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.     

Free-field response from inclined body waves in a viscoelastic random medium

This paper deals with the free-field response of the in-plane motion resulting from a combination of inclined incident body waves. The amplification of waves in a viscoelastic layer with stochastic changes in the elastic properties and density is investigated. The method used is that of Karal and Keller and is based on the idea of the fundamental matrix. The third order correlations are neglected. The resulting integro-differential equations for the average displacements are solved by the Laplace transform. Generally, analysis indicates that the stochastic changes in the shear modulus and density enhance the damping in a significant manner. However, increases in the waves' amplification can arise in the case of a small dimensionless frequency and uncorrelated stochastic changes of material parameters.     

Insight on 2D- versus 3D-modelling of surface foundations via strength-of-materials solutions for soil dynamics

To simplify the analysis, three-dimensional soil–structure interaction problems are often modelled by considering a two-dimensional slice without changing the material properties of the soil. This procedure, although convenient, is of questionable validity because two-dimensional modelling inherently overestimates the radiation damping for translational and rocking motions. To make matters worse, two-dimensional modelling always entails an underestimation of the dynamic-spring coefficient for the translational motions. The damping ratio of the two-dimensional case, which is proportional to the ratio of the damping coefficient to the spring coefficient, will thus be even larger. Thus, reliance upon a two-dimensional analysis based on an equivalent slice of a strip foundation may result in a dangerously non-conservative design. Valuable insights into the essence of radiation damping and the difference between two-dimensional and three-dimensional models may be obtained via approximate strength-of-materials solutions based on cone–wedge models and travel-time considerations. By examining the decay of the waves along the axes of the cone–wedge models, the essence of radiation damping can be grasped. The heuristic concept of more spreading of waves in three dimensions than in two is misleading. Indeed, just the opposite is true: The less the amplitude spreads and diminishes with distance, the greater is the radiation damping. Because the damping ratio is grossly overestimated, two-dimensional modelling of a three-dimensional case cannot be recommended for actual engineering applications. It is more feasible to take the opposite approach and idealize slender soil–structure interaction problems with a radially symmetric model. As an alternative, when defining the equivalent slice of the two-dimensional strip foundation, the impedance of the soil can be changed to achieve a much better agreement of the high-frequency limits of the damping coefficients. In the low-frequency range this modified two-dimensional model also overestimates radiation damping, although to a lesser extent. As a by-product, the dimensions of the equivalent slice of a two-dimensional strip foundation are discussed; and equations for the aspect ratios determining the opening angles of the corresponding wedges are derived. Also addressed is the quite separate but related topic of the transition from square to slender rectangular foundations.     

Influence of scattering of SH-waves in dynamic interaction of shear wall with soil layers

The dynamic soil-structure interaction of a shear wall embedded in elastic isotropic and homogeneous soil layers underlain by bedrock, subjected to SH waves, is modeled in the present article. The soil layers consist of irregular interfaces and it has been shown that the scattering due to the roughness of the layers has significant effect on the displacement of both the foundation and the shear wall. To demonstrate the phenomena indirect boundary element method(IBEM) has been used on the basis of its validation in previous problems of similar type. The system response is compared with the analytical solution of the same type of model for vertically propagating incident SH waves. It is observed that for the low frequency of wave, displacement is abruptly high, and as a result the combination of shear wall and foundation perceives resonance. The thickness of the soil layer, mass of the shear wall, stiffness of the bedrock and the soil layers all affects the system frequency and displacement.     

Displacement solutions for dynamic loads in transversely-isotropic stratified media

Solutions for the displacements caused by dynamic loads in a viscoelastic transversely-isotropic medium are derived. The medium extends horizontally to infinity, but is bounded below by a rigid base. Stratification of the medium presents no difficulties. The medium is discretized in the vertical direction only; discretization in the horizontal direction is obviated by use of analytical solutions to the equations of motion. Application of the displacement solutions to soil-structure interaction is illustrated. A soil flexibility matrix (and hence, a stiffness matrix) for a surface foundation follows directly from the displacement solutions. A simple modification to obtain the soil stiffness for an embedded foundation of arbitrary geometry is described. Stiffnesses of rigid surface and embedded foundations are computed and compared with previously published results. In addition, the dynamic stiffness of a rigid surface foundation on a soil layer with linearly increasing shear modulus is compared to that for a homogeneous soil layer. A reduction in radiation damping is found to result from the inhomogeneity.     

Estimation of soil degradation under prolonged cyclic loading,simulating storm wave effects on a structure

Integrated assessment of possible soil degradation under cyclic loading caused by storm waves includes the analysis of hydrological data; the calculation of anticipated parameters of storm-wave amplitude–frequency spectrum and the corresponding shear forces; strength prediction based on experiments performed using the calculated wave parameters and loads; and the final analysis of soil behavior with regard to the time distribution of waves with different heights during the accepted storm activity period. The soil foundation stability was estimated for a site in the North Caspian shelf.     

