It.(numax))go to 111
c*************************************************************
c find minimum beta value
c*************************************************************
betmin=100
do 7890 i=l,lnumbe
if(become(i).It.betmin)then
betmin=become(i)
itab=i
end if
7890 continue
c*************************************************************
c find mechanisms involved
c*************************************************************
mecoun=0
mcomb=l
do 7891 j=l,nucome-1
do 7895 jk=l,nucomb(j)
mecoun=l+mecoun
if(itab.eq.mecoun)then
nistar=mcomb
do 7893 1=1,j
locmec(l)=lct(nistar+l-l)
7893 continue
nummec=j
go to 7894
endif
mcomb=j +mcomb
7895 continue
7891 continue
7894 continue
c*************************************************************
c find elements involved
c*************************************************************
nk=l
numele=0
do 8000 m=l,nel
do 8001 k=l,numec
do 8002 l=l,nummec
if(abs(locmec(l)).eq.k)then
ml=2*m-l
m2=2*m
if(abs(theta(ml,k)).gt.0.0.
* or.abs(theta(m2,k)).gt.0.0)then
if(nk.gt.1)then
do 8003 lmi=l,nk


19
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.


231
(15) Watwood, V. B., Mechanism Generation for Limit
Analysis of Frames, Journal of Structural Division. ASCE,
Vol. 109, ST1, 1979, pg. 1-15.
(16) Thoft-Christensen, P., and Murotsu, Y., Application
of Structural Systems Reliability Theory. Springer-Verlag,
Berlin, 1986.
(17) Ditlevsen, 0., Narrow Reliability Bounds for
Structural Systems, Journal of Structural Mechanics. Vol. 7,
1979, pg. 435-451.
(18) Schmit, L. A., and Fox, R. L. Fox, An Integrated
Approach to Structural Synthesis and Analysis, AIAA Journal.
Vol. 3, No. 6, 1960, pg. 1104-1112.
(19) Grierson, D. E., and Schmit, L. A., Synthesis under
Service and Ultimate Performance Constraints, Computers and
Structures. Vol. 15, No. 4, 1982, pg. 405-417.
(20) Haftka, R. T., Simultaneous Analysis and Design, AIAA
Journal. Vol. 23, No. 7, 1985, pg. 1099-1103.
(21) Hughes, T. J. R., Winger, J., Levit, I., and Tezduar,
T. E., New Alternating Direction Procedures in Finite
Element Analysis Based on EBE Approximate Factorization,
Computer Method for Nonlinear Solids and Structural
Mechanics. AMD, Vol. 54, 1983, pg. 75-109.
(22) Burns, S. A., Simultaneous Design and Analysis Using
Geometric Programming and the Integrated Formulation.
Proceedings of the Swanson Analysis Systems, Newport Beach,
California, 1987.
(23) Marc I. Hoit, Alfredo V. Soeiro and Fernando E.
Fagundo, Integrated Structural Sizing Optimization,
Engineering Optimization. Vol. 12, No.3, 1987, pg. 207-218.
(24) Soeiro, A., and Hoit, M., Sizing Optimization.
Proceedings of the ASCE Structural Conference, Orlando,
Florida, 1987.
(25) Hoit, M., and Soeiro, A., Integrated Structural
Optimization. Proceedings of the International Conference on
Computational Engineering Science, Atlanta, 1988.
(26) Soeiro, A., Integrated Analysis and Optimal Design.
Thesis for the degree of Master of Engineering, University
of Florida, Gainesville, Florida, 1986.
(27) Avriel, M., Nonlinear Programming: Analysis and
Methods. Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
(28) Park, R., and Paulay, T., Reinforced Concrete
Structures. John Wiley and Sons, New York, 1975.


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.
ii

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 smoothe
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
LIST OF TABLES vi
LIST OF FIGURES V
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
iv

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
V

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
vi

LIST OF FIGURES
Figure Page
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 3 4
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.l. 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.
ix

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.
1

2
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.

3
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

4
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
Magne1's Diagram
Optimum Pair P-e
P Initial prestressing torce;
e Eccentricity of cable;
e*- maximum cable eccentricity;
a).b) minimum 1/P;
c).d) maximum 1/P.
Figure 1.1 Implicit optimization.

6
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:

7
2k+l =

where x is the vector of design variables, uk+1 is an
estimative of lagrangian multipliers and

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 ip 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,

8
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

9
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

10
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.

11
Minimize I LiAi
subject to
Fi < Fc
Fi < Ft
where
Li length of truss bar i
Ai area o truss bar i
Fi stress in truss bar i
Fc allowable compressive stress
Ft allowable tensile stress
Figure 1.3. Truss optimization.

12
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

13
Optimization of a Two span prestressed beam
XI
X6
XI to XB Section geometry
X7 to X9 Eccentricities of draped cable
X10 Prestressing force
Figure 1.4. System optimization.

14
OPTIMIZATION
OF
TRUSS GEOMETRY
Load
Initial Configuration
Load
Optimal Configuration
Figure 1.5. Geometry optimization.

15
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

16
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 equalities 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).

17
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.

18
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.

19
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.

20
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

22
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 Laqrancrian Function
The optimization technique of cycling unconstrained
minimizations of a penalty function, based on an pseudo
objective augmented lagrangian function, was chosen as the

23
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;

24
- R vector of static external loads;
- D vector of bounds of m;
Design variables
xk, k=l,3,...,2n-l area of element (k+l)/2;
Xj, j=2,4,...,2n inertia of element j/2;
xf, i=2n+l,2n+2,...,2n+m global displacements
Objective Function
f(x) =2 lpxk, 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

25
L(X,,V) = f(X) +UH+PHH+VG' +P G'G'
where
u, v lagrangian multipliers;
P penalty factor;
G' maximum of (G, -v/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

26
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.

27
HOOKE and JEEVES
l Initial Point 4/5 Pattern Move
6 Final Point
Figure 2.1. Pattern Search.

28
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.

29
XI area of beam
X2 inertia of beam
X3 horizontal tip displacement
X4 vertical tip displacement
X5 tip rotation
VARIABLE
INITIAL
FINAL
XI (in2)
i.O
6.65
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.

30
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

31
106Kin lOSKin
ELEMENT
INITIAL
FINAL
1
Area (in2)
1.0
25.4
Inertia (in4)
1.0
120224
2
Area (in2)
1.0
179
Inertia (in4)
1.0
5912
3
Area Cin2)
1.0
35.1
Inertia (in4)
1.0
17058
Figure 2.3. One bay frame.

32
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

33
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

34
STEEPEST DESCENT
1Initial Point
4-Final Point
Figure 2.4. Gradient method.

35
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

38
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

39
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

40
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

41
One-Component Model
Reinforced Concrete Element
Linear Elastic Element
Spring with Secant Stiffness
Figure 3.1. Element model.

42
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:

43
ts
Steel Stress-Strain Diagram
Figure 3.2. Material behavior.

44
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).
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
Cq = a fern b kd
where
*
eca
a = f c/( fcmeca)<^c
Jo
fem 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.

45
SECTION CHARACTERISTICS
Geometry
Strain
Diagram
Forces
Figure 3.3. Reinforced concrete section.

46
The force in the compressed steel is given by
where
cs ~ As fcs
As steel area;
fcs stress in compressed steel.
The force in the steel under tension is determined by
Ts As
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 Cl, that
defines the centroid of the concrete compressive stress
diagram, is introduced as
Cl = 1 -
ca
ec fc dec /(£ca
eca
fc dec)
0
These parameters, a and il, 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, Mji, may be defined as

47
where
Mi = Cs d'+ Cc n kd Ts d
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:
fty (ey + eca)A*
j^u = (esa + ecu)/^
where
tfy yielding curvature;
ey Es / fy;
Es 29X106 psi;
fy yielding stress of reinforcing steel;
eca maximum concrete compressive strain;
- 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

48
Figure 3.4. Element deformation diagrams.

49
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

50
MOMENT DIAGRAM
Mj Moment at node j
My Yielding moment
CURVATURE DIAGRAM
A
Mj
A
0U- Ultimate curvature
- Yielding curvature
- Curvature at node j
Figure 3.5. Curvature integration.

51
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
-j
1/Ksi 0
+ 3EI/L
1/3
-1/6
0 1/Kgj
-1/6
1/3
C a ]
-10 0 1
0 0
0 1/L 1 0 -1/L 0
0 1/L 0 0 -1/L 1
C Kmod ] = [ a ]t [ Ks* ] [ a ]
where
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;

52
Spring Moment-Rotation Diagram
Hu Ultimate moment
My Yielding moment
Kl 10e30
K2 (Hu My)/(Ou fly)
Ksec Spring stiffness for
M > My
Figure 3.6. Secant spring stiffness.

53
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

55
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).

56
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

57
different and is represented by the adopted respective
probability of failure.
For instance, in reinforced and prestressed concrete,
code checks for the ultimate limit states are based on
element forces, except in the plastic analysis where the
design variables are the loads. In cases where fatigue is
involved, stresses are also the control variables. The
service limit states are the cracking limit state and the
deformation limit state. In this work only the ultimate
flexural limit state and the global deformation limit state
are addressed since they are the more relevant for the
optimization study.
Acceptable risks of failure for any structure are
affected by the nature of the structure itself and its
expected application. These are dependent on social and
local variations. It is common for structural engineers to
balance the contradiction between the economy and safety of
the structure. This particular aspect is the main reason
why it is so appealing to combine reliability and
optimization in structural design.
Probabilities of failure used in limit state designs
vary with the risk of loss of human lives, the number of
lives affected and economic consequences. In ultimate limit
states the range of probability of failure adopted is
between 104 and 107 over a 50 year expected design life.
In serviceability limit states the probability of failure
varies between 101 and 103.

58
A criterion proposed is as follows (41):
pf = 105 U T / L
where
U 0.005 Places of public assembly, dams;
0.05 Domestic, office, industry, travel;
0.5 Bridges;
5 Towers, masts, offshore structures;
T life period of the structure(years);
L number of people involved.
These values must be interpreted carefully. For
example, the value of 103 means theoretically that, on the
average, out of 1000 nominally identical buildings, one will
crack or deform excessively. It is evident that in civil
engineering 1000 identical buildings rarely occur, even
neglecting the fact that a statistically significant number
require samples at least 10 to 20 times larger.
Moreover, the determination of these low probabilities
requires extrapolations of statistical properties that are
experimentally known only around the mean values of the
random quantities. For these reasons, the probabilities of
failure in civil engineering have no real statistical
significance and they must be considered not as
deterministic quantities but just as conventional
comparative values.

59
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

60
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

61
r
A
z = r-s = 0
0(z>0)
SAFE
A
A
A
/fif
Am
A
A
A
D'(z<0)
UNSAFE
Safe and Unsafe Design Regions
Figure 4.1. Design safety region.

62
where
z = r s
r resistance function;
s load function.
The domain D (z>0) is the safe domain and D'(z<0) 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 s as the mean values, and ctr 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 = (2R + o2s)^
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<0} = Fz(0)

63
A graphic representation of these functions is
presented in Figure 4.2. Introducing the standardized
variable u and the reliability index as
u = (Z 2m) / az
B = zm / oz = (rm sm) / (ct^p + a2g)^
then the probability of failure may be expressed as
Pf = Fu(-z"> / oz) = Fu(-fi)
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.

64
Probability Density Function
Cumulative Density Function
Figure 4.2. Probabilistic functions.

65
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 (Xi, 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(xi, 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) / CJXi
Measuring, in this space, the minimum distance 6 of the
transformed surface g (U]_, U2,...., un) from the origin of
the axes. A design is regarded reliable if 6 > fi*, where B*
is prescribed by an appropriate code provision.
In geometrical terms, the hypersphere having radius 6*
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

66
Ul 1-- O
02
Figure 4.3. Safety checks.

67
the hypersphere having radius R*, 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
R = min (2 u2i)h
In a great number of cases the safety boundary domain
is linear, and one can write an expression for z as follows:
z = g (xx, x2,...., xn) = b + 2 aixi
Then, R can be immediately determined as follows
g (U]_, u2,..., un) = b + 2 aixmi + 2 a^ax^ui = 0
and the distance of this hyperplane to the origin is
R = 2 (ai.xrai + b) / (2 a2icrx2j: )%
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

68
Ui
Figure 4.4. Reliability index.

69
the reliability index, B, are within certain approximations
related by
pf = 1 where

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
where
z = r- s = Mj_-Me
- 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 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

70
variables of the expression defining 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
batching, 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

71
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*CJ = 3500 + 1.34 550 = 4237 psi
or
f'cr = f'c + 2.33*cr 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.

72
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 & of each element can be calculated for
any given external moment, section and material properties.
Denoting the basic variables fc as xi, and Me as X2, and
eliminating the other parameters involved in the equation,
the flexural failure function takes the form
z = ax Xi + a2 x2 + b
where
ax = a n b (kd)2;
a2 = -1;
b = As fcs d'- As fy d.
Standardizing
replacement of the
the variables Xx and x2 leads to a
basic variables
ux = (xi m)/ai
U2 = (x2 M-2)/2

73
where
^1
- mean
value
of
fc?
^2
- mean
value
of
Me;
<*1
- standard deviation of fc;
cr2 standard deviation of Me.
Replacing the standard normal variables in the flexural
failure function the expression assumes the following form:
z = alalul + a2a2u2 + alM-i + a2p.2 + 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:
6 = (axm + a2u2 + b) /(a 1o1 + a2o2)%

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

75
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:
where
Minimize Ct = CQ + Cf Pf
Ct cost of the structure;
C0 initial cost of the structure;
Cf cost of failure;
Pf probability of structure failure.

76
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:
subject to
Minimize CD
Gi(X) < 0, i=l,m
Pu ^ puo
ps ^ pso

77
where
Gj_ 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.

SERIES SYSTEM
PARALLEL SYSTEM
Y
Figure 5.1. System models.

79
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 *i=in('l P(ei=0))
where
7T product;
ej_ = 0 if element is in a failure state,

80
ej_ = 1 if element i is in a non-failure state;
P(ej_=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

81
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.

82
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

83
Truss Bars
^ Load
FAILURE GRAPH
2F
F Bar Failure
Figure 5.2. Failure graph.

84
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, to Sg,
may be represented as in Figure 5.3. Element deformation
parameters may be defined by three independent quantities
S'i displacement about node i;
S'2 ~ rotation of node i;
S'3 rotation of node j.

85
Element
Displacements
Independent
Element
Displacements
Rigid Body
Displacements
Figure 5.3. Element displacements definition.

86
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' g 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].
where
{S> = [T] {S'}
[T] =
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0 0
1 0
0 1
0 0
1 -L
0 1
L element length.
For any elementary failure mechanism the element
deformations, s'^, S'2, S'3, must be zero. This is only for
elements that do not have plastic hinges. To materialize
this condition, a matrix 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
matrices as follows:
Ck
-10 0
0 -1/L 1
0 -1/L 0
10 0
0 1/L 0
0 1/L 1
Ci 0 0
0 C2 0
C = .
. *
* .
0 0 cn
Using the previous matrices and vectors the following
relation now holds
C {S} = {S'd}
where
S first element
S second element
S
nth element
and
(S'd)
S'l
S72
S'3
first element
S'l
S'2
S'3
nth element

88
Compatibility between the element displacements, {S}
and the structure global degrees of freedom {r} can be
established
where
CQ]
[A]
{S} = [Q] [A] {r}
rotation matrix;
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

89
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 Sg,
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} = [B1] {ra} = 0
where
([Q] [A])* matrix with extra 2n columns;
{ra} 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

90
case one can find a matrix [D], nonsingular with dimensions
p by p such that
[B'] [D] = [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.

91
mi.
mi. mi.
mi. mi.
Mode 3
Mode 4
Mode 5
mi.
Mode 6
mi. mi. mi.
Mode 7 Mode 8
Figure 5.4. System failure modes.

92
At Level 0 the estimation of the system reliability is
based on the failure of a single structural element. In
this case the system reliability is equal to the reliability
of the element with the higher probability of failure.
Level 1 gives more acceptable results. The concept is
that the structural system is modelled as a series system.
The system probability of failure is estimated as a function
of element probabilities of failure. The calculation of
this system probability can be made with acceptable accuracy
using only those elements with a low reliability index. The
interval where these significant or critical elements are
located is defined by [fimin/ fmin + 6]# where SB is chosen
adequately.
At Level 2, failure elements are grouped in pairs as
parallel systems. These significant pairs of failure
elements are obtained assuming failure of the significant
elements as defined in Level 1. For element i, the load
carrying capacity is added as fictitious loads if the
element is ductile. If the element is brittle, no
fictitious loads are added. Then new element reliability
indices of the modified structure are calculated and the
critical pairs are formed with element i and the new
significant elements.
The process can be analogously repeated for levels
greater than 2 creating critical groups of 3, 4, or more
elements. It is considered that above Level 3 there is no
practical benefit from the extra calculations. The method

93
could be used for ductile structures with the formation of
all collapse mechanisms but the computational effort of the
reanalysis is too great. It is preferable instead to use
the fundamental mechanisms and their linear combinations.
A structure with an elasto-plastic behavior and a given
static load configuration has a certain number of
fundamental failure mechanisms that can be determined using
the Watwood's method. Since they are one degree of freedom
mechanisms the failure function z for mechanism i can be
evaluated as follows:
Zf = S |aij| Rj 2 b^fc Pfc
where
aj rotation at hinge j in mechanism i;
bik ~ displacement of load k in mechanism i;
Rj strength of element j;
Pjc load number k.
Total number of collapse mechanisms is generally too
high and a significant portion of these have a low
probability of failure. For this reason, the Beta unzipping
method is used as it only considers the most critical
failure modes. The value found is a lower bound for the
exact probability of failure, since some mechanisms are
discarded. Once the identification of the fundamental
mechanisms and respective reliability indices are obtained,
the next step is to choose the elementary mechanisms that

94
will be the starting points. Ordering the reliability
indices as follows:
< 62 < < 3q
where
6^ reliability obtained using z.
A control value is selected and added to 8^. This
value and 3i define an interval [B]_, 8^ + £]_]. All
mechanisms outside this interval are discarded for future
combinations. The linear combinations to generate new
failure mechanisms are obtained through combinatorial
matching. First, elementary mechanism 1 is combined with
the mechanisms in the interval and their reliability indices
are evaluated. The same process is repeated for the rest of
elementary mechanisms with those in the interval. The
mechanisms are ordered in accordance with their reliability
indices, a new interval is defined and a new generation of
failure mechanisms is originated. The procedure is repeated
until a sufficient number of generations is accomplished.
Figure 5.5 exemplifies the failure tree creation.
To define the failure function zj for the combinations
of the pair of mechanisms i and j can determined as
zij = 2 j air ajrl ^r 2 (bj_s bjs)Ps,

95
where the sign + or is chosen to give the smallest
reliability index. A graphical illustration of these
combinations is shown in Figure 5.5.

96
Tree of Mechanism Combinations
Mechanism Combination with Reliability
in the Interval B, Rt+£
Figure 5.5. Combinatorial tree.

CHAPTER 6
PROCEDURE IMPLEMENTATION
Introduction
The optimization problem was tentatively solved using
two strategies. These were an Augmented Lagrangian method,
abridged by the class of penalty functions, and the
Generalized Reduced Gradient method, classified in the group
of gradient type techniques. The unconstrained minimization
techniques experimented with the Augmented Lagrangian
formulation are reported and performance is analyzed. Two
final versions for these two options are discussed, with
emphasis on the problems and decisions taken. Subroutines
are described and their essential characteristics
underlined.
Procedures for element and system reliability
evaluations are detailed. Subroutines involved in the
element reliability calculation are listed and their
specific task described. The system reliability
determination at the mechanism level is outlined with a
summarized description of the Beta unzipping method.
Particular problems, and respective solutions that arose
97

98
during the implementation and testing are presented and
discussed.
Augmented Lagrangian Formulation
The set of design variables is divided in two main
groups. These are the dimensions and steel area, defining
each element cross section, and the global displacements.
The objective function is the cost of the structure as a
function of the the volume of concrete and steel. Equality
constraints are defined by the global equilibrium equations.
Inequality constraints include the bounds on global
displacements and the minimum element reliability.
In a reinforced concrete portal frame the set of design
variables x having n elements and m global degrees of
freedom is partitioned as follows:
x, i=l,4,...,3n-2 base of rectangular element section;
Xj, j=2,5,...,3n-l height of rectangular element section;
xjr, k=3,6,..., 3n area of steel on one side of section;
xi, l=3n+l, ..., 3n+m global displacements.
The objective function used in this formulation was
defined using the average costs of cast in place concrete
for reinforced concrete frames and main reinforcing steel
(71). Combined relative cost function was obtained dividing

99
both unitary costs by the cost of concrete and the result is
as follows:
where
f(x) = (x Xj + xk 10) Lp
Lp length of element p, p = l,m.
Equality constraints, one for each global degree of
freedom, are defined as
hq 2r (kqr x3n+r) Rq/ <3"rl,...,m
where
kqr global stiffness coefficient;
x3n+r global displacement r;
Rq external force q.
Inequality constraints that control the maximum global
displacements and impose a minimum element reliability are
as follows
where
9i x3n+r dr i1,...,m
gj = relj betaj, < 0 j=l,...n
dr maximum absolute value of global displacement r;
relj reliability index of element j;
betaj minimum reliability index prescribed for
element j.

100
The minimization problem stated previously is highly
nonlinear. Consequently the adoption of the optimization
strategy was crucial and its characteristics played a big
role in the decision process. The Augmented Lagrangian
Multiplier method, or Augmented Lagrangian formulation, was
the first choice. It allows for an adaptation of the search
technique to the shape of the design surface since the dual
variables, or lagrangian multipliers, are updated at the end
of each minimization cycle in function of the constraints
violations.
The constrained problem is thus transformed into an
unconstrained function using the Augmented Lagrangian
formulation with the addition of dual variables u and y and
becomes
MK/U/Y) = f(x) + u^.h(x) + P.ht(x) .h(x) +
vt.g*(x) + P.g"1-* (x) .3* (x)
where
g*(x) = max {g(x) -v/(2P)};
P penalty factor.
Pseudo-objective function L is minimized with fixed
values of u and y and these are updated using the following
rules (72)
uk+l = uk + 2Ph(x)
vk+1 = vk + 2Pg*(x)

101
The minimization cycle is repeated until there is no
significant improvement of the objective function f(x).
Penalty parameters, P, contribute significantly to the
efficiency of the minimization procedure. Initially, there
was only one penalty parameter for equality and inequality
parameters alike. Since these two types of constraints have
different sensitivities to changes of the design variables,
different penalty parameters were introduced for the group
of equality constraints and the group of inequality
constraints. These starting values, and consequent updates,
were tuned to the optimization performance to improve the
procedure efficiency. Penalty parameters were obtained from
a set of experimental trials and assessment of the results.
Scaling of the constraints and variables also played an
important role in the optimization. The equality
constraints and objective function were scaled to the same
order of magnitude as the inequality constraints. This was
an attempt to regularize the magnitude of the different
functions composing the augmented lagrangian function with
the intent of smoothing the design surface.
Two methods were tested for the unconstrained
minimization of the augmented lagrangian function. These
were the Conjugate Gradient method, or Fletcher-Reeves, and
the Hooke and Jeeves method. The former one is a variation
of the Steepest Descent method, or Gradient method, and is
classified as a first order method since it is based on the
gradient of the function. The latter is defined as a zero

102
order method because it does not rely on information about
the shape of the function obtained from derivatives.
Methods based on the second derivatives were discarded since
the problem was highly nonlinear and inequality constraints
had discontinuous second derivatives.
Conjugate Gradient is based on obtaining consecutive
directions that are linearly independent, thus accelerating
the search. The algorithm for the method is summarized as
follows
Step 1: Calculate grad f(xk);
Step 2: dk = -grad f(x);
Step 3: Find ak so that f(xk + ak.dk) = rain;
Step 4: xk+1 = xk + ak.xk
Step 5: Check convergence. If converged, stop.
Step 6: dk+1 = dk + [grad f(xk+1)2/grad f(xk)2].dk
Go to 3.
Conjugate Gradient method proved to be unsuitable for
the type of function presented. Progress in the
minimization was minimal due to the ridge-type shape of the
function. Whenever the process started at any point where

103
the equality constraints were satisfied, the gradient method
was unable to progress to a better point. This happened
because any violation of the constraints created a high
increase of the augmented lagrangian function and the point
behaved as a local minimum. Otherwise, if the point did not
satisfy the equality constraints then the accuracy of the
derivatives obtained through forward difference, was not
good enough to converge to a better design point. As an
example of this abruptness the augmented lagrangian function
is represented as a function of displacements X2 and X3 of
the cantilever shown in Figure 6.1.
Several techniques were implemented to smooth the shape
of the augmented lagrangian to no avail. Scaling of the
variables, objective function and of the constraints were
performed. The displacements were scaled by the
multiplication of a constant regularizing the magnitude of
the set of variables. The scaling of the constraints and
objective function were already referred to as well as
another technique based on the evenness of the rate of
change of the constraints and objective function in terms of
the design variables (12).
These reasons justified the final adoption for
unconstrained minimization of the Hooke and Jeeves method.
This technique had provided acceptable results and
performances in the linear elastic formulation. The main
algorithm may be summarized as follows

104
Cross Section
Figure 6.1. Augmented lagrangian function plot.

)
105
Step 1: xik = xik-1*(l + a)
Step 2: If L(xk) < L(xk_1), a = a*inc;
Otherwise, a = a*dec;
Step 3: i = i+1; go to 1 if i < n;
Step 4: Try pattern move x* = xk + S(xk xk_1);
where
Step 5: Verify termination criteria. If not met, go
to 1. Otherwise, stop.
inc increase factor;
dec decrease factor;
6 stepsize parameter.
Flowchart of the final group of subroutines is
presented in Figure 6.2. The main program PRINCI, reads the
main input data, initializes the correct displacement
values, when solving the equilibrium equations, for the
starting dimensions, calls the optimizer subroutine and
writes the final results. Subroutine OPTIMI controls the
optimization process by verifying if the convergence
criteria is met, updating the lagrangian multipliers and
verifies the system reliability at the end of the
optimization cycle. Subroutine HOOJEE conducts the Hooke

Figure 6.2. Augmented Lagrangian version flowchart.

107
and Jeeves minimization operation. Subroutine LAGFUN
calculates the value of the augmented lagrangian function.
Subroutine DATINI initializes the values of the scaling
factors and lagrangian multipliers. ASSEMB is the
subroutine that creates the initial global stiffness matrix
with the starting values of the elements. Subroutine GLOSTI
solves the initial equilibrium equation system to obtain
good initial displacement values. INPUTD is the subroutine
that reads all the data concerning the definition of the
structure. CONSTR is the subroutine that reads the values
of the constraints. PARAME is the subroutine that inputs
all optimization parameters. Subroutine MECSYS defines all
elementary failure mechanisms of the structure and MULTI is
a related subroutine that multiplies matrices. SYSREL is
the subroutine that calculates the element reliability.
Subroutine LIM controls the maximum and minimum values of
the design variables excluding displacements. SOLCON is the
subroutine that obtains the global displacements with the
nonlinear global stiffness formulation. Subroutine EQUCON
evaluates equality constraints values, INECON calculates
inequality constraint values and VALOBF obtains the
objective function value. ELEY is the subroutine that
recovers the element forces with the current displacement
and element stiffness values. Subroutine MODSTI assembles
the nonlinear stiffness values using the actual spring
secant stiffness values. MUMY is the subroutine that
evaluates the ultimate and yielding values calling,

108
respectively, subroutines VALMU and COMCON. The coding of
the main subroutines is presented in Appendix A.
This method didn't provide acceptable performance for
the nonlinear material behavior. The convergence of the
method concerning the equality constraints was impossible to
obtain, probably because the variations of the equality
constraint values were severe whenever there was any change
of the design variables. For this reason, a mixed method of
integrated and cycling formulations was implemented. The
approximate displacements were obtained using only once a
Gauss type solution method of the equilibrium equations at
the end of each optimization cycle. The Hooke and Jeeves
search did not include the set of displacements although the
group of equality constraints remained in the augmented
lagrangian function definition. The main goal of this
modification was to improve the convergence for the equality
constraints while performing an optimization that would
remain in the vicinity of the previous design point.
Generalized Reduced Gradient
The optimization strategy is based on the iterative
solution of a system of nonlinear equalities. The method
was initially implemented as an extension of the
decomposition for linear programming problems (73). Several
variations and enhancements of this initial formulation

109
followed this work (74-76) but the general formulation of
the problem is
Minimize f(x)
subject to hi(x) = 0 i=l,...,m
x-*- < x < xu, x of Rn, m < n
Inequality constraints are handled as pseudo equality
constraints with the addition of slack variables. This
increase of the size of the variable set is balanced by the
implicit variable elimination generated by the following
relation between changes of design variables
dxb = -J-1 c dxnb
where
xb vector of basic variables;
xnb vector of nonbasic variables;
jl columns of jacobian matrix of equality
constraints corresponding to basic variables
[dh/dxb];
C other columns of jacobian matrix corresponding
to the nonbasic variables [dh/dxnb].
Nonbasic variables are thus calculated as a function of
the basic variables and eliminated from the gradient
calculation. The gradient is calculated whenever a feasible
point is obtained and a line search along that direction

110
tends to provide a better design point while maintaining
feasibility. The values of the nonbasic variables are
consequently evaluated and the process is restarted. If any
of the basic variables is at any bound, a search is performed
on the set of nonbasic variables to find a suitable
replacement. The method creates basically a succession of
feasible solutions x, x1, ..., xP, each one corresponding
to an improvement of the objective function from the
previous design point. The iteration is terminated whenever
the convergence criteria is satisfied or the maximum
prescribed number of iterations is exceeded.
The essence of this optimization technique seemed
adequate for the integrated optimization with nonlinear
constraints and nonlinear material behavior with a
considerable number of equality constraints. The number of
slack variables is not large, and as long as the initial
point is feasible, convergence should be quicker than in the
previous version.
To illustrate and assess the performance of the
generalized reduced gradient method, an example of a
cantilever beam with linear material behavior, submitted to
displacement and stress constraints, was solved. This
example is presented in Appendix B together with a flowchart
of the general algorithm. These preliminary results were
very promising and the following step was to extend the
method to the formulation previously tested with the
augmented lagrangian function.

Ill
The procedure was developed in two phases. First, a
linear material behavior was assumed followed by the
inclusion of material nonlinear behavior was included. The
computer program versions were created combining a public
domain software package and some subroutines already used in
the prior formulation (76). The coded version of the
Generalized Reduced Gradient method is a general purpose
program for constrained optimization and was changed
slightly when adapting to the present case.
Main modifications actually introduced in the software
were to increase the maximum number of variables and
constraints, the extension of the maximum number of Newton
iterations, and the modification of the number of times the
stepsize could be reduced when performing the line search.
The size of the problems tested caused the first alteration.
Although the authors had not tested the program with
examples as large as those described in the next chapter,
the computer code performed with no problems. The variation
of the maximum number of iterations was required due to the
material nonlinear behavior, which imposed a slower
computation of the basic variables when iteratively solving
the set of nonlinear equalities. The need for smaller step
sizes was due to the fact that the order of magnitude of the
change of variables in the vicinity of the design point is
very small compared to the corresponding changes of the
equilibrium equality constraints.

112
As specified before, some subroutines were directly
used from the Augmented Lagrangian formulation while others
had to be adapted or created. Since the data transfer
between subroutines in the Generalized Reduced Gradient
program was made through common data blocks, the same
methodology was used for most of the added subroutines. The
flowchart of this package is presented in Figure 6.3. An
example of the input data files and the listing of the new
or modified subroutines is presented in Appendix C. The
unmodified subroutines perform the same tasks as described
before.
Essential structure of this program is the same as
presented by the authors of the optimization package. There
is a program, OPTIMI, that calls the main subroutines
PRINCI, DATAIN, GRG and OUTRES. PRINCI reads the initial
data from file DATA1 that is not abridged by the typical
input data of the optimization package, which is read in
subroutine DATAIN. The subroutine GRG performs the problem
optimization calling other subroutines. The only subroutine
written for this implementation was GCOMP that computes the
values of the equality constraints, the inequality
constraints, and the objective function. The system
reliability was evaluated and the process was restarted if
the results were unsatisfactory. Subroutine OUTRES writes
the final results of each optimization run to a file RESULT.
This subroutine was also modified to include the relevant

113
Figure 6.3. Generalized Reduced Gradient version flowchart.

114
results of this type of problems, and to modify the output
format of the optimization information.
The particular structure of the optimization package
created some conceptual alterations in the group of
subroutines that evaluate the constraints and control the
variables bounds. For instance, the changes in the design
points are made simultaneously for all design variables.
Therefore the limits of the areas of reinforcement had to be
imposed as variable bounds instead of being controlled by a
specific subroutine.
On the other hand the limits imposed on the
displacements could be considered as upper and lower bounds
of the correspondent design variables, and consequently, the
total number of inequality constraints was substantially
reduced. The only scaling introduced in the problem was the
division of the equality constraints by the order of the
magnitude of the maximum external force.
Reliability
The only statistical parameters considered for the
probability of failure evaluation were the strength of the
concrete and the external loads. For that reason the
failure function is a linear function of these two basic
variables and the reliability index is calculated using the
formula referred in Chapter 4. Whenever there was a change
of the design variables the value of the ultimate moment for

115
each element was calculated in the subroutine VALMU, the
maximum element moment was evaluated in subroutine ELEY and
the element reliability index is determined.
Subroutine VALMU calculates the ultimate moment and
ultimate curvature for each configuration of the element
cross section using the assumptions and correspondent
formulas presented in Chapter 3. The ultimate curvature is
limited to a maximum of four times the yielding curvature
due to serviceability reasons. The curvatures of sections
beyond this point are so high that the corresponding
deformations will transform any regular frame to an
unserviceable structure. Furthermore for curvatures above
these values the strain hardening of high strength steel
would have to be considered. This upper bound for the
ultimate curvature is also a common value used in design of
structures with dynamic loads.
Recovery of the element moments at both ends is
performed in the subroutine ELEY as a function of the global
displacements and the current spring characteristics of the
element. The element stiffness matrix considered in this
evaluation results from the condensation of the element
elastic stiffness matrix and the spring stiffness. During
the optimization process the stiffness characteristics,
including the secant spring stiffness, are those defined in
the previous iteration.
All fundamental mechanisms of the initial structure are
determined in subroutine MECSYS at the beginning of the

116
optimization process using the methodology described in
Chapter 5. The sets of relative displacements
corresponding to the fundamental elasto-plastic mechanisms
are divided into two parts. The first one corresponds to
the virtual displacements of the external loads and the
second one to the added global degrees of freedom. Since
the first set corresponds to global displacements, the joint
mechanisms have to be transformed into element rotations.
Since usually there are no concentrated moments applied at
the nodes these mechanisms do not occur by themselves, they
are active in the linear combination with the fundamental
failure mechanisms that lead to the mechanisms with lower
reliability indices.
After the optimization process is finished, subroutine
SYSREL performs the evaluation of the system reliability at
the mechanism level. For that purpose the material behavior
is assumed plastic after the element rotation exceeds the
ultimate value as illustrated in Figure 6.4. The
combination with the elementary mechanisms is made in a
combinatorial type process. The first mechanism is linearly
combined with the remaining ones, the second mechanism with
the following mechanisms and so forth until the penultimate
is combined with the last one. The new mechanisms are
ordered in terms of the reliability index and those that
fall outside an acceptable interval are skipped from future
combinations. The process is repeated until all possible
combinations with fundamental mechanisms is performed. To

117
ASSUMED MOMENT ROTATION DIAGRAM
Vy- Yielding Rotation
A Ultimate Rotation
v u
M u- Ultimate Moment
My ~ Yielding Moment
Figure 6.4. Bilinear elastic-plastic diagram.

118
avoid possible precocious eliminations the fundamental
mechanisms are ordered such that those that involve external
work are placed before joint mechanisms. To facilitate the
combination of the mechanisms all virtual displacements are
scaled so that all hinge rotations are unitary. At the end,
if the mechanism with lower reliability index is not
acceptable, the elements with hinges that belong to this
failure mechanism have their required element reliability
indices increased. The optimization process is restarted
with these indices modified by the same percentage of the
system reliability violation.

CHAPTER 7
EXAMPLES
Introduction
Examples used to test the program versions are
described and the conditions for the tests are presented. A
one bay frame was used to debug the program during its
development and enhancement. For result comparison, an
available study in literature of a frame optimized using
limit equilibrium theory was used to compare results
obtained from the versions of the present optimization
program. The program was finally tested with a realistic
frame and loading configuration corresponding to an average
building frame.
Three versions of the element stiffness were
implemented and tested. The first one considered the
material behavior as elastic and that provides a high value
for the stiffness of the rotation springs. The second
formulation provided a spring stiffness equal to the ratio
between the element yielding moment and the yielding
rotation. The last version used the secant spring
stiffness. The final version was implemented both with the
119

120
Hooke and Jeeves and the Generalized Reduced Gradient
methods. The relevant results of these examples are
presented in Tables 7.1 through 7.8.
Result Verification
Validation of results from the three types of frames
described above was accomplished with a common strategy
implemented at three levels. These were element
reliability, compatibility with element moment rotation
diagram, and global structure equilibrium and compatibility.
Control of results was extensively performed for the debug
frame and carefully administered in the other two cases.
To evaluate the element reliability and the
compatibility of the moment rotation diagram at the end or
during the optimization process, a group of two programs was
used. These computer programs called YIEL and ELTES are
listed in Appendix C. The input data is composed of the
dimensions of the cross section, the steel area
reinforcement, the values of the secant stiffness of the
springs, the length of the element and the global
displacements of the element nodes. The output includes the
element moments at the ends, the yielding and ultimate
moments, the yielding and ultimate rotations, and the
element reliability. These values are compared with those
reported by the program results.

121
Global structural equilibrium and compatibility was
verified using the program SSTAN (77). The program is
prepared to handle linear elastic analysis. To verify the
nonlinear results some extra elements were added to the
initial structure simulating the nonlinear behavior. These
additional elements placed at the hinge locations normal to
the plane of the frame had a torsional rigidity equal to the
spring secant stiffness. An example of a transformed
structure used to test the accuracy of the displacements of
a debug frame output is presented in Figure 7.1.
Debug Frame
The structure used to verify and evaluate the
performance of the different versions of the program was a
one bay rectangular frame subjected to a horizontal and a
vertical load at the middle of the span. The material
properties, geometric layout, initial dimensions, loads,
reliability indices and other characteristics were
arbitrarily selected, with no intent of creating a practical
design. The global features of this frame are presented in
Figure 7.2.
Results of the optimization performed using the
Generalized Reduced Gradient and assuming linear behavior
are presented in Table 7.1. Performance and final results
were acceptable and satisfied the Kuhn-Tucker conditions.

