Structural hybrid reliability index and its convergent solving method based on random–fuzzy–interval reliability model

Abstract

Aiming to resolve the problems of a variety of uncertainty variables that coexist in the engineering structure reliability analysis, a new hybrid reliability index to evaluate structural hybrid reliability, based on the random–fuzzy–interval model, is proposed in this article. The convergent solving method is also presented. First, the truncated probability reliability model, the fuzzy random reliability model, and the non-probabilistic interval reliability model are introduced. Then, the new hybrid reliability index definition is presented based on the random–fuzzy–interval model. Furthermore, the calculation flowchart of the hybrid reliability index is presented and it is solved using the modified limit-step length iterative algorithm, which ensures convergence. And the validity of convergent algorithm for the hybrid reliability model is verified through the calculation examples in literature. In the end, a numerical example is demonstrated to show that the hybrid reliability index is applicable for the wear reliability assessment of mechanisms, where truncated random variables, fuzzy random variables, and interval variables coexist. The demonstration also shows the good convergence of the iterative algorithm proposed in this article.

Introduction

In engineering structure reliability analysis, it is common that various types of uncertainties coexist, such as probabilistic uncertainties, non-probabilistic uncertainties, and fuzziness. This is attributed to the existence of various uncertain variables with various data samples and various characters in modern complexity structure systems. For example, uncertain variables can be described as random variables when sample data are adequate, and an accurate probability distribution can be obtained. However, an accurate probability distribution cannot be obtained when the sample data are limited. Large differences can be found in the results of the probabilistic reliability for small changes in the probability distribution, for uncertain variables. ¹ Therefore, uncertainty variables cannot be treated as random variables when their sample data are not enough to obtain accurate probabilistic reliability analysis results. However, the boundaries of uncertain variables may be easy to be determined, such as structure dimensions whose tolerances, also boundaries, are easily obtained from the designer. Therefore, it is appropriate to use a non-probabilistic interval variable to describe these uncertain variables. Also, some uncertain variables are fuzzy and they should be described as fuzzy variables. Therefore, it is meaningful to present a hybrid reliability index, based on a random–fuzzy–interval reliability model, and its solving method for structural hybrid reliability analysis.

The reliability analysis of hybrid uncertain structures, where various uncertainty parameters coexist, has increasingly attracted the attention during the recent years. Some numerical methods have been proposed, such as the function approximation technique, ² the iterative rescaling method, ³ and the probability bounds approach. ⁴ Elishakoff and Pierluigi ⁵ considered the combination of probabilistic and convex models of uncertainty on acoustic excitation parameters. The optimization design and the solution algorithm, which are based on a hybrid reliability model with random variables and interval variables, were presented by Du et al. ⁶ A new hybrid model with random variables, multi-ellipsoidal variables, and an effective iterative algorithm were presented by Luo et al. ⁷ Gao et al. ⁸ presented a hybrid probabilistic and interval method for engineering problems, described by a mixture of random and interval variables. Wang and Qiu ⁹ evaluated the reliability of a probabilistic and interval hybrid structural system, based on the interval reliability model and probabilistic operation. Interval variables were treated as uniformly distributed random variables. The results were consistent with the results derived from a double-optimization model, which searched for the minimum of the limit-state function in the inner loop and for the probability reliability index in the outer loop. ¹⁰ Ni et al. ¹¹ presented a hybrid reliability model, which contains fuzzy random and non-probabilistic uncertainties. Jorge et al. ¹² proposed a Monte Carlo method that can be applied to probabilistic as well as to interval approaches for reliability analysis. There is a limited number of research articles focused on hybrid reliability analyses of engineering structures. Based on a hybrid model of random variables and multi-ellipsoidal variables, ⁷ a reliability optimization design for adhesive-bonded steel–concrete composite beams, with probabilistic and non-probabilistic uncertainties, has been performed by Luo et al. ¹³ The hybrid reliability of structural fractures, with random variables and interval variables, has been studied by Jiang et al., ¹⁴ and the failure probability interval of structural fractures was obtained using an effective iterative algorithm based on the response surface method. Based on the probabilistic reliability model and interval arithmetic, the interval of the probabilistic reliability index of structural systems was presented by Qiu and Wang. ¹⁵ Xia et al. ¹⁶ performed hybrid uncertain analysis in a structural–acoustic problem using random and interval parameters. Chen et al. ¹⁷ presented a hybrid perturbation method (HPM) for the prediction of the exterior acoustic field with interval variables and random variables.

From above references, we can see that hybrid reliability with two kinds of uncertain variables has been widely studied. However, hybrid reliability that involves three kinds of uncertain variables should be further studied. Moreover, the distributions of random variables are infinite in the above references. However, truncated random variables should be employed, instead of continuous random variables, for practical engineering applications because most of random variables are bounded. ¹⁸ Therefore, a hybrid reliability model that accommodates truncated random variables, fuzzy random variables, and interval variables should be considered.

In this article, a hybrid reliability index and its solving method, based on random–fuzzy–interval model, is presented. First, three various uncertainty variables and their models are introduced. Then, based on the interference relationship between the convex domain, formed by normalized interval variables, and the limit-state surface, the hybrid reliability index is defined. The calculation flowchart and solving method of the hybrid reliability index are presented, and the validity of solving method is verified through the calculation examples in literature. ⁷ Finally, a numerical example of wear reliability analysis of synchronizer cone surface is demonstrated.

Uncertain variables and their reliability models

Truncated random variables and truncated probability model

When accurate probability distributions of uncertain variables can be obtained, random variables and the probabilistic reliability model can be used to evaluate the reliability of engineering structures. However, design variables are generally bounded in engineering applications, and they are impossible to take the values of $-\infty$ or $+\infty$ . For example, most of the random variables in structure systems, such as the strength of a material or the modulus of elasticity, have sufficient sample data and are bounded. Thus, it is more appropriate that truncated probability distribution variables will be used for structure reliability analysis in engineering, instead of infinite probabilistic distribution variables.