A practical method for estimating dynamic soil stiffness on surface of multi‐layered soil

It is important to estimate the influence of layered soil in soil–structure interaction analyses. Although a great number of investigations have been carried out on this subject, there are very few practical methods that do not require complex calculations. In this paper, a simple and practical method for estimating the horizontal dynamic stiffness of a rigid foundation on the surface of multi‐layered soil is proposed. In this method, waves propagating in the soil are traced using the conception of the cone model, and the impulse response function can be calculated directly and easily in the time domain with a good degree of accuracy. The characteristics of the impedance, that is the transformed value to the frequency domain of the obtained impulse response, are studied using two‐ to four‐layered soil models. The cause of the fluctuation of impedance is expressed clearly from its relation to reflected waves from the lower layer boundary in the model. Copyright © 2005 John Wiley & Sons, Ltd.     

In-plane soil–structure interaction in layered,fluid-saturated,poroelastic half-space I: Structural response

Linear in-plane soil–structure interaction in two dimensions (2D) is studied in fluid-saturated, poroelastic, layered half-space using the Indirect Boundary Element Method (IBEM). The structure is a shear wall supported by a rigid embedded foundation. Exact stiffness matrices for the soil layer and half-space, and Green׳s functions of uniformly distributed loads and pore pressure on an inclined line are derived. Results of the system response in the frequency domain are presented for the special case of single soil layer over bedrock, semi-circular foundation and zero seepage force. The effects of water saturation, soil porosity, depth of soil layer, rigidity contrast between layer and bedrock are investigated in the frequency domain for incident plane P- and SV waves. The results suggest that water saturation may cause increase of the system frequency by more than 10%.     

Lumped-parameter model for a rigid cylindrical foundation embedded in a soil layer on rigid rock

To represent a cylindrical rigid foundation vibrating in horizontal, vertical, rocking or torsional motions embedded in a soil layer resting on rigid rock, a lumped-parameter model is described. The coupling between the horizontal and rocking degrees of freedom is considered. For each degree of freedom eight frequency-independent real coefficients determine the springs, dashpots and the mass of the lumped-parameter model with two internal degrees of freedom. These coefficients are specified for various ratios of the radius of the foundation to the depth of the layer and lateral contact ratios. To derive the mechanical properties of the lumped-parameter model a systematic procedure of curvefitting of the dynamic-stiffness coefficient up to, in general, twice the fundamental frequency of the layer is applied, capturing the fact that below the (horizontal) fundamental frequency (cutoff frequency) no radiation of energy occurs. The lumped-parameter model can be used to represent the soil in a standard finite-element program for structural dynamics working in the time domain, whereby the structure can exhibit non-linear behaviour. Stability of the unbounded soil-layer model and of the total system is guaranteed. A hammer foundation with partial uplift of the anvil is analysed for illustration.     

Antiplane response of a dike on flexible embedded foundation to incident SH-waves

The response of an elastic circular wedge on a flexible foundation embedded into a half-space is investigated in the frequency domain for incident pane SH-waves. The problem is solved by expansion of the motion in all three media (wedge, foundation and half-space) in cylindrical wave functions (Fourier-Bessel series). The structural model is simple, but accounts for both differential motions of the base and for the effects of soil-structure interaction. Usually, structural models in earthquake engineering consider either differential ground motion, but ignore soil-structure interaction, or consider soil-structure interaction, but for a rigid foundation, thus ignoring differential ground motion. The purpose of the study is to find how stiff the foundation should be relative to the soil so that the rigid foundation assumption in soil-structure interaction models is valid. The shortest wavelength of the incident waves considered in this study is one equal to the width of the base of the wedge. It is concluded that, for this model, a foundation with same mass density as the soil but 50 times larger shear modulus behaves as ‘rigid’. For ratio of shear moduli less than 16, the rigid foundation assumption is not valid. Considering differential motions is important because of additional stresses in structures that are not predicted by fixed-base and rigid foundation models.     

Effects of site dynamic characteristics on soil–structure interaction (II): Incident P and SV waves

In-plane, dynamic soil–structure interaction (SSI) for incident-plane P and SV waves is analyzed for a two-dimensional (2D) model of a shear wall on a rigid foundation that is embedded in a soil layer over bedrock. The indirect-boundary-element method (IBEM) and non-singular Green's functions of distributed loads on inclined lines are used to solve the problem. Although this in-plane, dynamic SSI problem displays characteristics similar to those of 2D, out-of-plane, dynamic SSI, which was studied in our previous work, there exist some significant differences. In analyses of the SSI of the full-scale structures, which recorded strong earthquake shaking, our interpretations are often based on the peaks in the transfer functions of observed structural response. It is shown in this paper how the amplitudes and the frequencies of those peaks are affected by the relative rigidity and thichness of the soil layer below the foundation.     

