Dynamic stiffness and kinematic response of single piles in inhomogeneous soil

The effect of soil inhomogeneity on dynamic stiffness and kinematic response of single flexural elastic piles to vertically-propagating seismic SH waves is explored. A generalized parabolic function is employed to describe the variable shear wave propagation velocity in the inhomogeneous stratum. A layered soil with piece-wise homogeneous properties is introduced to approximate the continuous inhomogeneity in the realm of a Beam-on-Dynamic-Winkler-Foundation model. The problem is treated numerically by means of a layer transfer-matrix (Haskell–Thompson) formulation, and validated using available theoretical solutions and finite-element analyses. The role of salient model parameters such as pile-head fixity conditions, pile-to-soil stiffness ratio, surface-to-base shear wave velocity ratio and rate of inhomogeneity is elucidated. A new normalization scheme for inertial and kinematic response of such systems is presented based on an average Winkler wavenumber. With reference to long piles in moderately inhomogeneous soils, results indicate that: (a) kinematic pile response is essentially governed by a single dimensionless frequency parameter accounting for pile-to-soil stiffness ratio, pile slenderness and soil inhomogeneity and (b) definition of a characteristic pile wavelength allows an approximate estimation of pile elastodynamic response for preliminary design or analysis. Issues related to domain discretization and Winkler moduli are discussed.     

Modeling shear rigidity of stratified bedrock in site response analysis

Where a distinct soil-rock interface exists, the bedrock medium is commonly treated as elastic half-space and the bedrock surface as the lower boundary of the soil-column model for site response analyses (or the lower boundary of the finite element model for soil-structure interaction analyses). While shear wave velocity in bedrock varies with depth, there has been no consensus amongst scientists and practitioners over the value of “effective depth” into bedrock at which the “half-space” shear wave velocity value should be taken for modeling purposes. This paper reports an interesting and important observation that the effective depth into bedrock is sensitive to the shear wave velocity profile of the overlying soil sediments. A simple and heuristic method, namely Resonant Period Equivalence (RPE) Method, is proposed herein for representing a stratified elastic bedrock of inhomogeneous properties by an equivalent homogeneous elastic half-space medium, which is characterized by a single equivalent shear wave velocity (V_R) value. The proposed calculation method has been verified by extensive comparative analyses involving the use of programs SHAKE and NERA and employing the complete shear wave velocity models of both the soil sediments and the underlying stratified bedrock.     

Effect of surface stress on magneto-elastic surface waves in finitely conducting media

This paper deals with the investigation of the effect of surface stress and conductivity on the propagation of surface wave in isotropic, homogeneous, elastic media under the action of a primary magnetic field. Formulation of the general surface wave propagation problem has been made, and the corresponding frequency equation has been derived. Frequency equations for Rayleigh wave, surface shear wave and Stoneley wave have been deduced from that of general surface wave as special cases. The effects of surface stress and magnetic field on the wave velocities and attenuation factors of Rayleigh wave and surface shear wave are shown by numerical calculations and graphs. Some important wave velocity equations, as obtained by other authors, have been deduced as special cases from the wave velocity equation for Stoneley wave. It is found that the combined effect of surface stress and magnetic field modulates the wave velocity ratios and attenuation factors of Rayleigh wave and surface shear wave to a considerable extent.     

层状半空间中洞室对平面SH波的放大作用

利用间接边界元法,求解了弹性层状半空间中洞室对入射平面SH波的放大作用问题,并以基岩上单一土层中洞室对入射平面SH波的放大作用为例进行了数值计算分析。本文模型的特点之一是考虑了层状场地的动力特性,因而更接近于实际工程;特点之二是计算精度非常高。研究表明,层状半空间中洞室对波的放大作用与均匀半空间中情况有着本质的差别;层状半空间中洞室附近地表动力响应由土层动力特性和洞室对波的散射二者共同决定。土层动力特性不仅影响洞室附近地表位移的幅值,还会影响地表位移的频谱。在土层的前几阶共振频率附近,随着基岩与土层剪切波速比的增大,土层的影响随之增大,而随着土层厚度的增加,土层的影响随之减小,并逐渐趋于均匀半空间情况。     

