Elsevier

Information Sciences

Volume 477, March 2019, Pages 466-489
Information Sciences

Equilibrium reliability measure for structural design under twofold uncertainty

https://doi.org/10.1016/j.ins.2018.10.059Get rights and content

Highlights

  • A new equilibrium reliability index is developed to evaluating structural reliability.

  • An ERSD optimization model with equilibrium reliability constraint is formulated to achieve the optimal design.

  • We show the equivalent representation of equilibrium reliability constraint, and discuss its convexity.

  • By introducing FS scheme, the feasible solutions generated by equilibrium reliability constraint are obtained.

  • All of numerical experiences clearly demonstrate the applicability and effectiveness of the equilibrium reliability modeling methods.

Abstract

In structural reliability engineering, designer may encounter twofold uncertainty that arises from the existence of the insufficient statistical data and the incorporated experts’ information. For this case, the randomness and fuzziness coexist in each uncertain element. This paper aims to develop a new approach to evaluating structural reliability and obtain the optimal structural design under twofold uncertainty. Specifically, we first introduce a metric termed equilibrium chance to formulate equilibrium reliability index (ERI) for modeling structural reliability under linear limit-state function and nonlinear limit-state function. Second, a new class of equilibrium reliability-based structural design (ERSD) optimization problem with equilibrium reliability constraint is developed. Under mild conditions, we provide the equivalent reformulation of the equilibrium reliability constraint and demonstrate its convexity. Finally, we introduce the fuzzy simulation (FS) scheme to obtain the feasible solution generated by equilibrium reliability constraint, and a hybrid heuristic algorithm by integrating the FS, genetic algorithm (GA) and local search (LS) is designed to solve the proposed ERSD optimization problem. Several numerical examples are carried out to illustrate the applicability and effectiveness of our proposed reliability modeling method.

Introduction

Structural reliability problems focus on quantifying the degree of structural reliability and obtaining the optimal design of structural system under the predetermined reliability level, which have attracted many researchers owing to its critical performance on mechanical structures. For the conventional methods, an underlying assumption is that all uncertain parameters are characterized by random variables, and reliability issues have been carried out with the classical probability theory. Melchers [30] investigated the classical structural reliability issues, and introduced various stochastic methods for predicting the structural safety, including first order second moment (FOSM) method and Monte Carlo techniques. Various studies have further expanded research along with the probabilistic framework. Based on the universal generating function technique, Huang and An [16] presented a discrete stress-strength interference model to evaluate structural reliability. Considering the random uncertainty in structural loading and material properties, Utkin and Kozine [38] developed the cautious structural reliability models based on the imprecise Bayesian inference, in which imprecise inferences are expressed in terms of posterior upper and lower probabilities. Chen et al. [6] studied a class of robust shape and topology optimization methodology by integrating the level set methods with the robust design formulation. To model the structural performance function, Dubourg et al. [11] proposed a hybrid approach to combining importance sampling and an adaptive metamodeling technique, and employed the Kriging surrogate model to evaluate the structural reliability. Under stationary stochastic excitations, Muscolino et al. [33] presented an efficient procedure for the evaluation of the bounds of the interval reliability function of the generic response process, and conducted the reliability analysis of linear discretized structural systems. On the basis of probabilistic reliability modeling, the reliability-based design optimization (RBDO) was formulated for structural design [50], and the readers may refer to [13] for an excellent review. Jensen et al. [18] investigated a model reduction technique combined with an appropriate optimization scheme to perform the design process in a reduced space of generalized coordinates.

When uncertain parameters are imprecise or vague, several fuzzy reliability modeling methods have been reported in the literature. On the basis of fuzzy set theory, Möller et al. [31] developed an improved assessment of load-bearing behavior in structural analysis where the existing uncertainty of subjective information is quantified and mapped onto the fuzzy value by means of the α-level optimization. Valdebenito et al. [39] presented an approximate representation of the structural displacement to perform fuzzy structural analysis, in which input parameters in the structural problem are modeled through fuzzy sets. Under the fuzzy possibility framework, Mourelatos and Zhou [32] proposed a possibility-based design optimization method where all design constraints are expressed in a possibilistic way, and demonstrated that this solution method can provide a conservative solution compared with conventional reliability-based designs. Tang et al. [35] established a possibility safe index (PSI) and developed the PSI-based design optimization model for solving structural optimization problems with fuzzy variables. By treating the membership level as random variable with uniform distribution, Li et al. [22] studied the fuzzy structural reliability problem based on the probability perspective. By means of interval ranking strategy, Wang et al. [45] investigated the heat transfer system under the subjective uncertainty, in which fuzzy variables are transformed into interval variables under different membership levels. Based on credibility theory, Marano and Quaranta [29] employed the fuzzy entropy as the global measure of variable dispersion, and proposed a robust structural optimization model that is able to give a set of Pareto optimal solutions in terms of structural efficiency and sensitivities regarding fuzzy uncertainty.

