## 4 Proposed Method of Sensitivity Analysis  
The proposed method utilizes the CS approximation to compute the sensitivity of a probabilistic constraint with respect to a vector of design variables. This section introduces procedures of the CS derivative approximation for gradients and sensitivity analysis. Consider the vector of random variables \(\mathbf{x}=(x_{1},x_{2},x_{3},\ldots,x_{n})^{\text{T}}\) and the vector of design variable \(\mathbf{d}=(d_{1},d_{2},d_{3},\ldots,d_{z})^{\text{T}}\). The sensitivity of a reliability index in the FORM with respect to a vector of design variables can be derived using the CS approximation as  
\[ \begin{array}{ll}\
abla_{\mathbf{d}}\beta(\mathbf{d},\mathbf{x}^{*})&=\frac{ 1}{\|\
abla\mathbb{G}(\mathbf{d},\mathbf{x}^{*})\|}\cdot\
abla_{\mathbf{d}} \beta(\mathbf{d},\mathbf{x}^{*})=\frac{1}{\|\
abla\mathbb{G}(\mathbf{d}, \mathbf{x}^{*})\|}\cdot\left[\frac{\partial\mathbb{G}(\mathbf{d},\mathbf{x}^{* })}{\partial\mathbf{d}_{1}},\ldots,\frac{\partial\mathbb{G}(\mathbf{d}, \mathbf{x}^{*})}{\partial\mathbf{d}_{k}},\ldots,\frac{\partial\mathbb{G}( \mathbf{d},\mathbf{x}^{*})}{\partial\mathbf{d}_{z}}\right]^{\text{T}}\\ &=\frac{1}{\|\
abla\mathbb{G}(\mathbf{d},\mathbf{x}^{*})\|}\cdot\frac{\Im[g( \mathbf{d}+\text{i}\text{i}\text{e}_{1},\mathbf{x}^{*}),\ldots,g(\mathbf{d}+ \text{i}\text{e}_{1},\mathbf{x}^{*})]\cdot\ldots,g(\mathbf{d}+\text{i}\text{ e}_{2},\mathbf{x}^{*})]^{\text{T}}}{h}\\ \end{array}  \tag{16}\]  
where \(\mathbf{e}_{k}\) denotes the \(k\)-th column of a \(z\)-th order identity matrix (\(\mathbf{I}_{z}\)). Because the limit-state function is defined in terms of the random variables \(\mathbf{x}\), it is necessary to carry out these calculations in the original space. Results are then transformed in the standard normal space to perform the FORM. The gradient vector \(\
abla G(\mathbf{u}^{*})\) can be rewritten as  
\[\
abla G(\mathbf{d},\mathbf{u}^{*})=\
abla_{\mathbf{x}}g(\mathbf{d},\mathbf{ x}^{*})\mathbf{J}_{\mathbf{u},\mathbf{x}}^{-1}(\mathbf{x}^{*}) \tag{17}\]  
where \(\mathbf{J}_{\mathbf{u},\mathbf{x}}^{-1}\) is the inverse of the Jacobian of the \(\mathbf{x}\) to \(\mathbf{u}\) transformation. Note that the Jacobian is dependent on the distribution types of random variables. For instance, the Jacobian of transformation \(\mathbf{J}_{\mathbf{u},\mathbf{x}}\) for statistically independent random variables \(\mathbf{x}\) is  
\[ \begin{array}{ll}\mathbf{J}_{\mathbf{u},\mathbf{x}}=diag[J_{ii}],&i=1,\ldots, n\\ J_{ii}=\frac{f_{i}(x_{i})}{\varphi\left(\Phi^{-1}\left[J_{\sim_{0}}^{x_{i}}f_{i}(x_{i}) dx_{i}\right]\right)}\end{array}  \tag{18}\]  
where \(f_{i}(x_{i})\) is the marginal PDF of \(x_{i}\) and \(\varphi(\cdot)\) is the univariate standard normal PDF. If the random variables \(\mathbf{x}\) are normally distributed with the correlation matrix \(\mathbf{R}=\mathbf{L}\mathbf{L}^{\text{T}}\), in which \(\mathbf{L}\) is a lower-triangular matrix obtained by the Cholesky decomposition of \(\mathbf{R}\), the Jacobian is  
\[\mathbf{J_{u,x}}=\mathbf{L}^{-1}\mathbf{D}^{-1} \tag{19}\]  
where \(\mathbf{D}\) is the diagonal matrix of standard deviations. Furthermore, the Jacobian matrix of dependent nonnormal random variables can be derived [49] as  
\[ \begin{array}{ll}f_{11}&=\frac{f_{1}(x_{1})}{g\left(\[PERSON]^{-1}[f_{-\infty}^{ \mathrm{T}_{1}}f_{1}(x_{1})dx_{1}]\right)}\\ f_{ij}&=0\\ &=\frac{f_{1}(x_{1}|x_{1},\ldots,x_{i-1})}{\varphi\left(\[PERSON]^{-1}[F_{i}(x_{1}|x _{1},\ldots,x_{i-1})]\right)}&i<j\\ &=\frac{1}{\varphi\left(\[PERSON]^{-1}[F_{i}(x_{1}|x_{1},\ldots,x_{i-1})]\right)} &i=j>1\\ &=\frac{1}{\varphi\left(\[PERSON]^{-1}[F_{i}(x_{1}|x_{1},\ldots,x_{i-1})]\right)} \frac{\partial F_{i}(x_{i}|x_{1},\ldots,x_{i-1})}{\partial x_{j}}&i>j\\ \end{array}  \tag{20}\]  
where \(f_{i}(x_{i}|x_{1},\ldots,x_{i-1})\) denotes the conditional PDF of \(x_{i}\) for given \(x_{1},\ldots,x_{i-1}\).