Love wave dispersion: A comparison of results for a semi-infinite medium with inhomogeneous layers and for its approximation by homogeneous layers

Love wave dispersion in various semi-infinite media consisting of inhomogeneous layers is discussed. The phase and group velocities are computed when shear wave velocity and density in each inhomogeneous layer are varying exponentially with depth. At the beginning one or two inhomogeneous layers over a homogeneous semi-infinite medium are considered. The dispersion results for these structures are compared with those for their approximations with homogeneous layers. Comparisons show that differences of phase and group velocities for the original models from those for their approximated models (i) increase with the increase of wave number and (ii) are larger for group velocity than for phase velocity. The difference is approximately proportional to the rate of change of parameters in the layers. Finally, dispersion curves are obtained for model IP3MC, which consists of many inhomogeneous and homogeneous layers over a homogeneous semi-infinite medium. The results are compared with the observed group velocity data across the Indian Peninsula.     

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     

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．     

土体参数对地表加速度峰值和反应谱的影响

运用一维土层地震反应的等效线性化波动方法,研究了单层均质土密度、动剪切模量比和阻尼比、土层厚度和土层剪切波速等参数变化对土层地表加速度峰值和反应谱的影响,所得到的结论对于指导地震安全性评价中的勘察试验工作、提高土层地震反应的计算精度具有一定的参考价值。     

回填土不确定性对场地地震反应的影响

选取某核电场地控制性钻孔的厚度、剪切波速、密度等实际勘探数据,通过改变回填土剪切波速,分析了回填土不确定性对场地地震动参数的影响。研究结果表明:在回填土层厚度不变和模型总厚度不变的情况下,地表的水平向峰值加速度随着回填土剪切波速的增大而减小,但水平向峰值加速度增幅逐渐减小;回填土剪切波速到达一定的波速就不再影响地表水平峰值加速度;随着回填土剪切波速的增加,整个反应谱的谱值都普遍减小。     

Characterisation of shear wave velocity profiles of non-uniform bi-layer soil deposits: Analytical evaluation and experimental validation

A crucial aspect of physical geotechnical model tests (under both 1-g and n-g conditions) is the evaluation of the initial (low-strain) stiffness of the soil layers of the sample test deposit, especially in the case of coarse materials. While for uniform soil deposits this issue can be addressed in a straightforward manner, e.g. by determining the fundamental frequency through the transfer function of an applied white-noise excitation, the problem becomes cumbersome for multi-layered deposits. After reviewing a number of available theoretical solutions, this paper illustrates a simplified yet reliable analytical procedure for determining the shear wave velocity profile (V_s) in a single or bi-layer deposit, taking into account the inhomogeneity of the individual soil layers, under the hypothesis of vanishing shear modulus at ground surface. The fundamental natural frequency of the inhomogeneous bi-layer deposit is analysed using the Rayleigh quotient procedure. The associated shape function is evaluated by considering the equilibrium of the soil column under a pseudo-static lateral inertial excitation imposed at its base, accounting for both layering and inhomogeneity. A validation of the proposed method is provided by comparing numerical results obtained from both time- and frequency- domain analyses against experimental data on Leighton Buzzard sand, from a recently-completed research project conducted on the shaking table facility at BLADE Laboratory, University of Bristol (UK).     

Cone models for nearly incompressible soil

Cones can be used to model soil in a unified strength-of-materials approach. For the vertical and rocking motions involving predominantly compressional-extensional deformation, the corresponding dilatational wave velocity tends to infinity for Poisson's ratio approaching 1/2. Based on the rigorous solution for the dynamic stiffness of a rigid disk for all frequencies, whereby the partition of the power among P-, S- and Rayleigh waves is also discussed, two special features are necessary for the vertical and rocking motions for nearly incompressible soil with Poisson's ratio between 1/3 and 1/2: (1) The appropriate wave velocity is selected as twice the shear wave velocity and not as the dilatational wave velocity; (2) A trapped mass which increases linearly with Poisson's ratio is introduced. The trapped mass can be assigned to the base mat, allowing the cone model to be constructed in the same way for all Poisson's ratios. The realization of cone models for surface foundations on a homogeneous half-space and on a layer on a flexible half-space and for embedded and pile foundations is addressed.     

