Series–Parallel System Analysis and Reliability Research of Grid Structures Considering Adaptive Dynamic Bounding Threshold

Abstract

There are many bars in grid structures, and their topological relationships are complex, so the workload of calculations involved in researching their reliability is large. In this paper, based on the geometric topological relationship within a grid structure, a series–parallel system analysis of the structure is carried out by means of node analysis, constraint grade division, and hierarchical combination analysis, and the series–parallel analysis results for the structure are used to judge the failure of the structure. In order to avoid the problem of missing important components due to unreasonable selection of the bounding threshold when identifying failed components, an adaptive dynamic bounding threshold method with high accuracy is introduced. The failure probability value of the failure path is calculated using the direct numerical integration method, and the reliability of the whole structure is obtained. Finally, the validity of this method is verified using an example of a square pyramid grid structure. The results show that the method is feasible for series–parallel system analysis, structural failure judgment, and reliability analysis of a grid structure.

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 $\beta$-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 $\beta$-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 $\beta$-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 $\beta$-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 $\beta$-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 $\beta$-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 $\beta$-unzipping method and identifies the failure path of the structure. Finally, by calculating the failure probability of the series–parallel failure path, the $\beta$-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.

2. Judgment of Structural Series–Parallel System

Considering the topological characteristics of the grid structure itself, based on the mechanism criterion method [12] and the local failure theory [13,14], it is proposed to realize the series–parallel system analysis through three steps: node analysis, constraint grade division, and hierarchical combination analysis; find out if the failure will lead to the structure becoming a local important component of the variable mechanism; and then judge the failure of the structure according to whether the local important component fails.

2.1. Node Analysis

(1) Under normal circumstances, the bars around the bearing nodes in a grid structure bear the largest force. Under the condition of the same resistance level of each member, the bars around the bearing nodes fail first [15]. When the number of failed components at each node reaches a certain number, the node fails due to a lack of constraints. In the grid structure, the support supports and constrains the function of the whole structure. When failure occurs at the support, it often leads to the destruction of the whole structure. The node analysis method is based on the situation that the failure of the nodes will lead to the destruction of the entire structure. The nodes at the support are regarded as important nodes, and a large number of nodes in the grid structure are screened via the node analysis method to reduce the analysis scope, reduce the workload, and improve the efficiency of the analysis.

(2) In addition, because the number of bearing nodes in the grid structure only accounts for a small part of the number of nodes in the whole structure, the screening of important nodes will affect the accuracy of the analysis if it is limited to the position of the bearing. For some nodes, there are many components connected to the non-boundary position of the whole structure, which can generally be regarded as important nodes [16].

2.2. Constraint Level Division

(1) In a grid structure, there is a certain constraint relationship between the connected components. This constraint relationship will generate a constraint level according to the distance between the topological relationship and the position of the support. For example, in the ten-bar truss structure in Figure 1, the support nodes (1, 2) play the role of supporting and constraining the entire superstructure, so the constraint level of the node at the support is the highest, and the components connected to the support nodes (①, ②, ③, ④) and the support nodes (1, 2) are mutually constrained. The relationship is a first-level constraint bar. The nodes (3, 4) are connected with the ends of the first-level constraint bars, which are regarded as the second-level constraint node. The member ⑤ between the nodes (3, 4) represents the second-level constraint bars. The members (⑥, ⑦, ⑧, ⑨) are connected with the second-level constraint nodes and away from the supports, which are the third-level constraint bars. The intersection points (5, 6) of the third-level constraint bars are the third-level constraint nodes, and the component ⑩ between the third-level constraint nodes is the third-level constraint bar.

(2) The divided constraint levels are sorted from high to low into levels one, two, and three. Through the different constraint levels, components that effectively constrain the analyzed components can be screened out, that is, components with a constraint level higher than or equal to the constraint level of the analyzed components. For example, in Figure 1, although the three-level constraint bars ⑥ and ⑦ are connected to the second-level constraint node 3, they do not impose constraints on the second-level constraint node 3. The constraint condition of the second-level constraint node 3 is only related to the first-level constraint bars (①, ②) and the second-level constraint bar 5.

