System reliability analyses of static and dynamic structures via direct probability integral method

Highlights

• System reliability analyses of static and dynamic structures are attacked uniformly.

• Probability density integral equation for joint PDF of performance functions is established.

• Reliability formulas of series, parallel and mixed systems are analytically derived.

• Extreme value mapping is utilized to calculate static and dynamic system reliabilities.

• Examples indicate high efficiency and accuracy of direct probability integral method.

Abstract

Structural system reliability analysis is of important significance for evaluating the safety of structures with many components. Since a structural system can be acted by static or dynamic load, a unified and efficient method is required to assess the system reliability of static or dynamic load induced structures. In this study, the direct probability integral method (DPIM) is proposed to uniformly attack system reliability problems of static and dynamic structures. Firstly, the static and first-passage dynamic reliability formulas of the series, parallel and mixed systems are established by the joint probability density function (PDF) of multiple performance functions. Based on the probability density integral equation (PDIE) of performance functions, the DPIM is proposed along the two approaches, i.e., DPIM-S and DPIM-H. The former computes the system reliability using the PDF of extreme value mapping of performance functions, which is obtained by smoothing Dirac delta function. In the latter, the system reliability formulas with Heaviside function are analytically derived by the PDIE of multiple performance functions. Specially, the role of smoothing of Dirac delta function in DPIM for stochastic response and reliability analyses is revealed. Finally, five typical examples, including two mathematical examples and three static and dynamic structural systems, demonstrate high efficiency and accuracy of the DPIM for system reliability computation. Because of omitting the smoothing of Dirac delta function, the DPIM-H takes less CPU time for solving system reliabilities of static and dynamic structures than DPIM-S, while the DPIM-S has a significant advantage of obtaining the PDF of performance function.

Introduction

In general, engineering structures are consisted of multiple components [1], [2], [3]. The safety of a structural system needs to be evaluated in a global manner to guarantee the service performance in life cycle. System reliability analysis of structure is devoted to assess the safety of structure considering multiple failure modes in the framework of probability theory.

Beginning with the seminal work of Freudenthal [4], the structural reliability is defined as the multiple dimensional integral of joint probability density function (PDF) in the safe domain. Given the difficulty solving this integral, the approximate methods were developed, which include first-order reliability method, second-order reliability method, and moment method [5], [6], [7], [8], [9]. The surrogate model of performance function is another approximate approach, and can be efficiently incorporated with direct Monte Carlo simulation (MCS) [10], [11], [12], [13], [14], [15], [16], [17], [18]. However, the accuracy of these methods depends on that of calculated moments or approximation of performance function.

When a structure is subjected to random excitation, the time-variant performance functions can be expressed as a series of functions of stochastic dynamic responses [1], [3], [19]. The level-crossing method was early adopted to deal with first-passage problem [20]. Since the number of zero-crossing is assumed as specific probability distribution, the computational accuracy is problematic. Along the path of random vibration analysis, the diffusion process method was also employed to obtain the first-passage dynamic reliability [21], [22]. Whereas, the difficulty of solving the Kolmogorov backward equation limits the application to engineering structures with more degrees of freedom. Stochastic sampling approaches with variance reduction techniques were proposed to compute time-variant reliability [23], [24], [25]. In the past two decades, the probability density evolution method (PDEM) was developed by Li and Chen [26], [27]. Starting from the random event description of the principle of probability conservation, the generalized density evolution equation (GDEE), a hyperbolic partial differential equation, was constructed. By incorporating the absorbing boundary condition in GDEE or constructing the equivalent extreme-value event, the PDEM can be utilized to solve structural dynamic reliability, and perform static reliability analysis by constructing a virtual stochastic process [26]. With the increasing demand of reliability-based design optimization (RBDO) [28], [29], [30], the highly efficient reliability method, especially for dynamic reliability, needs to be developed urgently. From the new perspective of integral equation, in the previous work of authors [31], the probability density integral equations (PDIEs) of static and dynamic systems were derived based on the principle of probability conservation. Actually, the dynamic PDIE is equivalent to the GDEE in the sense of probability conservation. By introducing the advanced numerical integration technique, i.e., the partition of probability space [32], the direct probability integral method (DPIM) was established to perform stochastic response and reliability analyses in a unified and efficient framework [33]. Note that both the PDEM and DPIM can obtain the PDF of performance function from the original distribution of random variables, which are consistent with the essential definition of structural reliability.

Structural system reliability aims to evaluate the overall or global reliability of a structural system. The researches on structural system reliability can be categorized as the component-level method and structure-level method. In general, the calculation of system reliability can be transformed into the standard normal space, being a high-dimensional integral problem. To solve this integral, the failure probability and the correlation coefficients of all failure modes need to be first determined. For structural system with many components, the computation of correlation coefficients among all failure modes is unpractical. The early studies focused on the bound estimation of system reliability [34], [35]. Since the correlations between failure modes were not considered, the estimated bound presents a wide range. Ditlevsen [36] obtained a narrow second-order bound by taking into account the joint PDF of any two failure modes. On this basis, the three- and high-order bounds for series system were also suggested [37], [38]. Due to the dependence of failure modes in high-order bound, Song and Der Kiureghian [39] developed the linear programming (LP)-based method to determine the narrowest possible bounds of the series and parallel systems. However, the computational effort of LP-based method increases exponentially with the number of the failure modes of system. To overcome this limitation, Wei et al. [40] established a small-scale LP by using the addition laws of failure probability. Byun and Song [41] advised an iterative procedure of binary integer programming based on the inclusion relationship between the considered events to alleviate the memory requirement in LP-based method.