Assumed that ${\overline{X}}_i$ is a truncated probability variable with an interval of $[a_i,b_i]$ , and its probability density function, $f_X({\overline{X}}_i)$ , and cumulative probability distribution function, $F_X({\overline{X}}_i)$ , are expressed as follows

$$f_X\left({\overline{X}}_i\right)=\frac{f_X\left(X_i\right)}{F_X\left(b_i\right)-F_X\left(a_i\right)}$$

(1)

$$F_X\left({\overline{X}}_i\right)=\frac{1}{F_X\left(b_i\right)-F_X\left(a_i\right)}(F_X\left(x\right)-F_X\left(a_i\right))$$

(2)

where $f_X(X_i)$ is the probability density function of ith continuous random variable $X_i$ , and $F_X(X_i)$ is the cumulative probability distribution functions of ith continuous random variable $X_i$ . The truncated probability reliability is expressed as

$$P_r=P[M(\overline{\mathit{X}})>0]=\underset{{\Omega }_r}{\int }{\overline{f}}_X(\overline{\mathit{X}})\mathrm{dx}=\Phi \left(\overline{\beta }\right)$$

(3)

where $\overline{\mathit{X}}=({\overline{X}}_1,\dots ,{\overline{X}}_n)$ , ${\Omega }_r$ is a safe region, $\overline{\beta }$ is the truncated probability reliability index, which is similar to the probabilistic reliability index $\beta$ . $\overline{\beta }$ can also be solved by first-order second-moment (FOSM), second-order second-moment (SOSM), and Monte Carlo methods, and the only difference with $\beta$ is that most probable failure point (MPP) is being searched within a certain interval. ¹⁹ When the iteration algorithm is employed to calculate $\overline{\beta }$ , the truncated random variables ${\overline{X}}_i$ need to be transformed to the standard normal space variables $u_i$

$$u_i=\frac{{\overline{X}}_i-{\mu }_{{\overline{X}}_i}}{{\sigma }_{{\overline{X}}_i}}$$

(4)

where ${\mu }_{{\overline{X}}_i}$ is the mean value of the truncated random variable ${\overline{X}}_i$ , and ${\sigma }_{{\overline{X}}_i}$ is the standard deviation of the truncated random variable ${\overline{X}}_i$ .

Fuzzy random variables and fuzzy random reliability model

Fuzzy random variables are common in engineering.^20–22 In many cases, since it is difficult to provide specific definitions and limits, the uncertainty of variable will be caused. Especially, for the multiple failure problems caused by fatigue, wear, corrosion, and creep, occurred in a mechanical system, there is no specific limit for their safety and failure and it is a gradual transitional procedure. In addition, due to the insufficient reliability data and the imperfection of the subjective cognizance of human being, a large number of fuzzy information exists in mechanical designs. Thus, the fuzzy random variables can be employed to describe above two kinds of fuzzy information. The formula of the fuzzy random reliability model is as follows ¹⁴

$$P_r\left(\tilde{\mathit{X}}\right)=P[M(\tilde{\mathit{X}})>0]=\underset{{\Omega }_r}{\int }{\mu }_{\tilde{\mathit{X}}}(\mathit{x})\mathrm{dP}=E[{\mu }_{\tilde{\mathit{X}}}(\mathit{x})]$$

(5)

where $\tilde{\mathit{X}}=({\tilde{X}}_1,\dots ,{\tilde{X}}_n)$ is the fuzzy random vector. Based on formula (1) and formula (5), the fuzzy random reliability formula of structural is as follows

$$P_r\left(\tilde{\mathit{X}}\right)=E[{\mu }_{\tilde{\mathit{X}}}(x)]=\underset{{\Omega }_r}{\int }{\mu }_{\tilde{\mathit{X}}}(\mathit{x})f_X\left(\overline{\mathit{X}}\right)\mathrm{dx}$$

(6)

where $f_X(\overline{\mathit{X}})$ is probability density function of the truncated random vector $\overline{\mathit{X}}$ , and ${\mu }_{\tilde{X}}(x)$ is the membership of $\tilde{X}$ . There are two common membership functions: the linear and the normal membership functions. The formula of the linear membership function is as follows ²²

$${\mu }_{\tilde{\mathit{X}}}(\mathit{x})=\{\begin{array}{c}L(x)=1-\frac{m-x}{\alpha }m-\alpha <x<m\hfill \\ \multicolumn{1}{c}{R(x)=1-\frac{x-m}{\beta }m<x<m+\beta }\\ \multicolumn{1}{c}{0\mathrm{others}}\end{array}$$

(7)

where m, α, and β are the coefficients to determine the membership function ${\mu }_{\tilde{\mathit{X}}}(\mathit{x})$ .

The formula of normal membership function is as follows

$$\begin{array}{c}{\mu }_{\tilde{\mathit{X}}}(\mathit{x})=\{\begin{array}{c}L(x)=\exp [-\frac{{(x-m)}^2}{{\alpha }^2}]x\le m\hfill \\ \multicolumn{1}{c}{R(x)=\exp [-\frac{{(x-m)}^2}{{\beta }^2}]x>m}\end{array}\hfill \end{array}$$

(8)

Interval variables and non-probabilistic reliability model

When data samples of structure are not sufficient in engineering problems, the accurate probability distribution cannot be obtained, and a small error of the probability distribution can cause large calculation errors in the structure reliability. ¹ However, the upper and lower boundaries of uncertainty variables are easy to be obtained. For example, accurate probability distribution of structure dimensions is difficult to obtain because its data samples are not sufficient at the early stage of product development; however, its tolerance can be easily provided by the designer. Therefore, the non-probabilistic interval variable can be employed to describe the uncertainty as follows ²³

$$Y_i\in Y_i^I=[Y_i^l,Y_i^u]$$

(9)

where $Y_i^I$ is the interval of interval variable $Y_i$ , $Y_i^l$ is the lower bound of $Y_i$ , and $Y_i^u$ is the upper bound of $Y_i$ . The interval variable $Y_i$ is standardized as follows

$$\begin{array}{c}{\delta }_{Y_i}=\frac{Y_i-Y_i^c}{Y_i^r}\hfill \\ Y_i^c=\frac{Y_i^l+Y_i^u}{2}{Y}_i^r=\frac{Y_i^u-Y_i^l}{2}\hfill \end{array}$$

(10)

where ${\delta }_{Y_i}$ is the standardized interval variable of $Y_i$ and ${\delta }_{Y_i}\in {\mathrm{\Delta }}^I=[-1,1]$ , $Y_i^c$ and $Y_i^r$ are the nominal value and deviation value of $Y_i$ , respectively.

