Chapter 5

RELIABILITY BASED OPTIMISATION

5,1 Introduction

The important objective in engineering design is the assurance of

structural safety and reliability. Factor of safety or load resistance factors are

commonly used to ensure structural safety rather than consistent probabilistic

analysis, However, it is generally recognized that there is always some

uncertainty involved in any structural system due to the variations in material

properties, improper definition of loading environment and the manufacturing

tolerances, In the design of structures, the strength is a random variable since it

varies considerably from sample to sample. Similarly, in the design of mechanical

systems the dimensions are random since the dimensions may lie anywhere

within the specified tolerance bands. Even the loads acting on the structure are

also random. All these factors stimulated a search for consistent and

mathematically correct solutions of structural safety problems. The solution is

achieved by taking the advantage of probabilistic methods which can be used to

handle the random character of structural parameters as well as uncertainties

arising in the formulation of design problems.

Recent developments in rapid growth of computing power have resulted

in high performance computing at relatively low cost. So the researchers are

attracted towards realistic optimal design modeling by minimizing the

approximations and assumptions. In general optimum structural design aims at

arriving at a design such that its weight or cost is minimum. The factors that

affect optimal design of discrete structures are cross sectional properties of

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members, configuration defined by the position of joints and topology of the

structure. Major part of the work is being carried out in the field of size

optimization. In this field the cross sectional properties are allowed to vary with

constant configuration and topology. Configuration optimization and topology

optimization are less popular because of the difficulty in selecting proper

mathematical programming techniques to handle different types of design

variables.

The objectives of this chapter are

~ To optimize truss structure considering the random character of

structural parameters.

a To develop technique which can be used to handle probability

based design problems

• To validate the method by comparing the results with classical

optimization methods.

5.2 Literature review

Deterministic optimization techniques have been successfully applied to a

large number of structural optimization problems during the last decades. The

main difficulties in dealing with nondeterministic problems are lack of

information about the variability of the system parameters and the high cost of

calculating their statistics. These difficulties were circumvented with the

introduction of probabilistic design where the mean and covariance of the

random parameters influencing the design alone are considered. A formulation

was suggested by Charm~s and Cooper(1959) by converting the stochastic

problem in to an equivalent deterministic one using chance constrained

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programming technique. The objective and constraint functions that depend on

random variables are expanded about the corresponding mean value.

The studies conducted by S.EJozwiak(1988) to optimize the truss

structures, the mean value of the structural mass is taken as the objective

function and the capacity as the constraint. The random character of the

structural parameters is considered by estimating the value of the coefficient of

variation of element strength.

Most structural optimization programs use deterministic criteria for

optimization which ignore the statistical properties of structural loads, materials

and performance models. To counter these shortcomings Y. W. Uu and F. Moses

(1992) presented a risk -oriented optimization formulation. Constraints for the

initial installed structure and system residual reliability corresponding to the

damaged structure were considered.

}. }. Chen and B.Y. Duan (1994) presented an approach for structural

optimization design by means of displaying the reliability constraints. The non

normal loads acting on the structure are transformed to normal loads by using

normal tail transformations. The displacements and stresses, reliability

constraints under random loads, are transformed in to constraints of

conventional forms. This method is suitable for truss structures subjected to one

or multiple random loads in any types of distribution.

M.V.Reddy et aL, (1994) developed a probabilistic analysis tool suitable

for optimization based on second moment method. Improved safety index

method is used for minimum weight design and optimization is done by

extended interior penalty method.

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Reliability based optimum design procedure for transmission line towers

was the objective of the study conducted by K. Natarajan and A. R. Santhakumar

(1995). A realistic and unified reliability-based optimum design procedure is

formulated, Studies were also conducted on the relationship between (i) weight

and system reliability of the tower and (ii) co efficient of variation of variables

and system reliability,

Chareles Camp, et aL, (1998) conducted studies on two dimensional

structures, GA based design procedure is developed as a module in Finite

Element Analysis program, The special features include discrete design

variables, multiple loading conditions and design checking using American

Institute of Steel Construction Allowable Stress Design, The results were

compared with classical optimization methods and found that this method can

design structures satisfying AISC- ASD specifications and construction

constraints while minimizing the overall weight of the structure

According to CoK. Prasad Varma Thampan and COS, Krishnamoorthy

(2001), for optimization of structures, it is essential to consider the probability