122
i j
O- plastic
hinge
location
i.j element
node
location
A
Element AB
Ks = GJ/L
where
Ks Secant spring stiffness;
G Shear modulus;
J Torsional moment of inertia;
L Element length.
Figure 7.1. Displacement verification.

123
5.000 lb
Coefficients
of
variation
fc 0-15
loads -0.15
Materials
fc = 3,000 psi
fy = 40.000 psi
Hinge location
Figure 7.2. Debug frame.

124
Table 7.1. Debug frame (GRG): linear version results.
Element Initial Final Reliability
Section Index
Base Height Area Base Height Area
(in) (in) (in2) (in) (in) (in2)
1 5.0 10.0 1.0 2.0* 6.0* 0.20 2.0*
2 5.0 10.0 1.0 2.0* 6.0* 0.23 2.0*
3 5.0 10.0 1.0 2.0* 11.1 0.73 2.0*
4 5.0 10.0 1.0 2.0* 12.4 0.81 2.0*
* lower bounds.
Total Initial Cost. 18,000
Total Final Cost 6,890
Global Displacements
1 23 4 567 89
(in) (in) (rad) (in) (in) (rad) (in) (in) (rad)
Initial .50 .06 0.0 .90 .81 .07 .94 0.0 -.09
Final .54 0.0 -.005 .53 -.087 .003 .53 0.0 -.003

125
The displacements satisfied the global equilibrium equations
and all the constraints were satisfied.
The optimization problem with material nonlinear
behavior was solved using two minimization techniques
described in Chapter 6. Results of the version using Hooke
and Jeeves method are listed in Table 7.2. However,
evaluation of element moments did not correspond to the
location of the hinges, i.e., the element moment values in
some hinges were below the yielding moment value. There was
no force equilibrium in some of the nodes. An extensive set
of initial design points and optimization parameters were
tested with negative results. For that reason the
optimization technique was tentatively replaced by the
Generalized Reduced Gradient.
First attempt to optimize with the Generalized Reduced
Gradient assuming nonlinear material behavior, was with a
secant spring stiffness equal to the ratio between the
yielding moment and the yielding rotation. The option of
using this spring stiffness had the advantage of avoiding
the oscillation of the spring stiffness between the rigid
and lower values. The implementation resulting from this
choice was named yielding stiffness. Although providing
incorrect displacements as the spring stiffness values did
not represent the true material behavior, the yielding
stiffness would model a situation somewhere between the
linear and the nonlinear material behavior. The results are
presented in Table 7.3. After adequate analysis it was

Table 7.2. Debug frame: Augmented Lagrangian version
Element
Section
1
2
3
4
Total
Total
Initial
Final
Initial
Final
Reliability
Base
Height
Area
Base
Height
Area
Index
(in)
(in)
(in2)
(in)
(in)
(in2)
5.0
10.0
1.0
2.0*
6.0*
0.06*
4.4
5.0
10.0
1.0
2.0*
6.8
0.07*
4.3
5.0
10.0
1.0
2.1*
8.4
0.09*
5.2
5.0
10.0
1.0
2.0*
9.4
0.09*
5.7
Initial
Cost..
* -
lower
bounds.
Final Cost....
Global Displacements
1
2
3
4
5
6
7
8
9
(in)
(in)
(rad)
(in)
(in)
(rad)
(in)
(in)
(rad)
.23
0.0
-.002
.23
-.041'
0.0
.23
-.003
0.0
1.1*
0.0
-.009
1.1*
-.126
.004
1.1*
-.008
-.006
Secant Spring Stiffness
(lb.in/rad)
1, 3, 4, 5, 6, 7, 8
30
Hinge Number
Spring Stiffness
10x10
2
6.7X108

127
Table 7.3.Debug frame (GRG): yielding stiffness results.
Element Initial Final Reliability
Section Index
Base Height Area Base Height Area
(in) (in) (in2) (in) (in) (in2)
1 2.0 6.0 0.41 2.0* 6.0* 0.21 0.1*
2 2.0 6.0 0.41 2.0* 6.0* 0.21 0.1*
3 2.0 9.74 0.64 2.0* 9.75 0.64 0.1*
4 2.0 10.9 0.71 2.0* 10.9 0.71 0.1*
* lower bounds.
Total Initial Cost 6,599
Total Final Cost 6,296
Global Displacements
123 4567 89
(in) (in) (rad) (in) (in) (rad) (in) (in) (rad)
Initial .77 0.0 -.008 .77 -.115 .004 .76 -.007 -.004
Final .82 0.0 -.008 .81 -.118 .004 .81 -.006 -.004
Yielding Spring Stiffness
(lb.in/rad)
Hinge Number 12 345678
Spring Stiffness 6.9 6.9 49 49 83 83 43 43

128
verified that the equilibrium of the moments at the nodes
and the compliance with the moment rotation diagrams were
satisfied. While this solution converged in some cases, it
did not in others. There seemed to be no general pattern to
the problem.
The next step was to test the formulation using the
secant stiffness spring values obtained from the element
moment rotation diagram. The results of one of these
problems are presented in Table 7.4. In this case, a
situation similar to the Hooke and Jeeves minimization was
encountered. The node equilibrium and the element moment
rotation diagram were not in accordance with the final
values. To illustrate these discrepancies of the final
results, the values of the yielding moments, ultimate
moments, and moments at the nodes recovered using the
condensed stiffness matrix are shown in Table 7.5.
To improve the performance of the optimization with a
nonlinear material behavior, the use of better estimates of
starting design points was tried. For that purpose the
frame was optimized in the linear version having the
ultimate moment set as the yielding moment, i.e, no element
was allowed to yield. These solutions of the linear
behavior would theoretically provide the best starting
points. The optimization problem was thus transformed to a
two stage process: a linear solution with the elements close
to the yielding situation followed by a nonlinear
optimization having as starting values the results of the

129
Table 7.4. Debug frame (GRG): secant stiffness results.
Element
Initial
Final
Reliability
Section
Base
Height
Area
Base Height Area
Index
(in)
(in)
(in2)
(in) (in) (in2)
1
2.0
6.0
0.41
2.0* 6.0* 0.38
0.1*
2
2.0
6.0
0.41
2.0* 6.0* 0.25
0.7
3
2.0
9.74
0.64
2.0* 9.81 0.64
0.1*
4
2.0
10.9
0.71
2.0* 10.9 0.72
0.1*
* lower
bounds.
Total Initial Cost 6,599
Total Final Cost 6,504
Global Displacements
12 3 456 78 9
(in) (in) (rad) (in) (in) (rad) (in) (in) (rad)
Initial
.77
0.0 -.008
.77 -.115
.004
.76 -.007
Final
CO
CO

0.0 -.007
.87 -.152
.005
.87 -.007
Secant Spring Stiffness
(lb.in/rad)
Hinge number 12345678
Spring Stiffness 0.9 0.9 29
(xlO?)
29 84 84
58
58

130
Table 7.5. Debug frame: element
Element
1
2
3
4
Yielding
(lb.in)
3.43e4
2.00e4
14.9e4
22.5e4
Element
1
2
3
4
Node i
(lb.in)
-4.15e4
1.21e4
-3.05e4
21.5e4
moments.
Ultimate
(lb.in)
6.27e4
4.28e4
20.3e4
28.5e4
Node j
(lb, in)
-5.46e4
3.27e4
2 0.3e4
26.2e4

131
first stage. The results were, however, similar to those
attained before.
Compared Frame
To complement the program testing with the one bay
frame and evaluate the capacity of the program to obtain
accurate and exact optimal solutions, a frame that was
optimized using the theory of the Optimal Limit Design was
also tested (78). This published example had the advantage
of considering the nonlinear behavior of reinforced concrete
elements at ultimate capacity. The resulting moment
redistribution at the nodes was limited to values assuring a
certain serviceability. The definition of the frame and
respective loads are presented in Figure 7.3.
This choice presented some disadvantages. Optimization
was carried out with the design variables as the steel
reinforcement areas, the elements in the reference were
singly reinforced, there were no reliability limits imposed,
and the moment redistribution was limited to a maximum of
30%. It follows a similar approach to the system
reliability in obtaining the optimal redistributed moment
diagram when evaluating the performance of the several
ultimate failure mechanisms. The failure mechanism with
lower external work is defined and in Table 7.6 the moment
redistribution coefficients, factored external moments and
redistributed moments are presented. The steel
reinforcement areas for the redistributed moments, or

132
ELEMENT DIMENSIONS
1.4,7,10 1Binx18in
others llinxllin
LOADS DEAD LIVE
W1 7.5K 15K
W2 5K
Original Sections
6 8 10 14 15
13 15
12
EQUIVALENT LOADS
W1 40K W2 10K
Figure 7.3. Compared frame.

133
Table 7.6. Compared frame: initial steel area reinforcement.
Section
cj
Mu (3c. ft)
I.8M1 + 1.5M
cj Mu
(K.ft)
Steel
Area (in2)
Comparison
with
elements
1
1.0
114
114
2
1.0
61
61
1
3
0.7
55
39
or
10
4
0.9
136
122
2.6
5
0.7
55
39
2
6
0.9
136
122
2.6
or
9
7
1.0
196
196
3.5
2 or 3
8
0.7
25
18
or
8 or 9
9
0.7
33
23
3
10
1.0
203
203
3.7
or
8
11
0.7
44
31
4
12
0.7
51
36
0.7
or
7
13
0.7
11
8
14
1.0
173
173
3.1
5
15
1.0
169
169
or
6
16
0.7
8
6
Legend:
indicates
sections
separation of
in the original
groups
study
of element
comparable with
element sections in present research frame;
cj redistribution coefficient;
Mu ultimate section moment;
Mi live load moment;
Md dead load moment.

134
optimal final moment diagram, are also presented in Table
7.6 and were calculated in accordance with ACI 318-63. This
code was used in the original published work to define
factored loads and ultimate section capacities.
The frame was initially tested assuming the linear
material behavior and the final results are presented in
Table 7.7. To simulate the same requirements the frame was
optimized using an equivalent set of loads. This set
resulted from the multiplication of the service loads by the
correspondent load factors and by the inverse of the
strength reducing factors prescribed in ACI 318-63. Final
values are very close to those obtained with the Optimal
Limit Design results and with a total lower cost.
Solution was then attempted with the yielding stiffness
and the secant stiffness versions. Results fell in two
unacceptable categories. The first category included the
results with some optimization but no convergence of the
equilibrium constraints. The other had very slight decrease
of objective function and verification of equality and
inequality constraints. Several starting points were tried,
including the design points obtained from the linear
solution, but no practical results were obtained.
Convergence and oscillation were again the key problems.

135
Table 7.7. Compared frame results.
Element 1
Steel Area
(in2)
Initial 3.0
Final 0.8
Reliability
Index 0.0*
Element 6
Steel Area
(in2)
Initial 3.0
Final 2.4
Reliability
Index 0.0*
2
3
4
5
3.0
3.0
3.0
3.0
2.9
3.0
0.6
2.5
0.0*
0.0*
0.0*
0.0*
*
- lower
bounds
7
8
9
10
3.0
3.0
3.0
3.0
0.3
2.5
2.5
2.5
0.0*
0.0*
0.0*
0.0*
* -
lower
bounds.
Initial Steel Cost 86,400
Final Steel Cost 54,720
OLD Steel Cost 63,360

136
Building Frame
To evaluate the performance of the program for a common
practical design a typical rectangular building frame with
two spans and three stories high was defined. Lateral and
vertical loads were calculated using the Standard Building
Code. Definition of the frame geometry, horizontal loads,
vertical loads, material properties and floor plan are
presented in Figure 7.4.
Vertical loads were applied at the midspan of each
beam. Values were equivalent to the distributed loads along
the adjacent slabs since this formulation does not handle
loading along the element. The pattern chosen for the
distribution of the vertical loads aims to create maximum
moments in the elements. For this reason the loading
combination includes the wind loads.
The major frame was analyzed using the linear version
of the Generalized Reduced Gradient method and the results
are summarized in Table 7.8. Kuhn-Tucker conditions were
verified and the final dimensions of the cross sections
corresponded to the lower bounds. The exception to this
last conclusion happened whenever the steel reinforcement
attained the upper limit. Testing of the nonlinear
versions, both with the yielding and the secant spring
stiffness formulations, provided no acceptable results in a
similar manner to that observed when testing the compared
frame.

137
5K
FLOOR
PLAN
J
u
u
L
1


C
1
n
n
:
h + ^ H
20ft 20ft 20ft
Figure 7.4. Building frame.

138
Table 7.8. Building frame results.
Element
1
2
3
4 5
6
7
8 9
Base
(in)
Initial
10
10
10
10 10
10
10
10 10
Final
8*
8*
8*
8* 8*
8*
8*
8* 8*
Height
(in)
Initial
25
25
25
30 30
30
25
25 25
Final
12*
12*
25
16* 16*
20
12*
12* 12*
Area
(in2)
Initial
3.0
3.0
3.0
3.0 3.0
3.0
3.0
3.0 3.0
Final
1.7
1.6
5.5
1.8 1.6
4.3
1.2
2.7 1.2
Reliability
Index
3.0*
3.0*
3.0*
3.0* 3.0*
3.0*
3.0*
3.0* 3.0*
Element
10
11
12
13
14
15
16
17
18
Base
(in)
Initial
10
10
10
10
10
10
10
10
10
Final
8*
8*
8*
8*
8*
8*
8*
8*
8*
Height
(in)
Initial
25
25
25
30
30
30
25
25
25
Final
16*
16*
16*
12*
12*
12*
16*
16*
16*
Area
(in2)
Initial
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
Final
1.7
2.2
1.1
.57
1.2
1.1
.69
.71
.77
Reliability
Index
3.0*
3.0*
3.0*
3.0*
3.0*
3.0
3.0*
3.0*
3.0
* lower bounds.
Total Initial Cost 515,400
Total Final Cost 401,274

CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS
Linear Material Behavior
This optimization approach with reliability constraints
proved to be a valuable formulation for reinforced concrete
frames with linear material behavior and static loading.
The formulation addresses a universal procedure for
obtaining optimal solutions independently of the local code
restrictions. The choices for the element and system
reliability indices are made by the user and may be chosen
as a function of the particular problem conditions.
The approach depends on initial choices and these have
a significant effect on final results. These effects can be
overcome by careful evaluation and planning by the designer.
Most relevant aspects are the choice of adequate element and
system reliability indices, the definition of the material
and of the load statistical values and the displacement
limits. Solutions provided by the current approach are not
definitive designs, since important aspects like axial
forces and shear forces are not included.
139

140
The results presented showed a perfect convergence,
even when the initial displacements were not those
corresponding to the starting physical properties. The
Kuhn-Tucker conditions were always verified unless the lower
bounds were active, as in the case of the building frame.
This implied that at least a local optimum was obtained.
For instance, a good indication of the quality of the
program performance was that in each case, the variables
representing the element bases always converged to the lower
bound. Another particular aspect of the program
capabilities was that at the end of the optimization the
displacement variables were always in the set of basic
variables of the Generalized Reduced Gradient method. This
meant that no improvement could be extracted from the
objective function, except iterating on the equilibrium
equations.
Integration of displacements in the set of design
variables was a valid option for optimization with
reliability considerations. Element reliability constraints
were always active unless there were conflicting lower
bounds. A good compromise was established between the
optimization and safety requirements. System reliability
was also satisfied every time required probabilities of
failure for the elements and the system were of the same
order of magnitude. The method proved to be adequate for
optimal predesign without code limitations.

141
Nonlinear Material Behavior
Tests performed with material nonlinear behavior were
not completely successful. The results of the debug frame
showed optimized solutions, mainly if the two stage
procedure was followed. However, as shown in Table 7.6
there was no complete node equilibrium. For the other types
of frames, independent of the technique and initial values
chosen, the results showed that simultaneous equilibrium
convergence and optimization did not occur. In certain
cases with these types of frames, there was satisfaction of
the constraints and little improvement of the objective
function. In other cases, the opposite results were
obtained.
The most probable reason for these failures is
attributed to the errors in the evaluation of the secant
spring stiffness. The element forces and the global
displacements are related to these values. On the other
hand, the changes in the element properties during the
optimization process create severe oscillations of the
secant spring stiffness values. The values of the spring
stiffness parameters oscillate abruptly between 1030 to
1010, approximately, when the moment exceeds the yielding
threshold. Also, after yielding, the spring stiffness
values oscillate between values of different order of
magnitude: the yielding stiffness and the ultimate
stiffness. The nonlinear analysis is a path dependent event

142
and using a secant approach relies upon the fact the exact
secant spring stiffness value is obtained. There are
certain approximations in the determination of the yielding
and ultimate rotations, that define the moment rotation
diagram from which the secant stiffness is evaluated. All
these instabilities and approximations may create the lack
of convergence that the results have shown.
Future Work
A good approach to improve the adequacy of the
formulation assuming linear material behavior would be the
determination of the proper values for the mean and standard
deviation values of the external loads and concrete
strength. Presently, there is a lack of information to
allow a practical choice of these parameters for each
particular design situation. More research should be done
to examine the influence of including other statistical
parameters such as the cross section dimensions, position of
reinforcing steel, steel strength and load characteristics.
Addition of other element effects will transform this
formulation into a more complete optimization package. The
main element force to be considered is the axial force that
is decisive for column design. This will transform the
system reliability evaluation and the element reliability
constraints. Fundamental failure mechanisms will include
axial failures coupled with flexural failures and there will

143
be additional element degrees of freedom in failure
mechanism sets. At the element level, the element failure
constraint would be replaced by a set of constraints
concerning also the axial failure and the interaction of
flexural and axial forces. Shear failure, important in
reinforced concrete elements, could also be added in a
similar fashion.
In the problem involving nonlinear material behavior
some alterations could provide a better performance in the
nonlinear optimization. These include the use of a mixed
approach of the integrated and the cycling formulation,
similar to that used in the Hooke and Jeeves version. A
possible improvement is the inclusion of an intermediate
stage where the solution for the exact displacements would
be calculated whenever the absolute violation of the
equality constraints exceeds an upper limit. This mixed
approach could improve the efficiency of this approach since
good displacements are essential for the definition of the
correct global and element nonlinear behavior.
Another possible improvement is the use of a different
model for the nonlinear reinforced concrete element. The
substitution of the one-component model by a model of an
element partitioned in several discrete elements. These
discrete elements, each with linear stiffness
characteristics, defined by the global element nonlinear
properties, would provide better accuracy for the element
behavior. This solution has the disadvantage of increasing

144
substantially the size of the problem. However, the benefits
of this change could be significant.
Changing the nonlinear analysis method from secant
stiffness approach to a tangent stiffness approach could be
another solution to the lack of convergence. In this case a
two stage process would be adopted. The first would consist
of a linear optimization up to the formation of a hinge
followed by a phase with a sequence of incremental loading
and optimization procedures until convergence was obtained.

APPENDIX A
AUGMENTED LAGRANGIAN SUBROUTINES

program princi
implicit double precision (a-h,o-z)
character*40 title
dimension x(100),r(60),cl(60),cosl(60),cos2(60),xol(l00),
* clah(80),clag(80),lm(6,50),vag(100),vah(80),
* ch(80),cg(80),xo(100);nol(80),no2(80),jm(6,80),alp(80),
* xc(80),yc(80),jdir(3),glk(80,80),d(1000),vinv(100),
* vahk(80),grad(100),xu(100),xl(100),xl(100),x2(100),vaho(80),
* vago(80),a(80,80),b(80,80),vahold(100),vjac(100,100),
* c(80),cm(80,80),qa(80,80),q(80,80),am(80,80),bl(80,80),
* theta(100,80),rv(80,100),beta(80),vmu(80),cvmu(80),
* cvload(80),become(100),lc(100),thesum(80),thesuml(80),
* dispsum(60),ni(60),nucomb(60),lct(100),bfal(100,100)
common /parr/ decfc,fcinc,cv,alpl,ec,rp,fc,es,ecm,relind
common /pari/ iter,numcy,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
open ( 8,file='finres',form='formatted' )
rewind 8
open ( 9,file='data',form='formatted' )
rewind 9
c********************************************** *************
c name of problem
c******** **************************** ************************
read ( 9,191 ) title
Q************************************************************
c # elements and # joints
c****************************************************** ******
read (9,*) n, nj
c* ********************************************************** *
c nodes per element
c*************************************************** **********
do 100 i = l,n
read (9,*) nol(i), no2(i)
100 continue
c*************************************************************
c initialize jm matrix
c**************^**^********************************** ********
do 200 kk = l,nj
jm ( l,kk ) =1
jm ( 2,kk ) = 2
jm ( 3,kk ) =3
200 continue
c*************************************************************
c support conditions and coordinates
c**************************************************************
do 300 j = l,nj
read ( 9,* ) jdir(l),jdir(2),jdir(3),xc(j),yc(j)
do 350 i =1,3
if (jdir(i).gt.0) then
jm (i, j) =0
endif
350 continue
300 continue
c************************************************************
146

147
c global degrees of freedom
g* ******************************************************* -k * h
iqh=0
do 500 j=l,nj
do 500 1 = 1,3
if (jm(l,j).ne.O) then
iqh=iqh+l
jm (1,j)=iqh
endif
500 continue
iqgn = iqh+n
iqg = iqh
ntot = iqgn+2*n
g**************************************************************
c input data
c**************************************************************
call inputd (cl,cosl,cos2,iqh,jdir,jm,lm,n,nj,nol,
* no2,r,xc,yc)
g**************************** ******************************
c reinforced concrete
c**********************************************************
read(9,*)fc,fy,co
ec=57000*sqrt(fc)
vn=29e6/ec
epsy=fy/29e6
do 987 ijh=l,n
vksi(ijh)=10e30
vksj(ijh)=10e30
987 continue
read(9,*)es,ecm
c************************************************************
c reinforcing steel guess
c********************************************* **************
n3=n*3
read( 9, *) (x( i), i=3 ,n3,3)
c******* *****************************************************
c constraint values
c***********************************************************
call constr (iqg,d)
c************************************************************
c determine bandwidth
c************************************************************
mband = 0
do 450 k = l,n
do 450 i = 1,6
if ( lm(i,k).eq.0) go to 450
do 440 j = i,6
if (lm(j,k).eq.0) go to 440
max = abs(lm(i,k) lm(j,k)) + 1
if (max.gt.mband) mband=max
440 continue
450 continue
C* * * * k * ie "k ic k * ic ic ic ic * ie * * * ic jc ie "k ie 1c ie * Jc * * *
c optimization parameters and initial guesses
g**************************************************************
call parame(toll,x,n21,glk,mband,cl,cosi,cos2,lm,

148
* r,delta,alpha,numec,rph)
c**************************************************************
c elementary mechanisms
c**************************************************************
call mecsys(n,iqh,cl,cosl,cos2,lm,numec,rv,theta,rv)
c**************************************************************
c coefficients of variation
c************************************************** ************
do 156 i=l,n
read(9,*)cvmu(i)
156 continue
do 157 i=l,iqh
read(9,*)cvload(i)
157 continue
c************************************************** **********
c lower bounds
c************************************************************
romin=200./fy
read(9,*)xll,xl2
c***********************************************************
c interval for generation of mechanisms
c*************************************************** ********
read(9,*)epsilo
c************************************************************
c write input data
c*************************************************************
write (8,190) title
write (8,110)
write (8,130) n
write (8,140) iqh
write (8,150) iqg
write (8,170) fc,fy
write (8,240)
do 501 k = l,iqh
write (8,250) k, r(k)
501 continue
write (8,260)
do 601 k = 1,iqg
write (8,270) k, d(k)
601 continue
write (8,351)
do 701 k = l,n
na = 3 k
ne = na 1
no = ne 1
write (8,360) k, cl(k), x(na), x(no), x(ne)
701 continue
write (8,650) rp
write (8,760) ga
write (8,860)
do 900 i=n21,ntot
k=i-3*n
write (8,870)k,x(i)
900 continue
c**************************************************************
c data initialization

149
c**************************************************************
call datini (clah,clag,ch,eg)
c***************************************************************
c subroutine optimization
c***************************************************************
call optimi (vlag,r,x,cl,cosl,cos2,lm,d,clah,xol,
* vag,toll,clag,vah,ch,eg,xo,vahk,grad,xu,xl,xl,x2,
* delta,alpha,alp,vaho,vago,vn,co,epsy,fy,ast,beta,theta,
* numec,vmu,cvmu,rv,cvload,become,1c,thesum,thesuml,
* dispsum,ni,nucomb,lct,xll,xl2,romin,rph,grad,vahold,
* vjac,vinv,bfal,epsilo)
c***************************************************************
c write final data
q*****************************************************************
write (8,880)
write (8,890) iter
write (8,910) vlag
write (8,840)
do 1000 k = l,n
na = 3 k
ne = na 1
no = ne 1
write (8,851) k, x(no), x(ne), x(na)
1000 continue
write (8,960)
do 1151 k = n21, ntot
i = k 3*n
write (8,970) i, x(k)
1151 continue
write (8,920)
do 1101 k = l,iqh
write (8,930) k, vah(k)
1101 continue
write (8,940)
do 1200 k = l,iqg
write (8,950) k, vag(k)
1200 continue
write (8,944)
do 1211 k=iqg+l,iqgn
write(8,946)k-iqh,beta(k-iqh)
1211 continue
write(8,945)
do 1241 k=l,n
write(8,947)k,vksi(k),vksj(k)
124 continue
write(8,*)
write(8,*)' LAGRANGIAN MULTIPLIERS EQUALITIES'
write(8,*)
do 1277 i=l,iqh
write(8,966)clah(i)
1277 continue
write(8,*)
write(8,*)' LAGRANGIAN MULTIPLIERS INEQUALITIES'
write(8,*)
do 1278 i=l,iqh+n
write(8,966)clag(i)

150
1278 continue
c****************************************************** ******
c output format
Q************************************************************
110
120
130
140
150
160
170
180
190
191
240
250
260
270
351
353
360
370
380
390
640
650
730
740
760
840
851
860
870
880
890
910
920
930
940
944
945
946
947
950
960
966
970
format ( //, lOx, ******* initial values ******,/ )
format ( a25 )
format ( /,lOx,'number of elements = ',i3 )
format ( /,lOx,'number of equality constraints = ',i3 )
format ( /,10x,'number of inequality constraints = ',i3 )
format ( /,lOx,'number of iterations per cycle = ',i5 )
format ( /,lOx,'fconcrete = ',el4.8,4x,'fsteel =',el4.8)
format ( /,lOx,'number of global degrees of freedom
= M2 )
format ( //,10x,a25,// )
format (a)
format(//,lOx,'global degree of freedom',lOx,
'external force')
format ( /,20x,i2,22x,el4.8 )
format ( //,lOx,'global degree of freedom',5x,
'displacement constraint' )
format ( //20x,i3,19x,el4.8 )
format ( //,'element',8x,'length',lOx,'steel',12x,
'base',12x,'height')
format ( 18x,i3,6x,el4.8,/)
format ( /,i3,6x/el4.8,3x,el4.8/4x,el4.8,4x,el4.8 )
format ( /,lOx,i3,14x,3i4,3x,3i4 )
format(//,lOx,'location matrix for global degrees
of freedom ')
format ( /,10x,' element ',10x,' node i',10x,'node j' )
format ( /,10x,'maximum number of cycles = ',i3 )
format ( /,lOx,'penalty factor = ',el4.8 )
format ( /,lOx,'factor of increase = ',el4.8 )
format ( /,10x,'factor of decrease = ',el4.8 )
format ( /,lOx,'penalty factor multiplier = ',el4.8 )
format ( /,12x,'element',llx,'base',17x,'height',lOx,
'steel')
format ( /,15x,i2,8x,el4.8,8x,el4.8,5x,el4.8)
format ( /,10x,'global degree of freedom',5x,
'initial guess' )
format ( /,20x,i2,18x,el4.8 )
format ( ////,lOx,'******* final values *******',/// )
format ( /,lOx,'number of iterations = ',i3 )
format ( /,10x,'value of lagrangian function = ',el4.8 )
format ( /,lOx,'equality',17x,'final value' )
format ( /,12x,i3,17x,el4.8 )
format ( /,lOx,'inequality',13x,'final value')
format ( /,5x,'element reliability',9x,'final value')
format (/,5x,'element',5x,'spring i',5x,'spring j')
format ( /,llx,i4,17x,el4.8)
format (/,8x,i5,8x,el4.8,3x,el4.8)
format ( /,12x,i3,17x,el4.8 )
format ( /,lOx,'displacement',14x,'final value' )
format (/,el4.8)
format ( /,12x,i3,17x,el4.8 )
stop
end

151
subroutine comcon(aste,fy,es,d,b,co,epsy,ecm,fc,vmy,phiy)
implicit double precision (a-h,o-z)
kll=0
node=0
epso=0.002
C
C EXC CONCRETE STRAIN
C EPCS COMPRESSIVE STEEL STRAIN
C EPSY YIELD STRAIN
C
C FIRST VALUE FOR A
C****************************************************************
al=d/2.
exc=al*epsy/(d-al)
epcs=exc*(al-co)/al
t=fy*aste
cs=epcs*es
eces=exc/epso
alpha=eces-eces*eces/3.
cc=alpha*fc*b*al
resl=cc+cs-t
C*********************************************************** ******
C SECOND VALUE FOR A
C*****************************************************************
a2=0.25*d
exc=a2*epsy/(d-a2)
epcs=exc*(a2-co)/a2
cc=fc*a2*alpha*b
eces=exc/epso
cs=epcs*es
res2=cc+cs-t
C* ************************************************************ **
C NEWTON ITERATION
C***************************************************************
100 a=a2-res2*(a2-al)/(res2-resl)
exc=a*epsy/(d-a)
if(exc.gt.epso) go to 200
c******************************************************** ********
C PARABOLIC SHAPE
C****************************************************** **********
epcs=exc*(a-co)/a
eces=exc/epso
alpha=eces-eces*eces/3.
cc=fc*alpha*b*a
cs=epcs*es
res=cc+cs-t
control=0.0001*b*d*fc
if (abs(res).gt.control) then
al=a2
a2=a

152
resl=res2
res2=res
kll=kll+l
if(kll.gt.100)then
stop
end if
go to 100
end if
gama=l.-(8.*epso-3.*exc)/(12.*epso-4.*exc)
arm=d-gama*a
vmy=cc*arm+epcs*es*aste*(d-co)
phiy=epsy/(d-a)
return
C CONCRETE STRAIN > EPSO
C****************************************************************
200 if(exc.gt.0.004) exc=0.004
xl=epso*a/exc
ccl=fc*xl*2./3*b
gama=3.6*exc*exc-200.*exc*exc*exc-0.0000128
gama=gama/((exc-epso)*(7.2*exc-300*exc*exc-0.0132))-l.
alpha=exc-50.*exc*exc+100.*exc*epso-0.0022
alpha=alpha/(exc-epso)
cc2=alpha*fc*(a-xl)*b
epcs=exc*(d-co)/a
t=fy*aste
cs=epcs*es
cc=ccl+cc2
res=cc+cs-t
control=0.0001*b*d*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
kll=kll+l
if(kll.gt.100)then
stop
endif
go to 100
endif
arml=d-a+2./3.*xl
arm2=d-gama*(a-xl)
vmy=ccl*arml+epcs*es*(d-co)*aste+cc2*ann2
phiy=epsy/(d-a)
return
end
subroutine eley(ec,tinert,cl,vki,vkj,u2,u3,u5,u6,fo3,f06)
implicit double precision(a-h,o-z)
ei=ec*tinert
w=cl/(3.*ei)+1./vki

153
y=cl/(3.*ei)+l./vkj
z=-cl/(6.*ei)
det=w*y-z*z
a=y/det
b=-z/det
c=b
d=w/det
fo3=(a+b)/cl*u2+a*u3-(a+b)/cl*u5+b*u6
fo6=(c+d)/cl*u2+c*u3-(c+d)/cl*u5+d*u6
return
end
subroutine equcon (x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,
* fy^tot^ql^vah^vahk^s^cir^beta^vm^cvload^l,
* vmu)
implicit double precision (a-h,o-z)
dimension lm(6,n),cosl(n),cos2(n),vmu(n)
dimension cl(n),x(ntot),vah(iqh),r(iqh)
dimension vahk(iqh),beta(n),cvmu(n),cvload(iqh)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
do 150 k = l,iqh
vahk(k)=0.0
150 continue
C****************************************************************
C GLOBAL DISPLACEMENTS PER ELEMENT
C****************************************************************
n3=n+n+n
do 100 kel = l,n
do 123 ipo=l,6
u(ipo)=0.0
123 continue
sigma2=0.0
do 200 i = 1,6
m=lm(i,kel)
if (m.eq.0) go to 200
if(cvload(m).gt.sigma2)sigma2=cvload(m)
1 = n3+m
u(i) = x(l)
200 continue
sigmal=cvmu(kel)
cl=cosl(kel)
c2=cos2(kel)
d2=-c2*u(l)+cl*u(2)
d3=U(3)
d5=-c2 *u(4)+cl*u(5)
d6=u(6)
C*****************************************************************
C ELEMENT FORCES
C****************************************************************

154
c = cl(kel)
45 continue
base = x(3*kel-2)
height = x(3*kel-l)
aste = x(3*kel)
area = base height
tinert = area*height*height/12.0
al = ec*tinert/(c*c*c)
fo3 = al*(6.0*c*(d2-d5) + 2.0*c*c*(2.0*d3+d6))
fo6 = al*(6.0*c*(d2-d5) + 2.0*c*c*(d3+2.0*d6))
C****************************************************************
C ULTIMATE AND YIELD MOMENTS
C****************************************************************
call mumy(base,height,aste,vn,co,epsy,ec,fc,fy,
* vksi(kel),vksj(kel),c,fo3,fo6,es,ecm,betak,
* sigmal,sigma2,tinert,kl,vmuk,vmy,ijflag)
if(vksi(kel).It.10e20.or.vksj(kel).It.l0e20)then
call eley(ec,tinert,c,vksi(kel),vksj(kel),
* d2,d3,d5,d6,fo3,fo6)
call mumy(base,height,aste,vn,co,epsy,ec,fc,fy,
* vksi(kel),vksj(kel),c,fo3,fo6,es,ecm,betak,
* sigmal,sigma2,tinert,kl,vmuk,vmy,ijflag)
endif
beta(kel)=betak
vmu(kel)=vmuk
C****************************************************************
C GLOBAL MODIFIED STIFFNESS
Q******************************************************* *********
call modsti(area,ec,vksi(kel),vksj(kel),c,tinert,cl,c2)
do 300 1 = 1,6
j = lm (l,kel)
if (j.eq.O) go to 300
do 400 11 = 1,6
m = lm (11,kel)
if (m.eq.0) go to 400
jj = n3+m
vahk(j)=vahk(j)+ck(l,ll)*x(jj)
400 continue
300 continue
100 continue
Q**********************************************************
C SUBTRACTION OF EXTERNAL GLOBAL FORCES
C***********************************************************
rmax=0.01
do 510 ilj=l,iqh
if(abs(r(ilj)).gt.rmax)rmax=abs(r(ilj))
510 continue
do 500 kpj = l,iqh
if(abs(r(kpj)).It.0.0001)then
vah(kpj)=vahk(kpj)/rmax
go to 500
endif
vah(kpj) = (vahk(kpj) r(kpj))/rmax
500 continue
return
end

155
subroutine hoojee(tvah,vlag,r,vah,vag,x,cl,cosl,cos2,
* lm,d,clah,clag,ch,eg,alp,xol,vahk,tol1,vn,co,
* epsy,fy,ast,beta,cvmu,cvload,xl1,xl2,romin,vmu,rph,
* grad,vahold,vjac,vinv,bfal)
implicit double precision (a-h,o-z)
dimension x(ntot),cl(n),cosl(n),cos2(n),lm(6,n),vahk(iqh),
* d(iqg),clah(iqh),clag(iqgn),r(iqh),vah(iqh),vag(iqgn),
* ch(iqh),cg(iqgn),alp(ntot),xol(ntot),beta(n),
* cvmu(n),cvload(iqh),vmu(n),grad(ntot),vahold(iqh),
* vjac(iqh,iqh),vinv(iqh),bfal(iqh,iqh)
common /parr/ decfc^cinc/cv^lpl^c^p^c^s^ecm/relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
C****************************************************************
C INITIALIZE LAGRANGIAN FUNCTION
C****************************************************************
k=0
kkl=0
n3 = n+n+n
romax=0.85*0.85*fc*87000./(fy*(87000.+fy))
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
vlago = vlag
C*************************** ****************** *******************
C LOOP ON VARIABLE # K FOR THE SPECIFIED # OF ITERATIONS
C************************************************ ****************
do 200 klj = 1,niter
do 450 k = l,ntot
450
alp(k) = alpl
do 150 k = l,ntot
150
xol(k) = x(k)
do 100 k = l,n3
xopt = x(k)
600
continue
kkl=kkl+l
C***************************************************************
C VALUE OF LAGRANGIAN FUNCTION FOR INITIAL VALUES
C****************************************************************
500 continue
x(k)= xopt+alp(k)*xopt
if (k.le.n3) call lim(x,n,ntot,xll,
* xl2,k,romin,romax)
C* ************************************************************** *
c
NEW VALUE OF LAG. FUNCTION
Q****************************************************************
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
if (vlago.gt.vlag) then
q******************************************************* ********

156
C VARIABLE INCREASE
C***************************************************************
alp(k)=alp(k)*fcinc
vlago=vlag
xopt=x(k)
go to 500
else
C***************************************************************
C DIRECTION REVERSED
C***************************************************************
alp(k)=-alp(k)
x (k) =xopt+alp(k)*xopt
end if
if (k.le.n3) call lim(x,n,ntot,xll,
* xl2,k,romin,romax)
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lin,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
c***************************************************************
C VARIABLE INCREASE
C***************************************************************
if (vlago.gt.vlag) then
alp(k)=alp(k)*fcinc
vlago=vlag
xopt=x(k)
go to 500
else
c***************************************************************
C VARIABLE DECREASE
C***************************************************************
alp(k)=alp(k)*decfc
x(k)=xopt
endif
if(kkl.lt.200)go to 600
100 continue
C************************************************************
C PATTERN MOVE
C***********************************************************
do 250 kp = l,ntot
x(kp) = 1.01*x(kp) 0.01*xol(kp)
if (kp.le.n3) call lim(x,n,ntot,xll,
* xl2/kp,romin/roinax)
250 continue
call lagfun (vlag,tvah,r,x,cl/cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,kp,vitiu,rph)
if (vlago.gt.vlag) then
vlago = vlag
else
do 300 kp = l,ntot
x(kp) = (x(kp)+0.01*xol(kp))/1.01
300 continue
endif
c***********************************************************
C SOLUTION OF EQUALITIES
C***********************************************************

