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]. Skrinar 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 Kisa [
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 Caddemi [
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. Kisa 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 Cannizzaro, 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 Cannizzaro [
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. Santoro 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 Santoro [
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:
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.
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.
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.
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]:
where
is the initial strength,
a and
C are material parameters. The relationship between the crack depth at the
j + 1st operational cycle and the
jth operational cycle is expressed as the function of the stress
s and the expansion rate
, as follows:
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 addition, a comparison between the reliability of the dependent system and that of the independent system is shown in
Figure 6.
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
and
in the case of considering MD, and those in the case without MD, are shown in
Table 2 and
Table 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
Figure 7 and
Figure 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. Moreover, the MD will make the FD more obvious. In the whole life-cycle assessment of beam structures, attention should be paid to the influence of repair behavior, especially the MD effect, on the time-varying reliability of the system.
Case 3: In order to analyze the effect of the dispersion of
on the FD and MD, the discrete PDFs of
and
are shown in
Table 4 and
Table 5, and the time-varying FDC and the time-varying MDC of the system are shown in
Figure 9 and
Figure 10, respectively.
As can be seen from
Figure 9 and
Figure 10, the trend of FD with and without considering MD is similar to that in case 2. However, as the dispersion of
decreases, the FDC significantly decreases, which reduces the effect of the FD on the time-varying reliability of the system to a larger extent. In addition, the decrease in the dispersion of
leads to a more pronounced decrease in MDC, which attenuates the MD effect and the extent to which the MD influences the FD.
Case 4: In order to analyze the effect of the dispersion of
on the FD and MD, the discrete PDFs of
and
are shown in
Table 6 and
Table 7, and the time-varying FDC and the time-varying MDC of the system are shown in
Figure 11 and
Figure 12, respectively.
As can be seen from
Figure 11 and
Figure 12, the dispersion of
has a pronounced effect on FDC and MDC. As the dispersion of
decreases, the influences of FD on the time-varying reliability of the system decreases, weakening the FD effect. In addition, the system MD effect also decreases when the dispersion of
decreases, weakening the influence of MD on FD as well as the system reliability. The results indicate that the proposed model can quantitatively evaluate the relationship between MD, FD and system time-varying reliability, which provide a theoretical basis for system fault diagnosis, reliability optimization design and maintenance strategy formulation.
It should be noted that the data in
Table 2,
Table 3,
Table 4,
Table 5,
Table 6 and
Table 7 provide discrete distributions empirically derived maintenance parameters that are appropriately assumed to characterize different maintenance effects. However, the methodology presented in this paper is not limited to specific distribution data, and any actual distribution data that are obtained can be used in the reliability calculation model presented in this paper to assess reliability.
5. Conclusions
In this paper, the time-varying reliability models of multi-cracked beam structures considering MD and FD are established. As the multiple cracks are subjected to common working loads, there are complex statistical correlations between different cracks and between the stresses of the crack elements. The multiple crack elements are regarded as a series system with FD. The time-varying reliability models of multi-cracked beam structures are further developed by the neural network method considering the complex FD which results from the stress dependence, crack extension dependence, and multi-failure mode dependence. In order to characterize the system FD effect, the system FDC index is proposed to measure the influence of FD on the time-varying reliability assessment of the system. On this basis, according to the working principle of the beam structure and the maintenance mechanism for the crack defects, the time-varying system reliability models considering the MD is proposed and the MDC index is further proposed. By establishing the relationship between the MDC and FDC, a method is proposed to quantitatively measure the interaction between the MD and the FD. In addition, the validity and correctness of the model are verified by the MCS method.
A numerical example is used to point at that the reliability models under the traditional independence assumption can result in an underestimation of time-varying reliability and service life. This error may cause excessive margin design in system reliability design, increasing design and maintenance costs. In addition, the MD will make the FD more pronounced. As the dispersion of and decrease, the FDC and MDC significantly decrease, weakening the MD effect and the influences of MD on the FD. The proposed models are capable of quantitatively evaluating the relationship between MD, FD and the time-varying reliability of the system, which provides a theoretical basis for the optimal design of reliability and the formulation of maintainability strategies.
In this paper, time-varying reliability models for crack-containing structural systems considering complex statistical correlations are proposed and the proposed models were validated using the MCS methodology. However, the reliability evaluation methodology based on the analysis of physical experiments is also an important value for their practical application in engineering, which is an important next step that will be carried out in future work. Moreover, different design and maintenance parameters, as well as BPNN parameters, have an impact on the system reliability calculations, which will be investigated in future work.