## 2 Response Surface Method  
The response surface methodology (RSM) was first proposed by [PERSON] and [PERSON] [31]. The method has been widely applied in the chemical, pharmaceutical, agricultural and mechanical engineering fields [31; 32; 33]. The idea underlying this method is the assumption that the real expression of \(Y\) is unknown, the relationship between the output response \(Y\) of the structural system and the random input parameter vector \(x=\)(\(x_{1},\ x_{2},\cdots,\ x_{NR}\)) is described by a quadratic function with cross terms, as shown in Equation (1). The reasons why RSM was adopted to analyze reliability sensitivity are follows: (1) It can be used when the limit state function i.e., the real analytical response expression of the mechanical system is unknown. (2) This limit state function is a simple quadratic polynomial, which facilitates the calculation of variance and partial derivatives, and makes the calculation of reliability sensitivity straightforward. (3) The limit state function includes the information of the first, second and cross terms, which greatly improves the accuracy of the reliability sensitivity calculation. The \(N\)s input samples of the random parameter vector are obtained by certain sampling methods, while a group of output values \((y_{1},\ y_{2},\cdots,\ y_{N_{S}})\) of the structural system are obtained by experimental design or numerical analysis based on the \(N\)s input values. The unknown coefficients of the response surface function are determined by the least square method in regression analysis, thus the response surface function is derived, which is used to replace the real response expression of the structural system in the following work. \[Y=C_{0}+\sum\
olimits_{i=1}^{N_{R}}C_{i}X_{i}+\sum\
olimits_{i=1}^{N_{R}} \sum\
olimits_{j=1}^{N_{R}}C_{ij}X_{i}X_{j} \tag{1}\]  
where: \(C_{0}\), \(C_{i}\), \(C_{ij}\) (\(i=1,\ \ldots,\ N_{R}\); \(j=i,\ \ldots,\ N_{R}\)) are unknown coefficients. To obtain \(N\)s samples of the random parameter vector, Box-Behnken sampling is used herein. This method not only reduces the number of samples but also improves the calculation accuracy of the response surface. According to the reference [31], three different level points are selected for each random variable, and then the center points and the edge midpoints are combined into samples according to certain rules. Figure 1 shows the Box-Behnken samples of three variables \((x_{1},\ x_{2},\ x_{3})\). For a random variable \(x_{s}\) which accords with arbitrarily distribution, its level value \(p_{n}\) is calculated from Equation (2). \[\int\
olimits_{-\infty}^{x_{s}}f(x)dx=p_{n},n=1,2,3 \tag{2}\]  
where: \(f(x)\) is the distribution density function of random variables; \(p_{n}\) are the random variable level, \(p_{1}=0.01\), \(p_{2}=0.50\), \(p_{3}=0.99\). For random variables of normal distribution  
\[x_{s}=\mu+\sigma\Phi^{-1}(p_{n}) \tag{3}\]  
where: \(\mu\) is the mean; \(\sigma\) is standard deviation; \(\Phi(^{\circ})\) is the standard normal distribution function. Figure 1: Box-Behnke samples of three variables. The \(Ns\) samples of random parameters are numerically calculated, then the \(Ns\) corresponding output values \((y_{1},\ y_{2},\cdots,\ y_{Ns})\) of the system response are obtained. These data are employed in a regression analysis using the least square method. \[s=\sum_{i=1}^{N_{s}}\varepsilon^{2}=\sum_{i=1}^{N_{s}}\left[C_{0}+\sum_{i=1}^{ N_{R}}C_{i}X_{i}+\sum_{i=1}^{N_{R}}\sum_{j=1}^{N_{R}}C_{ij}X_{i}X_{j}\right]^{2} \tag{4}\]  
where: \(Ns\) is the number of samples, \(N_{R}\) is the number of random input variables and \(\varepsilon\) is the error term. To minimize the sum of each error term, then:  
\[\left\{ \begin{array}{l}\frac{\partial s}{\partial C_{0}}=0\\ \frac{\partial s}{\partial C_{i}}=0\\ \frac{\partial s}{\partial C_{ij}}=0\end{array}