157
call jacequ(x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,
* fy,ntot, iqh,vah,r,vahk,es,ecm,beta,cvmu,cvload,k,
* vmu,vjac)
call sol(iqh,vjac,r,vinv)
do 5890 jgo=l,iqh
x(jgo+n3)=vinv(jgo)
5890 continue
200 continue
C*********************************************************
C REINITIALIZE VALUES
C********************************************************
k=0
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vo f,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
return
end
subroutine lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,cg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,kl,vmu,rph)
implicit double precision (a-h,o-z)
dimension x(ntot),cl(n),cosl(n),cos2(n),lm(6,n),d(iqg),
* clah(iqh), clag(iqgn), vag(iqgn), r(iqh), vah(iqh),
* ch(iqh), cg(iqgn), vahk(iqh), beta(n),
* cvmu(n), cvload(iqh),vmu(n)
common /parr/ decfc,fcinc,cv,alpl,ec,rp,fc,es,ecm,relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
tvah =0.0
tvag = 0.0
rp2 = 2.0*rp
C**************************************************************
C EQUALITY CONSTRAINTS
C**************************************************************
call equeon (x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,
* fy,ntot,iqh,vah,r,vahk,es,ecm,beta,cvmu,cvload,kl,vmu)
do 100 k=l,iqh
vc=vah(k)*ch(k)
tvah=tvah+clah(k)*vc+rph*vc*vc
100 continue
C***************************************************************
C DISPLACEMENT CONSTRAINTS
C**************************************************************
call inecon (iqg,n,ntot,vag,x,d,relind,beta)
do 200 k=l,iqgn
v=vag(k)*cg(k)
z=-clag(k)/rp2
psi=max(v,z)

158
tvag=tvag+clag(k)*psi+psi*psi*rp
200 continue
c***************************************************************
C VALUE OF LAGRANGIAN FUNCTION
C***************************************************************
call valobf (n,ntot,vof,x,cl)
avof = cv*vof
vlag = avof+tvah+tvag
return
end
subroutine modsti(area,ec,vki,vkj,cl,tinert,cosl,cos2)
implicit double precision (a-h,o-z)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
C* ************************************************************* *
C FLEXIBILITY MATRIX (2x2)
C********************************************************* ******
do 10 i=l,6
do 20 j=l,6
ck(i,j)=0.0
20 continue
10 continue
x=cl/(3*ec*tinert)+l./vki
y=cl/(3*ec*tinert)+l./vkj
z=-cl/(6*ec*tinert)
C***************************************************************
C INVERSION OF MATRIX
C******* ********************* ******************** ******* * * *
det=x*y-z*z
a=y/det
b=-z/det
c=b
d=x/det
C**************************************************************
C EXPANDED MATRIX (6x6)
Q************************************************************ *
ckll=ec*area/cl
ckl4=-ckll
ck41=-ckll
ck44=ckll
ck22=(a+b+c+d)/(cl*cl)
ck25=-ck22
ck52=ck25
Ck55=ck22
ck23=(a+c)/cl
Ck53=-Ck23
ck26=(b+d)/cl
Ck56=-Ck26
ck33=a

159
ck36=b
ck63=C
ck66=d
ck32=(a+b)/cl
ck35=-ck32
ck62=(c+d)/cl
ck65=-ck62
C***************************************************************
C ROTATED MATRIX
C***************************************************************
C2=COSl*COSl
s2=cos2*cos2
cs=cosl*cos2
ck(l,1)=ckll*c2+ck22*s2
ck(l,2)=ckll*cs-ck22*cs
ck(1,3)=-ck2 3 *cos2
ck(l,4)=-ckll*c2-ck22*s2
ck(l,5)=-ckll*cs+ck22*cs
ck(l,6)=-ck26*cos2
ck(2,1)=ckll*cs-ck22*cs
ck(2,2)=ckll*s2+ck22*c2
ck(2,3)=ck23*cosl
ck(2,4)=-ckll*cs+ck22*cs
ck(2,5)=-ckll*s2-ck22*c2
ck(2,6)=ck26*cosl
ck(3,1)=-ck32*cos2
ck(3,2)=ck32*cosl
ck(3,3)=ck33
ck(3,4)=-ck(3,1)
ck(3,5)=-ck(3,2)
ck(3,6)=ck36
ck(4,l)=-ck(l,1)
ck(4,2)=-ckll*cs+ck22*cs
ck(4,3)=ck23*cos2
ck(4,4)=ckll*c2+ck22*s2
ck(4,5)=ck(2,1)
ck(4,6)=ck26*cos2
ck(5,l)=ck(2,4)
ck(5,2)=ck(2,5)
ck(5,3)=-ck(2,3)
ck(5,4)=-ck(2,4)
ck(5,5)=-ck(2,5)
ck(5,6)=-ck26*cosl
ck(6,1)=-ck62*cos2
ck(6,2)=ck62 *cosl
ck(6,3)=ck36
ck(6,4)=-ck(6,l)
ck(6,5)=-ck(6,2)
ck(6,6)=ck66
return
end

160
subroutine mumy(b,h,aste,vn,co,epsy,ec,fc,fy,
* vki,vkj, elk,fo3,fo6,es,ecm,betak,sigmal,sigma2,
* tinert,kl,vmuk,vmy,ijflag)
implicit double precision (a-h,o-z)
nodel=0
node2=0
ijflag=0
d=h-co
C*************************************************************
C EVALUATION OF YIELDING MOMENT
C*************************************************************
call comcon(aste,fy,es,d,b,co,epsy,ecm,fc,vmy,phiy)
afo3=abs(fo3)
afo6=abs(fo6)
vki=10e30
vkj=10e30
vm=max(afo3,afo6)
C**********************************************************
C IDENTIFICATION OF HINGE NODE
C**********************************************************
if (afo3.gt.vmy) nodel=l
if (afo6.gt.vmy) node2=l
C*************************************************************
C ULTIMATE MOMENT AND RELIABILITY
C*************************************************************
call valmu(aste,b,betak,co,d,es,epsy,fc,fy,
* phiu,sigmal,sigma2,vm,vmuk,vmy,phiy)
C***************************************************************
C INTEGRATION OF CURVATURE
C***************************************************************
if(nodel.gt.0.or.node2.gt.0)then
if((fo3*fo6).gt.0) then
if(afo3.gt.afo6)then
zero=0.000001*afo3
if(abs(afo3-afo6).It.zero)then
vlp=clk
go to 145
endif
vlp=(afo3-vmy)/(afo3-afo6)*clk
endif
if(afo3.It.afo6)then
zero=0.000001*af06
if(abs(afo3-afo6).It.zero)then
vlp=clk
go to 145
endif
vlp=(afo6-vmy)/(afo6-afo3)*clk
endif
145 continue
tetay = phiy*clk*0.5
tetau = (phiu-phiy)*vlp*0.5+phiy*clk
endif
if((fo3*fo6).It.0) then
vlp=(vmuk-vmy)*clk/vmuk
tetay=clk*phiy*0.25
tetau=phiy*clk*0.25+(phiu-phiy)*vlp

161
endif
vksp=(vmuk-vmy)/(tetau-tetay)
C****************************************************************
C SPRING VALUES
C****************************************************************
if(nodel.eq.l) then
vki=vksp
vkj=10.0e20*ec
endif
if(node2.eq.1) then
vki=10.0e20*ec
vkj =vksp
endif
if(nodel.eq.l.and.node2.eq.l) then
vki=vksp
vkj =vksp
endif
endif
return
end
subroutine optimi(vlag,r,x,cl,cosl,cos2,lm,d,clah,xol,
* vag,toll,clag,vah,ch,eg,xo,vahk,grad,xu,xl,xl,x2,
* delta,alpha,alp,vaho,vago,vn,co,epsy,fy,ast,beta,theta,
* numec,vmu,cvmu,rv,cvload,become,1c,thesum,thesuml,
* dispsum,ni,nucomb,lct,xll,xl2,romin,rph,grad,vahold,
* vjac,vinv,bfal,epsilo)
implicit double precision (a-h,o-z)
dimension r(iqh),x(ntot),cl(n),cosl(n),cos2(n),d(iqg),
* alp(ntot), clag(iqgn), lm(6,n), vag(iqgn), vah(iqh),
* ch(iqh),xo(ntot),clah(iqh),eg(iqgn),beta(n),
* bfal(iqh,iqh),vahk(iqh),grad(ntot),xu(ntot),xl(ntot),
* xl(ntot),x2(ntot),xol(ntot),vaho(iqh),vago(iqgn),
* ast(n),theta(2*n,numec),vmu(n),cvmu(n),rv(iqh,numec),
* cvload(iqh),become(100),lc(100),thesum(n),thesuml(n),
* dispsum(iqh),ni(20),nucomb(20),let(100),vahold(iqh),
* vjac(iqh,iqh),vinv(iqh)
common /parr/ deefe,fcinc,cv,alpl,ec,rp,fc,es,ecm,relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
iter = 0
c***************************************************************
c initializing for scaling
c***************************************************************
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,d,clah,vag,
* clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,kl,vmu,rph)
c***************************************************************

162
c scaling objective function
c***************************************************************
cv = 5./vof
c***************************************************************
c hooke & jeeves
c***************************************************************
444 call hoojee (tvah,vlag,r,vah,vag,x,cl,cosl,cos2,lm,
* d,clah,clag,ch,eg,alp,xol,vahk,toll,vn,co,epsy,fy,ast,
* beta,cvmu,cvload,xll,xl2,romin,vmu,rph,grad,vahold,
* vjac,vinv,bfal)
iter=iter+l
write (8,1500)iter
1500 format ('end of loop =',i3)
Q***************************************************************
c control of maximum number of iterations
c***************************************************************
if (iter.gt.numcy) go to 99
rp2 = rp+rp
a****************************************************** **********
c updating lagrangian mult, equal.cons,
c****************************************************************
do 100 k = l,iqh
clah(k)=clah(k)+rp2*vah(k)*ch(k)
100 continue
c*************************************************************
c updating lagrangian mult. ineq. cons,
c*************************************************************
do 200 k=l,iqgn
v=vag(k)*cg(k)
z=-clag(k)/rp2
psi=max(v,z)
clag(k)=clag(k)+rp2*psi
200 continue
c*************************************************************
c updating penalty factor
c*************************************************************
rp=ga*rp2
rph=ga*ga*rph
Q*************************************************************
c system reliability evaluation
c*************************************************************
jflag=0
call sysrel(n,numec,iqh,theta,rv,vmu,cvmu,r,
* cvload,jflag)
if(jflag.gt.0)go to 444
return
end
subroutine sol(n,a,b,c)

163
implicit double precision(a-h,o-z)
dimension a(n,n),b(n),c(n)
do 50 ik=l,n
c(ik)=b(ik)
50 continue
do 100 k=l,n
vmax=abs(a(k,k))
krow=k
do 120 kj=k+l,n
if(abs(a(kj, k)).gt.vmax)then
vmax=abs(a(kj,k))
krow=kj
endif
120 continue
if (krow.gt.k)then
do 140 jj=k,n
temp=a(krow, j j)
a(krow,jj)=a(k,jj)
a(k,jj)=temp
140 continue
temp=c(krow)
c(krow)=c(k)
c(k)=temp
akk=a(k,k)
endif
do 200 ik+l,n
w=a(i,k)/akk
do 300 j=k+l,n
a(i / j)=a(i,j)-a(k,j)*w
300 continue
c(i)=c(i)-c(k) *w
200 continue
100 continue
do 400 k=l,n
i=n-k+l
do 600 j=i+i,n
c(i) =c(i) -a(i, j) *c(j)
600 continue
c(i)=c(i)/a(i,i)
500 continue
400 continue
return
end
subroutine valmu(aste,b,betak,co,d,es,epsy,fc,fy,
* phiu,sigmal,sigma2,vm,vmul,vmy,phiy)
implicit double precision (a-h,o-z)
c**************************************************************
C NEUTRAL AXIS

164
c*************************************************************
x=47./60.*b*fc
y=0.004*es*aste-aste*fy
z=-0.004*es*co*aste
if((y*y).It.(4.*x*z))then
y=sqrt(4.*x*z)
end if
vkd=(-y+sqrt(y*y-4.*x*z))/(2.*x)
epcs=0.004*(vkd-co)/vkd
if(epcs.ge.epsy)then
epcs=epsy
endif
c***************************************************************
C CONCRETE FORCE IN REGION AB
c***************************************************************
alphal=2./3.
ccab=alphal*b*0.5*vkd*fc
c* ************************************************************ *
c CONCRETE FORCE IN REGION BC
c****************************************************** *******
alpha2=0.9
ccbc=alpha2*b*0.5*vkd*fc
(^* ************************************************************* *
C DISTANCE OF CENTROID TO TOP IN AB
C**************************************************************
gama1=0.875*vkd
c******** *************************************** ****************
C DISTANCE OF CENTROID TO TOP IN BC
C***************************************************** *********
gama2=0.259255*vkd
c**********************************************************
C COEFFICIENTS FOR FAILURE FUNCTION
c************************************************************
al=(ccab*(dd-gamal)+ccbc*(dd-gama2))/fc
a2=-l.
c***********************************************************
c COSINE DIRECTORS
c******************************************** **************
tetal=al*sigmal*fc
teta2=a2*sigma2*vm
c************************************************************
C INDEPENDENT TERM
c***********************************************************
fps=0.004*es*(vkd-co)/vkd
bi=aste*fps*(dd-co)
c***********************************************************
C RELIABILITY INDEX
Q***********************************************************
beta(kel)=(al*fc+a2*vm+bi)/sqrt(tetal*tetal+teta2*teta2)
c************************************************************
C ULTIMATE MOMENT AND ROTATION
c************************************************************
vmu(kel)=al*fc+bi
phiu=0.004/vkd
if((4.*phiy).It.phiu)then
vmu(kel)=(vmu(kel)-vmy)/(phiu-phiy)*3.*phiy+vmy

165
phiu=4.*phiy
end if
return
end
subroutine assemb (e,iqh,n,ntot,x,cl,cosl,cos2,
* lm,glk)
implicit double precision (a-h,o-z)
dimension x(ntot),cl(n),cosl(n),cos2(n)
dimension lm(6,n),glk(iqh,iqh)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
do 100 k = l,iqh
do 150 j = l,iqh
150 glk (j ,k) =0.0
100 continue
c**********************************************************
c global stiffness evaluation
c**********************************************************
do 700 j = l,n
call glosti (e,j,n,ntot,cl,x,cosl,cos2)
do 300 1 = 1,6
k = lm (l,j)
if (k.eq.0) go to 300
do 200 11 = 1,6
m=lm(ll,j) -k+1
if (m.le.0) go to 200
glk(k,m)=ck(l,ll)+glk(k,m)
200 continue
300 continue
700 continue
return
end
subroutine constr (iqg,d)
implicit double precision (a-h,o-z)
dimension d(iqg)
c**********************************************************
c displacement constraints
c**********************************************************
do 201 k = l,iqg
read ( 9,* ) d(k)

201 continue
return
end
166
subroutine datini (clah,clag,ch,eg)
implicit double precision ( a-h,o-z )
dimension clah(iqh), clag(iqg), ch(iqh), cg(iqg)
common /parr/ deefe, fcinc,cv,alpl,ec,rp, fc/es,ecm,
* relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
c************************************************************
c lag. mult, of equality c.
c****************************************** ******************
do 100 k = l,iqh
100 clah(k) = 0.0
Q***********************************************************
c lag. mult, of inequality c.
c* ********************************************************* *
do 200 k = l,iqgn
clag(k) =0.0
200 continue
c***********************************************************
c scaling factors of equ. c.
c************************************************* **********
do 300 k = l,iqh
300 ch(k) = 1.0
c***********************************************************
c scaling factors of ineq. c.
q** ******************************************************** *
do 400 k = l,iqgn
400 eg(k) = 1.0
c*************************************************** *******
c scaling factor of objective fun.
c**********************************************************
cv = 1.0
return
end
subroutine glosti (e, j, n, ntot, cl, x, cosl, cos2)
implicit double precision ( a-h,o-z )
dimension cl(n), x(ntot), cosl(n), cos2(n)

167
common /esq/ u(6),ck(6,6);vksi(100),vksj(100)
cl = cosl(j)
c2 = cos2(j)
Cl2 = Cl Cl
C22 = c2 C2
area = x(j *31)*x(3*j-2)
tinert = area*x(3*j-l)*x(3*j-l)/l2.
ell = cl(j)
Cl2 = cll*cll
cl3 = cl2*cll
a = e*tinert/cl3
b = area*cl2/tinert
gl = a*(b*cl2+l2.*c22)
g2 = a*cl*c2*(b-12.)
g3 = a*(b*c22+12.*cl2)
g4 = -a*6.*cll*c2
g5 = a*6.*cll*cl
g7 = a*2
!.*C12
ge = g7
+
g7
ck(l,1)
=
gl
ck(2,1)
=
g2
ck(3,1)
=
g4
ck(4,1)
=
- gl
ck(5,1)
=
- g2
ck(6,1)
=
g4
ck(l,2)
=
g2
ck(2,2)
=
g3
ck(3,2)
=
gs
ck(4,2)
=
- g2
ck(5,2)
=
- g3
ck(6,2)
=
gs
ck(1,3)
=
g4
ck(2,3)
=
gs
Ck(3,3)
=
ge
Ck(4,3)
=
- g4
ck(5,3)
=
- g5
ck(6,3)
=
g7
ck(l,4)
=
- gi
Ck(2,4)
=
- g2
Ck(3,4)
=
- g4
Ck(4,4)
=
gi
Ck(5,4)
=
g2
Ck(6,4)
sr
- g4
ck(1,5)
=
- g2
ck(2,5)
=
- g3
Ck(3,5)
=
- g5
ck(4,5)
=
g2
ck(5,5)
=
gs
ck(6,5)
=s
- gs
ck(l,6)
=
g4
ck(2,6)
=
gs
ck(3,6)
=
g7
ck(4,6)
=
- g4
ck(5,6)
=
- g5
Ck(6,6)
=
ge
return

168
end
subroutine inecon (iqg,n,ntot,vag,x,d,relind,beta)
implicit double precision (a-h,o-z)
dimension vag(iqg+n),x(ntot),d(iqg),beta(n)
nel3=3*n
c* ******************************************************** *
c displacements
c**********************************************************
do 100 k l,iqg
j = nel3 + k
vag(k) = abs(x(j)) / d(k) 1.
100 continue
c************************************************* *********
c reliability
c**********************************************************
iqgpl=iqg+l
iqgn=iqg+n
do 200 k=iqgpl,iqgn
vag(k) = relind/beta(k-iqg)-1.
200 continue
return
end
subroutine inputd (cl,cosl,cos2,iqh,jdir,jm,lm,n,nj,
* nol,no2,r,xc,yc)
implicit double precision (a-h,o-z)
dimension nol(n),no2(n),jdir(3),xc(nj),yc(nj),
* jm(6,nj),lm(6,n),cl(n),cosl(n),cos2(n),r(iqh)
c**********************************************************
c filling lm matrix
c**********************************************************
do 600 i = l,n
j = nol(i)
k = no2(i)
do 600 1 = 1,3
lm(l,i) = jm (1, j)
lm(l+3,i) = jm(l,k)
600 continue
c**********************************************************
c geometric characteristics
Q* ***************** ************************************** *
do 800 ii = l,n

169
j = nol(ii)
k = no2(ii)
ell = xc(k) xc(j)
el2 = yc(k) yc(j)
cl(ii) = sqrt(ell*ell+el2*el2)
cosl(ii) = ell/cl(ii)
cos2(ii) = el2/cl(ii)
800 continue
c**********************************************************
c initialization of global forces
c**********************************************************
do 850 k=l,iqh
r(k) = 0.0
850 continue
c**********************************************************
c global forces
c**********************************************************
900 read (9 ,*) jnum,jdire, force
if(j num.ne.0)then
k=jm(jdire,jnum)
r(k)=force
go to 900
endif
return
end
subroutine 1im(x,n,ntot,xl1,xl2,k,r omin,romax)
implicit double precision (a-h,o-z)
dimension x(ntot)
n3=3*n
do 10 i=l,n3,3
ll=i
12=i+l
13=i+2
astma=romax*x(ll)*x(l2)
astm=romin*x(ll)*x(12)
if(k.eq.ll)then
if(x(k).It.xll)x(k)=xll
basemin=x(13)/(romax*x(12))
if(x(k).It.basemin)x(k)=basemin
basemax=x(13)/(romin*x(12))
if(x(k).gt.basemax)x(k)=basemax
i=n3
go to 10
endif
if(k.eq.12)then
if(x(k).It.xl2) x(k)=xl2
heightmin=x(l3)/(romax*x(ll))
if(x(k).lt.heightmin)x(k)=heightmin

170
heightmax=x(13)/(romin*x(ll))
if(x(k).gt.heightmax)x(k)=heightmax
i=n3
go to 10
endif
if(k.eq.13)then
if(x(k).gt.astma) x(k)=astma
if(x(k).lt.astm) x(k)=astm
i=n3
endif
10 continue
return
end
subroutine parame (toll,x,n21,glk,mband,cl,
* cos,cos2,lm,r,delta,alpha,numec,rph)
implicit double precision (a-h,o-z)
dimension x(ntot),glk(iqh,iqh),cl(n),cosl(n),
* cos2(n), lm(6,n), r(iqh)
common /parr/ decfc,fcinc,cv,alpl,ec,rp,fc,es,ecm,
* relind
common /pari/ iter,numcy,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
c***********************************************************
c penalty factor
c******************* ****************************************
read (9,*) rp,rph,alpl
Q* ********************************************************* *
c gamma,# of iterations, # of cycles
c***********************************************************
read (9,*) ga,niter,numcy
c************************************************** *********
c decrease and increase factors
c***********************************************************
read (9,*) decfc,fcinc
c* ********************************************************* *
c control tollerance
Q***********************************************************
read ( 9,* ) toll
c************************************************* **********
c initial guesses of dimensions
c********************************************************** *
n3=n+n+n
n21=n3+l
read (9,*) (x(i),x(i+l),i=l,n3,3)
c***************************************************** ******
c increment for slope evaluation
c***********************************************************
read(9,*)delta,alpha
c***********************************************************

171
c element reliability
c************************************************** *********
read(9,*)relind
Q************************************************************
c number of elementary mechanisms
c***************************************************** *******
read(9,*)numec
Q************************************************************
c generation of global stiffness
Q************************************************************
call assemb(ec,iqh,n,ntot,x,cl,cosl,cos2,lm,glk)
c*************************************************************
c initial displacements
c*************************************************************
call symsol(glk,r,x(n21),iqh,mband)
return
end
subroutine valobf (n,ntot,vof,x,cl)
implicit double precision (a-h,o-z)
dimension x(ntot), cl(n)
vof = 0.0
do 100 k = l,n
base = x(3*k-2)
height = x(3*k-l)
steel = x(3*k)
area = base height
vof=vof+(area+steel*10)*cl(k)
100 continue
return
end
subroutine sysrel(nel,numec,ndof,theta,r,vmu,cvmu,p,
* cvload,jflag)
implicit double precision (a-h,o-z)
dimension theta(200,100),p(100),rb(200,100),
* become(500),lc(300),thesum(100),temp(100,100),
* locmec(lOO),thesul(100),dispsu(100),ni(100),
* nucomb(lOO),let(300),elerel(100),cvmu(100),
* r(100),vmu(100),cvload(100)
c***************************************************************
c form fundamental mechanisms

172
C***************************************************************
ndof=iqh
do 20 j=l,ndof
P(j)=r(j)
20 continue
C** ****************** ******c***** ********************* ************
c ordering theta and r matrices
c***************************************************************
do 710 k=l,numec-l
jflag=0
do 720 i=l,ndof
if(abs(rb(i, k)).gt.0.0)j flag=i
720 continue
if (jflag.eq.0)then
do 730 l=k+l,numec
do 740 li=l,ndof
if(abs(rb(li,l)).gt.0.0)then
do 750 lj=l,ndof
temp(lj,1)=rb(lj,1)
rb(lj ,l)=rb(lj, jflag)
rb (1 j j flag) =temp (1 j 1)
750 continue
do 760 lj=l,2*nel
temp(1j,1)=theta(1j,1)
theta(lj,l)=theta(lj,jflag)
theta(1j,j flag)=temp(1j,1)
760 continue
go to 733
endif
740 continue
733 continue
730 continue
endif
710 continue
c***************************************************************
c normalizing theta and r vectors
c**************************************************************
do 810 i=l,numec
do 820 j=l,2*nel
if(abs(theta(j,i)).ne.1.O.and.
* theta(j,i).ne.0.0)then
fact=abs(1./theta(j,i))
do 830 jj=l,2*nel
theta(jj,i)=theta(jj,i)*fact
830 continue
do 840 jj=l,ndof
rb(jj ,i)=rb(j j ,i) *fact
840 continue
go to 734
endif
820 continue
734 continue
810 continue
c**************************************************************
c transpose theta and r matrices
c**************************************************************

173
do 25 j=l,numec
do 91 i=nel
temp(i,j)=theta(i,j)
91 continue
25 continue
do 56 i=l,numec
do 55 j=l,2*nel
theta(i,j)=temp(j,i)
55 continue
56 continue
do 28 j=l,numec
do 27 i=l,ndof
temp(i,j)=rb(i,j)
27 continue
28 continue
do 66 i=l,numec
do 65 j=l,ndof
rb(i,j)=temp(j,i)
65 continue
66 continue
c**************************************************************
c reliability of fundamental mechanisms
c**************************************************************
do 102 i=l,numec
vmeanr=0.0
stdevr=0.0
*
202
*
302
do 202 k=l,nel
j=2*k-l
theji=abs(theta(i,j))
thejil=abs(theta(i,j+1))
if(theji.It.0.0001.and.thejil.It.
0.0001)goto 202
term2=(theji+thejil)*cvmu(k)*vmu(k)
stdevr=stdevr+term2 *term2
vmeanr=term2/cvmu(k)+vmeanr
continue
vmeanl=0.0
stdevl=0.0
do 302 k=l,ndof
if(abs(p(k)),lt.0.001)go to 302
term=p(k)*rb(i,k)
vmeanl=vmeanl+term
stdevl=stdevl+term*term*(cvload(k)
*vload(k))
continue
vmean=vmeanr-vmeanl
stdev=stdevr+stdevl
become(i)=vmean/sqrt(stdev)
lnumbe=lnumbe+l
102 continue
c**************************************************************
c reliability of combined mechanisms
c (internal work)
c**************************************************************
do 50 l=l,numec
lc(l)=l

174
lct(l)=l
50 continue
lp=numec
ltemp=numec
ni(1)=1
numax=6
nucome=l
nucomb(1)=numec
lpt=numec
lpti=numec+l
111 continue
c***************************************************************
c loop over mechanisms in location vector
do 699 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
699 continue
do 698 klp=l,ndof
dispsu(kip)=0.0
698 continue
kmu=l
nic=ni(nucome)
nif=nic+nucomb(nucome)*nucome-l
nucomb(nucome+1)=0
c* ************************************************************ *
c define acceptable interval
c**************************************************************
if(nucome.gt.1)then
nifbet=lpti-l
nicbet=lpti-nucomb(nucome)
do 5544 ia=nicbet+l,nifbet
betaal=become(nicbet)+epsilo
if(become(ia).gt.betaal)then
do 5545 ib=ia,nifbet
become(ib)=1000.
5545
continue
go to 5541
endif
5544
continue
5541
continue
niccon=nicbet
endif
c**************************************************************
c reliability of combined mechanisms
c (internal work)
c******************************************************** ******
do 200 j=nic,nif,nucome
icontr=0
if(nucome.gt.1)then
if(become(niccon).gt.100.)icontr=l
niccon=niccon+l
if (icontr.eg.l)go to 5564
endif
do 300 1=1,nucome
lj=lc(j+l-l)

175
if(lj.It.-0.1)then
kmu=-l
lj=abs(lj)
endif
do 400 kk=l,nel
kkj=2*kk-l
thesum(kk)=thesum(kk)+theta(lj ,kkj) *kmu
thesul(kk)=thesul(kk)+theta(lj,kkj+l)*kmu
400 continue
kmu=l
300 continue
c**************************************************************
c reliability of combined mechanisms
c (external work)
c**************************************************************
do 372 l=l,nucome
lcl=lc(j+l-l)
if(lcl.It.-0.1)then
kmu=-l
lcl=abs(lcl)
endif
do 472 kk=l,ndof
if(abs(p(kk)).lt.0.001)go to 472
dispsu(kk)=dispsu(kk)+rb(lcl,kk)*kmu
472 continue
kmu=l
372 continue
5564 continue
c*************************************************************
c combination with fundamental mechanisms
c (internal work)
c*************************************************************
do 100 k=l,numec
vmeanr=0.0
vmeanl=0.0
stdevr=0.0
stdevl=0.0
vmanrm=0.0
vmanlm=0.0
stdvrm=0.0
stdvlm=0.0
do 499 lll=j,j+nucome-1
if(abs(lc(lll)).ge.k)go to 100
499 continue
if(nucome.gt.1)then
if (icontr.eq.1)then
becomi=400.
becopl=500.
go to 5574
endif
endif
thesu=0.0
thesu2=0.0
thesui=0.0
theslm=0.0
do 600 kk=l,nel

176
kkj=2*kk-l
thesu=thesum(kk)+theta(k,kkj)
thesu2=thesul(kk)+theta(k,kkj+1)
term=(abs(thesu)+abs(thesu2))*cvmu(kk)
* *vmu(kk)
vmeanr=term/cvmu (kk) +vmeanr
stdevr=stdevr+term*term
thesui=thesum(kk)-theta(k,kkj)
theslm=thesul(kk)-theta(k,kkj+1)
termm= (abs (thesui) +abs (theslm)) *cvmu(kk)
* *vmu(kk)
vmanrm=termm/cvmu(kk)+vmanra
stdvrm=stdvrm+termm*termm
600 continue
c************ *************************************************
c combination with fundamental mechanisms
c (external work)
Q*************************************************************
do 672 kk=l,ndof
if(abs(p(kk)),lt.0.001)go to 672
dispkk=(dispsu(kk)+rb(k,kk))*p(kk)
vmeanl=vmeanl+dispkk
stdevl=stdevl+dispkk*cvload(kk)*dispkk
*cvload(kk)
dispkm=(dispsu(kk)-rb(k,kk))*p(kk)
vmanlm=vmanlm+dispkm
stdvlm=stdvlm+dispkm*cvload(kk)*dispkm
*cvload(kk)
continue
becopl=(vmeanr-vmeanl)/sqrt(stdevr+stdevl)
becomi=(vmanrm-vmanlm)/sqrt(stdvrm+stdvlm)
continue
if(becomi.lt.becopl) then
do 138 lk=l,nucome
ltemp=ltemp+l
lc(ltemp)=lc(j+lk-1)
continue
ltemp=ltemp+l
lc(ltemp)=-k
lctlp=-k
becote=becomi
else
do 139 lk=l,nucome
ltemp=ltemp+l
lc(ltemp)=lc(j+lk-1)
139 continue
lctlp=k
ltemp=ltemp+l
lc(ltemp)=k
becote=becopl
endif
lpt=lpt+l
iflag=0
Q*************************************************************
c ordering the combined beta values in the same row
c*************************************************************
672
5574
138

177
if (lpti.gt.lpt-1)then
do 524 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
524 continue
lp=lp+l
let(lp)=lctlp
become(lpt)=becote
else
do 510 jkj=lpti,lpt-1
if(become(jkj).gt.becote)then
iflag=-l
do 511 kjk=jkj, lpt-1
itemp=lpt-l+jkj-kjk
become(itemp+1)=become(itemp)
511 continue
become(jkj)=becote
becote=become(lpt)
c*************************************************************
c moving lc array
c*************************************************************
movini=nic+nucomb(nucome)*nucome+
* (jkj-lpti)*(nucome+1)
movfin=(lpt-lpti)*(nucome+1)+nic+
* nucomb(nucome)*nucome-l
do 512 lmn=movini,movfin
lcou=movini+movfin-lmn
nucl=nucome+l
let(lcou+nucl)=lct(lcou)
512 continue
do 513 n=movini,movini+nucome-l
lptaa=n-movini+l
let(n)=lc(j+lptaa-1)
513 continue
let(movini+nucome)=lctlp
endif
510 continue
if(iflag.eq.0)then
do 124 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
124 continue
lp=lp+l
let(lp)=lctlp
become(lpt)=becote
else
lp=lp+nucome+l
endif
endif
nucomb(nucome+1)=nucomb(nucome+1)+1
lnumbe=lnumbe+l
100 continue
do 6991 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
6991 continue

178
do 6981 klp=l,ndof
dispsu(kip)=0.0
6981 continue
200 continue
c*************************************************************
c control of maximum number of tree rows
c* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
lpti=lpti+nucomb(nucome+1)
nucome=nucome+1
ni(nucome)=ni(nucome-1)+(nucome-1)*nucomb(nucome-1)
if(nucome.It.(numax))go to 111
c*************************************************************
c find minimum beta value
c*************************************************************
betmin=100
do 7890 i=l,lnumbe
if(become(i).It.betmin)then
betmin=become(i)
itab=i
end if
7890 continue
c*************************************************************
c find mechanisms involved
c*************************************************************
mecoun=0
mcomb=l
do 7891 j=l,nucome-1
do 7895 jk=l,nucomb(j)
mecoun=l+mecoun
if(itab.eq.mecoun)then
nistar=mcomb
do 7893 1=1,j
locmec(l)=lct(nistar+l-l)
7893 continue
nummec=j
go to 7894
endif
mcomb=j +mcomb
7895 continue
7891 continue
7894 continue
c*************************************************************
c find elements involved
c*************************************************************
nk=l
numele=0
do 8000 m=l,nel
do 8001 k=l,numec
do 8002 l=l,nummec
if(abs(locmec(l)).eq.k)then
ml=2*m-l
m2=2*m
if(abs(theta(ml,k)).gt.0.0.
* or.abs(theta(m2,k)).gt.0.0)then
if(nk.gt.1)then
do 8003 lmi=l,nk

179
*
8003
8004
if(invmec(lmi).eq.m)
go to 8004
continue
end if
nk=nk+l
numele=numele+l
invmec(n)=m
go to 8000
continue
end if
endif
8002 continue
8001 continue
8000 continue
c*************************************************************
c control of system reliability
c*************************************************************
write(8,*)
write(8,*)BETA MINIMAL FOR THE SYSTEM = ';betmin
write(8,*)
jflag=0
if(betmin.It.relind)then
delta=(relind-betmin)/relind
do 3891 i=l,nel
do 3892 j=l,numele
if(invmec(j).eq.i)then
elerel(i)=(1.+delta)*elerel(i)
endif
3892 continue
3891 continue
jflag=l
endif
return
end
subroutine mecsys(n,iqh,cl,cost,sint,lm,numec,r,theta)
implicit double precision (a-h,o-z)
dimension a(100,100),b(100,100),c(100),cm(100,100),
* cost(n),qa(100,100),sint(n),cl(n),q(100,100),
* lm(6,n),am(100,100),bl(100,100),theta(200,100),
* r(iqh,100)
c**************************************************************
c Constraint matrix for the structure
c**************************************************************
do 300 i=l,3*n
do 400 j=l,6*n
cm(i,j)=0.0
q(i,j)=0.0
400
continue

180
300 continue .
do 60 k=l,n
i=3*k
j=6*k
at=1.0/cl(k)
im2=i-2
iml=i-l
jml=j-l
jm2=j-2
jm3=j-3
jm4=j-4
jm5=j-5
cm(im2,j m5)=-l.0
cm(im2,jm2)=1.0
cm(iml,jm3)=1.0
cm(i,j)=1.0
cm(iml,jm4)=-at
cm(iml,jml)=at
cm(i,jm4)=-at
cm(i,jml)=at
60 continue
c**************************************************************
c Coordinate trnsformation matrix
c******************************* *******************************
do 70 k=l,n
co=cost(k)
si=sint(k)
j=6*k
do 80 i=l,2
jum=j-3*i+l
jdois=j-3*i+2
jtres=j-3*i+3
q(jum,jum)=co
q(jum,jdois)=si
q(jdois,jum)=-si
q(jdois,jdois)=co
q(j tres,j tres)=1.0
80 continue
70 continue
c**************************************************************
c Compatibilibity matrix from LM matrix
c**************************************************************
n6=6*n
do 500 i=l,n6
do 510 k=l,iqh
am(i,k)=0.0
510 continue
500 continue
do 520 i=l,6
do 530 k=l,n
iflag=lm(i,k)
if (iflag.gt.0) am(6*(k-1)+i,iflag)=1.0
530 continue
520 continue
c**************************************************************
c Rotation of basic compatibility matrix

181
c**************************************************************
mm=80
call multi(q,am,qa,n6, iqh,n6,mm)
c**************************************************************
c Expansion of QA matrix
c**************************************************************
do 585 i=l,n
i6=6*i
i3=i6-3
do 595 j=l,iqh
qa(i3,j)=0.0
qa(i6,j)=0.0
595 continue
i2=2*i
qa(i3,iqh+i2-l)=1.0
qa(i6,iqh+i2)=1.0
585 continue
c**************************************************************
c Matrix A = C QA (transformed)
c**************************************************************
m=3*n
nt=iqh+2*n
call multi(cm,qa,a,m,nt,n6,mm)
c**************************************************************
c Solution for virtual displacements
c**************************************************************
do 150 k=l,nt
do 175 1=1,nt
b(k,l)=0.0
175 continue
150 continue
do 160 k=l,nt
b(k,k)=1.0
160 continue
do 200 i=l,m
amax=0.0
iflag=0
do 250 j=i,nt
if (abs(a(i,j)).gt.amax)then
iflag=j
amax=abs(a(i,j))
endif
250 continue
do 305 k=l,m
c(k)=a(k,iflag)
a (k, iflag)=a(k,i)
a (k, i) =c (k)
305 continue
do 310 k=l,nt
c(k)=b(k,iflag)
b(k,iflag)=b(k,i)
b(k,i)=c(k)
continue
do 280 j=i+l,nt
if (abs(a(i,j)).gt.0.00001) then
fact=-a(i,j)/a(i,i)
310

182
do 290 kk=l,m
a(kk,j)=a(kk,j)+a(kk,i)*fact
290 continue
do 291 kk=l,nt
b(kk,j)=b(kk,j)+b(kk,i)*fact
291 continue
endif
280 continue
200 continue
numec=nt-3*n
c***************************************************** **********
c Forming bl
c***************************************************************
lcount=nt-numec+l
ki=l
do 800 i=lcount,nt
do 810 j=l,nt
bl(j ,ki)=b(j ,i)
810 continue
ki=ki+l
800 continue
q**************************************************************
c Creating Theta matrix
c*************************************************************
do 156 j=l,numec
do 157 i=l,2*n
k=iqh+i
theta(i,j)=bl(k,j)
157 continue
156 continue
c******** ********************************************** *********
c Creating virtual displacements
c****************************************************** ********
do 169 j=l,numec
do 158 i=l,iqh
r(i,j)=bl(i,j)
158 continue
c************************************************************
c Adding joint mechanisms
c******************************************************* *******
do 161 i=3,iqh/3
if (abs(r(i,j)).gt.0.000001)then
r(i,j)=0.0
do 162 k=l,n
if(lm(3,k).eq.i)then
lpo=2*k-l
theta(lpo,j)=l
endif
if(lm(6,k).eq.i)then
lpo=2*k
theta(lpo,j)=1
endif
162 continue
endif
161 continue
169 continue

return
end
183
subroutine multi(aa,bb,cc,1,m,n,k)
implicit double precision (a-h,o-z)
dimension aa(l,n),bb(n,m),cc(l,m)
do 10 i=l,1
do 20 j=l,m
d=0.0
do 30 kk=l,n
d=d+aa(i,kk)*bb(kk,j)
30
continue
cc(i/j)d
20
continue
10
continue
return
end
subroutine jacequ (x,n/cl,lm,cosl,cos2/fc,ec/vn/co,epsy/
* fy,ntot,iqh,vah,r,vahk,es,ecm,beta,cvmu,cvload,kl,
* vmu,vjac)
implicit double precision (a-h,o-z)
dimension lm(6,n),cosl(n),cos2(n),vmu(n),vjac(iqh,iqh)
dimension cl(n),x(ntot),vah(iqh),r(iqh)
dimension vahk(iqh),beta(n),cvmu(n),cvload(iqh)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
do 150 kme = l,iqh
do 160 kmo=l,iqh
vjac(kmo,kme)=0.0
160 continue
150 continue
do 100 k=l,n
cl=cosl(k)
c2=cos2(k)
c = cl(k)
base = x(3*k-2)
height = x(3*k-l)
aste = x(3*k)
area = base height
tinert = area*height*height/12.0
C**************************************************************
C GLOBAL MODIFIED STIFFNESS
C**************************************************************
call equcon (x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,fy,

184
ntot,iqh,vah,r,vahk, es,ecm,beta,cvmu,cvload,kl,vmu)
call modsti(area,ec,vksi(k),vksj(k),c,tinert,cl,c2)
do 300 1=1,6
j=lm(l,k)
if (j.eq.O) go to 300
do 400 11 = 1,6
m=lm(ll,k)
if (m.eq.0) go to 400
vjac(j ,m)=vjac(j ,m)+ck(l, 11)
400
continue
300
continue
100
continue
return
end

185
Example: Debug Frame
Input File: DATA
Debug frame
4,5
1,2
2.3
3.4
4.5
1,2,3,0,0
0,0,0,0,100
0,0,0,50,100
0,0,0,100,100
1,2,3,100,0
2,1,5000
3,2,-5000
0,0,0
3000.40000.1
29000000,0.004
1
1
1
1
1
1
1
1
1
1
1
1
1
5,0.0001,0.01
5.100.1
0.9,1.1
0.00001
3,12
3,12
3,12
3,12
2
4
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

186
in in in in in
H rl rl rl iI ID in
O O O O O CM rH

187
User's Manual
Augmented Lagrangian Formulation
Example: Debug Frame
Input File: DATA
Line 1
Problem title.
Line 2
Number of elements, number of nodes.
Line 3 to line 6
Node i, node j of element 1 through 4.
Line 7 to line 11
Boundary conditions of displacement in the horizontal
direction, vertical direction, in-plane rotation,
horizontal coordinate, vertical coordinate.
Line 12 and line 13
Node where force is applied, direction of load and
magnitude of load.
Line 14
Termination of force information.
Line 15
Flexural strength of concrete, yielding stress of steel
and reinforcement cover.
Line 16
Steel modulus of elasticity and concrete ultimate
strain.
Line 17 to line 20
Initial steel reinforcement areas.

