## 2 Failure-Related and Repair-Related Mechanism Analysis of Multi-Cracked Beams
### System Time-Varying Reliability Models Based on MD and FD Analyses
When the structure is repaired for the \(m\)th time, the correction factor for each element crack relative to the initial crack depth \(a_{0}=[a_{01},a_{02},\cdots,a_{0N}]\) is:  
\[\varphi_{m}=[\varphi_{1m},\varphi_{2m},\cdots,\varphi_{Nm}] \tag{19}\]  
Then, the initial crack depth in the next working period after the \(m\)th repair of the crack can be expressed as:  
\[a_{0m}=[\varphi_{1m}*a_{01},\varphi_{2m}*a_{02},\cdots,\varphi_{Nm}*a_{0N}] \tag{20}\]In addition, the correction factor for the crack expansion rate of each element relative to the initial crack expansion rate \(v_{0}=[v_{01},v_{02},\cdots,v_{0N}]\) is expressed as:  
\[\delta_{m}=[\delta_{1m},\delta_{2m},\cdots,\delta_{Nm}] \tag{21}\]  
Then, the initial crack expansion rate in the next working period after the \(m\)th repair of the crack can be expressed as:  
\[v_{0m}=[\delta_{1m}*v_{01},\delta_{2m}*v_{02},\cdots,\delta_{Nm}*v_{0N}] \tag{22}\]  
When the randomness of the repair effect is considered, from the probability density function (PDF) of \(\varphi_{m}\) and \(\delta_{m}\), denoted by \(f_{\varphi_{m}}(\varphi_{m})\) and \(f_{\delta_{m}}(\delta_{m})\), respectively, the time-varying reliability considering the FD in the next operating period after the \(m\)th repair is obtained as follows:  
\[R_{2m}(n)=\int_{-\infty}^{\infty}f_{\varphi_{m}}(\varphi_{m})\int_{-\infty}^{ \infty}f_{\delta_{m}}(\delta_{m})\prod\
olimits_{i=1}^{n}\int_{-\infty}^{ \infty}f_{im}(F_{m}(i))*\]  
\[\left(\prod\
olimits_{j=1}^{N}\int_{\eta_{j}(F_{m}(i),\mu_{1m}(i),\sigma_{2m} (i),\cdots,\sigma_{Nm}(i),\delta)}^{\infty}f_{fjm}(\epsilon_{j}(F_{m}(i),s_{ 1m},s_{2m},s_{j-1m},s_{j+1m},\cdots,s_{Nm},i))dr_{jm}dF_{m}(i)\right)d\varphi_ {m}d\delta_{m} \tag{23}\]  
The time-varying reliability under the independence assumption in the next working period after the \(m\)th repair is:  
\[R_{1m}(n)=\int_{-\infty}^{\infty}f_{\varphi_{m}}(\varphi_{m})\int_{-\infty}^{ \infty}f_{\delta_{m}}(\delta_{m})\prod\
olimits_{i=1}^{n}\prod\limits_{j=1}^ {N}\int_{0}^{\infty}f_{s_{jm}}(\tau_{jm}(F_{m},s_{1m},s_{2m},s_{j-1m},s_{j+1m },\cdots,s_{Nm},i))*\]  
\[\left(\int_{s_{jm}}^{\infty}f_{r_{jm}}(\epsilon_{jm}(F_{m},s_{1m},s_{2m},s_{ j-1m},s_{j+1m},\cdots,s_{Nm},i))dr_{jm}d{s_{jm}}\right)d\varphi_{m}d\delta_{m} \tag{24}\]  
The FDC of the system in the next working period after the \(m\)th repair can be expressed as:  
\[\beta_{m}(n)=\int_{-\infty}^{\infty}f_{\varphi_{m}}(\varphi_{m})\int_{- \infty}^{\infty}f_{\delta_{m}}(\delta_{m})\prod\
olimits_{i=1}^{n}\int_{- \infty}^{\infty}f_{jm}(F_{m}(i))*\]  
\[\left(\prod\limits_{j=1}^{N}\int_{\eta_{j}(F_{m}(i),\sigma_{1m}(i),\sigma_{2m} (i),\cdots,\sigma_{Nm}(i),i)}^{\infty}f_{r_{jm}}(\epsilon_{j}(F_{m}(i),s_{1m},s_{2m},s_{j-1m},s_{j+1m},\cdots,s_{Nm},i))dr_{jm}dF_{m}(i)\right)d\varphi_{ m}d\delta_{m}\]  
\[a_{0m}=[\varphi_{1m}*a_{01},\varphi_{2m}*a_{02},\cdots,\varphi_{Nm}*a_{0N}] \tag{20}\]0  
\[a_{0m}=[\varphi_{1m}*a_{01},\varphi_{2m}*a_{02},\cdots,\varphi_{Nm}*a_{0N}] \tag{20}\]1  
When considering the MD, \(\varphi_{m}\) and \(\delta_{m}\) will no longer be independent variables. Both will receive the common influence of maintenance personnel and maintenance equipment at the same time, which have a corresponding relationship. In this case, the initial crack depth and the initial crack expansion rate variables are expressed as a matrix with statistically correlative column elements, as follows:  
\[A_{0}= \begin{bmatrix}a_{01}&a_{02}&\cdots&a_{0N}\\ v_{01}&v_{02}&\cdots&v_{0N}\end{bmatrix}  \tag{26}\]  
The matrix of repair coefficients after the \(m\)th repair is expressed by:  
\[A_{1}= \begin{bmatrix}\varphi_{1m}(a_{01},\ v_{01})&\varphi_{1m}(a_{02},\ v_{02})&\cdots&\varphi_{1m}(a_{0N},\ v_{0N})\\ \delta_{1m}(a_{01},\ v_{01})&\delta_{1m}(a_{02},\ v_{02})&\cdots&\delta_{1m}(a_ {0N},\ v_{0N})\end{bmatrix}  \tag{27}\]Then, the crack depth and the initial crack expansion rate after the \(m\)th repair become:  
\[A_{2}=