1. Introduction
The grid structure is widely used in the roofs of large-scale building structures. Since the 1980s, a large number of grid structures have been built at a rate of hundreds of units per year. Usually, the life expectancy of a grid structure design is 50 years, so a large number of grid structures built in the early stages are facing or will soon face extended service work [
1,
2]. Whether the grid structure can continue to be used is the main concern at present. In the early stages, researchers conducted a lot of work on the identification of dynamic damage, the structural vulnerability, and the important component selection of grid structures. However, in view of their complex topological relationship, research on the reliability of grid structures has the following problems: (1) the establishment of structural failure criteria; (2) the determination of the main failure modes of the structure.
In practical engineering, due to the neglect of researchers or the simple structure, many researchers mainly judge the damage of a structure through subjective experience. Xiao et al. [
3] judged the structural damage of a specific hybrid structure through subjective experience. It is assumed that the failure of any component will lead to the overall damage of the structure. Song et al. [
4] relied on subjective experience to identify the first two failure stages using the
-unzipping method for structural damage. Yang et al. [
5] judged the failure of the structure through subjective experience. When the minimum reliability index of a certain stage is greater than 3, it is considered as structural failure. This method of relying only on subjective experience for judgment cannot accurately judge the failure of a grid structure with complex topological relations. Therefore, many researchers rely on failure criteria when judging the failure of structures. Liu et al. [
6] used the displacement criterion and mechanism criterion to judge the failure of a structure, that is, when the structure exhibited large deformation or became a variable mechanism, it was regarded as structural failure. Liu [
7] judged the failure of a structure through the mechanism criterion, and when the structure reached the mechanism level, it was regarded as structural damage. However, there are many problems in several widely used judgment methods. For example, the judgment of the displacement criterion needs to refer to the relevant design specifications. When the maximum deformation of the structure exceeds the safety limit, it is considered to be damaged. However, the maximum safety deformation in the design specification is a relatively conservative reference value, so the accuracy of the displacement criterion in judging structural damage is low. The mechanism criterion generally needs to be analyzed using simulation software. When the structure becomes a variable mechanism, it will lead to the singularity of the stiffness matrix, so as to judge whether the structure is damaged according to whether the stiffness matrix is singular. However, there are many factors that lead to the singularity of the stiffness matrix. For example, when a structure is analyzed using simulation software, the problem of imposing constraints on the structure and the selection of element types will affect the singularity of the stiffness matrix. Therefore, with the help of finite-element software, when the mechanism criterion method is used to determine the failure of a structure, it will often misjudge the failure of the structure because of the singularity of the stiffness matrix caused by other influencing factors. More recently, some researchers proposed to start from the topological relationship within the structure itself, and, through series–parallel system analysis of the structure, find out if the failure will lead to the structure becoming a local important component of the variable mechanism, and judge the failure of the whole structure by analyzing whether the local important component fails. Wan [
8] found the series system of a structure through series and parallel analyses of a plane truss structure and judged the failure of the structure through the series system. Lu et al. [
9] analyzed the series–parallel system of truss structures with simple topological relations and used the bearing part of the series system as the basis for structural damage to accurately determine the structural failure. However, the failure judgment of the current series–parallel system structure is only used in plane truss structures and structures with simple topological relationships. The remarkable characteristics of a grid structure are its large number of bars, large number of nodes, and complex topological relationships. Due to the lack of a perfect series–parallel system analysis method, it is not possible to judge the failure of grid structures through the series–parallel system.