distribution of random variables related to load and strength parameters, Also

system level reliability requirements are to be satisfied, They concluded that

better optimal solutions are obtained by genetic algorithm based RBSO of

frames,

Main objective of the study conducted by Tarek N Kudsi and Chung C Fu

(2002) was to develop a new methodology for redundancy analysis of structural

systems, The structural systems were modeled as a collection of structural

elements in series and paralleL The redundant element is assumed to be parallel

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with the rest of the system and the non redundant member is considered to be in

series with the rest of the system.

Claudia R Eboli and Luiz E. Vaz (2005) illustrated two different

approaches of the reliability-based optimization problem using reliability index

approach and performance measure approach. Both methods led to the same

optimization solution. It is observed that performance measure approach is more

reliable regarding the performance.

V.Kalatjari and P. Mansoorian (2009) have attempted to approximate the

probability of structural system failure. The optimization of the truss is

performed in two different levels using parallel genetic algorithm. The inefficient

chromosomes are discarded by the first level and an initial population is created

for the second level thereby saving considerable computational time. Faster

convergence is achieved by competitive distributed genetic algorithms

Todd W. Benazer, et aL, (2009) proposed s solution method for

minimizing the cost of a system maintaining the system reliability. The cost

efficient design was achieved by performing a reliability-based design

optimization using the statistical spread of structural properties as design

variables. The computational time was reduced by using meta models. Finite

elem~nt analysis was used to initialize the optimization problem and for each

ensuing iteration, the analysis was only performed if the desired point of

evaluation did not have two previous evaluations within the prescribed move

limits. These move limits ensure that an approximation was not used in an

unexplored region of the design space. In the present study the move limits were

set to a maximum change of any input variable of 10%.

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Reliability based optimization of two and three dimensional structures is

studied by M.RGhasemi and M Yousefi (2011), Applied load and yield stress

were the considered as probabilistic, the failure criterion was the violation of

interior forces from the member ultimate strength. Optimisation was done by GA

and the constraints were the failure probabilities and the objective was to

minimize the weight of the structure. Results obtained indicated that, for

prevention of nodal failure, one should define a nodal failure probability

constraint assuring that displacement in members and drift in floors do not

exceed from allowable values.

5.3 Structural Analysis under Multiple Random Loads.

In reliability-based analysis, uncertainties in numerical values are

modeled as random variables. Loads, material properties, element properties,

boundary conditions, dimensions, and finite element model discretization error

are the quantities modeled as random. If one or more quantities are modeled as

random, reliability-based analysis is needed. Each random variable is assigned a

probability distribution. Distribution can be defined by a mean, 11, a standard

deviation, and a distribution type.

If a linear elastic structure is subjected to S normal loads, the

displacements and stresses are also normally distributed because of the additive

property of normal distribution. Displacements vector 6(l), (l =1, 2, 00000' S ) can

be found from the finite element equation such that the elastic structure is

subjected to S normal random loads simultaneously,

(5.1)

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where

[K] is the structural stiffness matrix,

peL) ,(l = 1, 2, ... ,S) is the lth normal load.

Considering the relationship between stress and displacement, the

random vector of stress for an arbitrary element 'j 'is as given below.

[o"P) ,a?) '000. U?) ] = [1]] [0(1), 0(2), 000 •, o(S) ] (5.2)

[ap) ,a?) ,.... a?) ]=[1]] [Kt1 [p(1),p(2), .... ,p(S)] (5.3)

j =1,2, .... ,NE

where

[1j] is the matrix of the relationship between the jth joint displacement

and the jth element stress.

oW and afL) ,(l = 1,2, ... ,S)are the random vectors of joint

displacement and the jth element stress, respectively, under the l th normal load.

NE is the total number of structural elements.

