Approximate Absorption Corrections for Complete SH Seismograms

An approximate absorption correction is proposed that can be applied retroactively to the response of a perfectly elastic medium, not decomposed into elementary waves, for example in the finite-difference method. As the duration of excitation decreases, the approximate correction approaches the exact, and becomes quite exact for impulsive excitation. The correction makes use of the perfectly elastic response and dissipation operator, but their relation is not simply convolutory. The method is restricted to a spatially constant Q with an arbitrary power-law frequency dependence. The advantage of the approximate solutions, as compared to the exact, in which the absorption is included in the equation of motion, is in a considerable saving of computer time if several absorption models (or several Q values) are repeatedly applied to the same structure. The simplest computations are for Q linearly dependent of the frequency, where the absorption correction reduces to an exponential time windowing.     

THE PERFECTLY REFLECTING WEDGE USED AS A CONTROL MODEL IN SEISMIC DIFFRACTION MODELLING*

The numerical modelling of seismic diffraction, e.g., at faults and other discontinuities, generally requires the use of fast approximate methods. The geophysicist responsible for the development of such numerical methods has a real need of exact solutions to certain ideal geometries to check the accuracy of his calculations. One such exact solution, which is available, is the acoustic wave solution to the perfectly reflecting wedge. The solution is three-dimensional and the source is an explosive point source. This model is ideal for seismic diffraction; the solution has the advantage of being exact, truly three-dimensional and of being in the convenient form of the temporal and spatial impulse response. More complicated sources which are extended in either space or time can, therefore, be modelled exactly by numerical integration. This paper presents some examples of the use of the perfectly reflecting wedge as a control model for an asymptotic high frequency diffraction modelling method. This control model has revealed that certain survey and wedge configurations can yield significant disagreement with, e.g., the Kirchhoff approximation. Such configurations could occur during VSP modelling when the survey lies in the near field or in the shadow zone of a high contrast fault. This control model has also been instructive in demonstrating why the high frequency, asymptotic, approximation is generally very good and has indicated a possible improvement to the Kirchhoff approximation for edge diffraction.     

基于精确权函数的仪器响应失真校正方法与算例

当前,经仪器响应失真校正后,加速度计有效观测频带通常为0至1倍自振频率,这限制了强震仪应用领域的延拓。有鉴于此,本文提出一个基于精确权函数的仪器响应失真校正方法并给出一个验证算例。分析表明:较之现有方法,新方法兼有高精度和高效率的优点,并可以将力平衡式加速度计的有效观测频带至少延拓3倍,对于拓展强震仪的应用领域具有重要意义。     

完全匹配层吸收边界在孔隙介质弹性波模拟中的应用

模拟弹性波在孔隙介质中传播,需要稳定有效的吸收边界来消除或尽可能的减小由人工边界引起的虚假反射. 本文在前人工作基础上,首次建立了弹性孔隙介质情况下完全匹配层吸收边界的高阶速度-应力交错网格有限差分算法,并详细讨论了完全匹配层的构建及其有限差分算法实现. 首先,本文通过均匀孔隙模型的数值解与解析解的对比,验证所提出的数值方法的正确性;然后,本文考察了完全匹配层对不同入射角度入射波和自由表面上的瑞利波的吸收性能,将完全匹配层与廖氏和阻尼吸收边界进行了对比,研究了这三种吸收边界在不同吸收厚度情况下对弹性波吸收能力. 数值结果表明,在孔隙介质中,完全匹配层作为吸收边界能十分有效地吸收衰减外行波,无论对体波还是面波,是一种高效边界吸收算法.     

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.     

