Reliability Sensitivity Analysis Method for Mechanical Components

Based on the univariate dimension-reduction method (UDRM), Edgeworth series, and sensitivity analysis, a new method for reliability sensitivity analysis of mechanical components is proposed. The univariate dimension-reduction method is applied to calculate the response origin moments and their sensitivity with respect to distribution parameters (e.g., mean and standard deviation) of fundamental input random variables. Edgeworth series is used to estimate failure probability of mechanical components by using first few response central moments. The analytic formula of reliability sensitivity can be derived by calculating partial derivative of the failure probability 
 
 
 
 P
 
 
 f
 
 
 
 with respect to distribution parameters of basic random variables. The nonnormal random parameters need not to be transformed into equivalent normal ones. Three numerical examples are employed to illustrate the accuracy and efficiency of the proposed method by comparing the failure probability and reliability sensitivity results obtained by the proposed method with those obtained by Monte Carlo simulation (MCS).


Introduction
e responses of mechanical components or engineering structures are often random due to random inputs of them including loads, material properties, and geometry. Many reliability analysis methods have been developed to calculate the failure probabilities of these random structures [1][2][3][4]. In reliability analysis, reliability sensitivity is defined as the partial derivative of the failure probability with respect to the distribution parameters (e.g., mean and standard deviation) of fundamental random input variables. Reliability sensitivity provides information about the importance of each input random variable to a structure's failure probability. Reliability sensitivity analysis methods can be divided into the numerical simulation method and approximate analytic calculation method. e numerical simulation method can be divided into many different methods due to different sampling methods including importance sampling, direction sampling, line sampling, subset simulation, and low-discrepancy sampling [5][6][7][8][9][10]. ese different simulation methods are all based on Monte Carlo simulation but have different sampling methods. Although the program of these methods is not complicated, lots of samples are needed to get precise results because the structural failure probability is often very small and the computational efficiency is therefore reduced.
Analytic methods for reliability sensitivity analysis are always based on analytic reliability methods. ese reliability analysis methods are the mean-value first-order reliability method (MVFORM)/mean-value second-order reliability method (MVSORM) [11,12], JC method, mean-value firstorder saddlepoint approximation (MVFOSA) [13], and moment method. FORM/SORM needs to expand the performance function into first-order or second-order Taylor series at the most likely failure point. e expansion is widely used because of its relatively moderate precision and efficiency. e corresponding reliability sensitivity calculation methods have been derived from these reliability methods [14][15][16][17][18][19][20][21][22][23]. Analytic methods for reliability sensitivity analysis always have higher calculation efficiency than MCS.
In this study, a feasible method to compute the sensitivity of failure probability with respect to distribution parameters of basic random input variables is proposed. e method can be applied to estimate failure probability and reliability sensitivity based on Edgeworth series. e proposed reliability sensitivity analysis method includes the following steps. (1) Choose random input variables and determine their distributions and distribution parameters.
(2) Calculate the first four response origin moments and their sensitivity with respect to distribution parameters of random input variables by UDRM. (3) Calculate failure probability by the reliability method based on Edgeworth series. (4) Calculate partial derivatives of failure probability with respect to response central moments. (5) Calculate partial derivatives of response central moments with respect to response origin moments. (6) Calculate reliability sensitivity defined by partial derivatives of failure probability with respect to distribution parameters of input random variables.

Response Origin Moments by UDRM.
Structures subject to random input vector X � [X 1 , X 2 , Λ, X N ] T ∈ R N , which characterizes uncertainty in loads, material properties, and geometry. Let Y(X) represent a response of interest that depends on independent random variables X � [ X 1 , X 2 , Λ, X N ] T ; then, the l th order origin moment of Y(X) can be written as follows: where m l is the l th order origin moment of Y(X), f X (x)is the joint probability density function of X, and E(·)is the expectation operator. According to the univariate dimension reduction method proposed by Rahman and Xu [24][25][26], Y(X) can be approximately written as follows: where en, the l th order origin moment m l of Y(X) can be written as follows: Applying the binomial formula on the right-hand side of equation (3), the l th response origin moment m l can be written as follows: where C(·) is the combination operator. Define Equation (5) can be expressed using the recursive formula as follows [23]: 2 Mathematical Problems in Engineering N).
e q th origin moment m q Y j in equation (8) can be calculated out by the numerical integration method and can be written as follows: where is the k th integration point, and n is the number of integration points.
If x j has different probability density (e.g., normal, lognormal, and Weibull), the corresponding integral weights and integration points in equation (9) are different. e numerical integration points and the corresponding orthogonal polynomial for different probability densities are listed in Table 1 [27].

