Axisymmetric interaction of a rigid disc with a transversely isotropic half‐space

A theoretical formulation is presented for the determination of the interaction of a vertically loaded disc embedded in a transversely isotropic half‐space. By means of a complete representation using a displacement potential, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth‐order partial differential equation. With the aid of Hankel transforms, a relaxed treatment of the mixed‐boundary value problem is formulated as dual integral equations, which, in turn, are reduced to a Fredholm equation of the second kind. In addition to furnishing a unified view of existing solutions for zero and infinite embedments, the present treatment reveals a severe boundary‐layer phenomenon, which is apt to be of interest to this class of problems in general. The present solutions are analytically in exact agreement with the existing solutions for a half‐space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy and compared with existing solutions. Further numerical examples are also presented to elucidate the influence of the degree of the material anisotropy on the response. Copyright © 2009 John Wiley & Sons, Ltd.     

The Reissner–Sagoci problem for a transversely isotropic half‐space

A transversely isotropic linear elastic half‐space, z?0, with the isotropy axis parallel to the z‐axis is considered. The purpose of the paper is to determine displacements and stresses fields in the interior of the half‐space when a rigid circular disk of radius a completely bonded to the surface of the half‐space is rotated through a constant angle θ₀. The region of the surface lying out with the circle r?a, is free from stresses. This problem is a type of Reissner–Sagoci mixed boundary value problems. Using cylindrical co‐ordinate system and applying Hankel integral transform in the radial direction, the problem may be changed to a system of dual integral equations. The solution of the dual integral equations is obtained by an approach analogous to Sneddon's (J. Appl. Phys. 1947; 18 :130–132), so that the circumferential displacement and stress fields inside the medium are obtained analytically. The same problem has already been approached by Hanson and Puja (J. Appl. Mech. 1997; 64 :692–694) by the use of integrating the point force potential functions. It is analytically proved that the present solution, although of a quite different form, is equivalent to that given by Hanson and Puja. To illustrate the solution, a few plots are provided. The displacements and the stresses in a soil deposit due to a rotationally symmetric force or boundary displacement may be obtained using the results of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Rigid cylinder in a transversely isotropic half‐space under lateral loads

A boundary integral equation method is presented for a rigid cylindrical pipe‐pile of finite length embedded in a transversely isotropic half‐space under lateral loads. In the framework of three‐dimensional elastostatics, the complicated soil‐structure interaction problem is shown to be reducible to three coupled Fredholm integral equations. Through an analysis of the associated Cauchy singular kernels, the intrinsic singular characteristics of the radial, angular, and vertical interfacial load transfers are rendered explicit. By means of a complicated numerical procedure, detailed results on the three‐dimensional load–transfer process are provided for benchmark comparison and practical applications. Copyright © 2011 John Wiley & Sons, Ltd.     

Elastostatic response of a pile embedded in a transversely isotropic half‐space under transverse loading

In the framework of elastostatics, a mathematical treatment is presented for the boundary value problem of the interaction of a flexible cylindrical pile embedded in a transversely isotropic half‐space under transverse loadings. Taking the pile region as a stiffened subdomain of an extended half‐space, the formulation of the interaction problem is reduced to a Fredholm integral equation of the second kind. The necessary set of Green's functions for the transversely isotropic half‐space is obtained by means of a method of potentials. The resulting Green's functions are incorporated into a numerical procedure for the solution of the integral equation. The theoretical response of the pile is presented in terms of bending moment, displacement and slope profiles for a variety of transversely isotropic materials so that the effect of different anisotropy parameters can be meaningfully discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Rocking vibrations of a rigid embedded foundation in a poroelastic half‐space

