Non-Probabilistic Reliability Bounds Method for Series Structural Systems Considering Redundant Failure Modes

Abstract

Non-probabilistic structural reliability analysis is based on the convex model and more applicable for practical engineering problems with limited samples. Recently, the authors proposed a non-probabilistic reliability bounds method (NRBM) for series structural systems as an effective means for the assessment of systems’ non-probabilistic reliability. A novel non-probabilistic reliability bounds method considering redundant failure modes is proposed in this paper for series structural systems to further improve the efficiency and accuracy of the NRBM. By decomposing the system into several subsystems with two or three failure modes, three identification criteria for redundant failure modes are developed for these subsystems. A bounding formula for the system’s non-probabilistic failure degree is then derived after removing the redundant failure modes. An investigation of three numerical examples indicates that the proposed method has a higher efficiency and at least equivalent accuracy compared to the NRBM.

1. Introduction

The probabilistic model, known as a powerful tool for dealing with uncertainties in practical engineering problems, has been successfully applied in various fields in the past decades [1]. Probabilistic modeling generally requires sufficient sample data of an uncertain variable. However, these samples are not always available in many engineering problems [2]. In view of this, the non-probabilistic convex model has been developed to deal with problems with fewer sample data as an alternative but useful supplement to the probabilistic model. The non-probabilistic convex model described herein includes a series of convex models such as an interval model [3,4], ellipsoid model [5,6,7], parallelepiped model [8,9], and super ellipsoid model [10,11].

In recent years, many reliability methods based on the non-probabilistic convex model have been reported as effective means for evaluating reliability. A non-probabilistic robust reliability index and its variants were first developed [12,13,14]. Another non-probabilistic reliability index was then suggested by using an interval rather than a deterministic value [15]. Inspired by their probabilistic counterparts, the non-probabilistic reliability index and reliability were also proposed. A non-probabilistic reliability index for the interval model and its several computation methods were proposed [16,17,18]. A non-probabilistic reliability index was then presented for the ellipsoid model, whereby its two approximate computation methods—namely, the mean value method, and the design point method—were formulated [19,20]. Two non-probabilistic reliability indices were also constructed for the parallelepiped and super ellipsoid models [21,22]. A non-probabilistic reliability and its computation methods were further proposed for the interval model [23,24]. A non-probabilistic reliability and its two computation methods—namely, the first- and second-order approximation methods—were advanced for the ellipsoid model [25,26].

The abovementioned reliability methods [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] underline the reliability assessment of a component. Moreover, several system non-probabilistic reliability methods have also been developed. Two system non-probabilistic reliability indices were proposed for the interval model, in which the optimal criterion method and the branch-and-bound method were used to enumerate the main failure modes of the structural system [27,28]. An efficient method was further developed for calculating a system non-probabilistic reliability index [29]. A Monte Carlo simulation (MCS) method was also developed to calculate system non-probabilistic reliability [26]. The MCS can provide a theoretical reference solution but generally has a lower efficiency. In our recent work [30], the NRBM was proposed to estimate the above system non-probabilistic reliability. This method is expected to provide an efficient means for the assessment of system reliability due to advantages such as convenient use and a good balance between accuracy and efficiency. However, in its current state, the NRBM still has a major drawback: Due to the ordering dependency problem existing in the NRBM, all of the possible ordering alternatives need to be considered to yield the narrowest bounds. The number of these alternatives, however, becomes enormously large when a system involves more failure modes. This size problem requires a higher computational cost and, hence, becomes a hindrance for practical applications of this method.

In view of the size problem, a novel non-probabilistic reliability bounds method is proposed for series systems by considering redundant failure modes. The proposed method has two advantages over the NRBM [30]: Firstly, the proposed method significantly reduces the number of the possible ordering alternatives by identifying and removing the redundant failure modes, leading to a higher efficiency. Secondly, the proposed method has equal or greater accuracy than the NRBM, making it more suitable for practical applications.

The remainder of this paper is structured as follows: Section 2 briefly reviews the NRBM; Section 3 presents the proposed method and its implementation; Section 4 provides three examples to demonstrate the effectiveness of the proposed method; and Section 5 presents the conclusions.

2. Review of the NRBM

Consider a series structural system consisting of $m$ failure modes, each of which can be defined by a performance function:

$$g_j\left(X\right),j=1,2,\cdots ,m$$

(1)

where $X={\left(X_1,X_2,\dots ,X_n\right)}^{\mathrm{T}}$ is an n-dimensional vector; $X_i$ is a variable associated with geometric dimensions, material properties, loads, etc., and all of its possible values form a marginal interval $X_i^{\mathrm{I}}$; and $g_j\left(X\right)=0$ denotes the failure surface. For the sake of simplicity, the performance functions involved are assumed to be linear, since a weak nonlinear performance function can be replaced by its linear approximation at the design point. The interval model and the ellipsoid model have been widely used to deal with uncertain variables. In the following text, the ellipsoid model is only considered to describe the uncertainty domain.

$$E_X=\left\{\left.X\right|{\left(X-X^{\mathrm{c}}\right)}^{\mathrm{T}}\Omega \left(X-X^{\mathrm{c}}\right)\le 1\right\}$$

(2)

where $E_X$ and $X^{\mathrm{c}}$ denote the ellipsoid and its midpoint, respectively; and $\Omega$ is a characteristic matrix determining the size and orientation of the ellipsoid.