All the random loads are assumed to be normal loads, therefore

P(l)-N [E(P(O), D(P(O)] ,l = 1,2, .... ,S (5.4)

According to the principle of invariance of the responses in a linear elastic

structure to the normal loads, the random vectors of displacement and stress will

satisfy the following normal distribution.

o(O-N [E(8(l)), D(c(l))] , l =1,2, .... ,S

a(O-N [E(af°), D(afO)] ,

= 1,2,>... ,5; j = 1,2,.".,NE

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(5.5)

(5.6)

where

He) and DC) are the operators of the expectation and variance

separately.

Applying the superposition principle of the linear elastic structures,

which are subjected to S normal loads simultaneously, gives the following form

of random variable Zk for the kth response element

Z ~s Z(i)k = ~i=l k

where

(5.7)

Zki) is the response parameter of the k th element in the structure under

the lthnormal random load.

Considering the formulae 4.5 and 4.6 and according to the reproductive

characteristics of the linear combination of normal variables, the distribution of

Zk can be expressed as

(5.8)

The expectation and variance of the variable Zk can be described as

(5.9)

(5.10)

where

J.l~) and u~Z) (l =1,2, .... ,S) denote the expectation and the variance of

the response to the k th element of the structure subjected to the lth normal load;

Plr(l ,r = 1,2, .... ,S) is the relation coefficient between the lth and the

r th normal loads.

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Equation (5,10) is a general expression ofthe variance of variable Zk' If S

loads are independent of each other, I.e, PlT =o(I. r =1,2, ".. ,5) , the variance

ofvariable Zk will become

(5.10a)

If S loads are dependent of each other completely, I.e, PlT =1(l , r =

1,2, ",.,5, the variance ofZk holds

(~~ ,..(1))2u!=l uk (5.10b)

5,4 Normal Transformation of Non Normal Random Loads

If the load to which the structure is subjected is not the normal one, it can

be transformed in to an equivalent normal load by means of the following

technique.

The normal variates are generated by Box and Muller technique,

(Ranganathan,1990), Standard normal deviates are obtained by generating two

uniform random numbers Vl and V2 in the range 0 to 1 at a time. The desired

normal deviates are given by

Ul = [2 In 1/ vlf~ COS(21l"V2 )

U2= [21n l/vJIh sin(2nv2)

(5.11a)

(5.11b)

Standard normal variate is connected to the normal variate Yas follows:

y- J.L-=u

u

where

U is the standard normal variate.

Hence we can get two normal variates Yl and Y2 using the equations (5.11a) and

(5.11b).

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Yi = a Ui + 11 (5.12a)

Yz = (1 Uz+ 11 (5.12b)

That is

Yi = It + (J [21n l/Vi]* COS(21l"Vz) (5.13a)

Yz = 11 + C1 [21n 1/Vi]*sin(2nv2) (5.13b)

5.5 Reliability Optimization by Reliability Constraints

Structural reliability optimization is to find sizes of all the members of a

structure to minimize the objective function such as the weight of the structure

while satisfying the reliability of structural displacement and element strength.

Suppose that the reliability constraint to which the response parameter

(displacement or stress) in the structure subjected to Snormal loads is

(5.14)

where

R;k is the predefined reliability of k th response;

RZk is the value of reliability by which the reaction Zk is less than

or equal to its allowable value.

Applying first order second moment theory, the reliability

RZk =cfJ(f3)

where

f3 is called safety index.

f3 = [x:ll]

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(5.15)

(5.16)

(5.17)

(5.18)

(5.19)

5.6 GA-Based Methodologies for Optimal Design of Trusses

Genetic algorithm suggested by Goldberg (1989) and Krishna Moorthy

and Rajeev (1991) is a different search algorithm used to optimize the truss

system involving area of members as discrete design variables. In genetic

algorithm based methodologies, the design space is transformed to genetic

space. This transformation is achieved by appropriate genetic coding schemes.

Binary coding scheme is the most popular one and is used to code the design

variables.

It is required to optimize the weight of the pin jointed truss subjected to

stress and displacement constraints. The objective function is to minimize the

weight and the weight function [(x) is written as

(5.20)

where

Ai is the cross sectional area of the i th member,

Li is the length of the i th member,

P is the weight density of material

NE is the number of elements

Area of cross section of the members of the truss is taken as variable. The

available sections are given as input.

(5.21)

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where

{A} is the cross sectional area of elements.