The non-probabilistic interval reliability index $\eta$ is defined as the shortest distance from the origin of coordinates to the failure surface in the expanding space of the standard interval variables, which can be measured using the infinite norm as follows ²³

$$\begin{array}{c}\eta =\min \left({\Vert {\mathbf{\delta }}_{\mathit{Y}}\Vert }_{\infty }\right)=\min \left(\max (\left|{\delta }_{Y_1}\right|,\left|{\delta }_{Y_2}\right|,\dots ,\left|{\delta }_{Y_n}\right|)\right)\hfill \\ \mathrm{s}.\mathrm{t}.M=g\left(\mathit{Y}\right)=g\prime \left({\mathbf{\delta }}_{\mathit{Y}}\right)=0\hfill \end{array}$$

(11)

where ${\mathbf{\delta }}_{\mathit{Y}}$ is the standardized interval vector, and $g\prime ({\mathbf{\delta }}_{\mathit{Y}})=0$ is the limit-state equation in the standardized interval space. The optimization model described in formula (11) is the min–max optimization problem.

The hybrid reliability model with random variables, fuzzy random variables, and interval variables

Considering that truncated random variables, fuzzy random variables, and interval variables usually coexist in engineering designs, the safety margin equation M can be written by employing the stress–strength interference model as follows

$$M(\overline{\mathit{X}},\tilde{\mathit{X}},\mathit{Y})=R(\overline{\mathit{X}},\tilde{\mathit{X}},\mathit{Y})-S(\overline{\mathit{X}},\tilde{\mathit{X}},\mathit{Y})$$

(12)

where $R(\overline{\mathit{X}},\tilde{\mathit{X}},\mathit{Y})$ is the strength, $S(\overline{\mathit{X}},\tilde{\mathit{X}},\mathit{Y})$ is the stress, $\overline{\mathit{X}}$ , $\tilde{\mathit{X}}$ , and $\mathit{Y}$ are the truncated random variables, fuzzy random variables, and non-probabilistic interval variables, respectively.

First, the fuzzy random variable ${\tilde{x}}_j$ is transformed into equivalent random variable $x{\prime }_j$ , and the probability density function of equivalent random variable is ²⁴

$$f_X\left({X\prime }_j\right)=\frac{{\mu }_{{\tilde{X}}_j}({\tilde{x}}_j)f_{X_j}(X_j)}{\underset{-\infty }{\overset{\infty }{\int }}{\mu }_{{\tilde{X}}_j}({\tilde{x}}_j)f_{X_j}(X_j)d{x}_j}$$

(13)

$$F_X\left({X\prime }_j\right)=\underset{-\infty }{\overset{x}{\int }}{f}_X\left(X_j\right)d{x}_j$$

(14)

The transformation from fuzzy random variable to random variable is the equivalent mathematic transition, and the possibility distribution of the fuzzy random variable has not been changed. ²⁵

Second, the probability density function and cumulative probability distribution functions of the truncated random variable $\overline{x}{\prime }_j$ are obtained by substituting the equivalent random variable $x{\prime }_j$ into formula (1) and formula (2). The transformation principle is the cumulative probability distribution in interval $[a_i,b_i]$ equals to 1; that is, the parts of probability density function before a and after b are added into the interval $[a_i,b_i]$ . Because the parts outside of the interval $[a_i,b_i]$ are very small, the change of probability density function in interval $[a_i,b_i]$ can be neglected. There are only truncated random variables and interval variables in formula (12), namely

$$M(\overline{\mathit{X}},\overline{\mathit{X}\prime },\mathit{Y})=R(\overline{\mathit{X}},\overline{\mathit{X}\prime },\mathit{Y})-S(\overline{\mathit{X}},\overline{\mathit{X}\prime },\mathit{Y})$$

(15)

Following, the stress–strength interference problems in the structural safety margin equation that contain the truncated random variables and interval variables are discussed. Two different cases correspond to these problems, whether the limit-state surfaces interfere with the normalized square convex region or not, and it is explained by the simplest safe margin equation with only strength R and stress S, as shown in Figure 1.

Figure 1.

Two different cases: (a) $\eta >1$ and (b) $0\le \eta \le 1$ .

It is clear that when the strength R does not interfere with the stress S, as presented in Figure 1(a), the strength R is always higher than stress S and $\eta >1$ . Here, the reliability always equals to 1. In this situation, the hybrid reliability index $\beta$ , which is presented in Luo et al., ⁷ is theoretically equal to $\infty$ and is difficult to be obtained by the iterative method. To solve this problem, truncated probabilistic variables can be transferred into interval variables with only the information of upper and lower in this case. Under non-interference conditions of strength and stress, the distribution information of the truncated probabilistic variables cannot inference reliability results. Hence, it is useless in this case. Therefore, the non-probabilistic reliability index $\eta$ is fitted to be used, when the limit-state equation does not interfere with the normalized square convex region.

However, the non-probabilistic reliability index $\eta$ does no longer fit for reliability measurements, when the strength R interferes with the stress S, that is, $0\le \eta \le 1$ , as presented in Figure 1(b). The term $\eta$ only contains the information of one point, which is the most probabilistic failure point on the limit-state surface. It is appropriate to be used for measuring the structure absolute safety degree, when there is no interference between the limit-state surface and the normalized square convex region in the standard interval variable space. When interference takes place, it is not appropriate to employ $\eta$ for measuring the reliability because structure absolute safety does not exist, and the information of the MPP is not enough to evaluate the reliability. Therefore, it is appropriate to employ $\beta$ . The interference domain and also the distribution information of variables should be considered in this case.

