An improved response spectrum method for calculating seismic design response. Part 1: Classically damped structures

A good response spectrum method, used for calculating the seismic design response for design inputs defined in terms of response spectra, should be able to include the effect of the higher modes, even though they may be truncated and not explicitly considered in the analysis. It should also be able to include the intercorrelation of the retained modes as well as their correlation with truncated modes. Herein one such method based on a suitable combination of the mode displacement and mode acceleration formulations of structural dynamics is proposed. Several numerical examples are presented to demonstrate the efficiency of the proposed approach vis-a-vis some other commonly used response spectrum approaches.     

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.     

Effects of high modes on the wind-induced response of super high-rise buildings

For super high-rise buildings,the vibration period of the basic mode is several seconds,and it is very close to the period of the fluctuating wind.The damping of super high-rise buildings is low,so super high-rise buildings are very sensitive to fluctuating wind.The wind load is one of the key loads in the design of super high-rise buildings.It is known that only the basic mode is needed in the wind-response analysis of tall buildings.However,for super high-rise buildings,especially for the acceleration response,because of the frequency amplification of the high modes,the high modes and the mode coupling may need to be considered.Three typical super high-rise projects with the SMPSS in wind tunnel tests and the random vibration theory method were used to analyze the effect of high modes on the wind-induced response.The conclusions can be drawn as follows.First,for the displacement response,the basic mode is dominant,and the high modes can be neglected.Second,for the acceleration response,the high modes and the mode coupling should be considered.Lastly,the strain energy of modes can only give the vibration energy distribution of the high-rise building,and it cannot describe the local wind-induced vibration of high-rise buildings,especially for the top acceleration response.     

Mode acceleration-based response spectrum approach for nonclassically damped structures

The paper describes the development of a mode accleration-based response spectrum approach for calculating the seismic design response of the nonclassically damped structures. The response is divided into a pseudo-static part and a dynamic part. The pseudo-static part is calculated by a simple static analysis of the structure for the inertial forces, induced by a unit ground acceleration, applied statically. The dynamic part, of course, pends upon the dynamic characteritics of the structure which are defined in terms of the complex-valued modal characteristics. The correlation between the pseudo-static and dynamic components is properly considered. The design ground input in this approach is defined in terms of the relative acceleration and relative velocity response spectra. The proposed approach has the desired attribute of the mode acceleration approach as the response can be accurately calculated even if only a first few modes are used in the analysis. The approach is computationally more efficient than the convenionally used mode displacement approach. The applicability of the approach is verified by numerical simulation results.     

TMD多点控制体系随机地震响应分析的虚拟激励法

对于频率分布密集或受频带较宽的地震激励的结构，其响应不再以某一单一振型为主，须考虑采用多点控制。本文对受TMD多点控制的结构进行了研究。文中建立了带有多个子结构系统的以模态坐标和子结构自由度为未知量的统一运动方程。针对所得方程为非对称质量、非对称刚度、非经典阻尼的情况，本文给出了使用直接法求解的格式。地震随机响应分析采用了虚拟激励法，可以考虑各振型之间的耦合项，计算量小且精度高。本文的方法适用于带有多个子结构的系统的一般性问题，具有广泛的应用价值。     

考虑高振型影响的结构层间位移能力谱分析方法

工程界普遍采用的能力谱分析方法是建立在结构单自由度简化假设基础上，利用结构基本振型来进行能力分析。虽然此简化假设可以为工程界提供一种简便的分析方法，但从其分析效果来看，简化方法并不能很好地反映结构真实地震反应。高振型对结构局部变形的影响在分析结构动力反应特性中是很重要的因素，不应忽略掉。基于上述对现有能力谱分析方法不足之处的考虑，从提高结构能力谱分析方法的准确性出发，本文在Chopra能力谱分析方法的基础之上，根据我国振型分解反应谱思想，在能力谱分析方法中引入了结构高振型的影响分析，并提出了考虑结构高振型影响的结构层间位移能力谱分析方法的一般步骤。通过平面框架结构体系的算例分析，说明上述推荐方法在改进能力谱分析精度上的有效性。     

土中Love波弥散特性

利用有限单元法及解析法建立和求解了土中Love波特征方程以及位移计算公式．计算结果表明,这一计算方法比纯解析法优越,可以用来分析均质和非均质上中Love波弥散性．本文利用这一方法详细讨论了Love波在上软下硬地基及软夹层地基中的传播特性和弥散特性．上软下硬地基Love波具有弥散性,土层的剪切波及厚度对Love波弥散曲线影响较大,而质量密度的相对变化对Love彼弥散曲线影响较小．软夹层地基中低频时Love波以第一模态波为主,现场所测为第一模态波波速;高频时存在多个高模态波,土中传播的波为这几个高模态波的叠加波,现场所测波速随两传感器的位置不同而有波动．     

