## 6 Numerical Examples
Six independent normal random variables, their means and standard deviations are given in Table 5._  
Figure 5: Maximum relative errors of sensitivity and failure probability when \(N=100\) and \(1000\) in Example 2. (a) Maximum relative errors of sensitivity (%); (b) Maximum relative errors of failure probability (%). Figure 6: Roof truss structure in Example 3. The results of failure probability, its sensitivity and standard deviation using different quasi random sequences with \(N=50\), \(1000\) are compared to those results obtained by the LHSIT with correlation elimination at \(N=10^{6}\) as the accurate answer, as is shown in Tables 6 and 7, respectively. The maximum relative errors of sensitivity of reliability and failure probability yielded by different quasi random sequences and transformations at the sample size \(N=50\) and \(1000\) are shown in Figure 7. Similarly, compared to the results of SMC with \(N=5\times 10^{7}\), the results of the quasi-Monte Carlo method are much closer to the accurate answer, especially in the sensitivity estimation of failure probability. The SMC produces more than \(11\%\) relative errors in \(\partial p_{f}/\partial u_{q}\), \(\partial p_{f}/\partial u_{t}\), etc. When \(N=50\), the maximum and minimum relative error in failure probability estimation come from the combination SobolBM and HaltonBM, producing \(4.576\%\) and \(0.701\%\) errors, respectively. In the meantime, the SobolBM and HaltonBM produce \(3.797\%\) and \(0.399\%\) errors in sensitivity estimation of failure probability, respectively. As the sample size increases to \(1000\), the accuracy of the sensitivity is further enhanced, especially in the SobolBM, which has the minimum relative error in both failure probability and its sensitivity estimation, reducing to \(0.048\%\) and \(0.080\%\) relative errors, respectively. The LHSIT has the maximum relative error in failure probability and its  
**Example 4**. : _A more complex 23-bar truss with 30 independent normal random variables is considered (shown in Figure 8), including cross-sectional area for members, applied loads, and modulus of elasticity, whose distribution parameters are listed in Table 8. The limit state function is given based on vertical displacement of the middle node \(D(X)\), as follows_  
\[G(X)=0.14-D(X) \tag{40}\]  
_where\(D(X)\)is the vertical displacement of middle node 7. The failure probability and reliability index calculated by the FORM are \(1.44\times 10^{-4}\) and \(3.6266\), respectively._  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **HaltonBM** & **HaltonIT** & **SobolBM** & **SobolIT** & **LHSBM** & **LHSIT** & **CLHSIT (N=\(10^{6}\))** & **SMC (N=\(5\times 10^{7}\))** \\ \hline \(\mu\) & \(-2.432\) & \(-2.4175\) &  
The failure probability and reliability index obtained by the SMC are \(1.8709\times 10^{-4}\) and \(3.5684\) with 1,000,000 samples, respectively. The failure probability reliability index calculated by the importance LHS are \(1.82\times 10^{-4}\) and \(3.5577\) with 1,000,000 samples. Ref. [28] shows that these methods are consistent with each other in terms of the failure probability, while the SMC obtains sensitivity estimations with a large error and the sensitivities of standard deviations obtained by the common importance LHS are inaccurate. The trend of maximum relative error of sensitivity with sample size \(N\) are shown in Figure 9. When \(N=50\), the maximum and minimum relative error in sensitivity estimation of failure probability result from the combination HaltonIT and LHSIT, producing 44.465% and 0.374% errors, respectively. In the meantime, the SobolBM, SobolII and HaltonBM produce 1.956%, 1.326% and 20.915% errors in sensitivity estimation of failure probability, respectively. The accuracy of the HaltonIT and HaltonBM are unsatisfactory with more than 20% relative errors. As the sample size increases, the accuracy of all the combinations is improved.