非匀质场地地震反应的模态叠加法解

将模态叠加法应用于刚性基岩上、受竖直向上传播的稳态剪切地震波激振下剪切刚度沿深度按指数规律增长的单层非匀质场地土模型的地震反应分析,求得有效振型参与系数的解.在算例分析中将此类非匀质场地与剪切波速取为平均值的匀质场地的有效振型参与系数进行了比较,并验证了第一阶固有频率的工程估算式仍可用于此类非匀质场地,且其对估算高阶固有频率是个很重要的参数.     

Surface waves in isotropic,laterally inhomogeneous media

Summary Two different viewpoints of the phase velocities of the elastic surface waves in isotropic, laterally inhomogeneous media have led to inconsistent results. Arguments in terms of surface wave modes give the conclusion that the phase velocity is independent of the propagation direction, while the outcome of calculations based on a constructive interference of body waves in a surface layer is that the phase velocity is dependent on the propagation direction. Both arguments are summarized and an error in the calculations giving dependence is pointed out. The calculations and observations of surface wave amplitude changes in laterally inhomogeneous media are also summarized.     

Seismic earth pressures on rigid and flexible retaining walls

While limiting-equilibrium Mononobe–Okabe type solutions are still widely used in designing rigid gravity and flexible cantilever retaining walls against earthquakes, elasticity-based solutions have been given a new impetus following the analytical work of Veletsos and Younan [23]. The present paper develops a more general finite-element method of solution, the results of which are shown to be in agreement with the available analytical results for the distribution of dynamic earth pressures on rigid and flexible walls. The method is then employed to further investigate parametrically the effects of flexural wall rigidity and the rocking base compliance. Both homogeneous and inhomogeneous retained soil is considered, while a second soil layer is introduced as the foundation of the retaining system. The results confirm the approximate convergence between Mononobe–Okabe and elasticity-based solutions for structurally or rotationally flexible walls. At the same time they show the beneficial effect of soil inhomogeneity and that wave propagation in the underlying foundation layer may have an effect that cannot be simply accounted for with an appropriate rocking spring at the base.     

多年冻土区桥梁的地震反应

首先研究了冻土区桥梁的抗震计算方法,对波动法和惯性力法的计算结果进行厂对比分析。对比结果表明惯性力法的计算结果偏大,但当基岩的剪切波速很大时,两种方法的结果接近。然后,采用波动法对季节性冻土区和多年冻土区桥梁结构进行了地震反应分析,研究了冻土层的变化对桥梁结构地震反应的影响,得到了不同冻土厚度、不同墩高时桥墩地震内力分布的规律。计算结果表明,冻土层的存在对桥梁的地震反应具有显著影响,桥墩的地震反应在冬夏两季具有显著区别。     

Love waves in inhomogeneous and anisotropic earth — I

Summary The frequency equation is derived for the propagation of Love waves in the earth's crust, composed of transversely isotropic layers and overlying anisotropic and inhomogeneous mantle. The exact boundary value problem is solved for a single layer and extended to multilayered media by generalizing theHaskell's technique. In fact the problem of deriving the frequency equation has been reduced to finding out the solution of the equation of motion subject to the appropriate boundary conditions. To illustrate the method, the author has derived frequency equations of Love waves for linear, exponential and generalized power law variation of vertical shear wave velocity with depth in the half space overlain by transversely isotropic inhomogeneous stratum.     

