Homogeneous Elastic Half-space Research Articles

Transient Response of an Elastic Homogeneous Half-Space to Suddenly Applied Rectangular Loading

A closed-form solution of transient response to suddenly applied loading distributed over a rectangular area on the surface of an elastic homogeneous half-space is developed for special purposes such as analysis of dynamic soil-structure interaction or contact problems. The solution is obtained using Laplace transform with respect to time and Fourier transform with respect to space. Inverse Laplace transform is implemented analytically. As extreme cases of rectangular loading, the solutions for a point force or finite line load can also be obtained. The advantages of this solution over most other solutions by numerical analyses are that the multiple integrations are reduced by one order, the singularity is removed from the integral kernel, and no additional discretization in the vicinity of the region of interest is required.

Rolling/sliding of a linear elastic body on a deformable layered substrate

We analyze a three-dimensional frictional contact problem between a homogeneous and isotropic linear elastic body and a substrate consisting of a homogeneous and isotropic linear elastic half-space coated with a transversely isotropic elastic layer. The problem formulation includes the kinematics of the body, including its rolling/sliding on the substrate. Expressions relating the displacements of a point on the contact surface to the surface tractions acting there have been derived. We have also proposed a variational problem for the determination of surface tractions at points of the contact surface. An example problem involving the motion of a homogeneous spherical rigid ball on a substrate made of a homogeneous thin layer of a linear elastic material and bonded to a rigid base is studied. It is shown that there can be at most one adhesion zone followed by a slipping zone. Numerical results illustrating the dependence of the slip velocity upon the resultant frictional force are also presented graphically.

Stresses in an elastic plate lying over a base due to strip-loading

The closed-form analytic expressions for the stresses at any point of an elastic plate coupling in different ways to a base as a result of a two-dimensional shear strip-loading are obtained. The contact between the horizontal layer and the base is either smooth-rigid or rough-rigid or welded. The variations of the shear stresses with the horizontal distance have been studied numerically. It is found that the effect of different boundary conditions on the stress field is significant and the stresses for an elastic layer lying over an elastic half-space differ considerably from those of an entire homogeneous elastic half-space.

Surface Waves in Vertically Heterogeneous and Transversely Isotropic Layer Overlying a Homogeneous Isotropic Half-Space and under a Uniform Layer of Liquid.

The effect of heterogeneity and anisotropy on the dispersion of Rayleigh-type surface waves in a heterogeneous anisotropic layer of elastic medium overlying a homogeneous elastic solid half-space and lying under a homogeneous liquid layer of uniform depth is discussed. The variation of phase velocity with the wavenumber is obtained numerically for the fundamental mode. The results are compared with the homogeneous case. The results show that the heterogeneity of the layer decreases the phase velocity for small values of wavenumber. The problem of Gogna (1979) has been reduced as a special case.

Effects of the wave passage and the embedment depth for in-plane building-soil interaction

Building foundation-soil interaction is studied in the frequency domain using a two-dimensional analytical model. The building is represented by an infinitely long shear wall resting on a circular foundation, embedded into an elastic homogeneous half-space. Deep and shallow foundations are considered (with depth-to-half-width ratios of 1 and 0·5). Both the dynamic interaction and the wave passage effects are included. The excitation is a plane P- or SV-wave,or a surface Rayleigh wave. The results show that for incident waves which are long relative to the width of the foundation, the foundation driving forces are larger when the embedment is deeper. For shorter incident waves, the input base rotation is larger for shallow foundations and, therefore, the relative building response may then be larger. It is also shown that the input base rotation may contribute significantly to the building excitation and that neglecting it may cause nonconservative estimates for the forces in the building.

In-plane foundation-soil interaction for embedded circular foundations

Foundation soil interaction is studied using an analytical two-dimensional model, for circular foundations embedded in a homogeneous elastic half-space and for incident plane P- and SV- and for surface Rayleigh waves. The scattered waves are expanded in complete series of cyclindrical wave functions. A detailed analysis is presented of the foundation response to unit amplitude incident waves as a function of the type of incident waves and angle of incidence, the depth of the embedment and the foundation mass per unit length. It is shown that free-field translations and point rotation approximate well the foundation input motion only for very long incident waves. For shorter incident waves, those in general overestimate the foundation input motion. Neglecting the rotation of the foundation input motion (which is usually done in practice) may eliminate a major contribution to the base excitation of buildings and may cause nonconservative estimates of the forces in these buildings. Incident waves appear as ‘longer’ to a shallow foundation than to a deeper foundation. Therefore, deeper foundations are more effective in reflecting and scattering the short incident waves.

