Non-stationary Seismic Excitations Research Articles

Survival probability surfaces of hysteretic fractional order structures exposed to non-stationary code-compliant stochastic seismic excitation

A novel first-passage probability stochastic incremental dynamics analysis (SIDA) methodology tailored for hysteretic fractional order structural systems under a fully non-stationary seismic excitation vector consistently designated with contemporary aseismic codes provisions (e.g., Eurocode 8) is developed. Specifically, the vector of the imposed seismic excitations is characterised by evolutionary power spectra that stochastically align with aseismic codes elastic response acceleration spectra, defined for specified modal damping ratios and scaled ground accelerations. Leveraging the concepts of stochastic averaging and statistical linearization, the approximative non-stationary response displacement joint probability density function (PDF) is derived, retaining the particularly coveted attribute of computational efficacy. Subsequently, the coupling with the survival probability model allows for the efficient determination of the response first-passage time probability density surfaces and the survival probability surfaces across various limit-state rules and scalable intensity measures. The first-passage time probability serves as a robust engineering demand parameter, effectively monitoring structural behaviour by considering both intensity and timing information, while inherently aligned with pertinent limit-state requirements. Notably, the associated low computational cost and the ability to handle a wide range of complex nonlinear/hysteretic structural behaviours, coupled with its compliance with modern aseismic codes, underscore its potential for applications in the fields of structural and earthquake engineering. A nonlinear system endowed with fractional derivative elements is used to exemplify the method’s reliability. The accuracy of the proposed method is validated in a Monte Carlo-based context, conducting nonlinear response time–history analyses with an extensive ensemble of accelerograms compatible with Eurocode 8 response acceleration spectra.

Soil-dependent stochastic optimization of base isolation system with negative stiffness inerter damper subjected to non-stationary seismic excitation

The negative stiffness inerter damper (NSID) has demonstrated excellent performance in mitigating vibrations in frame structures. However, analysis and optimization of BIS-NSID considering both soil conditions and non-stationary seismic excitation are rarely reported. This study proposes a multi-objective performance optimization procedure (MPOP) to investigate the optimal parameters of BIS-NSID under non-stationary seismic excitation considering firm, medium and soft soil conditions. The non-stationary seismic excitation is expressed as Clough-Penzien model with time modulation function. The analytical solution for the root mean square responses of BIS-NSID are derived considering non-stationary seismic excitation. The pareto optimal fronts is utilized to minimize the BIF displacement and superstructure acceleration. The optimal design is further validated under artificial seismic records. The results illustrate the validity of the MPOP and highlight the advantage of the BIS-NSID for all soil conditions. The demand for inertance and damping values are greater in soft soil conditions, compared to firm and medium soil condition.

An efficient approach for the equivalent linearization of frame structures with plastic hinges under nonstationary seismic excitations

An efficient approach for the equivalent linearization of frame structures with plastic hinges under nonstationary seismic excitations

Estimation of Extreme Value Distribution and Probability of Minor Failures in Rockfill Dam Response to Non-Stationary Seismic Excitation

Estimation of Extreme Value Distribution and Probability of Minor Failures in Rockfill Dam Response to Non-Stationary Seismic Excitation

Multi-Performance Economical Optimization of Base Isolation System with Tuned Inerter Negative Stiffness Damper Subjected to Non-Stationary Seismic Excitation

Base isolation proves to be an effective vibration control strategy for buildings. However, the base-isolation floor (BIF) may undergo substantial displacements. To improve the seismic performance of base-isolation system, this paper proposes a multi-performance economical optimization procedure (MPEOP) for exploring the optimal parameters of base isolation system with tuned inerter negative stiffness damper (BIS-TINSD) subjected to non-stationary seismic excitation. The simplified analysis model for BIS-TINSD is developed, followed by the formulation of the state-space representation. The non-stationary seismic excitation is represented as a stationary Gaussian process with a time-modulating function based on the Clough–Penzien spectrum, and combining the equations of motion with seismic excitation results in the augmented state-space representation of structure-damper-excitation, followed by the formulation of differential Lyapunov equation. The multi-objective performance economical index is proposed, in which the performance optimization objective is defined as a weighted combination of base isolation floor displacement and superstructure acceleration, and the economic optimization objective is set as the support stiffness. The Pareto optimal fronts (POF) are adopted to deal with optimal objectives. The examination of optimal design parameters is extended under real earthquake records. The results demonstrate the effectiveness of the MPEOP, and indicate the advantages of the BIS-TINSD compared with the inerter amplify damper (IAD) and negative stiffness inerter damper (NSID).

