## 6 Numerical Examples  
In this section, four examples are illustrated below to demonstrate the computational efficiency and the accuracy of the method. The LHS, Halton and Sobol quasi-random sequences and the inverse transform and Box-Muller transformation are respectively applied and compared with each other for failure probability and its sensitivity estimation. HaltonIT denotes the Halton quasi-random sequence employed to generate the uniform sequence together with the inverse transformation to obtain the target distribution, LHSBM represents LHS together with the Box-Muller transformation, and so on. SMC refers to the standard Monte Carlo method. Figure 3: Procedure of structural reliability and its sensitivity analysis. **Example 1**. : _Consider the following performance function_  
\[\mathrm{g}(\mathrm{X})=X_{1}X_{2}-X_{3}=0 \tag{37}\]  
_where \(X_{1},X_{2},X_{3}\) are independent random variable with normal distribution. The basic random variables and their distribution characteristic parameters are listed in Table 1. The results of reliability, its sensitivity and variance using different methods with the sample size \(N=15\) are compared to those results obtained by the HaltonBM method at \(N=10^{5}\) as the accurate answer, as is shown in Table 2. The maximum relative errors of sensitivity of reliability and failure probability yielded by different methods at sample sizes \(N=15\) and \(100\) are shown in Figure 4, respectively. From these, we can conclude that the results of the quasi-Monte Carlo method with the sample size \(15\) are in agreement with those of the HaltonBM method with the sample size \(N=10^{5}\). The maximum relative error comes from the Sobol sequence; the combination SobolIT and SobolBM respectively produce 2.124% and 2.795% errors in failure probability estimation, and 1.925% and 2.528% errors in sensitivity estimation of failure probability, respectively. The minimum relative error stems from the Halton sequence, the combination HaltonIT and HaltonBM produce 0.068% and 1.368% errors in failure probability estimation, respectively, and 0.096% and 1.247% errors in sensitivity estimation of failure probability, respectively. As the sample size increases to \(N=100  
**Example 2**. : _A highly non-linear performance function_  
\[g(\mathrm{X})=X_{1}X_{2}X_{3}X_{4}-\frac{X_{5}X_{6}^{2}}{8}=0 \tag{38}\]  
_which involves both normal and non-normal basic random variables and their distribution parameters is given in Table 3. The results of reliability, its sensitivity and variance using different methods with the sample size \(N=100\) are compared to those results obtained by the HaltonBM method at \(N=10^{6}\) as the accurate answer, as is shown in Table 4. The maximum relative errors of sensitivity of reliability and failure probability yielded by different methods at the sample size \(N=100\) and \(1000\) are shown in Figure 5 respectively. It can be found that, similar to Example 1, compared to the results of SMC with \(N=10^{7}\), the result of the quasi-Monte Carlo method with \(N=100\) are much closer to the HaltonBM method with \(N=10^{6}\). Moreover, the relative error of failure probability is less than \(5\%\) and sensitivity error of failure probability is more than \(10\%\) when \(N=100\). The maximum and minimum relative error in failure probability estimation come from the combination HaltonIT and LHSBM, producing 4.423\(\%\) and 0.864\(\%\) errors, respectively. The relative errors of the SobolBM are 0.911\(\%\) and 34.305\(\%\) in failure probability and its sensitivity estimation, respectively. Nevertheless, HaltonIT and LHSBM produce 47.020\(\%\) and 39.761\(\%\) errors in sensitivity estimation of failure probability, respectively. Even the minimum relative error exceeds \(15\%\), arriving at 16.816\(\%\) obtained by the LHSIT. As the sample size increases to 1000, the accuracy of the sensitivity is greatly enhanced, while the relative error of the LHS is still more than 15\(\%\), 15.338\(\%\) and 23.155\(\%\) in the LHSIT and LHSBM, respectively.