The constraints equations are given below.

[RZkls S [RZkls

k = 1,2, ...... ,N]

where

N] is the number of displacements

[RZk]q S [RZkt

k =1,2, ... ... ,NE

where

NE is the number of elements.

where

(5.22)

(5.23)

(5.24)

R**zk is the predefined reliability of the structural system.

Binary individual strings are generated using genetics to represent the

variables. Number of strings in each generation or the population size is also

varied. Stresses in the members and deflection at various joints were obtained

using finite element program. The violation coefficient which is the sum of values

of all violated constraints is calculated using Eqn. (5.25) and then the fitness

function F for each generation. The fitness function has to be converted in to

corresponding fitness values. The best population is the one which has maximum

fitness value.

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C = I:Bjfor Bj > 0

F = f(x)(l + C)

(5.25)

(5.26)

The populations are mated randomly and crossed at random lengths of

the full string and thus individuals for next generation are obtained. The

variables for each population are obtained by decoding the strings. The process

is repeated until minimum weight is obtained without violation of the

constraints. The optimal areas of members are the values of variables for which

the weight is minimum and satisfies the constraints.

5.7 Optimization of 10 bar truss Using Reliability Method and Genetic

Algorithm

A ten bar truss shown in Fig. 5.1 is to be designed. The acting loads are

Pl and pz which are random variables. The geometry of the truss is as shown in

Fig. 5.1. The strength of steel is a random variable with a mean value

Fy =25kN/ cmzand coefficient of variation (Vs ) = 10%.

The cross sectional areas of all the ten members are to be determined.

The design requirements are

Weight of the structure should be minimum.

Maximum allowable probability of displacement exceeding limiting value 0.015.

Maximum allowable probability of stress exceeding limiting value 0.001.

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Fig. 5.1 Ten bar truss

x

L=360 "(914.4cm)

Fy = 25 kN/cm2

E=20000 kN/ cm2

p = 0.1Ib/in3(2.72 X10-5 kN/cm3)

Pl and Pz are normal loads with the following parameters.

E(Pl), E(Pz) = 100 kips( 445.374 kN)

Displacement constraint

R8i ::: prob {(-2.0 in < 8 ~ 2.0 in)} ~ R6 ::: 0.985,

i =1,2, ..... ,10

Stress constraint

miniSj:~ao{Ruj} = miniSjSl0 {prob(O'j ~ [0" tn} ~ R~ = 0.999

Aj~ Amin = 0.10 inZ,j =1,2, ..... ,10

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The first step is to convert the non normal load to normal loads by Box

and Muller technique. Truss analysis is carried out by direct stiffness method for

each set of loads. Mean and variance of member stresses and displacements

corresponding to the random loads were then calculated. Reliability index P is

found out and using this value the reliability is obtained from the normal

distribution curve. Optimization of the structure using genetic algorithm is done

using the reliability constraints.

The problem is run with the following genetic parameters.

String length =40

Population size = 20,30,40

Probability of cross over =0.7

Probability of mutation = 0.001

Convergence parameter =85%

Number of simulations =200

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Table 5.1 Results of 10 bar truss problem with population size 20 and stringlength 40

Method With reliability Without reliabilityconstraints constraints **

Weight (kN) 22.8001 22.72604

Ai (cm2) 128.39 176.29

A2 (cm2) 16.97 0.65

A3 (cm2) 128.39 154.85

A4(cm2) 89.68 100.01

As (cm2) 11.61 0.65

A6(cm2) 16.9 0.65

A7 (cm2) 109.03 54.84

As (cm2) 109.03 135.49

A9(cm2) 128.39 135.49

AlO (cm2) 18.58 0.65

**Rajeev, S. (1993)

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Table 5.2 Results of 10 bar truss problem with population size 30 and stringlength 40

Method With reliability Without reliabilityconstraints constraints **

Weight (kN) 22.3911 22.70995

Ai (cmZ ) 128.39 183.88

Az (cmZ ) 16.90 0.65

A3 (cmZ ) 128.39 167.75

A4(cmZ ) 89.68 96.78

As (cmZ ) 12.84 0.65

A6(cmZ ) 13.74 3.23

A7(cmZ ) 91.61 51.62

As (cmZ ) 109.03 132.27

Ag (cmZ ) 128.39 141.94

AlO (cmZ ) 20.19 0.65

**Rajeev, S.(1993)

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Table 53 Results of 10 bar truss problem with population size 40 and stringlength 40