Sensitivity of Response Moment with Respect to Distribution Parameters of Inputs.
Calculate the partial derivative on both side of equation (1) with respect to distribution parameters T i j of random input variables X � [X 1 , X 2 , Λ, X N ] T . e partial derivative can be written as follows: Mathematical Problems in Engineering , is the i th distribution parameter of the j th random input variable (i � 1, represents the mean; i � 2 represents standard deviation), f j (x) is the probability density function of the j th random variable X j , and Applying the binomial formula on the right-hand side of equation (10), zm l /zT i j can be written as follows: where zm l /zT i j can be calculated out by equation (11) and recursive formula similar to equations (5) and (6). Suppose where k i j is the kernel function of the i th distribution parameter T i j of j th the random input variable. e kernel function in equation (12) of X j with different probability density functions can be obtained as follows: here, x j is denoted by x.
If X is a normal distributed variable, its probability density function can be written as follows: where μ is the mean of X, and σ is the standard deviation of X. e kernel function for normal variable X with respect to its distribution parameters (mean and standard deviation) can be derived from equation (12) directly and can be written as follows: where k μ and k σ are the kernel functions with respect to mean and standard deviation, respectively. μ and σ are the mean and standard deviation of normal variable X, respectively.
If X is a lognormal distributed variable, its probability density function can be written as follows: where μ 1 is the mean of ln X, and σ 1 is the standard deviation of ln X. e kernel function of lognormal variable X with respect to μ 1 and σ 1 can be derived from equation (12) directly and can be written as follows: where k μ 1 and k σ 1 are the kernel functions with respect to μ 1 and σ 1 , respectively. Distribution parameters μ 1 and σ 1 of the lognormal distributed variable x can be written as follows: where μ and σ are the mean and standard deviation of the lognormal distributed variable x, respectively. Calculate partial derivative of μ 1 and σ 1 with respect to μ and σ, respectively, according to equation (16); then,

Distribution
Orthogonal polynomial Integral formula Normal Hermite Jacobi matrix J can be written as follows: e kernel function of the lognormal distributed variable X with respect to its mean and standard deviation can be written as follows: where k μ and k σ are the kernel functions with respect to mean and standard deviation of the lognormal distributed variable x, respectively.
If x is two parameter Weibull distributed variable, its probability density function can be written as follows: where β is the shape parameter, and α is the scale parameter. e kernel function of the Weibull distributed variable X with respect to scale parameter α and shape parameter β can be derived from equation (12) directly, which can be written as follows: e mean and standard deviation μ and σof X can be written as follows: where μ and σ are the mean and standard deviation of X, respectively, and Γ(·) is the gamma function. Calculate partial derivative of μ and σ with respect to α and β, respectively, from equation (21); then, where Ψ(·) is the psi function. Jacobi matrix J can be written as follows: J � zμ/zα zσ/zα zμ/zβ zσ/zβ ; then, the inverse matrix of J can be written as J − 1 � zα/zμ zβ/zμ zα/zσ zβ/zσ . e kernel function of the Weibull distributed variable X with respect to its mean and standard deviation can be written as follows: where k α and k β are the kernel functions with respect to scale parameter and shape parameter, respectively. α and β are the scale parameter and shape parameter of the two parameter Weibull distributed X, respectively.

Reliability Based on Edgeworth
Series. e failure probability of a mechanical component or a random structure can be calculated out by the following multidimensional integral: where P f is the failure probability of a random structure, X � [X 1 , X 2 , Λ, X N ] T ∈ R N represents the N-dimensional random input variables of the random structure, f X (x) is the joint probability density function of random input variable X, g(X) is the performance function, g(X) < 0 represents the failure domain, g(X) > 0 represents the safety domain, and g(X) � 0 represents the limit state.

