STRUCTURAL FUZZY RELIABILITY ANALYSIS USING THE CLASSICAL RELIABILITY THEORY

This study is focused on a novel approach for calculating structural fuzzy reliability by using the classical reliability theory. In order to handle the structural fuzzy reliability problem, the formulae for establishing normal random variables equivalent to symmetric triangular fuzzy number are presented. From these equivalent random ones, the original problem is converted to the basic structural reliability problems, then the methods of the classical reliability theory should be applied to calculate. Moreover, this study proposes two notions in terms of central fuzzy reliability and standard deviation of fuzzy reliability as well as a calculation procedure to define them. Lastly, the ultimate fuzzy reliability of the proposed method is established and utilized to compare the allowable reliability in the design codes. Numerical results are supervised to verify the accuracy of the proposed method.


INTRODUCTION
In engineering structures, most of the input data, such as load characteristics, material properties, boundary conditions, geometric dimensions, load-carrying capacities, contain non-deterministic quantities, which are described as uncertainty variables. In [1], uncertainties present in a structural system can be categorized as either aleatory or epistemic. If the input data are random parameters, the limit state function is a random parameter, is expressed as follows M=R-S (1) where M is the limit state function; R is the resistance function; S is the load effect function. The structural reliability is defined as follows [2,3] s P = Prob M> ( 0) Due to using the probability theory, which is the most complete theory, assessing structures by the reliability index is prescribed in the structural design code [4]. When the input data in structural systems are epistemic uncertainties, depending on how to describe the uncertainties, the structural reliability can be derived by the different approaches, as using either the fuzzy sets theory [5÷12] or the fuzzy random theory [13,14]. In this study, we only consider that the epistemic ones are represented as fuzzy numbers, which are always interested in the reality problems. In this case, the structural reliability had the different names, such as the safety possibility of structures [6], the fuzzy reliability index [7,8], the fuzzy reliability [9,10,11]. The terminologies are named the fuzzy reliability later.This fuzzy reliability can be classified into three classes: the resistance function R and the load effect S are fuzzy numbers, the resistance R is a random parameter and the load effect S is a fuzzy number, the resistance R is a fuzzy number and the load effect S is a random parameter. These approaches [5÷12] for the fuzzy reliability are detailed analyzed below. According to Dong et al. [5], the fuzzy failure is determined based on the two α -cuts M α and R α (Fig.1a) as follows In Fig. 1a, the fuzzy failure (FF) and the fuzzy reliability (FR) are expressed as follows, respectively  (4), one realizes that the fuzzy failure is the average of the possibility measure and the necessity measure of event A (Fig. 1b) according to the possibility theory [15]. Hence, the method [6] is an approximation of the average, and the transformation from logical expression (3) to formula (4) is an intuitive formula. Besides, the effect of the spread of fuzzy numbers R~ and S~ aren't considered absolutely. Differed from Dong et al. [5], Sherstha and Duckstein [6] considered directly M, which is the subtraction of the fuzzy resistance number and the fuzzy load effect number, and the fuzzy reliability is defined as the ratio of the membership function of M which is greater than 0 to the total area of the membership funtion of M ( Fig. 2) Figure 2: The method [6,7] where μM(x) is the membership function of fuzzy number MF igure 3: The interference model [10,11] The formula (5) is approriate for the geometrical definition of probabiliy in [16]. Nevertheless, it should be realized that the membership functions in the fuzzy sets theory and the density distribution functions in the probability theory are not equivalent representations of uncertainty. Hence, the fuzzy reliability in the formula (5) is only similiar to the probabilistic reliability in the classical reliability theory. Based on the formula (5), Park et al. [7] determined the fuzzy reliability when the membership funtion μ M (x) is triangular and trapezoidal. Rezaei et al. [8] extended formula (5) in case fuzzy variables formed the pyramid to asess fuzzy reliability of the wing flutter speed.
In [9], Li et al. proposed a formula for fuzzy reliability analysis of mechanical structures when the stress S was modeled as a fuzzy number with given membership function μ S (x) and the strength R was modeled as a random variable with given distribution function f R (x) (Fig. 3). The authors idealized that fuzzy reliability in the fuzzy stress -random strength interference model was a real value, and the Zadeh's notion for the probability of a fuzzy event [17] is used to calculate. The fuzzy reliability is given as follows The drawback of this method is that two functions are under integral in the formula (6) have different measurements, the area between the curve f R (x) and the abscissa is unit but for the fuzzy number μS(x) isn't. In order to overcome this drawback, Jiang and Chen [10] supposed that the α-cuts of a fuzzy number were the linear distributions, then computed the conventional probability at various α-cuts and the fuzzy reliability was the average of the reliabilities at the α-cuts. Tang et al. [11] also utilized this assumption and computed the fuzzy failure probability by applying the Gauss-Legendre quadrature formula. However, it should be seen that different distribution assumptions of the α-cuts lead to totally different fuzzy reliability results. From the above analysis, one realizes that it had diversity approaches for evaluating the structural fuzzy reliability. Nevertheless, the meaning of the existing formulas is less evident than the reliability in the classical reliability theory. In order to overcome this drawback, using the traditional reliability analysis theory as a basis for assessing fuzzy reliability is a reasonable approach, due to the fact that the traditional probabilistic methods remain dominant in the field of measurements [18] and are well established in the decision making problems [19]. Hence, the aim of this study is to establish a novel approach for calculating structural fuzzy reliability by using the classical reliability theory. Firstly, the transformation from fuzzy number to equivalent normal random variables is presented. As a consequence, a fuzzy reliability problem is replaced by the traditional reliability problems, and the classical methods could be utilized to solve. Additionally, two notions including the central fuzzy reliability and the standard deviation of fuzzy reliability are proposed. While the central fuzzy reliability is the mean value of fuzzy reliability, the remain notion represents a measure of the spread of fuzzy reliability around its own central. Lastly, the ultimate fuzzy reliability is constituted based on three sigma rule, and is easily compared with the allowable reliability in the structural design codes. Numerical results are thoroughly explored to demonstrate the accuracy of the proposed method.

