## 2 Failure-Related and Repair-Related Mechanism Analysis of Multi-Cracked Beams
### Interaction between MD and FD
#### 3.1.3 Reliability for Dependent System  
When the BPNN fitting results of the element stresses under different crack states and operating loads are obtained using the above method, the reliability of the system considering the FD can be calculated based on the distribution of random operating loads over different operating cycles, as follows:  
\[R_{2}(n)=\prod_{i=1}^{n}\int_{-\infty}^{\infty}f_{i}(F(i))\prod_{j=1}^{N}\int _{\eta_{j}(F(i),a_{1}(i),a_{2}(i),\cdots,a_{N}(i),i)}^{\infty}f_{\gamma}( \epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i))dr_{j}dF(i) \tag{16}\]  
The corresponding system failure rate can be expressed as:  
\[\begin{split}\psi_{1}(n)=\left(\prod\limits_{i=1}^{n}\int_{- \infty}^{\infty}& f_{i}(F(i))\prod\limits_{j=1}^{N}\int_{\eta_{j}( F(i),a_{1}(i),a_{2}(i),\cdots,a_{N}(i),i)}^{\infty}f_{\gamma}(\epsilon_{j}(F,s_{1},s_{ 2},s_{j-1},s_{j+1},\cdots,s_{N},i))dr_{j}dF(i)\\ &-\prod\limits_{i=1}^{n+1}\int_{-\infty}^{\infty}f_{i}(F(i))\prod \limits_{j=1}^{N}\int_{\eta_{j}(F(i),a_{1}(i),a_{2}(i),\cdots,a_{N}(i),i)}^{ \infty}f_{\gamma}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i)) dr_{j}dF(i)\\ &/\left(\prod\limits_{i=1}^{n}\int_{-\infty}^{\infty}f_{i}(F(i)) \prod\limits_{j=1}^{N}\int_{\eta_{j}(F(i),a_{1}(i),a_{2}(i),\cdots,a_{N}(i),i )}^{\infty}f_{\gamma}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i))dr_{j}dF(i)\right)\end{split} \tag{17}\]  
The above reliability and failure rate calculations consider the FD between the working load, each unit stress and crack extension, avoiding the traditional assumption of element independence. However, there are fewer quantitative measures of FD in the existing literature, which can directly reveal the influence of different materials and design parameters on the FD and can guide the reliability design of mechanical components. In order to quantify the FD and analyze the impact of FD on reliability, the failure correlation coefficient (FDC) is proposed as follows:  
Figure 3: BPNN for stress analysis.  
\[\beta(n) \tag{18}\] \[=\left(\prod\limits_{i=1}^{\mathrm{n}}\int_{-\infty}^{\infty}f_{i}(F( i))\prod\limits_{j=1}^{\mathrm{N}}\int_{\eta_{j}(F(i),a_{1}(i),a_{2}(i),\cdots,a_{N}(i),j)}^{ \infty}f_{r_{j}}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i))dr_ {j}dF(i)\right)\] \[/\left(\prod\limits_{i=1}^{\mathrm{n}}\prod\limits_{j=1}^{\mathrm{ N}}\int_{0}^{\infty}f_{s_{j}}(\tau_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i)) \int_{s_{j}}^{\infty}f_{r_{j}}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1}, \cdots,s_{N},i))dr_{j}ds_{j}\right)\]  
FDC is an important indicator that can quantitatively characterize the FD of a system, and it can characterize the degree of system correlation and its impact on the time-varying reliability of the system. By setting different input parameters to calculate FDC, it can reflect the influence of different parameters on the FD. When the value of FDC tends toward 1, the classical reliability independent hypothesis model is consistent with the actual reliability, indicating that the FD effect is not obvious and the traditional independent hypothesis model can be used for the evaluation of the reliability of the system. The larger the value of FDC, the more obvious the FD effect. Therefore, it is necessary to pay attention to the factors that may cause the FD. The elements with a strong correlation should be analyzed in groups and modeled independently as FD subsystems in the system reliability logic block diagram. In addition, FDC can also quantitatively reflect the FD effects of a system in different operational cycles and during different tasks, so that the elements with a strong FD can be paid attention to during the specific time periods where the FDC value is larger. The proposed reliability model overcomes the shortcomings of traditional mechanical system reliability models in which independent assumptions are made about the components or empirical FD parameters that are used.