剪切波速对场地地表地震动参数的影响

本文以江淮地区典型场地资料为原型,将土层剪切波速实测值按照一定比例进行增减,构造多种场地土层地震反应分析模型,选择Taft、E1centro和Kobe三条强震记录作为地震输入,采用一维频域等效线性化波动方法进行了土层地震反应分析.研究结果表明,剪切波速的变异性与场地地表地震动的影响程度与输入基岩地震动的频谱特性、幅值、土层结构等因素有关.地表峰值加速度随着剪切波速的增大而逐渐增大,地表加速度反应谱的特征周期随着剪切波速的增大而逐渐减小.     

Centrifuge modeling of seismic response of layered soft clay

Centrifuge modeling is a valuable tool used to study the response of geotechnical structures to infrequent or extreme events such as earthquakes. A series of centrifuge model tests was conducted at 80g using an electro-hydraulic earthquake simulator mounted on the C-CORE geotechnical centrifuge to study the dynamic response of soft soils and seismic soil–structure interaction (SSI). The acceleration records at different locations within the soil bed and at its surface along with the settlement records at the surface were used to analyze the soft soil seismic response. In addition, the records of acceleration at the surface of a foundation model partially embedded in the soil were used to investigate the seismic SSI. Centrifuge data was used to evaluate the variation of shear modulus and damping ratio with shear strain amplitude and confining pressure, and to assess their effects on site response. Site response analysis using the measured shear wave velocity, estimated modulus reduction and damping ratio as input parameters produced good agreement with the measured site response. A spectral analysis of the results showed that the stiffness of the soil deposits had a significant effect on the characteristics of the input motions and the overall behavior of the structure. The peak surface acceleration measured in the centrifuge was significantly amplified, especially for low amplitude base acceleration. The amplification of the earthquake shaking as well as the frequency of the response spectra decreased with increasing earthquake intensity. The results clearly demonstrate that the layering system has to be considered, and not just the average shear wave velocity, when evaluating the local site effects.     

广西柳州地区常见土类剪切波速与埋深之间的相关性

基于广西柳州地区地震安全性评价中实测所获的钻孔资料,利用线性模型、幂函数模型和二次函数模型分别对该地区土层剪切波速与埋深之间的相关性进行了拟合分析,通过对比发现幂函数模型为二者间相关性拟合的最优选择,同时探讨了土体状态对二者相关性的影响。结果表明:除人工填土外,柳州地区内常见土层剪切波速与埋深之间具有较强的相关性,区域性对其相关性也具有影响。最后以实测钻孔为例,验证了本文模型的预测精度和可靠性,而且模型的预测精度可以通过区分土体状态得到明显提高。      

Seismic response of multi-layered basins with velocity gradients upon incidence of plane shear waves

A boundary integral scheme based on boundary-integral discrete-wave-number approach has been developed to compute the seismic response of two-dimensional irregular-shaped basins with horizontal soil layers. Each layer exhibits a linear gradient of shear wave velocity with depth. The approach combines the boundary-integral representation of the seismic wave field outside the basin with plane wave representation of the seismic wave field inside the basin. The propagation throughout the layers is performed by matrix propagators in which the effect of the vertical variation of the velocity is incorporated by using confluent hyper-geometric functions of the Whittaker type. Our method is tested against otherwell-accepted solutions for the case of a circular basin with excellent agreement. Test of the ground response for a semi-circular basin with radius a shows that stable solutions are obtained if the chosen model parameters satisfy following conditions: (1) the distance from the sources to the interface is greater than 0·1a; (2) the distance between the sources is smaller than a quarter of the incident wavelength; and (3) the discrete wave-number step is smaller than 2π/4a. The computation of ground response of basins with a sharp interface and several horizontal deposits leads to the following main results: (1) the amplification of a basin with velocity gradients is larger than that of a basin with homogeneous layers; (2) the frequencies of the second- and third-order harmonics for a basin with velocity gradients are lower than those of a basin with homogeneous layers; and (3) the response amplitude of the basin with velocity gradients attenuates more slowly in time domain than when layers are homogeneous. Since these results have been obtained for realistic values of basin geometrical and mechanical consideration, they should find some interest in earthquake engineering or seismic microzonation studies. © 1998 John Wiley & Sons, Ltd.