The formula for computing the deviation of the equivalent normal random variable
The transformation from fuzzy numbers into random quantities and conversely should be taken into acount in any problem where heterogeneous uncertain and imprecise data appear together (e.g. information deficit, lingustic variables, statistical data). The representative transformation principles, such as insufficient reason, maximum specify, uncertainty invariance, are proposed by Dubois et al. [18,20], Dubois [21] and Klir [15]. Based T on combining the principles of insufficient reason and maximum specify, Tuan and Huynh [22] proposed an innovation transformation to calculate the deviation of the equivalent normal random variable. For the transformation from standardized symmetric triangular fuzzy number ( Fig. 4) into random quantity, the error of probability measure between equivalent probability density function p(x) of standardized symmetric triangular fuzzy number got by the principle of insufficient reason and normal random variable p 1 (x) is expressed by the following formula ( ) where p(x), p 1 (x), P(A), P 1 (A) are determined as follows Due to the different domain between probability density function p(x) and normal random variable p 1 (x), in order that probability of density function p 1 Combine (9) and (10), we have For the reverse transformation from normal random variable into equivalent fuzzy number, the error of possibility measure between equivalent fuzzy number of normal random variable got by the principle of maximum specificity and standardized fuzzy number is expressed by the following formula where π 1 (x) is the equivalent fuzzy number of normal random variable determined as follows In order to solve the multiobjective optimization problem (11) and (12), transforms multiple objectives into a scalar objective function by multiplying each objective function by a weighting factor and summing up all contributors where γ∈[0,1]. For the mathematical meaning, formula (14) is an extension which modifies the equivalent characteristic according to two principles: the principle of insufficient reason when going from fuzzy number to random variable, and the principle of maximum specificity when going from radom variable to fuzzy number. For solving (14), Genetic algorithm (GA) [23] is applied using the built-in functions in Matlab. The relation between weighting factor γ and deviation σ is detailed depicted in Fig. 6. Indeed, the deviation of normal random variable using uncertainty invariance principle is of 0.3989 [24], as the same as the result attains according to the proposed transformation when weighting factor is of 0.728. Hence, the constrains generated in the proposed transformation are more flexible than Klir's method.
For a symmetric triangular fuzzy number X=(a,l) LR (Fig.  6), the relation between it and the standardized fuzzy variable x=(0,1) LR is defined as follows [22] X-a x= l

