Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation

Nonlinear viscous dampers are supplemental devices widely used for enhancing the performance of structural systems exposed to seismic hazard. A rigorous evaluation of the effect of these damping devices on the seismic performance of a structural system should be based on a probabilistic approach and take into account the evolutionary characteristics of the earthquake input and of the corresponding system response. In this paper, an approximate analytical technique is proposed for studying the nonstationary stochastic response characteristics of hysteretic single degree of freedom systems equipped with viscous dampers subjected to a fully nonstationary random process representing the seismic input. In this regard, a stochastic averaging/linearization technique is utilized to cast the original nonlinear stochastic differential equation of motion into a simple first‐order nonlinear ordinary differential equation for the nonstationary system response variance. In comparison with standard linearization schemes, the herein proposed technique has the significant advantage that it allows to handle realistic seismic excitations with time‐varying frequency content. Further, it allows deriving a formula for determining the nonlinear system response evolutionary power spectrum. By this way, ‘moving resonance’ effects, related to both the evolutionary seismic excitation and the nonlinear system behavior, can be observed and quantified. Several applications involving various system and input properties are included. Furthermore, various response parameters of interest for the seismic performance assessment are considered as well. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed technique. Copyright © 2014 John Wiley & Sons, Ltd.     

Bivariate Gaussian mixture–based equivalent linearization method for stochastic seismic analysis of nonlinear structures

To address challenges in stochastic seismic analysis of nonlinear structures, this paper further develops a recently proposed Gaussian mixture–based equivalent linearization method (GM‐ELM). The GM‐ELM uses a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. Using properties of the Gaussian mixture model, GM‐ELM can decompose the non‐Gaussian response of a nonlinear system into multiple Gaussian responses of linear single–degree of freedom oscillators. With the set of the equivalent linear systems identified by GM‐ELM, response statistics as crossing rate and first‐passage probability can be computed conveniently using theories of linear random vibration analysis. However, the original version of GM‐ELM may lead to an inaccurate estimate because of the heuristic parameters of the linear system introduced to supplement insufficient information. To overcome this limitation and define unique equivalent linear systems, this paper proposes a further developed version of GM‐ELM, which uses a mixture of bivariate Gaussian densities instead of univariate models. Moreover, to facilitate the use of elastic response spectra for estimating the mean peak responses of a nonlinear structure, a new response spectrum combination rule is proposed for GM‐ELM. Two numerical examples of hysteretic structural systems are presented in this paper to illustrate the application of the bivariate GM‐ELM to nonlinear stochastic seismic analysis. The analysis results obtained by the bivariate GM‐ELM are compared with those obtained by the univariate GM‐ELM, the conventional equivalent linearization method, the tail equivalent linearization method, and Monte Carlo simulation. The supporting source code and data are available for download at https://github.com/yisangri/GitHub‐bGM‐ELM‐code.git     

Gaussian mixture–based equivalent linearization method (GM-ELM) for fragility analysis of structures under nonstationary excitations

Gaussian mixture–based equivalent linearization method (GM-ELM) is a recently developed stochastic dynamic analysis approach which approximates the random response of a nonlinear structure by collective responses of equivalent linear oscillators. The Gaussian mixture model is employed to achieve an equivalence in terms of the probability density function (PDF) through the superposition of the response PDFs of the equivalent linear system. This new concept of linearization helps achieve a high level of estimation accuracy for nonlinear responses, but has revealed some limitations: (1) dependency of the equivalent linear systems on ground motion intensity and (2) requirements for stationary condition. To overcome these technical challenges and promote applications of GM-ELM to earthquake engineering practice, an efficient GM-ELM-based fragility analysis method is proposed for nonstationary excitations. To this end, this paper develops the concept of universal equivalent linear system that can estimate the stochastic responses for a range of seismic intensities through an intensity-augmented version of GM-ELM. Moreover, the GM-ELM framework is extended to identify equivalent linear oscillators that could capture the temporal average behavior of nonstationary responses. The proposed extensions generalize expressions and philosophies of the existing response combination formulations of GM-ELM to facilitate efficient fragility analysis for nonstationary excitations. The proposed methods are demonstrated by numerical examples using realistic ground motions, including design code–conforming nonstationary ground motions.     

