Reliability Analysis of Structures Using Modified FA-PSO Algorithm

Journal of Soft Computing in Civil Engineering 2-2 (2018) 89-101
How to cite this article: Shahmoradi Qomi H, Voitenko P, Taheri M. Reliability analysis of structures using modified FA-PSO
algorithm. J Soft Comput Civ Eng 2018;2(2):89–101. https://doi.org/10.22115/scce.2018.114034.1045.
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Journal of Soft Computing in Civil Engineering
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Reliability Analysis of Structures Using Modified FA-PSO
Algorithm
H. Shahmoradi Qomi1
, P. Voitenko2,*
, M. Taheri1
1. Ph.D. Student, Faculty of Civil Engineering, Semnan University, Semnan, Iran
2. Ph.D. Candidate, Graduate Research Assistant, Department of Civil Engineering, Auburn University, Auburn,
United States
Corresponding author: pzv0011@auburn.edu
https://doi.org/10.22115/SCCE.2018.114034.1045
ARTICLE INFO ABSTRACT
Article history:
Received: 08 January 2018
Revised: 23 January 2018
Accepted: 25 January 2018
Designing buildings with a very high safety factor is one of the
main purposes of a civil engineer. Since in the structural design
process, there are several no-confidence; we cannot achieve a
perfect safe design. In these cases, we face amount of the
probability of failure. So the theory of reliability used to assess
the uncertainty. This theoretical for expression the safety of a
system uses the reliability index, so it can be said that the
calculation of reliability index is an important part of the theory.
By the theory of structural reliability, uncertainties arising from
the nature of the statistical parameters can be written
mathematical equations and considerations of safety and
performance of the structure into the design process. Since
classical methods are not capable of solving complex functions,
metaheuristic algorithms used. In fact, a metaheuristic algorithm
is a set of concepts, which significantly able to solve many
complex issues, which they can reach an optimal solution in a
short time. In this paper, the particle swarm algorithm combined
with Firefly and to assess the reliability theory has been used.
Reliability index is calculated by searching the shortest distance
between the origin and the closed point of Limit State Surface in
the Standard normalized space.Mathematical and engineering
studies on the issues indicated; Hybrid Firefly and particle
swarm algorithm are with great accuracy and speed.
Keywords:
Reliability Index;
Engineering problems;
Limit state function;
Modified FA-PSO algorithm.

90 H. Shahmoradi et al./ Journal of Soft Computing in Civil Engineering 2-2 (2018) 89-101
1. Introduction
Reliability theory is the science of estimating the probability of system failure due to uncertainty
in design parameters. The definition of reliability varies with the context, but according to the
most commonly used definition, reliability is the probability that a component will function
properly for a specified period of time under specified conditions. The performance of a building
can be described as a function of its safety, serviceability, and cost-effectiveness. However, the
information available about input variables such as these is rarely, accurate, complete and
definite. With such uncertainties, it is impossible to acquire an absolute estimation about building
safety. The structural systems reliability theory allow us to use statistical concepts to formulate
the uncertainty of parameters and complete the design with safety and performance
considerations accounted for quantitatively. According to this theory, one should carefully study
not only the probability of failure in each single element but also the probability of failure of the
entire structural system as a whole. To do so, after ensuring the safety of every single element,
one needs to examine the combined effect of their interactions on the system safety.
Engineering problems are typically associated with a wide range of uncertainties. The nature of
these uncertainties varies with the nature of the problem and the type of analysis. The most
common types of uncertainty that engineers encounter during structural design and construction
include physical uncertainty, uncertainty in modeling, and uncertainty in statistics. These
uncertainties are particularly important for metal structures such as bridges, tanks, and
spaceframe structures, as uncertainties in the features of the site, nature of the applied loads, and
durability of the materials are more likely to lead to their failure or unserviceability. Structural
design should be able to account for a variety of uncertainties in -for example- the loads,
strengths, and geometric properties. This issue can be viewed as a statistical problem subject to
multiple uncertainties, which can be analyzed based on the principles of probability and
reliability theories. Such reliability analysis assigns each structural member with a certain
probability of failure, which allows us to evaluate its functionality or non-functionality and
decide when it should be repaired or replaced accordingly [1].
A question that regularly arises in regard to existing structures -even those that are in operation
for a long time is that how safe this building is? Clearly, this question is completely different
from the question that a designer must answer when planning a new building. Evidence also
suggests that checking the safety uncertainties of an existing structure with the building codes by
which it has been designed may not be helpful. This is because design codes and regulation have
been devised such as to account for certain uncertainties in the design and construction, but for a
built structure, these uncertainties are all identified (i.e., completely determined) and can no
longer be considered uncertainty. Also, determining the actual value of structural parameters of
an existing building is never easy and can have its own uncertainties [2].
The uncertainty in the safety of structures can be addressed more logically with the help of
probabilistic methods. Naturally, the main objective of structural safety assessments and
undertakings is to ensure the proper functionality of the building over its useful life. Some of the
parameters involved in this assessments are not constant but rather a sample from a set of