Mathematical Problems in Engineering
In practice, it is difficult to obtain the analytical solution of the multidimensional integral in equation (25) because of the nonlinear integration boundary g(X) � 0 and high dimension of X. ere are two classes of approaches available for estimating the failure probability P f in equation (25), the analytical method and numerical simulation method. e numerical simulation method is the Monte Carlo simulation as known. Analytical methods (e.g., first-order reliability method (FORM)/second-order reliability method (SORM)), higher-order moment method, and saddlepoint approximation) can always be applied to calculate failure probability P f because of high computational efficiency and accuracy of them.
Suppose the first four central moments of performance function g(X) are μ g , v g , θ g , and η g . e failure probability P f in equation (27) can be approximately estimated by Edgeworth series which can be written as follows: where Φ(·) is the standard normal distribution function, φ(·) is the standard normal probability density, μ g , v g , θ g , and η g are the mean, deviation, the third central moment, and the fourth central moment of performance function g(X), σ g is the standard deviation of g(X), ς � − μ g /σ h is the negative reliability index, and H j (·) is the j th Hermite polynomial and can be written as follows:

Reliability Sensitivity Based on Edgeworth Series.
Partial derivative of failure probability P f in equation (28) with respect to the first four central moments μ g , v g , θ g , and η g of performance function g(X) can be written as follows: where e first four central moments μ g , v g , θ g , and η g of performance function g(X) can be written as follows:

Example 1: Nonlinear Performance Function with Normal
Variables. is example considers a nonlinear performance function which is the stress limit state function of a multileaf spring written as follows: where basic random variables X � [r, p, l, b, h] are the independent and identically distributed Gaussian random variables, r is the material strength of the leaf spring, p is the load, and b, h, and l are the geometric dimension width, thickness, and span of the leaf spring, respectively. e mean and standard deviation of random variables X � [r, p, l, b, h] are listed in Table 2 [22]. e proposed method and MCS are applied to analyze the failure probability and reliability sensitivity, respectively, and their results are listed in Table 3 for comparison. e MCS results can be considered as accurate results and N is the number of samples.
In Table 3, the results obtained by the proposed method have small errors compared with those obtained by MCS. Example 1 indicates that the proposed method is suitable for problems with nonlinear performance functions.

Example 2: Linear Performance Function with Six Lognormal Variables.
is example considers a linear performance function, a plastic collapse mechanism of a one-bay frame which has been used as example 5 by Der et al. [28] and Zhao and Ono [29], written as follows: where basic random variables X � [x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ] are the independent and identically distributed lognormal random variables. e mean and standard deviation of random variables X � [x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ] are listed in Table 4. e proposed method and MCS are applied to analyze the failure probability and reliability sensitivity, respectively, and their results are listed in Table 5 for comparison.
In Table 5, the results obtained by the proposed method have small errors compared with those obtained by MCS. Example 2 indicates that the proposed method is suitable for problems with nonnormal random input variables.

Example 3: Nonlinear Performance Function with Nonnormal Variables.
is example considers a nonlinear performance function which describes the displacement response of a I section cantilever beam shown in Figure 2 written as follows: where basic random variables X � [a, t, b, h, l, F, d, E] are the independent random variables. Here, a is the thickness of web, t is the thickness of flange, b is the width of I-beam section, h is the height of I-beam section, l is the length of cantilever beam, f is the load force, d is the allowable maximum deformation, and E is the elastic modulus. e    Table 6. e scale parameter and shape parameter of random variable F are α � 5.197 × 10 4 and β � 310.36, respectively. e proposed method and MCS are applied to analyze the failure probability and reliability sensitivity, respectively, and their results are listed in Table 7 for comparison. In Table 7, the results obtained by the proposed method have small errors compared with those obtained by MCS. Example 3 indicates that the proposed method is suitable for problems with nonlinear performance functions with nonnormal random input variables. e nonnormal variables need not to be transformed into equivalent normal ones. But the coefficient of variation of the input random variables should be small (≤0.1 for Weibull distribution variables).

Conclusions
e univariate dimension-reduction method, reliability analysis based on Edgeworth series, and reliability sensitivity analysis are employed to present a computational procedure for estimating failure probability sensitivity of random engineering structures.
ree numerical examples are employed to illustrate the performance of the proposed method. To illustrate the feasibility of the proposed method, failure probability and failure probability sensitivity results obtained by the new method are compared with those obtained by direct MCS. It is concluded as follows: e proposed method is suitable for models of which the input random variables have small coefficient of variation (≤0.1). e nonnormal input variables need not to be transformed into equivalent normal ones, so the new method can be conveniently applied to solve problems with nonnormal input variables.
Data Availability e data and models used to support the findings of the study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.