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 63
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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.