H. Shahmoradi et al./ Journal of Soft Computing in Civil Engineering 2-2 (2018) 89-101 91
probable values. In other words, these parameters can be considered as stochastic variables.
Basically, the variable nature of these parameters is not completely known, and it is suggested
that safety factors should be employed to account for their variations. Structural reliability
analysis is a method for achieving this objective [3].
Progress and development of the reliability theory have about 80 years of history consisting of
multiple periods. The first period from 1920 to 1960 was the era of the gradual emergence of the
structural reliability theory. In 1924, Forsell described the optimal design as a design that
minimizes the sum of expected costs, including the cost of initial construction and the expected
cost of failure [4].
In 1947, the principles of structural safety of members under random loads were introduced, and
this provided a foundation for future works of structural engineers [5]. In 1953, Johnson
published the first comprehensive formulation of structural reliability and economic design
theory, including the statistical theories of strength [6]. In 1974, Hasofer & Lind introduced an
efficient method for determination of reliability index through a search for the design point, that
is, the point of minimum distance from the limit state function to the origin of the standard
normal coordinate system [7]. In 1991, reliability problems were analyzed with the help of
gradient method [8]. Then, Melchers suggested a probabilistic model for assessing the reliability
of steel structures against marine corrosion [9]. In 2007, a series of studies investigated the utility
of genetic algorithms for structural reliability analysis by solving complex reliability functions
[10]. In 2011, a study on the reliability of wooden structures with ductile behavior introduced a
new evaluation method for this purpose [11]. In 2013, Jahani et al. developed an extremely
effective sampling methodology for solving the reliability problems [12]. In the same year, a
method was proposed for estimating the reliability of RC beams against fire [13]. Then, in a
study by Xiao et al., they proposed an efficient and effective method based on truncated random
variables for estimating the reliability of structural systems [14]. In 2015, several studies
investigated the utility of artificial neural network modeling in the reliability analysis of steel
structures [15]. Recently, a new generation of the reliability index has been defined by Ghasemi
and Nowak (2016-2017). The advantage of Ghasemi-Nowak formula is to measure the safety
level of a non-normal limit state function using the simplified convolution concepts [16,17]. For
evaluation of the required reliability index of structures, there is a need for expression of the
optimum safety level, which is known as target reliability. Ghasemi and Nowak (2017) and
Yanaka et al. (2016) proposed several approaches to estimate the target reliability for bridges
with respect to the minimization of the cost [18,19].
2. Structural reliability
Reliability relationship is defined based on two major parameters: structural strength and load.
Accordingly, the limit state function is written as follows [3]:
 
G R, Q R Q
 
(1)
Where G (R, Q) is the limit state function of structural strength and load. In the above equation,
strength function R and load function Q both consists of several random variables with different