Uncertainty analysis of strain modal parameters by Bayesian method using frequency response function

Structural strain modes are able to detect changes in local structural performance, but errors are inevitably intermixed in the measured data. In this paper, strain modal parameters are considered as random variables, and their uncertainty is analyzed by a Bayesian method based on the structural frequency response function (FRF). The estimates of strain modal parameters with maximal posterior probability are determined. Several independent measurements of the FRF of a four-story reinforced concrete frame structural model were performed in the laboratory. The ability to identify the stiffness change in a concrete column using the strain mode was verified. It is shown that the uncertainty of the natural frequency is very small. Compared with the displacement mode shape, the variations of strain mode shapes at each point are quite different. The damping ratios are more affected by the types of test systems. Except for the case where a high order strain mode does not identify local damage, the first order strain mode can provide an exact indication of the damage location.     

On the accuracy of mode superposition analysis in structural dynamics

It is pointed out that the number of modes which should be included in a mode superposition dynamic response analysis depends on both the frequency content and the distribution of the loading. If the loading frequency is low the effect of the higher modes can be approximated by a static analysis. A technique is described for calculating this static contribution from the higher modes; the total response is then represented by the sum of the lower mode dynamic response and the higher mode static effects. The effectiveness of the procedure is demonstrated by a numerical example.     

地震动永久位移对结构地震反应影响

本文推导了基于位移激励计算单自由度体系拟速度谱公式,通过构造的脉冲位移时程对公式精度进行了验证;之后利用小波变换去除强震记录噪声而保留地震动永久位移,再基于去趋势项方法和滤波方法去除永久位移后,计算拟速度谱。算例结果表明:短周期段内,不保留永久位移的位移激励拟速度谱值与保留永久位移的位移激励拟速度谱值相差很小;中长周期段内,不保留永久位移的位移激励拟速度谱值总体上小于保留永久位移的位移激励拟速度谱值,且不保留永久位移时,滤波方法引起的拟速度谱降幅大于去趋势项方法所引起的拟速度谱降幅。因此,基于位移激励计算中长周期结构的地震反应时,应保留地震动永久位移,或基于去趋势项方法去除永久位移。      

Axisymmetric seismic response of a thick circular plate supporting many rods by modal synthesis

Dynamic response of a thick, horizontal, circular plate supporting a large number of slender rods subjected to uniform boundary motion in the vertical direction has been studied by synthesizing component modes of continuous substructures. The excitation considered corresponds to the vertical component of boundary movement produced by earthquake disturbances and the axisymmetric response problem was solved. Mindlin theory was used to formulate the component equations of the plate which is treated as the main component in a modal synthesis technique. The slender rods, which are attached vertically to the plate, are handled as branch components. Vibration modes of a classical thin plate were used as the initial displacement functions for the Mindlin plate. These functions were subsequently modified by a component mode improvement process to obtain plate modes. System modes were generated by combining the improved plate modes with component modes of rods. Numerical results for the natural frequencies and time-history response of the coupled system are compared with those given by a three-dimensional finite element method.     

Sloshing response of layered liquids in rigid tanks

The sloshing action of layered liquids in rigid cylindrical and long rectangular tanks is investigated, considering both their free vibrational characteristics and their response to a horizontal component of base shaking. Special attention is given to the maximum surface displacement induced by the base motion. The analysis is formulated for systems with N superimposed layers of different thicknesses and densities, and is illustrated by a numerical example. In addition, comprehensive numerical data are presented for two-layered and some three-layered systems which elucidate the underlying response mechanisms and the effects and relative importance of the numerous parameters involved. It is shown that for each horizontal natural mode of vibration, there are N distinct vertical modes, the frequencies of which are lower than the natural frequency of a homogeneous liquid of the same total depth. It is further shown that the maximum surface sloshing displacement of the base-excited layered system is typically larger than that of the corresponding homogeneous system, and that the results for the long rectangular and cylindrical tanks are quite similar.     

Dynamic response of an elastic-viscoplastic system in modal co-ordinates

A dynamic analysis of elastic–viscoplastic systems, incorporating the modal co-ordinate transformation technique, is presented. The formulation results in uncoupled incremental equations of motion with respect to the modal co-ordinates. The elastic–viscoplastic model adopted allows the analysis not to involve yielding regions and loading/unloading processes. An implicit Runge–Kutta scheme together with the Newton–Raphson method are used to solve the non-linear constitutive equations. Stability and accuracy of the numerical solution are improved by utilizing a local time step sub-incrementing procedure. Applications of the analyses to multi-storey shear buildings show that good results can be obtained for the maximum displacement response by including only a few lower modes in the computation, but the prediction of the ductility factor response tends to underestimate the peak values when too few modes are used. In addition, stable and valid results can be obtained even with a sizable time step increment.     