This paper presents an analysis of the rocking vibrations of a rigid cylindrical foundation embedded in poroelastic soil. The foundation is subjected to time‐harmonic rocking excitation and is perfectly bonded to the surrounding soil. The soil underlying the foundation base is represented by a homogeneous poroelastic half‐space, whereas the soil along the side of the foundation is modeled as an independent poroelastic stratum composed of a series of infinitesimally thin layers. The behavior of the soil is governed by Biot's poroelastodynamic theory. The contact surface between the foundation base and the poroelastic soil is assumed to be smooth and either fully permeable or impermeable. The dynamic interaction problem is solved by employing a simplified analytical method. Some numerical results for the nondimensional rocking dynamic impedance and nondimensional angular displacement amplitude of the foundation are presented to show the effect of nondimensional frequency of excitation, poroelastic material parameters, hydraulic boundary condition, depth ratio and mass ratio of the foundation. Copyright © 2009 John Wiley & Sons, Ltd.     

Tensionless–frictionless interaction of flexible annular foundation with a transversely isotropic multi‐layered half‐space

A transversely isotropic multi‐layered half‐space, with axis of material symmetry perpendicular to the free surface, supports a flexible either annular or solid circle foundation. The contact area of the foundation and the half‐space is considered to be both frictionless and tensionless. The foundation is assumed to be affected by a vertical static axisymmetric load. Detailed analysis of the interaction of these two systems with different thickness of layers is the target of this paper. With the use of ring load Green's functions for both the foundation and the continuum half‐space, an integral equation accompanied with some inequalities is introduced to model the complex BVP. With the incorporation of ring‐shape FEM, we are capable of capturing both regular and singular solution smoothly. The validity of the combination of the analytical and numerical method is proved with comparing the results of this paper with a number of benchmark cases of both linear and nonlinear interaction of circular and annular foundation with half‐space. Some new illustrations are presented to portray the aspect of the anisotropy and layering of the half‐space. Copyright © 2014 John Wiley & Sons, Ltd.     

Forced vertical and horizontal movements of a rectangular rigid foundation on a transversely isotropic half‐space

An analytical investigation of a half‐space containing transversely isotropic material under forced vertical and horizontal displacements applied on a rectangular rigid foundation is presented in this paper. With the goal of a rigorous solution to the shape‐ and rigidity‐ induced singular mixed boundary value problem, the formulation employs scalar potential representation, the Fourier expansion and the Hankel integral transforms method to obtain the surface arbitrary point‐load solution in cylindrical coordinate system. The obtained Green's functions are rewritten in rectangular coordinate system, allowing the response of the half‐space because of an arbitrary distributed load on a rectangular surface area be given in terms of a double integral. The numerical evaluations of stresses are done with the use of an element, which is singular at the edge and the corner of the rectangle. Upon the imposition of the rigidity displacement boundary condition for a rigid foundation and the use of a set of two‐dimensional adaptive‐gradient elements, which can capture the singular behavior in the contact stress effectively, a set of new numerical results are presented to illustrate the effect of transverse isotropy on the foundation response. Copyright © 2012 John Wiley & Sons, Ltd.     

Impedances of rigid cylindrical foundations embedded in transversely isotropic soils

A complete formulation and implementation for assessment of the response to dynamic loads of cylindrical rigid structures embedded in transversely isotropic elastic half‐spaces is presented. The analysis is performed in the frequency domain and the steady‐state structure response is obtained. The method is based on a non‐singular version of the indirect boundary element method which uses influence functions, instead of Green's functions, as fundamental solutions. These influence functions are the response of an elastic half‐space to distributed, internally applied loads. The proposed method imposes full bonding contact between the foundation and the surrounding soil. Numerical results for displacement (vertical and horizontal) and rotation (twisting and rocking) impedances, showing the influence of the soil anisotropy, are presented. Results for the soil–structure interface tractions and for the displacement field throughout the half‐space are also shown. Copyright © 2006 John Wiley & Sons, Ltd.     