In order to prove that the above constraint level is higher than or equal to the constraint level of the analyzed component, an effective constraint conclusion must be generated. Through the verification of the ten-bar truss structure, for any component in the plane truss structure, it is only necessary to ensure that the two directions of x and y are completely constrained to achieve the effect of complete constraint. Therefore, for any node, as long as it is constrained by any two bars with different constraint directions, it can be guaranteed to be completely constrained. Taking the second-level constraint node 3 as an example, the node is connected with first-level constraint members (①, ②), a second-level constraint member ⑤, and third-level constraint members (⑥, ⑦). In order to verify the conclusion that the low-level constraint member cannot constrain the high-level constraint member, the five bars are removed one by one, and only two bars are retained each time to be connected to the second-level constraint node 3. The demolition results are shown in Figure 2.

According to the split results of Figure 2, when any two bars of the first-level constraint members (①, ②) and the second-level constraint member ⑤ fail, the second-level constraint node 3 will fail due to the lack of constraints. However, in this process, whether the third-level constraint members ⑥ and ⑦ fail or not will not affect the second-level constraint node 3. Therefore, it can be proved that the component with a constraint level higher than or equal to the constraint level of the analyzed component will impose effective constraints.

2.3. Hierarchical Combination Analysis

(1) In order to simplify the analysis of the constraints of important nodes, the structure is divided into an upper chord layer, a middle web layer, and a lower chord layer.

(2) If a component has reached complete constraint in a certain layer, the result of the component in the combination analysis must also be completely constrained. Only when the component being analyzed lacks component constraints in each layer is it necessary to combine different layers together to perform combination analysis on the component to determine whether there is a lack of constraints.

Taking the simplified model in Figure 3 as an example, the hierarchical combination analysis is carried out. In Figure 3, nodes 2, 3, 4, 5, 6, 7, 8, and 9 are fixed supports, and the structure is completely constrained. (1) The structure is divided into an upper chord layer (dark blue part), a middle web member layer (green part), and a lower chord layer (purple part). (2) A constraint analysis of node 1 (red node) is carried out. Because the upper chord layer does not directly constrain node 1, it is not necessary to consider the upper chord layer in the analysis, while the middle web member layer and the lower chord layer can completely constrain node 1 alone without the failure of the member. Therefore, when there are failed bars in both layers, node 1 may lack constraints, and it is necessary to conduct a specific combination analysis according to the number and locations of failure bars.

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 $\beta$-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 $n$ stages. It is assumed that in the $k$ ($0 < k < n$) stage of the failure process, the member $r_k$ satisfying the condition ${\beta }_k^e\in [{\beta }_k^{\min },{\beta }_k^{\min }+\Delta {\beta }_k]$ will become the candidate failure member in this stage. Where ${\beta }_k^{\min }=\min ({\beta }_k^e)$ and $\Delta {\beta }_k$ are the bounding thresholds, the basic steps of identifying the main failure modes using the $\beta$-unzipping method are as follows.

Assuming that the two basic variables of the external load and the structural resistance of the component $e$$(e=1,2,\dots ,M)$ are normally distributed, the reliability index of the component can be calculated according to the following formula in the $k$ failure stage [19]:

$${\beta }_k^e=\frac{{\mu }_R-{\mu }_S}{\sqrt{{\sigma }_R^2+{\sigma }_S^2}}$$

(1)

where $M$ is the number of components, and ${\beta }_k^e$ is the reliability index of component $e$ in stage $k$; ${\mu }_S$ and ${\sigma }_S$ are the mean value and standard deviation of the component effect; ${\mu }_R$ and ${\sigma }_R$ are the mean value and standard deviation of the component resistance.