Reliability-based topology optimization of fractionally-damped structures under nonstationary random excitation

This contribution proposes an effective reliability-based topology optimization (RBTO) framework for the layout of viscoelastic dampers of energy-dissipating structures under nonstationary seismic excitation, in which the constitutive behavior of viscoelastic damper is described by the novel fractional derivative models. To formulate the RBTO problem, the installation number of fractional viscoelastic dampers is minimized under the constraints of dynamic reliability, and the design variables are chosen as the damper parameters and the existence variables of the potential fractional viscoelastic dampers. The explicit responses are first established for the fractionally-damped structure, and the explicit response sensitivities with respect to the design variables are further derived through the adjoint variable method (AVM). On this basis, an explicit time-domain method (ETDM) is developed for explicit formulation of the statistical moments of responses and the moment sensitivities of the fractionally-damped structure, which are further employed to analytically derive the first-passage dynamic reliabilities and the reliability sensitivities of the structure based on the classical level-crossing process theory. Finally, the RBTO problem for the layout of fractional viscoelastic dampers is solved with the gradient-based method of moving asymptotes (MMA), in which the existence variables of fractional viscoelastic dampers can be driven to binary solutions by the solid isotropic material with penalization (SIMP) technique. An engineering application involving a building frame structure installed with fractional viscoelastic dampers is presented to validate the feasibility of the proposed ETDM-based RBTO framework.

An efficient time-variant reliability analysis strategy embedding the NARX neural network of response characteristics prediction into probability density evolution method

Stochastic response analysis and first-passage reliability evaluation of multi-degree of freedom nonlinear systems subject to non-stationary seismic excitation have been a critical issue in the field of compound random vibrations, yet remain unsolved. This study extends the time-varying reliability analysis method based on the probability density evolution method, making it applicable to the problem of stochastic response analysis and first-passage reliability evaluation of complex engineering structures. To efficiently represent the response of representative samples, a nonlinear autoregressive with exogenous inputs neural network model is introduced in this study. After comparing three different optimization methods, the neural network model based on the Bayesian regularization algorithm is selected as the predictor for the response of complex nonlinear systems, enabling efficient stochastic response analysis and time-varying reliability evaluation. To this end, leveraging the high efficiency of the probability density evolution method, the analysis of time-varying reliability of large-scale structures has been successfully conducted. Furthermore, the proposed method can be easily expanded to dynamic reliability evaluation problems represented by extreme value theory. Three numerical examples are quoted to demonstrate the applicability and advantages of the proposed method in the field of first-passage and extreme-value problems of systems, which are also compared with PCE, SVR, and Kriging. The results from these numerical examples indicate that the proposed method can effectively reduce the required calculational cost for the reliability analysis of complex structures while maintaining high analytical accuracy.

Bayes-informed mixture distribution for the EVD estimation and dynamic reliability analysis

In this paper, a Bayes-informed mixture distribution is proposed to capture the underlying implicit extreme value distribution of a nonlinear dynamic system subjected to stochastic seismic excitation. First, three possible candidate distributions considering the properties of the extreme value distribution are selected, i.e., Gumbel, inverse Gaussian and Gamma distributions. Then, the mixture of these possible candidate distributions is conducted from Bayesian perspective by using the proposed moment-generating function-guided likelihood function. The benchmark data for constructing the likelihood function is the function value of the moment-generating function related to the unknown extreme value distribution, which can be calculated by the Latinized partially stratified sampling technique. The predicted data can be calculated by the mixed moment-generating function of the mixture distribution. Finally, the mixture distribution’s unknown parameters can be calibrated via the Bayesian inference and transitional Markov chain Monte Carlo so that the calibrated mixture distribution can be a representation of the implicit extreme value distribution. Regarding the dynamic reliability analysis, the first passage failure probability can be readily obtained from the mixture distribution. Three nonlinear dynamic systems subjected to the fully non-stationary seismic excitations, i.e., a single-degree-of-freedom system, a 3-D reinforced concrete frame structure and a practical highway bridge are investigated to validate the efficiency and accuracy of the proposed method for extreme value distribution estimation.