The uncertain vector $X$ is first mapped into the normalized vector $\delta ={\left[{\delta }_1,{\delta }_2,\cdots ,{\delta }_n\right]}^{\mathrm{T}}$ by using a linear transformation. This linear transformation generally consists of a parallel shift and a rotation of the coordinate system, and the reader can refer to Refs. [19,26] for more information. Through such a treatment, the subsequent system reliability analysis becomes easy to perform. Correspondingly, the ellipsoid and the performance function become the unit sphere ${\delta }^{\mathrm{T}}\delta \le 1$ and the normalized performance function $G_j\left(\delta \right)$, respectively.

For a series system, the structural system will fail when any one of the failure modes is in a failure state. The failure surface of the system can be mathematically expressed as follows:

$$G_{\mathrm{series}}\left(\delta \right)=0$$

(3)

In Figure 1, a system with two failure modes for a two-dimensional problem is taken as an example. Here, the failure surface $G_{\mathrm{series}}\left(\delta \right)=0$ represents a geometrically continuous but unsmooth surface. The entire ellipsoid is divided into two different parts by the failure surface, one of which is in the safe domain ($G_{\mathrm{series}}\left(\delta \right)\ge 0$), while the other is in the failure domain ($G_{\mathrm{series}}\left(\delta \right)<0$).

Intuitively, the non-probabilistic failure degree $f_{\mathrm{s}}$ and non-probabilistic reliability $R_{\mathrm{s}}$ of a series system can be given as follows [26]:

$$\left\{\begin{array}{c}{f}_{\mathrm{s}}=\frac{A_{\mathrm{s}}}{A_{\mathrm{w}}}\\ R_{\mathrm{s}}=\frac{A_{\mathrm{w}}-A_{\mathrm{s}}}{A_{\mathrm{w}}}\end{array}\right.$$

(4)

where $A_{\mathrm{w}}=\frac{{\pi }^{\frac{n}{2}}}{\mathsf{\Gamma }\left(\frac{n+2}{2}\right)}$ denotes the volume of the whole ellipsoid, with $\mathsf{\Gamma }\left(•\right)$ denoting a gamma function; $A_{\mathrm{s}}$ denotes the volume of the ellipsoid falling into the failure domain, which can be called the failure volume of the system.

The authors proposed the NRBM to estimate the system’s non-probabilistic failure degree when $A_{\mathrm{s}}$ is not always available, i.e.,

$$\frac{A_1+{\displaystyle \sum _{j=2}^m\max }\left[\left(A_j-{\displaystyle \sum _{k=1}^{j-1}{A}_{jk}}\right),0\right]}{A_{\mathrm{w}}}\le \frac{A_{\mathrm{s}}}{A_{\mathrm{w}}}\le \frac{{\displaystyle \sum _{j=1}^m{A}_j}-{\displaystyle \sum _{j=2}^m\max \left(\underset{k<j}{A_{jk}}\right)}}{A_{\mathrm{w}}}$$

(5)

where $A_j$ is the failure volume of the $j_{\mathrm{th}}$ failure mode and $A_{jk}$ is the joint failure volume of the $j_{\mathrm{th}}$ and $k_{\mathrm{th}}$ failure modes. The reader can refer to Refs. [26,30] for more information on $A_j$ and $A_{jk}$.

3. The Proposed Method

As stated in the introduction, the NRBM may have a size problem. In this section, a novel non-probabilistic reliability bounds method is proposed to overcome the above problem. The proposed method consists of two parts: (a) identifying redundant failure modes, and (b) a bounding formula after removing redundant failure modes.

3.1. Identifying Redundant Failure Modes

By treating each of the failure volumes involved as a set, $G_i$ is a redundant failure mode when its corresponding set $A_i$ satisfies

$$A_i\subseteq {\displaystyle \underset{j=1}{\overset{q}{\cup }}{A}_j}\left(i\ne j,q=1 or 2,\dots ,or m-1\right)$$

(6)

where ${\displaystyle \cup A_j}$ denotes the union of the sets corresponding to other failure modes, and $q$ denotes the number of other failure modes. Figure 2 shows a redundant failure mode in a system with two failure modes, from which the set $A_2$ is a subset of the set $A_1$ and, therefore, the failure mode $G_2$ is a redundant one.

