## 4 Numerical Examples  
Case 1: Consider a cantilever beam structure containing two cracked elements, as shown in Figure 2, with the left end of the cantilever beam being fixed. The material and dimensional parameters of the beam structure are listed in Table 1. The rightmost end of the beam structure is subjected to a random working load, \(F\). The stochastic statistical characteristics of \(F\) are shown in Table 1. The mean values of the variables are indicated in Table 1. The residual strength of each crack element is expressed as [22]:  
\[\mathrm{r}(\mathrm{n})=r_{0}(1-\frac{\mathrm{n}\int_{0}^{\infty}\mathrm{s}^{ \mathrm{e}}\mathrm{f}_{\mathrm{s}}(\mathrm{s})\mathrm{d}s}{\mathrm{C}})^{g} \tag{33}\]  
where \(r_{0}\) is the initial strength, \(a\) and \(C\) are material parameters. The relationship between the crack depth at the \(j+1\)st operational cycle and the \(j\)th operational cycle is expressed as the function of the stress \(s\) and the expansion rate \(v\), as follows:  
\[a_{j+1}=a_{j}+s^{*}v \tag{34}\]  
The reliability of the Monte Carlo simulation (MCS) is compared with the time-varying reliability calculated by the models proposed in this paper, as shown in Figure 5. The MCS simulates the system working process without being affected by any analytic reliability model. Thus, it can be used to verify the validity and correctness of the proposed models. The time-consuming problem of MCS has now been widely raised in the field of reliability engineering, which is the main reason the method is heavily used to validate analytical models in reliability analyses. The variability between analytical models and MCS in terms of computation time is affected by many factors, including the number of random variables, the statistical characteristics of random variables, the system operating principles and the system operating life, and is sensitive to these parameters. Overall, the analytic model is able to provide accurate computational results while saving computational time. In  
Figure 4: System reliability calculation flowchart. As can be seen in Figure 6, the MCS results are in good agreement with the results of the reliability models proposed in this paper. In addition, the FD has a large impact on the multi-cracked beam structure. The reliability model, when used under the traditional independence assumption, can result in a underestimation of the time-varying reliability. This error may cause excessive margin design in the system reliability design, resulting in wasted costs, such as material costs. Meanwhile, the effect of FD on the beam structure is mainly concentrated in the middle stage of the working period. The reliability error caused by FD increases continuously with time and disappears when the beam structure is close to complete failure. In addition, the FD causes an underestimation of the life of the beam structure, which results in the early maintenance of the structure and affects the system's fault diagnosis and maintenance strategy development. Case 2: In order to analyze the effect of maintenance behavior on the time-varying system reliability, the discrete PDFs of \(\varphi_{m}\) and \(\delta_{m}\) in the case of considering MD, and those in the case without MD, are shown in Tables 2 and 3, respectively. The time-varying reliability of the system after repair is shown in Figure 7 and the time-varying system FDC is shown in Figure 8.  
As can be seen from Figures 7 and 8, due to the imperfect maintenance of the system, the system performance still cannot be restored to the original statistics. The time-varying reliability of the system under both the independent and FD assumptions decreases compared to that before maintenance. However, the effect of FD is similar to that before repair, which results in an underestimation of the time-varying system reliability, affecting the reliability-based optimal design and the formulation of maintenance strategies. In addition, the FDC enhancement after repair is more obvious, indicating that the repair behavior increases the degree of FD.