Different from the bound estimation, the point estimation methods were also developed to calculate system reliability. Ang and Ma [42] advanced the probabilistic network evaluation technique to compute the system reliability by selecting the representative failure modes and adopting the independent assumption for these failure modes. Due to the difficulty of obtaining the joint PDF of multiple performance functions, system reliability was defined as multivariate normal integral, and thus the computation of correlation coefficients of components becomes a challenging problem. To tackle the different correlations of multiple components approximately, Thoft-Christensen and Sørensen [43] proposed the average and equivalent correlation coefficient method. The moments method was also extended to the calculation of system reliability by Zhao and Ang [44]. In addition, Kang et al. [45], [46] established the Matrix-based system reliability (MRS) method. In the context of MRS method, system reliability is transformed as a numerical integration in the space of the common source dependent random variables, while the common source random variables need to be identified by fitting the correlation coefficient matrix using generalized Dunnett–Sobel model. The sequential compounding method was suggested by Kang and Song [47], in which the numerical integration of multivariate normal integral was reduced to a sequential single equivalent compound events of two components and the remaining components. To achieve structural system reliability and perform RBDO, Xing et al. [48] extended the sequential compounding method and derived a new approximate formula for solving the correlation coefficients of failure modes. Roscoe et al. [49] presented the equivalent planes method of system reliability by transforming the calculation of correlation coefficients between failure modes into the autocorrelation and influence coefficients. Meanwhile, the topic on time-varying system reliability has drawn a lot of attention. Based on the level-crossing method, Song and Der Kiureghian [50] obtained the joint first-passage reliability of structural system subjected to stationary random excitation, where the joint first-passage probability was recursively calculated by using the addition rule for the marginal probability and the out-crossing probability of the vector process. Yu et al. [51] investigated the time-variant system reliability based on the extreme value moment method and improved the maximum entropy method. The MCS and variance reduction techniques [52], [53], [54], [55] and the various surrogate models [56], [57], [58], [59], [60] were utilized to tackle time-invariant and time variant system reliabilities. Song et al. [61] reviewed the existing methods for system reliability of time-invariant and time-invariant structures. However, most of the studies above mentioned were classified as component-level method, which cannot obtain the true joint PDF of performance functions. Under the assumption of joint normal distribution, the correlation coefficients between failure modes had to be estimated approximately.

The component-level method is devoted to evaluate the system reliability from the failure probability of each component, which causes a combinatorial explosion problem, even if the branch and bound method was proposed to investigate main failure modes [5]. In fact, once the joint PDF of multiple performance functions is achieved, the calculation of correlation coefficients of failure modes would be unnecessary. Besides, in above-mentioned point estimation methods, the input probability space is required to be transformed into standard normal space by using a nonlinear mapping (e.g., Rosenblueth transformation), which produces some numerical error. Li et al. [62] proved that the inherent correlation of failure modes can be reflected in the extreme value event, and carried out the structural system reliability by solving the PDF of extreme value of performance functions in the context of PDEM. Further, the PDEM-based physical synthesis method was established to evaluate the global safety of structural system [63]. Specifically, the PDEM-based reliability analysis is based on the original probability space, eliminating the error in the process of probability space transformation. The PDEM belongs to a structure-level method of system reliability analysis, in which the correlations among failure modes are embodied in the physical system. The difference between the structure-level and component-level methods lies in the sequence of calculations of correlation coefficients of multiple components and failure probability. In the latter, the first step is to calculate the failure probability of each component, and then solve the system failure probability by combining the components, while the former naturally considers the correlations between the components by structural analysis at first and then calculates of system failure probability without solving the correlation coefficients of components explicitly.

Since the joint PDF of multiple performance functions contains the whole probability information, the joint PDF-based system reliability analysis essentially fulfills the uncertainty propagation and quantification of structural system. Nevertheless, it is intractable to obtain joint PDF of multiple performance functions by using the existing approaches. In author’s previous works [31], [33], the DPIM was proposed to achieve the PDF of stochastic static and dynamic responses, and then extended to calculate static and dynamic reliabilities of structures with single failure mode based on the obtained PDF of time-invariant and time-variant performance functions. In DPIM [31], the PDIE is solved by means of the partition of probability space and smoothing of Dirac delta function. In the framework of DPIM, this study proposes the two approaches, i.e., DPIM with Heaviside function (DPIM-H) and DPIM with smoothing of Dirac delta function (DPIM-S), for reliability analyses of series, parallel and mixed systems. The PDIEs describing the joint PDF of performance functions of static and dynamic structural systems include the multiple-dimensional Dirac delta function, in which the physical correlation between failure modes is inherently taken into account, and the correlation coefficients do not need to be computed explicitly. The DPIM-S is the extension of original version of DPIM in [31], [33] to system reliability analysis, which can calculate the PDF of extreme value mapping of performance functions using the above two techniques. In fact, the system reliability of structures can also be calculated by integrating the joint PDF of performance functions. In this way, the Dirac delta functions in multivariant PDIE can be analytically integrated as the Heaviside functions, which passes by the smoothing of Dirac delta function. Such a strategy can fulfill reliability analysis with higher efficiency, and is termed as DPIM-H approach in this study.

The remainder of this paper is organized as follows. In Section 2, the system reliabilities for static and dynamic systems are formulated from the joint PDF of performance functions and extreme value mapping. The DPIM-S and DPIM-H approaches for system reliability analyses of static and dynamic structures are proposed in Section 3, in which the role of smoothing Dirac function and the application scenarios of two ways of DPIM are emphasized. Section 4 illustrates the five representative examples, namely, four static series, parallel and mixed systems and a two-span 12-story hysteretic frame under nonstationary random excitation, to demonstrate the merits of proposed approaches. Section 5 presents the main conclusions.