It is well known that in probabilistic reliability analysis the redundant failure mode may occur when and only when two failure planes are parallel to one another. However, as shown in Figure 2, in non-probabilistic reliability analysis the redundant failure mode may occur when two failure planes are not only in a parallel state but also in an intersecting state because of the limitation of the ellipsoid. This means that the redundant failure mode is more likely to occur in non-probabilistic reliability analysis and, therefore, needs to be further considered.

The question arises as to how to identify the redundant failure modes in a system. For a practical problem, this is enormously complex and, hence, is a challenging or even impossible issue because of the high dimensionality and multiple failure modes. Considering the feasibility of performing the identification, in the following text a complex structural system with multiple failure modes is decomposed into several simple subsystems with two and three failure modes. For better understanding, we first focus on a subsystem with two failure modes, and then we deal with a subsystem with three failure modes. It should be pointed out that, here, an approximate but effective means instead of a precise one is presented for the identification of redundant failure modes.

3.1.1. Identification of Redundant Failure Mode for a Subsystem with Two Failure Modes

Consider a subsystem consisting of any two failure modes of the system. Without loss of generality, the performance functions of the subsystem are given as follows:

$$\left\{\begin{array}{l}{G}_1\left(\delta \right)=a^{\mathrm{T}}\delta +a_0\hfill \\ G_2\left(\delta \right)=b^{\mathrm{T}}\delta +b_0\hfill \end{array}\right.$$

(7)

where $a={\left(a_1^{},a_2^{},\dots ,a_n^{}\right)}^{\mathrm{T}}$ and $b={\left(b_1^{},b_2^{},\dots ,b_n^{}\right)}^{\mathrm{T}}$ denote the coefficient vectors, while $a_0$ and $b_0$ are given constants. The correlation angle ${\gamma }_{12}$ between two failure modes can be obtained as follows:

$${\gamma }_{12}=\arccos \left(\frac{{\alpha }^{\mathrm{T}}b}{‖\alpha ‖‖b‖}\right)$$

(8)

where $‖•‖$ denotes the norm of a vector.

Let $d_i$ denote the minimum distance from the origin to the failure surface in the normalized variable space. In this study, $d_i$ is limited by $0\le d_i\le 1$. This is primarily because $d_i<0$ means that the system has an unacceptable reliability for practical problems, and secondly because $d_i>1$ means that the corresponding failure volume is equal to zero and has no effect on system reliability.

As shown in Figure 3, the identification criterion for a subsystem with two failure modes is that the correlation angle ${\gamma }_{12}$ satisfies [30]

$$0\le {\gamma }_{12}\le \left|\arccos \left(\frac{\left|a_0\right|}{a^{\mathrm{T}}a}\right)-\arccos \left(\frac{\left|b_0\right|}{b^{\mathrm{T}}b}\right)\right|$$

(9)

$G_1$ is the redundant failure mode when $\frac{\left|a_0\right|}{a^{\mathrm{T}}a}$ is greater than $\frac{\left|b_0\right|}{b^{\mathrm{T}}b}$; otherwise, $G_2$ is the redundant failure mode, as shown in Figure 3b. This conclusion is also applicable for three- and n-dimensional problems. For the sake of simplicity, those identified redundant failure modes do not participate in the subsequent identification process. Thus, the number of the above subsystems does not exceed $\frac{m\left(m-1\right)}{2}$.

3.1.2. Identification of Redundant Failure Modes for a Subsystem with Three Failure Modes

A subsystem consisting of any three failure modes of the system is considered, among which the two performance functions that have the maximum correlation angle are also given by Equation (7), while the third is given by

$$G_3\left(\delta \right)=c^{\mathrm{T}}\delta +c_0$$

(10)

where $c={\left(c_1^{},c_2^{},\dots ,c_n^{}\right)}^{\mathrm{T}}$ is the coefficient vector and $c_0$ is a given constant. The corresponding correlation angles, ${\gamma }_{13}$ and ${\gamma }_{23}$, are given as follows:

$$\left\{\begin{array}{c}{\gamma }_{13}=\arccos \left(\frac{a^{\mathrm{T}}c}{‖a‖‖c‖}\right)\\ {\gamma }_{23}=\arccos \left(\frac{b^{\mathrm{T}}c}{‖b‖‖c‖}\right)\end{array}\right.$$

(11)

For this subsystem, the redundant failure mode $G_3$ can be identified by the three special points $P_{12}$, $P_{\mathrm{u}}$, and $P_{123}$. Herein, $P_{12}$ is the point located at the intersection of the failure surfaces $G_1\left(\delta \right)=0$ and $G_2\left(\delta \right)=0$, and it has the minimum distance to the origin; $P_{\mathrm{u}}=\frac{P_{12}}{‖P_{12}‖}$ is the point located on the unit sphere and has the same direction as $P_{12}$. $P_{123}$ is the point located at the intersection of the failure surfaces $G_1\left(\delta \right)=0$, $G_2\left(\delta \right)=0$, and $G_3\left(\delta \right)=0$, and it has the minimum distance to the origin.

