An iterative equivalent linearization approach for stochastic sensitivity analysis of hysteretic systems under seismic excitations based on explicit time-domain method

doi:10.1016/j.compstruc.2020.106396

Computers & Structures

Volume 242, 1 January 2021, 106396

https://doi.org/10.1016/j.compstruc.2020.106396 Get rights and content

Highlights

•
A novel approach is proposed for nonlinear stochastic sensitivity analysis.
•
Explicit formulation for sensitivities of response statistics is developed.
•
The approach is high-efficient for nonstationary random vibration problems.
•
The proposed method is feasible for stochastic optimization of base-isolated systems.

Abstract

The sensitivity analysis of hysteretic systems under nonstationary random excitations is of great concern to the stochastic optimal design and control of structures. The equivalent linear equation of motion is first constructed for the hysteretic system by the equivalent linearization method (ELM), and the sensitivity equation of the equivalent linear system is then derived by the direct differentiation technique. These two equations are further combined into an overall equivalent linear equation, which depends on the statistical moments and sensitivities of responses of the system and should be solved on an iterative basis. The analysis problem is thus transformed to a series of nonstationary random vibration analyses of the iterative linearized systems, which can be solved with the explicit time-domain method (ETDM) recently proposed for linear random vibration problems. The ETDM has high computational efficiency owing to the explicit formulations of statistical moments of responses. Therefore, a numerical algorithm is developed by combining ELM and ETDM for efficient stochastic sensitivity analysis of hysteretic systems. A 100-degree-of-freedom hysteretic system is investigated to illustrate the accuracy and efficiency of the proposed method, and a 10-degree-of-freedom base-isolated system is further analyzed to show the feasibility of the proposed method for stochastic structural optimization.

Introduction

Sensitivity analysis of responses with respect to structural parameters is of great importance to structural optimization, structural health monitoring and model updating, etc. The early works [1], [2], [3] on the sensitivity analysis of structures mainly concentrated on deterministic dynamic loads. However, the deterministic sensitivity analysis cannot capture reasonable sensitivity results from a probabilistic point of view when structures are subjected to random excitations such as earthquake, wind and wave. Therefore, in this sense, sensitivity analysis methods should take into account the uncertainties of dynamic excitations.

The concept of stochastic sensitivity analysis, which is aimed at finding the sensitivities of statistical moments of structural responses with respect to structural parameters, was proposed by Szopa [4] and Socha [5]. Afterwards, Benfratello et al. [6] developed a time-domain approach for the sensitivity evaluation of linear structural systems subjected to stationary random excitations. Cacciola et al. [7] proposed a semi-analytical approach to investigate the sensitivities of random responses of both classically and non-classically damped linear systems under nonstationary random excitations. Chaudhuri and Chakraborty [8] dealt with the response sensitivity evaluation of linear structures in the frequency domain under nonstationary random excitations. Based on the pseudo-excitation method [9], Xu et al. [10] presented a new approach for the first and second order sensitivity analysis of the nonstationary random responses of coupled vehicle-bridge linear systems, and Liu [11] developed a numerical method for calculation of the sensitivity of the evolutionary response power spectral density functions of linear structures subjected to nonstationary random excitations. Motivated by the explicit time-domain method (ETDM) proposed by Su and Xu [12] for nonstationary random response analysis of linear structures, Hu et al. [13], [14] further extended this method to the sensitivity analysis of variance responses and dynamic reliability of linear structures subjected to nonstationary random excitations. More recently, Chun et al. [15], Zhu et al. [16] and Gomez and Spencer [17] derived the sensitivities of stationary random responses of linear systems in different ways and further employed the sensitivity results for topology optimization. More research on the sensitivity analysis of linear systems under random excitations can be found in [18], [19].

As compared with the research on the sensitivity analysis of linear systems under random excitations, only a few works were presented in the literatures on the sensitivity analysis of stochastically excited nonlinear systems. As a pioneer in the field of stochastic sensitivity analysis of nonlinear systems, Socha [5] studied the response sensitivity of a single-degree-of-freedom (SDOF) Duffing oscillator and a SDOF Van der Pol oscillator under stationary white noise and colored noise excitations based on the moment equation method. Later, Socha and Zasucha [20] derived the approximate analytical sensitivities with respect to model parameters for the stationary responses of a SDOF nonlinear hysteretic system subjected to Gaussian white noise excitation via the statistical linearization approach. Socha and Soong [21] discussed the relationship between sensitivity analysis and equivalent linearization as applied to an oscillator with an odd order polynomial stiffness under stationary Gaussian white noise excitation. As can be seen from the above literatures, these works are limited to SDOF nonlinear systems with stationary random excitations.