In the $k$ failure stage, the member whose reliability index meets the boundary condition ${\beta }_k^e\in [{\beta }_k^{\min },{\beta }_k^{\min }+\Delta {\beta }_k]$ will become the candidate failure member in the $k$ failure stage and then enter the $k+1$ failure stage. In the $k+1$ stage, the $k$ 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 $\beta$-unzipping method depends largely on the value of the bounding threshold $\Delta {\beta }_k$. The selection of $\Delta {\beta }_k$ 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 $d_k$ 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 $d_k$ of the $k$ failure stage is defined as follows:

$$d_k=\frac{{\overline{\beta }}_k+{\beta }_k^{\min }}{{\overline{\beta }}_k+{\beta }_k^{\max }}$$

(2)

where ${\overline{\beta }}_k$, ${\beta }_k^{\min }$, and ${\beta }_k^{\max }$ 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:

$$\Delta {\beta }_k=({\beta }_k^{\max }+{\beta }_k^{\min })\cdot d_k$$

(3)

The bar that satisfies the boundary condition ${\beta }_k^e\in [{\beta }_k^{\min },{\beta }_k^{\min }+\Delta {\beta }_k]$ will become the candidate failure bar in the first failure stage.

The failure path of the grid structure identified through the $\beta$-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 $P_{fi}$, and the single component failure probability formula [26] is as follows:

$$P_{fi}=\Phi [-{\beta }_i]=1-\Phi [{\beta }_i]$$

(4)

where $\Phi [•]$ is the cumulative distribution function of one-dimensional standard normal distribution.

For the series system composed of $n$ failure modes $E_1$, $E_2$, …, $E_n$, if the failure probability of the system is $P_{fs}$, then there is

$$P_{fs}=(P_{f{E}_1}\cup P_{f{E}_2}\cup \dots \cup P_{f{E}_n})$$

(5)

For the parallel system composed of $n$ failure modes $E_1$, $E_2$, …, $E_n$, the failure probability of the system is as follows:

$$P_{fs}={\Phi }_n(-\beta ;{\rho }_e)$$

(6)

where $\beta =\{{\beta }_{E_1},{\beta }_{E_2},\dots ,{\beta }_{E_n}\}$ and ${\rho }_e$ are the equivalent correlation coefficients, and ${\Phi }_n(•)$ is the cumulative distribution function of $n$-dimensional standard normal distribution.

$${\rho }_e=\frac{2}{n(n-1)}\overline{\rho }=\frac{2}{n(n-1)}\left[\begin{array}{cccc}1& {\rho }_{12}& \cdots & {\rho }_{1n}\\ {\rho }_{21}& 1& \cdots & {\rho }_{2n}\\ ⋮& ⋮& ⋮& ⋮\\ {\rho }_{n1}& {\rho }_{n2}& \cdots & 1\end{array}\right]$$

(7)

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.

4. Example Analysis

The size of the square pyramid grid studied here is shown in Figure 5. The nodes 43, 45, 53, and 55 are placed with supports, and the bars are circular steel tubes. The models are the upper chord member Φ68 mm × 6 mm, the web member Φ50 mm × 2.5 mm, and the lower chord member Φ63.5 mm × 4.5 mm. Supposing that the unit of vertically concentrated load acts on each node of the upper chord, the direction of action is vertical and downward. In this study, the unit load is 31.3 KN and the bar resistance is 313 KN.

4.1. Analysis of Series–Parallel System Structure

Because the load of the square pyramid grid is evenly distributed on the upper chord and the structure is symmetrical, the lower-left quarter of the whole structure is taken for series–parallel system analysis.

4.1.1. Node Analysis

(1) Node 43 at the bearing position is an important node.

(2) The important nodes are judged according to the number of bars connected to each node and their position in the structure. All eight-bar nodes with the largest number of connecting bars and at the non-boundary positions of the structure are regarded as important nodes, and these important nodes are selected (their node numbers are 8, 9, 14, 15, 43, 44, 48, and 49). Figure 6 shows a schematic diagram of five kinds of nodes with different numbers of connecting bars in the grid.

4.1.2. Constraint Level Division

In Figure 7, node 43 is located at the support, which is a first-order constraint node, and the blue bar connected with it is a first-order constraint bar. Nodes 8, 9, 14, 15, 44, and 48 are secondary constrained nodes, and the green bar is a secondary constrained bar. Although nodes 38 and 42 are also secondary nodes, they are not important nodes in the node analysis, so they are not within the scope of the subsequent analysis. The black bars and nodes in Figure 7 are three-level constraint bars and three-level constraint nodes.