$P_{12}$ can be obtained by creating the following optimization problem:

$$\begin{array}{cc}\min & {\delta }^{\mathrm{T}}\delta \\ s.t.& \left\{\begin{array}{c}{G}_1\left(\delta \right)=0\\ G_2\left(\delta \right)=0\end{array}\right.\end{array}$$

(12)

A Lagrange function $L\left(\delta \right)$ is then constructed to obtain the analytical solution of the above problem:

$$L\left(\delta \right)={\delta }^{\mathrm{T}}\delta -\lambda \left(a^{\mathrm{T}}\delta +a_0\right)-\mu \left(b^{\mathrm{T}}\delta +b_0\right)$$

(13)

where $\lambda$ and $\mu$ are Lagrange multipliers. According to the necessity condition of an extreme value, we have

$$\left\{\begin{array}{c}2\delta -\lambda a-\mu b=0\\ a^{\mathrm{T}}\delta +a_0=0\\ b^{\mathrm{T}}\delta +b_0=0\end{array}\right.$$

(14)

and then

$$\left\{\begin{array}{c}{a}^{\mathrm{T}}\left(\frac{\lambda a+\mu b}{2}\right)=-a_0\\ b^{\mathrm{T}}\left(\frac{\lambda a+\mu b}{2}\right)=-b_0\end{array}\right.$$

(15)

from which $\lambda$ and $\mu$ can be obtained.

$$\left\{\begin{array}{c}\lambda =\frac{2\left(b_0\cos {\gamma }_{12}-a_0\right)}{1-{\cos }^2{\gamma }_{12}}\\ \mu =\frac{2\left(a_0\cos {\gamma }_{12}-b_0\right)}{1-{\cos }^2{\gamma }_{12}}\end{array}\right.$$

(16)

By substituting Equation (16) into Equation (14), we can obtain the optimal ${\delta }^{\ast }$:

$${\delta }^{\ast }=\frac{\left(b_0\cos {\gamma }_{12}-a_0\right)}{{\sin }^2{\gamma }_{12}}a+\frac{\left(a_0\cos {\gamma }_{12}-b_0\right)}{{\sin }^2{\gamma }_{12}}b$$

(17)

$P_{12}$ is equal to ${\delta }^{\ast }$ due to the fact that ${\delta }^{\ast }$ is the most extreme possible unique point and that there must be the shortest distance from the origin to a particular plane. $P_{\mathrm{u}}$ can then be obtained by the expression $\frac{P_{12}}{‖P_{12}‖}$.

Similar to $P_{12}$, $P_{123}$ can be obtained as follows:

$$\begin{array}{c}{P}_{123}=\frac{a_0-b_0\cos {\gamma }_{12}-c_0\cos {\gamma }_{13}-a_0\cos {\gamma }_{23}{}^2+c_0\cos {\gamma }_{12}\cos {\gamma }_{23}+b_0\cos {\gamma }_{13}\cos {\gamma }_{23}}{\cos {\gamma }_{12}{}^2+\cos {\gamma }_{13}{}^2+\cos {\gamma }_{23}{}^2-1-2\cos {\gamma }_{12}\cos {\gamma }_{13}\cos {\gamma }_{23}}a\\ +\frac{b_0-a_0\cos {\gamma }_{12}-c_0\cos {\gamma }_{23}-b_0\cos {\gamma }_{13}{}^2+c_0\cos {\gamma }_{12}\cos {\gamma }_{13}+a_0\cos {\gamma }_{13}\cos {\gamma }_{23}}{\cos {\gamma }_{12}{}^2+\cos {\gamma }_{13}{}^2+\cos {\gamma }_{23}{}^2-1-2\cos {\gamma }_{12}\cos {\gamma }_{13}\cos {\gamma }_{23}}b\\ +\frac{c_0-a_0\cos {\gamma }_{13}-b_0\cos {\gamma }_{23}-c_0\cos {\gamma }_{12}{}^2+b_0\cos {\gamma }_{12}\cos {\gamma }_{13}+a_0\cos {\gamma }_{12}\cos {\gamma }_{23}}{\cos {\gamma }_{12}{}^2+\cos {\gamma }_{13}{}^2+\cos {\gamma }_{23}{}^2-1-2\cos {\gamma }_{12}\cos {\gamma }_{13}\cos {\gamma }_{23}}c\end{array}$$

(18)

Figure 4 gives two cases where $G_3$ is a redundant failure mode for a three-dimensional problem. In Figure 4a, $P_{123}$ is outside the unit sphere, while $P_{12}$ and $P_{\mathrm{u}}$ are located in the safe domain and the failure domain of $G_3$, respectively. Thus, the identification criterion for this case is given as follows:

$$\left\{\begin{array}{l}‖P_{123}‖>1\\ G_3\left(P_{12}\right)>0\\ G_3\left(P_{\mathrm{u}}\right)<0\end{array}\right.$$