As mentioned above, structural reliability issues generally stem from the presence of two types of uncertainty, one is probabilistic uncertainty, the other is fuzzy uncertainty. In real-life applications, several studies have paid attention to such hybrid uncertainty that randomness and fuzziness coexist in one system, such as chance measure for the uncertain events [20], trip distribution problem [23], feedstock flow planning for waste-to-energy systems [47], hazardous materials transportation [48], robust granular optimization [42], [43], multi-period portfolio selection [41], and budget-portfolio investment optimization [46]. In structural reliability modeling, many researchers focus on reliability modeling under hybrid uncertainty of combining randomness and fuzziness. For the structural system with fuzzy variables as well as random variables, Chakraborty and Sam [5] developed a class of structural reliability models based on the transformations of equivalent entropy and scaling the membership function, in which the fuzzy variables are transformed into the equivalent random variables. Based on probability model and non-probabilistic set-based model, Ni and Qiu [34] formulated a class of hybrid reliability model which contains fuzzy random variables and interval variables of structural parameters. Balu and Rao [2] proposed a multicut-high dimensional model representation technique to estimate the reliability bounds of structural systems under the hybrid uncertainty. Beer et al. [3] provided the imprecise probabilistic method for predicting the reliability of structure systems which involve both probabilistic and fuzzy information. Li and Lu [21] designed a hybrid algorithm to obtain the membership function of fuzzy reliability, in which the bounds of structural reliability are determined by the interval optimization in the presence of random variables and fuzzy variables. Jafari and Jahani [17] proposed a fuzzy random reliability sensitivity measure of the failure probability to consider the effect of the epistemic and aleatory uncertainties. Wang et al. [44] developed the interval ranking strategy for the thermal structure design with random, interval and fuzzy uncertainties, where random variables are used to quantify the probabilistic uncertainty, and interval variables and fuzzy variables are adopted to model the non-probabilistic uncertainty. In view of the existing studies, many researchers adopt random variable, fuzzy variable, interval variable, and fuzzy random variable to characterize the hybrid uncertainty of combining randomness and fuzziness, such as structural reliability model with the combining of fuzzy variable, random variable and probability measure [5], [21], hybrid reliability model with the combining of interval variable, fuzzy random variable and probability measure [34], reliability sensitivity measure constructed by fuzzy random variable and probability measure [17], thermal structure design model established by random variable, fuzzy variable, interval variable and interval possibility measure [44], multicut-high dimensional model developed by random variable, fuzzy variable, and probability measure [2], imprecise probability model formulated by random variable, fuzzy variable, interval variable, fuzzy random variable and probability measure [3], minimum risk model constructed by fuzzy random variable [19].

