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.     

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.     

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.     

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。     

Non-stationary random seismic responses of non-uniform beams

In this paper, a direct Ritz method is presented for analysing the responses of non-uniform shear beams subjected to the action of an evolutionary random seismic excitation. Because of the use of a pseudo ground acceleration, time-dependent random responses can conveniently be calculated by solving a set of deterministic equations of motion. These responses include power spectral densities, variances and higher spectral moments of any required responses of the beam. A non-uniform beam example was solved and the time-dependent variances and second spectral moments of both the beam-top displacement and the beam-base shear force are given to show the simplicity and effectiveness of the method, which can also easily be applied to other continuous elastic structures.     

以反应谱和功率谱密度函数表征的强震运动的统计特性

文中利用我国海城和唐山地震１９０条强震记录和美国西部１３８条强震记录研究了以反应谱和功率谱密度函数表征的强震运动的统计特性。反应谱用最大地面加速度归一化，即表为谱放大系数，功率谱密度函数用金井清谱描述。文中研究了参数的统计值和依赖性，以及谱放大系数和金井清谱参数与场地条件、震源和震中距等之间的相关性，分析比较了中美两国强震运动谱的统计特性，并对我国地震工程应用的强震运动谱的统计参数值的选取提出了建议。     

Seismic spectra on soil with uncertain properties

Expected spectral ordinates and their standard deviations are computed for ground motions produced by SH waves at the surface of a homogeneous soil formation resting on a half-space of rock. Wave transmission is idealized as linear and one-dimensional. Uncertainties about soil properties are taken into account using various approximations: a Taylor series expansion of the Fourier spectrum magnification factors with different numbers of terms and different intervals for numerical differentiation; discretization of the joint probability density function of the soil properties so as to match its moments up to those of third order including the crossed moments; a discretization that matches up to some fifth-order moments; and a fine numerical integration for the marginal density function of the modulus of rigidity, discretizing the rest of the density function. The latter approach gives sufficiently accurate results for practical purposes. Effects of even moderate uncertainty about soil properties on the statistics of spectral ordinates are found to be decisive.     

Statistical Characteristics of Strong Ground Motion Specified by Response Spectrum and Power Spectral Density Function

The statistical characteristics of strong ground motion specified by response spectrum and power spectral density function are studied using 190 strong-motion records of the Haicheng and Tangshan earthquakes in China and 138 earthquakes in the western United States.The response spectrum is normalized by the peak ground acceleration(i.e.,represented as spectral magnification factor),and the power spectral density function is described by the Kanai-Tajimi spectrum.The statistics and dependence of parameters are evaluated,and correlations between the spectral magnification factor or Kanai-Tajimi spectral parameters and the site condition,epicentral distance,or local magnitude are investigated.The statistical characteristics of spectra China and the U.S.A.are compared.Based on the results obtained the values of the statistics on spectral parameters for earthquake engineering applications in China are suggested.     

Response of multi-span beams to spatially incoherent seismic ground motions

The response of continuous two- and three-span beams of various lengths subjected to spatially varying seismic ground motions is evaluated. Stochastic representations of the seismic ground motions are used as input at the supports of the structures, and sensitivity analyses of the response with respect to the degree of correlation between the support motions are performed. The validity of the commonly used assumption of equal support motion is examined. Square-roots of mean-square values of total displacements, bending moments and shear forces are outputs of the analyses. The results indicate that fully correlated motions may produce higher or lower response than partially correlated motions, depending on the dynamic characteristics of the structure, the response quantity (bending moment or shear force) that is evaluated, the position along the axis of the beam where the response quantity is evaluated, the separation between the natural frequencies of the structure and the dominant frequencies of the input motions, and on the degree of partial correlation between the support motions.     

Methods of detection and estimation errors in ST radar studies

The classical theory of detection using the Neyman-Pearson principle is applied to stratosphere-troposphere (ST) radar signals. It is extended to provide information regarding the detection of weak signals which complements the detectability method usually employed in ST radar studies. It is shown that for ST radar signals of low amplitude and a detectability around 3 (a value commonly invoked in literature), the probability of detection is about equal to the probability of false alarm. The question of threshold detectability is also discussed.Spectral moments errors are evaluated by a method which is an extension of the analytical method of estimation developed by Miller and Rochwarger and the results compared to other statistical and analytical models. As already known, three factors can affect the error on the estimated parameters: the signal-to-noise ratio, the spectral width and the incoherent integration number. For high signal-to-noise ratios, analytical results are in good agreement with Barrick’s and Denenberg’s theoretical models and with Yamamoto’s statistical one. For low signal-to-noise ratios, the spectral parameters are more sensitive to the selected model but overall variability is similar.     