The main purpose of this paper is to present an explicit time-domain approach for sensitivity analysis of multi-degree-of-freedom (MDOF) nonlinear systems under nonstationary seismic excitations. The recently proposed ETDM [12], [13], [14] has proven to be an efficient method for nonstationary stochastic response and sensitivity analysis of linear systems owing to the use of the explicit time-domain expressions of dynamic responses and their sensitivities. And for nonlinear systems, the ETDM has been used in conjunction with the equivalent linearization method (ELM) for the nonstationary response analysis of Duffing systems and hysteretic systems with high efficiency [22]. On this basis, in the present study, the ETDM-based ELM is further extended to the sensitivity analysis of nonstationary stochastic responses of hysteretic systems under seismic excitations. For each time instant, an overall equivalent linear equation consisting of the equivalent linear equation of motion and the corresponding sensitivity equation as well for the original hysteretic system is first derived based on the statistical linearization technique. Then, the above overall equivalent linear equation is solved on an iterative basis, in which the ETDM is used to conduct the series of nonstationary stochastic sensitivity analyses of the iterative linearized systems. A 100-degree-of-freedom hysteretic system is investigated to illustrate the accuracy and efficiency of the proposed method, and a 10-degree-of-freedom base-isolated system is further analyzed to show the feasibility of the proposed method for stochastic structural optimization.

Section snippets

Equivalent linear system of hysteretic system under nonstationary seismic excitation

Various methods have been developed for random vibration analysis of nonlinear systems, such as the Fokker-Planck-Kolmogorov equation method [23], the stochastic averaging method [24], the moment equation method [25], the ELM [26], [27], the probability density evolution method [28] and the Wiener path integral method [29]. Among them, the ELM has gained wide popularity for its versatility in engineering application, while most of the other methods are limited to the size of the problem and

ETDM-based ELM for stochastic sensitivity analysis of hysteretic systems

Given ${\tilde{C}}_{e} (τ)$ and ${\tilde{K}}_{e} (τ)$ with previous iterative values of statistical moments and sensitivities, Eq. (15) is in the same form as the equation of motion for a linear system. Therefore, Eq. (15) can be solved using linear random vibration methods. The ETDM has been found to be an efficient approach for nonstationary random vibration problems of linear systems owing to the use of explicit time-domain expressions of dynamic responses [12], [22]. In this section, the ETDM is incorporated into the

Numerical examples

Two numerical examples are investigated in this section. In the first example, different methods, including the ETDM-based ELM, the PSM-based ELM and the Monte-Carlo simulation (MCS), are used to obtain the statistical moments and sensitivities of responses, through which the accuracy and efficiency of the proposed method are validated. In the second example, the proposed ETDM-based ELM is further used to conduct the stochastic optimal design of base isolators so as to show the feasibility of

Conclusions

An efficient time-domain approach has been proposed for the stochastic sensitivity analysis of hysteretic systems subjected to nonstationary seismic excitations in the frame of ELM with fusion of ETDM. For each time instant, the equivalent linear equation of motion is first established by the statistical linearization technique, and the corresponding sensitivity equation is then determined analytically through direct differentiation. For convenience of formulation and direct use of the existing

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research is funded by the National Natural Science Foundation of China (51678252) and the Science and Technology Program of Guangzhou, China (201804020069).

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An iterative equivalent linearization approach for stochastic sensitivity analysis of hysteretic systems under seismic excitations based on explicit time-domain method

Highlights

Abstract

Introduction

Section snippets

Equivalent linear system of hysteretic system under nonstationary seismic excitation

ETDM-based ELM for stochastic sensitivity analysis of hysteretic systems

Numerical examples

Conclusions

Declaration of Competing Interest

Acknowledgements

Comput Methods Appl Mech Eng

Int J Solids Struct

J Sound Vib

J Sound Vib

Comput Struct

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

Comput Methods Appl Mech Eng

Comput Struct

Finite Elem Anal Des

J Sound Vib

Struct Saf

Soil Dyn Earthquake Eng

Probab Eng Mech

J Sound Vib

Int J Non Linear Mech

J Sound Vib

Struct Saf

Probab Eng Mech

Probab Eng Mech