In addition, in the process of identifying the failure mode of grid structures via the bounding method, the size of the bounding threshold has a very important influence on the identification of the failure path of the structure. When Xiao et al. [
3] identified the failure mode of a structure using the
-unzipping method, the semi-empirical formula was used to set the bounding threshold, and reliability analysis of a specific hybrid structure was carried out. Song et al. [
4] set the boundary threshold of the
-unzipping method according to subjective experience, identified the failure mode of the structure, and analyzed the reliability of the structure. Liu et al. [
7] set the number of failed bars to be selected at different failure stages, that is, three failed bars were selected in the first stage of failure, and only one member was selected in each subsequent stage. Liu [
8] only selected one or two failure bars to identify the failure path of the structure at each stage of the
-unzipping method. Lu et al. [
9] set the bounding threshold to only select two failed bars at each stage. Najafi et al. [
10] only selected the bar with the smallest reliability index when identifying the failure path of the series structure. In the above research, the use of the
-unzipping method has the problem of a lack of reasonable criteria for the selection of the bounding threshold. In recent years, many researchers have paid attention to the selection of the bounding threshold, studied the selection criteria of the bounding threshold, and achieved good results. When identifying the failure path, Yang et al. [
5] used the difference in failure probability of several special bars in each stage as the boundary threshold and set different selection ranges according to the different failure probabilities of the member, which improved the accuracy of selecting the failure member. Liu et al. [
11] proposed an adaptive dynamic bounding threshold method which can automatically adjust the bounding threshold according to the reliability index of the bar. This method effectively avoids the problem of identifying the wrong failure path due to an unreasonable setting of the boundary threshold when the traditional
-unzipping method identifies the failure path.
In summary, this paper first establishes an analysis method for the series–parallel system of a grid structure, then standardizes the bounding threshold of the -unzipping method and identifies the failure path of the structure. Finally, by calculating the failure probability of the series–parallel failure path, the -unzipping method, considering the adaptive dynamic bounding threshold and the important component selection, and reliability analysis method for the grid structure of the series–parallel system are established.
3. Research on the Reliability of Grid Structures Considering Adaptive Dynamic Bounding Threshold
After one or more components in the grid structure fail, the remaining structure can usually still complete the specified functions. Only when the number of failed components reaches a certain number can the entire structure fail [
17]. In order to accurately select the important components of each failure stage in the structure, the failure process of the structure is divided into different failure stages using the
-unzipping method [
18]. Each failure stage sets a boundary threshold to identify the failure components of different failure stages and finally obtain all of the possible failure paths of the structure.
The failure is divided into stages. It is assumed that in the () stage of the failure process, the member satisfying the condition will become the candidate failure member in this stage. Where and are the bounding thresholds, the basic steps of identifying the main failure modes using the -unzipping method are as follows.
Assuming that the two basic variables of the external load and the structural resistance of the component
are normally distributed, the reliability index of the component can be calculated according to the following formula in the
failure stage [
19]:
where
is the number of components, and
is the reliability index of component
in stage
;
and
are the mean value and standard deviation of the component effect;
and
are the mean value and standard deviation of the component resistance.
In the failure stage, the member whose reliability index meets the boundary condition will become the candidate failure member in the failure stage and then enter the failure stage. In the stage, the stage step is repeated until the cumulative failure of the component conforms to the combination form of any series part in the series–parallel system. When the structure is destroyed, the failure component is identified, and a main failure path of the structure is determined.
The number of candidate failure components in the
-unzipping method depends largely on the value of the bounding threshold
. The selection of
mainly depends on experience, and it is difficult to ensure the efficiency and accuracy of the main failure pattern recognition. To this end, Liu [
20] introduced the reliability index uniformity
and defined the adaptive dynamic boundary threshold to achieve efficiency and accuracy of the main failure pattern recognition. The threshold size set through this method is generally larger than the threshold originally set in the main judgment, which avoids omission, but the threshold size will not identify most of the members in the grid structure, so the method improves the efficiency and accuracy. The reliability index uniformity
of the
failure stage is defined as follows:
where
,
, and
are the average, minimum, and maximum values of the reliability indexes of all components in this stage.
Then, the failure bound threshold is defined as follows:
The bar that satisfies the boundary condition will become the candidate failure bar in the first failure stage.