Analysis of structural fuzzy random seismic response

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Dynamic response analysis of nonlinear secondary oscillators to idealised seismic pulses

The paper deals with the seismic response analysis of nonlinear secondary oscillators. Bilinear, sliding and rocking single-degree-of-freedom dynamic systems are analysed as representative of a wide spectrum of secondary structures and nonstructural components. In the first stage, the equations governing their full dynamic interaction with linear multi-degree-of-freedom primary structures are formulated, and then conveniently simplified using primary-secondary two-degree-of-freedom systems and dimensionless coefficients. In the second stage, the cascade approximation is applied, whereby the feedback action of the secondary oscillator on the primary structure is neglected. Owing to the piecewise linearity of the secondary systems being considered, efficient semi-analytical and step-by-step numerical solutions are presented. The semi-analytical solutions allow the direct evaluation of the seismic response under pulse-type ground excitations and are also used to validate step-by-step numerical schemes, which in turn can be used for general-type seismic excitations. In the third stage, a set of decoupling criteria are proposed for the pulse-type base excitations, identifying the conditions under which a cascade analysis is admissible from an engineering standpoint. Finally, the influence and relative dependencies between the input parameters of the ground motion and the primary-secondary assembly are quantified on the response of the secondary systems through nonlinear floor response spectra, and general trends are identified and discussed.     

A new model for spectral velocity ordinates at long periods

The estimation of peak linear response via elastic design (response) spectra continues to form the basis of earthquake‐resistant design of structural systems in various codes of practice all over the world. Many response spectrum‐based formulations of peak linear response require an additional input of the spectral velocity (SV) ordinates consistent with the specified seismic hazard. SV ordinates have been conventionally approximated by pseudo spectral velocity (PSV) ordinates, which are close to the SV ordinates only over the intermediate frequency range coinciding with the velocity‐sensitive region. At long periods, PSV ordinates underestimate the SV ordinates, and this study proposes a formulation of a correction factor (>1) that needs to be multiplied by the PSV ordinates in order to close the gap between the two sets of ordinates. A simple model is proposed in the form of a power function in oscillator period to estimate this factor in terms of two governing parameters which are in turn estimated from two single‐parameter scaling equations. The parameters considered for the scaling equations are (1) the period at which the PSV spectrum is maximized and (2) the rate of decay of the pseudo spectral acceleration (PSA) amplitudes at long periods. For a given damping ratio, four regression coefficients are determined for the scaling equations with the help of 205 ground motions recorded in western USA. A numerical study undertaken with the help of several design PSA spectra and ensembles of spectrum‐compatible ground motions illustrates the effectiveness of the proposed correction factor, together with the proposed scaling models, in comparison with the PSV approximation in a variety of design situations. Both the input parameters mentioned above can be easily obtained from the specified design spectrum, and thus the proposed model is convenient to use.     

A statistical approach to determine 2D optimal equivalent linear parameters with application to earthquake engineering

The equivalent linearization method approximates the maximum displacement response of nonlinear structures through the corresponding equivalent linear system.By using the particle swarm optimization technique,a new statistical approach is developed to determine the key parameters of such an equivalent linear system over a 2D space of period and damping ratio.The new optimization criterion realizes the consideration of the structural safety margin in the equivalent linearization method when applied to the performance-based seismic design/evaluation of engineering structures.As an application,equations for equivalent system parameters of both bilinear hysteretic and stiffness degrading single-degree-offreedom systems are deduced with the assumption of a constant ductility ratio.Error analyses are also performed to validate the proposed approach.     

