## 2 Failure-Related and Repair-Related Mechanism Analysis of Multi-Cracked Beams
### Interaction between MD and FD
Reliability for Independent System  
The classical series system reliability models need to be obtained by synthesizing the structure function and the reliability of each component, expressed in the following form:  
\[R=R_{1}R_{2}\cdots R_{n} \tag{9}\]  
In order to use each component strength and stress as inputs to the component reliability function, the component reliability can be calculated as follows:  
\[R_{j}=P\big{(}s_{j}<r_{j}\big{)}\]  
\[=\int_{0}^{\infty}f_{s_{j}}(\tau_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N }))\int_{s_{j}}^{\infty}f_{r_{j}}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1}, \cdots,s_{N}))dr_{j}ds_{j} \tag{10}\]  
Hence, the reliability of the system can be expressed as:  
\[R=\prod_{j=1}^{\text{N}}\int_{0}^{\infty}f_{s_{j}}(\tau_{j}(F,s_{1},s_{2},s_{ j-1},s_{j+1},\cdots,s_{N}))\int_{s_{j}}^{\infty}f_{r_{j}}(\epsilon_{j}(F,s_{1},s_{2}, s_{j-1},s_{j+1},\cdots,s_{N}))dr_{j}ds_{j} \tag{11}\]  
The above model can only be considered for static system reliability calculations. If the time effects and the time-varying effects of stresses and residual strengths of individual cracked elements during different operational cycles are taken into account, the probability that the system will not fail within \(n\) operational cycles, i.e., the reliability, can be calculated as follows:  
\[R_{1}(n)=\prod_{i=1}^{\text{n}}\prod_{j=1}^{\text{N}}\int_{0}^{\infty}f_{s_{j} }(\tau_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i))\int_{s_{j}}^{\infty} f_{r_{j}}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i))dr_{j}ds_{j} \tag{12}\]  
where \(\tau_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i)\) and \(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i)\) denote the stress and residual strength of the \(j\)th element at the \(i\)th operational cycle, respectively. The corresponding system failure rate is:  
Figure 2: Failure dependence and maintenance dependence. \[\psi(n) \tag{13}\] \[=\left(\prod\limits_{i=1}^{\text{n}}\prod\limits_{j=1}^{\text{N}} \int_{0}^{\infty}f_{s_{j}}(\tau_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N}, i))\int_{s_{j}}^{\infty}f_{\tau_{j}}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1}, \cdots,s_{N},i))dr_{j}ds_{j}\right.\] \[-\prod\limits_{i=1}^{n+1}\prod\limits_{j=1}^{\text{N}}\int_{0}^{ \infty}f_{s_{j}}(\tau_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s_{N},i))\int_{s _{j}}^{\infty}f_{\tau_{j}}(\epsilon_{j}(F,s_{1},s_{2},s_{j-1},s_{j+1},\cdots,s _{N},i))dr_{j}ds_{j}\right)\] \[R=R_{1}R_{2}\cdots R_{n} \tag{9}\]0  
The above equation is a model used to calculate the reliability index obtained according to the classical system reliability theory. However, it should be pointed out that, as shown by the analysis in Section 2, the stresses of each element in the system are statistically correlated and receive the joint influence of crack depth and working load. Moreover, the changes in the stresses in different operational cycles also affect each other. Therefore, it is necessary to develop system reliability models that can take into account a complex FD. Before establishing a system reliability model considering a complex FD, it is necessary to solve the stress solution problem. Since the stresses in each element are influenced by each other, it is unrealistic to try to directly obtain the common probability density function for different operating cycles and different elements. Meanwhile, by separately counting the probability distribution function of the stresses in each element, the correlation between the stresses in each element is neglected, which may result in a large error in the reliability assessment. Due to the need to consider the material parameters, crack characteristics and the randomness of the working load, the analytical expression of the element stresses is difficult, and a particularly large sample size is required in the statistical analysis, which is impossible to afford in practice.