A faster simulation method for the stochastic response of hysteretic structures subject to earthquakes

A semi-analytical forward-difference Monte Carlo simulation procedure is proposed for the determination of the lower order statistical moments and the joint probability density function of the stochastic response of hysteretic non-linear multi-degree-of-freedom structural systems subject to nonstationary gaussian white noise excitation, as an alternative to conventional direct simulation methods. The method generalizes the so-called Ermak-Allen algorithm developed for simulation applications in molecular dynamics to structural hysteretic systems. The proposed simulation procedure rely on an assumption of local gaussianity during each time step. This assumption is tantamount to various linearizations of the equations of motion. The procedure then applies an analytical convolution of the excitation process, hereby reducing the generation of stochastic processes and numerical integration to the generation of random vectors only. Such a treatment offers higher rates of convergence, faster speed and higher accuracy. The procedure has been compared to the direct Monte Carlo simulation procedure, which uses a fourth-order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process. The considered system was a multi-dimenensional hysteretic shear frame, where the constitutive equation of the hysteretic shear forces are described by a bilinear hysteretic model. The comparisons show that significant savings in computer time and accuracy can be achieved.     

Earthquake response of hysteretic mass-column using non-Gaussian closure

A non-Gaussian closure scheme based on the Edgeworth expansion of the probability density function is used to study the response of a hysteretic structure under random parametric excitation. The system considered consists of a weightless mass supporting a concentrated mass and it is subjected to the vertical and horizontal components of the ground acceleration modeled as nonstationary Gaussian white noise processes. The material of the structure exhibits bilinear hysteretic behaviour. The equation governing the motion of the system is transformed into an Itô stochastic differential equation. A set of ordinary differential equations governing the response statistics are obtained. These form an infinite hierarchy of equations which must be truncated in order to solve for moments of any order. The Edgeworth expansion of the joint density is used to truncate this infinite hierarchy. Such a closure scheme appears desirable since for hysteretic systems an explicit expression of the probability density is required. A frequently used closure scheme based on Gaussian assumption underestimates the response. The non-Gaussian density can be used in reliability studies.     

A damage model for structures with degrading response

The paper presents a hysteretic damage model for the response simulation of structural components with strength and stiffness deterioration under cyclic loading. The model is based on 1D continuum damage mechanics and relates any 2 work‐conjugate response variables such as force‐displacement, moment‐rotation, or stress‐strain. The strength and stiffness deterioration is described by a continuous damage variable. The formulation uses a criterion based on the hysteretic energy and the maximum or minimum deformation for damage initiation with a cumulative probability distribution function for the damage evolution. A series of structural component response simulations showcase the ability of the model to describe different types of hysteretic behavior. The relation of the model's damage variable to the Park‐Ang damage index is also discussed. Because of its consistent and numerically robust formulation, the model is suitable for the large‐scale seismic response simulation of structural systems with strength and stiffness deterioration.     

Choice between series and parallel connections of hysteretic system and viscous damper for seismic protection of structures

Supplemental viscous damping devices are generally envisioned to be connected in parallel to the inelastic parent structure or hysteretic damping devices. This gives rise to higher base shear, and often greater ductility demand of the hysteretic system. The series connection of the viscous and hysteretic system (the inelastic structure or a damper) is an alternative approach. In this paper, comparisons between the series and parallel connections of the hysteretic system and viscous dampers are done through response spectra analyses of single degree of freedom structures. Ductility demand of the hysteretic system and the total base shear are chosen as the response quantities. For the series model, a semi‐implicit solution scheme for classical Maxwell model is modified to include the inelasticity of the time‐independent hysteretic spring. It is observed that the series connection of the 2 dampers gives lower base shear than does the parallel connection. For long‐period and low‐damping structures, the ductility demand of the hysteretic system in series connection is higher than that in parallel connection. Increasing the viscous damping in series connection reduces the ductility demand substantially, lower than that obtained in parallel connection. Practical methods for implementing the series and parallel connections, in line with roof isolation, are also suggested.     