The central fuzzy reliability, the standard deviation of fuzzy reliability
As a result of accomplishing a family of normal random variables, the original structural fuzzy reliablity problem is converted to the basic reliability problems using the classical reliability theory to solve. By this we mean that the calculated reliablity will vary from value to value. Therefore, the attained fuzzy reliability can be treat as a random variable with its own mean, standard deviation. Based on the relation between the deviation of equivalent nornal random variable and weighting factor explored in the previous section, three especial values of the weighting factor γ are considered: When γ = 0.5 we get σ = 0.476. (16.a) When γ = 1 we get σ = 0.288.
(16.b) When γ = 0 we get σ = 0.640. (16.c) The value σ = 0.476 with γ = 0.5 in formula (16.a) means to choice the equilibrium for the weighting factors of objective function F(σ) and objective function G(σ). Due to this reason, the value σ = 0.476 is utilized to calculate the central fuzzy reliablity FR c being the mean value of fuzzy reliability. Two remain values in formula (16.b) and (16.c) are the minimum and the maximum values of the deviation of equivalen normal random variable, respectively. Therefore, we will utilize them to calculate the standard deviation of fuzzy reliability σ FR . Following that, without loss of generality, one considers the limit state function is given by where x F =(x 1 ,x 2 ,......,x n ) are independent fuzzy variables, as symmetric triangular fuzzy number x i =(a i ,l i ) LR ; x R is random parametter of the resistance function or the load effect.
In order to determine reliability according to the traditional reliability theory, the transformation from fuzzy variable x i =(a i ,l i ) LR to normal random variable x i~N (μ i , σ i ) is accomplished based on formulas (15) and (16) When γ = 0.0 : μ i = a i , σ i = 0.640 l i (18.c) For calculating the central fuzzy reliability FRc, the mean and the deviation of the equivalent normal random variable determined by formula (18.a) are applied. Then, can utilize any techniques in the classical reliability theory, such as First-order reliability method (FORM), Second-order reliability method (SORM), Cornell reliability index, Hasofer-Lind reliability index, Monte Carlo method, so on, to estimate structural reliability. Due to the fact that a large amount of the means and the deviations of the equivalent normal random variables are determined by solving the formulae (14) with weighting fator γ is fixed, there should be accomplished a large number of problems to determine the standard deviation σ FR of fuzzy reliability. In order to reduce the number of computations, the central composite design in the response surface method [25] is applied and the standard devition of fuzzy reliability is calculated as follows where FR sk is the reliability value at the kth experimental of the central composite design, with its own mean, standard deviation are defined by formulas (18.b) and (18.c); FR c is the central fuzzy reliability; m is the total number of test runs in the central composite design: m = 2 n +2n, with n is the number of fuzzy variables.

The ultimate fuzzy reliability
In order to take into account the bias of fuzzy reliability, the ultimate fuzzy reliability FRu is presented and used to compare the allowable reliability in the structural design codes. Based on three sigma rule in the probability theory, the ultimate fuzzy reliability is computed as follows When the input data has only one fuzzy variable (n=1), the ultimate fuzzy reliability is calculated as follows where FR 1 = P s (x 1 , x R ), FR 2 = P s (x 2 , x R ) with x 1 and x 2 are equivalent normal random variables, which have mean μi and deviation σi is defined by formulas (18.b) and (18.c), respectively.

NUMERICAL EXAMPLES AND DISCUSSIONS
In order to verify the proposed method, the illustrative examples including mathematical examples and engineering examples are given. The limit state functions in the illustrative examples are explicit functions, in order to get to a resemblance to structural analysis methods.

Example 1
Hypothetical limit state function with three variables is indicated as follows~~~Õ -sin x x   2 2  3  1  2  2  3  1 3 2 0 0 32 1 0 2 (22) where x 1 , x 2 , x 3 are assumed to be symmetric triangular fuzzy numbers (0,1) LR . The fuzzy reliability FR of the method [5] is: FR D =0.89053. The fuzzy reliability FR of the method [6] is: FR S =0.918638. The relative difference ε between FR D and FR S is: The Monte Carlo method [2,3] with a number of trials N s =106 is used to calculate in the proposed method. Tab. 1 displays the results of the central and the ultimate fuzzy reliability of the proposed method and the relative difference ε of the fuzzy reliability between the proposed method and the method [6].

The fuzzy reliability
The fuzzy reliability FR of the proposed method

Example 2
Consider a two-story frame structural system in Fig. 7, two girders are infinitely rigid. Elastic modulus E, ceiling height H, loads P 1 and P 2 are assumed to be symmetric triangular fuzzy numbers: E=(2x10 7 , 1.  The fuzzy reliability FR of the method [5] is: FR D =0.892432.
The fuzzy reliability FR of the method [6] is: FR S 0.970360. The relative difference ε between FR S and FR D is: The second-order reliability method SORM [27,28] is used to calculate the central and the ultimate fuzzy reliability in the proposed method. The derived results are compared with the method [6] and represented in Tab. 2.

The fuzzy reliability FR
The proposed method

Example 3
Determine the fuzzy reliability in terms of plastic collapse of the steel beam is shown in Fig. 8. Two cases are considered as follows: The steel beam is statically indeterminate to the second degree, therefore three hinges are required to change it from a girder structure to a mechanism, as shown in Fig. 9.
Then, the principle of virtual work is applied to determine the limit state function in terms of plastic collapse.
Equating (27) and (28): where M p =σ y .W p , with W p -the plastic moment of resistance. Hence, the limit state function in terms of plastic collapse is given as follows Tab. 3 and Tab. 4 display the results of the central and the ultimate fuzzy reliability of the proposed method and the relative difference ε of the fuzzy reliability between the proposed method and the method [10] in Case 1 and the method [6] in Case 2. The Cornell reliability index is applied in terms of Case 1 and the SORM is applied in terms of Case 2 in the proposed method.