Reliability‐based control algorithms for nonlinear hysteretic systems based on enhanced stochastic averaging of energy envelope

Current reliability‐based control techniques have been successfully applied to linear systems; however, incorporation of stochastic nonlinear behavior of systems in such control designs remains a challenge. This paper presents two reliability‐based control algorithms that minimize failure probabilities of nonlinear hysteretic systems subjected to stochastic excitations. The proposed methods include constrained reliability‐based control (CRC) and unconstrained reliability‐based control (URC) algorithms. Accurate probabilistic estimates of nonlinear system responses to stochastic excitations are derived analytically using enhanced stochastic averaging of energy envelope proposed previously by the authors. Convolving these demand estimates with capacity models yields the reliability of nonlinear systems in the control design process. The CRC design employs the first‐level and second‐level optimizations sequentially where the first‐level optimization solves the Hamilton–Jacobi–Bellman equation and the second‐level optimization searches for optimal objective function parameters to minimize the probability of failure. In the URC design, a single optimization minimizes the probability of failure by directly searching for the optimal control gain. Application of the proposed control algorithms to a building on nonlinear foundation has shown noticeable improvements in system performance under various stochastic excitations. The URC design appears to be the most optimal method as it reduced the probability of slight damage to 8.7%, compared with 11.6% and 19.2% for the case of CRC and a stochastic linear quadratic regulator, respectively. Under the Kobe ground motion, the normalized peak drift displacement with respect to stochastic linear quadratic regulator is reduced to 0.78 and 0.81 for the URC and CRC cases, respectively, at comparable control force levels. Copyright © 2017 John Wiley & Sons, Ltd.     

非一致激励条件下非线性工程场地地震动相干函数研究

将土非线性分析的等效线性化方法与随机工程波动散射问题的求解方法相结合，建立了开放系统中非一致激励条件下考虑岩土介质非线性的工程场地地震动随机场数值模拟方法并研究了介质非线性对场地地震动相干性的影响。分析结果表明：与不考虑非线性相比，在地震过程中，场地土介质的非线性改变了复杂场地的局部特性从而导致场地地震动空间相干性的变化，因此有必要在场地地震动相干函数的分析中考虑场地介质非线性特性对地震动相干性的影响。     

Artificial ground motion compatible with specified ground shaking peaks and target response spectrum

1 Introduction Time history analysis is usually needed not only for the seismic design of important structures, such as nuclear plants, large concrete dams and super high-rise buildings, but also for seismic performance based design. Performance-based design requires tha knowledge of the entire process of nonlinear seismic responses of structures be obtained, which depends on the ground motion time history as input for this analysis. However, because the observed strong-motion recordings are l…     

Cyclic‐demand spectrum

The seismic response spectrum defines the amplitude of the load, but it does not specify the number of cycles for which the load must be resisted by the structure. The amplitude by itself is not sufficient to evaluate the seismic resistance of a structure, because the structure's strength, stiffness and energy‐dissipation capacity reduce with an increase in the number of load cycles. This paper presents a cyclic‐demand spectrum, which, in conjunction with the amplitude spectrum, provides a more complete definition of the seismic load, hence a way to consider the degradation in strength, stiffness and energy‐dissipation capacity in a rational manner. Similarly to three amplitude parameters (peak ground acceleration, peak ground velocity, and peak ground displacement), three cyclic‐demand parameters are introduced for stiff, moderately stiff, and flexible systems. A design example is presented to illustrate the use of the cyclic‐demand spectrum. Copyright © 2002 John Wiley & Sons, Ltd.     

Nonlinear surface ground analysis via statistical approach

A statistical approach is proposed for nonlinear surface ground analysis. In contrast to the conventional method which deals with only a single ground motion for equivalent linearization of soil properties, a design response spectrum defined at the upper level (bottom of the surface ground) of an engineering bedrock can be handled as the target design earthquake in the present paper. The effective shear strain in each soil layer is evaluated by means of a statistical procedure in which the mean peak shear strain is computed in terms of its standard deviation and the corresponding peak factor. The stiffness and damping ratio of each soil layer are obtained iteratively from the nonlinear relation of stiffness reduction factors and damping ratios with respect to the strain level. After the evaluation of the equivalent stiffness and damping ratio of every soil layer, the ground surface response spectrum is transformed from the design response spectrum defined at the upper level of the engineering bedrock via the one-dimensional wave propagation theory. The reliability and accuracy of the proposed analysis method is examined through the comparison with the results by the conventional method (represented by the program) for many simulated spectrum-compatible ground motions.     

