Modal response analysis of multi‐support structures using a random vibration approach

A formulation is developed for modal response analysis of multi‐support structures using a random vibration approach. The spectral moments of the structural response are rigorously decomposed into contributions from spectral moments of uncoupled modal responses. An advantage of the proposed formulation is that the total dynamic response can be obtained on the basis of mode by mode uncoupled analyses. The contributions to the total response from modal responses under individual support ground motions and under cross‐correlated pairs of support ground motions can be recognized explicitly. The application and performance of the formulation is illustrated by means of an example using a well‐established coherency spectrum model and widely known power spectra models, such as white noise and Kanai–Tajimi. The first three spectral moments of displacement, shear, and bending moment responses are computed, showing that the formulation produces the same results as the exact solution. Copyright © 2015 John Wiley & Sons, Ltd.     

超长混凝土框架梁弹性内力行波效应研究

鉴于超长混凝土框架结构行波效应规律尚缺乏系统研究,借助建筑结构通用有限元软件MI-DAS/GEN,采用一致与非一致两种激励方式对165～1100m/s地震波速作用下长度为33～264m的3层钢筋混凝土框架结构进行弹性时程分析。通过对比一致与非一致激励下的梁弯矩、剪力和轴力,详细讨论了其随波速、长度以及不同部位的变化规律。研究表明:波速越慢,模型越长,行波效应越明显。框架梁弯矩的增大作用主要集中在264m模型第1层两端,减小作用表现在各模型第1层中部和第2、3层的梁上;剪力与弯矩有相似的变化规律;梁中存在拉力与压力,中部各梁的拉力和压力较两端的梁要大。     

Non-stationary random vibrations of a shear beam under high frequency seismic effects

Band-limited, non-stationary random vibrations of a shear beam are studied in order to investigate high frequency seismic effects on building structures. A solution for the evolutionary spectral density of the shear beam response to a time segment of band-limited white noise is given in a closed form. The root mean square (rms) and peak response of the shear beam are studied for two characteristic frequency bands: the conventional 1–4 Hz and higher frequency 4–16 Hz, characteristic for rockburst ground motion. Applying the criterion of equal excitation intensity with constant rms velocity, both responses are analyzed in detail and compared. The “switching off” fundamental mode for high frequency excitations results in characteristic overshoot of the stationary response level by the non-stationary rms response and an amplification of the response in the upper part of the shear beam.     

The complete SRSS modal combination rule

The complete Square‐Root‐of‐Sum‐of‐Squares (c‐SRSS) modal combination rule is presented. It expresses the structural response in terms of uncoupled SDOF modal responses, yet accounting fully for modal response variances and cross‐covariances. Thus, it is an improvement over the classical SRSS rule which neglects contributions from modal cross‐covariances. In the c‐SRSS rule the spectral moments of the structural response are expressed rigorously in terms of the spectral moments of uncoupled modal responses and of some coefficients that can be computed straightforwardly as a function of modal frequencies and damping, without involving the computation of cross‐correlation coefficients between modal responses. An example shows an application of the c‐SRSS rule for structural systems with well separated and closely spaced modal frequencies, subjected to wide‐band and narrow‐band excitations. Comparisons with response calculations using the SRSS and the Complete Quadratic Combination rules are given and discussed in detail. Based on the c‐SRSS rule a response spectrum formulation is introduced to estimate the maximum structural response. An example considering a narrow‐band excitation from the great Mexico earthquake of September 19, 1985, is given and the accuracy of the response spectrum formulation is examined. Copyright © 2010 John Wiley & Sons, Ltd.     

错位转换层位置对高层建筑竖向构件受力性能影响的研究

作为一种新型转换层结构——错位转换层结构,其竖向位置的移动对高层结构在水平地震作用下竖向构件受力性能有何影响目前尚未见文献报道。采用有限元程序对高层带错位转换层结构进行了水平地震作用下的时程反应和反应谱分析．分析了上部转换层和下部转换层相对位置保持不变的情况下．整体改变错位转换层位置对结构地震作用、剪力及竖向构件内力的影响。分析研究发现。错位转换层整体位置的竖向移动对结构整体剪力、上部转换层下承托墙肢内力、上部转换层框支剪力墙内力影响不大．但对落地剪力墙、上部转换层下框支柱和下部转换层梁托柱内力有较大影响。     