probability distribution functions that depend on the structural dimensions, the type of materials,
and the loads exerted on the structure.
In the reliability analysis, the failure is expressed by the concept of limit state function, which
represents the extent of functionality. A limit state function that is equal to zero (g(x) = 0)
represents a state that lies on the boundary of functionality (safety) and dysfunctionality
(unsafety). Accordingly, g (x)> 0 has been defined as the safety region and g (x) <0 as the
unsafety region.
2.1. Classification of reliability methods
There are various methods for idealizing the models of structural reliability, and various ways to
combine the idealized models for a design problem. Given this high degree of diversity, these
methods have been classified as follows.
Level-I methods: In these methods, only one characteristic value (usually the mean value) of
each uncertain parameter is used. The allowed stress methods are examples of this group.
Level-II methods: these methods rely on two characteristic values (usually the mean and
variance) and the correlation between them (usually covariance). The reliability index methods
are examples of this group.
Level-III methods: These methods use the probability of failure as a measure, and thus require
the joint distribution of all uncertain parameters.
Level-IV methods: these methods analyze our expectations from the structure with a measure of
acceptability defined based on the probabilistic conditions and in accordance with the principles
of engineering economics analysis, i.e., the cost and benefit of construction, maintenance, repair,
failure consequences, etc. This approach is suitable for structures of great economic importance
such as highway bridges, power transmission towers, nuclear installations, offshore platforms
and ship and airplanes structures, but is still in development.
2.2. Cornell reliability index
In 1969, Cornell defined the reliability index as follows:
[ ]
[ ]
c
E M
D M
 
(2)
Cornell’s reliability index provides a reasonable estimation of reliability by measuring the
distance from E[M] to the failure surface as a multiple of uncertainty parameter D[M] (E[M] and
D[M] are the mean and standard deviation of the safety margin).
In Cornell’s initial formulation, the failure function was expressed as the difference between
strength R from the effect of load S.
 
,
G R S R S
 
(3)
The safety margin corresponding to the above relationship is:

M R S
  (4)
Assuming that R and S are independent, the reliability index can be calculated as follows:
[ ] [ ]
[ ] [ ]
c
E R E S
Var R Var S




(5)
If the limit state function is a first-order function, we have:
0
1
( ) ( )
n
i i i
G x a a x
  
(6)
Thus, the reliability index is given by:
[ ]
T
T
x
a a E X
a c a



(7)
Where E[16] denotes the expected value of the main variables, and the matrix Cx is their
covariance. This reliability index will remain constant under any linear transformation. When the
limit state function is not linear, it is impossible to formulate linear safety margin based on the
main variables, because the mean and variance of the main variables are not enough to derive the
expected value and variance of the safety margin.
2.3. Hasofer & Lind reliability index
As mentioned earlier, Hasofer & Lind [7] have introduced an efficient method for determination
of reliability index. In this method, reliability index is obtained through a search for the point of
minimum distance from the limit state function to the origin of the standard normal coordinate
system. The distance of the failure surface from the origin is called the reliability index function
and has been defined as follows.
0.5
( ) min( . )
T
c
U U U
 
(8)
 
U G U 0
 
(9)
Where G(U) is the limit state function in the standard normal domain of the base random
variables (U). Given the above relation and the definition of HL, the reliability index for has been
defined as follows.
0.5
( ) min( . )
T
c
U U U
 
(10)
 
Subject to : G U 0

(11)
In the above equation, which is known as the first-order reliability equation, the point
U*(u*
1,u*
2,…,u*
n) on the failure surface, G(U)=0 is called the design point.

2.4. Reliability Index & Probability of Failure
The most important application of reliability index is the estimation of failure probability. The
relationship between the two can be expressed as follows:
1 1
(1 ) ( ) ( )
f f f
p p p
    
 
      
(12)
Where Pf is the failure probability,  is the safety index, and Ø is the standard normal
distribution function. As mentioned,  can be determined by various methods, and then, the
probability of failure can be calculated by the above equation.
3. Proposed algorithm
Firefly algorithm is a meta-heuristic optimization algorithm inspired by the simple
communication behavior of fireflies. A firefly can understand its environment through its sensors
and comminute with the environment using the light produced from its lower abdomen. Studies
have shown that fireflies are social insects living in colonies and that their behavior is governed
by the desire for the survival of the self as well as the colony. One of the most interesting facts
about fireflies is their behavior during the search for food, and in particular, how they find the
shortest route between food and the nest. For this purpose, each individual firefly moves to the
point where it has had the best experience of finding food. The light produced by Firefly acts as a
signal that attracts other fireflies or preys. Firefly colony can be view as a natural manifestation
of swarm intelligence, in which the cooperation (or competition) of simpler and less intelligent
members generates a higher degree of intelligence unattainable by any individual member.
Recently, the social behavior of fireflies and their swarm intelligence have fascinated and
inspired scientists working in other fields.
Inspired by the aforementioned social behavior, in 2009 Yang developed the firefly algorithm
with the following assumptions [17]:
1- Fireflies are unisexual, therefore all fireflies will be attracted to each other. The attraction
intensity (I) has been defined as follows:
2
r
0
I r I
( e
)