(19)

In Figure 4b, $P_{123}$ is inside the unit sphere, and $P_{12}$ coincides with $P_{123}$. The identification criterion for this case can be given as follows:

$$\left\{\begin{array}{l}‖P_{123}‖\le 1\\ P_{123}=P_{12}\end{array}\right.$$

(20)

These above criteria are also suitable for two- and n-dimensional problems. Again, those redundant failure modes identified in this subsection also do not participate in the subsequent identification process, and the number of the above subsystems is not greater than $\frac{m\left(m-1\right)\left(m-2\right)}{6}$.

3.2. A Bounding Formula after Removing the Redundant Failure Modes

Assume that there are $r\left(r<m\right)$ redundant failure modes in the system. These redundant failure modes can be removed due to the fact that their failure surfaces have no contribution to that of the system. After removing the redundant failure modes, the remaining failure modes in the system can be renumbered in ascending order according to their original number. A novel bounding formula can then be obtained as follows:

$$\frac{A_1+{\displaystyle \sum _{j=2}^{m-r}\max }\left[\left(A_j-{\displaystyle \sum _{k=1}^{j-1}{A}_{jk}}\right),0\right]}{A_{\mathrm{w}}}\le \frac{A_{\mathrm{s}}}{A_{\mathrm{w}}}\le \frac{{\displaystyle \sum _{j=1}^{m-r}{A}_j}-{\displaystyle \sum _{j=2}^{m-r}\max \left(\underset{k<j}{A_{jk}}\right)}}{A_{\mathrm{w}}}$$

(21)

The above bounding formula also encounters the ordering dependency problem and, hence, needs $\left(m-r\right)!$ possible ordering alternatives to obtain the narrowest reliability bounds. In general, the proposed method requires significantly fewer alternatives than the NRBM. For example, for a system with 10 failure modes, the NRBM has 3,628,800 alternatives while the proposed method has 40,320 alternatives when 2 redundant failure modes are involved. On the other hand, the proposed method incurs a slightly greater computational cost for the identification of redundant failure modes. In a nutshell, the proposed method normally has a higher efficiency than the NRBM.

In terms of the accuracy, the proposed bounding formula can also provide efficient reliability bounds to estimate the system’s non-probabilistic failure degree. This estimation is not inferior to that of the NRBM, as demonstrated in Section 4.

3.3. Implementation Steps of the Proposed Method

Specific steps of the proposed method are listed in

Table 1. Implementation steps of the proposed method.

Step 1	Define the performance function $g_j\left(X\right)$. Transform the variables from the original space to the $\delta$ space, and thereby obtain $G_j\left(\delta \right)$.
Step 2	Take any two performance functions $G_i\left(\delta \right)$ and $G_j\left(\delta \right)$ to constitute a subsystem. Identify a redundant failure mode in the above subsystem using Equation (9). The identified redundant failure mode does not participate in the subsequent steps.
Step 3	Repeat Step 2 to obtain all the redundant failure modes in subsystems with two failure modes.
Step 4	Take any three performance functions $G_i\left(\delta \right)$, $G_j\left(\delta \right)$, and $G_k\left(\delta \right)$ to form a subsystem. Identify a redundant failure mode in the above subsystem using Equation (19) or (20). The identified redundant failure mode does not participate in the subsequent steps.
Step 5	Repeat Step 4 to identify all of the redundant failure modes in subsystems with three failure modes.
Step 6	Remove all of the redundant failure modes from the system. After renumbering the remaining failure modes, calculate the non-probabilistic failure degree of the system using Equation (21).

.

4. Numerical Examples

In this section, three numerical examples are investigated, including a mathematical example and two structural examples. Through these three examples, the proposed method is compared to the NRBM [30] to evaluate its accuracy and efficiency, where the reliability results from an MCS [26] with $1\times {10}^8$ samples are treated as the data.

4.1. Numerical Example 1

Consider a series system consisting of six performance functions:

$$\begin{array}{l}{G}_1\left(\delta \right)=-\cos \left(\frac{9\pi }{20}\right){\delta }_1-\sin \left(\frac{9\pi }{20}\right){\delta }_2+0.35\left(1+\alpha \right)\hfill \\ G_2\left(\delta \right)=\cos \left(\frac{9\pi }{20}\right){\delta }_1-\sin \left(\frac{9\pi }{20}\right){\delta }_2+0.35\left(1+\alpha \right)\hfill \\ G_3\left(\delta \right)=\cos \left(\frac{\pi }{2}\right){\delta }_1-\sin \left(\frac{\pi }{2}\right){\delta }_2+0.38\left(1+\alpha \right)\hfill \\ G_4\left(\delta \right)=\cos \left(\frac{\pi }{25}\right){\delta }_1-\sin \left(\frac{\pi }{25}\right){\delta }_2+0.45\left(1+\alpha \right)\hfill \\ G_5\left(\delta \right)=-\cos \left(\frac{\pi }{40}\right){\delta }_1-\sin \left(\frac{\pi }{40}\right){\delta }_2+0.38\left(1+\alpha \right)\hfill \\ G_6\left(\delta \right)=-\cos \left(\frac{\pi }{20}\right){\delta }_1-\sin \left(\frac{\pi }{20}\right){\delta }_2+0.45\left(1+\alpha \right)\hfill \end{array}$$