From the above analysis, we can see that $\beta$ is not suitable for measuring reliability when there is no interference, and $\eta$ is not suitable for measuring reliability when there is interference. Therefore, the two reliability indexes can be combined into a new hybrid reliability index $\kappa$ , which can be expressed as follows

$$\kappa =\{\begin{array}{cc}\eta (\overline{\mathit{X}},\overline{\mathit{X}\prime },\mathit{Y})\hfill & \eta >1\hfill \\ \multicolumn{1}{c}{\Phi \left(\beta (\overline{\mathit{X}},\overline{\mathit{X}\prime },\mathit{Y})\right)}& 0\le \eta \le 1\hfill \end{array}$$

(16)

Comparing with the hybrid reliability model proposed in literature, ⁷ the hybrid reliability model proposed in this article has the following advantages:

1. It can resolve the reliability analysis problems of mechanical structures, which include truncated random variables, non-probability variables, and fuzzy random variables at the same time. However, literature ⁷ can only resolve the problems of random variables and non-probability variables and cannot consider the cases when the fuzzy random variables exist.

2. The distributed region of random variables considered in literature ⁷ is infinite. However, the distributed region of random variables in engineering applications should be limited and it will be more appropriate to use the truncated random variables instead of the random variables.

3. When employing the truncated random variables, the situation shown in Figure 1 will occur. There is no way to measure the reliability degree by employing the reliability model of literature, ⁷ which is based on the traditional probability concept, which can be solved very well using the hybrid reliability model proposed in this article.

Following normalization of the original uncertain variables X and Y into u and v, the limit-state function $G(\overline{\mathit{X}},\overline{\mathit{X}\prime },\mathit{Y})$ is mapped into the corresponding normalized limit-state function $g(\overline{\mathit{u}},\overline{\mathit{u}}\prime ,\mathit{v})$ .

When $\eta >1$ , the truncated random variables are considered as interval variables. The standardized interval vector $\mathrm{\Delta }\mathit{v}\prime$ is introduced, which is transferred from the truncated random vector $\overline{\mathit{u}}$ and the equivalent fuzzy truncated random vector $\overline{\mathit{u}}\prime$ . According to equation (11), the non-probabilistic reliability index $\eta$ can be expressed as follows

$$\begin{array}{c}\eta =\mathrm{sgn}(g(0,0))·\underset{(\mathrm{\Delta }\mathit{v}\prime ,\mathrm{\Delta }\mathit{v})}{\min }\hfill \\ \multicolumn{1}{c}{\left(\underset{\genfrac{}{}{0ex}{}{i=1,2,\cdots m}{j=1,2,\cdots n}}{\max }({\delta }_i=\sqrt{\mathrm{\Delta }{\mathit{v}\prime }_i^T\mathrm{\Delta }{\mathit{v}\prime }_i},{\delta }_j=\sqrt{\mathrm{\Delta }{\mathit{v}}_j^T\mathrm{\Delta }{\mathit{v}}_j})\right)}\\ \mathrm{s}.\mathrm{t}.g(\mathrm{\Delta }\mathit{v}\prime ,\mathrm{\Delta }\mathit{v})=g(\mathit{Y}\prime ,\mathit{Y})=0\hfill \end{array}$$

(17)

where $\mathrm{\Delta }\mathit{v}\prime =[\overline{\mathit{u}},\overline{\mathit{u}}\prime {]}^T$ .

When $|\eta |\le 1$ , based on the hybrid reliability index $\beta$ , as presented in Luo et al., ⁷ the upper and lower boundaries of the truncated random variables are added, and its expression is as follows

$$\begin{array}{c}\beta =\mathrm{sgn}\left(g(0,\mathrm{\Delta }\mathit{v})\right)\underset{u,v}{\min }\sqrt{{\mathit{u}}^T\mathit{u}}\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{g}(\mathit{u},\mathrm{\Delta }\mathit{v})=0\hfill \\ \mathrm{\Delta }{\mathit{v}}_j^T\mathrm{\Delta }{\mathit{v}}_j\le 1(i=1,2,\dots ,n)\hfill \\ {\mathit{u}}_L\le \mathit{u}\le {\mathit{u}}_R\hfill \end{array}$$

(18)

where $\mathit{u}=[\overline{\mathit{u}},\overline{\mathit{u}}\prime {]}^T$ is the standardized combined random vector, consisting of the truncated random vector $\overline{\mathit{u}}$ and the equivalent fuzzy truncated random vector $\overline{\mathit{u}}\prime$ ; $\mathrm{\Delta }\mathit{v}$ is the standardized interval vector; and ${\mathit{u}}_l$ , ${\mathit{u}}_u$ are the lower and the upper boundaries of $\mathit{u}$ , respectively. The reliability index $\beta$ denotes the shortest distance from the original point to the limit-state surface in standard normal probability space, which can be solved using the iteration method first-order reliability method (FORM).

To solve $\eta$ using the iterative method FORM, as the shortest distance problem is solved in formula (18), the min–max optimization problem in formula (17) should be transferred into a shortest distance problem. Based on the expression of $\eta$ , as presented in Zhou et al., ²³ the equivalent standard interval variable $\mathrm{\Delta }v\prime$ of the truncated random variables and the fuzzy truncated random variables are added, and the formula is expressed as

$$\begin{array}{l}\eta =\mathrm{sgn}(g(0,0))\cdot \frac{1}{\sqrt{m+n}}\underset{\left(\mathrm{\Delta }{\mathit{v}}^{\prime },\mathrm{\Delta }\mathit{v}\right)}{\min }\sqrt{{\displaystyle \sum _{i=1}^m{{\delta }^{\prime }}_i^2}+{\displaystyle \sum _{j=1}^n{\delta }_j^2}}\\ \mathrm{s}.\mathrm{t}.G\left(\mathrm{\Delta }{\mathit{v}}^{\prime },\mathrm{\Delta }\mathit{v}\right)=\frac{1}{2}\{g{\left(\mathrm{\Delta }{v}^{\prime },\mathrm{\Delta }v\right)}^2+C[{\displaystyle \sum _{i=1}^{m-1}{\left({{\delta }^{\prime }}_i-{{\delta }^{\prime }}_{i+1}\right)}^2}\\ +{\left({\delta }_1-{{\delta }^{\prime }}_m\right)}^2+{\displaystyle \sum _{j=1}^{n-1}{\left({\delta }_j-{\delta }_{j+1}\right)}^2}+{\left({\delta }_n-{{\delta }^{\prime }}_1\right)}^2]\}=0\end{array}$$

(19)

where C is a weight coefficient.