随机振动下建筑工程抗震参数的测试与实验分析

随机振动对建筑工程的破坏严重,随机性振动的破坏程度很难量化,抗震参数测试较为困难,给相关建筑工程的稳定性带来极大威胁。研究随机振动下建筑工程抗震测试方法。结合随机振动理论,获取随机振动下建筑工程功率谱密度函数,分析建筑随机振动下响应功率;依照响应功率谱密度,分析随机振动下建筑工程各阶谱矩、均方差值等抗震性能概率特征,检测建筑工程在随机地震振动下的抗震性。以我国某省建筑工程为实验对象,验证测试方法的抗震参数检测性能,结果表明:提出的方法在随机振动作用下,能有效获取建筑工程的功率谱密度,和振动波动情况,且检测失效概率与可靠度与实际情况更为相近。     

考虑行波效应的车桥系统地震响应分析

借助虚拟激励法,本文研究了非一致水平地震作用下三维车桥耦合时变系统的非平稳随机振动问题。通过轮轨间位移协调关系将列车和桥梁运动方程进行耦合,建立了考虑行波效应的多点地震激励下列车-桥梁时变系统运动方程。水平地震激励假设为均匀调制多点异相位完全相干随机激励;轨道不平顺激励假设为多点异相位完全相干平稳随机激励。采用时变系统的虚拟激励法将这些随机激励分别转化为一系列虚拟的确定性激励,然后通过数值积分求解相应的虚拟响应,进而方便地得到系统随机响应的时变功率谱密度函数和标准差等。数值算例中,通过Monte Carlo法验证了本文随机振动分析方法的正确性和有效性,并讨论了地震波视波速、场地土条件等对系统随机响应的影响。     

非平稳地震激励下隔震曲线梁桥振动控制研究

曲线梁桥由于其平面不规则性导致结构在地震激励下产生弯扭耦合效应,使得隔震曲线梁桥的地震响应更加复杂。目前常用的控制方法是将隔震技术与附加减震装置相结合对曲线梁桥进行控制。本文将地震动考虑为一均匀调制非平稳随机过程,针对隔震曲线梁桥长周期、低频率的特点,选取Clough-Pension平稳地震动功率谱模型作为随机地震动输入模型,对无控(NON-C)、经典线性最优控制(COC)以及序列最优控制算法(SOC)三种状态下的曲线桥梁进行随机响应分析。通过建立曲线梁桥在随机地震动作用下的运动方程,求出减震控制结构的位移谱密度、加速度谱密度响应及时变方差。分析结果表明:序列最优控制算法(SOC)在使隔震层位移得到减小的同时,可以更有效地控制上部结构的地震响应,具有更好的控制效果。     

Longitudinal integral response deformation method for the seismic analysis of a tunnel structure

For the longitudinal seismic response analysis of a tunnel structure under asynchronous earthquake excitations, a longitudinal integral response deformation method classified as a practical approach is proposed in this paper. The determinations of the structural critical moments when maximal deformations and internal forces in the longitudinal direction occur are deduced as well. When applying the proposed method, the static analysis of the free-field computation model subjected to the least favorable free-field deformation at the tunnel buried depth is performed first to calculate the equivalent input seismic loads. Then, the equivalent input seismic loads are imposed on the integral tunnel-foundation computation model to conduct the static calculation. Afterwards, the critical longitudinal seismic responses of the tunnel are obtained. The applicability of the new method is verified by comparing the seismic responses of a shield tunnel structure in Beijing, determined by the proposed procedure and by a dynamic time-history analysis under a series of obliquely incident out-ofplane and in-plane waves. The results show that the proposed method has a clear concept with high accuracy and simple progress. Meanwhile, this method provides a feasible way to determine the critical moments of the longitudinal seismic responses of a tunnel structure. Therefore, the proposed method can be effectively applied to analyze the seismic response of a long-line underground structure subjected to non-uniform excitations.     

Wavelet‐based non‐stationary seismic rocking response of flexibly supported tanks

The non‐stationary rocking response of liquid storage tanks under seismic base excitations including soil interaction has been developed based on the wavelet domain random vibration theory. The ground motion has been characterized through statistical functionals of wavelet coefficients of the ground acceleration history. The tank–liquid–foundation system is modelled as a multi‐degree‐of‐freedom (MDOF) system with both lateral and rocking motions of vibration of the foundation. The impulsive and convective modes of vibration of the liquid in the tank have been considered. The wavelet domain coupled dynamic equations are formulated and then solved to get the expressions of instantaneous power spectral density function (PSDF) in terms of functionals of input wavelet coefficients. The moments of the instantaneous PSDF are used to obtain the stochastic responses of the tank in the form of coefficients of hydrodynamic pressure, base shear and overturning base moment for the largest expected peak responses. Parametric variations are carried out to study the effects of various governing parameters like height of liquid in the tank, height–radius ratio of the tank, ratio of total liquid mass to mass of foundation, and shear wave velocity in the soil medium, on the responses of the tank. Copyright © 2003 John Wiley & Sons, Ltd.     

