RELIABILITY BASED OPTIMISATION -...
Transcript of RELIABILITY BASED OPTIMISATION -...
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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