(22)

where $\alpha$ is a control parameter. The uncertainty domain is given as follows:

$${\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\end{array}\right]\le 1$$

(23)

The case of $\alpha =0.5$ is used as an example to verify the proposed identification criteria. Six failure modes are given in Figure 5, and the identification criteria and redundant failure modes are listed in

Table 2. Identification criteria and redundant failure modes for numerical example 1.

Subsystems	Identification Criteria	Redundant Failure Modes
$\left(G_5,G_6\right)$	$0\le {\gamma }_{56}=0.0785\le \left|\arccos \left(0.57\right)-\arccos \left(0.675\right)\right|=0.1345$	$G_6$
$\left(G_1,G_2,G_3\right)$	$\left\{\begin{array}{l}‖P_{123}‖=\infty >1\\ G_3\left(P_{12}\right)=0.0889>0\\ G_3\left(P_{\mathrm{u}}\right)=-0.43<0\end{array}\right.$	$G_3$

. It can be seen from Figure 5 that $G_3$ and $G_6$ are redundant failure modes, which coincide with the results listed in Table 2.

With the control parameter ranging from 0.1 to 1, $G_3$ and $G_6$ are always redundant failure modes. In this process, the variations in the absolute errors of the proposed method and the NRBM [30] with the control parameter are as given in Figure 6. The absolute error is computed by $f-f_{\mathrm{exact}}$, where $f$represents the results from the proposed method or the NRBM and $f_{\mathrm{exact}}$ is the exact solution that can be computed analytically.

As shown in Figure 6, the proposed method has the same absolute error of the lower bound as the NRBM, which shows a trend of first decreasing and then increasing (the peak occurs at $\alpha =0.5$). However, the proposed method has a smaller absolute error of the upper bound than the NRBM. That is, the former has an absolute error that remains constant at zero, while the latter has an absolute error greater than zero, which displays a trend of first increasing and then decreasing (the peak occurs at $\alpha =0.8$). Through the above analysis, one can conclude that both methods can provide the reliability bounds containing the exact solution, and that the proposed method has narrower reliability bounds than the NRBM. This indicates that the proposed method has a higher accuracy than the NRBM. In addition, the proposed method also has a higher efficiency than the NRBM, since the former involves 24 possible ordering alternatives while the latter involves 720. In summary, for this problem, the proposed method is superior to the NRBM in terms of accuracy and efficiency.

4.2. Numerical Example 2

In Figure 7, a gate-shaped frame problem [31] is modified and used as example 2. Here, the yield moments $R_i(i=1,2,\cdots 5)$ and the load $F$ are uncertain parameters with marginal intervals $R_1^{\mathrm{I}}=\left[58\mathrm{kNm},78\mathrm{kNm}\right]$, $R_2^{\mathrm{I}}=\left[62\mathrm{kNm},84\mathrm{kNm}\right]$, $R_3^{\mathrm{I}}=\left[62\mathrm{kNm},88\mathrm{kNm}\right]$, $R_4^{\mathrm{I}}=\left[62\mathrm{kNm},84\mathrm{kNm}\right]$, $R_5^{\mathrm{I}}=\left[55\mathrm{kNm},75\mathrm{kNm}\right]$, and $F^{\mathrm{I}}=\left[32\mathrm{kN},48\mathrm{kN}\right]$, respectively. The structure is treated as a series system consisting of four performance functions determined by four failure paths [31], i.e.,

$$\begin{array}{l}{g}_1=R_1+2R_3+R_4-lF\\ g_2=R_1+2R_3+R_5-lF\\ g_3=R_2+2R_3+R_4-lF\\ g_4=R_2+2R_3+R_5-lF\end{array}$$