Optimum design and performance of a base-isolated structure with tuned mass negative stiffness inerter damper

In order to increase the efficiency of the structures to resist seismic excitation, combinations of inerter, negative stiffness, and tuned mass damper are used. In the present work, the optimum tuning frequency ratio and damping of the tuned mass negative stiffness damper-inerter (TMNSDI) for the base-isolated structure were determined by employing the numerical searching technique under filtered white-noise earthquake excitation and stationary white noise. The energy dissipation index, the absolute acceleration, and the relative displacement of the isolated structure were considered as the optimum parameters, obtained by their maximization. Evaluations of base-isolated structures with and without TMNSDI under non-stationary seismic excitations were investigated. The efficiency of the optimally designed TMNSDI for isolated flexible structures in controlling seismic responses (pulse-type, and real earthquakes) were evaluated in terms of acceleration and displacement. A dynamic system was used for deriving the tuning frequency and tuned mass negative stiffness damper inerter (TMNSDI) for white noise excitation by using explicit formulae of the curve fitting method. The proposed empirical expressions, for design of base-isolated structures with supplementary TMNSDI, showed lesser error. Fragility curve results and story drift ratio indicate reduction in seismic response by 40% and 70% in base-isolated structure using TMNSDI.

Stochastic Optimization and Sensitivity Analysis of the Combined Negative Stiffness Damped Outrigger and Conventional Damped Outrigger Systems Subjected to Nonstationary Seismic Excitation

In recent years, various kinds of damped outrigger systems, particularly with the negative stiffness devices, have been shown to effectively suppress excessive vibration induced by earthquake and wind. To gain insight into such systems, this paper proposes a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. The simplified analysis model of various combinations of damped outriggers is developed, and the state-space representation of outrigger systems is then formulated. The nonstationary seismic excitation is modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following Clough–Penzien spectrum, and combining equations of motions of outrigger systems and seismic excitation gives rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. The optimal designs of damped outriggers are defined via the solution of the differential Lyapunov equation. The multiobjective optimization with Pareto optimal fronts is adopted to deal with conflicting objectives between harmful interstory drift and floor absolute acceleration. The sensitivity of negative stiffness devices on optimal objectives is also investigated. The optimal designs are further examined under real typical and near-fault earthquake records. The results demonstrate the efficacy of the proposed method and provide insight into the systems subjected to nonstationary seismic excitation. While the purely NSDO systems could have better performance with adequate negative stiffness devices, the proposed combined NSDO and CDO systems prove to be more effective with limited negative stiffness devices and thus provide more design flexibilities tailored to different application situations.

Stochastic Analysis of a Large-Span Continuous Girder High-Speed Railway Bridge under Fully Non-Stationary Earthquake

The main objective of the present work is to evaluate the stochastic seismic response of a large-span continuous girder high-speed railway (CGHSR) bridge subjected to fully non-stationary seismic excitation. The stochastic ground motion is determined by a fully non-stationary ground acceleration model with both the time-varying intensity and frequency content. The stochastic seismic analysis of the large-span CGHSR bridge under a tri-directional earthquake, taking into consideration the wave passage effect, is performed using the pseudo excitation method (PEM) in frequency domain and the Monte Carlo sampling (MCS) method in time domain. A framework is proposed to determine the pseudo static displacement for the PEM under fully non-stationary seismic excitation, using the proper orthogonal decomposition (POD). A comparative study on the stochastic seismic response of the bridge is conducted using both the fully and modulated non-stationary ground acceleration. The results show that the structural nodal displacements estimated by the PEM agree well with those obtained by the MCS method. The proposed POD-based method works well in estimating the structural quasi-static displacement. Ignoring the frequency non-stationarity of the earthquake may underestimate the quasi-static displacement caused by the surface waves.