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.     

Seismic response estimation method for earthquake‐damaged RC buildings

This study proposes a procedure for identifying spectral response curves for earthquake‐damaged areas in developing countries without seismic records. An earthquake‐damaged reinforced concrete building located in Padang, Indonesia was selected to illustrate the identification of the maximum seismic response during the 2009 West Sumatra earthquake. This paper summarizes the damage incurred by the building; the majority of the damage was observed in the third story in the span direction. The damage was quantitatively evaluated using the damage index R according to the Japanese guidelines for post‐earthquake damage evaluation. The damage index was also applied to the proposed spectral response identification method. The seismic performance of the building was evaluated by a nonlinear static analysis. The analytical results reproduced a drift concentration in the third story. The R‐index decreased with an increase in the story drift, which provided an estimation of the maximum response of the building during the earthquake. The estimation was verified via an earthquake response analysis of the building using ground acceleration data, which were simulated based on acceleration records of engineering bedrock that considered site amplification. The maximum response estimated by the R‐index was consistent with the maximum response obtained from the earthquake response analysis. Therefore, the proposed method enables the construction of spectral response curves by integrating the identification results for the maximum responses in a number of earthquake‐damaged buildings despite a lack of seismic records. Copyright © 2016 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.     

用于塔里木地壳构造成像的重力场谱矩方法研究

现代地壳探测取得的各种地球物理场中包含有大量地壳构造信息,如果提取这些信息,可以在数学物理普适规律指导下计算出关于地壳构造的图件.由于区域重力场的高阶谱矩包含了场轮廓面的几何信息,可用于地质构造要素的分析计算,为地质构造要素选取和定位的计算机自动制图提供理论依据.由地面布格重力场的二阶谱矩的统计不变量计算,可导出表征地壳变形带的脊形化系数,用于计算机自动制作区域地壳变形带分布图.脊形化系数强弱区的边界代表了活动的和稳定的构造单元的边界.在计算脊形化系数的基础上,还可进一步计算其二阶谱矩的统计不变量并作进一步增强处理,计算构造单元的边界系数,对地壳构造单元边界作连续和精细的追踪.文中介绍了脊形化系数和边界系数的计算方法,以及用在塔里木盆地进行地壳构造分析的结果.圈定了塔里木盆地深层满中高密度扰动带和满南、满北低密度扰动带,为深层油气勘探提供重要基础资料.     

Inverse modelling of elastic thickness by convolution method - the eastern Alps as a case example

An unconventional scheme is used to estimate the flexural rigidity, or equivalently the elastic thickness of the lithosphere, given the topography and gravity data. The flexural rigidity is the parameter that governs the flexural response of the lithosphere in the frame of the thin plate flexure model. The scheme is an alternative to the widely used calculation of admittance of topography (sea-floor or continental topography) and gravity, bearing some advantages which are explained in the paper. The scheme involves the inversion of the gravity data in order to formulate a model of the crust-mantle interface (CMI) undulations. In a second step the flexure parameter is then evaluated from the relation between topography and CMI variations. Instead of calculating the admittance function using a spectral analysis, a set of point-load response functions are used in order to retrieve the optimal flexure parameter. This has two main advantages: instabilities of the numerical admittance evaluation at wavenumbers with low spectral energy in the topography are overcome and the analysis can be made over an area which is not necessarily rectangular, as required for the spectral analysis. The proposed method allows a higher space resolution of elastic thickness than any spectral method. For validation, the numerical strategy is applied to the situation of a realistic synthetic model, where all inputs and outputs are known a priori. Finally the spatial variations of the elastic thickness are studied in an area across the Eastern Alps.     

Kinematic bending moments in pile foundations

In this paper the kinematic seismic interaction of single piles embedded in soil deposits is evaluated by focusing the attention on the bending moments induced by the transient motion. The analysis is performed by modeling the pile like an Euler–Bernoulli beam embedded in a layered Winkler-type medium. The excitation motion is obtained by means of a one-D propagation analysis. A comprehensive parametric analysis is carried out by varying the main parameters governing the dynamic response of piles like the soil properties, the bedrock location, the diameter and embedment in the bedrock of piles. On the basis of the parametric analysis, a new design formula for predicting the kinematic bending moments for both the cross-sections at the deposit–bedrock interface and at the pile head is proposed.     