多点激励下结构随机地震反应分析的反应谱方法

基于随机振动理论,提出了多点激励作用下线性系统随机地震反应分析的均值反应谱方法,给出了结构峰值反应的均值、标准差以及反应平均频率的反应谱组合公式。这可以将反应谱方法推广应用到多点激励结构的抗震可靠度分析中。鉴于组合公式中谱参数和相关系数需要由烦琐的数值积分得到,本文进一步针对它们给出合理的简化计算式,从而使得建议的反应谱方法的计算效率大大增加。最后,以一个双塔斜拉桥为例,对本文方法进行了验证。基于建议方法的计算结果与Monte Carlo模拟结果吻合较好。与经典的多点激励反应谱方法(MSRS法)比较,本文方法具有其无法比拟的计算效率。     

Wavelet-based response spectrum compatible synthesis of accelerograms—Eurocode application (EC8)

An integrated approach for addressing the problem of synthesizing artificial seismic accelerograms compatible with a given displacement design/target spectrum is presented in conjunction with aseismic design applications. Initially, a stochastic dynamics solution is used to obtain a family of simulated non-stationary earthquake records whose response spectrum is on the average in good agreement with the target spectrum. The degree of the agreement depends significantly on the adoption of an appropriate parametric evolutionary power spectral form, which is related to the target spectrum in an approximate manner. The performance of two commonly used spectral forms along with a newly proposed one is assessed with respect to the elastic displacement design spectrum defined by the European code regulations (EC8). Subsequently, the computational versatility of the family of harmonic wavelets is employed to modify iteratively the simulated records to satisfy the compatibility criteria for artificial accelerograms prescribed by EC8. In the process, baseline correction steps, ordinarily taken to ensure that the obtained accelerograms are characterized by physically meaningful velocity and displacement traces, are elucidated. Obviously, the presented approach can be used not only in the case of the EC8, for which extensive numerical results/examples are included, but also for any code provisions mandated by regulatory agencies. In any case, the presented numerical results can be quite useful in any aseismic design process dominated by the EC8 specifications.     

Simplified non-linear seismic displacement demand prediction for low period structures

The prediction of non-linear seismic demand using linear elastic behavior for the determination of peak non-linear response is widely used for seismic design as well as for vulnerability assessment. Existing methods use either linear response based on initial period and damping ratio, eventually corrected with factors, or linear response based on increased equivalent period and damping ratio. Improvements to the original EC8 procedure for displacement demand prediction are proposed in this study. Both propositions may be graphically approximated, which is a significant advantage for practical application. A comparison with several other methods (equal displacement rule, EC8 procedure, secant stiffness and empirical equivalent period methods) is performed. The study is based on non-linear SDOF systems subjected to recorded earthquakes, modified to match design response spectra of different ground types, and focuses on the low frequency range that is of interest for most European buildings. All results are represented in the spectral displacement/fundamental period plane that highlights the predominant effect of the fundamental period on the displacement demand. This study shows that linearized methods perform well at low strength reduction factors but may strongly underestimate the displacement demand at strength reduction factors greater than 2. This underestimation is an important issue, especially for assessment of existing buildings, which are often related with low lateral strength. In such cases, the corresponding strength reduction factors are therefore much larger than 2. The new proposals significantly improve the reliability of displacement demand prediction for values of strength reduction factors greater than 2 compared to the original EC8 procedure. As a consequence, for the seismic assessment of existing structures, such as unreinforced masonry low-rise buildings, the current procedure of EC8 should be modified in order to provide accurate predictions of the displacement demand in the domain of the response spectrum plateau.     