The solving method of hybrid reliability index

Calculation flowchart of the hybrid reliability index

The calculation flowchart of the hybrid reliability index $\kappa$ is given in Figure 2.

Figure 2.

Flow graph of calculating hybrid reliability index.

The modified limit-step length iterative algorithm (MLSLIA) is provided in order to solve $\beta$ , and the solving process of $\eta$ is similar. The iterative formulae in standard normal space are expressed as follows ²⁶

$${\mathbf{\alpha }}_Z^{(k+1)}=\frac{{\mathit{z}}^{\left(k\right)}-{\lambda }^{\left(k\right)}\nabla g_Z\left({\mathit{z}}^{\left(k\right)}\right)}{\Vert {\mathit{z}}^{\left(k\right)}-{\lambda }^{\left(k\right)}\nabla g_Z\left({\mathit{z}}^{\left(k\right)}\right)\Vert }$$

(20)

$${\beta }^{(k+1)}=-\frac{g_Z\left({\mathit{z}}^{\left(k\right)}\right)-\nabla g_Z{\left({\mathit{z}}^{\left(k\right)}\right)}^T{\mathit{z}}^{\left(k\right)}}{{\left({\mathbf{\alpha }}_Z^{(k+1)}\right)}^T\nabla g_Z\left({\mathit{z}}^{\left(k\right)}\right)}$$

(21)

$${\mathit{z}}^{(k+1)}={\mathbf{\alpha }}_Z^{(k+1)}{\beta }^{\left(k\right)}$$

(22)

where ${\mathit{z}}^{(k)}$ is the standard random vector, $g_Z({\mathit{z}}^{(k)})$ is the structure performance function, ${\mathbf{\alpha }}_Z^{(k+1)}$ is the sensitivity coefficient, ${\beta }^{(k+1)}$ is the reliability index, ${\lambda }^{(k)}$ is the step length, and superscript k indicates the iterative step number. The merit function is expressed as

$$m\left(\mathit{z}\right)=\Vert \mathit{z}-\frac{\nabla g_Z{\left(\mathit{z}\right)}^T\mathit{z}}{{\Vert \nabla g_Z\left(\mathit{z}\right)\Vert }^2}\nabla g_Z\left(\mathit{z}\right)+r{g}_Z\left(\mathit{z}\right)\nabla g_Z\left(z\right){\Vert }^2$$

(23)

Combining with searching the optimal step length ${\lambda }^{(k)}$ for minimizing the evaluation function $m(\mathit{z})$ , the iterative steps of the modified limit-step iterative method are as follows: ²⁷

• Step 1. Set the initial parameters, such as k = 0, ${\mathit{z}}^{(0)}=0$ , the positive real constants c, $\alpha$ , and the initial penalty coefficient $r^{(0)}$ . Then, initial searching value ${\lambda }_0$ and the initial searching step length ${\lambda }_h$ , for the interval search of the step length ${\lambda }^{(k)}$ , are hence determined, and the loop termination criterion $\epsilon$ is also determined.

• Step 2. First, the searching interval $[{\lambda }_{\mathrm{low}}^{(k)},{\lambda }_{\mathrm{up}}^{(k)}]$ of the optimal step length ${\lambda }^{(k)}$ is determined by ${\lambda }_0$ , ${\lambda }_h$ , and ${\mathit{z}}^{(k)}$ , and the optimal step length ${\lambda }^{(k)}$ is searched in the interval $[{\lambda }_{\mathrm{low}}^{(k)},{\lambda }_{\mathrm{up}}^{(k)}]$ by the golden section method. ${\beta }^{(k+1)}$ , ${\mathit{z}}^{(k+1)}$ , and $m({\mathit{z}}^{(k+1)})$ are obtained by the iterative formulae (20), (21), (22), and formula (23).

• Step 3. If $m({\mathit{z}}^{(k+1)})>m({\mathit{z}}^{(k)})$ , then set ${\lambda }^{(k)}={\lambda }^{(k-1)}/c$ . ${\beta }^{(k+1)}$ , ${\mathit{z}}^{(k+1)}$ , and $m({\mathit{z}}^{(k+1)})$ are calculated again.

• Step 4. When $\beta$ is solved, the iterative formula of $\mathrm{\Delta }{\mathit{v}}^{(k+1)}$ becomes as follows ⁷

$$\begin{array}{l}\Delta v^{\left(k+1\right)}=\left[{\left(\Delta v_1^{\left(k+1\right)}\right)}^T,{\left(\Delta v_2^{\left(k+1\right)}\right)}^T,\dots ,{\left(\Delta v_n^{\left(k+1\right)}\right)}^T\right]\\ =-[{\left(\frac{G_{\Delta v_1}^{\left(k\right)}}{\sqrt{{\left(G_{\Delta v_1}^{\left(k\right)}\right)}^T{G}_{\Delta v_1}^{\left(k\right)}}}\right)}^T,{\left(\frac{G_{\Delta v_2}^{\left(k\right)}}{\sqrt{{\left(G_{\Delta v_2}^{\left(k\right)}\right)}^T{G}_{\Delta v_2}^{\left(k\right)}}}\right)}^T\\ \dots ,{\left(\frac{G_{\Delta v_n}^{\left(k\right)}}{\sqrt{{\left(G_{\Delta v_n}^{\left(k\right)}\right)}^{\mathrm{T}}{G}_{\Delta v_n}^{\left(k\right)}}}\right)}^T]\end{array}$$

(24)

• Step 5. The penalty coefficient $r^{(k+1)}$ is determined by the following formula

$$r^{(k+1)}=\{\begin{array}{cc}\alpha r^{\left(k\right)}\hfill & \frac{g_Z\left({\mathit{z}}^{\left(k\right)}\right)}{g_Z\left({\mathit{z}}^{(k-1)}\right)}>0.25\hfill \\ \multicolumn{1}{c}{r^{\left(k\right)}}& \frac{g_Z\left({\mathit{z}}^{\left(k\right)}\right)}{g_Z\left({\mathit{z}}^{(k-1)}\right)}\le 0.25\hfill \end{array}$$

(25)

Then, set k = k + 1. If $||{\mathit{z}}^{(k+1)}-{\mathit{z}}^{(k)}||<\epsilon$ , the iterative loop is terminated. Otherwise, go back to step 2.

