Probability density evolution analysis of stochastic seismic response of structures with dependent random parameters

https://doi.org/10.1016/j.probengmech.2020.103032Get rights and content

Highlights

  • A random function model for dependent random variables.

  • A physically-guided data-driven methodology for uncertainty quantification.

  • Physical constraints for the relation between elastic modulus and strength.

  • Numerical examples demonstrating the importance of parametric dependence.

Abstract

Performance evaluation and reliability assessment of real-world structures under earthquakes is of paramount importance. Generally, different mechanical property parameters of a structure are usually not independent, nor completely dependent, but partly dependent or correlated. Therefore, how to reasonably characterize such partial dependency and whether such partial dependency real matters in the stochastic response and reliability of structures under earthquakes are crucial issues. For this purpose, in the present paper, a novel physically-guided data-driven methodology of capturing the correlation configuration of basic random variables and the probability density evolution method are synthesized. The physically-guided data-driven methodology is firstly outlined. In this methodology, the underlying physical mechanism between dependent random variables is firstly involved to establish a random function model, and then the available observed data are adopted to identify the parameters in this model. What is more, physical constraints are also revealed for the initial modulus of elasticity and compressive strength of concrete. The probability density evolution method is then adopted, and the point selection by minimizing the GF-discrepancy is adjusted according to the correlation configuration and physical constraints. A reinforced concrete frame structure subjected to earthquake input is studied. It is found that when the structure is in the strongly nonlinear stage, the correlation configuration has considerable effects on the standard deviation of the stochastic responses, by a factor of nearly 2. In addition, whether the mechanical parameters in different floors are independent or not has great effects on the stochastic responses as well. Problems to be further studied are also outlined.

Introduction

Reasonable quantification of mechanical properties of construction materials is crucial for the seismic performance evaluation and reliability analysis of structures. These mechanical properties are to be taken as random variables or random fields, and are usually correlated. For the sake of simplicity, however, in most practical applications, these random variables are either considered to be independent, or completely dependent. For instance, in the widely used orthogonal polynomial expansion method, the basic random variables are usually assumed to be independent [1], [2], [3]. In contrast, in engineering practice it is widely recommended that the initial modulus of elasticity of concrete is determined deterministically according to the compressive strength in an empirical relationship [4], which implies a perfect, though, nonlinear dependency between the two random variables. However, it was well recognized from experimental data that these two mechanical parameters of concrete are partly correlated in nature [5], [6]. In static reliability of structures it has been demonstrated that such correlation has great effect on structural reliability assessment [7], [8]. Then the following two questions arise: what is the reasonable probabilistic model that can capture such correlations? What is the effect of such correlation on the stochastic response and reliability of structures under earthquakes?

The methodology of capturing partially correlation between two random variables based on data can be broadly classified into two types: one is to find an appropriate joint probability density function (PDF) by hypothesis and test; and the other is to find a random function relationship between the two random variables so that the dependent random variables can be converted into independent random variables. To the former class belonging, for instance, the copula function has been applied in soil and rock engineering [9], [10] to find a joint PDF with known marginal PDFs and a deterministic copula function. Nevertheless, the selection of form of copula function is usually difficult and somewhat empirically based. On the other hand, to the second type belonging, e.g., the Rosenblatt transformation [11] and the polynomial chaos expansion [12], are widely employed in practice, where the dependency is represented by strong nonlinear transforms. Though converting dependent random variables to independent basic random variables, unfortunately such transforms will usually worsen, more or less, the well-posedness of the problem in many cases. For instance, due to such transformations, the accuracy of point estimate of moments of response may be considerably deteriorated [13].

Recently, based on the thought of physically-guided data-driven (PGDD) modeling methodology, Chen et al. [6] proposed a new random function model, which is of weak nonlinearity, to capture the correlation configuration of dependent random variables. Based on this random function model, the correlation between the modulus of elasticity Ec and the compressive strength fc of concrete was investigated. Remarkably, in this random function model the underlying physical background was advocated, and therefore, two physical constraints, as byproducts of modeling, were also obtained. These two physical constraints will guide and adjust the point selection, which is an important step in the probability density evolution method (PDEM, [14]). In this paper, the stochastic responses of a reinforced concrete (RC) frame structure subjected to earthquake acceleration are analyzed. The cases with completely independent (CI) basic random vector, completely dependent (CD) basic random vector [4], and partially dependent (PD) basic random vector [6] are studied and compared. The results indicate that the effect of correlation configuration of mechanical properties is unignorable for the stochastic response of structures.

Section snippets

Random function model for dependent random variables

For clarity, considering two dependent random variables X and Y. In Chen et al. [6], the dependency between X and Y is written as the following form of random function Y=g1X+ζg2Xwhere ζ is a random variable with zero mean and unity variance, and is independent to X. The two weak nonlinear functions g1 and g2 are to be determined, either by physical reasoning or data learning. Further, by conducting the conditional expectation on both sides of Eq. (1), we have EY|X=g1Xwhere E denotes the

Fundamentals of PDEM

For clarity, we start with an outline of the probability density evolution method [14]. Without loss of generality, consider a stochastic dynamical system Ẋ=GX,Θ,twhere X=X1,,XmT is an m-dimensional state vector with the initial condition Xt0=X0, G denotes a state mapping and Θ=Θ1,,ΘsT is an s-dimensional random vector with the joint PDF pΘθ. For well-posed problems, the solution of Eq. (20) can be denoted by X=HX0,Θ,t with its components X=HX0,Θ,t for =1,,m. The generalized velocity

Structural information

Consider a 10-story reinforced concrete frame structure as shown in Fig. 4. The external excitation is generated by a physical stochastic model for earthquake ground motion process [27], which is physically based on the seismic source-path-site physical mechanism. The basic parameters of this model are A0, τ, ξg and ωg, with respect to the amplitude, the Brune source factor, the equivalent damping ratio and the equivalent predominant circular frequency, respectively. For more details, see [27]

Concluding remarks

Practical mechanical properties of concrete structures are usually correlated, and may have unignorable effects on stochastic responses of structures. For this purpose, a physically-guided data-driven methodology of capturing correlation configuration of basic random variables and the probability density evolution method is synthesized in this paper to implement stochastic dynamic response of concrete structures with dependent random parameters. The main findings include:

(1) The

Acknowledgments

The supports of the National Natural Science Foundation of China (Grant Nos. 51725804, 11672209 and 51538010), the NSFC-DFG joint project (Grant No. 11761131014), the Committee of Science and Technology of Shanghai China (Grant No. 18160712800), and the Research Fund for State Key Laboratories of Ministry of Science and Technology of China (Grant No. SLDRCE19-B-23) are highly appreciated.

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