Local transmitting boundaries for transient elastic analysis

The aim of this paper is to investigate and develop alternative methods of analyzing problems in dynamic soil–structure-interaction (SSI). The interaction means that the amplitude of structural response is effected by additional energy dissipation through radiation and material damping in the soil. The surrounding soft soil behaves as a natural damper for a massive and stiff structure supported or embedded in it. The main focus is the major difficulty posed by such an analysis — the phenomena of waves that radiate outward from the excited structures towards infinity. In numerical calculations only a finite region of the foundation medium is analyzed and something is done to prevent the outgoing radiation waves from reflecting at the boundary region.Development of a simple and efficient finite element (FE) procedure for the solution directly in the time domain of transient SSI problems is the main concern. The central feature of the procedure is local absorbing boundaries used to render the computational domain finite. These boundaries are local in both time and space and are completely defined by a pair of symmetric stiffness and damping matrices. As the effort for implementing them is the same as for the impedance boundary condition (BC) considering the angle of incidence, standard assembly procedure can be used. Due to the local nature they also preserve the overall structure of the global equations of motion. Even though the focus is in the time domain the same equations of motions can be used to determine the solution under time-harmonic excitation directly in the frequency domain. Explicit formulae for the element matrices are included in the paper and numerical examples for transient radiation model problems to illustrate the validity and accuracy of the new procedures, are given.     

黄土丘陵河谷场地地震动特征研究

研究黄土丘陵河谷场地在地震作用下强地面运动特征的变化情况,可以揭示强震对该类场地上震害的触发机理。结合黄土高原的地貌特征,建立具有代表性的动力数值分析模型,通过输入不同幅值、频谱特性和持续时间的地震波,对起伏地形和覆盖黄土层共同影响下的黄土河谷场地进行地震反应分析。结果表明:黄土层和地形耦合作用控制了地表的PGA变化,使其趋于复杂,在同一输入波不同振幅作用下,与基岩河谷各测点相比,黄土覆盖河谷场地的地震动频谱幅值均有所增加,并且频谱主峰均向高频移动。在不同地震波输入下,场地不同部位的固有频率受地形高程和土层影响;而地震动大小和频谱幅值不仅与场地的基本频谱和地形起伏有关,也与输入地震波的频谱成分相关。输入波PGA与地震频谱特征都不变时,同一场地输出的地震频谱形状具有相似的特征,随着地震持时增长,能量向场地基本频率附近集中,从而可能导致场地上相应频率建筑物震动幅值增加,造成累积破坏。     

Effects of strong lateral discontinuity on ground motion characteristics and aggravation factor

This paper presents the effects of strong lateral discontinuity (SLD) in basins on the ground motion characteristics, differential ground motion (DGM) and aggravation factor (aggravation factor is simply the extra spectral amplification due to complex 2D site effects over the 1D response of the soil column). The seismic responses of open- and closed-basin models with SLD were simulated using SH-wave finite difference algorithm with fourth-order spatial accuracy. Simulated seismic responses, DGM and its spectra revealed that SLD induces Love waves, and the lower cutoff frequency for the same is equal to the fundamental frequency (f₀) of the soil layer. The maximum average aggravation factor (AAF) and DGM were obtained near the edge of open basin and at the centre of closed basin. A decrease of amplitude of Love wave, DGM level and AAF with offset from SLD was observed in an open basin. On the other hand, in closed basin, spatial variation of AAF and DGM level was highly variable. Duration of shaking, AAF and DGM level was more in the closed basin than in an open basin. Increase of DGM level and AAF with decrease of the width of basin was observed.     

循环荷载下液化对土层水平往返变形的影响

采用多工况振动台实验研究液化对土层水平往返变形的影响.以干砂实验为参照,分析孔压增长与土层加速度和土层往返变形之间的关系.结果表明:液化将引起土表加速度显著降低,减小惯性力传递,但同时会引起土层往返剪应变明显增大.对往返变形而言,液化土层往返剪应变就可达到1%～5%的大变形状态,且液化土层往返剪应变沿深度呈下大上小分布.土层中孔压比0.4～0.8是往返变形出现放大的敏感段,在孔压比0.8左右而不是在1.0达到最大.作为其结果,土层液化将对刚性上部结构振动起减震作用,但同时增大的往返剪应变也易导致基础和地下结构破坏,特别是对液化层与下部非液化层交界处的构件更敏感.     