4.1.3. Hierarchical Combination Analysis

(1) Firstly, the grid is divided into three layers: the upper chord layer, the middle web member layer, and the lower chord layer. The local failure of the eight-bar joint is discussed without considering the constraint level. The constraints of the corresponding nodes after the separate failure of the web member and the chord are shown in

Table 1. Analysis of constraints of different types of failure bars on joints after delamination.

Number of Webs Failed	Constrained Direction of the Nodes	Number of Chords Failed	Constrained Direction of the Nodes
0	X, Y, Z	0	X, Y, Z
1	X, Y, Z	1	X, Y, Z
2 adjacent bars failed	X or Y	2 adjacent bars failed	X, Y
2 relative bars failed	Z	2 relative bars failed	X, Y, Z
3	No constraints	3	No constraints
4	No constraints	4	No constraints

. In Figure 8, the blue dots show the eight-bar node positions.

(2) After the layered analysis of the chord and web is completed, a combination analysis is carried out. The upper chord layer, the middle web layer, and the lower chord layer can completely constrain the directly constrained nodes alone without the failure of the bars. Therefore, in the combined analysis, when the failure bars exist in different layers of the node constraint, the constrained nodes may lack constraints. The constraints of the joints after the failure of the chord and web combination in different situations are shown in

Table 2. Analysis results of web and chord combinations.

Situation Number	Number of Webs Failed	Number of Chords Failed	Constrained Direction of the Nodes	Does Local Failure Occur?
1	2 adjacent bars failed	2 adjacent bars failed	X, Y, Z	No
2	2 relative bars failed	2 adjacent bars failed	X, Y, Z	No
3 *	2 adjacent bars failed	3	At least one direction is not constrained	Yes
4	2 relative bars failed	3	X, Y, Z	No
5	3	2 adjacent bars failed	X, Y, Z	No
6	3	2 relative bars failed	X, Y, Z	No
7	≥3	≥3	At least one direction is not constrained	Yes

* In situation 3, the one remaining unfiled chord and two unfiled web bars are in the same plane.

.

Finally, it is concluded that two kinds of local failure will lead to the failure of the whole structure: (1) when two adjacent web bars and three chord bars fail at the important nodes and the non-failure bars at the nodes are in the same plane, the nodes will lack constraints, resulting in the failure of the structure; (2) when more than three (including three) web bars and more than three (including three) chord bars fail, the structure will be damaged.

According to the three steps of node analysis, constraint grade division, and hierarchical combination analysis, the series–parallel system of the structure is obtained. The whole structure is divided into two main parts: I and II. The two main parts are connected in series, and each main part is connected in parallel. The series–parallel system of the grid structure is shown in Figure 9.

In the process of identifying the failure path via the $\beta$-unzipping method, when the bar condition contained in the identification path is the same as any series part of the series–parallel system in Figure 9, it can be determined that the structure is damaged, the path identification is over, and the failure path is formed.

4.2. Structural Reliability

4.2.1. Failure Path Identification

Firstly, the reliability index of 200 bars is calculated using Equation (1) and sorted from small to large. The minimum value of 0.7623, the maximum value of 10, and the average value of 6.7881 are selected. Substituted into the reliability index uniformity calculation (Equation (5)), the reliability index uniformity is $d_k=4.843$, and the corresponding ${\beta }_k^2\in [0.7623,5.6053]$ is obtained. Because this structure is a symmetrical structure, the lower-left quarter of the grid is taken as the initial research object, and all of the bars within the boundary threshold are regarded as failure bars, and the failure bars in the first stage are obtained.

The failure bars identified in the first stage are used as the initial failure bars of each failure path to calculate the subsequent failure paths. When the failure path contains a bar that conforms to any of the series parts in Figure 9, the path identification is completed and the failure path is formed.