188
Line 21 to 29
Displacement limits.
Line 30
Penalty factor, equality penalty factor and stepsize
Line 31
Factor of penalty increase, maximum number of
iterations and maximum number of cycles.
Line 32
Decrease factor and increase factor.
Line 33
Convergence tollerance.
Line 34
Element and system reliability index.
Line 35
Number of elementary mechanisms.
Line 36 to line 39
Coefficient of variation of concrete strength.
Line 40 to line 48
Coefficient of variation of external loads.
Line 49
Lower bounds of cross section dimensions.
Line 50
Value of interval gap in the Beta unzipping method.

APPENDIX B
GENERALIZED REDUCED GRADIENT EXAMPLE

The example and correspondent optimization conditions
chosen to illustrate the performance of the Generalized
Reduced Gradient method using the integrated formulation are
presented in Figure B.l. The maximum flexural stress,
compression or tension, is 1,000 psi. The problem is solved
in separate steps presented below.
Step A Problem Formulation
Objective Function
Minimize f(x) = IOX1X2
Equality Constraints
hl(x) = 0.03xiX23X3 0.015xiX23X4 +1=0
h2(x) = -0.15XiX23X3 + X1X23 = 0
Inequality Constraints
h3(x) = 60/(X!X22) 1 + x5 = 0
where X5 slack variable;
Variable bounds
xi > 0.5 in
X2 > 0*5 in
I X3 I <0.5 in
I X4 I <0.5 rad
Step B Explicit Derivatives
df/dxi = 10x2 df/dX2 = 10xi df/dX3 = .... =0;
dhi/dxi = X23(0.03x3 0.15x4);
dhi/dx2 = xiX23(0.09x3 0.45x4);
dhi/dX3 = 0.03XiX23;
dhi/dx4 = -0.15xiX23;
dhi/dxs = 0;
190

191
Figure B.l. Integrated optimization example.

192
dh2/dX!
dh2/dx2
dh2/dx3
dh2/dx4
dh2/dx5
dh3/dxi
dh3/dx3
dh3/dx5
x23 (-0.15x3 + x4);
XiX22(-0.45x3 + 3x4);
-0.15x^x23;
xix23;
0;
dh3/dx2 = dh3/dx4 = 0;
-0.45x3;
1;
Step c Initial Design Point and Initial Values
Dependent variables d^t = {xi,x2,x3};
Independent variables dj^ = {X4,xs};
ddfc = {1,10,-0.1333} difc = {-0.02,0.4}
grad = {100,10,0} grad fit = {0,0}
where grad f is gradient of f;
H = [ J I C ]
where H is Hessian matrix of the equalities
J
-l -0.3 30
0 0 -150
-0.6-0.12 0
C
-150 0
1000 0
0 1
Step D Recurrence Formulas
dkt = {ddk I dik};

193
dj_k+1 = grad fk + (J 1C) grad fdk;
ddk+1 = j-lc dj_k+1;
Step E Iterations
First iteration:
xot = {1 10 -0.133 -0.02 0.4};
dlt = {-501667 2505556 33333 5000 -333};
a1 = 106, because x^ > 0.5;
= {0.5 12.506 -0.1 -0.015 0.3997};
Second iteration:
Change of variables X4 replaces x^ that is at a
lower bound;
d2t = {0 -332.09 -7.9664 -1.195 40.75};
a2 = 0.3997/40.75, because o < 1000;
x2t = {0.5 9.249 -0.178 -0.0267 0};
Independent variables, {x^, X5} are at their lower
bounds;
Iteration is performed on the set of dependent
variables {X2, X3, X4};
Third iteration:
-J_1 h(x2)t = {1.3228 -0.082 -0.0124};
X3t = {10.58 -0.26 -0.0391};
d
Fourth iteration:
-J_1 h(x3)t = {0.358 0.061 0.009};
x3t = {10.935 -0.199 -0.0301};
d
Fifth iteration:
-J-1 h(x4)t = {0.019 -0.004 -0.003};

194
x3t = {10.954 -0.203 -0.0304};
d
Stop.
Optimum design
X*t = {0.5 10.954 -0.203 -0.304 0}.

APPENDIX C
GENERALIZED REDUCED GRADIENT SUBROUTINES

program optim
call datain
call grg
call outres
stop
end
subroutine gcomp(g,x)
implicit real*8 (a-h,o-z)
dimension g(l), x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
call equcon(x)
call inecon(x)
call valobf(x,vof)
do 100 i=l,iqh
g(i)=vah(i)
100 continue
c***************************************************************
C ELEMENT RELIABILITY LIMITS
c***************************************************************
do 200 i=iqh+l,iqh+nel
g(i)=-vag(i-iqh)
200 continue
c*********************************************** ****** *********
C REINFORCEMENT LIMITS
c****************************************************** *********
do 300 i=iqh+nel+l,iqh+2*nel
kj=i-iqh-nel
g(i)=1000.*x(3*kj)/(x(3*kj-2)*x(3*kj-l))
300 continue
g(iqh+2*nel+l)=vof
return
end
subroutine princi
implicit double precision (a-h,o-z)
196

197
*
*
common /vgeom/ cl(100),cos(100),cos2(100),lm(6,100),
cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
open ( 9,file='data',form='formatted' )
rewind 9
c****************************************************************
c # ELEMENTS AND # JOINTS
c***************************************************************
read (9,*)nel,nj
c***************************************************************
C NODES PER ELEMENT
C***************************************************************
do 100 i=l,nel
read(9,*)nol(i),no2(i)
100 continue
c***************************************************************
C JM MATRIX
c***************************************************************
do 200 kk=l,nj
jm(l,kk)=l
jm(2,kk)=2
jm(3,kk)=3
200 continue
c***************************************************************
C SUPPORT CONDITIONS AND COORDINATES
c***************************************************************
do 300 j=l,nj
read(9,*)jdir(1),jdir(2),jdir(3),xc(j),yc(j)
do 350 i =1,3
if (jdir(i).gt.0) then
jm (i,j) = 0
endif
350 continue
300 continue
c***************************************************************
c GLOBAL DEGREES OF FREEDOM
C***************************************************************
iqh=0
do 510 j=l,nj
do 500 1=1,3
if (jm(l,j).ne.0) then
iqh=iqh+l
jm (1,j)=iqh
endif
500 continue
510 continue
iqgn=iqh+nel
iqg=iqh
ntot=iqgn+2 *nel
c***************************************************************
C FILLING LM MATRIX
c***************************************************************

198
do 600 i = l,nel
j = nol(i)
k = no2(i)
do 600 1 = 1,3
lm(l,i) = jm(l,j)
lxn(l+3,i) = jm(l,k)
600 continue
c***************************************************************
C GEOMETRIC CHARACTERISTICS
C***************************************************************
do 800 ii = l,nel
j = nol(ii)
k = no2(ii)
ell = xc(k) xc(j)
el2 = yc(k) yc(j)
cl(ii) = sqrt(ell*ell+el2*el2)
cosl(ii) = ell/cl(ii)
cos2(ii) = el2/cl(ii)
800 continue
c*************************************************************
C INITIALIZATION OF GLOBAL FORCES
c*************************************************************
do 850 k=l,iqh
r(k) = 0.0
850 continue
c***********************************
C GLOBAL FORCES
c***********************************
900 read(9,*)jnum,jdire,force
if(j num.ne.0)then
k=jm(jdire,jnum)
r(k)=force
go to 900
endif
c***************************************************************
C REINFORCED CONCRETE
c***************************************************************
read(9,*)fc,fy,co
ec=57000*sqrt(fc)
vn=29e6/ec
epsy=fy/29e6
read(9,*)es,ecm
c***************************************************************
c ELEMENT RELIABILITY
c***************************************************************
read(9,*)relind
c***************************************************************
c COEFFICIENTS OF VARIATION
C***************************************************************
do 156 i=l,nel
read(9,*)cvmu(i)
156 continue
do 157 i=l,iqh
read(9,*)cvload(i)
157 continue
return
**************************
**************************

199
end
subroutine valobf(x,vof)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(100)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
vof = 0.0
do 100 k = l,nel
base = x(3*k-2)
height = x(3*k-l)
steel = x(3*k)
area = base height
vof=vof+(area+steel*10)*cl(k)
100 continue
return
end
subroutine inecon(x)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100) ,lm(6,100) ,
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
nel3=3*nel
c***************************************************************
C RELIABILITY
c***************************************************************
do 200 k=l,iqg
vag(k) = relind-beta(k)
200 continue
return
end

200
subroutine modsti(kel,tinert,area,vksi,vksj)
implicit double precision (a-h,o-z)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir (3)
common /parr/ cv, ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
c***********************************************************
c FLEXIBILITY MATRIX (2x2)
C************************************************* **********
n=nel
do 10 i=l,6
do 20 j=l,6
ck(i,j)=0.0
20 continue
10 continue
xd=cl(kel)/(3*ec*tinert)+l./vki
y=cl(kel)/(3*ec*tinert)+i./vkj
z=-cl(kel)/(6*ec*tinert)
c***************************************************************
C INVERSION OF MATRIX
c***************************************************************
det=xd*y-z*z
a=y/det
b=-z/det
c=b
dd=xd/det
c************************************************ **************
C EXPANDED MATRIX (6x6)
c************************************************** ***********
ckll=ec*area/cl(kel)
ckl4=-ckll
ck41=-ckll
ck44=ckll
ck22=(a+b+c+dd)/(cl(kel)*cl(kel))
ck25=-ck22
ck52=ck25
ck55=ck22
ck23=(a+c)/cl(kel)
ck53=-ck23
ck26=(b+dd)/cl(kel)
ck56=-ck26
ck33=a
ck36=b
ck63=c
ck66=dd
ck32=(a+b)/cl(kel)
ck35=-ck32
ck62=(c+dd)/cl(kel)

201
ck65=-ck62
c***************************************************************
c ROTATED MATRIX
c***************************************************************
csl=cosl(kel)
cs2=cos2(kel)
c2=csl*csl
S2=CS2*CS2
cs=csl*cs2
ck(l,1)=ckll*c2+ck22*s2
ck(l,2)=ckll*cs-ck22*cs
ck(1,3)=-ck2 3 *cs2
ck(l,4)=-ckll*c2-ck22*s2
ck(1,5)=-ckll*cs+ck2 2 *cs
ck(l,6)=-ck26*cs2
ck(2,1)=ckll*cs-ck22*cs
ck(2,2)=ckll*s2+ck22*c2
ck(2,3)=ck23*csl
ck(2,4)=-ckll*cs+ck2 2 *cs
ck(2,5)=-ckll*s2-ck22*c2
ck(2,6)=ck26*csl
ck(3,l)=-ck32*cs2
ck(3,2)=ck32*csl
Ck(3,3)=ck33
ck(3,4)=-ck(3,1)
ck(3,5)=-ck(3,2)
ck(3,6)=ck36
ck(4,l)=-ck(l,1)
ck(4,2)=-ckll*cs+ck22*cs
ck(4,3)=ck23*cs2
ck(4,4)=ckll*c2+ck22*s2
ck(4,5)=ck(2,1)
ck(4,6)=ck2 6*cs2
ck(5,1)=ck(2,4)
ck(5,2)=ck(2,5)
ck(5,3)=-ck(2,3)
ck(5,4)=-ck(2,4)
ck(5,5)=-ck(2,5)
ck(5,6)=-ck26*csl
ck(6,1)=-ck62*cs2
ck(6,2)=ck62*csl
ck(6,3)=ck36
ck(6,4)=-ck(6,l)
ck(6,5)=-ck(6,2)
ck(6,6)=ck66
return
end
subroutine sysrel(jflag)
implicit double precision (a-h,o-z)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),

202
720
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100) ,
* no2(100)
common /pari/ iqh,iqg,nel,ntot,iqgn
common /sysr/ betmin,epsilo/elerel(100),invmec(lOO)
common /equal/ vah(lOO),vmu(100)
dimension theta(200,100),p(100),rb(200,100),
* become(500),lc(300),thesum(100),temp(100,100),
* locmec(lOO),thesul(100),dispsu(100),ni(100),
* nucomb(lOO),lct(300)
c***************************************************************
c form fundamental mechanisms
Q***************************************************************
cal1 mecsys(nel,iqh,cl,cos1,cos2,lm,numec,rb,theta)
ndof=iqh
do 20 j=l,ndof
P(j)=r(j)
20 continue
o***************************************************************
c ordering theta and r matrices
0***************************************************************
do 710 k=l,numec-l
jflag=0
do 720 i=l,ndof
if(abs(rb (i, k)).gt.0.0)j flag=i
continue
if (jflag.eq.O)then
do 730 l=k+l,numec
do 740 li=l,ndof
if(abs(rb(li,l)).gt.0.0)then
do 750 lj=l,ndof
temp(1j,1)=rb(1j,1)
rb(lj ,l)=rb(lj, jflag)
rb(lj,jflag)=temp(lj,1)
continue
do 760 lj=l,2*nel
temp(1j,1)=theta(1j,1)
theta(lj,1)=theta(lj,jflag)
theta(lj,jflag)=temp(lj,1)
continue
go to 733
endif
continue
continue
continue
endif
710 continue
0**************************************************************
c normalizing theta and r vectors
0**************************************************************
do 810 i=l,numec
do 820 j=l,2*nel
if(abs(theta(j,i)).ne.1.0.and.
* theta(j,i).ne.0.0)then
fact=abs(1./theta(j,i))
do 830 jj=l,2*nel
750
760
740
733
730

203
theta(j j,i)=theta(j j,i)* fact
830 continue
do 840 jj=l,ndof
rb(jj,i)=rb(jj,i)*fact
840 continue
go to 734
endif
820 continue
734 continue
810 continue
c**************************************************************
c transpose theta and r matrices
c**************************************************************
do 25 j=l,numec
do 91 i=nel
temp(i,j)=theta(i,j)
91 continue
25 continue
do 56 i=l,numec
do 55 j=l,2*nel
theta(i,j)=temp(j,i)
55 continue
56 continue
do 28 j=l,numec
do 27 i=l,ndof
temp(i, j)=rb(i,j)
27 continue
28 continue
do 66 i=l,numec
do 65 j=l,ndof
rb(i,j)=temp(j,i)
65 continue
66 continue
c**************************************************************
c reliability of fundamental mechanisms
c**************************************************************
do 102 i=l,numec
vmeanr=0.0
stdevr=0.0
do 202 k=l,nel
j=2*k-l
theji=abs(theta(i,j))
thejil=abs(theta(i,j+1))
if(theji.It.0.0001.and.thejil.lt.
* 0.0001)goto 202
term2=(theji+thejil)*cvmu(k)*vmu(k)
stdevr=stdevr+term2*term2
vmeanr=term2/cvmu(k)tvmeanr
202 continue
vmeanl=0.0
stdevl=0.0
do 302 k=l;ndof
if(abs(p(k)),lt.0.001)go to 302
term=p(k)*rb(i,k)
vmeanl=vmeanl+term
stdevl=stdevl+term*term*(cvload(k)

204
* *vload(k))
302 continue
vmean=vmeanr-vmeanl
stdev=stdevr+stdevl
become(i)=vmean/sqrt(stdev)
lnumbe=lnumbe+l
102 continue
c**************************************************************
c reliability of combined mechanisms
c (internal work)
c**************************************************************
do 50 l=l/numec
lc(l)=l
lct(l)=l
50 continue
lp=numec
ltemp=numec
ni(1)=1
numax=6
nucome=l
nucomb(1)=numec
lpt=numec
lpti=numec+l
111 continue
c***************************************************************
c loop over mechanisms in location vector
c***************************************************************
do 699 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
699 continue
do 698 klp=l,ndof
dispsu(kip)=0.0
698 continue
kmu=l
nic=ni(nucome)
nif=nic+nucomb(nucome)*nucome-l
nucomb(nucome+1)=0
. c**************************************************************
c define acceptable interval
c**************************************************************
if(nucome.gt.1)then
nifbet=lpti-l
nicbet=lpti-nucomb(nucome)
do 5544 ia=nicbet+l,nifbet
betaal=become(nicbet)+epsilo
if(become(ia).gt.betaal)then
do 5545 ib=ia,nifbet
become(ib)=1000.
5545
continue
go to 5541
endif
5544
continue
5541
continue
niccon=nicbet
end i f

205
C***************************************** *********************
c reliability of combined mechanisms
c (internal work)
c**************************************************************
do 200 j=nic,nif,nucome
icontr=0
if(nucome.gt.1)then
if(become(niccon).gt.100.)icontr=l
niccon=niccon+l
if (icontr.eq.1)go to 5564
endif
do 300 l=l,nucome
lj=lc(j+l-l)
if(lj.It.-0.1)then
kmu=-l
lj=abs(1j)
endif
do 400 kk=l,nel
kkj=2*kk-l
thesum(kk)=thesum(kk)+theta(lj,kkj)*kmu
thesul(kk)=thesul(kk)+theta(lj,kkj+l)*kmu
400 continue
kmu=l
300 continue
c*************************************************** ***********
c reliability of combined mechanisms
c (external work)
c**************************************************************
do 372 l=l,nucome
lcl=lc(j+l-i)
if(lcl.lt.-0.1)then
kmu=-l
lcl=abs(lcl)
endif
do 472 kk=l,ndof
if(abs(p(kk)).It.0.001)go to 472
dispsu(kk)=dispsu(kk)+rb(lcl,kk)*kmu
472 continue
kmu=l
372 continue
5564 continue
c******************************************************* ******
c combination with fundamental mechanisms
c (internal work)
c***************************************** ********************
do 100 k=l,numec
vmeanr=0.0
vmeanl=0.0
stdevr=0.0
stdevl=0.0
vmanrm=0.0
vmanlm=0.0
stdvrm=0.0
stdvlm=0.0
do 499 lll=j,j+nucome-1
if(abs(lc(lll)).ge.k)go to 100

206
499 continue
if(nucome.gt.1)then
if (icontr.eq.l)then
becomi=400.
becopl=500.
go to 5574
endif
endif
thesu=0.0
thesu2=0.0
thesui=0.0
theslm=0.0
do 600 kk=l,nel
kkj=2*kk-l
thesu=thesum(kk)+theta(k,kkj)
thesu2=thesul(kk)+theta(k,kkj+1)
term=(abs(thesu)+abs(thesu2))*cvmu(kk)
* *vmu(kk)
vmeanr=term/cvmu (kk) +vmeanr
stdevr=stdevr+term*term
thesui=thesum(kk)-theta(k,kkj)
theslm=thesul(kk)-theta(k,kkj+1)
termm=(abs(thesui)tabs(theslm))*cvmu(kk)
* *vmu(kk)
vmannu=tennm/cvmu (kk) +vmannn
stdvrm=stdvrm+tennin*termm
600 continue
c*************************************************************
c combination with fundamental mechanisms
c (external work)
Q*************************************************************
do 672 kk=l,ndof
if(abs(p(kk)).lt.0.001)go to 672
dispkk=(dispsu(kk)+rb(k,kk))*p(kk)
vmeanl=vmeanl+dispkk
stdevl=stdevl+dispkk*cvload(kk)*dispkk
* *cvload(kk)
dispkm=(dispsu(kk)-rb(k,kk))*p(kk)
vmanlm=vmanlm+dispkm
stdvlm=stdvlm+dispkm*cvload(kk)*dispkm
* *cvload(kk)
672
5574
138
continue
becopl=(vmeanr-vmeanl)/sqrt(stdevr+stdevl)
becomi=(vmanrm-vmanlm)/sqrt(stdvrm+stdvlm)
continue
if(becomi.lt.becopl) then
do 138 lk=l,nucome
ltemp=ltemp+l
lc(ltemp)=lc(j+lk-1)
continue
1temp=ltemp+1
lc(ltemp)=-k
lctlp=-k
becote=becomi
else
do 139 lk=l,nucome

207
ltemp=ltemp+l
le(ltemp)=lc(j+lk-1)
139 continue
lctlp=k
ltemp=ltemp+l
lc(ltemp)=k
becote=becopl
endif
lpt=lpt+l
iflag=0
c*************************************************************
c ordering the combined beta values in the same row
c*********************************************** **************
if (lpti.gt.lpt-1)then
do 524 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
continue
lp=lp+l
lct(lp)=lctlp
become(lpt)=becote
do 510 jkj=lpti,lpt-1
if(become(jkj).gt.becote)then
iflag=-l
do 511 kjk=jkj,lpt-1
itemp=lpt-l+j kj-kj k
become(itemp+1)=become(itemp)
continue
become(jkj)=becote
becote=become(lpt)
C*************************************************** **********
c moving lc array
c***************************** ********************************
movini=nic+nucomb(nucome)*nucome+
* (jkj-lpti)*(nucome+1)
movfin=(lpt-lpti)*(nucome+1)+nic+
* nucomb(nucome)*nucome-l
do 512 lmn=movini,movfin
lcou=movini+movfin-lmn
nucl=nucome+l
let(lcou+nucl)=lct(lcou)
512 continue
do 513 n=movini,movini+nucome-l
lptaa=n-movini+l
let(n)=lc(j+lptaa-1)
513 continue
let(movini+nucome)=lctlp
endif
510 continue
if(iflag.eq.0)then
do 124 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
continue
lp=lp+l
524
else
511
124

208
lct(lp)=lctlp
become(lpt)=becote
else
lp=lp+nucome+i
endif
endif
nucomb(nucome+1)=nucomb(nucome+1)+1
1numbe=1numbe+1
100 continue
do 6991 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
6991 continue
do 6981 klp=l,ndof
dispsu(kip)=0.0
6981 continue
200 continue
c*************************************************************
c control of maximum number of tree rows
c*************************************************************
lpti=lpti+nucomb(nucome+1)
nucome=nucome+l
ni(nucome)=ni(nucome-1)+(nucome-1)*nucomb(nucome-1)
if(nucome.It.(numax))go to 111
c*************************************************************
c find minimum beta value
c*************************************************************
betmin=100
do 7890 i=l,lnumbe
if(become(i).It.betmin)then
betmin=become(i)
itab=i
endif
7890 continue
c*************************************************************
c find mechanisms involved
c*************************************************************
mecoun=0
mcomb=l
do 7891 j=l,nucome-1
do 7895 jk=l,nucomb(j)
mecoun=l+mecoun
if(itab.eq.mecoun)then
nistar=mcomb
do 7893 1=1,j
locmec(l)=lct(nistar+1-1)
continue
nummec=j
go to 7894
endif
mcomb=j +mcomb
continue
continue
continue
c*************************************************************
c find elements involved
7893
7895
7891
7894

209
C*************************************************************
nk=l
numele=0
do 8000 m=l,nel
do 8001 k=l,numec
do 8002 l=l,nummec
if(abs(locmec(l)).eq.k)then
*
*
8003
8004
endif
8002 continue
8001 continue
8000 continue
ml=2*m-l
m2=2*m
if(abs(theta(ml,k)).gt.0.0.
or.abs(theta(m2,k)).gt.0.0)then
if(nk.gt.1)then
do 8003 lmi=l,nk
if(invmec(lmi).eq.m)
go to 8004
continue
endif
nk=nk+l
nume1e=nume1e+1
invmec(n)=m
go to 8000
continue
endif
c*************************************************************
c control of system reliability
c*************************************************************
write(3,*)
write(3,*)'BETA MINIMAL FOR THE SYSTEM = ',betmin
write(3,*)
jflag=0
if(betmin.It.relind)then
delta=(relind-betmin)/relind
do 3891 i=l,nel
do 3892 j=l,numele
if(invmec(j).eq.i)then
elerel(i)=(1.+delta)*elerel(i)
endif
3892 continue
3891 continue
jflag=l
endif
return
end
subroutine mecsys(n,iqh,cl,cost,sint,lm,numec,r,theta)

210
implicit double precision (a-h,o-z)
dimension a(100,100),b(100,100),c(100),cm(100,100),
* cost(n),qa(100,100),sint(n),cl(n),q(100,100),
* lm(6,n),am(100,100),bl(100,100),theta(200,100),
* r(iqh,100)
£* ************************************************************ *
c Constraint matrix for the structure
q**************************************************** **********
do 300 i=l,3*n
do 400 j=l,6*n
cm(i,j)=0.0
q(i,j)=0.0
400 continue
300 continue
do 60 k=l,n
i=3*k
j=6*k
at=l.0/cl(k)
im2=i-2
iml=i-l
jml=j-1
jm2=j-2
jm3=j-3
jm4=j-4
jm5=j-5
cm(im2,j m5)=-l.0
cm(im2,jm2)=1.0
cm(iml,jm3)=l.0
cm(i,j)=1.0
cm(iml,jm4)=-at
cm(iml,jml)=at
cm(i,jm4)=-at
cm(i,jml)=at
60 continue
a*************************************************************it
c Coordinate trnsformation matrix
c**************************************************************
do 70 k=l,n
co=cost(k)
si=sint(k)
j=6*k
do 80 i=l, 2
jum=j-3*i+l
jdois=j-3*i+2
jtres=j-3*i+3
q( jum,jum)=co
q(jum, jdois)=si
q(jdois,jum)=-si
q(jdois,jdois)=co
q(jtres,jtres)=1.0
80 continue
70 continue
Q* ************************************************************ *
c Compatibilibity matrix from LM matrix
c**************************************************************
n6=6*n

211
do 500 i=l,n6
do 510 k=l,iqh
am(i,k)=0.0
510 continue
500 continue
do 52 0 i=l, 6
do 530 k=l,n
iflag=lm(i,k)
if (iflag.gt.0) am(6*(k-l)+i,iflag)=1.0
530 continue
520 continue
c**************************************************************
c Rotation of basic compatibility matrix
c**************************************************************
mm=80
call multi(q,am,qa,n6,iqh,n6,mm)
c**************************************************************
c Expansion of QA matrix
c**************************************************************
do 585 i=l,n
i6=6*i
i3=i6-3
do 595 j=l,iqh
qa(i3,j)=0.0
qa(i6,j)=0.0
595 continue
i2=2*i
qa(i3,iqh+i2-l)=l. 0
qa(i6,iqh+i2)=1.0
585 continue
c**************************************************************
c Matrix A = C QA (transformed)
c**************************************************************
m=3*n
nt=iqh+2*n
call multiCcmjqa^mintjnejmm)
c**************************************************************
c Solution for virtual displacements
c**************************************************************
do 150 k=l,nt
do 175 1=1,nt
b(k,l)=0.0
175 continue
150 continue
do 160 k=l,nt
b(k,k)=1.0
160 continue
do 200 i=l,m
amax=0.0
iflag=0
do 250 j=i,nt
if (abs(a(i,j)).gt.amax)then
iflag=j
amax=abs(a(i,j))
endif
continue
250

212
do 305 k=l,m
c(k)=a(k,iflag)
a(k,iflag)=a(k,i)
a(k,i)=c(k)
305 continue
do 310 k=l,nt
c(k)=b(k,iflag)
b(k,iflag)=b(k,i)
b(k,i)=c(k)
310 continue
do 280 j=i+l,nt
if (abs(a(i,j)).gt.0.00001) then
fact=-a(i,j)/a(i,i)
do 290 kk=l,m
a(kk,j)=a(kk,j)+a(kk,i)*fact
290 continue
do 291 kk=l,nt
b(kk,j)=b(kk,j)+b(kk,i)*fact
291 continue
endif
280 continue
200 continue
numec=nt-3*n
c* ************************************************************* *
c Forming bl
c***************************************************** **********
lcount=nt-numec+l
ki=l
do 800 i=lcount,nt
do 810 j=l,nt
bl(j/ki)=b(j,i)
810 continue
ki=ki+l
800 continue
c**************************************************************
c Creating Theta matrix
c******************* ******************************************
do 156 j=l,numec
do 157 i=l,2*n
k=iqh+i
theta(i,j)=bl(k,j)
157 continue
156 continue
c***************************************************************
c Creating virtual displacements
c***************************************************** *********
do 169 j=l,numec
do 158 i=l,iqh
r(i,j)=bl(i,j)
158 continue
c************************************************************
c Adding joint mechanisms
c**************************************************************
do 161 i=3,iqh,3
if (abs(r(i,j)).gt.0.000001)then
r(i,j)=0.0

do 162 k=l,n
if (lm(3,k).eq.i)then
lpo=2*k-l
theta(lpo,j)=l
endif
if(lm(6,k).eq.i)then
lpo=2*k
theta(lpo,j)=l
endif
continue
endif
continue
continue
return
end
subroutine multi(aa,bb,cc,1,m,n,k)
implicit double precision (a-h,o-z)
dimension aa(l,n),bb(n,m),cc(l,m)
do 10 i=l,l
do 20 j=l,m
d=0.0
do 30 kk=l,n
d=d+aa(i,kk)*bb(kk,j)
continue
cc(i,j)d
continue
continue
return
end
subroutine equcon(x)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
common /springs/ vksi(100),vksj(100)

214
do 121 ikl=l,iqh
vahk(ikl)=0.0
121 continue
n=nel
n3=n+n+n
do 100 k = l,n
sigma2=0.0
do 111 ipo=l,6
u(ipo)=0.0
111 continue
do 200 i = 1,6
m=lm(i,k)
if (m.eq.O) go to 200
if(cvload(m).gt.sigma2)sigma2=cvload(m)
1 = n3 + m
u(i) = x(l)
200 continue
cl=cosl(k)
c2=cos2(k)
d2=-c2*u(1)+cl*u(2)
d3=u(3)
d5=-c2*U(4)+Cl*U(5)
d6=U(6)
c***************************************************************
c element forces
c***************************************************************
c = cl(k)
base = x(3*k-2)
height = x(3*k-l)
aste = x(3*k)
area = base height
tinert = area*height*height/12.0
al = ec*tinert/(c*c*c)
call eley (ec,tinert,c,vksi(k),vksj(k),d2,d3,d5,d6,
* fo3,fo6)
sigmal=cvmu(k)
c***************************************************************
c ultimate and yield moments
c***************************************************************
call newmum(k,x,sigmal,sigma2,fo3,fo6,vksi(k),vksj(k),
* d2,d5,d3,d6)
c***************************************************************
c global modified stiffness
c***************************************************************
call modsti(k,tinert,area,vksi(k),vksj(k))
do 300 1=1,6
j=lm(l,k)
if (j.eq.0) go to 300
do 400 11=1,6
m = lm(ll,k)
if (m.eq.O) go to 400
jj = n3+m
vahk(j)=vahk(j)+ck(l,11)*x(jj)
400 continue
300 continue
100 continue

215
C***************************************************************
c subtraction of external global forces
G***************************************************************
rmax=0.01
do 510 i=l,iqh
if(abs(r(i)).gt.rmax)rmax=abs(r(i))
510 continue
do 500 k=l,iqh
if(abs(r(k)).It.0.0001)then
vah(k)=vahk(k)/rmax
go to 500
endif
vah(k)=(vahk(k) r(k))/rmax
500 continue
return
end
subroutine mumy(kel/x/sigmal,sigma2/fo3,fo6,vki,vkj,displ,
* disp2,rotl,rot2)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg/nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
common /springs/ vksi(100),vksj(100)
n=nel
nodel=0
node2=0
b=x(3*kel-2)
h=x(3*kel-l)
dd=h-co
aste=x(3*kel)
clk=cl(kel)
ei=57000.*sqrt(ec)*h*h*h*b/12.
c***************************************************************
c evaluation of yielding moment
g****************************************************************
call comcon(aste,dd,b/vmy,phiy)
afo3=abs(fo3)
afo6=abs(fo6)
vm=max(afo3,af06)
c*************************************************************
c ultimate moment and reliability
G*************************************************************
hl2v0s0b84X1^blilia
216
vmuk=vmu(kel)
c****************************************************************
c integration of curvature
c****************************************************************
if(vmy.lt.afo3)nodel=l
if(vmy.lt.afo6)node2=l
if(nodel.eq.l.or.node2.eq.1)then
if((fo3*fo6).gt.0.) then
if(afo3.ge.afo6)then
tetay=(vmy/(3.*ei)+afo6/(6.*ei))*clk
vlp=(afo3-vmy)/(afo3-afo6)*clk
if(abs(vlp).gt.clk)vlp=clk
tetal=(vmuk/(3.*ei)+afo6/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
if(afo3.It.afo6)then
tetay=(vmy/(3.*ei)+afo3/(6.*ei))*clk
vlp=(afo6-vmy)/(afo6-afo3)*clk
if(abs(vlp).gt.clk)vlp=clk
tetal=(vmuk/(3.*ei)+afo3/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
endif
if((fo3*fo6).It.0.) then
if(afo3.ge.afo6)then
tetay=(vmy/(3.*ei)-afo6/(6.*ei))*clk
vlp=(afo3-vmy)/(afo3+afo6)*clk
tetal=(vmuk/(3.*ei)-afo6/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
if(afo3.It.afo6)then
tetay=(vmy/(3.*ei)-afo3/(6.*ei))*clk
vlp=(afo6-vmy)/(afo3+afo6)*clk
tetal=(vmuk/(3.*ei)-afo3/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
endif
endif
crotl=rotl-(-displ/clk+disp2/clk)
crot2=rot2-(-displ/clk+disp2/clk)
vksp=(vmuk-vmy)/(tetau-tetay)
c*****************************************************************
c spring rotation
c*****************************************************************
srotl=abs(crotl)-tetay
srot2=abs(crot2)-tetay
c******************************************************************
c new secant spring values
c******************************************************************
if(nodel.eq.1) then
if(srotl.le.O.)then
vki=vmy/tetay
go to 123
endif
vki=(vksp*srotl+vmy)/abs(crotl)
if(vm.gt.vmuk)vki=vmuk/tetau

