## 1 Introduction  
Due to the defects generated in the processing and manufacturing process, and the metal fatigue resulting from alternating stress during operation and overload conditions, cracks may exist in the components of mechanical systems. These cracks may result in a reduction in the mechanical properties of mechanical components, causing the components to be more likely to fail far earlier than under the design load, resulting in economic losses and threatening personal safety [1; 2; 3]. Although cracks are common defects in mechanical parts, it is important to take appropriate measures to suppress or improve the impact of crack defects on mechanical components and mechanical systems [4; 5; 6]. In this paper, time-varying reliability models for multi-cracked beam structure will be established. In the models, not only is the complex statistical correlation mechanism between system parameters during system operation considered, but also the maintenance dependence generated by the complex maintenance behavior. The interaction mechanism between the failure dependence during system operation and the maintenance dependence are also considered. In mechanical systems, many mechanical components can be viewed as beam structures from the perspective of mechanical analysis [7]. Therefore, the multi-cracked beam structure will be studied in this paper. A reliability and maintenance quantitative evaluation model will be developed in this paper. Some scholars have conducted in-depth research on the analysis of beam structures with multiple cracks [8; 9; 10]. [PERSON] presented a simplified computational model for the calculation of multi-cracked beams with linearly varying heights [11]. The model extended the utilization of the principle of virtual work to obtain a stiffness matrix and a load vector for a uniform load over the whole element. The model can be applied to analyze the response of multi-cracked beams that include an arbitrary number of transverse cracks. A novel technique for the vibration and stability analyses of axially loaded beams with multiple cracks was proposed by [PERSON] [12], which combines mode synthesis methods and the finite element method. The technique can be used to predict the crack status in the defected structures. A justification of the localized flexibility model of an open crack in a beam in bending deformation was proposed by [PERSON] [13]. The results showed that the formulation and solution of the bending problem for multi-cracked beams can be included in the classical formalism of the theory of distributions. [PERSON] proposed a modal analysis method of beams with a circular cross-section and containing multiple non-propagating open cracks by combining the finite element and component mode synthesis methods [14]. The method can be used to calculate the natural frequencies and analyze the mode shapes of a beam with an arbitrary number of cracks. During the process of system operation, the crack state parameters will show random characteristics. Uncertainty in the maintenance effect can also lead to new uncertainties in the crack state and the system's operating state after repair. Regarding the uncertainty analysis of multi-cracked beam structures, some in-depth research has been carried out. An explicit treatment of dynamic problems of damaged structures in the presence of cracks with variable intensities was developed by [PERSON], exploiting the generalised function approach [15]. The proposed models were applied to an analysis of structures with uncertain cracks. A method based on interval analysis to assess the dynamic response of damaged beams was presented by [PERSON] [16]. This method is capable of handling uncertainties in beam structures and inferring the upper and lower bounds of response parameters without introducing any probability content. By introducing uncertainty in the model, the proposed method effectively evaluates the dynamic response range of damaged beams while significantly reducing the computational burden.