Simultaneous layout and size optimization of nonlinear viscous dampers for frame buildings under stochastic seismic excitation

Viscous dampers, as effective energy-dissipation devices, have been widely used for seismic mitigation of building structures. Due to the intrinsic uncertainties of structural parameters and earthquake ground motions, to obtain an optimal performance of structural control, the stochastic design optimization of viscous dampers for buildings is essential. This paper establishes a new dynamic reliability-based optimization framework to address the simultaneous layout and size design of nonlinear viscous dampers in frame buildings, considering the dual randomness of both nonstationary seismic excitations and uncertain structural model parameters. Firstly, a mixed integer optimization problem for nonlinear viscous dampers, including discrete and continuous design variables, is formulated, and the design target is to minimize the cost of the dampers subjected to performance constraints on dynamic reliability of building structures. Then, sequential approximate mixed integer programming with trust region is proposed to solve the optimization problem. Moreover, direct probability integral method is suggested to assess the dynamic reliability and its sensitivity with respect to design variables. Finally, accounting for both stochastic nonstationary seismic excitations and random structural parameters, the optimized results of two numerical examples indicate that the proposed method is a competitive choice for realizing the layout and size optimization of limiting types of nonlinear viscous dampers in frame buildings. To ensure the same target performance of buildings in terms of the probability of exceeding the target value of inter-story drift, the larger peak ground acceleration of stochastic ground motions is, the more cost of dampers is required for seismic protection of buildings.

Stochastic dynamic analysis of nonlinear MDOF systems under combined Gaussian and Poisson noise excitation based on DPIM

Random vibration analysis of structures subjected to combined Gaussian and Poisson white noise excitation is a challenging issue. In this paper, a novel direct probability integral method (DPIM) is suggested to address stochastic responses and dynamic reliability of nonlinear multi-degree-of-freedom (MDOF) systems under combined excitation. Firstly, probability density integral equation (PDIE) of MDOF system under combined excitation is derived based on the principle of probability conservation. Then, DPIM is proposed to achieve the probability density function of stochastic response by solving deterministic motion equation of MDOF system and PDIE in sequence. To solve PDIE, two techniques, i.e., partition of input probability space and smoothing of Dirac delta function, are introduced. From the perspective of probability conservation, furthermore, the equivalent relationship between the PDIE and the corresponding probability density differential equation of a Markov system under combined Gaussian and Poisson noise is established. Since Dirac delta function is analytically integrated as Heaviside function, the first-passage dynamic reliability of MDOF system under combined excitation is readily evaluated by introducing the extreme value mapping of stochastic response. Finally, two nonlinear MDOF systems, including multiple-span bridge under combined vehicle load and non-stationary seismic excitation, are solved. Results demonstrate that the proposed DPIM is effective for random vibration and dynamic reliability analyses of MDOF structures excited by combined Gaussian and Poisson noise, and considering the randomness of vehicle load and seismic excitation simultaneously in the bridge design benefits the safe operation of bridge.

Spectral decomposition-based explicit integration method for fully non-stationary seismic responses of large-scale structures

This paper presents a spectral decomposition-based explicit integration method for random vibration analysis of large-scale linear structures subjected to fully non-stationary seismic excitations. In this method, the seismic excitation process is decomposed using the spectral representation method, leading to a discrete representation of the excitation process in terms of an orthogonal random vector. Next, an explicit expression for the state response vector of structures with respect to the orthogonal random vector is derived by direct discretization and integration of the structural state equation, in which the time-dependent coefficient matrix can be obtained in a recursive manner. Since the state response vector contains only the displacement and velocity responses of structures, the finite-difference formulas are further used to obtain the explicit expression for the structural acceleration responses. These explicit expressions can be used not only to directly calculate the correlation matrix and evolutionary power spectral density of stochastic responses of structures, but also to efficiently evaluate the response samples required in the Monte Carlo simulation. For demonstration purposes, the proposed method is applied to the random vibration analysis of a coupled building system and a large-scale space-frame structure excited by two kinds of fully non-stationary seismic inputs; other existing methods are employed to illustrate the high accuracy and efficiency of the proposed method.