Generally, the initial penalty coefficient $r^{(0)}>0$ is assigned, and the adjustment coefficients c and $\alpha$ are assigned between 1.2 and 1.5 in MLSLIA. When $\beta$ is solved by MLSLIA, the upper and lower values of the truncated random vector are confined. $g(u^{(k)})$ and $\partial g(u^{(k)})/\partial u$ are used in every iterative step, and according to $\varphi (u^{(k)})=F_{\overline{X}}(\overline{X})$ , the formula can be obtained as follows

$$g\left(u^{(k)}\right)=g\left(x^{(k)}\right)=g\left(F_X^{-1}\left(\varphi \left(u^{(k)}\right)\right)\right)$$

(26)

$${\frac{\partial g\left(u\right)}{\partial u}|}_{u=u^{(k)}}={\frac{\partial g\left(x\right)}{\partial x}\frac{\partial x}{\partial u}|}_{u=u^{(k)}}={\frac{\partial g\left(x\right)}{\partial x}\frac{\phi (u)}{f_X(x)}|}_{u=u^{(k)}}$$

(27)

If $\eta$ is solved by MLSLIA, standard normal space variable ${\mathit{z}}^{(k)}$ should be substituted by the standard hyper-sphere space variable $\mathrm{\Delta }\mathit{v}$ in formulas (20)–(23). Notice that $g(•)$ should be substituted by $G(•)$ which is expressed in formula (19). Also, step 4 should not be executed.

Validation example

The example presented in this section is used to validate MLSLIA when used to solve the hybrid reliability index. Here, the example of Luo et al. ⁷ is used, and its structural performance function only contains random variables and non-probabilistic variables. The iteration convergence criterion is set as $\epsilon =0.001$ .

The structural performance function of a cantilever beam can be expressed as follows ⁷

$$g(P,L,E,b,h)=0.15-\frac{4P{L}^3}{\mathrm{Eb}{h}^3}$$

(28)

The uncertainty characteristics of the variables are listed in Table 1. The iteration results in Table 2 are consistent with those presented in the literature. ⁷ Thus, MLSLIA can be considered as a valid method to calculate the hybrid reliability index.

Table 1.

Uncertain variables.

Uncertain variables	Mean (nominal) value	COV	Convex model description
E (psi.)	107	0.05	–
P (lb.)	100	0.1	–
L (lb.)	30	–	$-0.05\le {\delta }_L\le 0.05$
b (lb.)	0.8359	–	$[{\delta }_b,{\delta }_h]\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]\left[\begin{array}{c}{\delta }_b\hfill \\ \multicolumn{1}{c}{{\delta }_h}\end{array}\right]\le 0.{05}^2$
h (lb.)	2.5093	–

COV: coefficient of variation.

Table 2.

Iteration process.

k	E (106 psi.)	P (lb.)	L (lb.)	b (lb.)	h (lb.)	λ	β
0	10	100	30	0.836	2.509	1	–
1	9.304	127.836	31.5	0.823	2.390	10	3.1122
2	9.195	123.908	31.5	0.823	2.390	3071	2.883
3	9.194	123.928	31.5	0.823	2.390	2559	2.885
4	9.194	123.926	31.5	0.823	2.390	2133	2.885
Luo et al. 7	9.194	123.926	31.5	0.823	2.390	–	2.885

Numerical examples

In this section, a numerical example where random variables, fuzzy random variables, and interval variables coexist is presented, that is, the wear reliability evaluation of mechanisms. The wear of mechanisms relates to many factors, such as loads, velocity, lubricating conditions, ambient temperature, and surface hardness. These factors are considered fuzzy to some extent. Considering the effect to these fuzzy factors, it is difficult for the allowance wear loss, which is provided in the form of a constant, to determine the failure of wear. The wear failure of mechanisms is considered as fuzzy random variable. Recently, many researches have been focused on fuzziness of wear in literatures.^28–32 However, besides the fuzzy random variables, there are also other variables, such as the random variables and the non-probabilistic interval variables, in engineering applications, due to the absence of sample data. Moreover, the wear failure of kinematic pairs and parts of mechanisms account approximately for 30%–80% of the total failures of mechanisms. Therefore, it is meaningful to perform wear reliability analysis of mechanisms using the random–fuzzy–interval hybrid reliability model presented in this article.

The wear loss of mechanisms can be modeled by Archard’s law, and the mathematical expression is given as follows ³³

$$h=k\frac{\mathrm{PL}}{H}$$

(29)

where $h$ is the wear loss of mechanisms, $k$ is the wear coefficient, $P$ is pressure, L is the sliding distance, and H is the material hardness.

Before wear failure occurs, wear $h$ is lower than the allowance wear loss $h^{*}$ . So, the safe margin equation of wear is as follows

$$M=h^{*}-h=h^{*}-k\frac{\mathrm{PL}}{H}$$

(30)

Here, the allowance wear loss is considered as a fuzzy random variable, and the wear coefficient $k$ , pressure P, sliding distance L, and hardness of material H can be treated as truncated probabilistic variables or interval variables, according to their data samples are sufficient or not.

Here, it was selected to perform the wear hybrid reliability analysis of a synchronizer cone surface as a numerical example. In Figure 3, ${\delta }_2$ is the clearance between the gear sleeve and the meshing gear, ${\delta }_1$ is the clearance of the conical surface, $\alpha$ is the semi-cone angle of the wear cone surface, and allowance wear $h^{*}$ is expressed as follows ³³

$$h^{*}={\delta }_2\sin \alpha -{\delta }_1$$

(31)

Figure 3.

Cone surface wear of synchronizer.

After n connection shift, the wear loss $h$ is expressed as follows ³²

$$h=2n\frac{0.032{\left(\frac{F}{A\sin \alpha }\right)}^{1.312}{\upsilon }_0^{1.467}t\underset{0}{\overset{t}{\int }}{(1-\frac{t_1}{t})}^{1.467}d{t}_1}{\rho A}$$

(32)

where n is the number of connection shift, F is the applied force, ${\upsilon }_0$ is the initial velocity, A is the contact surface, $\alpha$ is the semi-cone angle of wear cone surface, t is the synchronization time, and $\rho$ is the density of the material.