Despite the considerable efforts made in the above studies, modeling structural reliability and optimization under hybrid uncertainty still remain challenging. In structural reliability engineering, statistical information is usually limited, and there are insufficient historical data available for generating the nondeterministic parameters of probability distributions. In such situations, the subjective judgement by experts could be incorporated into the limited statistical data in a complimentary manner. Due to the imprecise human knowledge in capturing historical data, the randomness and fuzziness are simultaneously in each uncertain parameter, we refer to this type of hybrid uncertainty as twofold uncertainty. On the one hand, this twofold uncertainty exists in many practical engineering problems. In aircraft design industry, the material strength of metallic structure is assumed to be normally distributed variable with distribution N(μ,σ2) and partially unknown parameters μ and/or σ. Often these parameters μ and/or σ can be estimated from historical data. Nevertheless, due to limitation of experiments and capacity, the historical data are usually insufficient to determine these parameters exactly. Instead, subjective inference is used to provide the estimations of these parameters, in which these unknown parameters are characterized by fuzzy variables μ(γ) and/or σ(γ), and the corresponding fuzzy possibility distributions determined from experts’ evaluation on μ, σ. Then, the material strength is a class of uncertain variable with twofold uncertainty of combining randomness and fuzziness, in which the fuzziness is embedded into the probabilistic distribution parameters, and the distribution of material strength is usually characterized by N(μ(γ),σ2), N(μ,σ2(γ)) or N(μ(γ),σ2(γ)). Random fuzzy variable [24] as a powerful tool is used to characterize the material strength with twofold uncertainty. As for more details and examples on this twofold uncertainty, the reader may refer to [24]. On the other hand, twofold uncertainty is a class of mixed uncertainty where randomness and fuzziness are in the state of affairs, and are required to be considered simultaneously. For this example mentioned above, the material strength from practical case is characterized by distribution N(μ(γ),σ2), which is given in the following form: material strength follows normal distribution N(10,22) with possibility 0.3, follows normal distribution N(20,22) with possibility 1, and follows normal distribution N(30,22) with possibility 0.7, and so on. In other words, the realizations of fuzzy parameter are not crisp numbers but normally distributed random variables. In fact, for any given γ0 ∈ Γ, the fuzzy variable μ(γ) reduces to the crisp value μ(γ0), and the material strength degenerates to random variable with normal distribution N(μ(γ0),σ2). For any γ1, γ2 ∈ Γ and γ1 ≠ γ2, the normal distribution N(μ(γ1),σ2) is different from normal distribution N(μ(γ2),σ2). Based on previous discussion, the ignorance of randomness or fuzziness will lead to the result which is logically inconsistent with modeling twofold uncertainty. In such a case, none of random variable, fuzzy variable, fuzzy random variable, and even the combining of random variable and fuzzy variable is available to describing this twofold uncertainty. Although these previous existing approaches are the closest to our research topic, none is well applicable to describing and studying such twofold uncertainty.

The aim of this study is to develop a new reliability modeling approach based on random fuzzy variable and equilibrium chance [26], and provide a new measure index and a class of structural optimization model under twofold uncertainty of combining randomness and fuzziness. This study differs from the above mentioned studies in two aspects. First, under the condition that subjective judgement is embedded into insufficient statistical data, we adopt random fuzzy variable and equilibrium chance to gauge the quantities with such twofold uncertainty, in which the nondeterministic parameters of probabilistic distribution is described by fuzzy distributions. Second, this study proposes a new class of structural optimization problem with equilibrium reliability constraint, in which the optimal design can be attained at the predetermined reliability level by combining the probability level and credibility level.

For this study, the main challenges are summarized in three aspects. First, under twofold uncertainty of combining randomness and fuzziness, we cannot obtain the exact probability distributions of uncertain parameters. For this case, using random variable or fuzzy variable, separately becomes invalid to characterize this twofold uncertainty. Second, by means of random fuzzy variable and equilibrium chance, evaluating structural reliability has not been well established in the literature. The complexity arises from the need to the exact expression of reliability index available to real application. Third, to obtain the optimal structural design under twofold uncertainty, it is essential to establish a class of structural optimization model subject to equilibrium reliability constraint. The complexity results from the treatment of equilibrium reliability constraint and achieving the feasible design solution. To the best of our knowledge, these issues have not yet been investigated in the literature. The contributions of this study are summarized as follows:

  • 1)

    Under twofold uncertainty, we formulate a new ERI to evaluating the structural reliability under linear limit-state function and nonlinear limit-state function. The ERI is an extension of probabilistic reliability index, and provides an effective and accurate value to characterize the degree of structural reliability under twofold uncertainty.

  • 2)

    We develop an ERSD optimization model with equilibrium reliability constraint, which is concerned with achieving the optimal design at the predetermined reliability level by combining probability level and credibility level. Furthermore, we show the equivalent representation of equilibrium reliability constraint and demonstrate its convexity under mild assumptions.

  • 3)

    To solve the proposed ERSD optimization problem, FS scheme is introduced to obtain the feasible solution generated by equilibrium reliability constraint, which enables a variety of meta-heuristic algorithms to be used to find the optimal structural design.

The remainder of this paper is organized as follows. In Section 2, the ERI is developed in detail to evaluate the structural reliability under twofold uncertainty. Section 3 is dedicated to formulating the ERSD optimization problem, and deriving the equivalent representation of equilibrium reliability constraint with its convexity. Section 4 focuses on obtaining the feasible solution generated by equilibrium reliability constraint, and designs the hybrid heuristic algorithm by integrating the FS, GA and LS to solve the ERSD optimization problem. Several illustrative examples are conducted to verify the applicability and effectiveness of the proposed methods in Section 5. Finally, conclusions are given in Section 6.