(13)
Where I0 is the initial light intensity, γ is the constant light absorption coefficient, and r is the
distance between the two fireflies.
Since attractiveness of a firefly is proportional to its light intensity as viewed by others, the
attractiveness function β(r) has been defined as follows:
 
2
r
0
r e

 


(14)
In this function, β0 is the attractiveness at r = 0.

2- The attractiveness of a firefly is proportional to its brightness. Thus for any two fireflies,
the less bright one will move toward the brighter one. However, the perceived brightness and
attractiveness is inversely proportional to distance. Thus, as distance increases, perceived
brightness decreases, and so does the attractiveness.
The distance between the fireflies i and j located at xi and xj is given by the following equation:
 
d
2
ij i j i ,k j ,k
k 1
r | x x | x x

   

(15)
Where xi,k is the k-th spatial coordinate of firefly i, and d is the number of problem variables.
3- The brightness of each firefly is determined by the objective function defined for the
problem.
Finally, each firefly (i) moves toward the most attractive firefly in its view (j) in accordance with
the following equation:
 
t t t t
i i j i
x x x x
2
i , j
r
1 t
0 i
e

 


   
(16)
In this equation, the second term represents the relocation resulting from attraction and the third
term introduces some randomness into the movement. The coefficient α is the motion
randomization parameter, and εi is a random vector with normal distribution in the interval [0, 1].
In most optimization problems, β0 = 1.
The value of the parameter γ plays a critical role in algorithm’s rate of convergence and its
behavior. Although in theory , in practice, γ must be between 0.1 and 10.
The primary merit of firefly algorithm (FA) is its very good accuracy in solving optimization
problems, but it is rather slow when implemented with large populations. In contrast, the particle
swarm optimization algorithm (PSO), which is inspired by the collective behavior of bird
swarms, exhibits higher speed in solving complex optimization problems.
In the proposed hybrid algorithm, first, firefly algorithm will be run for a specified number of
iterations (FAiter), and the best solution among all iterations will be registered. Then, this best
solution will be imported into PSO to calculate the velocity of each particle and update its
position accordingly. The output of PSO function for the new position will be compared with the
current best solution, and if a better solution is found, the best solution will be updated
accordingly (standard PSO procedure). This PSO will be run for a specified number of iterations
(PSOiter). Finally, all of the above steps will be repeated for a specified number of iterations
(HYBiter). This algorithm is designed such that the ratio of the combination of two algorithms
and their population are adjustable. In this algorithm, the best solution is the minimum distance
between the failure surface of the limit state function and the origin of the standard normal
coordinate system. In summary, the reliability index and the failure probability are obtained by
 
γ 0,
 

the use of the Hasofer-Lind definition of the reliability index and the search for the optimal
solution by the algorithm.
The pseudocode of the proposed FA-PSO hybrid algorithm is presented in Figure (1).
4. Problem
This section is divided into two parts, one dedicated to mathematical problems and the other
dedicated to reference engineering problems. In each part, we use the proposed hybrid algorithm
to solve a number of typical problems already solved through different methods and with
different algorithms.
Fig. 1. Proposed pseudocode FA-PSO.

4.1. Numerical examples
The accuracy and performance of the proposed algorithm are expressed by the following
mathematical functions. These functions are used for comparison with other methods according
to the number of involved statistical parameters and variables.
2
1 2 1 2
( , ) 5 0.5( 0.1)
G X X X X
   
(17)
1 2 1 2
( , ) exp (0.2 1.4)
G X X X X
    (18)
   
