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Abstract

Sensitivity analysis plays a crucial role in identifying the structure important parameters. In this article, a new non-probabilistic parameter sensitivity analysis method is proposed according to the ellipsoidal model. Meanwhile, an analytical solution of non-probabilistic parameter sensitivity analysis method based on the ellipsoidal model is derived for linear performance function, as well as an approximately analytical solution is obtained for nonlinear performance function using the first-order Taylor expansion to linearize the functions in design point. Finally, the proposed method is illustrated by three examples, which shows that it is reasonable and applicable.

Keywords

Non-probability reliability index ellipsoidal model sensitivity analysis first-order Taylor expansion

Introduction

Parameter sensitivity analysis can be used to understand the importance of variables for structural reliability, which provides the reference for structural analysis and optimization design.^1–3 In practical projects, probabilistic reliability analysis method has been fully applied in many fields, and accordingly, the corresponding probabilistic reliability sensitivity analysis gradually becomes more mature.^4,5 Wu⁶ and Wu and Mohanty⁷ defined the parameter sensitivity in the probability model. Melchers and Ahammed,⁸ Isabelle and Komlanvi,⁹ Jensen et al.,¹⁰ Geraci et al.,¹¹ Dubourg and Sudret,¹² Zhang et al.,¹³ and Yuan et al.¹⁴ have developed the methods solving the parameter sensitivity in probability models. Depending on the study of probabilistic reliability sensitivity, the non-probabilistic reliability index has been widely used in engineering, and the study of non-probabilistic reliability sensitivity analysis has also developed smoothly.^15–17 In the current research, however, the sensitivity of non-probabilistic parameters is confined to the interval model application, and the non-probabilistic parameter sensitivity analysis method under ellipsoidal model has not been constructed yet. We have to acknowledge that the variables in engineering problems are interrelated with each other, so the sensitivity of non-probabilistic parameter problems in the ellipsoid model has to be researched further in future studies.

In recent years, the non-probabilistic model has attracted wide attention by researchers. The main reason is it can effectively solve the reliability problems if there is fewer data as well as lack of probability density function situation.^18–21 Ben-Haim^22,23 proposed the concept of non-probabilistic reliability based on the convex model and put forward the idea that the reliability index can be measured by the maximum degree of uncertainty that systems can stand. Also, Elishakoff²⁴ applied the convex model to describe the uncertainty, from which the non-probabilistic index is considered as an interval rather than a specific value, and the boundary of the reliability index can be obtained by the interval calculation of the safety factor method. Generally, a non-probabilistic convex model can be divided into two models, such as interval model and ellipsoid model, respectively. Compared with the interval model, the ellipsoid model can describe the correlation in multiple uncertain variables. When the variables are irrelevant, the ellipsoidal model degrades into an interval model. Thus, interval model can be regarded as a special case of the ellipsoidal model.

Furthermore, many kinds of research have conducted research on the non-probabilistic reliability analysis under the ellipsoidal model in recent studies.^25–29 Qiao et al.²⁵ studied the structural non-probabilistic reliability based on the ellipsoidal convex model. Luo and Kang²⁶ deduced the explicit iterative formula of an optimization problem and solved the ellipsoidal model. Liu et al.²⁷ proposed a general robust reliability index that can be applied to linear and nonlinear systems with basic variables contained a variety of uncertain convex sets. Zhou et al.²⁸ solved the convex aggregate comprehensive index, which combined the improved finite step-size iterative method with Monte-Carlo method. Cao and Duan²⁹ researched the structural non-probabilistic reliability problems under hyper-ellipsoid convex set description and promoted a reliability index, which is used to measure the safety degree of structures when hyper-ellipsoid convex set model coexists with interval variables.

In this article, the ellipsoid model is combined with structural parameter sensitivity, from which the sensitivity of non-probabilistic parameters is defined and its formula is deduced. Then, for the case, the non-probabilistic performance function is the linear combinations of ellipsoidal variables, and the exact analytical solutions of the non-probabilistic parameter sensitivity are given in ellipsoidal model. Next, the nonlinear performance function is linearized with Taylor expansion based on an approximate analytical method which is proposed for solving parameter sensitivity. Finally, the proposed method is verified by numerical examples in the following part.