An identification methodology for a class of hysteretic structures

This paper presents a method of identification for determining non-linear dynamic models for certain hysteretic structures. Particular attention is given to modelling and identifying the hysteretic behaviour of structures from strong-motion earthquake data. In this method, the response is separated into mode-like components which are analogous to those of a linear system. Based on modelling of the generalized restoring force of each mode-like component, both non-hysteretic and hysteretic non-linear models are incorporated into the general methodology. A non-hysteretic model provides an initial estimate for a final hysteretic model. The approach is applicable even when data are available from only a small number of locations in the structure. The structural model identified from this method provides a means to predict the response to future events and, ultimately, to examine the damage to a structure as a result of an earthquake.     

Non-linear seismic behavior of structures with limited hysteretic energy dissipation capacity

This paper investigates the non-linear seismic behavior of structures such as slender unreinforced masonry shear walls or precast post-tensioned reinforced concrete elements, which have little hysteretic energy dissipation capacity. Even if this type of seismic response may be associated with significant deformation capacity, it is usually not considered as an efficient mechanism to withstand strong earthquakes. The objective of the investigations is to propose values of strength reduction factors for seismic analysis of such structures. The first part of the study is focused on non-linear single-degree-of-freedom (SDOF) systems. A parametric study is performed by computing the displacement ductility demand of non-linear SDOF systems for a set of 164 recorded ground motions selected from the European Strong Motion Database. The parameters investigated are the natural frequency, the strength reduction factor, the post-yield stiffness ratio, the hysteretic energy dissipation capacity and the hysteretic behavior model (four different hysteretic models: bilinear self-centring, with limited or without energy dissipation capacity, modified Takeda and Elastoplastic). Results confirm that the natural frequency has little influence on the displacement ductility demand if it is below a frequency limit and vice versa. The frequency limit is found to be around 2 Hz for all hysteretic models. Moreover, they show that the other parameters, especially the hysteretic behavior model, have little influence on the displacement ductility demand. New relationships between the displacement ductility demand and the strength reduction factor for structures having little hysteretic energy dissipation capacity are proposed. These relationships are an improvement of the equal displacement rule for the considered hysteretic models. In the second part of the investigation, the parametric study is extended to multi-degree-of-freedom (MDOF) systems. The investigation shows that the results obtained for SDOF systems are also valid for MDOF systems. However, the SDOF system overestimates the displacement ductility demand in comparison to the corresponding MDOF system by approximately 15%.     

Dynamic response of non-linear building-foundation systems

An analysis is made of the steady-state response of a bilinear hysteretic structure supported on the surface of a viscoelastic half-space. The method of equivalent linearization is used to solve the equations of motion, and simplified approximate formulas are obtained for the fundamental resonant frequency of the system and for an effective critical damping ratio. Numerical results indicate that for non-linear hysteretic structures compliance of the soil foundation may lead to larger displacements than would occur if the base were rigid. This behaviour differs from that generally observed for linear systems, for which the effect of soil-structure interaction is to reduce the rigid-base response.     

Correlation failure analysis of an uncertain hysteretic vibration system

In this paper, a numerical method for correlation sensitivity analysis of a nonlinear random vibration system is presented. Based on the first passage failure model, the probability perturbation method is employed to determine the statistical characteristics of failure modes and the correlation between them. The sensitivity of correlation between failure modes with respect to random parameters characterizing the uncertainty of the hysteretic loop is discussed. In a numerical example, a two-DOF shear structure with uncertain hysteretic restoring force is considered. The statistical characteristics of response, failure modes and the sensitivity of random hysteretic loop parameters are provided, and also compared with a Monte Carlo simulation.     

Nonstationary frequency effect of the ground motion upon the strength levels of the shear type RC structures

In order to investigate the influence of nonstationary frequency content variations, nonlinear response analysis for origin-oriented type hysteretic model is carried out and results are presented by applying two different S_I and S_II ensembles with varied frequency contents. The theoretical background of strong motion simulation is explained briefly depending on frequency parameters of stiff soil data. Based on models and applying Monte-Carlo's simulation mean values and standard deviations of displacement ductility demands for both ensembles are calculated. Required strength levels are investigated in probabilistic sense adopting light and heavy (but controlled) damage criteria. It is concluded that the nonlinear behaviour of the structure is greatly influenced by the nonstationary frequency content deviations as being shown that larger strength requirements for the S_I type strong ground motion ensemble than the S_II, although both ensembles have almost the same acceleration response spectra.     