The uncertainty parameters of the synchronizer cone surface are listed in Table 3. As the allowance wear loss is a fuzzy random variable, $h^{*}$ is treated as a fuzzy random variable. The applied force F and the velocity ${\upsilon }_0$ are treated as truncated random variables, as their data samples are sufficient, which can be obtained from the test or the previous operation record, and then, their exact distribution can be obtained. The contact area A and the semi-cone angle of the wear cone surface $\alpha$ are treated as interval variables, because their data samples are not sufficient in the early stage of product development; however, their boundaries or tolerances can be obtained from the designers. The rest of the variables are treated as deterministic variables.

Table 3.

Uncertainty parameters of synchronizer.

Parameters	Type of parameters	Mean value/interval	Standard deviation
Allowance wear $h^{*}$ (mm)	Truncated fuzzy variable	1.5 mm	0.06
Number of connection shift n (times)	Deterministic variable	2000–6000
Applied force F (N)	Truncated random variable	2100	10/3
Contact surface A (m2)	Interval variable	(0.055, 0.065)
Semi-cone angle of wear cone surface $\alpha$ (°)	Interval variable	(7.45, 7.55)
Initial velocity ${\upsilon }_0$ (m/s)	Truncated random variable	4.5	0.2
Synchronization time t (s)	Deterministic variable	1.5
Density of material $\rho$ (kg/m3)	Deterministic variable	7800

According to the experience of the designer engineer, the membership function of wear loss $h^{*}$ is expressed as follows

$${\mu }_{\tilde{\mathit{X}}}(\tilde{\mathit{x}})=\{\begin{array}{c}L(x)=\exp [-\frac{{(x-1.5)}^2}{{0.18}^2}]1.32<x\le 1.5\hfill \\ \multicolumn{1}{c}{R(x)=\exp [-\frac{{(x-1.5)}^2}{{0.18}^2}]1.5<x\le 1.68}\end{array}$$

(33)

When the number of meshing equals $n=2000$ times, the iterative process of $\eta$ in the standard interval variable space is listed in Table 4, the convergence result is $\eta =2.583>1$ , so $\kappa =\eta =2.583$ .

Table 4.

Iterative process of $\eta$ .

k	$\mathrm{\Delta }{v}_1$	$\mathrm{\Delta }{v}_2$	$\mathrm{\Delta }{v}_3$	$\mathrm{\Delta }{v}_4$	$\mathrm{\Delta }{v}_5$	$\eta$
0	0	0	0	0	0	–
1	−2.238	0.031	−0.952	−0.043	0.967	0.967
2	−0.445	−0.493	−0.241	0.290	−0.100	0.100
3	−0.052	0.127	−0.497	−0.143	−0.517	0.517
4	−0.104	0.049	0.055	−0.110	0.094	0.094
5	−2.409	1.177	0.052	−0.137	2.094	2.094
6	−0.383	−0.091	−0.393	0.640	0.445	0.445
7	−1.099	−1.175	−1.267	1.303	1.702	1.702
8	−1.503	−1.356	−1.468	1.607	1.677	1.677
9	−1.990	−1.923	−1.935	1.953	1.943	1.943
10	−2.298	−2.264	−2.275	2.263	2.294	2.294
11	−2.443	−2.431	−2.429	2.430	2.441	2.441
12	−2.517	−2.508	−2.509	2.506	2.516	2.516
13	−2.549	−2.546	−2.545	2.545	2.548	2.549
14	−2.567	−2.565	−2.565	2.564	2.567	2.567
15	−2.575	−2.574	−2.574	2.574	2.575	2.575
16	−2.580	−2.579	−2.579	2.579	2.580	2.580
17	−2.582	−2.582	−2.581	2.582	2.582	2.582
18	−2.583	−2.583	−2.583	2.583	2.583	2.583
19	−2.583	−2.583	−2.583	2.583	2.583	2.583

When n = 5000 times, the iterative process of $\eta$ in standard interval variable space is also listed in Table 5, the convergence result is $\eta =0.8076$ , so $\beta$ needs to be calculated. The iterative process in standard normal variable space is shown in Table 6, and the convergence result is $\kappa =\Phi (\beta )=\Phi (3.012)=0.9987$ . Notice that the lower and upper boundaries of the truncated random variables are constrained in the last three iterative steps of $x_5$ .

Table 5.

Iterative process of $\eta$ .

k	$\mathrm{\Delta }{v}_1$	$\mathrm{\Delta }{v}_2$	$\mathrm{\Delta }{v}_3$	$\mathrm{\Delta }{v}_4$	$\mathrm{\Delta }{v}_5$	$\eta$
0	0	0	0	0	0	–
1	−0.4255	0.0147	−0.4528	−0.0204	0.4596	0.4596
2	−0.0376	−0.0959	−0.0918	0.0985	0.0475	0.0475
3	−0.3141	0.1340	−0.2143	−0.1269	0.2765	0.2765
4	0.0788	0.1337	0.0472	−0.1221	−0.0203	0.0203
5	−0.1703	−0.2370	0.1051	0.2957	−0.0994	0.0994
6	−0.0244	0.0553	0.0627	−0.1073	−0.0465	0.0465
7	−0.4619	−0.0641	−0.1889	0.3179	0.1927	0.1927
8	−0.0148	0.0127	0.0019	0.0063	−0.0114	0.0114
9	−0.4564	−0.0302	−0.4338	−0.0196	0.5330	0.5330
10	−0.0370	0.0989	−0.0914	0.1111	0.0647	0.0647
11	−0.3775	0.4614	−0.6135	0.5225	0.4617	0.4617
12	−0.5928	0.6373	−0.7111	0.6972	0.6156	0.6156
13	−0.6596	0.6763	−0.7071	0.6874	0.6819	0.6819
14	−0.7330	0.7333	−0.7470	0.7445	0.7338	0.7338
15	−0.7685	0.7699	−0.7793	0.7750	0.7727	0.7727
16	−0.7876	0.7881	−0.7902	0.7901	0.7888	0.7888
17	−0.7977	0.7974	−0.7995	0.7985	0.7987	0.7987
18	−0.8028	0.8027	−0.8033	0.8033	0.8032	0.8032
19	−0.8055	0.8054	−0.8059	0.8057	0.8058	0.8058
20	−0.8068	0.8068	−0.8069	0.8069	0.8069	0.8069
21	−0.8075	0.8075	−0.8076	0.8075	0.8076	0.8076

Table 6.