(24)

where the height $l$ is 6 m. The uncertainty domain can be expressed as follows:

$${\left[\begin{array}{c}{R}_1-R_1^{\mathrm{c}}\\ R_2-R_2^{\mathrm{c}}\\ R_3-R_3^{\mathrm{c}}\\ R_4-R_4^{\mathrm{c}}\\ R_5-R_5^{\mathrm{c}}\\ F-F^{\mathrm{c}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cccccc}0.0116& -0.0001& -0.0017& -0.0019& -0.0021& 0\\ -0.0001& 0.0088& -0.0008& -0.0009& -0.0016& 0\\ -0.0017& -0.0008& 0.0065& -0.0005& -0.0005& 0\\ -0.0019& -0.0009& -0.0005& 0.0092& -0.0012& 0\\ -0.0021& -0.0016& -0.0005& -0.0012& 0.0113& 0\\ 0& 0& 0& 0& 0& 0.0156\end{array}\right]\left[\begin{array}{c}{R}_1-R_1^{\mathrm{c}}\\ R_2-R_2^{\mathrm{c}}\\ R_3-R_3^{\mathrm{c}}\\ R_4-R_4^{\mathrm{c}}\\ R_5-R_5^{\mathrm{c}}\\ F-F^{\mathrm{c}}\end{array}\right]\le 1$$

(25)

The redundant failure mode and the identification criterion are listed in

Table 3. Redundant failure mode and identification criterion for numerical example 2.

Subsystem	Identification Criterion	Redundant Failure Mode
$\left(G_3,G_4\right)$	$0\le {\gamma }_{34}=0.2280\le \left|\arccos \left(0.956\right)-\arccos \left(0.822\right)\right|=0.3081$	$G_3$

. Reliability analysis results for the proposed method, NRBM, and MCS are given in

Table 4. Reliability analysis results for numerical example 2.

Methods	Non-Probabilistic Failure Degree	Alternatives
NRBM [30]	0.0123710609 ≤ $f_{\mathrm{s}}$ ≤ 0.0123710611	4! = 24
The proposed method	0.0123710609 ≤ $f_{\mathrm{s}}$ ≤ 0.0123710611	3! = 6
MCS [26]	0.01237106107628	-

.

In Table 3, $G_3\left(g_3\right)$ is a redundant failure mode and, hence, can be removed. In Table 4, the proposed method yields the same reliability bounds containing the reference solution from the MCS as the NRBM. Therefore, the former has equivalent accuracy to the latter. For the number of possible ordering alternatives, as listed in Table 4, the proposed method and the NRBM have 6 and 24, respectively. Thus, the former has a higher efficiency than the latter. In summary, for this problem, the proposed method is superior to the NRBM.

4.3. Numerical Example 3

In Figure 8, a seven-bar truss problem [32] is modified and used as example 3. Each bar has the same length $L$, and their cross-section areas $A_i\left(i=1,2,\cdots ,7\right)$ are 1820 mm², 1641 mm², 1722 mm², 1695 mm², 2095 mm², 3275 mm², and 2678 mm², respectively. Three concentrated loads $F_1$, $F_2$, and $F_3$ are uncertain parameters whose marginal intervals are $F_1{}^{\mathrm{I}}=\left[2.7\times {10}^5\mathrm{N},3.3\times {10}^5\mathrm{N}\right]$, $F_2{}^{\mathrm{I}}=\left[0.855\times {10}^5\mathrm{N},1.045\times {10}^5\mathrm{N}\right]$, and $F_3{}^{\mathrm{I}}=\left[0.9\times {10}^5\mathrm{N},1.1\times {10}^5\mathrm{N}\right]$, respectively. Neglecting the buckling failure mode, this statically determinate structure can be regarded as a series system consisting of seven performance functions (details on the performance functions’ determination can be found in Appendix A):

$$\begin{array}{c}{g}_1=S-\left(\frac{F_1}{2\sqrt{3}}+\frac{3F_2}{4}+\frac{F_3}{4\sqrt{3}}\right)/A_1\\ g_2=S-\left(\frac{F_1}{2\sqrt{3}}+\frac{F_2}{4}+\frac{\sqrt{3}{F}_3}{4}\right)/A_2\\ g_3=S-\left(\frac{F_1}{\sqrt{3}}-\frac{F_2}{2}+\frac{F_3}{2\sqrt{3}}\right)/A_3\\ g_4=S-\left(\frac{F_1}{\sqrt{3}}-\frac{F_2}{2}+\frac{F_3}{2\sqrt{3}}\right)/A_4\\ g_5=S-\left(\frac{F_1}{\sqrt{3}}+\frac{F_2}{2}-\frac{F_3}{2\sqrt{3}}\right)/A_5\\ g_6=S-\left(\frac{F_1}{\sqrt{3}}+\frac{F_2}{2}+\frac{\sqrt{3}{F}_3}{2}\right)/A_6\\ g_7=S-\left(\frac{F_1}{\sqrt{3}}+\frac{F_2}{2}+\frac{F_3}{2\sqrt{3}}\right)/A_7\end{array}$$