Sensitivity and moment analyses of head in variably saturated regimes

A numerical approach for approximating statistical moments of hydraulic heads of variably saturated flows in multi-dimensional porous media is developed. The approximation relies on a first-order Taylor series expansion of a finite element flow model and an adjoint state numerical method for variably saturated flows to evaluate sensitivities. This approach can be employed to analyze uncertainties associated with predictions of head of steady-state or transient flows in variably saturated porous media, with any type of boundary and initial conditions. Limitations of stochastic analytical methods such as spectral/perturbation approaches and the time-consuming Monte Carlo simulation technique are thus alleviated. An example is given to demonstrate the utility of the approach and to investigate the temporal evolution of head variances in a variably saturated flow regime. Results show that the fluctuation of the water table can have significant impacts on the propagation of the head variance.     

Gaussian plume modeling of contaminant transport

A technique for modeling contaminant transport based on Markov process theory is developed. Transport is quantified by summing the first two moments of independent random displacements and applying the central limit theorem (CLT) to obtain solute distributions of a Gaussian nature. For non-uniform flow fields the CLT is applied in a streamfunction/equi-travel time space and transforms are used to give concentrations in Cartesian coordinates. Simulations in uniform, radially converging and circular flow fields show the method to be two to three orders of magnitude faster than modeling with the advection-dispersion equation, using a control volume technique.     

Non‐stationary seismic response of tanks with soil interaction by wavelets

A wavelet‐based random vibration theory has been developed for the non‐stationary seismic response of liquid storage tanks including soil interaction. The ground motion process has been characterized via estimates of statistical functionals of wavelet coefficients obtained from a single time history of ground accelerations. The tank–liquid–soil system has been modelled as a two‐degree‐of‐freedom (2‐DOF) system. The wavelet domain equations have been formulated and the wavelet coefficients of the required response state are obtained by solving two linear simultaneous algebraic equations. The explicit expression for the instantaneous power spectral density function (PSDF) in terms of the functionals of the input wavelet coefficients has been obtained. The moments of this PSDF are used to estimate the expected pseudo‐spectral acceleration (PSA) response of the tank. Parametric variations are carried out to study the effects of tank height, foundation natural frequency, shear wave velocity of soil and ratio of the mass of tank (including liquid) to the mass of foundation on the PSA responses of tanks. Copyright © 2001 John Wiley & Sons, Ltd.     

Modal decomposition method for stationary response of non-classically damped systems

The stationary response of multi-degree-of-freedom non-classically damped linear systems subjected to stationary input excitation is studied. A modal decomposition procedure based on the complex eigenvectors and eigenvalues of the system is used to derive general expressions for the spectral moments of response. These expressions are in terms of cross-modal spectral moments and explicitly account for the correlation between modal responses; thus, they are applicable to structures characterized with significant non-classical damping as well as structures with closely spaced frequencies. Closed form solutions are presented for the important case of response to white-noise input. Various quantities of response of general engineering interest can be obtained in terms of these spectral moments. These include mean zero-crossing rate and mean, variance and distribution of peak response over a specified duration. Examples point out several instances where non-classical damping effects become significant and illustrate the marked improvement of the results of this study over conventional analysis based on classical damping approximations.     

A generalization of the timoshenko beam model for coupled vibration analysis of thin-walled beams

A generalization of the Timoshenko beam model is presented which accounts for the influence of the shear strains, due to non-uniform bending and torsion, on the flexural-torsional vibrations of thin-walled cores with open or closed cross-section. The axial displacement field incorporates the torsion secondary warping as well as the warping terms depending on the shear resultants. It is shown that exact solutions for the interior domain problem can be obtained under proper load conditions. Moreover, a discrete model for the free-vibration analysis is derived by adopting a linear interpolation of the unknown functions and a reduced integration in order to avoid locking phenomena. Various applications are developed, including the case of the coupled vibrations of a shear-core-steel-frame building.     