Dynamic stiffness of foundation on layered soil half-space using cone frustums

Approximate dynamic-stiffness coefficients of a disk on the surface of a single layer on a half-space may be calculated using cone models. This concept is generalized to the case of a horizontally stratified site consisting of many layers on a homogeneous half-space. After constructing the so-called ‘backbone cone’ determining the radii of the disks at all interfaces, the dynamic-stiffness matrices of the layers (modelled as cone frustums) and the dynamic-stiffness coefficient of the underlying half-space (modelled as a cone) are assembled to that of the site. The dynamic-stiffness matrix of a layer is a complex-valued function of frequency because radiation of energy in the horizontal direction is considered. In this model of the layered half-space the properties of the cone reproduce themselves (cloning). The advantages of using cone models are also present for the layered half-space; in particular, no transformation to the wave-number domain is performed.     

Cone models for a soil layer on a flexible rock half-space

A modified truncated cone model is used to calculate approximately the dynamic response of a disk on the surface of a soil layer resting on flexible rock. The procedure is analogous to that for a layer on rigid rock, the only modification being that the reflection coefficient —α at the layer–rock interface is no longer equal to ?1. The modified value of α can be determined straightforwardly by considering one-dimensional wave propagation along the cone. The low- and high-frequency limits lead to a frequency-independent α, which allows the dynamic analysis to be performed directly in the familiar time domain. This cone represents a wave pattern with amplitude decay and also incorporates the reflection at the free surface and the reflection-refraction at the layer–rock interface. The results for the static stiffness of the disk are highly accurate for a wide range of geometrical and material properties of the layer and the rock. For the dynamic stiffness the agreement with the exact solution is satisfactory.     

SECONDARY SHEAR WAVES FROM SOURCE BOREHOLES1

Previous studies of radiation from point sources in fluid-filled boreholes have most often been based on far-field, stationary phase analysis. In these papers, the explicit contribution of the borehole itself acting as a waveguide has not been properly considered, with a few exceptions. In general, these studies accurately describe S-wave radiation in high-velocity rocks such as granites and limestones and P-wave radiation in most rocks, and experiments have confirmed this. However, tube waves directly influence the external wavefield and in fact create a shear-wave ‘wake’ outside the borehole due to constructive interference of tube-wave emission if a velocity condition is met. This constructive interference or wake is generated when the tube-wave velocity is greater than the shear-wave velocity. When this happens, a tube-wave complex pole invalidates the mathematical assumptions for stationary phase analysis and the stationary phase predictions do not agree with experimentally derived radiation patterns. Shales at shallow depths and other soft sediments characteristically have tube-wave velocities greater than shear-wave velocities. Because the tube-wave is of relatively high amplitude compared to body waves generated directly by the source, these secondary shear waves can be the highest amplitude arrivals on receiver arrays. The shape and properties of these secondary shear waves are calculated and shown to have identical properties to Mach waves of aerodynamics and seismology. For instance, these waves are geometrically conical and the aperture of the cone and the moveout velocity can be calculated. This paper also demonstrates the important effect that casing has on the Mach waves and provides predictions about when these waves are likely to be observed. Finally, evidence of Mach waves in data sets is examined and it is shown how these waves have been confused with receiver borehole tube waves. It is possible, though rare, that the tube-wave velocity of the borehole is greater than the compressional-wave velocity of the surrounding medium. In this case secondary compressional or compressional Mach waves would be generated although this problem is not addressed here.     

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.     

Correction factors for SSI effects predicted by simplified models: 2D versus 3D rectangular embedded foundations

The effects of soil‐structure interaction (SSI) are often studied using two‐dimensional (2D) or axisymmetric three‐dimensional (3D) models to avoid the high cost of the more realistic, fully 3D models, which require 2 to 3 orders of magnitude more computer time and storage. This paper analyzes the error and presents correction factors for system frequency, system damping, and peak amplitude of structural response computed using impedances for linear in‐plane 2D models with rectangular foundations, embedded in uniform or layered half‐space. They are computed by comparison with results for 3D rectangular foundations with the same vertical cross‐section and different aspect ratios. The structure is represented by a single degree‐of‐freedom oscillator. Correction factors are presented for a range of the model parameters. The results show that in‐plane 2D approximations overestimate the SSI effects, exaggerating the frequency shift, the radiation damping, and the reduction of the peak amplitude. The errors are larger for stiffer, taller, and heavier structures, deeper foundations, and deeper soil layer. For example, for a stiff structure like Millikan library (NS response; length‐to‐width ratio ≈ 1), the error is 6.5% in system frequency, 44% in system damping, and 140% in peak amplitude. The antiplane 2D approximation has an opposite effect on system frequency and the same effect on system damping and peak relative response. Linear response analysis of a case study shows that the NEHRP‐2015 provisions for reduction of base shear force due to SSI may be unsafe for some structures. The presented correction factor diagrams can be used in practical design and other applications.