## 4 Sensitivity Analysis of RPC-FST
### Sensitivity Analysis Based on Kriging  
To ensure the accuracy of the reliability sensitivity analysis for RPC-FST based on MCS, numerous repeatability calculations should be completed. However, the limit state function Equation (6) is highly complex and implicit. Therefore, the reliability sensitivity analysis for RPC-FST is limited by its high computational cost. To improve the calculation efficiency of the limit state function, the surrogate model Kriging is adopted to establish an approximation form for \(G(x)\). The surrogate Kriging model was created from an initial number of computational RPC-FST dynamic response calculations in a design of experiments (DoE) sampling plan. There exist several DoE techniques, and in this work, the Latin hypercube sampling (LHS) plan was used. The LHS plan divides each design parameter into \(N\) equally sized intervals, where the same value of a parameter can only occur once [39]. TNT weight, TNT distance, steel tube thickness, RPC-FST diameter, and RPC-FST height are five design parameters. An example of a randomly generated LHS plan can be seen in Figure 7.  
Figure 6: Failure probability \(P_{f}\) of RPC-FST at different allowable deflection angles \(\theta\). The basic idea of the Kriging method includes two parts: a regression model used to represent the global trend and a stochastic process used to represent the local behavior [40]. Therefore, the Kriging model can be expressed as Equation (10). \[g_{k}(X)=\sum_{i=1}^{p}f_{i}(X)\beta_{i}+z(X) \tag{10}\]  
where \(f(X)=\{f_{1}(X),f_{2}(X),\,\dots\,f_{p}(X)\}^{\rm T}\) is the vector of regression basis functions; \(\beta=\{\beta_{1},\,\beta_{2},\,\dots\,\,\beta_{p}\}^{\rm T}\) is the vector of the regression coefficients; \(p\) is the number of the basic functions in the regression model; and \(z(X)\) is assumed to be a Gaussian stationary process with a mean of zero and a standard deviation of \(\sigma\). The covariance matrix can be expressed as:  
\[\mathrm{cov}\big{[}z(X^{(i)}),z(X^{(j)})\big{]}=\sigma^{2}[R(X^{(i)},X^{(j)}, \theta)\big{]} \tag{11}\]  
where \(R(X^{(i)},X^{(j)})\) is the spatial auto-correlation function between input samples \(X^{(i)}\) and \(X^{(i)}\), which coordinates the smoothness of the Gaussian model. It can be expressed as:  
\[R[X^{(i)},X^{(j)},\theta]=\exp(-\sum_{k=1}^{m}\theta_{k}\Big{|}X^{(i)}_{k}-X^{ (j)}_{k}\Big{|}^{2}) \tag{12}\]  
where \(\theta=\{\theta_{1},\,\theta_{2}\,\dots\,\,\theta_{k}\}^{\rm T}\) is a hyperparameters vector defining the auto-correlation function. The hyperparameters vector \(\theta\) can be estimated as follows using the maximum likelihood method:  
\[\max F(\theta)=-\frac{m\ln(\hat{\sigma}^{2})+\ln(R)}{2}\;\theta_{k}\geq 0 \tag{13}\]  
where  
\[\left\{ \begin{array}{c}\hat{\sigma}^{2}=(g-F_{k}\hat{\beta})^{\rm T}R^{-1}( g-F_{k}\hat{\beta})/m\\ \hat{\beta}=(F_{k}^{\rm T}R^{-1}F_{k})^{-1}F_{k}^{\rm T}R^{-1}g\end{array} \right. \tag{14}\]  
where \(\hat{\sigma}^{2}\) is an estimate of \(\sigma\); \(\hat{\beta}\) is an estimate of \(\beta\); \(F_{k}\) is a matrix, which gathers the regression functions based on training points; and \(g\) is the response vector corresponding to training points. The predicted response vector of the Kriging model at a given unknown point \(X\) can be expressed as:  
\[g_{k}(X)=f^{T}(X)\hat{\beta}+r^{T}(X)R^{-1}(g-F\hat{\beta}) \tag{15}\]  
Figure 7: An example of a randomly generated LHS plan. where \(r(X)\) is the correlation vector between the training and predicting points.