Stochastic analysis of structures and piping systems subjected to stationary multiple support excitations

A stochastic method has been developed for seismic analysis of structures and piping systems subjected to multiple support excitations. In either the time or the frequency domain, mean and extreme values of structural and piping system response can be found, including the effects of cross-correlations of modal response and cross-correlations of multiple support excitations. Stationary white noise and stationary filtered white noise ground excitations are used. A computer program has been developed to carry out the stochastic seismic analysis. Results for a realistic nuclear power plant structure and piping system with and without modal cross-correlations and support excitation cross-correlations are compared. From these results, it is concluded that neglecting cross-correlations can lead to large errors. The stochastic method reported is shown to be more accurate than the response spectrum method and more economical than the time-history method; therefore, it is recommended for seismic analysis of nuclear power plants.     

Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.     

Dynamic analysis of track nonlinear energy sinks subjected to simple and stochastice excitations

Track nonlinear energy sinks (track NESs) have been shown to be an effective and applicable strategy to mitigate structural response in recent years. However, previous studies on track NESs has mainly focused on demonstrating the benefits of track NESs through numerical simulations and experiments, with relatively little attention paid to the analytical understanding of the unique dynamics of track NESs. This study analyzes the responses of a track NES when subjected to impulsive and harmonic excitations by the harmonic balance method. Special attention is given to the cause and effect of the peaking behavior that is a prominent characteristic of the track NES's restoring force–displacement relationship. Analytical results reveal that the special energy–frequency characteristics of track NESs can be, at times, utilized to enhance the energy robustness that is absent in the conventional cubic NESs. Based on the analytical response expression, an equivalent linearization method (ELM) for the track NESs is developed for stochastic analysis. This ELM is numerically validated on the systems with strong nonlinearities. Stochastic optimization built on the ELM is performed to obtain design parameters of the track NES that can lead to minimum response variances of the primary structure. In particular, the proposed optimization procedure can be applied to seismic optimum design in which the seismic excitations are modeled as filtered white-noise ground motions. The analytical techniques provided in this study lay the groundwork for the practical implementation of track NESs as a robust and effective control strategy for engineering structures.     

均质土坝非平稳随机地震反应分析

考虑地震荷载的随机性及强度、频率的非平稳性,基于作者提出的适用于非平稳随机过程的一般随机地震动模型,采用虚拟激励法,建立了非平稳随机地震反应分析方法,并将其应用于某实际均质土坝动力分析中。土石坝及坝基体系采用整体有限元离散,坝体和坝基材料的动力非线性性能以等效线性化方法考虑。首先,基于目标加速度时程的强度和能量信息,确定了作为输入的加速度时—频演变功率谱密度;其次,比较了确定性时程动力分析和非平稳随机分析的结果,探讨了频率非平稳随机地震激励下的土石坝地震反应特性;最后,比较了2种不同坝基条件下的土石坝非平稳随机地震反应,探讨了频率非平稳随机激励下的土石—坝基动力相互作用。分析结果表明：地震动的频率非平稳性对土石坝动力反应有一定影响;坝体—坝基动力相互作用在地震过程中的不同阶段表现有所不同,主震阶段的相互作用显著。     

Wavelet‐based characterization of design ground motions

With the recent emergence of wavelet‐based procedures for stochastic analyses of linear and non‐linear structural systems subjected to earthquake ground motions, it has become necessary that seismic ground motion processes are characterized through statistical functionals of wavelet coefficients. While direct characterization in terms of earthquake and site parameters may have to wait for a few more years due to the complexity of the problem, this study attempts such characterization through commonly available Fourier and response spectra for design earthquake motions. Two approaches have been proposed for obtaining the spectrum‐compatible wavelet functionals, one for input Fourier spectrum and another for input response spectrum, such that the total number of input data points are 30–35% of those required for a time‐history analysis. The proposed methods provide for simulating ‘desired non‐stationary characteristics’ consistent with those in a recorded accelerogram. Numerical studies have been performed to illustrate the proposed approaches. Further, the wavelet functionals compatible with a USNRC spectrum in the case of 35 recorded motions of similar strong motion durations have been used to obtain the strength reduction factor spectra for elasto‐plastic oscillators and to show that about ±20% variation may be assumed from mean to 5 and 95% confidence levels due to uncertainty in the non‐stationary characteristics of the ground motion process. Copyright © 2002 John Wiley & Sons, Ltd.