Tuned inerter dampers with linear hysteretic damping

This paper explores the influence of linear hysteretic damping on the performance of passive tuned-inerter devices. An inerter is a device that produces a force proportional to the relative acceleration across its two terminals; devices incorporating inerters have received widespread attention in the earthquake engineering community, because they offer the ability to improve the seismic response of structures. However, the majority of this research has assumed that the damping components within the tuned-inerter device exhibit viscous, rather than hysteretic, damping. This restriction imposes an essential question on how the hysteretic damping model will change the performance of the device compared with the viscous damping model. It is shown that the response of viscous and hysteretic inerter systems have significant differences in displacement amplitude due to the frequency dependency of the damping. Therefore, a new formulation for obtaining the optimum loss factor of the hysteretic damping in the inerter system is proposed. Next, the challenges associated with accurately predicting the time-response of a hysteretically damped system are discussed. A numerical time-integration method is extended to address these challenges, using a new formulation that has the benefit of being broadly applicable to multidegree-of-freedom hysteretic linear systems and nonstationary random signals. The results show that the earthquake responses from the hysteretic damping model can differ significantly from the ones obtained via the viscous model.     

A Wavelet-based Seismogram Inversion Algorithm for the In Situ Characterization of Nonlinear Soil Behavior

We present a full waveform inversion algorithm of downhole array seismogram recordings that can be used to estimate the inelastic soil behavior in situ during earthquake ground motion. For this purpose, we first develop a new hysteretic scheme that improves upon existing nonlinear site response models by allowing adjustment of the width and length of the hysteresis loop for a relatively small number of soil parameters. The constitutive law is formulated to approximate the response of saturated cohesive materials, and does not account for volumetric changes due to shear leading to pore pressure development and potential liquefaction. We implement the soil model in the forward operator of the inversion, and evaluate the constitutive parameters that maximize the cross-correlation between site response predictions and observations on ground surface. The objective function is defined in the wavelet domain, which allows equal weight to be assigned across all frequency bands of the non-stationary signal. We evaluate the convergence rate and robustness of the proposed scheme for noise-free and noise-contaminated data, and illustrate good performance of the inversion for signal-to-noise ratios as low as 3. We finally employ the proposed scheme to downhole array data, and show that results compare very well with published data on generic soil conditions and previous geotechnical investigation studies at the array site. By assuming a realistic hysteretic model and estimating the constitutive soil parameters, the proposed inversion accounts for the instantaneous adjustment of soil response to the level and strain and load path during transient loading, and allows results to be used in predictions of nonlinear site effects during future events.     

岩石滞后非线性弹性响应的物理模型

介绍了岩石滞后非线性弹性的几个物理模型。赫兹颗粒接触模型是具有多尺度和滞后特性的经典模型,它预测了岩石中强烈的非线性;软的粘结系统几乎决定了岩石的力学性质,粘结系统中的流体对非线性响应的贡献特别显著,但是目前还没有搞清楚粘结系统和孔隙流体究竟是如何影响非线性响应的;GL模型是一个基于金属位错的物理模型,这是滞后动力行为方面一个开拓性的微观模型;PM模型是一个基于岩石细观尺度的唯象模型,它对理解岩石滞后非线性的机制和尺度是很重要的。     