Stochastic sensitivity analysis of energy-dissipating structures with nonlinear viscous dampers by efficient equivalent linearization technique based on explicit time-domain method

Nonlinear fluid viscous dampers have been widely used in energy-dissipating structures due to their stable and high dissipation capacity and low maintenance cost. However, the literature on stochastic optimization of nonlinear viscous dampers under nonstationary excitations is limited. This paper is devoted to the stochastic response and sensitivity analysis of large-scale energy-dissipating structures equipped with nonlinear viscous dampers subjected to nonstationary seismic excitations. The analysis procedure is developed in the frame of the equivalent linearization method (ELM) in conjunction with the explicit time-domain method (ETDM). The equivalent linear system and the corresponding statistical moments of responses at a specific time instant are first obtained through a series of stochastic response analyses of the linearized systems. Then the sensitivities of the statistical moments of responses are determined via a series of stochastic sensitivity analyses of the equivalent linear system at the corresponding time instant. The above two iterative procedures are facilitated at high efficiency using ETDM with explicit formulations of the statistical moments of responses and the sensitivities of the statistical moments. This process is repeated for different time instants, and the time histories of the statistical moments and their sensitivities can be obtained. The stochastic response and sensitivity results are further utilized to conduct the stochastic optimal parametric design of the nonlinear viscous dampers. A one-storey building model equipped with a nonlinear viscous damper is analyzed to demonstrate the accuracy of the proposed method, and a suspension bridge with a main span of 1200 m equipped with 4 nonlinear viscous dampers is further investigated to illustrate the feasibility of the proposed method for stochastic optimal design of large-scale structures.

Topology optimisation of truss structures under non-stationary random seismic excitations with displacement and stress constraints

This paper studies topology optimisation problems of truss structures under non-stationary random seismic excitations considering displacement and stress constraints. The optimisation model of truss structures is formulated by taking volume minimisation with constraints on variances of displacements and stresses. The explicit time domain method (ETDM) is applied to analyse the stochastic responses of the structure and the sensitivities with respect to the design variables. Based on the sensitivity analysis, the nodal positions and the cross-sectional areas of the bar elements of the truss structures are optimised simultaneously by using the globally convergent method of moving asymptotes (GCMMA). Meanwhile, a scheme is proposed to delete zero bars and redundant nodes during the optimisation process. Finally, several topology optimisation examples of 2-D and 3-D truss structures illustrate the accuracy and computational efficiency of the proposed method.

An inerter‐equipped vibrating barrier for noninvasive motion control of seismically excited structures

The Vibrating Barrier (ViBa) is a large-scale oscillating mass-spring-damper unit contained in the ground and tuned to mitigate the motion of surrounding structures under earthquakeinduced ground shaking, without being directly in contact to them, through a structure-soilstructure interaction mechanism. Previous research showed that ViBa achieves significant seismic structural response reductions but, in doing so, it requires excessive vibrating mass, equal to the mass of the structure that aims to control or more, which hinders its practical applicability. This paper considers coupling ViBa with a grounded inerter acting as a mass amplifier to reduce ViBa mass/weight in suppressing seismically induced structural response. Attention is focused on structures amenable to modelling as single-degree-of freedom (SDOF) damped oscillators by establishing equations of motion of a 5-DOF dynamical system of a grounded inerter-equipped ViBa (IViBa) fused with a SDOF structure and accounting for soil structure interaction (SSI) effects due to soil compliance. Optimal closed-form H∞ and numerical H2 IViBa tuning are addressed minimizing the response of SDOF structure subject to harmonic resonant and to broadband/white base excitation, respectively. Numerical results pertaining to a small-scale physical ViBa prototype specimen are furnished quantifying the trade-off between IViBa mass and inertance considering non-rigid/compliant inerter-to-ground connectivity. Monotonic IViBa mass/weight reduction trend is found for fixed targeted structural performance under white stationary as well as colored non-stationary seismic excitation for increasing inertance and for sufficiently rigid inerter-to-ground connection. It is concluded that careful engineering design of the inerter-to-ground connection minimizing compliance is most critical in fully exploiting the beneficial effects of grounded inerter for mass/weight IViBa reduction facilitating, thus, its practical implementation.