217
123
124
125
126
if(abs(crotl).gt.tetau)vki=vmuk/tetau
continue
endif
if(node2.eg.1) then
if(srot2.le.0.)then
vkj=vmy/tetay
go to 124
endif
vkj=(vksp*srot2+vmy)/abs(crot2)
if(vm.gt.vmuk)vkj=vmuk/tetau
if(abs(crot2).gt.tetau)vkj=vmuk/tetau
continue
endif
if(nodel.eq.l.and.node2.eg.1) then
if(srotl.le.0.)then
vki=vmy/tetay
go to 125
endif
vki=(vksp*srotl+vmy)/abs(crotl)
if(vm.gt.vmuk)vki=vmuk/tetau
if(abs(crotl).gt.tetau)vki=vmuk/tetau
continue
if(srot2.le.0.)then
vkj =vmy/tetay
go to 126
endif
vkj = (vksp*srot2+vmy)/abs(crot2)
if(vm.gt.vmuk)vkj=vmuk/tetau
if(abs(crot2).gt.tetau)vkj=vmuk/tetau
continue
endif
return
end
subroutine valmu(aste,b,kel,dd,vm,vmy,phiy,phiu,
* sigmal,sigma2)
implicit double precision (a-h,o-z)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100);jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv^c^p^c^s^cmjrelind/co^fy^psy
common /pari/ igh,igg,nel,ntot,iggn
common /inequa/ vag(100),beta(100),u(6),vahk(100),
* ck(6,6)
common /equal/ vah(100),vmu(100)
n=nel
c*************************************************************

218
c neutral axis
c******************************************************* ******
x=47./60.*b*fc
y=0.004*es*aste-aste*fy
z=-0.004*es*co*aste
vkd=(-y+sqrt(y*y-4.*x*z))/(2.*x)
epcs=0.004*(vkd-co)/vkd
if(epcs.gt.epsy)then
epcs=epsy
endif
c***** *********************** *********'!(****** ******* ***********
c concrete force in region ab
c**************************************************************
alphal=2./3 ,
ccab=alphal*b*0.5*vkd*fc
c*************************************************************
c concrete force in region be
c*************************************************************
alpha2=0.9
ccbc=alpha2*b*0.5*vkd*fc
c******************************************************** ******
c distance of centroid to top in ab
c**************************************************************
gama1=0.875*vkd
c******************************************************* *******
c distance of centroid to top in be
**************************************************************
gama2=0.259255*vkd
Q**********************************************************
c coefficients for failure function
c************************************************************
al=(ccab*(dd-gamal)+ccbc*(dd-gama2))/fc
a2=-l.
c* ********************************************************* *
c cosine directors
c**********************************************************
tetal=al*sigmal*fc
teta2=a2*sigma2*vm
c****************************************************** ******
c independent term
c******************************************** ***************
fps=0.004*es*(vkd-co)/vkd
bi=aste*fps*(dd-co)
c***********************************************************
c reliability index
c***********************************************************
beta(kel)=(al*fc+a2*vm+bi)/sqrt(tetal*tetal+teta2*teta2)
c************************************************************
c ultimate moment and rotation
c**************************************************************
vmu(kel)=al*fc+bi
phiu=0.004/vkd
vmuk=vmu(kel)
if((4.*phiy).It.phiu)then
vmu(kel)=(vmuk-vmy)/(phiu-phiy)*3*phiy+vmy
phiu=3*phiy

219
endif
return
end
subroutine coxticon (aste, dd, b, vmy, phiy)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100)/cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cV/ec^pjfC/es^cm^elind^o^y^psy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),
* ck(6,6)
common /equal/ vah(100),vmu(100)
node=0
epso=0.002
c************************************************ **************
c
c exc concrete strain
c epcs compressive steel strain
c epsy yield strain
c
c**************************************************************
c first value for a
c**************************************************************
al=dd/2.
exc=al*epsy/(dd-al)
epcs=exc*(al-co)/al
t=fy*aste
cs=epcs*es*aste
eces=exc/epso
alpha=eces-eces*eces/3.
cc=alpha*fc*b*al
resl=cc+cs-t
c******************************************************** ******
c second value for a
c* ************************************************************ *
a2=0.25*dd
exc=a2*epsy/(dd-a2)
epcs=exc*(a2-co)/a2
cc=fc*a2*alpha*b
eces=exc/epso
cs=epcs*es*aste
res2=cc+cs-t
c***************************************************************
c newton iteration
c***************************************************************
100 a=a2-res2*(a2-al)/(res2-resl)

220
exc=a*epsy/(dd-a)
if(exc.gt.epso) go to 200
C***************************************************************
c parabolic shape
c***************************************************************
epcs=exc*(a-co)/a
eces=exc/epso
alpha=eces-eces*eces/3.
cc=fc*alpha*b*a
cs=epcs*es*aste
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
go to 100
endif
gama=l.-(8.*epso-3.*exc)/(12.*epso-4.*exc)
arm=dd-gama*a
vmy=cc*arm+epcs*es*aste*(dd-co)
phiy=epsy/(dd-a)
return
c*************************************************** ************
c concrete strain > epso
c* ************************************************************* *
200 if(exc.gt.0.004) exc=0.004
xl=epso*a/exc
cclfc*xl*2./3.*b
gama=3.6*exc*exc-200.*exc*exc*exc-0.0000128
gama=gama/((exc-epso)*(7.2*exc-300*exc*exc-0.0132))-1.
alpha=exc-50.*exc*exc+100.*exc*epso-0.0022
alpha=alpha/(exc-epso)
cc2=alpha*fc*(a-xl)*b
epcs=exc*(dd-co)/a
t=fy*aste
cs=epcs*es*aste
cc=ccl+cc2
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
go to 100
endif
arml=dd-a+2./3.*xl
arm2=dd-gama*(a-xl)
vmy=ccl*arxnl+epcs*es* (dd-co) *aste+cc2*arm2
phiy=epsy/(dd-a)
return
end

221
program eley
implicit double precision(a-h,o-z)
open(1,file='ydata',form='formatted1)
rewind 1
read(1,*)ec,tinert,cl,vki,vkj,u2,u3,u5,u6
ei=ec*tinert
w=cl/(3.*ei)+1./vki
y=cl/(3.*ei)+l./vkj
z=-cl/(6.*ei)
det=w*y-z*z
a=y/det
b=-z/dt
c=b
d=w/det
fo3=(a+b)/cl*u2+a*u3-(a+b)/cl*u5+b*u6
f06=(c+d)/cl*u2+c*u3-(c+d)/cl*u5+d*u6
write(*,*)'fo3 = 1,fo3,1 fo6 = ',fo6
stop
end
program yiel
implicit double precision (a-h,o-z)
open(1,file='yielm',form='formatted')
rewind 1
read(l,*)b,d,aste,epsy,es,co,fy,fc,ecm,vm,sigmal,sigma2
ec=3122019
node=0
epso=0.002
c**************************************************************
c
C EXC CONCRETE STRAIN
C EPCS COMPRESSIVE STEEL STRAIN
c EPSY YIELD STRAIN
c
C FIRST VALUE FOR A
c***************************************************************
dd=x(3*kel-l)-co
al=dd/2.
exc=al*epsy/(dd-al)
epcs=exc*(al-co)/al
t=fy*aste
cs=epcs*es*aste
eces=exc/epso
alpha=eces-eces*eces/3.
cc=alpha*fc*b*al

222
resl=cc+cs-t
c***************************************************************
C SECOND VALUE FOR A
q* ****************************************************** -k ****** *
a2=0.25*dd
exc=a2*epsy/(dd-a2)
epcs=exc*(a2-co)/a2
cc=fc*a2 *alpha*b
eces=exc/epso
cs=epcs*es*aste
res2=cc+cs-t
c***************************************************** **********
C NEWTON ITERATION
Q***************************************************************
100 a=a2-res2*(a2-al)/(res2-resl)
exc=a*epsy/(dd-a)
if(exc.gt.epso) go to 200
c******************************************************* ********
c PARABOLIC SHAPE
C**************************************************************ie
epcs=exc*(a-co)/a
eces=exc/epso
alpha=eces-eces*eces/3.
cc=fc*alpha*b*a
cs=epcs*es*aste
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
go to 100
end if
gama=l.-(8.*epso-3.*exc)/(12.*epso-4.*exc)
arm=dd-gama*a
vmy=cc*arm+epcs*es*aste*(dd-co)
phiy=epsy/(dd-a)
Q**********************************************************
C CONCRETE STRAIN > EPSO
c**********************************************************
200 if(exc.gt.0.004) exc=0.004
xl=epso*a/exc
ccl=fc*xl*2./3.*b
gama=3.6*exc*exc-200.*exc*exc*exc-0.0000128
gama=gaina/ ((exc-epso) (7.2*exc-300*exc*exc-0.0132)) -1.
alpha=exc-50.*exc*exc+i00.*exc*epso-0.0022
alpha=alpha/(exc-epso)
cc2=alpha*fc*(a-xl)*b
epcs=exc*(dd-co)/a
t=fy*aste
cs=epcs*es*aste
cc=ccl+cc2
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then

223
al=a2
a2=a
resl=res2
res2=res
go to 100
endif
arml=dd-a+2./3.*xl
arm2=dd-gama*(a-xl)
vmy=ccl*arail+epcs*es*(dd-co)*aste+cc2*arm2
phiy=epsy/(dd-a)
c************************************************ ********
c LINEAR APPROXIMATION
Q********************************************************
ro=aste/(b*d)
vn=es/ec
vk=sqrt(4.*ro*ro*vn*vn+2.*(ro+ro*co/d)*vn)-2.*ro*vn
vkd=vk*d
exc=epsy/(d-vkd)*vkd
cc=0.5*vkd*b*ec*exc
excs=exc/vkd*(vkd-co)
fps=excs*es
cs=fps*aste
vmy=cc*(d-vkd/3.)+cs*(d-co)
phiy=exc/vkd
write(*,*)'cc=1,cc,'cs=',cs,'exc=',exc,'excs=',exes
write(*,*)'linear approx','vmy=',vmy,'phiy=',phiy
stop
end

224
Example: Debug Frame
Input File: DATA
TESTE LINEAR BETAO ARMADO
21
17 9
8
1
2.0
2
6.0
4
2.0
5
6.0
7
2.0
8
6.0
10
2.0
11
6.0
0
4
14
27.8
15
27.8
16
27.8
17
27.8
5.00
10.00 1.00
5.00
10.00
1.00
5.0 10.0
1.0
.5
.805
.07 .94
.0
-.092
2
EPNE
0.1
EPST
0.000001
1
1500 500010000
1
0
0
4
2
0
1.00
. 059
5.
. 0
0 10.0
.900
5
8
11

225
User's Manual
Generalized Reduced Gradient Method
Example: Debug Frame
Input File: DATA
Line 1
Problem title.
Line 2
Number of variables, number of constraints, number of
equality constraints.
Line 3
Number of variables with lower bounds.
Line 4 to line 11
Variable number and respective lower bound.
Line 12
Number of constraints with upper bounds.
Line 13 to line 16
Constraint number and respective upper bound.
Line 17 to line 19
Initial values of design variables.
Line 20
Number of prescribed optimization parameters.
Line 21
Constraint tolerance.
Line 22
Convergence tolerance.
Line 23

226
Parameter indicating alteration of the limit of number
of iterations.
Line 24
Maximum number of consecutive iterations without
objective function improvement, maximum number of
consecutive Newton iterations, maximum number of
completed one dimensional searches.
Line 25
Parameter that controls the quantity of information in
the output file.
Line 26
Number indicating minimum printed information.
Line 27
Indication that tangent vector extrapolation should be
used for estimating initial values of basic variables.
Line 28
Number of design variables iniatially included in the
basis.
Line 29
Numbers of design variables of the initial basis.
Line 30
Parameter that indicates if new data should be read.

227
Example: Debug Frame
Input File: DATA1
4,5
1,2
2.3
3.4
4.5
1,1,1,0,0
0,0,0,0,100
0,0,0,50,100
0,0,0,100,100
1,1,1,100,0
2,1,5000
3,2,-5000
'0,0,0
3000,40000,1
29e6,0.004
2
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

228
User's Manual
Generalized Reduced Gradient Method
Example: Debug Frame
Input File: DATA1
Line 1
Number of elements, number of nodes.
Line 2 to line 5
Node i, node j of element 1 through 4.
Line 6 to line 10
Boundary conditions of displacement in the horizontal
direction, vertical direction, in-plane rotation,
horizontal coordinate, vertical coordinate.
Line 11 and line 12
Node where force is applied, direction of load and
magnitude of load.
Line 13
Termination of force information.
Line 14
Flexural strength of concrete, yielding stress of steel
and reinforcement cover.
Line 15
Steel modulus of elasticity and concrete ultimate
strain.
Line 16
Minimum element and system reliability index.
Line 17 to line 20
Coefficient of variation of flexural concrete strength.
Line 21 to line 29

229
Coefficient of variation of external global loads.
Line 30 to line 33
Coefficient of variation of element ultimate moment.
Line 34
Value of interval gap in the Beta unzipping method.

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230

231
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2 32
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Engineering. ASCE, Vol. 113, No. 6, 1987, pg. 1221-1235.
(32) Kayal, S., Finite Element Analysis of RC Frames,
Journal of Structural Engineering. ASCE, Vol. 110, No. 12,
1984., pg. 2891-2907.
(33) Otani, S., Inelastic Analysis of R/C Frame
Structures, Journal of Structural Division. ASCE, Vol. 100,
N. ST7, 1974, pg. 1433-1457.
(34) Tsay, J. J., and Arora, J. S., Variational Methods
for Design Sensitivity Analysis of Nonlinear Response with
History Dependent Effects. Proceedings of the International
Conference on Computational Engineering Science, Atlanta,
1988.
(35) Umerura, H., and Takizawa, H., Dynamic Response of
Reinforced Concrete Buildings. IABSE Structural
Engineering Documents, Number 2, Zurich, 1982.
(36) Kanaan, A. E., and Powell, G. H., DRAIN-2D. A
General Purpose Computer Program for Dynamic Analysis of
Inelastic Plane Structures. Report n EERC 73-6 and EERC
73-22, University of Berkeley, Berkeley, 1975.
(37) Breyse, D., and Mazars, J., Simplified Approach of
Nonlinearity in R/C Beams, Journal of Structural
Engineering. ASCE. Vol. 114, n. 2, 1988, pg. 251-268.
(38) Gedling, J. S., Mistry N. S., and Welch, A. K.,
Evaluation of Material Models for Reinforced Concrete
Structures, Computers and Structures. Vol. 24, n. 2, 1986,
pg. 225-232, 1986.
(39) Cauvin, A., Nonlinear Elastic Design and Optimization
of Reinforced Concrete Frames. CSCE-ASCE-ACI-CEB
International Symposium, Ontario, Canada, 1979.
(40) Charney, F. A., Correlation of the Analytical and
Experimental Seismic Response of a l/5th-Scale Seven-Storv
Reinforced Concrete Frame-Wall Structure. PhD Dissertation,
University of California, Berkeley, 1986.
(41) Augusti, G., Baratta, A., and Casciati, F.,
Probabilistic Methods in Structural Engineering. Chapman and
Hall, New York, 1984.

233
(42) Madsen, H. O., Krenk, S., and Lind, N. C., Methods of
Structural Safety. Prentice-Hall, Englewood Cliffs, New
Jersey, 1986.
(43) Regan, P. E., and Yu, P. E., Limit State Design of
Structural Concrete. John Wiley & Sons, New York, 1973.
(44) Leporati, E., The Assessment of Structural Safety.
Research Studies Press, Forest Grove, Oregon, 1977.
(45) Pahoheimo, E. and Hannus, M., Structural Design Based
on Weighted Fractiles, Journal of Structural Division. ASCE,
Vol. 100, ST7, 1974, pg. 1367-1378.
(46) Requlamento de Estruturas de Beto Armado e Pr-
Esforeado. Imprensa Nacional Casa da Moeda, Lisboa,
Portugal, 1983.
(47) Thoft-Christensen, P., and Baker, M. J.,Structural
Reliability Theory and its Applications. Springer-Verlag,
Heidelberg, West Germany, 1982.
(48) Neville, A. M., Properties of Concrete. Pitman, Bath,
United Kingdom, 1983.
(49) Recommended Practice for Evaluation of Strenghth Test
Results for Concrete, ACI 214-71, American Concrete
Institute, Detroit, 1976.
(50) Mindess, S. and Young, J. F., Concrete. Prentice-
Hall, Englewood Cliffs, New Jersey, 1981.
(51) Building Code Requirements for Reinforced Concrete.
(ACI 318-83), American Concrete Institute, Detroit, 1976.
(52) Hart, G., Uncertainty Analysis, Loads and Safety in
Structural Engineering. Prentice-Hall, Englewood Cliffs, New
Jersey, 1982.
(53) Chou, Karen C., and Corotis, Ross B., Nonlinear
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4th ASCE Specialty Conference, ASCE, New York, 1984.
(55) Blockley, D. I., The Nature of Structural Design and
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1980.
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of Structural Mechanics. Vol. 14, No. 4, 1986, pg. 437-453.

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Modes in System Reliability, Journal of Structural Division.
ASCE, Vol. 114, NO. 2, 1988, pg. 292-313.
(63) Bennett, R. M., and Ang, A. H-S., Investigation of
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Redundancy and Reliability of Structural Systems, Computers
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under Seismic Loading, Structural Optimization Developments.
ASCE, New York, 1986, pg. 1-16.
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Redundant Ductile Structural Systems. Structural Research
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M., Design and Testing of a Generalized Reduced Gradient
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Ontario, Canada, 1972.

BIOGRAPHICAL SKETCH
The author was born in Porto, Portugal in 1954.
He finished high school in D. Manuel II, Porto in 1971,
graduated from the University of Porto in Civil Engineering,
majoring in Structures and obtained the degree of Master in
Engineering from the University of Florida, United States of
America in 1986.
The writer taught introductory courses in the
University of Porto, College of Engineering, between 1976 and
1985, and is on leave to pursue his Ph.D. degree. He was a
structural consultant between 1977 and 1984.
He is a recipient of a scholarship from the binational
Fulbright Commission during his studies in the United States
of America. The author received a grant from Fundapao
Oriente to present a paper in the IV World Conference on
Continuing Engineering Education. He is a member of Phi
Beta Delta, ASCE and ACI.
236

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Mafc'I. HoCt,' Chair
Assistant Professor of Civil
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Fernando E. F^gjindo^Cochair
Associate Professor^of Civil
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Cl if fjord 0. Hays
Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
3hn M. Lybas
Associate Profes
Engineering
of Civil

I certify that I have read this studv and that in my
opinion it conforms to acceptable standard, of scholarly
, as
Associate Professor of
Aerospace Engineering,
Mechanics, and Engineering
Science
presentation and is fully adequate, in
a dissertation for the degree of Doc
scope and quality
lak Philosophy.
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August, 1989
Dean,//College flof Engineering
Dean, Graduate School

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TITLE: Optimization of reinforced concrete frames using integrated analysis
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PUBLICATION DATE: 1989
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75
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:
where
Minimize Ct = CQ + Cf Pf
Ct cost of the structure;
C0 initial cost of the structure;
Cf cost of failure;
Pf probability of structure failure.


90
case one can find a matrix [D], nonsingular with dimensions
p by p such that
[B'] [D] = [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.


59
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


167
common /esq/ u(6),ck(6,6);vksi(100),vksj(100)
cl = cosl(j)
c2 = cos2(j)
Cl2 = Cl Cl
C22 = c2 C2
area = x(j *31)*x(3*j-2)
tinert = area*x(3*j-l)*x(3*j-l)/l2.
ell = cl(j)
Cl2 = cll*cll
cl3 = cl2*cll
a = e*tinert/cl3
b = area*cl2/tinert
gl = a*(b*cl2+l2.*c22)
g2 = a*cl*c2*(b-12.)
g3 = a*(b*c22+12.*cl2)
g4 = -a*6.*cll*c2
g5 = a*6.*cll*cl
g7 = a*2
!.*C12
ge = g7
+
g7
ck(l,1)
=
gl
ck(2,1)
=
g2
ck(3,1)
=
g4
ck(4,1)
=
- gl
ck(5,1)
=
- g2
ck(6,1)
=
g4
ck(l,2)
=
g2
ck(2,2)
=
g3
ck(3,2)
=
gs
ck(4,2)
=
- g2
ck(5,2)
=
- g3
ck(6,2)
=
gs
ck(1,3)
=
g4
ck(2,3)
=
gs
Ck(3,3)
=
ge
Ck(4,3)
=
- g4
ck(5,3)
=
- g5
ck(6,3)
=
g7
ck(l,4)
=
- gi
Ck(2,4)
=
- g2
Ck(3,4)
=
- g4
Ck(4,4)
=
gi
Ck(5,4)
=
g2
Ck(6,4)
sr
- g4
ck(1,5)
=
- g2
ck(2,5)
=
- g3
Ck(3,5)
=
- g5
ck(4,5)
=
g2
ck(5,5)
=
gs
ck(6,5)
=s
- gs
ck(l,6)
=
g4
ck(2,6)
=
gs
ck(3,6)
=
g7
ck(4,6)
=
- g4
ck(5,6)
=
- g5
Ck(6,6)
=
ge
return


Ill
The procedure was developed in two phases. First, a
linear material behavior was assumed followed by the
inclusion of material nonlinear behavior was included. The
computer program versions were created combining a public
domain software package and some subroutines already used in
the prior formulation (76). The coded version of the
Generalized Reduced Gradient method is a general purpose
program for constrained optimization and was changed
slightly when adapting to the present case.
Main modifications actually introduced in the software
were to increase the maximum number of variables and
constraints, the extension of the maximum number of Newton
iterations, and the modification of the number of times the
stepsize could be reduced when performing the line search.
The size of the problems tested caused the first alteration.
Although the authors had not tested the program with
examples as large as those described in the next chapter,
the computer code performed with no problems. The variation
of the maximum number of iterations was required due to the
material nonlinear behavior, which imposed a slower
computation of the basic variables when iteratively solving
the set of nonlinear equalities. The need for smaller step
sizes was due to the fact that the order of magnitude of the
change of variables in the vicinity of the design point is
very small compared to the corresponding changes of the
equilibrium equality constraints.


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In reference to the following dissertation:
AUTHOR: Soeiro, Alfredo
TITLE: Optimization of reinforced concrete frames using integrated analysis
and reliability / (record number: 1515144)
PUBLICATION DATE: 1989
I, ? as copyright holder for the aforementioned
dissertation, hereby grant specific and limited archive and distribution rights to the Board of
Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize
and distribute the dissertation described above for nonprofit, educational purposes via the Internet
or successive technologies.
This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite
term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the
terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance
and preservation of a digital archive copy. Digitization allows the University of Florida to generate
image- and text-based versions as appropriate and to provide and enhance access using search
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This grant of permissions prohibits use of the digitized versions for commercial use or profit.
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Signature of Copyright Holder
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10-07-2008 12:0


187
User's Manual
Augmented Lagrangian Formulation
Example: Debug Frame
Input File: DATA
Line 1
Problem title.
Line 2
Number of elements, number of nodes.
Line 3 to line 6
Node i, node j of element 1 through 4.
Line 7 to line 11
Boundary conditions of displacement in the horizontal
direction, vertical direction, in-plane rotation,
horizontal coordinate, vertical coordinate.
Line 12 and line 13
Node where force is applied, direction of load and
magnitude of load.
Line 14
Termination of force information.
Line 15
Flexural strength of concrete, yielding stress of steel
and reinforcement cover.
Line 16
Steel modulus of elasticity and concrete ultimate
strain.
Line 17 to line 20
Initial steel reinforcement areas.


141
Nonlinear Material Behavior
Tests performed with material nonlinear behavior were
not completely successful. The results of the debug frame
showed optimized solutions, mainly if the two stage
procedure was followed. However, as shown in Table 7.6
there was no complete node equilibrium. For the other types
of frames, independent of the technique and initial values
chosen, the results showed that simultaneous equilibrium
convergence and optimization did not occur. In certain
cases with these types of frames, there was satisfaction of
the constraints and little improvement of the objective
function. In other cases, the opposite results were
obtained.
The most probable reason for these failures is
attributed to the errors in the evaluation of the secant
spring stiffness. The element forces and the global
displacements are related to these values. On the other
hand, the changes in the element properties during the
optimization process create severe oscillations of the
secant spring stiffness values. The values of the spring
stiffness parameters oscillate abruptly between 1030 to
1010, approximately, when the moment exceeds the yielding
threshold. Also, after yielding, the spring stiffness
values oscillate between values of different order of
magnitude: the yielding stiffness and the ultimate
stiffness. The nonlinear analysis is a path dependent event


85
Element
Displacements
Independent
Element
Displacements
Rigid Body
Displacements
Figure 5.3. Element displacements definition.


30
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


43
ts
Steel Stress-Strain Diagram
Figure 3.2. Material behavior.


194
x3t = {10.954 -0.203 -0.0304};
d
Stop.
Optimum design
X*t = {0.5 10.954 -0.203 -0.304 0}.


57
different and is represented by the adopted respective
probability of failure.
For instance, in reinforced and prestressed concrete,
code checks for the ultimate limit states are based on
element forces, except in the plastic analysis where the
design variables are the loads. In cases where fatigue is
involved, stresses are also the control variables. The
service limit states are the cracking limit state and the
deformation limit state. In this work only the ultimate
flexural limit state and the global deformation limit state
are addressed since they are the more relevant for the
optimization study.
Acceptable risks of failure for any structure are
affected by the nature of the structure itself and its
expected application. These are dependent on social and
local variations. It is common for structural engineers to
balance the contradiction between the economy and safety of
the structure. This particular aspect is the main reason
why it is so appealing to combine reliability and
optimization in structural design.
Probabilities of failure used in limit state designs
vary with the risk of loss of human lives, the number of
lives affected and economic consequences. In ultimate limit
states the range of probability of failure adopted is
between 104 and 107 over a 50 year expected design life.
In serviceability limit states the probability of failure
varies between 101 and 103.


return
end
183
subroutine multi(aa,bb,cc,1,m,n,k)
implicit double precision (a-h,o-z)
dimension aa(l,n),bb(n,m),cc(l,m)
do 10 i=l,1
do 20 j=l,m
d=0.0
do 30 kk=l,n
d=d+aa(i,kk)*bb(kk,j)
30
continue
cc(i/j)d
20
continue
10
continue
return
end
subroutine jacequ (x,n/cl,lm,cosl,cos2/fc,ec/vn/co,epsy/
* fy,ntot,iqh,vah,r,vahk,es,ecm,beta,cvmu,cvload,kl,
* vmu,vjac)
implicit double precision (a-h,o-z)
dimension lm(6,n),cosl(n),cos2(n),vmu(n),vjac(iqh,iqh)
dimension cl(n),x(ntot),vah(iqh),r(iqh)
dimension vahk(iqh),beta(n),cvmu(n),cvload(iqh)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
do 150 kme = l,iqh
do 160 kmo=l,iqh
vjac(kmo,kme)=0.0
160 continue
150 continue
do 100 k=l,n
cl=cosl(k)
c2=cos2(k)
c = cl(k)
base = x(3*k-2)
height = x(3*k-l)
aste = x(3*k)
area = base height
tinert = area*height*height/12.0
C**************************************************************
C GLOBAL MODIFIED STIFFNESS
C**************************************************************
call equcon (x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,fy,


201 continue
return
end
166
subroutine datini (clah,clag,ch,eg)
implicit double precision ( a-h,o-z )
dimension clah(iqh), clag(iqg), ch(iqh), cg(iqg)
common /parr/ deefe, fcinc,cv,alpl,ec,rp, fc/es,ecm,
* relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
c************************************************************
c lag. mult, of equality c.
c****************************************** ******************
do 100 k = l,iqh
100 clah(k) = 0.0
Q***********************************************************
c lag. mult, of inequality c.
c* ********************************************************* *
do 200 k = l,iqgn
clag(k) =0.0
200 continue
c***********************************************************
c scaling factors of equ. c.
c************************************************* **********
do 300 k = l,iqh
300 ch(k) = 1.0
c***********************************************************
c scaling factors of ineq. c.
q** ******************************************************** *
do 400 k = l,iqgn
400 eg(k) = 1.0
c*************************************************** *******
c scaling factor of objective fun.
c**********************************************************
cv = 1.0
return
end
subroutine glosti (e, j, n, ntot, cl, x, cosl, cos2)
implicit double precision ( a-h,o-z )
dimension cl(n), x(ntot), cosl(n), cos2(n)


163
implicit double precision(a-h,o-z)
dimension a(n,n),b(n),c(n)
do 50 ik=l,n
c(ik)=b(ik)
50 continue
do 100 k=l,n
vmax=abs(a(k,k))
krow=k
do 120 kj=k+l,n
if(abs(a(kj, k)).gt.vmax)then
vmax=abs(a(kj,k))
krow=kj
endif
120 continue
if (krow.gt.k)then
do 140 jj=k,n
temp=a(krow, j j)
a(krow,jj)=a(k,jj)
a(k,jj)=temp
140 continue
temp=c(krow)
c(krow)=c(k)
c(k)=temp
akk=a(k,k)
endif
do 200 ik+l,n
w=a(i,k)/akk
do 300 j=k+l,n
a(i / j)=a(i,j)-a(k,j)*w
300 continue
c(i)=c(i)-c(k) *w
200 continue
100 continue
do 400 k=l,n
i=n-k+l
do 600 j=i+i,n
c(i) =c(i) -a(i, j) *c(j)
600 continue
c(i)=c(i)/a(i,i)
500 continue
400 continue
return
end
subroutine valmu(aste,b,betak,co,d,es,epsy,fc,fy,
* phiu,sigmal,sigma2,vm,vmul,vmy,phiy)
implicit double precision (a-h,o-z)
c**************************************************************
C NEUTRAL AXIS


20
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


50
MOMENT DIAGRAM
Mj Moment at node j
My Yielding moment
CURVATURE DIAGRAM
A
Mj
A
0U- Ultimate curvature
- Yielding curvature
- Curvature at node j
Figure 3.5. Curvature integration.


202
720
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100) ,
* no2(100)
common /pari/ iqh,iqg,nel,ntot,iqgn
common /sysr/ betmin,epsilo/elerel(100),invmec(lOO)
common /equal/ vah(lOO),vmu(100)
dimension theta(200,100),p(100),rb(200,100),
* become(500),lc(300),thesum(100),temp(100,100),
* locmec(lOO),thesul(100),dispsu(100),ni(100),
* nucomb(lOO),lct(300)
c***************************************************************
c form fundamental mechanisms
Q***************************************************************
cal1 mecsys(nel,iqh,cl,cos1,cos2,lm,numec,rb,theta)
ndof=iqh
do 20 j=l,ndof
P(j)=r(j)
20 continue
o***************************************************************
c ordering theta and r matrices
0***************************************************************
do 710 k=l,numec-l
jflag=0
do 720 i=l,ndof
if(abs(rb (i, k)).gt.0.0)j flag=i
continue
if (jflag.eq.O)then
do 730 l=k+l,numec
do 740 li=l,ndof
if(abs(rb(li,l)).gt.0.0)then
do 750 lj=l,ndof
temp(1j,1)=rb(1j,1)
rb(lj ,l)=rb(lj, jflag)
rb(lj,jflag)=temp(lj,1)
continue
do 760 lj=l,2*nel
temp(1j,1)=theta(1j,1)
theta(lj,1)=theta(lj,jflag)
theta(lj,jflag)=temp(lj,1)
continue
go to 733
endif
continue
continue
continue
endif
710 continue
0**************************************************************
c normalizing theta and r vectors
0**************************************************************
do 810 i=l,numec
do 820 j=l,2*nel
if(abs(theta(j,i)).ne.1.0.and.
* theta(j,i).ne.0.0)then
fact=abs(1./theta(j,i))
do 830 jj=l,2*nel
750
760
740
733
730


100
The minimization problem stated previously is highly
nonlinear. Consequently the adoption of the optimization
strategy was crucial and its characteristics played a big
role in the decision process. The Augmented Lagrangian
Multiplier method, or Augmented Lagrangian formulation, was
the first choice. It allows for an adaptation of the search
technique to the shape of the design surface since the dual
variables, or lagrangian multipliers, are updated at the end
of each minimization cycle in function of the constraints
violations.
The constrained problem is thus transformed into an
unconstrained function using the Augmented Lagrangian
formulation with the addition of dual variables u and y and
becomes
MK/U/Y) = f(x) + u^.h(x) + P.ht(x) .h(x) +
vt.g*(x) + P.g"1-* (x) .3* (x)
where
g*(x) = max {g(x) -v/(2P)};
P penalty factor.
Pseudo-objective function L is minimized with fixed
values of u and y and these are updated using the following
rules (72)
uk+l = uk + 2Ph(x)
vk+1 = vk + 2Pg*(x)


109
followed this work (74-76) but the general formulation of
the problem is
Minimize f(x)
subject to hi(x) = 0 i=l,...,m
x-*- < x < xu, x of Rn, m < n
Inequality constraints are handled as pseudo equality
constraints with the addition of slack variables. This
increase of the size of the variable set is balanced by the
implicit variable elimination generated by the following
relation between changes of design variables
dxb = -J-1 c dxnb
where
xb vector of basic variables;
xnb vector of nonbasic variables;
jl columns of jacobian matrix of equality
constraints corresponding to basic variables
[dh/dxb];
C other columns of jacobian matrix corresponding
to the nonbasic variables [dh/dxnb].
Nonbasic variables are thus calculated as a function of
the basic variables and eliminated from the gradient
calculation. The gradient is calculated whenever a feasible
point is obtained and a line search along that direction


133
Table 7.6. Compared frame: initial steel area reinforcement.
Section
cj
Mu (3c. ft)
I.8M1 + 1.5M
cj Mu
(K.ft)
Steel
Area (in2)
Comparison
with
elements
1
1.0
114
114
2
1.0
61
61
1
3
0.7
55
39
or
10
4
0.9
136
122
2.6
5
0.7
55
39
2
6
0.9
136
122
2.6
or
9
7
1.0
196
196
3.5
2 or 3
8
0.7
25
18
or
8 or 9
9
0.7
33
23
3
10
1.0
203
203
3.7
or
8
11
0.7
44
31
4
12
0.7
51
36
0.7
or
7
13
0.7
11
8
14
1.0
173
173
3.1
5
15
1.0
169
169
or
6
16
0.7
8
6
Legend:
indicates
sections
separation of
in the original
groups
study
of element
comparable with
element sections in present research frame;
cj redistribution coefficient;
Mu ultimate section moment;
Mi live load moment;
Md dead load moment.


159
ck36=b
ck63=C
ck66=d
ck32=(a+b)/cl
ck35=-ck32
ck62=(c+d)/cl
ck65=-ck62
C***************************************************************
C ROTATED MATRIX
C***************************************************************
C2=COSl*COSl
s2=cos2*cos2
cs=cosl*cos2
ck(l,1)=ckll*c2+ck22*s2
ck(l,2)=ckll*cs-ck22*cs
ck(1,3)=-ck2 3 *cos2
ck(l,4)=-ckll*c2-ck22*s2
ck(l,5)=-ckll*cs+ck22*cs
ck(l,6)=-ck26*cos2
ck(2,1)=ckll*cs-ck22*cs
ck(2,2)=ckll*s2+ck22*c2
ck(2,3)=ck23*cosl
ck(2,4)=-ckll*cs+ck22*cs
ck(2,5)=-ckll*s2-ck22*c2
ck(2,6)=ck26*cosl
ck(3,1)=-ck32*cos2
ck(3,2)=ck32*cosl
ck(3,3)=ck33
ck(3,4)=-ck(3,1)
ck(3,5)=-ck(3,2)
ck(3,6)=ck36
ck(4,l)=-ck(l,1)
ck(4,2)=-ckll*cs+ck22*cs
ck(4,3)=ck23*cos2
ck(4,4)=ckll*c2+ck22*s2
ck(4,5)=ck(2,1)
ck(4,6)=ck26*cos2
ck(5,l)=ck(2,4)
ck(5,2)=ck(2,5)
ck(5,3)=-ck(2,3)
ck(5,4)=-ck(2,4)
ck(5,5)=-ck(2,5)
ck(5,6)=-ck26*cosl
ck(6,1)=-ck62*cos2
ck(6,2)=ck62 *cosl
ck(6,3)=ck36
ck(6,4)=-ck(6,l)
ck(6,5)=-ck(6,2)
ck(6,6)=ck66
return
end


130
Table 7.5. Debug frame: element
Element
1
2
3
4
Yielding
(lb.in)
3.43e4
2.00e4
14.9e4
22.5e4
Element
1
2
3
4
Node i
(lb.in)
-4.15e4
1.21e4
-3.05e4
21.5e4
moments.
Ultimate
(lb.in)
6.27e4
4.28e4
20.3e4
28.5e4
Node j
(lb, in)
-5.46e4
3.27e4
2 0.3e4
26.2e4


177
if (lpti.gt.lpt-1)then
do 524 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
524 continue
lp=lp+l
let(lp)=lctlp
become(lpt)=becote
else
do 510 jkj=lpti,lpt-1
if(become(jkj).gt.becote)then
iflag=-l
do 511 kjk=jkj, lpt-1
itemp=lpt-l+jkj-kjk
become(itemp+1)=become(itemp)
511 continue
become(jkj)=becote
becote=become(lpt)
c*************************************************************
c moving lc array
c*************************************************************
movini=nic+nucomb(nucome)*nucome+
* (jkj-lpti)*(nucome+1)
movfin=(lpt-lpti)*(nucome+1)+nic+
* nucomb(nucome)*nucome-l
do 512 lmn=movini,movfin
lcou=movini+movfin-lmn
nucl=nucome+l
let(lcou+nucl)=lct(lcou)
512 continue
do 513 n=movini,movini+nucome-l
lptaa=n-movini+l
let(n)=lc(j+lptaa-1)
513 continue
let(movini+nucome)=lctlp
endif
510 continue
if(iflag.eq.0)then
do 124 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
124 continue
lp=lp+l
let(lp)=lctlp
become(lpt)=becote
else
lp=lp+nucome+l
endif
endif
nucomb(nucome+1)=nucomb(nucome+1)+1
lnumbe=lnumbe+l
100 continue
do 6991 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
6991 continue


61
r
A
z = r-s = 0
0(z>0)
SAFE
A
A
A
/fif
Am
A
A
A
D'(z<0)
UNSAFE
Safe and Unsafe Design Regions
Figure 4.1. Design safety region.


