## 3 Construction of New Membership Function
### Analytical Expression of Failure Probability Based on Suggested Model  
In this paper, based on the fuzzy theory, the fatigue strength was considered as a fuzzy variable \(r\), and the working stress was considered as constant \(S_{0}\). Subsequently, according to the stress-strength interface model, its failure probability could be expressed by [24]:  
\[F_{\lambda}=P(S_{0}>\overline{r}_{\lambda})=\int_{-\infty}^{S_{0}}\overline{r _{\lambda}}(r)dr \tag{7}\]  
where \(P(S_{0}>\overline{r}_{\lambda})\) stood for probability of the invariable working stress \(S_{0}\) over the fatigue strength \(r\) which was uniform distribution during interval \([a_{\lambda},b_{\lambda}]\). \(a_{\lambda}\) and \(b_{\lambda}\) were relative with the membership function of fuzzy fatigue strength and they could be evaluated by the function of threshold \(\lambda\). \(f\overline{r}_{\lambda}(r)\) was probability density function of fuzzy fatigue strength.  
If the working stress was turned into random variable \(S_{i}\) and less than or equal to the average value \(m\), the failure probability could be evaluated by:  
\[F_{i}=P(S_{i}>\overline{r})=\int_{0}^{\mathcal{L}(S_{i})}\dfrac{S_{i}-a_{ \lambda}}{b_{\lambda}-a_{\lambda}}d\lambda \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Membership Function** & **Predicted Fatigue Life** & **Real Lifetime** & **Relative Error** \\ \hline
1 & Linear & \(9.17\times 10^{4}\) & \(7.78\times 10^{4}\) & 17.9\% \\
2 & Normal & \(9.38\times 10^{4}\) & \(7.78\times 10^{4}\) & 20.6\% \\
3 & Parabolic & \(9.39\times 10^{4}\) & \(7.78\times 10^{4}\) & 20.7\% \\
4 & Square root & \(9.06\times 10^{4}\) & \(7.78\times 10^{4}\) & 16.5\% \\ \hline \end{tabular}
\end{table}
Table 1: Predicted fatigue life from four traditional membership functions.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Membership Function** & **Predicted Fatigue Life** & **Real Lifetime** & **Relative Error** \\ \hline
1 & Linear & \(3.39\times 10^{7}\) & \(2.20\times 10^{7}\) & 54.1\% \\
2 & Normal & \(3.50\times 10^{7}\) & \(2.20\times 10^{7}\) & 59.1\% \\
3 & Parabolic & \(3.26\times 10^{7}\) & \(2.20\times 10^{7}\) & 48.2\% \\
4 & Square root & \(3.55\times 10^{7}\) & \(2.20\times 10^{7}\) & 61.4\% \\
5 & Suggested model & \(2.99\times 10^{7}\) & \(2.20\times 10^{7}\) & 35.9\% \\ \hline \end{tabular}
\end{table}
Table 2: Predicted fatigue life from five traditional membership functions.  
When the random working stress was more than the average value \(m\), the failure probability could be evaluated by:  
\[F_{i}=P(S_{i}>\overline{r})=\int_{0}^{R(S_{i})}\frac{S_{i}-a_{\lambda}}{b_{ \lambda}-a_{\lambda}}d\lambda+1-R(S_{i}) \tag{9}\]  
where \(L(\cdot)\) and \(R(\cdot)\) were the left and right reference function.  
Subsequently, after the definite integrals were conducted, Equations (8) and (9) could be expressed by:  
\[F_{i}=3h(\frac{S_{i}-m}{c-a})\left[-ln(1-\sqrt[3]{\frac{L(S_{i})}{h}})-\sqrt[3] {\frac{L(S_{i})}{h}}\right]-\frac{3h(S_{i}-a)}{2(c-a)}(\frac{L(S_{i})}{h})^{ \frac{2}{3}}+\frac{(m-a)}{(c-a)}L(S_{i}) \tag{10}\]  
\[F_{i}=3h(\frac{S_{i}-m}{c-a})\left[-ln(1-\sqrt[3]{\frac{R(S_{i})}{h}})-\sqrt[3] {\frac{R(S_{i})}{h}}\right]-\frac{3k(S_{i}-a)}{2(c-a)}(\frac{R(S_{i})}{h})^{ \frac{2}{3}}+\frac{(m-c)}{(c-a)}R(S_{i})+1 \tag{11}\]  
where \(h\) was the constant and equal to \((1/5K)^{3}\), and \(a\) and \(c\) were deterministic value during interval \([a_{\lambda},b_{\lambda}]\). Once the failure probability \(F_{i}\) was determined, the fatigue reliability \(R_{i}\) could be obtained by:  
\[R_{i}=1-F_{i} \tag{12}\]