Section snippets

Problem description

In the traditional structural reliability theory, structural performance is characterized by the limit-state function g(ξ) which is established by the failure mechanism, and all uncertain parameters of structure are treated as a random vector ξ=(ξ1,ξ2,,ξn), where each component ξi represents one uncertain element, such as external load, structural strength, material parameter, and so on. The failure surface is determined by the limit-state equation g(ξ)=0, which divides the performance space

Equilibrium reliability-based structural design (ERSD)

RBDO formulation has widely used in structural design engineering, which aims to search for the optimal design subject to the reliability constraint. A class of RBDO formulation devotes to minimize the total weight/cost of structural system, in which the reliability constraint is characterized by probability measure. However, under twofold uncertainty, the probability measure is invalid to gauge the quantities with twofold uncertainty, then the feasible solution generated by the probabilistic

Handling for reliability constraint

For the general ERSD optimization problem, we will provide the FS scheme [25] to tackle the equilibrium reliability constraint and obtain the feasible solution generated by this constraint. The basic idea of FS scheme is to approximate the credibility of reliability constraint and verify the feasibility of solution for the purpose of generating the feasibility set. Suppose that the output response function G(x, ξ) follows normal distribution N(m(x,μ(γ)),σ2) where the expectation m(x, μ(γ)) is a

Numerical study

In this section, we take several examples to examine the computational performance of the proposed reliability modeling methods. The computational experiments are divided into four subsections. In Section 5.1, we provide Example A to demonstrate the applicability of the proposed ERI evaluation in capturing randomness as well as fuzziness under twofold uncertainty. In 5.2 Example B: 9-box wing structure, 5.3 Example C: freely-supported beam structure, we provide Examples B, C to investigate the

Conclusions

In this paper, we developed a new measure tool to evaluate the structural reliability, and formulated a class of structural optimization model by considering the randomness and fuzziness. The obtained new results included the following four aspects:

  • (i)

    Based on equilibrium chance, the ERI was developed to evaluate structural reliability under the linear limit-state function and nonlinear limit-state function, which provides a scalar value to characterize the degree of structural reliability by the

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos.51675026, 71671009), and the National Basic Research Program of China (No.2013CB733002).

References (51)

  • G.C. Marano et al.

    Fuzzy-based robust structural optimization

    Int J Solids Struct

    (2008)
  • G. Muscolino et al.

    Reliability analysis of structures with interval uncertainties under stationary stochastic excitations

    Comput. Methods Appl. Mech. Eng.

    (2016)
  • Z. Ni et al.

    Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability

    Comput. Industr. Eng.

    (2010)
  • Z.C. Tang et al.

    An efficient approach for design optimization of structures involving fuzzy variables

    Fuzzy Sets Syst.

    (2014)
  • M. Tavana et al.

    Fuzzy stochastic data envelopment analysis with application to base realignment and closure (BRAC)

    Expert Syst. Appl.

    (2012)
  • L.V. Utkin et al.

    On new cautious structural reliability models in the framework of imprecise probabilities

    Struct. Saf.

    (2010)
  • M.A. Valdebenito et al.

    Approximate fuzzy analysis of linear structural systems applying intervening variables

    Comput. Struct.

    (2016)
  • C.H. Wang et al.

    Using quality function deployment for collaborative product design and optimal selection of module mix

    Comput. Industr. Eng.

    (2012)
  • B. Wang et al.

    Multi-period portfolio selection with dynamic risk/expected-return level under fuzzy random uncertainty

    Inf. Sci.

    (2017)
  • C. Wang et al.

    Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters

    Appl. Math. Model.

    (2017)
  • C. Wang et al.

    Novel fuzzy reliability analysis for heat transfer system based on interval ranking method

    Int. J. Therm. Sci.

    (2017)
  • G. Yang et al.

    Optimizing an equilibrium supply chain network design problem by an improved hybrid biogeography based optimization algorithm

    Appl. Soft Comput.

    (2017)
  • P. Yi et al.

    A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization

    Struct. Saf.

    (2008)
  • H. Zhai et al.

    Modeling two-stage UHL problem with uncertain demands

    Appl. Math. Model.

    (2016)
  • E.H. Aarts et al.

    Local Search in Combinatorial Optimization

    (1997)
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