论架空结构的震动

本文所謂架空結構是指搁置在較高的支墩上的連續梁結構,例如高架橋梁或水電站的架空輸水管道等。這類結構在承受震動荷載（例如地震）時,常會引起較大的變形和应力。如支墩較粗矮,則在計算較低的頻率時,其作用可近似的以一彈簧支座代替之,而忽略支墩本身的結構震動影響。這是一個常用的假定,可以使計算工作充分簡化。本文首先對這一情况作了詳盡討論,給出较完整的解答,包括起始條件影響在内,並提出用“載常數”及“形常數”的觀念來進行計算。然后本文給出考虑支墩的震動影響時的精確解答,分別就縱横方向的震動進行討論,並指出近似計算公式和精確公式间的關係,近似公式只能在支墩較剛固和計算较低階的頻率時才適用。文中列出了結構作横向諧和震動時的特別解答,可見其計算工作量將比近似法增加不少。     

A unified formulation of the piecewise exact method for inelastic seismic demand analysis including the P‐delta effect

The non‐linear analysis of single‐degree‐of‐freedom (SDOF) systems provides the essential background information for both strength‐based design and displacement‐based evaluation/design methodologies through the development of the inelastic response spectra. The recursive solution procedure called the piecewise exact method, which is efficiently used for the response analysis of linear SDOF systems, is re‐formulated in this paper in a unified format to analyse the non‐linear SDOF systems with multi‐linear hysteresis models. The unified formulation is also capable of handling the P‐delta effect, which generally involves the negative post‐yield stiffness of the hysteresis loops. The attractiveness of the method lies in the fact that it provides the exact solution when the loading time history is composed of piecewise linear segments, a condition that is perfectly satisfied for the earthquake excitation. Based on simple recursive relationships given for positive, negative and zero effective stiffnesses, the unified form of the piecewise exact method proves to be an extremely powerful and probably the best tool for the SDOF inelastic time‐history and response spectrum analysis including the P‐delta effect. A number of examples are presented to demonstrate the implementation of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

从反应谱求功率谱的精确方法及其应用

本文对由反应谱求功率谱的近似公式的误差进行了讨论,从而说明了精确求解地震动功率谱的必要性,并从构造函数的特征出发,提出了一类符合要求的构造函数族,同时对其参数的取值加以讨论。最后给出了一个算例,展示了从反应谱求功率谱的精确方法与近似方法在数值上的差别,以及对人造地震动的影响。     

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

Dynamic interaction between 3-D rigid surface foundations and comparison with the ATC-3 provisions

The dynamic behaviour of a system of three-dimensional, massless, rigid, surface foundations of arbitrary shape perfectly bonded to the elastic half-space is numerically studied with the frequency domain boundary element method. This method employs the dynamic Green's function for the surface of the half-space and this results in a discretization of only the soil-foundation interfaces. In addition, use of isoparametric quadratic quadrilateral boundary elements increases the accuracy of the method, which is confirmed by comparison with other known numerical solutions. Externally applied loads, harmonically varying with time, are considered. The through the soil coupling effect between the foundations as a function of distance and frequency is assessed through extensive parametric studies involving two and four rigid foundations being isolated or interconnected. It is found that the assertion of ATC-3 regulations that omission of coupling effects leads to conservative results is not always correct for all frequencies.     

内共振对斜拉索面内瞬时相频特性的影响

为研究内共振对斜拉索的瞬时相频特性的影响,利用多尺度法对拉索面内分布激励下的运动方程进行摄动求解,分别得到不计内共振和计内共振2种状态下面内主共振响应的近似解析式以及其稳态方程,最后采用Hilbert变换分别得到两者面内响应与激励的瞬时相位差及其幅值.同时在小垂度条件下,研究了不同索力下内共振对拉索面内响应与激励的瞬时...     

基于Pasternak地基模型和Adomian分解方法的扩底桩水平动力响应分析

工程上广泛采用基于Winkler模型的层状地基反力系数法对桩土水平动力响应进行分析,该方法忽略了地基土剪切作用的影响,与工程实际有一定偏差。另外,对桩土的非线性相互作用和如扩底桩、楔形桩等变截面桩问题常用的传递矩阵法或中心差分法,计算过程较为繁琐。基于Pasternak地基模型和Adomian分解方法,提出一种考虑地基土剪切作用的桩土水平动力相互作用近似计算方法,该方法计算简便且结果精度较高,对变截面桩问题有很好的适用性;并基于该方法,对扩底桩水平动力响应问题和影响因素进行分析。结果指出,扩底半径和上部桩周土弹性模量对扩底桩水平动力响应影响较大,随着扩底半径的增加和桩周土弹性模量的增大,扩底桩水平振动位移幅值逐渐减小。另外,在较低频率的荷载激励下,应考虑土层对桩的剪切作用。     