Method With reliability Without reliabilityconstraints constraints **

Weight (kN) 22.672 22.78431

Al (cm2) 141.94 122.59

A2(cm2) 18.9 0.65

A3 (cm2) 128.39 148.40

A4(cm2) 89.68 100.01

As (cm2) 13.74 0.65

A6(cm2) 13.74 0.65

A7(cm2) 89.67 51.62

As (cm2) 109.03 129.04

A9 (cm2) 128.39 132.27

AlO (cm2) 18.06 0.65

**Rajeev, S. (1993)

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Table 504 Comparison of optimal design results of a ten bar truss

With reliability Without reliability Classicalconstraints constraints * results**

Population 30 30sizeWeight (kN) 22.3911 22.70995 25.268

Al (cm2) 128.39 183.88 162.52

A2 (cm2) 16.90 18.9 0.65

A3 (cm2) 128.39 128.39 135.23

A4(cm2) 89.68 89.68 132.92

As (cm2) 12.84 13.74 0.65

A6(cm2) 13.74 13.74 2.45

A7 (cm2) 91.61 89.67 110.34

Aa(cm2) 109.03 109.03 134.30

A9 (cm2) 128.39 128.39 143.99

AlO (cm2) 20.19 18.06 0.65

*Rajeev, 5.(1993)

**Chen, J. J. and Duan, B. Y. (1994)

5.8 Effect of population size on the solution

Number of generations required for the desired convergence is influenced

by the population size. Population size is to be selected properly to have good

performance. Too small populations will require less number of iterations to give

better results. On the other hand, a population with higher number of individuals

will result in longer waiting time for significant improvements, since more

number of genetic operations are required to obtain convergence. The optimum

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solution obtained for the truss with the following parameters is presented in

Table. 5.5.

The problem is run with the following genetic parameters.

String length = 40

Population size = 20, 30, 40

Probability of cross over =0.7

Probability of mutation =0.001

Convergence parameter = 70

Number of simulations = 200

Table 5. 5 Effect of population size with reliability constraints

Population size Weight (kN) No. of No. ofgenerations evaluations

20 17.23928 24 307

30 16.26728 30 567

40 15.1216 76 1926

Table 5. 6 Effect of population size without reliability constraints*

Population size Weight (kN) No. of generations

20 22.72604 245

30 22.70995 237

40 22.78431 244

*Rajeev, S. (1993)

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5.9 Effect of string length on the solution

The problem is run with the following genetic parameters.

String length = 30,40

Population size =20

Probability of cross over =0.7

Probability of mutation = 0.001

Convergence parameter =70

Number of simulations =200

Table 5. 7 Effect of string length with reliability constraints

Population String Weight No. of No. ofsize length (kN) evaluations generations

20 30 22.72604 463 37

20 40 22.70995 160 13

Table 5. 8 Effect of string length - without reliability constraints**

Population size String length Weight (kg) No. of generations

20 30 22.68347 222

20 40 22.72604 245

** Rajeev, S. (1993)

It is seen that larger string length and larger populations require more

generations to converge. Increase in length of string leads to the increase in the

number of possibilities. Increase in the number of generations is due to the

increase in number of possibilities being tried. Hence, to get faster convergence

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and better solutions in less number of generations, the string length is to be

minimum.

When the constraints are in terms of reliability, faster convergence is

achieved with longer strings. With the increase in string length the number of

possibilities also increases. Better offsprings are selected considering the

reliability of the structure which leads to faster convergence with less number of

generations.

5.10 Summary

Reliability based optimization technique is explained in this chapter. The

constraints in terms of displacements and stresses are converted to reliability

constraints of displacements and stresses. Reliability is calculated in terms of p

index. A ten bar truss is considered for validating the proposed theory. It is

observed that the results obtained by considering the reliability constraints are

better compared to the results obtained by simple optimization by genetic

algorithm technique. Loads were random and the number of simulations used is

200.

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