Reliability indexes of the ellipsoidal model

The structural function is g( x ), where x = (x₁,…, x_n) is a n-dimensional variable. The ellipsoid model of x is shown in equation (1)

U (u, σ) = {x = (x_{1}, \dots, x_{n}) : \sum_{i = 1}^{n} \frac{{(x_{i} - u_{i})}^{2}}{σ_{i}^{2}} \leq 1}

(1)

where $u = (u_{1}, u_{2}, \dots, u_{n})$ are the location parameters of the ellipsoid model. $σ = (σ_{1}, σ_{2}, \dots, σ_{n})$ are the shape parameters of the ellipsoid model. The surface $g (x) = 0$ is the performance function, which divides the variable space D into the security domain D_s ( x : g( x ) > 0) and the failure domain D_f ( x : g( x ) < 0). η is the reliability index of the structure that can be expressed as equation (2)

\begin{matrix} Find η, x_{i} \\ η = λ \\ s . t . {\begin{matrix} M = g (x) = g (x_{1}, x_{2}, \dots, x_{n}) \leq 0 \\ \sum_{i = 1}^{n} \frac{{(x_{i} - u_{i})}^{2}}{σ_{i}^{2}} = λ (i = 1, 2, \dots, n) \end{matrix} \end{matrix}

(2)

From equation (2), it can be seen that the solution of reliability index η is a constrained optimization problem in real ellipsoidal model. The reliability index η and the optimal solution x ^* can be obtained from optimization method. x ^* is the corresponding design point in g( x ) = 0.

If n = 2, u_i = 0, $σ_{i} = 1 (i = 1, 2, \dots, n)$ , n-dimensional ellipsoid degenerates into two-dimensional circle, as shown in Figure 1. η is the shortest distance from the coordinate origin to the g( x ) = 0, and x ^* is the design point in the middle of g( x ) = 0.

Figure 1.

Diagram of reliability index based on two-dimensional ellipsoidal model.

In Wu,⁶ the parameter sensitivity of random model is defined as the partial derivative of failure probability P_f to the basic parameter variable, that is, $\partial P_{f} / \partial θ_{x}$ . So the definition of non-probabilistic parameter sensitivity is given as partial derivatives of reliability indexes with respect to their parameters in the ellipsoidal model. On the other hand, the partial derivative of the reliability index to the position parameter is denoted by $\partial η / \partial u_{i}$ in the ellipsoidal model, as well as the partial derivative of the reliability index to the shape parameter is denoted by $\partial η / \partial u_{i}$ . In the previous engineering analysis, the partial derivative problem can be solved by the finite difference method, such as the finite difference method, to obtain the convergence of the solution, which has to be processed by a large number of calculation and the step size has a great impact on the results. Therefore, for the non-probabilistic reliability problems of complex structures, the authors present an analytical solution for the sensitivity of non-probabilistic parameters in ellipsoidal variables in this paper.

Reliability sensitivity analysis of ellipsoid model

Reliability sensitivity analysis of linear function

When g( x ) is a linear function, as shown in the following equation (3)

M = g (x) = a_{0} + \sum_{i = 1}^{n} b_{i} x_{i}

(3)

the variable x satisfies equation (1).

$z_{i} = (x_{i} - u_{i}) / σ_{i}$ , suppose that $z = (z_{1}, z_{2} \dots z_{n})$ is super sphere in the n-dimensional unit, that is $z_{1}^{2} + z_{2}^{2} + \dots + z_{n}^{2} \leq 1$ , then we substitute $z_{i} (i = 1, 2, \dots, n)$ into equation (3), the function can be rewritten as equation (4)

g (z) = a_{0} + \sum_{i = 1}^{n} b_{i} u_{i} + \sum_{i = 1}^{n} b_{i} σ_{i} z_{i}

(4)

According to the definition of non-probabilistic reliability index, we can get the reliability index of the linear function in ellipsoidal model

η = \frac{a_{0} + \sum_{i = 1}^{n} b_{i} u_{i}}{{(\sum_{i = 1}^{n} {(b_{i} σ_{i})}^{2})}^{1 / 2}}

(5)

For the linear function g( x ), the analytical formula of the non-probabilistic parameter sensitivity in the ellipsoidal model is

\frac{\partial η}{\partial u_{i}} = \frac{b_{i}}{{(\sum_{i = 1}^{n} {(b_{i} σ_{i})}^{2})}^{1 / 2}}

(6)

\frac{\partial η}{\partial σ_{i}} = - b_{i}^{2} σ_{i} \frac{a_{0} + \sum_{i = 1}^{n} b_{i} u_{i}}{{(\sum_{i = 1}^{n} {(b_{i} σ_{i})}^{2})}^{3 / 2}}

(7)

Reliability sensitivity analysis of nonlinear function

In fact, it can be difficult to put the above-described solving method into the application directly if the function is nonlinear. Therefore, from this consideration, we aim to present an approximate analytic method of non-probabilistic parameter sensitivity. Its basic idea of this method is to linearize the nonlinear function at its design point and then use the previous paragraph method to obtain the approximately analytic solution of nonlinear and non-probabilistic parameter sensitivity in the ellipsoidal model.