基于小波变换的结构地震响应与能量计算分析

多分辨分析可将地震信号分解到不同的频段。本文推导了运用多分辨分析计算多自由度体系地震响应的基本公式,讨论了各频段地震信号及结构响应的能量分配。最后,通过多自由度体系的算例验证了弹性体系在原始地震作用下的动力响应可以由该地震作用在时域的各小波分量的动力响应叠加而得,同时将高频抑制后的重构信号应用于近似模型的计算,并利用能量分析明确了地震信号与结构响应在各频段的能量分配。     

Seismic response of base-isolated structures with vertical-rocking coupling

A base-isolated building is liable to have a small horizontal eccentricity between the centre of mass of the superstructure and the centre of rigidity of the supporting bearings. In seismic analysis, the structure is modelled as a rigid block with tributary masses supported on massless elastomeric rubber bearings placed at a constant elevation below the centre of mass. This simplified system has three degrees of freedom: two translations and one rotation in the vertical plane. The investigation of the dynamic behaviour of a base-isolated building is carried out for both the detuned and the perfectly tuned cases. In the detuned case, the natural frequencies of the system are assumed to be well separated. In the perfectly tuned case, the uncoupled rocking frequency is assumed to be identical to the vertical translational frequency, which may result from an unusual mass distribution and/or an extreme aspect ratio of the superstructure. Perturbation methods are implemented in finding the dynamic characteristics for both cases. However, the dynamic response of the perfectly tuned case is the major concern in this investigation. The Green's functions for the displacement response of the three-degree-of-freedom system are derived for both the undamped and damped conditions. The response spectrum modal superposition method is used in estimating the maximum acceleration response. A simple method, accounting for the effect of closely spaced modes, is proposed for combining modal maxima and results in an approximate solution corresponding to a single-degree-of-freedom system. This approximate solution may be used for the preliminary design of a base-isolated structure. Numerical results for a base-isolated building subjected to the vertical component of the El Centro earthquake of 1940 were carried out for comparison with these analytical results. The proposed modal combination method showed superiority over the conventional Square Root of the Sum of the Squares method in estimating maximum responses. The results also indicated that the approximate single-degree-of-freedom system yields accurate estimations. It is shown that the effect of rocking coupling on the vertical response of base-isolated structures subjected to transient loadings, such as earthquake motions, can generally be neglected as a result of the combined effects of the time lag between the maximum translational and rotational responses and the influence of damping in the isolation system, which for elastomeric bearings can be as high as 8 to 10 per cent of critical.     

Uniform half-plane elastodynamic problems by an approximate boundary element method

In this paper the applicability of an approximate Boundary Element Method to uniform half-plane elastodynamic problems is investigated. This method employs the concept of images to construct approximate fundamental solutions for the half-plane and does not require any half-plane surface discretization. The method is formulated in the frequency domain for the case of harmonic disturbances or the Laplace transform domain for the case of transient disturbances. In the latter case a numerical inversion of the transformed solution is necessary to obtain the time domain response. The proposed method can be used as an alternative to boundary element methods that either utilize the infinite plane fundamental solution and thus require a half-plane surface discretization, or employ the exact half-plane fundamental solution, which even though leads to no surface discretization, is of a very lengthy and complicated form. Two characteristics numerical examples are used to illustrate the proposed method and study its advantages and disadvantages.     