Nonlinear dynamic analysis of Meloland Road Overpass using three-dimensional continuum modeling approach

This paper presents a three-dimensional (3D) continuum nonlinear analysis of the Meloland Road Overpass (MRO) near El Centro, California. The modeling methodology and the computational tools are discussed in detail. The performance of the computational model is evaluated by comparing the computed responses with the responses recorded at the bridge site during the 1979 Imperial Valley and 2010 El Mayor-Cucapah earthquakes. Amongst the recorded earthquake events at the bridge site, these two events caused the strongest shaking. The comparison shows that the 3D model is potentially an effective tool for detailed analysis of a full bridge system including foundation soils, pile foundations, embankments, supporting columns, and the bridge structure itself in a unified system without relying on any ancillary models such as Winkler springs. Additional response parameters such as displacements, rockings, and bending moments are also evaluated although none of these was measured during the seismic events.     

An Incremental Response Spectrum Analysis Procedure Based on Inelastic Spectral Displacements for Multi-Mode Seismic Performance Evaluation

The so-called Nonlinear Static Procedure (NSP) based on pushover analysis has been developed in the last decade as a practical engineering tool to estimate the inelastic response quantities in the framework of performance-based seismic evaluation of structures. However NSP suffers from a major drawback in that it is restricted with a single-mode response and therefore the procedure can be reliably applied only to the two-dimensional response of low-rise, regular buildings. Recognizing the continuously intensifying use of the pushover-based NSP in the engineering practice, the present paper attempts to develop a new pushover analysis procedure to cater for the multi-mode response in a practical and theoretically consistent manner. The proposed Incremental Response Spectrum Analysis (IRSA) procedure is based on the approximate development of the so-called modal capacity diagrams, which are defined as the backbone curves of the modal hysteresis loops. Modal capacity diagrams are used for the estimation of instantaneous modal inelastic spectral displacements in a piecewise linear process called pushover-history analysis. It is illustrated through an example analysis that the proposed IRSA procedure can estimate with a reasonable accuracy the peak inelastic response quantities of interest, such as story drift ratios and plastic hinge rotations as well as the story shears and overturning moments. A practical version of the procedure is also developed which is based on the code-specified smooth response spectrum and the well-known equal displacement rule. This revised version was published online in July 2006 with corrections to the Cover Date.     

Selection of ground motion time series and limits on scaling

A procedure to select time series for use in non-linear analyses that are intended to result in an average response of the non-linear system is proposed that is not based simply on magnitude, distance, and spectral shape. A simple model of a yielding system is used as a proxy for the non-linear behavior of a more complicated yielding system. As an example, Newmark displacements are used as a proxy for more complex slope-stability models. The candidate scaled time series are evaluated to find those that yield a response of the simple non-linear system that is near the expected response for the design event. Those scaled time series with responses near the expected value are selected as the optimum time series for defining average response even if the scale factors are larger than commonly accepted (e.g. scale factors >factor of 2).     

Arch responses to correlated multiple excitations

This paper analyses the in-plane stochastic responses of incompressible circular arches to spatially correlated multiple excitations. The dynamic, quasi-static and total structural displacements, moments, shear and axial forces are calculated. The results are compared with those obtained by various simplified excitations. The effects of the ground motion spatial variations on arch responses are examined. The results indicate that the responses may be underestimated or overestimated by neglecting the ground motion spatial variations, depending upon the structure and ground motion properties, the locations where the responses are evaluated, and the response quantity under consideration.     

On the efficiency of viscous dampers in reducing various seismic responses of wall structures

This paper rigorously assesses the efficiency of viscous dampers connecting two walls to result in “viscously coupled shear walls”. This assessment also holds for viscous dampers in wall structures as they are mounted on frames parallel to the walls leading to “wall-viscous frame” systems. A continuum approach is adopted to model the structure so as to enable non-dimensional formulation of the governing equations. Those equations reveal that, under the approximations considered, the system damping ratio (defined here by 0.5 sqrt(c^2/(m*EI))) is a convenient compact single parameter controlling the response reduction w.r.t. the response of the corresponding undamped system. In contrast to coupled shear walls, this controlling parameter does not depend on the height of the building; therefore, the viscously damped system is efficient for low-rise buildings as well. The continuum approach also allows a semi-analytical solution of the eigenproblem in the complex domain followed by a complex modal spectral analysis. Those solutions reveal the efficiency of the added damping in reducing not only the displacements, inter-story drifts, and wall moments but also the absolute accelerations, wall shear, total shear, and total overturning moments. The results of the analyses and the non-dimensional tables and graphs developed for important response parameters lead to a simple method that could easily be implemented in practice for the purpose of initial design. Copyright © 2012 John Wiley & Sons, Ltd.