As for the design point x ^* of the relative ellipsoid model U( u , σ ), the function M = g( x ) is expanded linearly at the design point $x^{*} (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*})$ , as shown in equation (8)

\begin{matrix} M = g (x_{1}, x_{2}, \dots, x_{n}) \approx g (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}) \\ + \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} (x_{i} - x_{i}^{*}) \end{matrix}

(8)

As the design point x ^* on the surface g( x ) = 0, it substitutes $g (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}) = 0$ into equation (7). z_i is the normalized variable, and the linear limit state equation corresponding to the original function can be expressed as equations (9) and (10)

M = g (x_{1}, x_{2}, \dots, x_{n}) \approx \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} x_{i} - \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} x_{i}^{*}

(9)

\begin{matrix} M = G (z_{1}, z_{2}, \dots, z_{n}) = \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} u_{i} \\ + \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} σ_{i} z_{i} - \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} x_{i}^{*} \end{matrix}

(10)

According to the definition of non-probabilistic reliability index and equation (10), the approximate analytic solution of nonlinear limit state equation is shown as follows

η = \frac{\sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} u_{i} - \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} x_{i}^{*}}{{(\sum_{i = 1}^{n} {({(\frac{\partial g}{\partial x_{i}})}_{x^{*}} σ_{i})}^{2})}^{1 / 2}}

(11)

Through the definition of non-probabilistic reliability sensitivity, we can get the approximately analytical solution of reliability sensitivity in the case of nonlinear limit state equation as follows

\frac{\partial η}{\partial u_{i}} = \frac{{(\frac{\partial g}{\partial x_{i}})}_{x^{*}}}{{(\sum_{i = 1}^{n} {({(\frac{\partial g}{\partial x_{i}})}_{x^{*}} σ_{i})}^{2})}^{1 / 2}}

(12)

\frac{\partial η}{\partial σ_{i}} = - {(\frac{\partial g}{\partial x_{i}})}_{x^{*}}^{2} σ_{i} \frac{\sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} u_{i} - \sum_{i = 1}^{n} {(\frac{\partial g}{\partial x_{i}})}_{x^{*}} x_{i}^{*}}{{(\sum_{i = 1}^{n} {({(\frac{\partial g}{\partial x_{i}})}_{x^{*}} σ_{i})}^{2})}^{3 / 2}}

(13)

Based on the analysis above, we get two expressions of the reliability sensitivity indices for the linear and nonlinear functions, respectively. In nonlinear case, an approximate solution is illustrated, which can solve the problem of selecting the step size in the case of weak nonlinearity using the method of difference. Eventually, the sensitivity indices will be more simply and conveniently.

Computational analysis of examples

There, three examples have been used to demonstrate the rationality of this method in this article.

Example 1

A function $M (r_{1}, r_{2}, s)$ as shown in equation (14)

M = r_{1} + \frac{1}{\sqrt{2}} r_{2} - s

(14)

when M > 0, it is safe, and when M < 0, it is failure. The variables r₁, r₂, s meet following equation (15)

{(\frac{r_{1} - u_{r_{1}}}{σ_{r_{1}}})}^{2} + {(\frac{r_{2} - u_{r_{2}}}{σ_{r_{2}}})}^{2} + {(\frac{s - u_{s}}{σ_{s}})}^{2} \leq 1

(15)

The parameters of the variables are shown in Table 1.

Table 1.

Ellipsoidal parametric variable of Example 1.

The location parameter	$u_{r_{1}}$	$u_{r_{2}}$	$u_{s}$
Numerical value	200	300	200
The shape parameter	$σ_{r_{1}}$	$σ_{r_{2}}$	$σ_{s}$
Numerical value	20	30	40

The results of non-probabilistic parameter sensitivity analysis in ellipsoidal model are shown in Table 2.

Table 2.

Non-probabilistic parametric sensitivity analysis results of Example 1.