A new method for a realistic treatment of the sea bottom in the parabolic approximation

A new method for realistic treatment of the bottom in the parabolic approximation in underwater acoustics is presented. The method utilizes a finite difference scheme for the numerical implementation of the parabolic equation and the boundary condition is introduced to the problem in the form of an impedance condition. The impedance condition for the fluid half-space bottom can be implemented analytically, while total numerical implementation is necessary for a general type bottom. A characteristic type of bottoms that can be handled by this method is the homogeneous elastic half-space. The reliability of the method is confirmed by examples where the results obtained by the method are compared with well-known results of reference problems.

Surface roughness induced attenuation and changes in the propagation velocity of long Rayleigh-type waves

The propagation of long Rayleigh waves on a rough surface is studied theoretically by Rayleigh method and for the free rough surface of a homogeneous and isotropic elastic half-space. The frequency dependence of the propagation velocity and attenuation is analysed under the assumption that the frequency is real. Moreover, the changes in the propagation velocity induced by the surface roughness are expressed in terms of the square of the ratio of the root-mean-square departure of the surface from flatness to the correlation length of the distance between successive peaks and valleys in the rough-surface profile.

Finite element analysis of the contract stresses in an elastic coating on an elastic substrate

A finite element method has been used to investigate the contract stresses due to elastic indentation of coating/substrate systems consisting of a high modulus surface coating on a relativity low modulus substrate. Calculations have been made for two imposed pressure distribution identical to those arising during hertzian indentation of a homogeneous elastic half-space by spherical and cylindrical indenters respectively. For each distribution the stresses have been investigated in terms of the ratio of the Young moduli of the coating and substrate and the ratio of the coating thickness to the contact halfwidth. It has been shown that the stresses of both the outer surface of the coating and the coating-substrate interface are complicated functions of the two variables and that the stresses which could be responsible for coating failure cannot simultanously be minimized by appropriate choice of variables.

The effect of inhomogeneity and anisotropy on Stoneley waves

In this paper we study the dispersion equation of Stoneley waves that are travelling in an inhomogeneous elastic half-space over an anisotropic homogeneous elastic half-space. The phase velocity is calculated as a function of the wave number. The results indicate that the effect of anisotropy on such waves is small and can be neglected, while the effect of inhomogeneity is very pronounced. The results show that Stoneley waves do not exist after some cut-off value of the wave number.

A fault model for the 1989 Kilauea South Flank Earthquake from leveling and seismic data

The geometry of the fault that ruptured during the M6.1 south flank earthquake on Kilauea volcano in 1989 is determined from leveling data. The elastic dislocation, in a homogeneous elastic half‐space, that best fits the data is found using a nonlinear inversion procedure. The best fitting model is a gently dipping thrust fault that lies at 4 km depth. This is significantly shallower than the 9 km hypocentral depth determined from the local seismic network. Two‐dimensional finite‐element calculations indicate that at least part of this discrepancy can be attributed to the focusing of the surface deformation by the upper few kilometers of compliant, low‐density lavas. We conclude that it is important to include realistic elastic structure to estimate source geometry from geodetic data.

Dynamic response of underground structures by time domain SBEM and SFEM

The dynamic interaction problems of three-dimensional linear elastic structures with arbitrary shaped section embedded in a homogeneous, isotropic and linear elastic half space under dynamic disturbances are numerically solved. The numerical method employed is a combination of the time domain semi-analytical boundary element method (SBEM) used for the semi-infinite soil medium and the semi-analytical finite element method (SFEM) used for the three-dimensional structure. The two methods are combined through equilibrium and compatibility conditions at the soil-structure interface. Displacements, velocities, accelerations and interaction forces at the interface between underground structure and soil medium produced by the diffraction of wave by an underground structure for every time step are obtained. In dynamic soil-structure interaction problems, it is advantageous to combine the SBEM and the SFEM in an effort to produce an optimum numerical hybrid scheme which is characterized by the main advantages of the two methods. The effects of the thickness, the ratio of length and diameter of underground structure and the soil medium on dynamic responses are discussed.

The solution of Cagniard's problem for anSHline source in elastic and anelastic media, calculated using ω —kintegrals

AbstractComputations have been performed using an improper ω — k double integral representing the response to an SH pulse generated by a line source, acting in a model consisting of two homogeneous isotropic elastic or linear viscoelastic half-spaces separated by a plane boundary. The advantage of this method is its universality: complete seismograms for relevant time and frequency windows are produced by the program. The calculated synthetic seismograms are accurate enough even for a small number of frequencies and compare well with exact results for elastic models obtained using Cagniard's method.For the simple models of two anelastic or one elastic and one anelastic half-space in welded contact considered here, all the arrivals that are predicted by elasticity theory, including the incident, reflected, head, transmitted, and evanescent waves, are present. It seems that attenuation does not radically change the physical picture seen in the elastic case but rather modifies it: there is continuous variation with dispersion and attenuation. For viscoelastic media, the seismograms present features due to the nonplanar wavefronts propagating from a line source. Such features cannot be explained by the inhomogeneous plane wave theory.