   
1 2
1, 2 exp 0.4 2 6.2 exp 0.3 5 200
G X X X X
      (19)
The results concerning the reliability index and the failure probability function for the above
mathematical functions are presented in Table (1). For mathematical problems, the reliability
index is obtained through Hasofer-Lind method. Comparison of the results demonstrates the
good accuracy and performance as well as the high speed of the proposed algorithm.
Table 1
Comparison of results in numerical examples.
Limit Function Variable Algorithm U*
µ σ  Pf
Eqe. 17
X1
X2
PSO [20] (
0.9625
،
-2.7416
)
0 1
2.905 0.00183
CSS [21] (
0.9607
،
-2.7408
) 2.906 0.00183
FA-PSO (
0.9241
،
2.7551
) 2.905 0.00183
18
X1
X2
PSO [20] (-1.6882, 2.8931)
0 1
3.497 0.000405
CSS [21] (-1.6797, 2.8981) 3.35 0.000404
FA-PSO (-1.6826, 2.8964) 3.349 0.000404
19
X1
X2
PSO [20] (-2.5477, 0.9235)
0 1
2.709 0.00337
CSS [21] (-2.5396, 0.9453) 2.71 0.00336
FA-PSO (-2.5783, 0.8407) 2.709 0.00330
4.2. Engineering Problems
4.2.1. Simply-supported beam
In this problem instance, a simple beam is subjected to a point load (Figure 2). For this problem,
the limit state function is defined as:
2
( , , ) 48 100
G E I P EI PL
  (20)

Fig. 2. Simply-Supported Beam.
The beam has a length of 6 meters. The mean values and standard deviations of variables P, E,
and I are given in Table (2).
Table 2
Random variable for Simply-Supported Beam.
Variable Mean Standard Deviation Distribution
P (kN) 2 0.6 normal
E (kN/m2
) 2×107
3×106
normal
I (m4
) 2×105
2×10-6
normal
In Table (3), the optimum value of the reliability index β and the failure probability Pf as
calculated by the proposed algorithm are compared with the results of previous methods.
Table 3
Comparison between Risk Analysis and FA-PSO algorithms.
Method Optimum Point  Pf
Risk Analysis [20] - - - - 3.15 0.00081
FA-PSO
P = 3.12
E = 1.316×107
I = 1.77×105
3.148 0.00082
As shown in Table (3), the results obtained from the Modified algorithm are highly accurate and
give a conservative estimation of beam failure probability. The convergence diagram of the
proposed algorithm is plotted in Figure (3).

Fig. 3. The convergence diagram of the algorithm.
4.2.2. three-span continuous beam
In this problem instance, a three-span beam of 15 meters in length (5 meters per span) is
subjected to a distributed load (Figure 4).
Fig. 4. 3-span continuous beam subjected to a distributed load.
In accordance with AISC regulations, the allowable displacement in this beam is L/360, and its
limit state function is:
4
( , , ) 0.0069
360
L wL
G w E I
EI
  (21)
This problem involves three variables w, E, and I, which all have a normal distribution function.
Table 4
Random variable for a continuous beam.
Variable Mean Standard Deviation Distribution
w (kN/m) 10 0.4 normal
E (kN/m2
) 2×107
0.5×10-7
normal
I (m4
) 8×10-4
1.5×10-4
normal

The results obtained for this problem instance are presented in Table (5).
Table 5
Comparison between the HL-RF and FA-PSO algorithms.
Method Optimum Point  Pf
HL-RF [22]
w = 10.044
3.1805 0.000735
E = 4368200
I = 7.139×10-4
FA-PSO
w = 10.043
E = 4374200
I = 7.128×10-4
3.1805 0.000735
The convergence diagram of the proposed algorithm is shown in Fig. 5.
Fig. 5. the convergence diagram of the algorithm.
5. Conclusion
This paper introduces a hybrid algorithm consisting of firefly algorithm (FA) and particle swarm
optimization algorithm (PSO) for determination of reliability index and failure probability. The
good optimization performance of both FA and PSO has been acknowledged in many
publications. Thus, a combination of these two can be remarkably effective in reliability analysis.
In the proposed algorithm, FA-PSO combination ratio is adjustable, and can therefore, be fine-
tuned to optimize the runtime and accuracy according to problem requirements and
characteristics. The capability of the proposed algorithm in solving reliability theory problems
without the limitations of classical approach is verified with the help of a set of mathematical and
structural problems. The satisfactory performance and accuracy and acceptable speed of the
proposed algorithm are demonstrated by comparing its results with those of alternative methods.

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