	Method
	The presented analytical method	The finite difference method	Relative error (%)
$\partial η / \partial u_{r_{1}}$	0.020203	0.020184	0.0940
$\partial η / \partial σ_{r_{1}}$	−0.034985	−0.035062	0.2201
$\partial η / \partial u_{r_{2}}$	0.014286	0.014296	0.0700
$\partial η / \partial σ_{r_{2}}$	−0.026239	−0.026067	0.6555
$\partial η / \partial u_{s}$	−0.020203	−0.020285	0.4059
$\partial η / \partial σ_{s}$	−0.069971	−0.069861	0.1572

In algorithm case 1, the sensitivity of non-probabilistic parameters is analyzed in the case of linear function. As shown in Table 2 where the finite difference method is taken as the exact solution, the difference between the two methods is less than 1%. What is more, the results of the proposed method for the linear function are consistent with the finite difference theory, which shows the method proposed in this article is effective and feasible for solving linear problems.

Example 2

As shown in Figure 2, the distance from the point where the concentrated load p₁ is applied to the fixed end of a cantilever beam is b₁. The distance from the point where the concentrated load p₂ is applied to the fixed end of a cantilever beam is b₂. The function $M (p_{1}, p_{2}, b_{1}, b_{2}, M_{cr})$ is shown in equation (16)

M = M_{cr} - p_{1} b_{1} - p_{2} b_{2}

(16)

when M > 0, it is safe; when M < 0, it is failure. The variables $(M_{cr}, p_{1}, p_{2}, b_{1}, b_{2})$ satisfy following equation (17)

\begin{matrix} {(\frac{M_{cr} - u_{M_{cr}}}{σ_{M_{cr}}})}^{2} + {(\frac{p_{1} - u_{p_{1}}}{σ_{p_{1}}})}^{2} + {(\frac{p_{2} - u_{p_{2}}}{σ_{p_{2}}})}^{2} \\ + {(\frac{b_{1} - u_{b_{1}}}{σ_{b_{1}}})}^{2} + {(\frac{b_{2} - u_{b_{2}}}{σ_{b_{2}}})}^{2} \leq 1 \end{matrix}

(17)

Figure 2.

Diagram of the cantilever beam.

The parameters of the variables are shown in Table 3.

Table 3.

Ellipsoidal parametric variable of Example 2.

The location parameter	$u_{p_{1}}$	$u_{p_{2}}$	$u_{b_{1}}$	$u_{b_{2}}$	$u_{M_{cr}}$
Numerical value	5	2	2	5	35
The shape parameter	$σ_{p_{1}}$	$σ_{p_{2}}$	$σ_{b_{1}}$	$σ_{b_{2}}$	$σ_{M_{cr}}$
Numerical value	0.6	0.3	0.2	0.5	4

The results of non-probabilistic parameter sensitivities in the ellipsoidal model are shown in Table 4.

Table 4.

Non-probabilistic reliability sensitivity analysis results.

	Method
	The presented analytical method	The finite difference method	Relative error (%)
$\partial η / \partial u_{M_{cr}}$	0.2089	0.2090	0.0478
$\partial η / \partial σ_{M_{cr}}$	−0.5916	−0.5915	0.0169
$\partial η / \partial u_{p_{1}}$	−0.4506	−0.4505	0.0222
$\partial η / \partial σ_{p_{1}}$	−0.4129	−0.4130	0.0242
$\partial η / \partial u_{b_{1}}$	−1.1592	−1.1587	0.0432
$\partial η / \partial σ_{b_{1}}$	−0.9111	−0.9113	0.0219
$\partial η / \partial u_{p_{2}}$	−1.1311	−1.1310	0.0088
$\partial η / \partial σ_{p_{2}}$	−1.3010	−1.3009	0.0077
$\partial η / \partial u_{b_{2}}$	−0.4898	−0.4899	0.0204
$\partial η / \partial σ_{b_{2}}$	−0.4067	−0.4068	0.0246

Case 2 illustrated sensitivity analysis of the non-probabilistic parameters for nonlinear function. By comparing the approximate analytic solution with the result of the finite difference theory, it can be seen that the error results of the two kinds are less than 0.1%, which also certifies that the approximately analytical solution is effective to solve the non-probabilistic parameter sensitivity problem for nonlinear conditions.

Example 3

An aerofoil 9-box structure composed of 64-bar and 42-plate elements is shown in Figure 3. The material is aluminum alloy. The whole original data are taken from Song.³⁰ Both applied load and the strengths of every element are variables. The significant failure mode of structure system is enumerated by the optimum criterion method, which is

M = 4.0 R_{78} + 4.0 R_{68} - 3.9998 R_{77} - P

(18)

Figure 3.