A simple 2-D model for the evolution of an island-arc system

Slow seismotectonic movements along inclined deep fault planes under compressive horizontal stresses are supposed to be the principal mechanism controlling the structure and processes in island-arc systems. In order to treat the stress variations caused by this mechanism, a simple geomechanical model is investigated. We consider a shearing surface crack embedded in a homogeneous elastic half-space. The key element of the model is viscous interaction between the sides of the crack, the viscosity varying with depth. The model differs from the classical steady-state mode of subduction by nonstationary creep processes on deep faults and possibly by cyclical evolution of island-arc systems. The results of our numerical analysis are in good agreement with geological, geophysical and seismological data. (i) Vertical displacements of the free surface in the model fit well with the typical topography of a trench—arc-basement rise—back-arc basin system. (ii) The Benioff seismic zone is supposed to be formed due to the concentration of shear stresses near the fault plane. The characteristic patterns of seismicity, the fine geometry of Benioff zones, and their double-planed structure can be explained in terms of our model. (iii) A zone of considerable heat generation caused by viscous dissipation along the fault plane is found within a narrow area in the depth range 100–200 km. Moreover, the island-arc basement rise is characterized in the model by a relative tension of a few tens or even hundreds of bars, while at depths of 100–150 km below the surface, additional compression of the same order of magnitude acts. The magmatic plumbing system may be visualised as a “toothpaste tube” or a sponge filled with magma which is squeezed from the depths to the surface due to the redistribution of the tectonic stresses only. This can explain the physical origin of island-arc magmatism and the typical position of volcanic belts.

A possible explanation for foreland thrust propagation

A common feature of thin‐skinned fold and thrust belts is the sequential nature of foreland directed thrust systems. As a rule, younger thrusts develop in the footwalls of older thrusts, the whole sequence propagating towards the foreland in the transport direction. As each new younger thrust develops, the entire sequence is thickened; particularly in the frontal region. The compressive toe region can be likened to an advancing wave; as the mountainous thrust belt advanced the down‐surface slope stresses drive thrusts ahead of it much like a surfboard rider. In an attempt to investigate the stresses in the frontal regions of thrustsheets, a numerical method has been devised from the algorithm given by McTigue and Mei [1981]. The algorithm yields a quickly computed approximate solution of the gravity‐ and tectonic‐induced stresses of a two‐dimensional homogeneous elastic half‐space with an arbitrarily shaped free surface of small slope. A comparison of the numerical method with analytical examples shows excellent agreement. The numerical method was devised because it greatly facilitates the stress calculations and frees one from using the restrictive, simple topographic profiles necessary to obtain an analytical solution. The numerical version of the McTigue and Mei algorithm shows that there is a region of increased maximum resolved shear stress, τ, directly beneath the toe of the overthrust sheet. Utilizing the Mohr‐Coulomb failure criterion, predicted fault lines are computed. It is shown that they flatten and become horizontal in some portions of this zone of increased τ. Thrust sheets are known to advance upon weak decollement zones. If there is a coincidence of increased τ, a weak rock layer, and a potential fault line parallel to this weak layer, we have in place all the elements necessary to initiate a new thrusting event. That is, this combination acts as a nucleating center to initiate a new thrusting event. Therefore, thrusts develop in sequence towards the foreland as a consequence of the stress concentrating abilities of the toe of the thrust sheet. The gravity‐ and tectonic‐induced stresses due to the surface topography (usually ignored in previous analyses) of an advancing thrust sheet play a key role in the nature of shallow foreland thrust propagation.

A theoretical study of pile heave

Installation of a driven pile displaces the soil vertically and radially. The ensuing vertical soil movement causes the uplift or heave of piles already installed in the vicinity. The Paper describes a theoretical study of the vertical soil movement and the pile heave due to the installation of a driven pile in clay. The clay deposit is idealized as a homogeneous isotropic incompressible linear elastic half-space. Pile installation is simulated by the injection of material along the pile axis with a constant intensity. The resulting soil movement is obtained using the source-sink imaging technique. The heave of a pile already installed is obtained from the interaction between the pile and the vertical soil movement. Numerical results are presented showing how the soil movement is affected by the dimensions of the penetrating pile. The pile heave is affected by the ratio of the Young's moduli of the pile and soil, in addition to the pile dimensions and pile spacing. A number of reported case histories are analysed; in general, the theoretical and observed behaviour show an encouraging measure of agreement. L'installation d'un pieu battu déplace le sol verticalement et radialement. Le mouvement vertical du sol qui en résulte provoque le soulèvement des pieux adjacents déjà installés. L'article décrit une étude théorique du mouvement vertical du sol et le soulèvement des pieux provoqué par l'installation d'un pieu battu dans l'argile. Le dépôt d'argile est fictivement considéré comme un demi-espace élastique linéaire isotrope homogène et incompressible. L'installation du pieu est simulé par l'injection à intensité constante de matériau le long de 1'axe du pieu. Le mouvement du sol ayant lieu est défini à l'aide de la technique d'idealisation employant des images corrigées de la pointe d'origine du soulèvement par rapport à la surface du sol. Le soulevèment d'un pieu déjà installé est déterminé à partir de l'interaction entre le pieu et le mouvement vertical du sol. Des résultats numériques sont présentés qui démontrent la façon dont le mouvement du sol est influencé par les dimensions du pieu y pénétrant. Le soulèvement du pieu est influencé par le rapport des modules de Young du pieu et du sol ainsi que par les dimensions du pieu et l'écartement des pieux. Une analyse est présentée de quelques cas réels concernant des mouvements de sol et des mesures du soulèvement des pieux. En général le comportement observé s'accorde de façon satisfai-sante avec la théorie.

