## 6 Numerical Examples
The HaltonIT produces 7.555\(\%\) error in sensitivity estimation of failure probability. The relative errors of the SobolBM are 0.064\(\%\) and 2.371\(\%\) in failure probability and its sensitivity estimation, respectively. The relative errors of the SobolIT are 0.365\(\%\) and 1.798\(\%\) in failure probability and its sensitivity estimation, respectively. Taking the accuracy of reliability and its sensitivity into account, the combination of the SobolBM and SobolIT obtains a satisfactory accuracy with a relatively small number of samples._ \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **HaltonBM** & **HaltonIT** & **SobolBM** & **SobolIT** & **LHSBM** & **LHSIT** & **HaltonBM (N=10\({}^{5}\))** & **SMC (N=10\({}^{7}\))** \\ \hline \(\mu\) & \(-5.9807\) & \(-5.9121\) & \(-5.758\) & \(-5.7936\) & \(-5.8274\) & \(-5.8116\) & \(-5.9071\) & \(-5.95\) \\ \(\sigma\) & \(0.15559\) & \(0.11522\) & \(0.097977\) & \(0.098288\) & \(0.10587\) & \(0.13867\) & \(0.0017258\) & - \\  & & & & \(\partial p_{f}/\partial u_{x_{2}}\big{(}\times 10^{-4}\big{)}\) & & & & \\ \(\mu\) & \(-3.3503\) & \(-3.3119\) & \(-3.2256\) & \(-3.2455\) & \(-3.2644\) & \(-3.2556\) & \(-3.3091\) & \(-3.48\) \\ \(\sigma\) & \(0.087157\) & \(0.064547\) & \(0.054886\) & \(0.05506\) & \(0.05931\) & \(0.077681\) & \(0.0009668\) & - \\  & & & & \(\partial p_{f}/\partial u_{x_{3}}\big{(}\times 10^{-5}\big{)}\) & & & & \\ \(\mu\) & \(1.2256\) & \(1.2115\) & \(1.1799\) & \(1.1872\) & \(1.1941\) & \(1.1909\) & \(1.2105\) & \(1.13\) \\ \(\sigma\) & \(0.031883\) & \(0.023612\) & \(0.020078\) & \(0.020142\) & \(0.021696\) & \(0.028417\) & \(0.0003537\) & - \\  & & & & \(\partial p_{f}/\partial u_{x_{1}}\big{(}\times 10^{-3}\big{)}\) & & & & \\ \(\mu\) & \(1.3724\) & \(1.3587\) & \(1.3271\) & \(1.3345\) & \(1.3413\) & \(1.3378\) & \(1.3574\) & \(1.31\) \\ \(\sigma\) & \(0.031636\) & \(0.023361\) & \(0.020164\) & \(0.020186\) & \(0.021706\) & \(0.02819\) & \(0.0003490\) & - \\  & & & & \(\partial p_{f}/\partial v_{x_{2}}\big{(}\times 10^{-4}\big{)}\) & & & & \\ \(\mu\) & \(2.1534\) & \(2.1319\) & \(2.0823\) & \(2.0938\) & \(2.1046\) & \(2.0992\) & \(2.1299\) & \(2.60\) \\ \(\sigma\) & \(0.049638\) & \(0.036656\) & \(0.031638\) & \(0.031674\) & \(0.034058\) & \(0.044232\) & \(0.0005475\) & - \\  & & & & \(\partial p_{f}/\partial v_{x_{3}}\big{(}\times 10^{-5}\big{)}\) & & & & \\ \(\mu\) & \(2.3053\) & \(2.2822\) & \(2.292\) & \(2.2415\) & \(2.2531\) & \(2.2473\) & \(2.2801\) & \(2.12\) \\ \(\sigma\) & \(0.05314\) & \(0.039241\) & \(0.033869\) & \(0.033908\) & \(0.03646\) & \(0.047352\) & \(0.0005862\) & - \\  & & & & \(p_{f}/\{\times 10^{-5}\}\) & & & & \\ \(\mu\) & \(1.1930\) & \(1.1778\) & \(1.1442\) & \(1.1519\) & \(1.1593\) & \(1.1561\) & \(1.1769\) & \(1.175\) \\ \(\sigma\) & \(0.003407\) & \(0.025289\) & \(0.0212\) & \(0.02135\) & \(0.023031\) & \(0.030374\) & \(0.0003798\) & - \\  & & & & \(\beta\) & & & & \\  & & & & \(\beta\) & & & & \\  & & & & & & & & & \\  & & & & & & & & & \\ \(\mu\): the estimated value, \(\sigma\) : the standard deviation of estimate. & & & & & & & \\ \end{tabular}
\end{table} 
Table 2: Reliability and sensitivity results in Example 1 when \(N=15\). **Example 3**. : _As shown in Figure 6, a roof truss is considered, whose top chords and compression bars are made of steel-reinforced concrete, and the rest members are made of steel. The uniformly distributed load \(q\) applied on the roof truss is equivalent to the nodal load\(P=ql/4\). The limit state function is written as:_  
\[\mathrm{g}=0.03-\Delta_{\mathrm{C}}=0.03-\frac{ql^{2}}{2}\left(\frac{3.81}{A_{ \mathrm{C}}E_{\mathrm{C}}}+\frac{1.13}{A_{\mathrm{S}}E_{\mathrm{S}}}\right) \tag{39}\]  
_where \(l\) is the span length of the truss, \(A_{\mathrm{S}}\), \(A_{\mathrm{C}}\) are the cross-sectional areas of reinforce-concrete and steel bars, respectively, \(E_{\mathrm{S}}\), \(E_{\mathrm{C}}\) are the corresponding elastic modules.