## 1 Introduction
[PERSON] proposed an approach to compute the bounds of the response for multi-cracked beams with uncertain parameters [17]. According to the response function for each uncertain parameter, two different models were adopted to calculate the response bounds. This approach provided accurate bounds even for large uncertainties. Moreover, a non-probabilistic approach to evaluate the frequency response of multi-cracked beams was presented by [PERSON] [18]. The interval variables were used to describe the parameters of each crack, instead of the traditional probabilistic approach. Furthermore, a two-step method to evaluate the bounds of all response variables was presented in the proposed models. Time-varying reliability evaluation methods for crack-containing beam structures, considering maintainability, need further study. Although studies have been conducted on the performance and uncertainty analysis of multi-cracked structures, the following difficulties often occur during the evaluation of the time-varying reliability and maintainability of multi-cracked beam structures:  
1. Multiple cracks in a beam structure interact with each other. The stresses in cracks are statistically dependent under common operating loads. Furthermore, the stress dependence results in an interaction between the expansion rates of multiple cracks, which again results in a statistical correlation between the stresses in the vicinity of each crack at different moments. This complex correlation is a key challenge that needs to be addressed to ensure a quantitative assessment of the time-varying reliability and maintainability of beam structures. Although the stress-strength interference model and its extension can reflect the relationship between stress and strength, the working mechanism of cracked parts and the complex effects of crack defects on these parts cannot be considered in the model. 2. Cracks and their extension have a large impact on the system stiffness, and also generate large stresses, which affect the system strength degradation. Therefore, there is a statistical correlation between the two failure modes of stiffness degradation and the strength degradation of crack-containing beam structures. The complex correlation of the stresses mentioned above makes a correlation analysis of the two failure modes even more complex. 3. When repairing mechanical parts, the randomness of crack repair leads to varying operational states in multi-cracked parts after repair. Therefore, maintainability is closely related to the failure dependence (FD) of components. Maintainability and FD during work tend to be relatively independent in traditional models. Reliability modeling that considers the complex statistical correlations and mechanical mechanisms of factors such as stresses and cracks, and that is capable of quantitatively evaluating the relationship between maintainability dependence (MD) and system FD, faces great challenges. In order to solve the above problems, the complex mechanism of FD and MD is analyzed in detail in this paper. Moreover, quantitative time-varying reliability models for multi-cracked beam structures are developed considering MD and FD. In Section 2, the multi-cracked structure is regarded as a dependent series system. The complex statistical correlation mechanism among the working load, crack depth, crack expansion rate, stress, stiffness and strength is discussed. In Section 3, a time-varying reliability model for multi-cracked beam structures is developed, which considers the interaction of FD and MD. In Section 4, the correctness and validity of the model are verified through Monte Carlo simulations (MCS). Furthermore, the interaction between the FD and MD is illustrated through the examples. Finally, the conclusions are given in Section 5.