128
verified that the equilibrium of the moments at the nodes
and the compliance with the moment rotation diagrams were
satisfied. While this solution converged in some cases, it
did not in others. There seemed to be no general pattern to
the problem.
The next step was to test the formulation using the
secant stiffness spring values obtained from the element
moment rotation diagram. The results of one of these
problems are presented in Table 7.4. In this case, a
situation similar to the Hooke and Jeeves minimization was
encountered. The node equilibrium and the element moment
rotation diagram were not in accordance with the final
values. To illustrate these discrepancies of the final
results, the values of the yielding moments, ultimate
moments, and moments at the nodes recovered using the
condensed stiffness matrix are shown in Table 7.5.
To improve the performance of the optimization with a
nonlinear material behavior, the use of better estimates of
starting design points was tried. For that purpose the
frame was optimized in the linear version having the
ultimate moment set as the yielding moment, i.e, no element
was allowed to yield. These solutions of the linear
behavior would theoretically provide the best starting
points. The optimization problem was thus transformed to a
two stage process: a linear solution with the elements close
to the yielding situation followed by a nonlinear
optimization having as starting values the results of the


143
be additional element degrees of freedom in failure
mechanism sets. At the element level, the element failure
constraint would be replaced by a set of constraints
concerning also the axial failure and the interaction of
flexural and axial forces. Shear failure, important in
reinforced concrete elements, could also be added in a
similar fashion.
In the problem involving nonlinear material behavior
some alterations could provide a better performance in the
nonlinear optimization. These include the use of a mixed
approach of the integrated and the cycling formulation,
similar to that used in the Hooke and Jeeves version. A
possible improvement is the inclusion of an intermediate
stage where the solution for the exact displacements would
be calculated whenever the absolute violation of the
equality constraints exceeds an upper limit. This mixed
approach could improve the efficiency of this approach since
good displacements are essential for the definition of the
correct global and element nonlinear behavior.
Another possible improvement is the use of a different
model for the nonlinear reinforced concrete element. The
substitution of the one-component model by a model of an
element partitioned in several discrete elements. These
discrete elements, each with linear stiffness
characteristics, defined by the global element nonlinear
properties, would provide better accuracy for the element
behavior. This solution has the disadvantage of increasing


63
A graphic representation of these functions is
presented in Figure 4.2. Introducing the standardized
variable u and the reliability index as
u = (Z 2m) / az
B = zm / oz = (rm sm) / (ct^p + a2g)^
then the probability of failure may be expressed as
Pf = Fu(-z"> / oz) = Fu(-fi)
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.


72
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 & of each element can be calculated for
any given external moment, section and material properties.
Denoting the basic variables fc as xi, and Me as X2, and
eliminating the other parameters involved in the equation,
the flexural failure function takes the form
z = ax Xi + a2 x2 + b
where
ax = a n b (kd)2;
a2 = -1;
b = As fcs d'- As fy d.
Standardizing
replacement of the
the variables Xx and x2 leads to a
basic variables
ux = (xi m)/ai
U2 = (x2 M-2)/2


205
C***************************************** *********************
c reliability of combined mechanisms
c (internal work)
c**************************************************************
do 200 j=nic,nif,nucome
icontr=0
if(nucome.gt.1)then
if(become(niccon).gt.100.)icontr=l
niccon=niccon+l
if (icontr.eq.1)go to 5564
endif
do 300 l=l,nucome
lj=lc(j+l-l)
if(lj.It.-0.1)then
kmu=-l
lj=abs(1j)
endif
do 400 kk=l,nel
kkj=2*kk-l
thesum(kk)=thesum(kk)+theta(lj,kkj)*kmu
thesul(kk)=thesul(kk)+theta(lj,kkj+l)*kmu
400 continue
kmu=l
300 continue
c*************************************************** ***********
c reliability of combined mechanisms
c (external work)
c**************************************************************
do 372 l=l,nucome
lcl=lc(j+l-i)
if(lcl.lt.-0.1)then
kmu=-l
lcl=abs(lcl)
endif
do 472 kk=l,ndof
if(abs(p(kk)).It.0.001)go to 472
dispsu(kk)=dispsu(kk)+rb(lcl,kk)*kmu
472 continue
kmu=l
372 continue
5564 continue
c******************************************************* ******
c combination with fundamental mechanisms
c (internal work)
c***************************************** ********************
do 100 k=l,numec
vmeanr=0.0
vmeanl=0.0
stdevr=0.0
stdevl=0.0
vmanrm=0.0
vmanlm=0.0
stdvrm=0.0
stdvlm=0.0
do 499 lll=j,j+nucome-1
if(abs(lc(lll)).ge.k)go to 100


45
SECTION CHARACTERISTICS
Geometry
Strain
Diagram
Forces
Figure 3.3. Reinforced concrete section.


program optim
call datain
call grg
call outres
stop
end
subroutine gcomp(g,x)
implicit real*8 (a-h,o-z)
dimension g(l), x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
call equcon(x)
call inecon(x)
call valobf(x,vof)
do 100 i=l,iqh
g(i)=vah(i)
100 continue
c***************************************************************
C ELEMENT RELIABILITY LIMITS
c***************************************************************
do 200 i=iqh+l,iqh+nel
g(i)=-vag(i-iqh)
200 continue
c*********************************************** ****** *********
C REINFORCEMENT LIMITS
c****************************************************** *********
do 300 i=iqh+nel+l,iqh+2*nel
kj=i-iqh-nel
g(i)=1000.*x(3*kj)/(x(3*kj-2)*x(3*kj-l))
300 continue
g(iqh+2*nel+l)=vof
return
end
subroutine princi
implicit double precision (a-h,o-z)
196


173
do 25 j=l,numec
do 91 i=nel
temp(i,j)=theta(i,j)
91 continue
25 continue
do 56 i=l,numec
do 55 j=l,2*nel
theta(i,j)=temp(j,i)
55 continue
56 continue
do 28 j=l,numec
do 27 i=l,ndof
temp(i,j)=rb(i,j)
27 continue
28 continue
do 66 i=l,numec
do 65 j=l,ndof
rb(i,j)=temp(j,i)
65 continue
66 continue
c**************************************************************
c reliability of fundamental mechanisms
c**************************************************************
do 102 i=l,numec
vmeanr=0.0
stdevr=0.0
*
202
*
302
do 202 k=l,nel
j=2*k-l
theji=abs(theta(i,j))
thejil=abs(theta(i,j+1))
if(theji.It.0.0001.and.thejil.It.
0.0001)goto 202
term2=(theji+thejil)*cvmu(k)*vmu(k)
stdevr=stdevr+term2 *term2
vmeanr=term2/cvmu(k)+vmeanr
continue
vmeanl=0.0
stdevl=0.0
do 302 k=l,ndof
if(abs(p(k)),lt.0.001)go to 302
term=p(k)*rb(i,k)
vmeanl=vmeanl+term
stdevl=stdevl+term*term*(cvload(k)
*vload(k))
continue
vmean=vmeanr-vmeanl
stdev=stdevr+stdevl
become(i)=vmean/sqrt(stdev)
lnumbe=lnumbe+l
102 continue
c**************************************************************
c reliability of combined mechanisms
c (internal work)
c**************************************************************
do 50 l=l,numec
lc(l)=l


229
Coefficient of variation of external global loads.
Line 30 to line 33
Coefficient of variation of element ultimate moment.
Line 34
Value of interval gap in the Beta unzipping method.


77
where
Gj_ 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.


165
phiu=4.*phiy
end if
return
end
subroutine assemb (e,iqh,n,ntot,x,cl,cosl,cos2,
* lm,glk)
implicit double precision (a-h,o-z)
dimension x(ntot),cl(n),cosl(n),cos2(n)
dimension lm(6,n),glk(iqh,iqh)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
do 100 k = l,iqh
do 150 j = l,iqh
150 glk (j ,k) =0.0
100 continue
c**********************************************************
c global stiffness evaluation
c**********************************************************
do 700 j = l,n
call glosti (e,j,n,ntot,cl,x,cosl,cos2)
do 300 1 = 1,6
k = lm (l,j)
if (k.eq.0) go to 300
do 200 11 = 1,6
m=lm(ll,j) -k+1
if (m.le.0) go to 200
glk(k,m)=ck(l,ll)+glk(k,m)
200 continue
300 continue
700 continue
return
end
subroutine constr (iqg,d)
implicit double precision (a-h,o-z)
dimension d(iqg)
c**********************************************************
c displacement constraints
c**********************************************************
do 201 k = l,iqg
read ( 9,* ) d(k)


115
each element was calculated in the subroutine VALMU, the
maximum element moment was evaluated in subroutine ELEY and
the element reliability index is determined.
Subroutine VALMU calculates the ultimate moment and
ultimate curvature for each configuration of the element
cross section using the assumptions and correspondent
formulas presented in Chapter 3. The ultimate curvature is
limited to a maximum of four times the yielding curvature
due to serviceability reasons. The curvatures of sections
beyond this point are so high that the corresponding
deformations will transform any regular frame to an
unserviceable structure. Furthermore for curvatures above
these values the strain hardening of high strength steel
would have to be considered. This upper bound for the
ultimate curvature is also a common value used in design of
structures with dynamic loads.
Recovery of the element moments at both ends is
performed in the subroutine ELEY as a function of the global
displacements and the current spring characteristics of the
element. The element stiffness matrix considered in this
evaluation results from the condensation of the element
elastic stiffness matrix and the spring stiffness. During
the optimization process the stiffness characteristics,
including the secant spring stiffness, are those defined in
the previous iteration.
All fundamental mechanisms of the initial structure are
determined in subroutine MECSYS at the beginning of the


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233
(42) Madsen, H. O., Krenk, S., and Lind, N. C., Methods of
Structural Safety. Prentice-Hall, Englewood Cliffs, New
Jersey, 1986.
(43) Regan, P. E., and Yu, P. E., Limit State Design of
Structural Concrete. John Wiley & Sons, New York, 1973.
(44) Leporati, E., The Assessment of Structural Safety.
Research Studies Press, Forest Grove, Oregon, 1977.
(45) Pahoheimo, E. and Hannus, M., Structural Design Based
on Weighted Fractiles, Journal of Structural Division. ASCE,
Vol. 100, ST7, 1974, pg. 1367-1378.
(46) Requlamento de Estruturas de Beto Armado e Pr-
Esforeado. Imprensa Nacional Casa da Moeda, Lisboa,
Portugal, 1983.
(47) Thoft-Christensen, P., and Baker, M. J.,Structural
Reliability Theory and its Applications. Springer-Verlag,
Heidelberg, West Germany, 1982.
(48) Neville, A. M., Properties of Concrete. Pitman, Bath,
United Kingdom, 1983.
(49) Recommended Practice for Evaluation of Strenghth Test
Results for Concrete, ACI 214-71, American Concrete
Institute, Detroit, 1976.
(50) Mindess, S. and Young, J. F., Concrete. Prentice-
Hall, Englewood Cliffs, New Jersey, 1981.
(51) Building Code Requirements for Reinforced Concrete.
(ACI 318-83), American Concrete Institute, Detroit, 1976.
(52) Hart, G., Uncertainty Analysis, Loads and Safety in
Structural Engineering. Prentice-Hall, Englewood Cliffs, New
Jersey, 1982.
(53) Chou, Karen C., and Corotis, Ross B., Nonlinear
Response to Sustained Load Processes, Journal of Structural
Engineering. ASCE, Vol. Ill, No. 1, 1985, pg. 142-155.
(54) Probabilistic Mechanics and Structural Reliability.
4th ASCE Specialty Conference, ASCE, New York, 1984.
(55) Blockley, D. I., The Nature of Structural Design and
Safety. John Wiley and Sons, Chichester, United Kingdom,
1980.
(56) Feng, Y. S., and Moses, F., A Method of Structural
Optimization Based on Structural System Reliability, Journal
of Structural Mechanics. Vol. 14, No. 4, 1986, pg. 437-453.


110
tends to provide a better design point while maintaining
feasibility. The values of the nonbasic variables are
consequently evaluated and the process is restarted. If any
of the basic variables is at any bound, a search is performed
on the set of nonbasic variables to find a suitable
replacement. The method creates basically a succession of
feasible solutions x, x1, ..., xP, each one corresponding
to an improvement of the objective function from the
previous design point. The iteration is terminated whenever
the convergence criteria is satisfied or the maximum
prescribed number of iterations is exceeded.
The essence of this optimization technique seemed
adequate for the integrated optimization with nonlinear
constraints and nonlinear material behavior with a
considerable number of equality constraints. The number of
slack variables is not large, and as long as the initial
point is feasible, convergence should be quicker than in the
previous version.
To illustrate and assess the performance of the
generalized reduced gradient method, an example of a
cantilever beam with linear material behavior, submitted to
displacement and stress constraints, was solved. This
example is presented in Appendix B together with a flowchart
of the general algorithm. These preliminary results were
very promising and the following step was to extend the
method to the formulation previously tested with the
augmented lagrangian function.


67
the hypersphere having radius R*, 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
R = min (2 u2i)h
In a great number of cases the safety boundary domain
is linear, and one can write an expression for z as follows:
z = g (xx, x2,...., xn) = b + 2 aixi
Then, R can be immediately determined as follows
g (U]_, u2,..., un) = b + 2 aixmi + 2 a^ax^ui = 0
and the distance of this hyperplane to the origin is
R = 2 (ai.xrai + b) / (2 a2icrx2j: )%
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


114
results of this type of problems, and to modify the output
format of the optimization information.
The particular structure of the optimization package
created some conceptual alterations in the group of
subroutines that evaluate the constraints and control the
variables bounds. For instance, the changes in the design
points are made simultaneously for all design variables.
Therefore the limits of the areas of reinforcement had to be
imposed as variable bounds instead of being controlled by a
specific subroutine.
On the other hand the limits imposed on the
displacements could be considered as upper and lower bounds
of the correspondent design variables, and consequently, the
total number of inequality constraints was substantially
reduced. The only scaling introduced in the problem was the
division of the equality constraints by the order of the
magnitude of the maximum external force.
Reliability
The only statistical parameters considered for the
probability of failure evaluation were the strength of the
concrete and the external loads. For that reason the
failure function is a linear function of these two basic
variables and the reliability index is calculated using the
formula referred in Chapter 4. Whenever there was a change
of the design variables the value of the ultimate moment for


5
Magne1's Diagram
Optimum Pair P-e
P Initial prestressing torce;
e Eccentricity of cable;
e*- maximum cable eccentricity;
a).b) minimum 1/P;
c).d) maximum 1/P.
Figure 1.1 Implicit optimization.


185
Example: Debug Frame
Input File: DATA
Debug frame
4,5
1,2
2.3
3.4
4.5
1,2,3,0,0
0,0,0,0,100
0,0,0,50,100
0,0,0,100,100
1,2,3,100,0
2,1,5000
3,2,-5000
0,0,0
3000.40000.1
29000000,0.004
1
1
1
1
1
1
1
1
1
1
1
1
1
5,0.0001,0.01
5.100.1
0.9,1.1
0.00001
3,12
3,12
3,12
3,12
2
4
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15


16
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 equalities 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).


174
lct(l)=l
50 continue
lp=numec
ltemp=numec
ni(1)=1
numax=6
nucome=l
nucomb(1)=numec
lpt=numec
lpti=numec+l
111 continue
c***************************************************************
c loop over mechanisms in location vector
do 699 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
699 continue
do 698 klp=l,ndof
dispsu(kip)=0.0
698 continue
kmu=l
nic=ni(nucome)
nif=nic+nucomb(nucome)*nucome-l
nucomb(nucome+1)=0
c* ************************************************************ *
c define acceptable interval
c**************************************************************
if(nucome.gt.1)then
nifbet=lpti-l
nicbet=lpti-nucomb(nucome)
do 5544 ia=nicbet+l,nifbet
betaal=become(nicbet)+epsilo
if(become(ia).gt.betaal)then
do 5545 ib=ia,nifbet
become(ib)=1000.
5545
continue
go to 5541
endif
5544
continue
5541
continue
niccon=nicbet
endif
c**************************************************************
c reliability of combined mechanisms
c (internal work)
c******************************************************** ******
do 200 j=nic,nif,nucome
icontr=0
if(nucome.gt.1)then
if(become(niccon).gt.100.)icontr=l
niccon=niccon+l
if (icontr.eg.l)go to 5564
endif
do 300 1=1,nucome
lj=lc(j+l-l)


66
Ul 1-- O
02
Figure 4.3. Safety checks.


62
where
z = r s
r resistance function;
s load function.
The domain D (z>0) is the safe domain and D'(z<0) 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 s as the mean values, and ctr 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 = (2R + o2s)^
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<0} = Fz(0)


211
do 500 i=l,n6
do 510 k=l,iqh
am(i,k)=0.0
510 continue
500 continue
do 52 0 i=l, 6
do 530 k=l,n
iflag=lm(i,k)
if (iflag.gt.0) am(6*(k-l)+i,iflag)=1.0
530 continue
520 continue
c**************************************************************
c Rotation of basic compatibility matrix
c**************************************************************
mm=80
call multi(q,am,qa,n6,iqh,n6,mm)
c**************************************************************
c Expansion of QA matrix
c**************************************************************
do 585 i=l,n
i6=6*i
i3=i6-3
do 595 j=l,iqh
qa(i3,j)=0.0
qa(i6,j)=0.0
595 continue
i2=2*i
qa(i3,iqh+i2-l)=l. 0
qa(i6,iqh+i2)=1.0
585 continue
c**************************************************************
c Matrix A = C QA (transformed)
c**************************************************************
m=3*n
nt=iqh+2*n
call multiCcmjqa^mintjnejmm)
c**************************************************************
c Solution for virtual displacements
c**************************************************************
do 150 k=l,nt
do 175 1=1,nt
b(k,l)=0.0
175 continue
150 continue
do 160 k=l,nt
b(k,k)=1.0
160 continue
do 200 i=l,m
amax=0.0
iflag=0
do 250 j=i,nt
if (abs(a(i,j)).gt.amax)then
iflag=j
amax=abs(a(i,j))
endif
continue
250


228
User's Manual
Generalized Reduced Gradient Method
Example: Debug Frame
Input File: DATA1
Line 1
Number of elements, number of nodes.
Line 2 to line 5
Node i, node j of element 1 through 4.
Line 6 to line 10
Boundary conditions of displacement in the horizontal
direction, vertical direction, in-plane rotation,
horizontal coordinate, vertical coordinate.
Line 11 and line 12
Node where force is applied, direction of load and
magnitude of load.
Line 13
Termination of force information.
Line 14
Flexural strength of concrete, yielding stress of steel
and reinforcement cover.
Line 15
Steel modulus of elasticity and concrete ultimate
strain.
Line 16
Minimum element and system reliability index.
Line 17 to line 20
Coefficient of variation of flexural concrete strength.
Line 21 to line 29


201
ck65=-ck62
c***************************************************************
c ROTATED MATRIX
c***************************************************************
csl=cosl(kel)
cs2=cos2(kel)
c2=csl*csl
S2=CS2*CS2
cs=csl*cs2
ck(l,1)=ckll*c2+ck22*s2
ck(l,2)=ckll*cs-ck22*cs
ck(1,3)=-ck2 3 *cs2
ck(l,4)=-ckll*c2-ck22*s2
ck(1,5)=-ckll*cs+ck2 2 *cs
ck(l,6)=-ck26*cs2
ck(2,1)=ckll*cs-ck22*cs
ck(2,2)=ckll*s2+ck22*c2
ck(2,3)=ck23*csl
ck(2,4)=-ckll*cs+ck2 2 *cs
ck(2,5)=-ckll*s2-ck22*c2
ck(2,6)=ck26*csl
ck(3,l)=-ck32*cs2
ck(3,2)=ck32*csl
Ck(3,3)=ck33
ck(3,4)=-ck(3,1)
ck(3,5)=-ck(3,2)
ck(3,6)=ck36
ck(4,l)=-ck(l,1)
ck(4,2)=-ckll*cs+ck22*cs
ck(4,3)=ck23*cs2
ck(4,4)=ckll*c2+ck22*s2
ck(4,5)=ck(2,1)
ck(4,6)=ck2 6*cs2
ck(5,1)=ck(2,4)
ck(5,2)=ck(2,5)
ck(5,3)=-ck(2,3)
ck(5,4)=-ck(2,4)
ck(5,5)=-ck(2,5)
ck(5,6)=-ck26*csl
ck(6,1)=-ck62*cs2
ck(6,2)=ck62*csl
ck(6,3)=ck36
ck(6,4)=-ck(6,l)
ck(6,5)=-ck(6,2)
ck(6,6)=ck66
return
end
subroutine sysrel(jflag)
implicit double precision (a-h,o-z)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),


APPENDIX B
GENERALIZED REDUCED GRADIENT EXAMPLE


171
c element reliability
c************************************************** *********
read(9,*)relind
Q************************************************************
c number of elementary mechanisms
c***************************************************** *******
read(9,*)numec
Q************************************************************
c generation of global stiffness
Q************************************************************
call assemb(ec,iqh,n,ntot,x,cl,cosl,cos2,lm,glk)
c*************************************************************
c initial displacements
c*************************************************************
call symsol(glk,r,x(n21),iqh,mband)
return
end
subroutine valobf (n,ntot,vof,x,cl)
implicit double precision (a-h,o-z)
dimension x(ntot), cl(n)
vof = 0.0
do 100 k = l,n
base = x(3*k-2)
height = x(3*k-l)
steel = x(3*k)
area = base height
vof=vof+(area+steel*10)*cl(k)
100 continue
return
end
subroutine sysrel(nel,numec,ndof,theta,r,vmu,cvmu,p,
* cvload,jflag)
implicit double precision (a-h,o-z)
dimension theta(200,100),p(100),rb(200,100),
* become(500),lc(300),thesum(100),temp(100,100),
* locmec(lOO),thesul(100),dispsu(100),ni(100),
* nucomb(lOO),let(300),elerel(100),cvmu(100),
* r(100),vmu(100),cvload(100)
c***************************************************************
c form fundamental mechanisms


197
*
*
common /vgeom/ cl(100),cos(100),cos2(100),lm(6,100),
cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
open ( 9,file='data',form='formatted' )
rewind 9
c****************************************************************
c # ELEMENTS AND # JOINTS
c***************************************************************
read (9,*)nel,nj
c***************************************************************
C NODES PER ELEMENT
C***************************************************************
do 100 i=l,nel
read(9,*)nol(i),no2(i)
100 continue
c***************************************************************
C JM MATRIX
c***************************************************************
do 200 kk=l,nj
jm(l,kk)=l
jm(2,kk)=2
jm(3,kk)=3
200 continue
c***************************************************************
C SUPPORT CONDITIONS AND COORDINATES
c***************************************************************
do 300 j=l,nj
read(9,*)jdir(1),jdir(2),jdir(3),xc(j),yc(j)
do 350 i =1,3
if (jdir(i).gt.0) then
jm (i,j) = 0
endif
350 continue
300 continue
c***************************************************************
c GLOBAL DEGREES OF FREEDOM
C***************************************************************
iqh=0
do 510 j=l,nj
do 500 1=1,3
if (jm(l,j).ne.0) then
iqh=iqh+l
jm (1,j)=iqh
endif
500 continue
510 continue
iqgn=iqh+nel
iqg=iqh
ntot=iqgn+2 *nel
c***************************************************************
C FILLING LM MATRIX
c***************************************************************


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.
ix


65
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 (Xi, 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(xi, 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) / CJXi
Measuring, in this space, the minimum distance 6 of the
transformed surface g (U]_, U2,...., un) from the origin of
the axes. A design is regarded reliable if 6 > fi*, where B*
is prescribed by an appropriate code provision.
In geometrical terms, the hypersphere having radius 6*
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


151
subroutine comcon(aste,fy,es,d,b,co,epsy,ecm,fc,vmy,phiy)
implicit double precision (a-h,o-z)
kll=0
node=0
epso=0.002
C
C EXC CONCRETE STRAIN
C EPCS COMPRESSIVE STEEL STRAIN
C EPSY YIELD STRAIN
C
C FIRST VALUE FOR A
C****************************************************************
al=d/2.
exc=al*epsy/(d-al)
epcs=exc*(al-co)/al
t=fy*aste
cs=epcs*es
eces=exc/epso
alpha=eces-eces*eces/3.
cc=alpha*fc*b*al
resl=cc+cs-t
C*********************************************************** ******
C SECOND VALUE FOR A
C*****************************************************************
a2=0.25*d
exc=a2*epsy/(d-a2)
epcs=exc*(a2-co)/a2
cc=fc*a2*alpha*b
eces=exc/epso
cs=epcs*es
res2=cc+cs-t
C* ************************************************************ **
C NEWTON ITERATION
C***************************************************************
100 a=a2-res2*(a2-al)/(res2-resl)
exc=a*epsy/(d-a)
if(exc.gt.epso) go to 200
c******************************************************** ********
C PARABOLIC SHAPE
C****************************************************** **********
epcs=exc*(a-co)/a
eces=exc/epso
alpha=eces-eces*eces/3.
cc=fc*alpha*b*a
cs=epcs*es
res=cc+cs-t
control=0.0001*b*d*fc
if (abs(res).gt.control) then
al=a2
a2=a


186
in in in in in
H rl rl rl iI ID in
O O O O O CM rH


53
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.


234
(57) Frangopol, D. M., Sensitivity of Reliability-Based
Optimum Design, Journal of Structural Engineering. ASCE,
Vol. Ill, NO. 8, 1985, pg. 1703-1721.
(58) Moses, F., Structural System Reliability and
Optimizatiom, Computers and Structures. Vol. 7, 1977, pg.
283-290.
(59) Sexsmith, R., and Mau, S.-T., Reliability Design with
Expected Cost Optimization, Safety and Reliability of Metal
Structures, ASCE, New York, 1977, pg. 427-443.
(60) Yao, J. T. P., Safety and Reliability of Existing
Structures, Pitman Publishing, London, 1985.
(61) Frangopol, D. M., Structural Optimization under
Conditions of Uncertainty, with Reference to Serviceability
and Ultimate Limit States, Structural Optimization
Developments. ASCE, New York, 1986, pg. 54-71.
(62) Rashedi, R., and Moses, F., Identification of Failure
Modes in System Reliability, Journal of Structural Division.
ASCE, Vol. 114, NO. 2, 1988, pg. 292-313.
(63) Bennett, R. M., and Ang, A. H-S., Investigation of
Methods for Structural System Reliability. Structural
Research Series No. 510, University of Illinois, Urbana,
Illinois, 1983.
(64) Spencer, B. F., and Elishakoff, I., Reliability of
Uncertain Linear and Nonlinear Systems, Journal of
Engineering Mechanics. ASCE, Vol. 114, No. 1, 1988, pg. 135-
148.
(65) Yuansheng, F., and Moses, F., Optimum Design,
Redundancy and Reliability of Structural Systems, Computers
and Structures. Vol. 24, No.24, 1986, pg. 239-251.
(66) Kam, T., Corotis, R. B., and Rossow, E. C.,
Reliability of Nonlinear Framed Structures, Journal of
Structural Engineering. ASCE, Vol. 109, No. 7, 1983, pg.
1585-1601.
(67) Austin, M. A., Pister, K. S., and Mahin, S. A.,
Probabilistic Limit States Design of Moment Resistant Frames
under Seismic Loading, Structural Optimization Developments.
ASCE, New York, 1986, pg. 1-16.
(68) Ma, H-F., and Ang, A. H-S., Reliability Analysis of
Redundant Ductile Structural Systems. Structural Research
Series No. 494, University of Illinois, Urbana, Illinois,
1981.
(69) Advances in Structural Reliability. Proceedings of
the Advanced Seminar on Structural Reliability, D. Reidel
Publishing Company, Dordrecht, West Germany, 1987.


LIST OF FIGURES
Figure Page
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 3 4
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.l. Integrated optimization example... 191
vii


104
Cross Section
Figure 6.1. Augmented lagrangian function plot.


152
resl=res2
res2=res
kll=kll+l
if(kll.gt.100)then
stop
end if
go to 100
end if
gama=l.-(8.*epso-3.*exc)/(12.*epso-4.*exc)
arm=d-gama*a
vmy=cc*arm+epcs*es*aste*(d-co)
phiy=epsy/(d-a)
return
C CONCRETE STRAIN > EPSO
C****************************************************************
200 if(exc.gt.0.004) exc=0.004
xl=epso*a/exc
ccl=fc*xl*2./3*b
gama=3.6*exc*exc-200.*exc*exc*exc-0.0000128
gama=gama/((exc-epso)*(7.2*exc-300*exc*exc-0.0132))-l.
alpha=exc-50.*exc*exc+100.*exc*epso-0.0022
alpha=alpha/(exc-epso)
cc2=alpha*fc*(a-xl)*b
epcs=exc*(d-co)/a
t=fy*aste
cs=epcs*es
cc=ccl+cc2
res=cc+cs-t
control=0.0001*b*d*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
kll=kll+l
if(kll.gt.100)then
stop
endif
go to 100
endif
arml=d-a+2./3.*xl
arm2=d-gama*(a-xl)
vmy=ccl*arml+epcs*es*(d-co)*aste+cc2*ann2
phiy=epsy/(d-a)
return
end
subroutine eley(ec,tinert,cl,vki,vkj,u2,u3,u5,u6,fo3,f06)
implicit double precision(a-h,o-z)
ei=ec*tinert
w=cl/(3.*ei)+1./vki


39
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


169
j = nol(ii)
k = no2(ii)
ell = xc(k) xc(j)
el2 = yc(k) yc(j)
cl(ii) = sqrt(ell*ell+el2*el2)
cosl(ii) = ell/cl(ii)
cos2(ii) = el2/cl(ii)
800 continue
c**********************************************************
c initialization of global forces
c**********************************************************
do 850 k=l,iqh
r(k) = 0.0
850 continue
c**********************************************************
c global forces
c**********************************************************
900 read (9 ,*) jnum,jdire, force
if(j num.ne.0)then
k=jm(jdire,jnum)
r(k)=force
go to 900
endif
return
end
subroutine 1im(x,n,ntot,xl1,xl2,k,r omin,romax)
implicit double precision (a-h,o-z)
dimension x(ntot)
n3=3*n
do 10 i=l,n3,3
ll=i
12=i+l
13=i+2
astma=romax*x(ll)*x(l2)
astm=romin*x(ll)*x(12)
if(k.eq.ll)then
if(x(k).It.xll)x(k)=xll
basemin=x(13)/(romax*x(12))
if(x(k).It.basemin)x(k)=basemin
basemax=x(13)/(romin*x(12))
if(x(k).gt.basemax)x(k)=basemax
i=n3
go to 10
endif
if(k.eq.12)then
if(x(k).It.xl2) x(k)=xl2
heightmin=x(l3)/(romax*x(ll))
if(x(k).lt.heightmin)x(k)=heightmin


191
Figure B.l. Integrated optimization example.


93
could be used for ductile structures with the formation of
all collapse mechanisms but the computational effort of the
reanalysis is too great. It is preferable instead to use
the fundamental mechanisms and their linear combinations.
A structure with an elasto-plastic behavior and a given
static load configuration has a certain number of
fundamental failure mechanisms that can be determined using
the Watwood's method. Since they are one degree of freedom
mechanisms the failure function z for mechanism i can be
evaluated as follows:
Zf = S |aij| Rj 2 b^fc Pfc
where
aj rotation at hinge j in mechanism i;
bik ~ displacement of load k in mechanism i;
Rj strength of element j;
Pjc load number k.
Total number of collapse mechanisms is generally too
high and a significant portion of these have a low
probability of failure. For this reason, the Beta unzipping
method is used as it only considers the most critical
failure modes. The value found is a lower bound for the
exact probability of failure, since some mechanisms are
discarded. Once the identification of the fundamental
mechanisms and respective reliability indices are obtained,
the next step is to choose the elementary mechanisms that


217
123
124
125
126
if(abs(crotl).gt.tetau)vki=vmuk/tetau
continue
endif
if(node2.eg.1) then
if(srot2.le.0.)then
vkj=vmy/tetay
go to 124
endif
vkj=(vksp*srot2+vmy)/abs(crot2)
if(vm.gt.vmuk)vkj=vmuk/tetau
if(abs(crot2).gt.tetau)vkj=vmuk/tetau
continue
endif
if(nodel.eq.l.and.node2.eg.1) then
if(srotl.le.0.)then
vki=vmy/tetay
go to 125
endif
vki=(vksp*srotl+vmy)/abs(crotl)
if(vm.gt.vmuk)vki=vmuk/tetau
if(abs(crotl).gt.tetau)vki=vmuk/tetau
continue
if(srot2.le.0.)then
vkj =vmy/tetay
go to 126
endif
vkj = (vksp*srot2+vmy)/abs(crot2)
if(vm.gt.vmuk)vkj=vmuk/tetau
if(abs(crot2).gt.tetau)vkj=vmuk/tetau
continue
endif
return
end
subroutine valmu(aste,b,kel,dd,vm,vmy,phiy,phiu,
* sigmal,sigma2)
implicit double precision (a-h,o-z)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100);jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv^c^p^c^s^cmjrelind/co^fy^psy
common /pari/ igh,igg,nel,ntot,iggn
common /inequa/ vag(100),beta(100),u(6),vahk(100),
* ck(6,6)
common /equal/ vah(100),vmu(100)
n=nel
c*************************************************************


8
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


137
5K
FLOOR
PLAN
J
u
u
L
1


C
1
n
n
:
h + ^ H
20ft 20ft 20ft
Figure 7.4. Building frame.


9
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


64
Probability Density Function
Cumulative Density Function
Figure 4.2. Probabilistic functions.


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.


103
the equality constraints were satisfied, the gradient method
was unable to progress to a better point. This happened
because any violation of the constraints created a high
increase of the augmented lagrangian function and the point
behaved as a local minimum. Otherwise, if the point did not
satisfy the equality constraints then the accuracy of the
derivatives obtained through forward difference, was not
good enough to converge to a better design point. As an
example of this abruptness the augmented lagrangian function
is represented as a function of displacements X2 and X3 of
the cantilever shown in Figure 6.1.
Several techniques were implemented to smooth the shape
of the augmented lagrangian to no avail. Scaling of the
variables, objective function and of the constraints were
performed. The displacements were scaled by the
multiplication of a constant regularizing the magnitude of
the set of variables. The scaling of the constraints and
objective function were already referred to as well as
another technique based on the evenness of the rate of
change of the constraints and objective function in terms of
the design variables (12).
These reasons justified the final adoption for
unconstrained minimization of the Hooke and Jeeves method.
This technique had provided acceptable results and
performances in the linear elastic formulation. The main
algorithm may be summarized as follows


221
program eley
implicit double precision(a-h,o-z)
open(1,file='ydata',form='formatted1)
rewind 1
read(1,*)ec,tinert,cl,vki,vkj,u2,u3,u5,u6
ei=ec*tinert
w=cl/(3.*ei)+1./vki
y=cl/(3.*ei)+l./vkj
z=-cl/(6.*ei)
det=w*y-z*z
a=y/det
b=-z/dt
c=b
d=w/det
fo3=(a+b)/cl*u2+a*u3-(a+b)/cl*u5+b*u6
f06=(c+d)/cl*u2+c*u3-(c+d)/cl*u5+d*u6
write(*,*)'fo3 = 1,fo3,1 fo6 = ',fo6
stop
end
program yiel
implicit double precision (a-h,o-z)
open(1,file='yielm',form='formatted')
rewind 1
read(l,*)b,d,aste,epsy,es,co,fy,fc,ecm,vm,sigmal,sigma2
ec=3122019
node=0
epso=0.002
c**************************************************************
c
C EXC CONCRETE STRAIN
C EPCS COMPRESSIVE STEEL STRAIN
c EPSY YIELD STRAIN
c
C FIRST VALUE FOR A
c***************************************************************
dd=x(3*kel-l)-co
al=dd/2.
exc=al*epsy/(dd-al)
epcs=exc*(al-co)/al
t=fy*aste
cs=epcs*es*aste
eces=exc/epso
alpha=eces-eces*eces/3.
cc=alpha*fc*b*al


do 162 k=l,n
if (lm(3,k).eq.i)then
lpo=2*k-l
theta(lpo,j)=l
endif
if(lm(6,k).eq.i)then
lpo=2*k
theta(lpo,j)=l
endif
continue
endif
continue
continue
return
end
subroutine multi(aa,bb,cc,1,m,n,k)
implicit double precision (a-h,o-z)
dimension aa(l,n),bb(n,m),cc(l,m)
do 10 i=l,l
do 20 j=l,m
d=0.0
do 30 kk=l,n
d=d+aa(i,kk)*bb(kk,j)
continue
cc(i,j)d
continue
continue
return
end
subroutine equcon(x)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
common /springs/ vksi(100),vksj(100)


220
exc=a*epsy/(dd-a)
if(exc.gt.epso) go to 200
C***************************************************************
c parabolic shape
c***************************************************************
epcs=exc*(a-co)/a
eces=exc/epso
alpha=eces-eces*eces/3.
cc=fc*alpha*b*a
cs=epcs*es*aste
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
go to 100
endif
gama=l.-(8.*epso-3.*exc)/(12.*epso-4.*exc)
arm=dd-gama*a
vmy=cc*arm+epcs*es*aste*(dd-co)
phiy=epsy/(dd-a)
return
c*************************************************** ************
c concrete strain > epso
c* ************************************************************* *
200 if(exc.gt.0.004) exc=0.004
xl=epso*a/exc
cclfc*xl*2./3.*b
gama=3.6*exc*exc-200.*exc*exc*exc-0.0000128
gama=gama/((exc-epso)*(7.2*exc-300*exc*exc-0.0132))-1.
alpha=exc-50.*exc*exc+100.*exc*epso-0.0022
alpha=alpha/(exc-epso)
cc2=alpha*fc*(a-xl)*b
epcs=exc*(dd-co)/a
t=fy*aste
cs=epcs*es*aste
cc=ccl+cc2
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
go to 100
endif
arml=dd-a+2./3.*xl
arm2=dd-gama*(a-xl)
vmy=ccl*arxnl+epcs*es* (dd-co) *aste+cc2*arm2
phiy=epsy/(dd-a)
return
end


160
subroutine mumy(b,h,aste,vn,co,epsy,ec,fc,fy,
* vki,vkj, elk,fo3,fo6,es,ecm,betak,sigmal,sigma2,
* tinert,kl,vmuk,vmy,ijflag)
implicit double precision (a-h,o-z)
nodel=0
node2=0
ijflag=0
d=h-co
C*************************************************************
C EVALUATION OF YIELDING MOMENT
C*************************************************************
call comcon(aste,fy,es,d,b,co,epsy,ecm,fc,vmy,phiy)
afo3=abs(fo3)
afo6=abs(fo6)
vki=10e30
vkj=10e30
vm=max(afo3,afo6)
C**********************************************************
C IDENTIFICATION OF HINGE NODE
C**********************************************************
if (afo3.gt.vmy) nodel=l
if (afo6.gt.vmy) node2=l
C*************************************************************
C ULTIMATE MOMENT AND RELIABILITY
C*************************************************************
call valmu(aste,b,betak,co,d,es,epsy,fc,fy,
* phiu,sigmal,sigma2,vm,vmuk,vmy,phiy)
C***************************************************************
C INTEGRATION OF CURVATURE
C***************************************************************
if(nodel.gt.0.or.node2.gt.0)then
if((fo3*fo6).gt.0) then
if(afo3.gt.afo6)then
zero=0.000001*afo3
if(abs(afo3-afo6).It.zero)then
vlp=clk
go to 145
endif
vlp=(afo3-vmy)/(afo3-afo6)*clk
endif
if(afo3.It.afo6)then
zero=0.000001*af06
if(abs(afo3-afo6).It.zero)then
vlp=clk
go to 145
endif
vlp=(afo6-vmy)/(afo6-afo3)*clk
endif
145 continue
tetay = phiy*clk*0.5
tetau = (phiu-phiy)*vlp*0.5+phiy*clk
endif
if((fo3*fo6).It.0) then
vlp=(vmuk-vmy)*clk/vmuk
tetay=clk*phiy*0.25
tetau=phiy*clk*0.25+(phiu-phiy)*vlp


13
Optimization of a Two span prestressed beam
XI
X6
XI to XB Section geometry
X7 to X9 Eccentricities of draped cable
X10 Prestressing force
Figure 1.4. System optimization.


