## 5 Case Studies
### Engineering Experiment B: A Turbine Disc
The structure fails when the maximum stress at the dangerous point exceeds the allowable value in Table 3, 703.84 MPa; the corresponding limit state function can be expressed as follows:  
\[N_{life}=K(\sigma_{\text{max}}-703.84)^{-m}=K(f(E_{1},\varepsilon_{1},\rho_{1},E _{2},\varepsilon_{2},\rho_{2},\omega,F)-703.84)^{-m}, \tag{15}\]  
where \(N_{life}\) denotes the fatigue life, and its unit is a cycle. \(K\) and \(m\) represent disc material parameters, \(K=6.2784\times 10^{15}\) and \(m=4.736\). \(S_{\text{max}}\) represents the stress field strength, which is a comprehensive function of roulette elastic modulus, roulette Poisson ratio, roulette density, semi-hollow pin elastic modulus, semi-hollow pin Poisson ratio, semi-hollow pin density, speed and assembly force. The failure probability of the third-stage roulette of a low-pressure compressor can be calculated referring to the limit state equation, the characteristic parameters of random variables and the finite element model established above, utilizing the proposed method. The Kriging model is firstly fitted, and the convergence condition is satisfied after four iterations. The total sample number is 68 and the accuracy requirement is \(10^{-3}\). The first iteration puts 17 sample points into the sample pool; 17 sample points are added to the sample pool in the second iteration; 16 sample points are added in the third iteration; and in the final iteration, 16 sample points are added. Finally, the values of some sample points are shown in Table 4.  
The comparison outcome of several approaches for reliability analysis is indicated in Table 5, which shows that the classical response surface-importance sampling and the proposed method reach the convergence condition after 256 and 68 structural analyses, respectively. Furthermore, the number of iterations increases greatly with the increase in input parameters, but the proposed approach still maintains high efficiency, which demonstrates superior computation performance and robustness. In addition, the error corresponding to the response surface-importance sampling method is 3.9702%, while the calculation error via the proposed method is 2.6055%. Consequently, the proposed approach possesses higher computational accuracy and efficiency compared with the response surface-importance sampling method, which further demonstrates the superiority of dealing with implicit engineering issues. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Iteration**} & \multicolumn{2}{c}{
\begin{tabular}{c} **Number of** \\ **Sample Points** \\ \end{tabular} } & \multirow{2}{*}{\(\mathbf{E_{1}}\)} & \multirow{2}{*}{\(\mathbf{\varepsilon_{1}}\)} & \multirow{2}{*}{\(\mathbf{\rho_{1}}\)} & \multirow{2}{*}{\(\mathbf{E_{2}}\)} & \multirow{2}{*}{\(\mathbf{\varepsilon_{2}}\)} & \multirow{2}{*}{\(\mathbf{\rho_{2}}\)} & \multirow{2}{*}{\(\mathbf{n}\)} & \multirow{2}{*}{\(\mathbf{F}\)} & \multirow{2}{*}{\(\mathbf{S_{\text{max}}}\)} \\ \hline \multirow{4}{*}{1} & 1 & 123,000 & 0.330 & 4.480 & 219,000 & 0.300 & 7.760 & 1000 & 21.928 & 801.16 \\  & 2 & 120,000 & 0.330 & 4.480 & 219,000 & 0.300 & 7.760 & 1000 & 21.928 & 801.25 \\  & 3 & 123,000 & 0.315 & 4.480 & 219,000 & 0.300 & 7.760 & 1000 & 21.928 & 800.81 \\  &... &... &... &... &... &... &... &... &... \\  & 18 & 121,149.492 & 0.347 & 4.521 & 218,251.976 & 0.288 & 7.746 & 1127.9 & 91.934 & 837.75 \\
2 & 19 & 118,149.492 & 0.347 & 4.521 & 218,251.976 & 0.288 & 7.746 & 1127.9 & 91.934 & 838.21 \\  &... &... &... &... &... &... &... &... &... \\  & 35 & 121,152.780 & 0.351 & 4.531 & 217,800.072 & 0.292 & 8.047 & 1137.6 & 21.932 & 840.61 \\
3 & 36 & 118,152.780 & 0.351 & 4.531 & 217,800.072 & 0.292 & 8.047 & 1137.6 & 21.932 & 843.50 \\  &... &... &... &... &... &... &... &... &... \\  &... &... &... &... &... &... &... &... &... \\
4 & 67 & 121,065.224 & 0.354 & 4.554 & 216,893.977 & 0.295 & 7.745 & 1159.5 & 21.931 & 847.23 \\  & 68 & 121,065.224 & 0.354 & 4.554 & 216,893.977 & 0.295 & 7.745 & 1159.5 & 21.941 & 844.44 \\ \hline \hline \end{tabular}
\end{table}