The fuzzy reliability FR
The proposed method The fuzzy reliability FR J according to the method [10] The relative difference ε of the fuzzy reliability FR  Table 3: The results of the proposed method and that of the method [10] in Case 1 The fuzzy reliability FR The proposed method The fuzzy reliability FR S according to the method [6] The relative difference ε of the fuzzy reliability FR  Table 4: The results of the proposed method and that of the method [6] in Case 2

Example 4
Determine reliability in terms of stability of the structural system is shown in Fig.10. Two cases are considered as follows: Case 1: Inertia moment I = 808cm4, length l = 5m. Elastic modulus E is considered as a normal distribution with the mean value of 2.1x104 kN/cm 2 and the standard deviation of 2.5x103 kN/cm 2 . Concentrated load P (units:kN) is considered as symmetric triangular fuzzy numbers given as following: P=(110,10) LR . Case 2: Inertia moment I = 808cm 4 . Length l (units: m), elastic modulus E (units: kN/cm 2 ) and concentrated load P (units:kN) are assumed as symmetric triangular fuzzy numbers given as following: l=(5,0.5) LR , E=(2.1x10 4 ,2.5x10 3 ) LR , P=(110,10) LR . In order to calculate in terms of stability, the structural system is transformed to the continuous beam of two equal spans in Fig. 11 According to formula (33), it is evident that the value of k<16π 2 EI/(2l) 3 corresponds to stability loss in the symmetrical mode.
The equation in terms of stability is represented in the following form [26] ( ) The limit state function in terms of stability is given as follows ( Tab. 5 and Tab. 6 display the results of the central and the ultimate fuzzy reliability of the proposed method and the relative difference ε of the fuzzy reliability between the proposed method and the method [10] in Case 1 and the method [6] in Case 2. The Cornell reliability index is applied in terms of Case 1 and the SORM is applied in terms of Case 2 in the proposed method.

The fuzzy reliability FR
The proposed method The fuzzy reliability FR J according to the method [    In the proposed method, triangular membership fuction can be assigned to the allowable load P a , with the belief value of the membership function equals to the mean value, and the zero α-cut of fuzzy number is created by the extreme values [1350, 1800] kN. Therefore, the allowable load P a is the symmetric triangular fuzzy number: P a =[1575,225] LR (unit: kN). Then, the limit state function in terms of the design bearing capacity of the pile is given as follows Tab. 8 displays the results of the central and the ultimate fuzzy reliability of the proposed method and the relative difference ε of the fuzzy reliability between the proposed method and the classical reliability method based on assumption for distribution of uncertain variable.
The fuzzy reliability FR The proposed method The classical reliability method P S

DISCUSSIONS
Analysis the results of the above examples, the following discussions are given: 1. The results of the central and the ultimate fuzzy reliability of the proposed method has an only small relative difference in comparison with that of fuzzy reliability of the method [10]. In the all of the examples, the central fuzzy reliability FRc of the proposed method produces results closed to the results of the method [10]. It is found that the fuzzy reliability of the method [10] is the average of the reliabilities at the α-cuts, which is similar to the notion of the central fuzzy reliability of the proposed method. 2. The results of the ultimate fuzzy reliability of the proposed method has an only small relative difference in comparison with that of fuzzy reliability of the method [6]. The relative differences have been occured due to the different meaning between the fuzzy reliability in the method [6] and the reliability in the classical reliability theory. 3. The result of the ultimate fuzzy reliability of the proposed method approximates that of the reliability based on the classical approach. In addition, the proposed method need'n any assumption for distribution of uncertainty variables when information deficit appears. This pointed out that the proposed method is suitable to handle the problems of reality.

CONCLUSIONS
A novel method for calculating structural fuzzy reliability is proposed in this study. In order to apply the traditional reliability theory, a family of normal random variables equalling symmetric triangular fuzzy number are presented and explored in detail. From these equivalent random ones, two novel notions of the central fuzzy reliability and the ultimate fuzzy reliability are formulated and a procedure for calculating them is defined. In order to handle the bias of fuzzy reliability, the proposed ultimate reliability need to be compared with the allowable reliability in the structural design codes. Numerical results manifest the equivalence between the ultimate reliability in the proposed method and the "exact" reliability in the classical reliability theory.