Iterative process of $\beta$ .

k	$x_1$	$x_2$	$\mathrm{\Delta }{v}_3$	$\mathrm{\Delta }{v}_4$	$x_5$	$\beta$
0	0	0	0	0	0	–
1	0.063	−1.673	−1	−1	1.432	2.203
2	0.088	−2.164	−1	−1	1.973	2.930
3	0.126	−0.432	−1	−1	2.504	2.545
4	0.005	−0.153	−1	−1	2.576	2.580
5	0.004	−0.133	−1	−1	2.614	2.618
6	0.004	−0.125	−1	−1	2.648	2.651
7	0.004	−0.118	−1	−1	2.679	2.681
8	0.004	−0.112	−1	−1	2.707	2.709
9	0.003	−0.107	−1	−1	2.732	2.734
10	0.003	−0.103	−1	−1	2.756	2.758
11	0.003	−0.099	−1	−1	2.778	2.779
12	0.003	−0.096	−1	−1	2.798	2.800
13	0.003	−0.093	−1	−1	2.818	2.819
14	0.003	−0.090	−1	−1	2.836	2.838
15	0.003	−0.088	−1	−1	2.854	2.855
16	0.003	−0.086	−1	−1	2.870	2.872
17	0.003	−0.084	−1	−1	2.886	2.887
18	0.003	−0.082	−1	−1	2.901	2.903
19	0.003	−0.080	−1	−1	2.916	2.917
20	0.003	−0.079	−1	−1	2.930	2.931
21	0.002	−0.077	−1	−1	2.943	2.944
22	0.002	−0.076	−1	−1	2.956	2.957
23	0.002	−0.075	−1	−1	2.969	2.970
24	0.002	−0.074	−1	−1	2.981	2.982
25	0.002	−0.073	−1	−1	2.993	2.994
26	0.002	−0.072	−1	−1	3.000	3.005
27	0.002	−0.071	−1	−1	3.000	3.012
28	0.002	−0.071	−1	−1	3.000	3.012

The change in the wear reliability indexes $\kappa$ , $\eta$ , and $\beta$ of the synchronizer cone surface is provided with the increasing of the number of connecting shifts in Figure 4. It is shown from Figure 4 that all the wear reliability indexes $\kappa$ , $\eta$ , and $\beta$ gradually reduce. Here, when $\eta >1$ and $\Phi (\beta )\equiv 1$ , the hybrid reliability index $\kappa =\eta$ can be used to assess the non-probabilistic reliability of mechanisms. When $\eta <1$ , $\kappa =\Phi (\beta )$ is used to assess the probabilistic reliability of mechanism, while $\eta$ has no clear physical meaning. Therefore, the hybrid reliability index $\kappa$ should be considered as more appropriate for the two cases, whether the limit-state surface corresponds to the safety margin equation, which contains the truncated random variables, fuzzy random variables, and interval variables interferes with the normalized square convex region or not in standard interval variable space. This shows that the hybrid reliability index $\kappa$ is more suitable than $\eta$ or $\beta$ .

Figure 4.

The change of hybrid reliability index.

The iterative process of the reliability indexes $\eta$ and $\beta$ is presented in Figure 5. As the expression of the limit-state equation is complex and highly nonlinear, which contains a variety of uncertainty variables, the oscillation can be observed in the iterative process of the reliability indexes $\eta$ and $\beta$ , but the results are convergent at last. This shows that the iterative method proposed in this article has a good convergence.

Figure 5.

The iteration of the hybrid reliability index: (a) the iteration process of $\eta$ and (b) the iteration process of $\beta$ .

When $n>5000\mathrm{times}$ , $\eta <1$ , so $\kappa =\Phi (\beta )$ . The detailed altered trends of the reliability index $\beta$ at the conditions of $n=5000~6000$ are presented in Figure 6. It can be observed from Figure 6 that the wear reliability index $\beta$ declines with the increase in the number of connecting shifts of the synchronizer, which can provide reliability evaluation for the wear resistance design of the synchronizer cone surface in engineering applications. For example, the demand of the wear resistance design is that the wear reliability index equals $\beta =3.0$ after 5000 times of connecting shifts. According to Figure 6, $\beta =3.0121$ when $n=5000\mathrm{times}$ , which shows that wear reliability of the design satisfies this demand. Also, the number of connecting shifts can be set to correspond to $\beta <2$ , as the time needs to be repaired; that is, the repaired time corresponds to the number of connecting shift $n=5400$ in Figure 6.

Figure 6.

The change of the hybrid reliability index $\beta$ .

Conclusion

This article presented a new hybrid reliability model and its solving method, based on the random–fuzzy–interval model. The hybrid reliability index $\kappa$ is more suitable for the two cases, whether the limit-state surface corresponds to the safety margin equation, which contains truncated random variables, fuzzy random variables, and interval variables interfering with the normalized square convex region, or not in standard interval variable space. MLSLIA is proposed to solve the hybrid reliability index $\kappa$ , which can guarantee the convergence of results. The wear reliability analysis of a synchronizer cone surface was performed, and the results show that the wear hybrid reliability index is decreased with the increase in the number of connecting shifts. Also, the reliability evaluation of wear resistance design and the preventive maintenance time of the synchronizer cone surface have been provided. The model proposed in this article is not only limited to the reliability analysis of mechanical wear. It will be able to be applied to all hybrid reliability analyses of mechanical structure, only if three types of uncertain variables, namely, truncated random variable, non-probability variable, and fuzzy random variable, are simultaneously included in the reliability analysis of mechanical structure. The model can also be applied when there is only one type of variable, out of the above-mentioned three variables, in the reliability analysis of mechanical structures.