162
c scaling objective function
c***************************************************************
cv = 5./vof
c***************************************************************
c hooke & jeeves
c***************************************************************
444 call hoojee (tvah,vlag,r,vah,vag,x,cl,cosl,cos2,lm,
* d,clah,clag,ch,eg,alp,xol,vahk,toll,vn,co,epsy,fy,ast,
* beta,cvmu,cvload,xll,xl2,romin,vmu,rph,grad,vahold,
* vjac,vinv,bfal)
iter=iter+l
write (8,1500)iter
1500 format ('end of loop =',i3)
Q***************************************************************
c control of maximum number of iterations
c***************************************************************
if (iter.gt.numcy) go to 99
rp2 = rp+rp
a****************************************************** **********
c updating lagrangian mult, equal.cons,
c****************************************************************
do 100 k = l,iqh
clah(k)=clah(k)+rp2*vah(k)*ch(k)
100 continue
c*************************************************************
c updating lagrangian mult. ineq. cons,
c*************************************************************
do 200 k=l,iqgn
v=vag(k)*cg(k)
z=-clag(k)/rp2
psi=max(v,z)
clag(k)=clag(k)+rp2*psi
200 continue
c*************************************************************
c updating penalty factor
c*************************************************************
rp=ga*rp2
rph=ga*ga*rph
Q*************************************************************
c system reliability evaluation
c*************************************************************
jflag=0
call sysrel(n,numec,iqh,theta,rv,vmu,cvmu,r,
* cvload,jflag)
if(jflag.gt.0)go to 444
return
end
subroutine sol(n,a,b,c)


136
Building Frame
To evaluate the performance of the program for a common
practical design a typical rectangular building frame with
two spans and three stories high was defined. Lateral and
vertical loads were calculated using the Standard Building
Code. Definition of the frame geometry, horizontal loads,
vertical loads, material properties and floor plan are
presented in Figure 7.4.
Vertical loads were applied at the midspan of each
beam. Values were equivalent to the distributed loads along
the adjacent slabs since this formulation does not handle
loading along the element. The pattern chosen for the
distribution of the vertical loads aims to create maximum
moments in the elements. For this reason the loading
combination includes the wind loads.
The major frame was analyzed using the linear version
of the Generalized Reduced Gradient method and the results
are summarized in Table 7.8. Kuhn-Tucker conditions were
verified and the final dimensions of the cross sections
corresponded to the lower bounds. The exception to this
last conclusion happened whenever the steel reinforcement
attained the upper limit. Testing of the nonlinear
versions, both with the yielding and the secant spring
stiffness formulations, provided no acceptable results in a
similar manner to that observed when testing the compared
frame.


218
c neutral axis
c******************************************************* ******
x=47./60.*b*fc
y=0.004*es*aste-aste*fy
z=-0.004*es*co*aste
vkd=(-y+sqrt(y*y-4.*x*z))/(2.*x)
epcs=0.004*(vkd-co)/vkd
if(epcs.gt.epsy)then
epcs=epsy
endif
c***** *********************** *********'!(****** ******* ***********
c concrete force in region ab
c**************************************************************
alphal=2./3 ,
ccab=alphal*b*0.5*vkd*fc
c*************************************************************
c concrete force in region be
c*************************************************************
alpha2=0.9
ccbc=alpha2*b*0.5*vkd*fc
c******************************************************** ******
c distance of centroid to top in ab
c**************************************************************
gama1=0.875*vkd
c******************************************************* *******
c distance of centroid to top in be
**************************************************************
gama2=0.259255*vkd
Q**********************************************************
c coefficients for failure function
c************************************************************
al=(ccab*(dd-gamal)+ccbc*(dd-gama2))/fc
a2=-l.
c* ********************************************************* *
c cosine directors
c**********************************************************
tetal=al*sigmal*fc
teta2=a2*sigma2*vm
c****************************************************** ******
c independent term
c******************************************** ***************
fps=0.004*es*(vkd-co)/vkd
bi=aste*fps*(dd-co)
c***********************************************************
c reliability index
c***********************************************************
beta(kel)=(al*fc+a2*vm+bi)/sqrt(tetal*tetal+teta2*teta2)
c************************************************************
c ultimate moment and rotation
c**************************************************************
vmu(kel)=al*fc+bi
phiu=0.004/vkd
vmuk=vmu(kel)
if((4.*phiy).It.phiu)then
vmu(kel)=(vmuk-vmy)/(phiu-phiy)*3*phiy+vmy
phiu=3*phiy


180
300 continue .
do 60 k=l,n
i=3*k
j=6*k
at=1.0/cl(k)
im2=i-2
iml=i-l
jml=j-l
jm2=j-2
jm3=j-3
jm4=j-4
jm5=j-5
cm(im2,j m5)=-l.0
cm(im2,jm2)=1.0
cm(iml,jm3)=1.0
cm(i,j)=1.0
cm(iml,jm4)=-at
cm(iml,jml)=at
cm(i,jm4)=-at
cm(i,jml)=at
60 continue
c**************************************************************
c Coordinate trnsformation matrix
c******************************* *******************************
do 70 k=l,n
co=cost(k)
si=sint(k)
j=6*k
do 80 i=l,2
jum=j-3*i+l
jdois=j-3*i+2
jtres=j-3*i+3
q(jum,jum)=co
q(jum,jdois)=si
q(jdois,jum)=-si
q(jdois,jdois)=co
q(j tres,j tres)=1.0
80 continue
70 continue
c**************************************************************
c Compatibilibity matrix from LM matrix
c**************************************************************
n6=6*n
do 500 i=l,n6
do 510 k=l,iqh
am(i,k)=0.0
510 continue
500 continue
do 520 i=l,6
do 530 k=l,n
iflag=lm(i,k)
if (iflag.gt.0) am(6*(k-1)+i,iflag)=1.0
530 continue
520 continue
c**************************************************************
c Rotation of basic compatibility matrix


28
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.


55
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).


224
Example: Debug Frame
Input File: DATA
TESTE LINEAR BETAO ARMADO
21
17 9
8
1
2.0
2
6.0
4
2.0
5
6.0
7
2.0
8
6.0
10
2.0
11
6.0
0
4
14
27.8
15
27.8
16
27.8
17
27.8
5.00
10.00 1.00
5.00
10.00
1.00
5.0 10.0
1.0
.5
.805
.07 .94
.0
-.092
2
EPNE
0.1
EPST
0.000001
1
1500 500010000
1
0
0
4
2
0
1.00
. 059
5.
. 0
0 10.0
.900
5
8
11


235
(70) Krajcinovic, D., Limit Analysis of Structures,
Journal of Structural Division. ASCE, Vol. 95, ST9, 1969,
pg. 1901-1909.
(71) Building Construction Cost Data. Robert Snow Means
Company, Duxbury, Massachusetts, 1985.
(72) Schuldt, S. B., A Method of Multipliers for
Mathematical Programming Problems with Equality and
Inequality Constraints, Journal of Optimization Theory and
Applications. V. 17, 1975, pg. 155-161.
(73) Abadie, J. and Carpentier, J., Generalization of the
Wolfe Reduced Gradient Method to the Case of Nonlinear
Constraints. Optimization, Academic Press, London, 1969.
(74) Smeers, Y., Generalized Reduced Gradient Method as an
Extension of Feasible Direction Methods, Journal of
Optimization Theory and Applications. Vol. 22, 1977, pg.
209-226.
(75) Gabriele, G. A., and Ragsdell, K. M., The Generalized
Reduced Gradient Method: A Reliable Tool for Optimal Design,
ASME Journal of Engineering. Vol. 99, 1977, pg. 384-400.
(76) Lasdon, L. S., Waren, A. D., Jain, A., and Ratner,
M., Design and Testing of a Generalized Reduced Gradient
Code for Nonlinear Programming, ACM Transactions of
Mathematical Software. Vol. 4, 1978, pg. 34-50.
(77) Hoit, M. I., SSTAN-Simple Structural Analysis
Program. University of Florida, Gainesville, Florida, 1987.
(78) Cohn, M. Z., Analysis and Design of Inelastic
Structures, Vol.2: Problems. University of Waterloo Press,
Ontario, Canada, 1972.


150
1278 continue
c****************************************************** ******
c output format
Q************************************************************
110
120
130
140
150
160
170
180
190
191
240
250
260
270
351
353
360
370
380
390
640
650
730
740
760
840
851
860
870
880
890
910
920
930
940
944
945
946
947
950
960
966
970
format ( //, lOx, ******* initial values ******,/ )
format ( a25 )
format ( /,lOx,'number of elements = ',i3 )
format ( /,lOx,'number of equality constraints = ',i3 )
format ( /,10x,'number of inequality constraints = ',i3 )
format ( /,lOx,'number of iterations per cycle = ',i5 )
format ( /,lOx,'fconcrete = ',el4.8,4x,'fsteel =',el4.8)
format ( /,lOx,'number of global degrees of freedom
= M2 )
format ( //,10x,a25,// )
format (a)
format(//,lOx,'global degree of freedom',lOx,
'external force')
format ( /,20x,i2,22x,el4.8 )
format ( //,lOx,'global degree of freedom',5x,
'displacement constraint' )
format ( //20x,i3,19x,el4.8 )
format ( //,'element',8x,'length',lOx,'steel',12x,
'base',12x,'height')
format ( 18x,i3,6x,el4.8,/)
format ( /,i3,6x/el4.8,3x,el4.8/4x,el4.8,4x,el4.8 )
format ( /,lOx,i3,14x,3i4,3x,3i4 )
format(//,lOx,'location matrix for global degrees
of freedom ')
format ( /,10x,' element ',10x,' node i',10x,'node j' )
format ( /,10x,'maximum number of cycles = ',i3 )
format ( /,lOx,'penalty factor = ',el4.8 )
format ( /,lOx,'factor of increase = ',el4.8 )
format ( /,10x,'factor of decrease = ',el4.8 )
format ( /,lOx,'penalty factor multiplier = ',el4.8 )
format ( /,12x,'element',llx,'base',17x,'height',lOx,
'steel')
format ( /,15x,i2,8x,el4.8,8x,el4.8,5x,el4.8)
format ( /,10x,'global degree of freedom',5x,
'initial guess' )
format ( /,20x,i2,18x,el4.8 )
format ( ////,lOx,'******* final values *******',/// )
format ( /,lOx,'number of iterations = ',i3 )
format ( /,10x,'value of lagrangian function = ',el4.8 )
format ( /,lOx,'equality',17x,'final value' )
format ( /,12x,i3,17x,el4.8 )
format ( /,lOx,'inequality',13x,'final value')
format ( /,5x,'element reliability',9x,'final value')
format (/,5x,'element',5x,'spring i',5x,'spring j')
format ( /,llx,i4,17x,el4.8)
format (/,8x,i5,8x,el4.8,3x,el4.8)
format ( /,12x,i3,17x,el4.8 )
format ( /,lOx,'displacement',14x,'final value' )
format (/,el4.8)
format ( /,12x,i3,17x,el4.8 )
stop
end


I certify that I have read this studv and that in my
opinion it conforms to acceptable standard, of scholarly
, as
Associate Professor of
Aerospace Engineering,
Mechanics, and Engineering
Science
presentation and is fully adequate, in
a dissertation for the degree of Doc
scope and quality
lak Philosophy.
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August, 1989
Dean,//College flof Engineering
Dean, Graduate School


2 32
(29) Ali, M. M., and Grierson, D., Nonlinear Design of
Reinforced Concrete Frameworks, Journal of Structural
Engineering. ASCE, Vol. 112, No.10, 1986, pg. 2216-2233.
(30) Darvall, P. L., and Mendis, P. A., Elastic-Plastic
Softening Analysis of Plane Frames, Journal of Structural
Engineering. ASCE, Vol. Ill, No. 4, 1985, pg. 871-887.
(31) Chajes, A. and, Churchill, J. E., Nonlinear Frame
Analysis by Finite Element Methods, Journal of Structural
Engineering. ASCE, Vol. 113, No. 6, 1987, pg. 1221-1235.
(32) Kayal, S., Finite Element Analysis of RC Frames,
Journal of Structural Engineering. ASCE, Vol. 110, No. 12,
1984., pg. 2891-2907.
(33) Otani, S., Inelastic Analysis of R/C Frame
Structures, Journal of Structural Division. ASCE, Vol. 100,
N. ST7, 1974, pg. 1433-1457.
(34) Tsay, J. J., and Arora, J. S., Variational Methods
for Design Sensitivity Analysis of Nonlinear Response with
History Dependent Effects. Proceedings of the International
Conference on Computational Engineering Science, Atlanta,
1988.
(35) Umerura, H., and Takizawa, H., Dynamic Response of
Reinforced Concrete Buildings. IABSE Structural
Engineering Documents, Number 2, Zurich, 1982.
(36) Kanaan, A. E., and Powell, G. H., DRAIN-2D. A
General Purpose Computer Program for Dynamic Analysis of
Inelastic Plane Structures. Report n EERC 73-6 and EERC
73-22, University of Berkeley, Berkeley, 1975.
(37) Breyse, D., and Mazars, J., Simplified Approach of
Nonlinearity in R/C Beams, Journal of Structural
Engineering. ASCE. Vol. 114, n. 2, 1988, pg. 251-268.
(38) Gedling, J. S., Mistry N. S., and Welch, A. K.,
Evaluation of Material Models for Reinforced Concrete
Structures, Computers and Structures. Vol. 24, n. 2, 1986,
pg. 225-232, 1986.
(39) Cauvin, A., Nonlinear Elastic Design and Optimization
of Reinforced Concrete Frames. CSCE-ASCE-ACI-CEB
International Symposium, Ontario, Canada, 1979.
(40) Charney, F. A., Correlation of the Analytical and
Experimental Seismic Response of a l/5th-Scale Seven-Storv
Reinforced Concrete Frame-Wall Structure. PhD Dissertation,
University of California, Berkeley, 1986.
(41) Augusti, G., Baratta, A., and Casciati, F.,
Probabilistic Methods in Structural Engineering. Chapman and
Hall, New York, 1984.


80
ej_ = 1 if element i is in a non-failure state;
P(ej_=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


CHAPTER 7
EXAMPLES
Introduction
Examples used to test the program versions are
described and the conditions for the tests are presented. A
one bay frame was used to debug the program during its
development and enhancement. For result comparison, an
available study in literature of a frame optimized using
limit equilibrium theory was used to compare results
obtained from the versions of the present optimization
program. The program was finally tested with a realistic
frame and loading configuration corresponding to an average
building frame.
Three versions of the element stiffness were
implemented and tested. The first one considered the
material behavior as elastic and that provides a high value
for the stiffness of the rotation springs. The second
formulation provided a spring stiffness equal to the ratio
between the element yielding moment and the yielding
rotation. The last version used the secant spring
stiffness. The final version was implemented both with the
119


117
ASSUMED MOMENT ROTATION DIAGRAM
Vy- Yielding Rotation
A Ultimate Rotation
v u
M u- Ultimate Moment
My ~ Yielding Moment
Figure 6.4. Bilinear elastic-plastic diagram.


170
heightmax=x(13)/(romin*x(ll))
if(x(k).gt.heightmax)x(k)=heightmax
i=n3
go to 10
endif
if(k.eq.13)then
if(x(k).gt.astma) x(k)=astma
if(x(k).lt.astm) x(k)=astm
i=n3
endif
10 continue
return
end
subroutine parame (toll,x,n21,glk,mband,cl,
* cos,cos2,lm,r,delta,alpha,numec,rph)
implicit double precision (a-h,o-z)
dimension x(ntot),glk(iqh,iqh),cl(n),cosl(n),
* cos2(n), lm(6,n), r(iqh)
common /parr/ decfc,fcinc,cv,alpl,ec,rp,fc,es,ecm,
* relind
common /pari/ iter,numcy,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
c***********************************************************
c penalty factor
c******************* ****************************************
read (9,*) rp,rph,alpl
Q* ********************************************************* *
c gamma,# of iterations, # of cycles
c***********************************************************
read (9,*) ga,niter,numcy
c************************************************** *********
c decrease and increase factors
c***********************************************************
read (9,*) decfc,fcinc
c* ********************************************************* *
c control tollerance
Q***********************************************************
read ( 9,* ) toll
c************************************************* **********
c initial guesses of dimensions
c********************************************************** *
n3=n+n+n
n21=n3+l
read (9,*) (x(i),x(i+l),i=l,n3,3)
c***************************************************** ******
c increment for slope evaluation
c***********************************************************
read(9,*)delta,alpha
c***********************************************************


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


24
- R vector of static external loads;
- D vector of bounds of m;
Design variables
xk, k=l,3,...,2n-l area of element (k+l)/2;
Xj, j=2,4,...,2n inertia of element j/2;
xf, i=2n+l,2n+2,...,2n+m global displacements
Objective Function
f(x) =2 lpxk, 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


96
Tree of Mechanism Combinations
Mechanism Combination with Reliability
in the Interval B, Rt+£
Figure 5.5. Combinatorial tree.


188
Line 21 to 29
Displacement limits.
Line 30
Penalty factor, equality penalty factor and stepsize
Line 31
Factor of penalty increase, maximum number of
iterations and maximum number of cycles.
Line 32
Decrease factor and increase factor.
Line 33
Convergence tollerance.
Line 34
Element and system reliability index.
Line 35
Number of elementary mechanisms.
Line 36 to line 39
Coefficient of variation of concrete strength.
Line 40 to line 48
Coefficient of variation of external loads.
Line 49
Lower bounds of cross section dimensions.
Line 50
Value of interval gap in the Beta unzipping method.


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.
1


108
respectively, subroutines VALMU and COMCON. The coding of
the main subroutines is presented in Appendix A.
This method didn't provide acceptable performance for
the nonlinear material behavior. The convergence of the
method concerning the equality constraints was impossible to
obtain, probably because the variations of the equality
constraint values were severe whenever there was any change
of the design variables. For this reason, a mixed method of
integrated and cycling formulations was implemented. The
approximate displacements were obtained using only once a
Gauss type solution method of the equilibrium equations at
the end of each optimization cycle. The Hooke and Jeeves
search did not include the set of displacements although the
group of equality constraints remained in the augmented
lagrangian function definition. The main goal of this
modification was to improve the convergence for the equality
constraints while performing an optimization that would
remain in the vicinity of the previous design point.
Generalized Reduced Gradient
The optimization strategy is based on the iterative
solution of a system of nonlinear equalities. The method
was initially implemented as an extension of the
decomposition for linear programming problems (73). Several
variations and enhancements of this initial formulation


124
Table 7.1. Debug frame (GRG): linear version results.
Element Initial Final Reliability
Section Index
Base Height Area Base Height Area
(in) (in) (in2) (in) (in) (in2)
1 5.0 10.0 1.0 2.0* 6.0* 0.20 2.0*
2 5.0 10.0 1.0 2.0* 6.0* 0.23 2.0*
3 5.0 10.0 1.0 2.0* 11.1 0.73 2.0*
4 5.0 10.0 1.0 2.0* 12.4 0.81 2.0*
* lower bounds.
Total Initial Cost. 18,000
Total Final Cost 6,890
Global Displacements
1 23 4 567 89
(in) (in) (rad) (in) (in) (rad) (in) (in) (rad)
Initial .50 .06 0.0 .90 .81 .07 .94 0.0 -.09
Final .54 0.0 -.005 .53 -.087 .003 .53 0.0 -.003


56
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


49
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


147
c global degrees of freedom
g* ******************************************************* -k * h
iqh=0
do 500 j=l,nj
do 500 1 = 1,3
if (jm(l,j).ne.O) then
iqh=iqh+l
jm (1,j)=iqh
endif
500 continue
iqgn = iqh+n
iqg = iqh
ntot = iqgn+2*n
g**************************************************************
c input data
c**************************************************************
call inputd (cl,cosl,cos2,iqh,jdir,jm,lm,n,nj,nol,
* no2,r,xc,yc)
g**************************** ******************************
c reinforced concrete
c**********************************************************
read(9,*)fc,fy,co
ec=57000*sqrt(fc)
vn=29e6/ec
epsy=fy/29e6
do 987 ijh=l,n
vksi(ijh)=10e30
vksj(ijh)=10e30
987 continue
read(9,*)es,ecm
c************************************************************
c reinforcing steel guess
c********************************************* **************
n3=n*3
read( 9, *) (x( i), i=3 ,n3,3)
c******* *****************************************************
c constraint values
c***********************************************************
call constr (iqg,d)
c************************************************************
c determine bandwidth
c************************************************************
mband = 0
do 450 k = l,n
do 450 i = 1,6
if ( lm(i,k).eq.0) go to 450
do 440 j = i,6
if (lm(j,k).eq.0) go to 440
max = abs(lm(i,k) lm(j,k)) + 1
if (max.gt.mband) mband=max
440 continue
450 continue
C* * * * k * ie "k ic k * ic ic ic ic * ie * * * ic jc ie "k ie 1c ie * Jc * * *
c optimization parameters and initial guesses
g**************************************************************
call parame(toll,x,n21,glk,mband,cl,cosi,cos2,lm,


15
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


)
105
Step 1: xik = xik-1*(l + a)
Step 2: If L(xk) < L(xk_1), a = a*inc;
Otherwise, a = a*dec;
Step 3: i = i+1; go to 1 if i < n;
Step 4: Try pattern move x* = xk + S(xk xk_1);
where
Step 5: Verify termination criteria. If not met, go
to 1. Otherwise, stop.
inc increase factor;
dec decrease factor;
6 stepsize parameter.
Flowchart of the final group of subroutines is
presented in Figure 6.2. The main program PRINCI, reads the
main input data, initializes the correct displacement
values, when solving the equilibrium equations, for the
starting dimensions, calls the optimizer subroutine and
writes the final results. Subroutine OPTIMI controls the
optimization process by verifying if the convergence
criteria is met, updating the lagrangian multipliers and
verifies the system reliability at the end of the
optimization cycle. Subroutine HOOJEE conducts the Hooke


181
c**************************************************************
mm=80
call multi(q,am,qa,n6, iqh,n6,mm)
c**************************************************************
c Expansion of QA matrix
c**************************************************************
do 585 i=l,n
i6=6*i
i3=i6-3
do 595 j=l,iqh
qa(i3,j)=0.0
qa(i6,j)=0.0
595 continue
i2=2*i
qa(i3,iqh+i2-l)=1.0
qa(i6,iqh+i2)=1.0
585 continue
c**************************************************************
c Matrix A = C QA (transformed)
c**************************************************************
m=3*n
nt=iqh+2*n
call multi(cm,qa,a,m,nt,n6,mm)
c**************************************************************
c Solution for virtual displacements
c**************************************************************
do 150 k=l,nt
do 175 1=1,nt
b(k,l)=0.0
175 continue
150 continue
do 160 k=l,nt
b(k,k)=1.0
160 continue
do 200 i=l,m
amax=0.0
iflag=0
do 250 j=i,nt
if (abs(a(i,j)).gt.amax)then
iflag=j
amax=abs(a(i,j))
endif
250 continue
do 305 k=l,m
c(k)=a(k,iflag)
a (k, iflag)=a(k,i)
a (k, i) =c (k)
305 continue
do 310 k=l,nt
c(k)=b(k,iflag)
b(k,iflag)=b(k,i)
b(k,i)=c(k)
continue
do 280 j=i+l,nt
if (abs(a(i,j)).gt.0.00001) then
fact=-a(i,j)/a(i,i)
310


216
vmuk=vmu(kel)
c****************************************************************
c integration of curvature
c****************************************************************
if(vmy.lt.afo3)nodel=l
if(vmy.lt.afo6)node2=l
if(nodel.eq.l.or.node2.eq.1)then
if((fo3*fo6).gt.0.) then
if(afo3.ge.afo6)then
tetay=(vmy/(3.*ei)+afo6/(6.*ei))*clk
vlp=(afo3-vmy)/(afo3-afo6)*clk
if(abs(vlp).gt.clk)vlp=clk
tetal=(vmuk/(3.*ei)+afo6/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
if(afo3.It.afo6)then
tetay=(vmy/(3.*ei)+afo3/(6.*ei))*clk
vlp=(afo6-vmy)/(afo6-afo3)*clk
if(abs(vlp).gt.clk)vlp=clk
tetal=(vmuk/(3.*ei)+afo3/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
endif
if((fo3*fo6).It.0.) then
if(afo3.ge.afo6)then
tetay=(vmy/(3.*ei)-afo6/(6.*ei))*clk
vlp=(afo3-vmy)/(afo3+afo6)*clk
tetal=(vmuk/(3.*ei)-afo6/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
if(afo3.It.afo6)then
tetay=(vmy/(3.*ei)-afo3/(6.*ei))*clk
vlp=(afo6-vmy)/(afo3+afo6)*clk
tetal=(vmuk/(3.*ei)-afo3/(6.*ei))*clk
tetau=tetal+(phiu-phiy)*vlp
endif
endif
endif
crotl=rotl-(-displ/clk+disp2/clk)
crot2=rot2-(-displ/clk+disp2/clk)
vksp=(vmuk-vmy)/(tetau-tetay)
c*****************************************************************
c spring rotation
c*****************************************************************
srotl=abs(crotl)-tetay
srot2=abs(crot2)-tetay
c******************************************************************
c new secant spring values
c******************************************************************
if(nodel.eq.1) then
if(srotl.le.O.)then
vki=vmy/tetay
go to 123
endif
vki=(vksp*srotl+vmy)/abs(crotl)
if(vm.gt.vmuk)vki=vmuk/tetau


32
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


44
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).
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
Cq = a fern b kd
where
*
eca
a = f c/( fcmeca)<^c
Jo
fem 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.


121
Global structural equilibrium and compatibility was
verified using the program SSTAN (77). The program is
prepared to handle linear elastic analysis. To verify the
nonlinear results some extra elements were added to the
initial structure simulating the nonlinear behavior. These
additional elements placed at the hinge locations normal to
the plane of the frame had a torsional rigidity equal to the
spring secant stiffness. An example of a transformed
structure used to test the accuracy of the displacements of
a debug frame output is presented in Figure 7.1.
Debug Frame
The structure used to verify and evaluate the
performance of the different versions of the program was a
one bay rectangular frame subjected to a horizontal and a
vertical load at the middle of the span. The material
properties, geometric layout, initial dimensions, loads,
reliability indices and other characteristics were
arbitrarily selected, with no intent of creating a practical
design. The global features of this frame are presented in
Figure 7.2.
Results of the optimization performed using the
Generalized Reduced Gradient and assuming linear behavior
are presented in Table 7.1. Performance and final results
were acceptable and satisfied the Kuhn-Tucker conditions.


153
y=cl/(3.*ei)+l./vkj
z=-cl/(6.*ei)
det=w*y-z*z
a=y/det
b=-z/det
c=b
d=w/det
fo3=(a+b)/cl*u2+a*u3-(a+b)/cl*u5+b*u6
fo6=(c+d)/cl*u2+c*u3-(c+d)/cl*u5+d*u6
return
end
subroutine equcon (x,n,cl,lm,cosl,cos2,fc,ec,vn,co,epsy,
* fy^tot^ql^vah^vahk^s^cir^beta^vm^cvload^l,
* vmu)
implicit double precision (a-h,o-z)
dimension lm(6,n),cosl(n),cos2(n),vmu(n)
dimension cl(n),x(ntot),vah(iqh),r(iqh)
dimension vahk(iqh),beta(n),cvmu(n),cvload(iqh)
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
do 150 k = l,iqh
vahk(k)=0.0
150 continue
C****************************************************************
C GLOBAL DISPLACEMENTS PER ELEMENT
C****************************************************************
n3=n+n+n
do 100 kel = l,n
do 123 ipo=l,6
u(ipo)=0.0
123 continue
sigma2=0.0
do 200 i = 1,6
m=lm(i,kel)
if (m.eq.0) go to 200
if(cvload(m).gt.sigma2)sigma2=cvload(m)
1 = n3+m
u(i) = x(l)
200 continue
sigmal=cvmu(kel)
cl=cosl(kel)
c2=cos2(kel)
d2=-c2*u(l)+cl*u(2)
d3=U(3)
d5=-c2 *u(4)+cl*u(5)
d6=u(6)
C*****************************************************************
C ELEMENT FORCES
C****************************************************************


132
ELEMENT DIMENSIONS
1.4,7,10 1Binx18in
others llinxllin
LOADS DEAD LIVE
W1 7.5K 15K
W2 5K
Original Sections
6 8 10 14 15
13 15
12
EQUIVALENT LOADS
W1 40K W2 10K
Figure 7.3. Compared frame.


135
Table 7.7. Compared frame results.
Element 1
Steel Area
(in2)
Initial 3.0
Final 0.8
Reliability
Index 0.0*
Element 6
Steel Area
(in2)
Initial 3.0
Final 2.4
Reliability
Index 0.0*
2
3
4
5
3.0
3.0
3.0
3.0
2.9
3.0
0.6
2.5
0.0*
0.0*
0.0*
0.0*
*
- lower
bounds
7
8
9
10
3.0
3.0
3.0
3.0
0.3
2.5
2.5
2.5
0.0*
0.0*
0.0*
0.0*
* -
lower
bounds.
Initial Steel Cost 86,400
Final Steel Cost 54,720
OLD Steel Cost 63,360


101
The minimization cycle is repeated until there is no
significant improvement of the objective function f(x).
Penalty parameters, P, contribute significantly to the
efficiency of the minimization procedure. Initially, there
was only one penalty parameter for equality and inequality
parameters alike. Since these two types of constraints have
different sensitivities to changes of the design variables,
different penalty parameters were introduced for the group
of equality constraints and the group of inequality
constraints. These starting values, and consequent updates,
were tuned to the optimization performance to improve the
procedure efficiency. Penalty parameters were obtained from
a set of experimental trials and assessment of the results.
Scaling of the constraints and variables also played an
important role in the optimization. The equality
constraints and objective function were scaled to the same
order of magnitude as the inequality constraints. This was
an attempt to regularize the magnitude of the different
functions composing the augmented lagrangian function with
the intent of smoothing the design surface.
Two methods were tested for the unconstrained
minimization of the augmented lagrangian function. These
were the Conjugate Gradient method, or Fletcher-Reeves, and
the Hooke and Jeeves method. The former one is a variation
of the Steepest Descent method, or Gradient method, and is
classified as a first order method since it is based on the
gradient of the function. The latter is defined as a zero


2
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.


129
Table 7.4. Debug frame (GRG): secant stiffness results.
Element
Initial
Final
Reliability
Section
Base
Height
Area
Base Height Area
Index
(in)
(in)
(in2)
(in) (in) (in2)
1
2.0
6.0
0.41
2.0* 6.0* 0.38
0.1*
2
2.0
6.0
0.41
2.0* 6.0* 0.25
0.7
3
2.0
9.74
0.64
2.0* 9.81 0.64
0.1*
4
2.0
10.9
0.71
2.0* 10.9 0.72
0.1*
* lower
bounds.
Total Initial Cost 6,599
Total Final Cost 6,504
Global Displacements
12 3 456 78 9
(in) (in) (rad) (in) (in) (rad) (in) (in) (rad)
Initial
.77
0.0 -.008
.77 -.115
.004
.76 -.007
Final
CO
CO

0.0 -.007
.87 -.152
.005
.87 -.007
Secant Spring Stiffness
(lb.in/rad)
Hinge number 12345678
Spring Stiffness 0.9 0.9 29
(xlO?)
29 84 84
58
58


203
theta(j j,i)=theta(j j,i)* fact
830 continue
do 840 jj=l,ndof
rb(jj,i)=rb(jj,i)*fact
840 continue
go to 734
endif
820 continue
734 continue
810 continue
c**************************************************************
c transpose theta and r matrices
c**************************************************************
do 25 j=l,numec
do 91 i=nel
temp(i,j)=theta(i,j)
91 continue
25 continue
do 56 i=l,numec
do 55 j=l,2*nel
theta(i,j)=temp(j,i)
55 continue
56 continue
do 28 j=l,numec
do 27 i=l,ndof
temp(i, j)=rb(i,j)
27 continue
28 continue
do 66 i=l,numec
do 65 j=l,ndof
rb(i,j)=temp(j,i)
65 continue
66 continue
c**************************************************************
c reliability of fundamental mechanisms
c**************************************************************
do 102 i=l,numec
vmeanr=0.0
stdevr=0.0
do 202 k=l,nel
j=2*k-l
theji=abs(theta(i,j))
thejil=abs(theta(i,j+1))
if(theji.It.0.0001.and.thejil.lt.
* 0.0001)goto 202
term2=(theji+thejil)*cvmu(k)*vmu(k)
stdevr=stdevr+term2*term2
vmeanr=term2/cvmu(k)tvmeanr
202 continue
vmeanl=0.0
stdevl=0.0
do 302 k=l;ndof
if(abs(p(k)),lt.0.001)go to 302
term=p(k)*rb(i,k)
vmeanl=vmeanl+term
stdevl=stdevl+term*term*(cvload(k)


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
V


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


CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS
Linear Material Behavior
This optimization approach with reliability constraints
proved to be a valuable formulation for reinforced concrete
frames with linear material behavior and static loading.
The formulation addresses a universal procedure for
obtaining optimal solutions independently of the local code
restrictions. The choices for the element and system
reliability indices are made by the user and may be chosen
as a function of the particular problem conditions.
The approach depends on initial choices and these have
a significant effect on final results. These effects can be
overcome by careful evaluation and planning by the designer.
Most relevant aspects are the choice of adequate element and
system reliability indices, the definition of the material
and of the load statistical values and the displacement
limits. Solutions provided by the current approach are not
definitive designs, since important aspects like axial
forces and shear forces are not included.
139


CHAPTER 6
PROCEDURE IMPLEMENTATION
Introduction
The optimization problem was tentatively solved using
two strategies. These were an Augmented Lagrangian method,
abridged by the class of penalty functions, and the
Generalized Reduced Gradient method, classified in the group
of gradient type techniques. The unconstrained minimization
techniques experimented with the Augmented Lagrangian
formulation are reported and performance is analyzed. Two
final versions for these two options are discussed, with
emphasis on the problems and decisions taken. Subroutines
are described and their essential characteristics
underlined.
Procedures for element and system reliability
evaluations are detailed. Subroutines involved in the
element reliability calculation are listed and their
specific task described. The system reliability
determination at the mechanism level is outlined with a
summarized description of the Beta unzipping method.
Particular problems, and respective solutions that arose
97


10
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.


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


BIOGRAPHICAL SKETCH
The author was born in Porto, Portugal in 1954.
He finished high school in D. Manuel II, Porto in 1971,
graduated from the University of Porto in Civil Engineering,
majoring in Structures and obtained the degree of Master in
Engineering from the University of Florida, United States of
America in 1986.
The writer taught introductory courses in the
University of Porto, College of Engineering, between 1976 and
1985, and is on leave to pursue his Ph.D. degree. He was a
structural consultant between 1977 and 1984.
He is a recipient of a scholarship from the binational
Fulbright Commission during his studies in the United States
of America. The author received a grant from Fundapao
Oriente to present a paper in the IV World Conference on
Continuing Engineering Education. He is a member of Phi
Beta Delta, ASCE and ACI.
236


68
Ui
Figure 4.4. Reliability index.


176
kkj=2*kk-l
thesu=thesum(kk)+theta(k,kkj)
thesu2=thesul(kk)+theta(k,kkj+1)
term=(abs(thesu)+abs(thesu2))*cvmu(kk)
* *vmu(kk)
vmeanr=term/cvmu (kk) +vmeanr
stdevr=stdevr+term*term
thesui=thesum(kk)-theta(k,kkj)
theslm=thesul(kk)-theta(k,kkj+1)
termm= (abs (thesui) +abs (theslm)) *cvmu(kk)
* *vmu(kk)
vmanrm=termm/cvmu(kk)+vmanra
stdvrm=stdvrm+termm*termm
600 continue
c************ *************************************************
c combination with fundamental mechanisms
c (external work)
Q*************************************************************
do 672 kk=l,ndof
if(abs(p(kk)),lt.0.001)go to 672
dispkk=(dispsu(kk)+rb(k,kk))*p(kk)
vmeanl=vmeanl+dispkk
stdevl=stdevl+dispkk*cvload(kk)*dispkk
*cvload(kk)
dispkm=(dispsu(kk)-rb(k,kk))*p(kk)
vmanlm=vmanlm+dispkm
stdvlm=stdvlm+dispkm*cvload(kk)*dispkm
*cvload(kk)
continue
becopl=(vmeanr-vmeanl)/sqrt(stdevr+stdevl)
becomi=(vmanrm-vmanlm)/sqrt(stdvrm+stdvlm)
continue
if(becomi.lt.becopl) then
do 138 lk=l,nucome
ltemp=ltemp+l
lc(ltemp)=lc(j+lk-1)
continue
ltemp=ltemp+l
lc(ltemp)=-k
lctlp=-k
becote=becomi
else
do 139 lk=l,nucome
ltemp=ltemp+l
lc(ltemp)=lc(j+lk-1)
139 continue
lctlp=k
ltemp=ltemp+l
lc(ltemp)=k
becote=becopl
endif
lpt=lpt+l
iflag=0
Q*************************************************************
c ordering the combined beta values in the same row
c*************************************************************
672
5574
138


26
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.