Reliability analysis of nonlinear dynamic system with epistemic uncertainties using hybrid Kriging-HDMR

The small first passage probability of nonlinear dynamic structures under nonstationary seismic excitation implies a prohibitive computational demand because of the epistemic uncertainties rooted in structure system. In this work, a hybrid Kriging-high dimensional model representation (Kriging-HDMR) methodology for drastically simplifying this task is proposed. The epistemic uncertainties with well-defined bounds but no concrete distribution forms are addressed by interval model. Therefore, the first passage probability of the generic response process that the epistemic uncertainties involved, is represented by the conditional univariate extreme value distribution (EVD) of structural parameters with interval form. Then, a Kriging assisted HDMR together with third moment saddle point approximation (TMSA) is proposed to alleviate the computational burden and provides an accurate depiction of possible model outcome. The differential evolution (DE) interval optimization strategy is performed to accelerate the post process of searching the lower bound (LB) and upper bound (UB) of first passage probability with interval form. Finally, a nine-story shear frame with Bouc–Wen model is presented to demonstrate the accuracy and efficiency of proposed method.

Stochastic optimal design of nonlinear viscous dampers for large-scale structures subjected to non-stationary seismic excitations based on dimension-reduced explicit method

Nonlinear fluid viscous dampers have been widely used in energy-dissipation structures. This paper is devoted to the stochastic optimal design of viscous dampers for large-scale structures under non-stationary random seismic excitations. The optimization problem is formulated as the minimization of the standard deviation of a target displacement component subjected to the constraint on the standard deviations of damping forces of viscous dampers, and the method of moving asymptotes (MMA), a gradient-based optimization method, is employed to solve the optimization problem involved. An effective dimension-reduced explicit method is first proposed for fast nonlinear time-history analysis of structural responses and the corresponding sensitivity analysis with respect to the parameters of viscous dampers, in which only a small number of degrees of freedom associated with the viscous dampers need to be considered in the iteration scheme, leading to extremely low computational cost in the nonlinear analysis. Then the proposed dimension-reduced explicit method is further used to conduct sample analyses with high efficiency in Monte-Carlo simulation (MCS) so as to obtain the statistical moments of critical responses and the relevant moment sensitivities required in the process of optimal design. To demonstrate the feasibility of the proposed method, the stochastic optimal design of viscous dampers is carried out for a large-scale suspension bridge with a main span of 1688 m, and the mean peak values of critical responses with the optimal parameters of viscous dampers are finally obtained for the design purpose of the bridge.

Inelastic response analysis of bridges subjected to non-stationary seismic excitations by efficient MCS based on explicit time-domain method

This paper is devoted to stochastic response analysis of large-scale bridges with local plastic deformations under non-stationary random seismic excitations. A force-based fiber beam–column element is employed to simulate the inelastic behavior of bridges. By use of the explicit time-domain expressions of seismic responses, an effective dimension-reduced method is developed for fast inelastic time-history analysis of a structure with local nonlinearities. Only the nodal displacements that are associated with the inelastic elements need to be solved through iteration process, which leads to a significant reduction in computational cost for inelastic time-history analysis. With the merit of high efficiency, the dimension-reduced method is further used as a computational tool for repetitive inelastic time-history analyses in Monte Carlo simulation, and the statistical moments and mean peak values of critical responses can be readily achieved under random seismic excitations. To demonstrate the feasibility of the present approach, the nonlinear random vibration analysis of a long-span suspension bridge with a main span of 1200 m is carried out under rare earthquakes, in which the inelastic behaviors of the main towers are investigated with the use of inelastic fiber elements.