A response spectrum method for random vibration analysis of mdf systems

A response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed. The method is based on the assumption that the input excitation is a wide-band, stationary Gaussian process and the response is stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the system. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the system. The formulation includes the cross-correlation between modal responses, which is shown to be significant for modes with closely spaced natural frequencies. The proposed procedure is demonstrated for an example structure that is subjected to an ensemble of earthquake-induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method that neglects the modal correlations are also demonstrated.     

关于结构振型参与系数和振型贡献的分析

采用振型分解反应谱法求解多自由度弹性体系的地震反应时,为了在满足所需计算精度的前提下减少工作量,需要对振型数量进行合理的选择,而振型数的确定主要取决于结构各阶振型对总体反应的贡献。通过数学推导,对振型贡献及振型数量的选择问题进行了研究。首先,讨论了振型参与系数的性质,在此基础上给出了能够反映结构振型贡献参数的数学表达式,对这些参数的力学含义进行了解释,并给出了相关证明;其次,对有效质量法、振型位移控制法等基于不同振型贡献标准的确定振型数的方法进行了分析比较,指出了其合理性和不足。本文研究对进一步理解结构振型贡献和振型数的选择问题具有一定的理论意义。     

Seismic design response by an alternative SRSS rule

The square root of the sum of the squares (SRSS) procedure and its modified forms are often used to obtain seismic design response. The design inputs for such procedures are usually defined in terms of pseudo velocity or acceleration response spectra. Erroneous results have been obtained with these existing SRSS procedures, especially in the calculation of responses where high frequency effects dominate. Here an alternative SRSS procedure is developed using the so-called mode acceleration approach of structural dynamics. The design input in this procedure is defined in terms of relative acceleration and relative velocity spectra. The relative spectra can be related to pseudo spectra. For a given number of modes to be included in the analysis the new SRSS rule proposed here will reduce the error due to the so-called ‘missing mass’ effect and predict a more accurate response value than the rules which use pseudo spectra as input, for systems either with or without dominant high frequency mode effects.     

Development of basic technique to improve seismic response accuracy of tributary area-based lumped-mass stick models

Although a detailed finite element(FE) model provides more precise results, a lumped-mass stick(LMS) model is preferred because of its simplicity and rapid computational time. However, the reliability of LMS models has been questioned especially for structures dominated by higher modes and seismic inputs. Normally, the natural frequencies and dynamic responses of a LMS model based on tributary area mass consideration are different from the results of the FE model. This study proposes a basic updating technique to overcome these discrepancies; the technique employs the identical modal response, D(t), to the detailed FE model. The parameter D(t) is a time variable function in the dynamic response composition and it depends on frequency and damping ratio for each mode, independent of the structure's mode shapes. The identical response D(t) for each mode is obtained from the frequency adaptive LMS model; the adaptive LMS model which can provide identical modal frequencies as the detailed FE model. Theoretical backgrounds and formulations of the updating technique are proposed. To validate the updating technique, two types of structures(a symmetric straight column and an unsymmetric T-shaped structure) are considered. From the seismic response results including base shear and base moment, the updating technique considerably improves the seismic response accuracy of the tributary area-based LMS model.     

卵形消化池-桩-土相互作用地震反应分析

本文对卵形消化池-桩-土相互作用进行地震反应分析。从分枝模态——二步分析法的基本概念入手,将卵形消化池简化为振动特性和实际结构相同的简化结构,求得上、下部结构之间的耦合项,将耦合项和地震动一起组成修正地震动,以此作为刚性地基上卵形消化池地震反应分析的地震输入,从而求得考虑相互作用影响的卵形消化池地震反应。计算结果表明,在罕遇地震作用下,虽然部分土体进入塑性,池体仍处于弹性工作阶段,并且考虑相互作用影响后得到的卵形消化池地震反应,无论是池体位移还是池壁内力均要小于不考虑相互作用的。本文为解决复杂的卵形消化池-桩-土相互作用问题提供了一种较为实用的分析方法。     

A method for stochastic seismic response analysis of non-classically damped structures

A method to calculate the stationary random response of a non-classically damped structure is proposed that features clearly-defined physical meaning and simple expression. The method is developed in the frequency domain, The expression of the proposed method consists of three terms, i.e., modal velocity response, modal displacement response, and coupled (between modal velocity and modal displacement response), Numerical results from the parametric study and three example structures reveal that the modal velocity response term and the coupled term are important to structural response estimates only for a dynamic system with a tuned mass damper. In typical cases, the modal displacement term can provide response estimates with satisfactory accuracy by itself, so that the modal velocity term and coupled term may be ignored without loss of accuracy, This is used to simplify the response computation of non-classically damped structures. For the white noise excitation, three modal correlation coefficients in closed form are derived. To consider the modal velocity response term and the coupled term, a simplified approximation based on white noise excitation is developed for the case when the modal velocity response is important to the structural responses. Numerical results show that the approximate expression based on white noise excitation can provide structural responses with satisfactory accuracy~