## 3 Construction of New Membership Function  
### Modeling of Membership Function  
The fatigue damage calculation that is based on the traditional stress/strain approach only took the stress level over the fatigue limit into account and it ignored the one below the fatigue limit, which  
Figure 1: Mine road surface. Figure 2: Cracks in welded A-type frame. will result in the inaccuracy of this evaluation. The fuzzy linear fatigue damage accumulation could be evaluated by [15; 16; 17; 18; 19; 20] in order to give consideration to the stress level over and below the fatigue limit  
\[\sum_{i=1}^{k}\frac{n_{i}}{N_{i}}+\sum_{i=k+1}^{m}U(S_{i})\frac{n_{i}}{N_{0}}=1 \tag{1}\]  
where \(m\) is the number of total stress level, \(S_{i}\) is every stress level, \(n_{i}\) is the number of each stress level, \(k\) is the number of stress level over the fatigue limit, \(N_{i}\) is the number of cycles at each stress level over the fatigue limit, the number of stress level below the fatigue limit is varying from \(k+1\) to \(m\), \(N_{0}\) is the number of cycles at each stress level below the fatigue limit, and \(U(S_{i})\) is the membership function. The fuzziness of variables is usually described by the membership function of fuzzy sets. According to the fuzzy theory, there are four kinds of distribution forms: the linear distribution, the normal distribution, the parabolic distribution, and the square root distribution, which are successively expressed by:  
\[U(S)= \begin{cases}1&S\geq S_{r}\\ \dfrac{S-S_{r}}{S_{r}-a}&a\leq S<S_{r}\\ 0&S<a\end{cases}  \tag{2}\]  
\[U(S)= \begin{cases}1&S\geq S_{r}\\ \exp\left[-\left(\dfrac{S-S_{r}}{\sigma_{c}}\right)^{2}\right]&S<S_{r}\end{cases}  \tag{3}\]  
\[U(S)= \begin{cases}1&S\geq S_{r}\\ \left(\dfrac{S-S_{r}}{S_{r}-a}\right)^{2}&a\leq S<S_{r}\\ 0&S<a\end{cases}  \tag{4}\]  
\[U(S)= \begin{cases}1&S\geq S_{r}\\ \dfrac{1}{S_{r}-a}\dfrac{1}{2}&a\leq S<S_{r}\\ 0&S<a\end{cases}  \tag{5}\]  
where \(S\) is the stress value, \(S_{r}\) is the fatigue limit, \(a\) is the fuzzy damage boundary, and \(\sigma_{c}\) is the standard deviation of the fatigue limit. In our previous research [21], the fatigue expectancy of the welded A-type frame was predicted on the basis of the membership function that is mentioned above. According to the estimated results that are listed in the Table 1, the minimum relative error was from the square root distribution, and was around 16.5%. The others were around 20%. Even though they were close to the real lifetime, its prediction accuracy still needed to be improved. If there was enough sample data to determine the characteristic parameters in distribution functions, then the estimated results could be more precise. However, the samples of this engineering structure were limited; how to precisely characterize the distribution features of fuzzy variables is the key point to explore engineering fatigue reliability. Therefore, the authors attempted to create a new membership function to describe the distribution feature of fuzzy variables. According to the statistical data of different stress levels and the proposed membership function given in reference [21; 22], a new membership function was expressed by:  
\[U(S)= \begin{cases}1&S\geq S_{r}\\ \left(\dfrac{S-a}{KS_{r}}\right)^{3}&a\leq S<S_{r}\\ 0&S<a\end{cases}  \tag{6}\]  
where \(K\) is the constant for correcting membership degree of fuzzy variable. In this paper, it was taken as 0.164. Subsequently, this membership function was utilized to predict the fatigue lifespan of the welded A-type frame. The calculated result was \(8.25\times 10^{4}\) and its relative error was only 5.99%, which was more accurate than the ones based on the traditional membership functions. The fatigue lives that were obtained from different membership functions for the round bar with transverse holes in order to further verify the accuracy of this model were predicted and are listed in Table 2[23]. The suggested model had the minimum relative error in this case, which was 35.9% and far less than the others.