Some observations on time domain and frequency domain boundary elements

AbstractThe numerical solution of problems in elastodynamics involving infinite media calls for the use of discrete techniques such as the boundary element method and the finite element method. These techniques can, in turn, be formulated in the time or frequency domains, and have each relative merits and drawbacks. This paper presents a comparative study of the accuracy and limitations of three different implementations of these methods.The problem studied is that of transient loads on the surface of homogeneous elastic halfspaces, and of finite depth strata. In each case, the response is computed first for an uninterrupted (continuous) medium, and then for a medium that includes a trench (or cavity).Three independent computer programs were used that incorporated the following methods: (i) frequency domain boundary element method (FD‐BEM), using a discrete fundamental solution; (ii) time domain boundary element method (TD‐BEM) using an analytical fundamental solution; and (iii) a coupled time domain boundary element‐finite element model (BEM/FEM).It is found that for convex domains (halfspace or stratum without a trench) the three independent implementations are in excellent agreement, while for non‐convex domains (trench in the path of the waves), numerical errors associated with non‐causal behaviour become evident in some cases.

Dynamic soil stiffness in lateral vibrations of a floating pile

The authors' earlier investigation on the vertical and torsional modes of vibrations of a floating pile, is extended to the case of lateral vibrations. As before, the soil is considered to be a homogeneous elastic half-space and the slenderness aspect of the pile is used to assume the pile action on it to be due to a line distribution of horizontal forces. The near-field is determined by assuming that the displacement at a certain depth is predominantly effected by the force at that very level. It is found that it consists of only horizontally polarized coupled shear and compressional waves. The corresponding displacement and stresses yield a relation between the soil impedance to lateral vibration and the horizontal displacement of any point of the pile. When the pile length is made infinite, the result reduces to a form attributed to Baranov. It is found that finiteness of the length of the pile significantly alters the result, but is comparatively insensitive to variation with depth of a point of the pile.

Coupling of boundary and finite elements for soil‐structure interaction problems

AbstractThe investigation of complex soil‐structure interaction problems is usually carried out with numerical solution procedures such as the finite element or the boundary element method. It must be noted, however, that the choice of one or the other of these approaches is not just a matter of preferences; depending on the type of the problem under consideration, either boundary or finite elements may be more advantageous. A considerable expansion in the computational power can be obtained, on the other hand, if one resorts to hybrid schemes which retain the main advantages of the two methods and eliminate their respective disadvantages.This paper presents results obtained with a boundary element‐finite element coupling procedure, and discusses its applicability to some representative soil‐structure interaction problems. The structures considered are elastic systems, such as foundations, tunnels and filled trenches (modelled by finite elements), which are coupled with homogeneous elastic halfspaces (modelled by boundary elements). The examples demonstrate the importance of using a model that includes wave radiation effects.The coupling approach is formulated entirely in the time domain so that an extension of the algorithm to non‐linear analyses seems to present no further difficulties.

Rocking Damping of Arbitrarily Shaped Embedded Foundations

A simple analytical model is developed for estimating the rocking radiation damping coefficients of arbitrarily shaped rigid foundations embedded in an elastic homogeneous halfspace. The model, based on sound physical approximations that respect fundamental principles of dynamics and wave propagation, is calibrated with the help of rigorous numerical results from boundary element and finite element formulations. Results and comparisons are presented for a variety of basemat shapes (including the circle, rectangles with aspect ratio of up to ten, the strip, and T‐shapes), for a wide range of embedment depths, and for complete, as well as partial, symmetric and nonsymmetric contact between the vertical sidewalls and surrounding soil. Valuable insight is gained into the mechanics of radiation damping, and it is demonstrated quantitatively that, in practice, separation and slippage between side‐walls and soil would appreciably affect damping for rocking only in the lateral direction (i.e., about the long axis of the basemat).