The failure path of the grid structure identified through the
-unzipping method is generally more than one. It is generally believed that the failure path with the largest failure probability fails first [
21]. Therefore, it is necessary to screen out the path with the largest failure probability in these paths through appropriate methods. Through analysis of the series–parallel system of the structure, the series–parallel form of the failure path can be judged, and then the multi-mode joint failure probability can be calculated with the direct numerical integration method.
The multi-mode joint failure probability is a calculation method for the comprehensive analysis of the correlation between each failure mode and the failure value. Under the multi-failure mode, there is a correlation between modes. Therefore, the analysis of the reliability of the grid structure needs to integrate multiple failure modes, calculate the comprehensive joint failure probability, and then judge the structural reliability. At present, the commonly used methods for calculating multi-mode joint failure probability are divided into three categories, namely the Monte Carlo method [
22], the approximate numerical calculation method [
23], and the direct numerical integration method [
24,
25]. The multi-mode joint failure probability is actually a numerical integration problem, so the failure probability can be calculated with the direct numerical integration method.
When calculating the multi-mode joint failure probability, it is necessary to calculate the failure probability of a single component and the correlation coefficient between multiple modes. The single failure probability is
, and the single component failure probability formula [
26] is as follows:
where
is the cumulative distribution function of one-dimensional standard normal distribution.
For the series system composed of
failure modes
,
, …,
, if the failure probability of the system is
, then there is
For the parallel system composed of
failure modes
,
, …,
, the failure probability of the system is as follows:
where
and
are the equivalent correlation coefficients, and
is the cumulative distribution function of
-dimensional standard normal distribution.
The basic process of important component selection and reliability analysis of grid structures based on the adaptive dynamic bounding threshold and series–parallel system structure is shown in
Figure 4.
5. Discussion and Conclusions
Ref. [
14] analyzed the important members of the grid structure analyzed in this paper using the importance coefficient method, and the analysis results show that the importance of the members in the non-boundary region of the grid is greater, and the importance of the web members is greater than that of the chord. This is in line with the positions of the important members analyzed through the node analysis and constraint classification methods in this paper.
Ref. [
27] analyzed the important members of the grid examples analyzed in this paper by using the multiple response analysis method, and their analysis results show that the importance of the web at the support position is the greatest, and the importance of the upper chord at the support is less than that of the web at the support. This conclusion is consistent with our results stating that the nodes at the support are important nodes, and the constraint level of the web members at the support is higher than that of the upper chord.
In this paper, considering the topological characteristics of the grid structure itself, based on the mechanism criterion method and local failure theory, a three-step analysis of the series–parallel system of the structure is proposed, involving node analysis, constraint level division, and hierarchical combination analysis, so as to simplify the complex grid structure, and apply the obtained series–parallel system to the failure judgment of the structure. In addition, on the basis of the -unzipping method, an adaptive dynamic boundary threshold is introduced, which avoids the problem of missing important members caused by setting the limit threshold through subjective judgment, and ensures the accuracy of failure path identification via the limit method.
Finally, a square pyramid example is used to verify the feasibility of the application of this method in practical problems. The -unzipping method for identifying failure paths by introducing an adaptive dynamic bound threshold is formed; the structural failure is judged by relying on the series–parallel system structure; the multi-mode joint failure probability is calculated by using the direct numerical integration method; and, finally, the largest failure probability value of the failure path is taken as the overall failure probability to evaluate the reliability of the whole structure, forming a grid structure reliability analysis method with a clear technical route and more accurate analysis and calculation links.
It should be noted that, in addition to the orthogonal quadrangular pyramid-type grid considered in this paper, there are other types of grids such as two-way orthogonal pyramid grids, two-way orthogonal oblique grids, etc., and the topological characteristics of these grids are more complex than those of the forward-placed quadrangular pyramid grid. Thus, future research is needed to improve this method using different types of grids with more complex topological relationships.