Probabilistic critical excitation for MDOF elastic–plastic structures on compliant ground

Earthquake ground motions and their effects on structural responses are very uncertain even with the present knowledge. It is therefore desirable to develop a robust structural design method taking into account these uncertainties. Critical excitation approaches are promising and a new random critical excitation method is proposed for MDOF elastic–plastic shear‐building structures on compliant ground. The power (area of power spectral density (PSD) function) and the intensity (magnitude of PSD function) are fixed and the critical excitation is found under these restrictions. In contrast to linear elastic structures, transfer functions and simple expressions for response evaluation cannot be defined in elastic–plastic structures and difficulties arise in describing the peak responses except by laborious elastic–plastic time‐history response analysis. Statistical equivalent linearization is used to estimate the elastic–plastic stochastic peak responses approximately. The critical excitation responses are obtained for several examples and compared with those of the corresponding recorded earthquake ground motion. Copyright © 2001 John Wiley & Sons, Ltd.     

Stochastic dynamic analysis of a layered half-space

A stochastic thin-layer method is developed for the analysis of wave propagation in a layered half-space. A random field of shear moduli in the layered system is considered in terms of multiple correlated random variables. Expanding the random moduli and uncertain responses by means of Hermite polynomial chaos expansions and applying the Galerkin method in the spatial as well as stochastic domains, stochastic versions of thin-layer methods for a layered half-space in plane strain and antiplane shear are obtained. In order to represent the infinite half-space, continued-fraction absorbing boundary conditions are included in the thin-layer models of the half-space. Using these stochastic methods, dynamic responses of a layered half-space subjected to line loads are examined. Means, coefficients of variance, and probability density functions of the half-space responses with a varying correlation coefficient of the shear moduli are computed and verified by comparison with Monte Carlo simulations. It is demonstrated that accurate probabilistic dynamic analysis is possible using the developed stochastic thin-layer methods for a layered half-space.     

土层对地震的随机反应分析

利用随机振动理论,本文研究了非均匀土层对地震的动力反应问题,假定土层的剪切模量随深度呈指数函数形式分布,关于基岩输入地震动加速度的功率谱密度函数,考虑了两种形式：白噪声谱和有色谱,为了比较起见,文中考虑了均匀土层的情形,数值计算表明;（１）在基岩输入地震动加速度的功率谱密度为白谱的情况下,土层的最大期望反应（相对位移,绝对加速度,剪应变）大于基岩输入地震动的功率谱为有色谱的情形;（２）在土层的上部     

Elastically supported cantilever beam subjected to nonstationary seismic excitation

The dynamic behaviour of bridge piers under seismic load is studied here in the context of random vibration. The earthquake excitation is modelled as white noise filtered by the Clough–Penzien filter in cascade with modulation accounting for intensity non-stationarity. The bridge pier modelled as an elastically supported cantilever beam witha lumped mass at the top. An analytical solution is presented for the response statistics, which may be used to develop probabilistic seismic response spectra for design. It is found that the first two modes of the pier approach to rigid-body motion when the stiffness of the elastic support decreases. Seismic responses increase with the top mass, resultingin significantly high displacement and shear but negligible moment at the top, and higher shear and moment at thebase. Lower stiffness of the elastic support increases the pier top displacement and moment responses, but may increase or reduce shear responses. The probabilistic spectrum of the relative displacement between the bridge superstructure and the pier top may depend on the two systems’ relative modal properties. © 1998 John Wiley & Sons, Ltd.     

基于SVM与SRM耦合的抗剪强度参数随机场模拟

实验数据表明土体参数具有很大的空间变异性,而随机场理论为模拟土体参数空间变异性提供了有效途径。因为传统的谱表示法(SRM)无法正确模拟多维多元随机场参数间的互相关性,提出支持向量机法(SVM)与SRM耦合的方法。SVM是基于统计学习理论和结构风险最小化原理基础上的通用机器学习方法,它在解决小样本、非线性和高维模式识别问题中表现出诸多优势。以土体抗剪强度参数:黏聚力c和内摩擦角φ为例,通过实验证明二者之间存在天然负相关性,即为二维二元随机场。结果表明,在样本数量较少的条件下,基于耦合算法模拟随机场不仅能有效地描述变量的自相关性,而且能够准确地描述变量间的互相关性,为解决小样本条件下模拟多维多元随机场提供了一种有效的方法。