Diagram of aerofoil 9-box structure.

When M > 0, the structure is safe, and when M < 0, the structure is failed. The variables R₆₈, R₇₇, R₇₈, P have to satisfy the following

\begin{matrix} {(\frac{R_{68} - u_{R_{68}}}{σ_{R_{68}}})}^{2} + {(\frac{R_{77} - u_{R_{77}}}{σ_{R_{77}}})}^{2} \\ + {(\frac{R_{78} - u_{R_{78}}}{σ_{R_{78}}})}^{2} + {(\frac{P - u_{P}}{σ_{P}})}^{2} \leq 1 \end{matrix}

(19)

The parameters of the variables are shown in Table 5.

Table 5.

Ellipsoidal parametric variable of Example 3.

The location parameter	$u_{R_{68}}$	$u_{R_{77}}$	$u_{R_{78}}$	$u_{P}$
Numerical value (kg)	83.5	83.5	83.5	150
The shape parameter	$σ_{R_{68}}$	$σ_{R_{77}}$	$σ_{R_{78}}$	$σ_{P}$
Numerical value (kg)	10.02	10.02	10.02	37.5

The results of non-probabilistic parameter sensitivity calculation in ellipsoidal model are shown in Table 6.

Table 6.

Non-probabilistic parametric sensitivity analysis results of Example 3.

	Method
	The presented analytical method	The finite difference method	Relative error (%)
$\partial η / \partial u_{R_{78}}$	0.0507	0.0506	0.1972
$\partial η / \partial σ_{R_{78}}$	−0.0601	−0.0600	0.1664
$\partial η / \partial u_{R_{68}}$	0.0507	0.0506	0.1972
$\partial η / \partial σ_{R_{68}}$	−0.0601	−0.0600	0.1664
$\partial η / \partial u_{R_{77}}$	−0.0507	−0.0506	0.1972
$\partial η / \partial σ_{R_{77}}$	−0.0601	−0.0600	0.1664
$\partial η / \partial u_{P}$	−0.0127	−0.0126	0.7874
$\partial η / \partial σ_{P}$	−0.0140	−0.0141	0.7143

For the example mentioned above, the sensitivity of non-probabilistic parameters is analyzed by an aerofoil 9-box structure. As shown in Table 6 where the finite difference method is taken as the exact solution, the difference between the two methods is less than 1%, from which the linear function is consistent with the finite difference theory. The proposed method in this paper has demonstrated its effectiveness and feasibility meaning, which could effectively solve relate issues in this field.

Conclusion

In conclusion, non-probabilistic parameter sensitivity analysis based on the ellipsoidal model can be applied to quantitatively describe the importance of non-probabilistic reliability under bounded parameters. Therefore, it has important theoretical and practical meanings for research and engineering application. In this article, the non-probabilistic parameter sensitivity index in the ellipsoidal model is defined, and the analytic solution of the sensitivity of non-probabilistic parameters is deduced as well. At the same time, an exact analytical solution is deduced for the linear function. For this non-linear function case, an approximately analytic method for sensitivity analysis with non-probabilistic parameters under the ellipsoidal model is obtained through combined with the first-order Taylor expansion. Meanwhile, the correctness and effectiveness of the analytical method are illustrated by three numerical examples, which also show this method simplifies the calculation process compared with the finite difference method. And finally, the results of this article are of great significance for the optimization and exploration in non-probabilistic reliability models, which verifies the method proposed in this article benefits for the approximate linearization function. Rather, for high-nonlinear function problem, it is probably to explore further in the future.

Footnotes

Handling Editor: Shun-Peng Zhu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Authors gratefully appreciate the support of China Scholarship Council, Top International University Visiting Program for Outstanding Young scholars of Northwestern Polytechnical University, the fundamental research fund for the central universities (NPU-FFR-3102015BJ(II)JL04), natural science foundation of Shaanxi province (2016JQ5109), and the 2017 National Undergraduate Training Programs for Innovation and Entrepreneurship.

ORCID iD

Feng Zhang

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Non-probabilistic parameter sensitivity analysis for structures based on ellipsoidal model

Abstract

Keywords

Introduction

Reliability indexes of the ellipsoidal model

Reliability sensitivity analysis of ellipsoid model

Reliability sensitivity analysis of linear function

Reliability sensitivity analysis of nonlinear function

Computational analysis of examples

Example 1

Example 2

Example 3

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References