On planar seismic wavefront modeling for estimating rotational ground motions: Case of 2-D SH line-source

At large hypocentral distances, it is convenient to approximate the curved transient seismic wavefronts as planar to estimate rotational ground motions from the single-station recordings of translational ground motions. In this paper, we investigate whether and when this approximation, referred to as the ‘plane-wave’ approximation, can be considered adequate close to the source. For this, we consider a simplistic source model comprising a two-dimensional, kinematic shear dislocation SH line-source buried in a homogenous, elastic half-space and assume this to be an equivalent representation of a finite-sized fault. The ‘plane-wave’ rotational motion is then synthesized from the exact translational motion solution to the assumed model and compared with the exact rotational motion solution for this model. The comparison between the two sets of rotational amplitudes in frequency domain suggests that the plane-wave approximation may be adequate, when the wavelengths of the seismic waves are much smaller than the source depth. When this is not true, the plane-wave approximation is seen to underestimate the Fourier amplitudes close to the source by several orders, particularly when the fault planes are vertically oriented. A similar comparison in the time domain indicates that a severe underestimation may also occur when the source rise time is longer than the shear-wave arrival time at the epicenter. Significant discrepancies are also observed between the waveforms of the exact and plane-wave rotational motions.     

Investigation of the dynamic response of rigid footings on tensionless Winkler foundation

The dynamic response of a rigid footing resting on an elastic tensionless Winkler foundation is examined. A parametric investigation, concerning the effect of the main parameters on the response, is performed for harmonic excitation. The parameters examined include the stiffness and the damping of the foundation, the excitation frequency and the superstructure characteristics and loads. The maximum rocking response, the minimum length of contact after uplift, the maximum stress developed at the soil and the factor of safety with respect to the bearing capacity of the soil are used to measure the effect of each dimensionless parameter. An example for earthquake excitation is also given for a plane frame. The results are compared to the ones of a simplified static approach based on the maximum values of the applied loads, similarly to the procedure that is usually applied in practice. The results show that the static approach can predict the response satisfactorily if resonance does not happen, if the stiffness of the foundation is not large compared to the stiffness of the superstructure and if the dynamic part of the axial force of the column is not large; in these cases, it may underestimate or overestimate the response significantly, depending on the sign of the dynamic axial force that is considered.     

Seismic dynamic response by approximate methods

Current seismic safety evaluation for earth dams relies on approximate methods of analysis for prediction of non-linear, transient, dynamic response. One of these approximate methods uses a Strain Reduction Factor (SRF), and has been widely applied in a variety of one-dimensional and two-dimensional soil structural analyses. A second method considered, Equivalent Temporal Damping (ETD), has been previously applied to several seismic dynamic analyses of earth dams. The relative accuracy of the two methods is assessed by comparing them with incremental plasticity, nearly exact numerical solutions. For a simple shear element subjected to deterministic and non-stationary random input accelerations, a serious overprediction of the maximum peak shear response occurs by the SRF method, whereas the ETD results agree very closely with the incremental plasticity solutions. The SRF and ETD methods are also applied to seismic dynamic response analysis of a Vertical Soil Column system, and the same trends as established in the previous case are observed. It is concluded that, in lieu of combined incremental plasticity and finite difference or finite element numerical solutions, the ETD approach is the more accurate of the methods tested for seismic dynamic response analysis of earth dams.     

粘弹性VTI介质地震波模拟特征分析

本文首先利用有限差分法分别对弹性和粘弹性VTI介质进行地震波传播数值模拟,并针对波场快照和波场记录特征,分析不同品质因子组合对波场能量衰减和频率吸收作用的影响.结果表明：对应于膨胀滞弹性形变的品质因子变化主要影响qP波的能量衰减;对应于剪切滞弹性形变的品质因子变化主要影响qSV波的能量衰减;对于qSH波,两个品质因子分别对应于垂直和水平方向的能量衰减;品质因子较小时,qSV波和qSH波的频率向低频方向移动,qP波频率变化不明显.     

基于小波变换的层间隔震结构地震响应分析

利用小波多分辨率分析将地震动加速度分解为多频段小波分量,并运用复模态方法推导其计算层间隔震体系在地震作用下的动力响应公式,讨论各频段地震信号及结构响应的能量分配。同时利用小波时频工具分析地震动能量在时频域内的分布对层间隔震结构响应的影响,进而为考察地震动非平稳性对层间隔震结构非线性分析的影响提供方法。利用小波分析的以上优势,对一典型层间隔震结构分别进行弹性和弹塑性分析,结果表明弹性体系在地震作用下的响应可由该地震波各小波分量的响应叠加而得,地震动能量在时间上的集中会对层间隔震结构响应产生不利影响。