Probability density evolution analysis of stochastic seismic response of structures with dependent random parameters
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 and the compressive strength 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 and . In Chen et al. [6], the dependency between and is written as the following form of random function where is a random variable with zero mean and unity variance, and is independent to . The two weak nonlinear functions and 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 where 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 where is an m-dimensional state vector with the initial condition , denotes a state mapping and is an s-dimensional random vector with the joint PDF . For well-posed problems, the solution of Eq. (20) can be denoted by with its components for . 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 , , and , 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.
References (33)
- et al.
An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis
Probab. Eng. Mech.
(2010) - et al.
A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas
Probab. Eng. Mech.
(2019) - et al.
Bivariate distribution of shear strength parameters using copulas and its impact on geotechnical system reliability
Comput. Geotech.
(2015) - et al.
Copula-based approaches for evaluating slope reliability under incomplete probability information
Struct. Saf.
(2015) - et al.
Discussion on: moment methods for structural reliability
Struct. Saf.
(2003) - et al.
Stochastic damage model for concrete based on energy equivalent strain
Int. J. Solids Struct.
(2009) - et al.
A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters
Struct. Saf.
(2016) - et al.
Stochastic modeling of engineering dynamic excitations for stochastic dynamics of structures
Probab. Eng. Mech.
(2012) - et al.
The principle of preservation of probability and the generalized density evolution equation
Struct. Saf.
(2008) - et al.
Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures
Probab. Eng. Mech.
(2009)