(26)

where the yield strength $S$ is $100\mathrm{Mpa}$. The uncertainty domain can be expressed as follows:

$${\left[\begin{array}{c}{F}_1-F_1^{\mathrm{c}}\\ F_2-F_2^{\mathrm{c}}\\ F_3-F_3^{\mathrm{c}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{ccc}1.1317\mathrm{e}-9& -3.2489\mathrm{e}-10& -3.0864\mathrm{e}-10\\ -3.2489\mathrm{e}-10& 1.0186\mathrm{e}-8& -9.7466\mathrm{e}-10\\ -3.0864\mathrm{e}-10& -9.7466\mathrm{e}-10& 1.0185\mathrm{e}-8\end{array}\right]\left[\begin{array}{c}{F}_1-F_1^{\mathrm{c}}\\ F_2-F_2^{\mathrm{c}}\\ F_3-F_3^{\mathrm{c}}\end{array}\right]\le 1$$

(27)

The redundant failure modes and the identification criteria are listed in

Table 5. Redundant failure modes and identification criteria for numerical example 3.

Subsystems	Identification Criteria	Redundant Failure Modes
$\left(G_2,G_6\right)$	$0\le {\gamma }_{26}=0\le \left|\arccos \left(0.982\right)-\arccos \left(0.949\right)\right|=0.1307$	$G_2$
$\left(G_3,G_4\right)$	$0\le {\gamma }_{34}=0\le \left|\arccos \left(0.983\right)-\arccos \left(0.832\right)\right|=0.4034$	$G_3$
$\left(G_1,G_4,G_7\right)$	$\left\{\begin{array}{l}‖P_{147}‖=4.68\mathrm{e}+28>1\\ G_7\left(P_{14}\right)=0.0554>0\\ G_7\left(P_{\mathrm{u}}\right)=-0.1329<0\end{array}\right.$	$G_7$

. Reliability analysis results for the proposed method, NRBM, and MCS are given in

Table 6. Reliability analysis results for numerical example 3.

Methods	Non-Probabilistic Failure Degree	Alternatives
NRBM [30]	0.0441088331 ≤ $f_{\mathrm{s}}$ ≤ 0.0447715588	7! = 5040
The proposed method	0.0441088331 ≤ $f_{\mathrm{s}}$ ≤ 0.0447645259	4! = 24
MCS [26]	0.04451163	-

.

In Table 5, $G_2\left(g_2\right)$, $G_3\left(g_3\right)$, and $G_7\left(g_7\right)$ are redundant failure modes and, hence, can be removed. In Table 6, both the proposed method and the NRBM yield reliability bounds containing the reference solution provided by the MCS, and the former has narrower bounds than the latter. For the number of possible ordering alternatives, as listed in Table 6, the proposed method and the NRBM have 24 and 5040, respectively. Thus, for this problem, the proposed method is superior to the NRBM in terms of accuracy and efficiency.

4.4. Discussion of the Proposed Method

It can be concluded that the proposed method generally has higher efficiency and at least equivalent accuracy compared to the NRBM. However, the proposed method still has a major drawback, i.e., it can only identify most of the redundant failure modes instead of all them. For example, for a subsystem with four failure modes, the proposed method cannot identify the redundant failure mode when its corresponding set satisfies $A_i\in {\displaystyle \underset{j=1}{\overset{4}{\cup }}{A}_j},A_i\notin A_j,A_i\notin A_j\cup A_k(i\ne j\ne k,k=1,2,3,4)$. This drawback inevitably reduces the efficiency of the proposed method. Therefore, it appears to indicate that other efficient techniques are required for the identification of redundant failure modes.

It should be noted that the non-probabilistic convex model represents a series of convex models instead of a single model, and the interval model has also been widely used to quantify the uncertainty; the latter needs to be further considered to widen the applicability of the proposed method. On the one hand, the interval model utilizes a multidimensional box to describe the uncertainty domain, and the redundant failure modes are also more likely to occur. This implies that the proposed method is also suitable for the interval model only after some minor modifications. On the other hand, the multidimensional box and the multidimensional ellipsoid have different geometric shapes and properties, implying that the identification criteria of the proposed method need to be modified for the interval model. However, a more detailed modification of the interval model is beyond the scope of this paper.

5. Conclusions

In this paper, a novel non-probabilistic reliability bounds method for series structural systems is presented to overcome the size problem of the NRBM. The system is first divided into several simple subsystems with two and three failure modes. Three identification criteria for redundant failure modes are then developed for these subsystems. A bounding formula after removing redundant failure modes is further formulated to estimate the system’s non-probabilistic failure degree. A performance comparison between the proposed method and the NRBM through three numerical examples indicates that the proposed method has higher efficiency and at least equivalent accuracy compared to the NRBM. Therefore, the proposed method provides comparable results with lower computational costs.

The proposed method overcomes the size problem of the NRBM to some extent by identifying the redundant failures modes of the system. However, as an alternative, the linear programming technique can also be used to eliminate the size problem. Thus, one potential direction of work is to develop a linear-programming-based non-probabilistic reliability bounds method.