Time‐domain viscoelastic analysis of earth structures

In this paper two causal models that approximate the nearly frequency‐independent cyclic behaviour of soils are analysed in detail. The study was motivated by the need to conduct time‐domain viscoelastic analysis on soil structures without adopting the ad hoc assumption of Rayleigh damping. First, the causal hysteretic model is introduced in which its imaginary part is frequency independent the same way that is the imaginary part of the popular non‐causal constant hysteretic model. The adoption of an imaginary part that is frequency independent even at the zero‐frequency limit, in conjunction with the condition that the proposed model should be causal, yields a real part that is frequency dependent and singular at zero frequency. The paper shows that the causal hysteretic model, although pathological at the static limit, is the mathematical connection between the non‐causal constant hysteretic model and the physically realizable Biot model. The mathematical structure of the two causal models is examined and it is shown that the causal hysteretic model is precisely the high‐frequency limit of the Biot model. Although both models have a closed‐form time‐domain representation, only the Biot model is suitable for a time‐domain viscoelastic analysis with commercially available computer software. The paper demonstrates that the simplest, causal and physically realizable linear hysteretic model that can approximate the cyclic behaviour of soil is the Biot model. The proposed study elucidates how the dynamic analysis of soil structures can be conducted rigorously in terms of the viscoelastic properties of the soil material and not with the ad hoc Rayleigh damping approach which occasionally has been criticized that tends to overdamp the higher vibration modes. The study concludes that under pulse‐type motions the Rayleigh damping approximation tends to overestimate displacements because of the inappropriate viscous type of dissipation that is imposed. Under longer motions that induce several cycles, the concept of equivalent viscous damping is more appropriate and the Rayleigh damping approximation results to a response that is comparable to the response computed with a rigorous time‐domain viscoelastic finite element analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Seismic performance of cable-sliding friction bearing system for isolated bridges

During past strong earthquakes, highway bridges have sustained severe damage or even collapse due to excessive displacements and/or very large lateral forces. For commonly used isolation bearings with a pure friction sliding surface, seismic forces may be reduced but displacements are often unconstrained. In this paper, an alternative seismic bearing system, called the cable-sliding friction bearing system, is developed by integrating seismic isolation devices with displacement restrainers consisting of cables attached to the upper and lower plates of the bearing. Restoring forces are provided to limit the displacements of the sliding component. Design parameters including the length and stiffness of the cables, friction coefficient, strength of the shear bolt in a fixed-type bearing, and movements under earthquake excitations are discussed. Laboratory testing of a prototype bearing subjected to vertical loads and quasi-static cyclic lateral loads, and corresponding numerical finite element simulation analysis, were carried out. It is shown that the numerical simulation shows good agreement with the experimental force-displacement hysteretic response, indicating the viability of the new bearing system. In addition, practical application of this bearing system to a multi-span bridge in China and its design advantages are discussed.     

An analytical solution for the probabilistic response of sdof non-linear random systems subjected to variable amplitude cyclic loading

Design for a specific ductile failure mode is assuming a rǒle of increasing importance for earthquake-resistant structures. This necessitates an accurate assessment of the distribution of overstrength in the structure, in order that the predefined failure mode can be realized. Consequently, the variability of the response for a given variability in the salient material properties, such as yield strength for steel structures, should be assessed and accounted for. In this paper an analytical method is proposed for the evaluation of the probability density function of the response of a single-degree-of-freedom hysteretic system with random parameters subject to a variable amplitude cyclic load history. A simple algorithm is derived which may be used to obtain the system response as a function of the system parameters. This response function may then be used to evaluate the displacement response probability density function when given the probability density function of the system parameters. Results derived from this procedure are verified against Monte Carlo simulation. It is shown that accurate response statistics are obtained at a fraction of the computing cost of simulation techniques.     

Response of a non-linear system with restoring forces governed by fractional derivatives—Time domain simulation and statistical linearization solution

In this paper, the random response of a non-linear system comprising frequency dependent restoring force terms is examined. These terms are accurately modeled in seismic isolation and in many other applications using fractional derivatives. In this context, an efficient numerical approach for determining the time domain response of the system to an arbitrary excitation is first proposed. This approach is based on the Grunwald–Letnikov representation of a fractional derivative and on the well-known Newmark numerical integration scheme for structural dynamic problems. Next, it is shown that for the case of a stochastic excitation, in addition to the time domain solutions, a frequency domain solution can be readily determined by the method of statistical linearization. The reliability of this solution is established in a Monte Carlo simulation context using the herein adopted time domain solution scheme. Furthermore, several related parameter studies are reported.     