131
first stage. The results were, however, similar to those
attained before.
Compared Frame
To complement the program testing with the one bay
frame and evaluate the capacity of the program to obtain
accurate and exact optimal solutions, a frame that was
optimized using the theory of the Optimal Limit Design was
also tested (78). This published example had the advantage
of considering the nonlinear behavior of reinforced concrete
elements at ultimate capacity. The resulting moment
redistribution at the nodes was limited to values assuring a
certain serviceability. The definition of the frame and
respective loads are presented in Figure 7.3.
This choice presented some disadvantages. Optimization
was carried out with the design variables as the steel
reinforcement areas, the elements in the reference were
singly reinforced, there were no reliability limits imposed,
and the moment redistribution was limited to a maximum of
30%. It follows a similar approach to the system
reliability in obtaining the optimal redistributed moment
diagram when evaluating the performance of the several
ultimate failure mechanisms. The failure mechanism with
lower external work is defined and in Table 7.6 the moment
redistribution coefficients, factored external moments and
redistributed moments are presented. The steel
reinforcement areas for the redistributed moments, or


95
where the sign + or is chosen to give the smallest
reliability index. A graphical illustration of these
combinations is shown in Figure 5.5.


94
will be the starting points. Ordering the reliability
indices as follows:
< 62 < < 3q
where
6^ reliability obtained using z.
A control value is selected and added to 8^. This
value and 3i define an interval [B]_, 8^ + £]_]. All
mechanisms outside this interval are discarded for future
combinations. The linear combinations to generate new
failure mechanisms are obtained through combinatorial
matching. First, elementary mechanism 1 is combined with
the mechanisms in the interval and their reliability indices
are evaluated. The same process is repeated for the rest of
elementary mechanisms with those in the interval. The
mechanisms are ordered in accordance with their reliability
indices, a new interval is defined and a new generation of
failure mechanisms is originated. The procedure is repeated
until a sufficient number of generations is accomplished.
Figure 5.5 exemplifies the failure tree creation.
To define the failure function zj for the combinations
of the pair of mechanisms i and j can determined as
zij = 2 j air ajrl ^r 2 (bj_s bjs)Ps,


134
optimal final moment diagram, are also presented in Table
7.6 and were calculated in accordance with ACI 318-63. This
code was used in the original published work to define
factored loads and ultimate section capacities.
The frame was initially tested assuming the linear
material behavior and the final results are presented in
Table 7.7. To simulate the same requirements the frame was
optimized using an equivalent set of loads. This set
resulted from the multiplication of the service loads by the
correspondent load factors and by the inverse of the
strength reducing factors prescribed in ACI 318-63. Final
values are very close to those obtained with the Optimal
Limit Design results and with a total lower cost.
Solution was then attempted with the yielding stiffness
and the secant stiffness versions. Results fell in two
unacceptable categories. The first category included the
results with some optimization but no convergence of the
equilibrium constraints. The other had very slight decrease
of objective function and verification of equality and
inequality constraints. Several starting points were tried,
including the design points obtained from the linear
solution, but no practical results were obtained.
Convergence and oscillation were again the key problems.


222
resl=cc+cs-t
c***************************************************************
C SECOND VALUE FOR A
q* ****************************************************** -k ****** *
a2=0.25*dd
exc=a2*epsy/(dd-a2)
epcs=exc*(a2-co)/a2
cc=fc*a2 *alpha*b
eces=exc/epso
cs=epcs*es*aste
res2=cc+cs-t
c***************************************************** **********
C NEWTON ITERATION
Q***************************************************************
100 a=a2-res2*(a2-al)/(res2-resl)
exc=a*epsy/(dd-a)
if(exc.gt.epso) go to 200
c******************************************************* ********
c PARABOLIC SHAPE
C**************************************************************ie
epcs=exc*(a-co)/a
eces=exc/epso
alpha=eces-eces*eces/3.
cc=fc*alpha*b*a
cs=epcs*es*aste
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then
al=a2
a2=a
resl=res2
res2=res
go to 100
end if
gama=l.-(8.*epso-3.*exc)/(12.*epso-4.*exc)
arm=dd-gama*a
vmy=cc*arm+epcs*es*aste*(dd-co)
phiy=epsy/(dd-a)
Q**********************************************************
C CONCRETE STRAIN > EPSO
c**********************************************************
200 if(exc.gt.0.004) exc=0.004
xl=epso*a/exc
ccl=fc*xl*2./3.*b
gama=3.6*exc*exc-200.*exc*exc*exc-0.0000128
gama=gaina/ ((exc-epso) (7.2*exc-300*exc*exc-0.0132)) -1.
alpha=exc-50.*exc*exc+i00.*exc*epso-0.0022
alpha=alpha/(exc-epso)
cc2=alpha*fc*(a-xl)*b
epcs=exc*(dd-co)/a
t=fy*aste
cs=epcs*es*aste
cc=ccl+cc2
res=cc+cs-t
control=0.0001*b*dd*fc
if (abs(res).gt.control) then


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


156
C VARIABLE INCREASE
C***************************************************************
alp(k)=alp(k)*fcinc
vlago=vlag
xopt=x(k)
go to 500
else
C***************************************************************
C DIRECTION REVERSED
C***************************************************************
alp(k)=-alp(k)
x (k) =xopt+alp(k)*xopt
end if
if (k.le.n3) call lim(x,n,ntot,xll,
* xl2,k,romin,romax)
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lin,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
c***************************************************************
C VARIABLE INCREASE
C***************************************************************
if (vlago.gt.vlag) then
alp(k)=alp(k)*fcinc
vlago=vlag
xopt=x(k)
go to 500
else
c***************************************************************
C VARIABLE DECREASE
C***************************************************************
alp(k)=alp(k)*decfc
x(k)=xopt
endif
if(kkl.lt.200)go to 600
100 continue
C************************************************************
C PATTERN MOVE
C***********************************************************
do 250 kp = l,ntot
x(kp) = 1.01*x(kp) 0.01*xol(kp)
if (kp.le.n3) call lim(x,n,ntot,xll,
* xl2/kp,romin/roinax)
250 continue
call lagfun (vlag,tvah,r,x,cl/cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,kp,vitiu,rph)
if (vlago.gt.vlag) then
vlago = vlag
else
do 300 kp = l,ntot
x(kp) = (x(kp)+0.01*xol(kp))/1.01
300 continue
endif
c***********************************************************
C SOLUTION OF EQUALITIES
C***********************************************************


116
optimization process using the methodology described in
Chapter 5. The sets of relative displacements
corresponding to the fundamental elasto-plastic mechanisms
are divided into two parts. The first one corresponds to
the virtual displacements of the external loads and the
second one to the added global degrees of freedom. Since
the first set corresponds to global displacements, the joint
mechanisms have to be transformed into element rotations.
Since usually there are no concentrated moments applied at
the nodes these mechanisms do not occur by themselves, they
are active in the linear combination with the fundamental
failure mechanisms that lead to the mechanisms with lower
reliability indices.
After the optimization process is finished, subroutine
SYSREL performs the evaluation of the system reliability at
the mechanism level. For that purpose the material behavior
is assumed plastic after the element rotation exceeds the
ultimate value as illustrated in Figure 6.4. The
combination with the elementary mechanisms is made in a
combinatorial type process. The first mechanism is linearly
combined with the remaining ones, the second mechanism with
the following mechanisms and so forth until the penultimate
is combined with the last one. The new mechanisms are
ordered in terms of the reliability index and those that
fall outside an acceptable interval are skipped from future
combinations. The process is repeated until all possible
combinations with fundamental mechanisms is performed. To


14
OPTIMIZATION
OF
TRUSS GEOMETRY
Load
Initial Configuration
Load
Optimal Configuration
Figure 1.5. Geometry optimization.


140
The results presented showed a perfect convergence,
even when the initial displacements were not those
corresponding to the starting physical properties. The
Kuhn-Tucker conditions were always verified unless the lower
bounds were active, as in the case of the building frame.
This implied that at least a local optimum was obtained.
For instance, a good indication of the quality of the
program performance was that in each case, the variables
representing the element bases always converged to the lower
bound. Another particular aspect of the program
capabilities was that at the end of the optimization the
displacement variables were always in the set of basic
variables of the Generalized Reduced Gradient method. This
meant that no improvement could be extracted from the
objective function, except iterating on the equilibrium
equations.
Integration of displacements in the set of design
variables was a valid option for optimization with
reliability considerations. Element reliability constraints
were always active unless there were conflicting lower
bounds. A good compromise was established between the
optimization and safety requirements. System reliability
was also satisfied every time required probabilities of
failure for the elements and the system were of the same
order of magnitude. The method proved to be adequate for
optimal predesign without code limitations.


86
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' g 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].
where
{S> = [T] {S'}
[T] =
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0 0
1 0
0 1
0 0
1 -L
0 1
L element length.
For any elementary failure mechanism the element
deformations, s'^, S'2, S'3, must be zero. This is only for
elements that do not have plastic hinges. To materialize
this condition, a matrix is introduced for each element


22
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 Laqrancrian Function
The optimization technique of cycling unconstrained
minimizations of a penalty function, based on an pseudo
objective augmented lagrangian function, was chosen as the


193
dj_k+1 = grad fk + (J 1C) grad fdk;
ddk+1 = j-lc dj_k+1;
Step E Iterations
First iteration:
xot = {1 10 -0.133 -0.02 0.4};
dlt = {-501667 2505556 33333 5000 -333};
a1 = 106, because x^ > 0.5;
= {0.5 12.506 -0.1 -0.015 0.3997};
Second iteration:
Change of variables X4 replaces x^ that is at a
lower bound;
d2t = {0 -332.09 -7.9664 -1.195 40.75};
a2 = 0.3997/40.75, because o < 1000;
x2t = {0.5 9.249 -0.178 -0.0267 0};
Independent variables, {x^, X5} are at their lower
bounds;
Iteration is performed on the set of dependent
variables {X2, X3, X4};
Third iteration:
-J_1 h(x2)t = {1.3228 -0.082 -0.0124};
X3t = {10.58 -0.26 -0.0391};
d
Fourth iteration:
-J_1 h(x3)t = {0.358 0.061 0.009};
x3t = {10.935 -0.199 -0.0301};
d
Fifth iteration:
-J-1 h(x4)t = {0.019 -0.004 -0.003};


17
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.


The example and correspondent optimization conditions
chosen to illustrate the performance of the Generalized
Reduced Gradient method using the integrated formulation are
presented in Figure B.l. The maximum flexural stress,
compression or tension, is 1,000 psi. The problem is solved
in separate steps presented below.
Step A Problem Formulation
Objective Function
Minimize f(x) = IOX1X2
Equality Constraints
hl(x) = 0.03xiX23X3 0.015xiX23X4 +1=0
h2(x) = -0.15XiX23X3 + X1X23 = 0
Inequality Constraints
h3(x) = 60/(X!X22) 1 + x5 = 0
where X5 slack variable;
Variable bounds
xi > 0.5 in
X2 > 0*5 in
I X3 I <0.5 in
I X4 I <0.5 rad
Step B Explicit Derivatives
df/dxi = 10x2 df/dX2 = 10xi df/dX3 = .... =0;
dhi/dxi = X23(0.03x3 0.15x4);
dhi/dx2 = xiX23(0.09x3 0.45x4);
dhi/dX3 = 0.03XiX23;
dhi/dx4 = -0.15xiX23;
dhi/dxs = 0;
190


207
ltemp=ltemp+l
le(ltemp)=lc(j+lk-1)
139 continue
lctlp=k
ltemp=ltemp+l
lc(ltemp)=k
becote=becopl
endif
lpt=lpt+l
iflag=0
c*************************************************************
c ordering the combined beta values in the same row
c*********************************************** **************
if (lpti.gt.lpt-1)then
do 524 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
continue
lp=lp+l
lct(lp)=lctlp
become(lpt)=becote
do 510 jkj=lpti,lpt-1
if(become(jkj).gt.becote)then
iflag=-l
do 511 kjk=jkj,lpt-1
itemp=lpt-l+j kj-kj k
become(itemp+1)=become(itemp)
continue
become(jkj)=becote
becote=become(lpt)
C*************************************************** **********
c moving lc array
c***************************** ********************************
movini=nic+nucomb(nucome)*nucome+
* (jkj-lpti)*(nucome+1)
movfin=(lpt-lpti)*(nucome+1)+nic+
* nucomb(nucome)*nucome-l
do 512 lmn=movini,movfin
lcou=movini+movfin-lmn
nucl=nucome+l
let(lcou+nucl)=lct(lcou)
512 continue
do 513 n=movini,movini+nucome-l
lptaa=n-movini+l
let(n)=lc(j+lptaa-1)
513 continue
let(movini+nucome)=lctlp
endif
510 continue
if(iflag.eq.0)then
do 124 lk=l,nucome
lp=lp+l
let(lp)=lc(j+lk-1)
continue
lp=lp+l
524
else
511
124


11
Minimize I LiAi
subject to
Fi < Fc
Fi < Ft
where
Li length of truss bar i
Ai area o truss bar i
Fi stress in truss bar i
Fc allowable compressive stress
Ft allowable tensile stress
Figure 1.3. Truss optimization.


7
2k+l =

where x is the vector of design variables, uk+1 is an
estimative of lagrangian multipliers and

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 ip 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,


3
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


200
subroutine modsti(kel,tinert,area,vksi,vksj)
implicit double precision (a-h,o-z)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir (3)
common /parr/ cv, ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
c***********************************************************
c FLEXIBILITY MATRIX (2x2)
C************************************************* **********
n=nel
do 10 i=l,6
do 20 j=l,6
ck(i,j)=0.0
20 continue
10 continue
xd=cl(kel)/(3*ec*tinert)+l./vki
y=cl(kel)/(3*ec*tinert)+i./vkj
z=-cl(kel)/(6*ec*tinert)
c***************************************************************
C INVERSION OF MATRIX
c***************************************************************
det=xd*y-z*z
a=y/det
b=-z/det
c=b
dd=xd/det
c************************************************ **************
C EXPANDED MATRIX (6x6)
c************************************************** ***********
ckll=ec*area/cl(kel)
ckl4=-ckll
ck41=-ckll
ck44=ckll
ck22=(a+b+c+dd)/(cl(kel)*cl(kel))
ck25=-ck22
ck52=ck25
ck55=ck22
ck23=(a+c)/cl(kel)
ck53=-ck23
ck26=(b+dd)/cl(kel)
ck56=-ck26
ck33=a
ck36=b
ck63=c
ck66=dd
ck32=(a+b)/cl(kel)
ck35=-ck32
ck62=(c+dd)/cl(kel)


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
matrices as follows:
Ck
-10 0
0 -1/L 1
0 -1/L 0
10 0
0 1/L 0
0 1/L 1
Ci 0 0
0 C2 0
C = .
. *
* .
0 0 cn
Using the previous matrices and vectors the following
relation now holds
C {S} = {S'd}
where
S first element
S second element
S
nth element
and
(S'd)
S'l
S72
S'3
first element
S'l
S'2
S'3
nth element


155
subroutine hoojee(tvah,vlag,r,vah,vag,x,cl,cosl,cos2,
* lm,d,clah,clag,ch,eg,alp,xol,vahk,tol1,vn,co,
* epsy,fy,ast,beta,cvmu,cvload,xl1,xl2,romin,vmu,rph,
* grad,vahold,vjac,vinv,bfal)
implicit double precision (a-h,o-z)
dimension x(ntot),cl(n),cosl(n),cos2(n),lm(6,n),vahk(iqh),
* d(iqg),clah(iqh),clag(iqgn),r(iqh),vah(iqh),vag(iqgn),
* ch(iqh),cg(iqgn),alp(ntot),xol(ntot),beta(n),
* cvmu(n),cvload(iqh),vmu(n),grad(ntot),vahold(iqh),
* vjac(iqh,iqh),vinv(iqh),bfal(iqh,iqh)
common /parr/ decfc^cinc/cv^lpl^c^p^c^s^ecm/relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
C****************************************************************
C INITIALIZE LAGRANGIAN FUNCTION
C****************************************************************
k=0
kkl=0
n3 = n+n+n
romax=0.85*0.85*fc*87000./(fy*(87000.+fy))
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
vlago = vlag
C*************************** ****************** *******************
C LOOP ON VARIABLE # K FOR THE SPECIFIED # OF ITERATIONS
C************************************************ ****************
do 200 klj = 1,niter
do 450 k = l,ntot
450
alp(k) = alpl
do 150 k = l,ntot
150
xol(k) = x(k)
do 100 k = l,n3
xopt = x(k)
600
continue
kkl=kkl+l
C***************************************************************
C VALUE OF LAGRANGIAN FUNCTION FOR INITIAL VALUES
C****************************************************************
500 continue
x(k)= xopt+alp(k)*xopt
if (k.le.n3) call lim(x,n,ntot,xll,
* xl2,k,romin,romax)
C* ************************************************************** *
c
NEW VALUE OF LAG. FUNCTION
Q****************************************************************
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,
* d,clah,vag,clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,k,vmu,rph)
if (vlago.gt.vlag) then
q******************************************************* ********


47
where
Mi = Cs d'+ Cc n kd Ts d
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:
fty (ey + eca)A*
j^u = (esa + ecu)/^
where
tfy yielding curvature;
ey Es / fy;
Es 29X106 psi;
fy yielding stress of reinforcing steel;
eca maximum concrete compressive strain;
- 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


41
One-Component Model
Reinforced Concrete Element
Linear Elastic Element
Spring with Secant Stiffness
Figure 3.1. Element model.


92
At Level 0 the estimation of the system reliability is
based on the failure of a single structural element. In
this case the system reliability is equal to the reliability
of the element with the higher probability of failure.
Level 1 gives more acceptable results. The concept is
that the structural system is modelled as a series system.
The system probability of failure is estimated as a function
of element probabilities of failure. The calculation of
this system probability can be made with acceptable accuracy
using only those elements with a low reliability index. The
interval where these significant or critical elements are
located is defined by [fimin/ fmin + 6]# where SB is chosen
adequately.
At Level 2, failure elements are grouped in pairs as
parallel systems. These significant pairs of failure
elements are obtained assuming failure of the significant
elements as defined in Level 1. For element i, the load
carrying capacity is added as fictitious loads if the
element is ductile. If the element is brittle, no
fictitious loads are added. Then new element reliability
indices of the modified structure are calculated and the
critical pairs are formed with element i and the new
significant elements.
The process can be analogously repeated for levels
greater than 2 creating critical groups of 3, 4, or more
elements. It is considered that above Level 3 there is no
practical benefit from the extra calculations. The method


199
end
subroutine valobf(x,vof)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(100)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
vof = 0.0
do 100 k = l,nel
base = x(3*k-2)
height = x(3*k-l)
steel = x(3*k)
area = base height
vof=vof+(area+steel*10)*cl(k)
100 continue
return
end
subroutine inecon(x)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100) ,lm(6,100) ,
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg,nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
nel3=3*nel
c***************************************************************
C RELIABILITY
c***************************************************************
do 200 k=l,iqg
vag(k) = relind-beta(k)
200 continue
return
end


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 smoothe
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


70
variables of the expression defining 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
batching, 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


204
* *vload(k))
302 continue
vmean=vmeanr-vmeanl
stdev=stdevr+stdevl
become(i)=vmean/sqrt(stdev)
lnumbe=lnumbe+l
102 continue
c**************************************************************
c reliability of combined mechanisms
c (internal work)
c**************************************************************
do 50 l=l/numec
lc(l)=l
lct(l)=l
50 continue
lp=numec
ltemp=numec
ni(1)=1
numax=6
nucome=l
nucomb(1)=numec
lpt=numec
lpti=numec+l
111 continue
c***************************************************************
c loop over mechanisms in location vector
c***************************************************************
do 699 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
699 continue
do 698 klp=l,ndof
dispsu(kip)=0.0
698 continue
kmu=l
nic=ni(nucome)
nif=nic+nucomb(nucome)*nucome-l
nucomb(nucome+1)=0
. c**************************************************************
c define acceptable interval
c**************************************************************
if(nucome.gt.1)then
nifbet=lpti-l
nicbet=lpti-nucomb(nucome)
do 5544 ia=nicbet+l,nifbet
betaal=become(nicbet)+epsilo
if(become(ia).gt.betaal)then
do 5545 ib=ia,nifbet
become(ib)=1000.
5545
continue
go to 5541
endif
5544
continue
5541
continue
niccon=nicbet
end i f


program princi
implicit double precision (a-h,o-z)
character*40 title
dimension x(100),r(60),cl(60),cosl(60),cos2(60),xol(l00),
* clah(80),clag(80),lm(6,50),vag(100),vah(80),
* ch(80),cg(80),xo(100);nol(80),no2(80),jm(6,80),alp(80),
* xc(80),yc(80),jdir(3),glk(80,80),d(1000),vinv(100),
* vahk(80),grad(100),xu(100),xl(100),xl(100),x2(100),vaho(80),
* vago(80),a(80,80),b(80,80),vahold(100),vjac(100,100),
* c(80),cm(80,80),qa(80,80),q(80,80),am(80,80),bl(80,80),
* theta(100,80),rv(80,100),beta(80),vmu(80),cvmu(80),
* cvload(80),become(100),lc(100),thesum(80),thesuml(80),
* dispsum(60),ni(60),nucomb(60),lct(100),bfal(100,100)
common /parr/ decfc,fcinc,cv,alpl,ec,rp,fc,es,ecm,relind
common /pari/ iter,numcy,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
open ( 8,file='finres',form='formatted' )
rewind 8
open ( 9,file='data',form='formatted' )
rewind 9
c********************************************** *************
c name of problem
c******** **************************** ************************
read ( 9,191 ) title
Q************************************************************
c # elements and # joints
c****************************************************** ******
read (9,*) n, nj
c* ********************************************************** *
c nodes per element
c*************************************************** **********
do 100 i = l,n
read (9,*) nol(i), no2(i)
100 continue
c*************************************************************
c initialize jm matrix
c**************^**^********************************** ********
do 200 kk = l,nj
jm ( l,kk ) =1
jm ( 2,kk ) = 2
jm ( 3,kk ) =3
200 continue
c*************************************************************
c support conditions and coordinates
c**************************************************************
do 300 j = l,nj
read ( 9,* ) jdir(l),jdir(2),jdir(3),xc(j),yc(j)
do 350 i =1,3
if (jdir(i).gt.0) then
jm (i, j) =0
endif
350 continue
300 continue
c************************************************************
146


34
STEEPEST DESCENT
1Initial Point
4-Final Point
Figure 2.4. Gradient method.


184
ntot,iqh,vah,r,vahk, es,ecm,beta,cvmu,cvload,kl,vmu)
call modsti(area,ec,vksi(k),vksj(k),c,tinert,cl,c2)
do 300 1=1,6
j=lm(l,k)
if (j.eq.O) go to 300
do 400 11 = 1,6
m=lm(ll,k)
if (m.eq.0) go to 400
vjac(j ,m)=vjac(j ,m)+ck(l, 11)
400
continue
300
continue
100
continue
return
end


51
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
-j
1/Ksi 0
+ 3EI/L
1/3
-1/6
0 1/Kgj
-1/6
1/3
C a ]
-10 0 1
0 0
0 1/L 1 0 -1/L 0
0 1/L 0 0 -1/L 1
C Kmod ] = [ a ]t [ Ks* ] [ a ]
where
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;


35
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


58
A criterion proposed is as follows (41):
pf = 105 U T / L
where
U 0.005 Places of public assembly, dams;
0.05 Domestic, office, industry, travel;
0.5 Bridges;
5 Towers, masts, offshore structures;
T life period of the structure(years);
L number of people involved.
These values must be interpreted carefully. For
example, the value of 103 means theoretically that, on the
average, out of 1000 nominally identical buildings, one will
crack or deform excessively. It is evident that in civil
engineering 1000 identical buildings rarely occur, even
neglecting the fact that a statistically significant number
require samples at least 10 to 20 times larger.
Moreover, the determination of these low probabilities
requires extrapolations of statistical properties that are
experimentally known only around the mean values of the
random quantities. For these reasons, the probabilities of
failure in civil engineering have no real statistical
significance and they must be considered not as
deterministic quantities but just as conventional
comparative values.


148
* r,delta,alpha,numec,rph)
c**************************************************************
c elementary mechanisms
c**************************************************************
call mecsys(n,iqh,cl,cosl,cos2,lm,numec,rv,theta,rv)
c**************************************************************
c coefficients of variation
c************************************************** ************
do 156 i=l,n
read(9,*)cvmu(i)
156 continue
do 157 i=l,iqh
read(9,*)cvload(i)
157 continue
c************************************************** **********
c lower bounds
c************************************************************
romin=200./fy
read(9,*)xll,xl2
c***********************************************************
c interval for generation of mechanisms
c*************************************************** ********
read(9,*)epsilo
c************************************************************
c write input data
c*************************************************************
write (8,190) title
write (8,110)
write (8,130) n
write (8,140) iqh
write (8,150) iqg
write (8,170) fc,fy
write (8,240)
do 501 k = l,iqh
write (8,250) k, r(k)
501 continue
write (8,260)
do 601 k = 1,iqg
write (8,270) k, d(k)
601 continue
write (8,351)
do 701 k = l,n
na = 3 k
ne = na 1
no = ne 1
write (8,360) k, cl(k), x(na), x(no), x(ne)
701 continue
write (8,650) rp
write (8,760) ga
write (8,860)
do 900 i=n21,ntot
k=i-3*n
write (8,870)k,x(i)
900 continue
c**************************************************************
c data initialization


138
Table 7.8. Building frame results.
Element
1
2
3
4 5
6
7
8 9
Base
(in)
Initial
10
10
10
10 10
10
10
10 10
Final
8*
8*
8*
8* 8*
8*
8*
8* 8*
Height
(in)
Initial
25
25
25
30 30
30
25
25 25
Final
12*
12*
25
16* 16*
20
12*
12* 12*
Area
(in2)
Initial
3.0
3.0
3.0
3.0 3.0
3.0
3.0
3.0 3.0
Final
1.7
1.6
5.5
1.8 1.6
4.3
1.2
2.7 1.2
Reliability
Index
3.0*
3.0*
3.0*
3.0* 3.0*
3.0*
3.0*
3.0* 3.0*
Element
10
11
12
13
14
15
16
17
18
Base
(in)
Initial
10
10
10
10
10
10
10
10
10
Final
8*
8*
8*
8*
8*
8*
8*
8*
8*
Height
(in)
Initial
25
25
25
30
30
30
25
25
25
Final
16*
16*
16*
12*
12*
12*
16*
16*
16*
Area
(in2)
Initial
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
Final
1.7
2.2
1.1
.57
1.2
1.1
.69
.71
.77
Reliability
Index
3.0*
3.0*
3.0*
3.0*
3.0*
3.0
3.0*
3.0*
3.0
* lower bounds.
Total Initial Cost 515,400
Total Final Cost 401,274


113
Figure 6.3. Generalized Reduced Gradient version flowchart.


4
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.


208
lct(lp)=lctlp
become(lpt)=becote
else
lp=lp+nucome+i
endif
endif
nucomb(nucome+1)=nucomb(nucome+1)+1
1numbe=1numbe+1
100 continue
do 6991 kmo=l,nel
thesul(kmo)=0.0
thesum(kmo)=0.0
6991 continue
do 6981 klp=l,ndof
dispsu(kip)=0.0
6981 continue
200 continue
c*************************************************************
c control of maximum number of tree rows
c*************************************************************
lpti=lpti+nucomb(nucome+1)
nucome=nucome+l
ni(nucome)=ni(nucome-1)+(nucome-1)*nucomb(nucome-1)
if(nucome.It.(numax))go to 111
c*************************************************************
c find minimum beta value
c*************************************************************
betmin=100
do 7890 i=l,lnumbe
if(become(i).It.betmin)then
betmin=become(i)
itab=i
endif
7890 continue
c*************************************************************
c find mechanisms involved
c*************************************************************
mecoun=0
mcomb=l
do 7891 j=l,nucome-1
do 7895 jk=l,nucomb(j)
mecoun=l+mecoun
if(itab.eq.mecoun)then
nistar=mcomb
do 7893 1=1,j
locmec(l)=lct(nistar+1-1)
continue
nummec=j
go to 7894
endif
mcomb=j +mcomb
continue
continue
continue
c*************************************************************
c find elements involved
7893
7895
7891
7894


88
Compatibility between the element displacements, {S}
and the structure global degrees of freedom {r} can be
established
where
CQ]
[A]
{S} = [Q] [A] {r}
rotation matrix;
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


23
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;


107
and Jeeves minimization operation. Subroutine LAGFUN
calculates the value of the augmented lagrangian function.
Subroutine DATINI initializes the values of the scaling
factors and lagrangian multipliers. ASSEMB is the
subroutine that creates the initial global stiffness matrix
with the starting values of the elements. Subroutine GLOSTI
solves the initial equilibrium equation system to obtain
good initial displacement values. INPUTD is the subroutine
that reads all the data concerning the definition of the
structure. CONSTR is the subroutine that reads the values
of the constraints. PARAME is the subroutine that inputs
all optimization parameters. Subroutine MECSYS defines all
elementary failure mechanisms of the structure and MULTI is
a related subroutine that multiplies matrices. SYSREL is
the subroutine that calculates the element reliability.
Subroutine LIM controls the maximum and minimum values of
the design variables excluding displacements. SOLCON is the
subroutine that obtains the global displacements with the
nonlinear global stiffness formulation. Subroutine EQUCON
evaluates equality constraints values, INECON calculates
inequality constraint values and VALOBF obtains the
objective function value. ELEY is the subroutine that
recovers the element forces with the current displacement
and element stiffness values. Subroutine MODSTI assembles
the nonlinear stiffness values using the actual spring
secant stiffness values. MUMY is the subroutine that
evaluates the ultimate and yielding values calling,


69
the reliability index, B, are within certain approximations
related by
pf = 1 where

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
where
z = r- s = Mj_-Me
- 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 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


198
do 600 i = l,nel
j = nol(i)
k = no2(i)
do 600 1 = 1,3
lm(l,i) = jm(l,j)
lxn(l+3,i) = jm(l,k)
600 continue
c***************************************************************
C GEOMETRIC CHARACTERISTICS
C***************************************************************
do 800 ii = l,nel
j = nol(ii)
k = no2(ii)
ell = xc(k) xc(j)
el2 = yc(k) yc(j)
cl(ii) = sqrt(ell*ell+el2*el2)
cosl(ii) = ell/cl(ii)
cos2(ii) = el2/cl(ii)
800 continue
c*************************************************************
C INITIALIZATION OF GLOBAL FORCES
c*************************************************************
do 850 k=l,iqh
r(k) = 0.0
850 continue
c***********************************
C GLOBAL FORCES
c***********************************
900 read(9,*)jnum,jdire,force
if(j num.ne.0)then
k=jm(jdire,jnum)
r(k)=force
go to 900
endif
c***************************************************************
C REINFORCED CONCRETE
c***************************************************************
read(9,*)fc,fy,co
ec=57000*sqrt(fc)
vn=29e6/ec
epsy=fy/29e6
read(9,*)es,ecm
c***************************************************************
c ELEMENT RELIABILITY
c***************************************************************
read(9,*)relind
c***************************************************************
c COEFFICIENTS OF VARIATION
C***************************************************************
do 156 i=l,nel
read(9,*)cvmu(i)
156 continue
do 157 i=l,iqh
read(9,*)cvload(i)
157 continue
return
**************************
**************************


215
C***************************************************************
c subtraction of external global forces
G***************************************************************
rmax=0.01
do 510 i=l,iqh
if(abs(r(i)).gt.rmax)rmax=abs(r(i))
510 continue
do 500 k=l,iqh
if(abs(r(k)).It.0.0001)then
vah(k)=vahk(k)/rmax
go to 500
endif
vah(k)=(vahk(k) r(k))/rmax
500 continue
return
end
subroutine mumy(kel/x/sigmal,sigma2/fo3,fo6,vki,vkj,displ,
* disp2,rotl,rot2)
implicit double precision (a-h,o-z)
dimension x(l)
common /vgeom/ cl(100),cosl(100),cos2(100),lm(6,100),
* cvmu(lOO)
common /vload/ r(100),cvload(100),jm(6,100),nol(100),
* no2(100)
common /xcord/ xc(100),yc(100),d(100),jdir(3)
common /parr/ cv,ec,rp,fc,es,ecm,relind,co,fy,epsy
common /pari/ iqh,iqg/nel,ntot,iqgn
common /inequa/ vag(100),beta(100),u(6),vahk(100),ck(6,6)
common /equal/ vah(100),vmu(100)
common /springs/ vksi(100),vksj(100)
n=nel
nodel=0
node2=0
b=x(3*kel-2)
h=x(3*kel-l)
dd=h-co
aste=x(3*kel)
clk=cl(kel)
ei=57000.*sqrt(ec)*h*h*h*b/12.
c***************************************************************
c evaluation of yielding moment
g****************************************************************
call comcon(aste,dd,b/vmy,phiy)
afo3=abs(fo3)
afo6=abs(fo6)
vm=max(afo3,af06)
c*************************************************************
c ultimate moment and reliability
G*************************************************************
hl2v0s0b84X1^blilia

12
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


40
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


48
Figure 3.4. Element deformation diagrams.


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Mafc'I. HoCt,' Chair
Assistant Professor of Civil
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Fernando E. F^gjindo^Cochair
Associate Professor^of Civil
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Cl if fjord 0. Hays
Professor of Civil Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
3hn M. Lybas
Associate Profes
Engineering
of Civil


REFERENCES
(1) U. Kirsch, Optimum Structural Pesian. McGraw-Hill, New
York, 1981.
(2) Brandt, A. M., Criteria and~Methods of Structural
Optimization. Warszawa/Martinus Nijhoff Publishers, The
Hague, 1984.
(3) Spillers, W. R., Iterative Structural Design. North-
Holland Publishing Company, Amsterdam, 1975, Appendix A.
(4) Magnel, G., Prestressed Concrete. McGraw-Hill, New
York, 1954.
(5) Schmit, L. A., Structural Design bv Systematic
Approach. Proceedings of the Second National Conference on
Electronic Computation, Structural Division of ASCE,
Pittsburgh, 1960.
(6) Morris, A. J., Foundations of Structural Optimization:
a Unified Approach. John Wiley and Sons, New York, 1982.
(7) Fleury, C., and Geradin, M., Optimality Criteria and
Mathematical Programming in Structural Weight Design,
Computers and Structures. Vol. 8, no. 1, 1978, pg. 7-18.
(8) Haftka, R. T., and Kamat, M. P., Elements of
Structural Optimization. Martinus Nijhoff, Amsterdam, 1985.
(9) Farshi, B., and Schmit, L. A., Minimum Weight Design
of Stress Limited Trusses. Journal of Structural Division,
ASCE, Vol. 100, no. ST1, 1974, pg. 97-107.
(10) Gallagher, R. H., and Zienkiwicz, 0. C., Optimum
Structural Design. John Wiley and Sons, New York, 1977.
(11) Grierson, D. E., and Cameron, G. E., Computer-
Automated Synthesis of Building Frameworks, Canadian Journal
of Civil Engineering. Vol. 11, no. 4, 1984, pg. 863-874.
(12) Vanderplaats, G. N., Numerical Optimization
Techniques for Engineering Design. McGraw-Hill, New York,
1984.
(13) Wellen, H., and Bartholomew, P., Structural
Optimization in Aircraft Construction. Computer Aided
Optimal Design: Structural and Mechanical Systems, Springer-
Verlag, Berlin, 1986.
(14) Avriel, M., Nonlinear Programming. Prentice-Hall,
Englewood Cliffs, New Jersey, 1976.
230


127
Table 7.3.Debug frame (GRG): yielding stiffness results.
Element Initial Final Reliability
Section Index
Base Height Area Base Height Area
(in) (in) (in2) (in) (in) (in2)
1 2.0 6.0 0.41 2.0* 6.0* 0.21 0.1*
2 2.0 6.0 0.41 2.0* 6.0* 0.21 0.1*
3 2.0 9.74 0.64 2.0* 9.75 0.64 0.1*
4 2.0 10.9 0.71 2.0* 10.9 0.71 0.1*
* lower bounds.
Total Initial Cost 6,599
Total Final Cost 6,296
Global Displacements
123 4567 89
(in) (in) (rad) (in) (in) (rad) (in) (in) (rad)
Initial .77 0.0 -.008 .77 -.115 .004 .76 -.007 -.004
Final .82 0.0 -.008 .81 -.118 .004 .81 -.006 -.004
Yielding Spring Stiffness
(lb.in/rad)
Hinge Number 12 345678
Spring Stiffness 6.9 6.9 49 49 83 83 43 43


161
endif
vksp=(vmuk-vmy)/(tetau-tetay)
C****************************************************************
C SPRING VALUES
C****************************************************************
if(nodel.eq.l) then
vki=vksp
vkj=10.0e20*ec
endif
if(node2.eq.1) then
vki=10.0e20*ec
vkj =vksp
endif
if(nodel.eq.l.and.node2.eq.l) then
vki=vksp
vkj =vksp
endif
endif
return
end
subroutine optimi(vlag,r,x,cl,cosl,cos2,lm,d,clah,xol,
* vag,toll,clag,vah,ch,eg,xo,vahk,grad,xu,xl,xl,x2,
* delta,alpha,alp,vaho,vago,vn,co,epsy,fy,ast,beta,theta,
* numec,vmu,cvmu,rv,cvload,become,1c,thesum,thesuml,
* dispsum,ni,nucomb,lct,xll,xl2,romin,rph,grad,vahold,
* vjac,vinv,bfal,epsilo)
implicit double precision (a-h,o-z)
dimension r(iqh),x(ntot),cl(n),cosl(n),cos2(n),d(iqg),
* alp(ntot), clag(iqgn), lm(6,n), vag(iqgn), vah(iqh),
* ch(iqh),xo(ntot),clah(iqh),eg(iqgn),beta(n),
* bfal(iqh,iqh),vahk(iqh),grad(ntot),xu(ntot),xl(ntot),
* xl(ntot),x2(ntot),xol(ntot),vaho(iqh),vago(iqgn),
* ast(n),theta(2*n,numec),vmu(n),cvmu(n),rv(iqh,numec),
* cvload(iqh),become(100),lc(100),thesum(n),thesuml(n),
* dispsum(iqh),ni(20),nucomb(20),let(100),vahold(iqh),
* vjac(iqh,iqh),vinv(iqh)
common /parr/ deefe,fcinc,cv,alpl,ec,rp,fc,es,ecm,relind
common /pari/ iter,numey,niter,ga,iqh,iqg,n,ntot,iqgn
common /esq/ u(6),ck(6,6),vksi(100),vksj(100)
iter = 0
c***************************************************************
c initializing for scaling
c***************************************************************
call lagfun (vlag,tvah,r,x,cl,cosl,cos2,lm,d,clah,vag,
* clag,vah,ch,eg,vof,vahk,vn,co,epsy,fy,beta,
* cvmu,cvload,kl,vmu,rph)
c***************************************************************