When identifying the third stage of the failure path, the failure paths are $\overline{)7}$-$\overline{)85}$-$\overline{)68}$ and $\overline{)17}$-$\overline{)85}$-$\overline{)82}$, and the bars on both paths are the connecting bars at node 8. Failure analysis of node 8 is carried out through the series–parallel system.

(1) Node 8 is an important node of eight bars, so the destruction of this node will lead to the destruction of the whole structure.

(2) Node 8 is a two-level constraint node. In Figure 10, the first level is represented in blue, the second level is represented in green, and the third level is represented in yellow, so the three-level constraint bar (the yellow part) is deleted. At this time, only bars $\overline{)17}$, $\overline{)68}$, $\overline{)82}$, and $\overline{)85}$ are left at node 8. When $\overline{)7}$-$\overline{)85}$-$\overline{)68}$ fails, only chord $\overline{)17}$ and web $\overline{)82}$ are left at node 8, which conforms to a series part in Figure 9, that is, the parallel connection II, so the structure is destroyed. The same method determines that path $\overline{)17}$-$\overline{)85}$-$\overline{)82}$ is also damaged. At this time, the identification of the $\beta$-unzipping method stops, forming a failure path. The relevant parameters are shown in

Table 3. Related parameters for the failure paths.

Path	The First Stage			The Second Stage			The Third Stage
	Bar Numbers	Internal Force (KN)	$\mathit{\beta }$	Bar Numbers	Internal Force (KN)	$\mathit{\beta }$	Bar Numbers	Internal Force (KN)	$\mathit{\beta }$
1	$\overline{)7}$	5.0187	1.5471	$\overline{)7}$-$\overline{)85}$	5.6187	0.7623	$\overline{)7}$-$\overline{)85}$-$\overline{)68}$	6.4848	−0.2494
2	$\overline{)17}$	5.0187	1.5471	$\overline{)17}$-$\overline{)85}$	5.6187	0.7623	$\overline{)17}$-$\overline{)85}$-$\overline{)68}$	6.4848	−0.2494

.

4.2.2. Failure Probability Calculation

The parallel structure example uses the parallel failure from Equation (6) to calculate the failure probability. Firstly, the correlation coefficient of each path is calculated according to the performance function $Z=R-a_{i}S$ of each stage. The reliability index of the third stage of the two failure paths is less than 0, which is regarded as a 100% failure of the third stage. This means that after the failure of the bars in the current two stages, bars $\overline{)68}$ and $\overline{)82}$ in the third stage will fail. Therefore, it is only necessary to calculate the failure probability of the first two stages. The correlation coefficient matrix of the first failure path $\overline{)7}$-$\overline{)85}$-$\overline{)68}$ is as follows:

$${\rho }_{7\text{-}85\text{-}68}=\left[\begin{array}{cc}1& 0.5623\\ 0.5623& 1\end{array}\right]$$

The correlation coefficient matrix of the second failure path $\overline{)17}$-$\overline{)85}$-$\overline{)82}$ is as follows:

$${\rho }_{17\text{-}85\text{-}82}=\left[\begin{array}{cc}1& 0.5623\\ 0.5623& 1\end{array}\right]$$

We thus obtain the correlation coefficient ${\overline{\rho }}_{7\text{-}85\text{-}68}={\overline{\rho }}_{17\text{-}85\text{-}82}=0.5623$.

The multi-mode joint failure probability of two failure paths is obtained using Equation (6):

$$P_{fs}=P_{7\text{-}85\text{-}68}=P_{17\text{-}85\text{-}82}=0.0401$$

Since the multi-mode joint failure probability of the two paths is equal, the failure probability of the whole structure is 0.0401. According to the standard [19], assuming that a grid structure is designed to have a service life of 50 years, and considering that damage to the structure could have a great impact on human life, the economy, society, or the environment, then its failure probability value should not be greater than $1.078\times {10}^{-4}$ to remain in the safe state. However, under the current load, the failure probability of the structure is much greater than the safe value, so the grid is in an unreliable state.

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 $\beta$-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 $\beta$-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.