A new methodology for energy-based seismic design of steel moment frames

A procedure is proposed whereby input and hysteretic energy spectra developed for single-degree-of-freedom (SDOF) systems are applied to multi-degree-of-freedom (MDOF) steel moment resisting frames. The proposed procedure is verified using four frames, viz., frame with three-, five-, seven- and nine-stories, each of which is subjected to the fault-normal and fault-parallel components of three actual earthquakes. A very good estimate for the three- and five-story frames, and a reasonably acceptable estimate for the seven-, and nine-story frames, have been obtained. A method for distributing the hysteretic energy over the frame height is also proposed. This distribution scheme allows for the determination of the energy demand component of a proposed energy-based seismic design (EBSD) procedure for each story. To address the capacity component of EBSD, a story-wise optimization design procedure is developed by utilizing the energy dissipating capacity from plastic hinge formation/rotation for these moment frames. The proposed EBSD procedure is demonstrated in the design of a three-story one-bay steel moment frame.     

Seismic response of self‐centring hysteretic SDOF systems

The seismic response of single‐degree‐of‐freedom (SDOF) systems incorporating flag‐shaped hysteretic structural behaviour, with self‐centring capability, is investigated numerically. For a SDOF system with a given initial period and strength level, the flag‐shaped hysteretic behaviour is fully defined by a post‐yielding stiffness parameter and an energy‐dissipation parameter. A comprehensive parametric study was conducted to determine the influence of these parameters on SDOF structural response, in terms of displacement ductility, absolute acceleration and absorbed energy. This parametric study was conducted using an ensemble of 20 historical earthquake records corresponding to ordinary ground motions having a probability of exceedence of 10% in 50 years, in California. The responses of the flag‐shaped hysteretic SDOF systems are compared against the responses of similar bilinear elasto‐plastic hysteretic SDOF systems. In this study the elasto‐plastic hysteretic SDOF systems are assigned parameters representative of steel moment resisting frames (MRFs) with post‐Northridge welded beam‐to‐column connections. In turn, the flag‐shaped hysteretic SDOF systems are representative of steel MRFs with newly proposed post‐tensioned energy‐dissipating connections. Building structures with initial periods ranging from 0.1 to 2.0s and having various strength levels are considered. It is shown that a flag‐shaped hysteretic SDOF system of equal or lesser strength can always be found to match or better the response of an elasto‐plastic hysteretic SDOF system in terms of displacement ductility and without incurring any residual drift from the seismic event. Copyright © 2002 John Wiley & Sons, Ltd.     

Frequency‐dependent principal component analysis of multicomponent earthquake ground motions

Stochastic ground motion simulation techniques are becoming increasingly popular because of enhanced computation power enabling direct simulation of complex response quantities. Priestley process assumption is the most general approach for stochastic modeling of earthquake ground motion. However, a framework for multicomponent ground motion simulation using the general Priestley process assumption is not available. Multicomponent motions are useful especially when the correlation structure between them significantly influences the response. The present study proposes a framework for frequency‐dependent principal component analysis (PCA), which facilitates Priestley process–based simulation of multicomponent ground motions. The study focuses only on the frequency‐dependent PCA part, and the results show high dependency of the principal components/directions on the frequency bands of the signals. The present work also advocates that the frequency‐dependent PCA should be preferred to the conventional PCA as the former can address the issues related to the frequency‐independent uniform modulation associated with the latter.     

Midbroken reinforced concrete shear frames due to earthquakes. A hysteretic model to quantify damage at the storey level

A nonlinear hysteretic model for the response and local damage analyses of reinforced concrete shear frames subject to earthquake excitation is proposed, and, the model is applied to analyse midbroken reinforced concrete (RC) structures due to earthquake loads. Each storey of the shear frame is represented by a Clough and Johnston hysteretic oscillator with degrading elastic fraction of the restoring force. The local damage is numerically quantified in the domain [0,1] using the maximum softening damage indicators which are defined in closed form based on the variation of the eigenfrequency of the local oscillators due to the local stiffness and strength deterioration. The proposed method of response and damage analyses is illustrated using a sample 5 storey shear frame with a weak third storey in stiffness and/or strength subject to sinusoidal and simulated earthquake excitations for which the horizontal component of the ground motion is modeled as a stationary Gaussian stochastic process with Kanai-Tajimi spectrum, multiplied by an envelope function.