Time‐dependent behaviour of a rigid foundation on a transversely isotropic soil layer

An analytical solution is presented in this paper to study the time‐dependent settlement behaviour of a rigid foundation resting on a transversely isotropic saturated soil layer. The governing equations for a transversely isotropic saturated soil, within Biot's poroelasticity framework, are solved by means of Laplace and Hankel transforms. The problem is subsequently formulated in the Laplace transform domain in terms of a set of dual integral equations that are further reduced to a Fredholm integral equation of the second kind and solved numerically. The developed analytical solution is validated via comparison with the existing analytical solution for an isotropic saturated soil case, and adopted as a benchmark to examine the sensitivities of the mesh refinement and the locations of truncation boundaries in the finite element simulations using ABAQUS. Particular attention is paid to the influences of the degree of soil anisotropy, boundary drainage condition, and the soil layer thickness on the consolidation settlement and contact stress of the rigid foundation. Copyright © 2009 John Wiley & Sons, Ltd.     

横观各向同性饱和地基中埋置刚性圆柱基础的扭转振动

考虑埋置基础侧面和底面地基与基础的相互作用,研究了埋置于横观各向同性饱和地基中刚性圆柱基础的扭转振动问题。假设基础侧壁与地基完全黏结,沿侧面对剪应力积分得到了侧壁地基对基础所产生的反扭矩;通过研究位于横观各向同性饱和地基上刚性圆形基础的扭转振动,求得基础底面地基作用于基础的反扭矩。结合基础底面位移和应力连续等条件,根据基础振动的动力平衡方程,求解了相应的动力响应问题。给出了基础的角位移幅值,地基的等效刚度系数和等效阻尼系数的表达式。通过算例研究了相关参数对基础扭转振动的影响。     

Elastic solutions for stresses in a transversely isotropic half‐space subjected to three‐dimensional buried parabolic rectangular loads

In many areas of engineering practice, applied loads are not uniformly distributed but often concentrated towards the centre of a foundation. Thus, loads are more realistically depicted as distributed as linearly varying or as parabola of revolution. Solutions for stresses in a transversely isotropic half‐space caused by concave and convex parabolic loads that act on a rectangle have not been derived. This work proposes analytical solutions for stresses in a transversely isotropic half‐space, induced by three‐dimensional, buried, linearly varying/uniform/parabolic rectangular loads. Load types include an upwardly and a downwardly linearly varying load, a uniform load, a concave and a convex parabolic load, all distributed over a rectangular area. These solutions are obtained by integrating the point load solutions in a Cartesian co‐ordinate system for a transversely isotropic half‐space. The buried depth, the dimensions of the loaded area, the type and degree of material anisotropy and the loading type for transversely isotropic half‐spaces influence the proposed solutions. An illustrative example is presented to elucidate the effect of the dimensions of the loaded area, the type and degree of rock anisotropy, and the type of loading on the vertical stress in the isotropic/transversely isotropic rocks subjected to a linearly varying/uniform/parabolic rectangular load. Copyright © 2002 John Wiley & Sons, Ltd.     

Elastic solutions of displacements for a transversely isotropic full space with inclined planes of symmetry subjected to a point load

This work presents analytical solutions for displacements caused by three‐dimensional point loads in a transversely isotropic full space, in which transversely isotropic planes are inclined with respect to the horizontal loading surface. In the derivation, the triple Fourier transforms are employed toyield integral expressions of Green's displacement; then, the triple inverse Fourier transforms and residue calculus are performed to integrate the contours. The solutions herein indicate that the displacements are governed by (1) the rotation of the transversely isotropic planes (?), (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′), (3) the geometric position (r, φ, ξ) and (4) the types of loading (P_x, P_y, P_z). The solutions are identical to those of Liao and Wang (Int. J. Numer. Anal. Methods Geomechanics 1998; 22 (6):425–447) if the full space is homogeneous and linearly elastic and the transversely isotropic planes are parallel to the horizontal surface. Additionally, a series of parametric study is conducted to demonstrate the presented solutions, and to elucidate the effect of the aforementioned factors on the displacements. The results demonstrate that the displacements in the infinite isotropic/transversely isotropic rocks, subjected to three‐dimensional point loads could be easily determined using the proposed solutions. Also, these solutions could realistically imitate the actual stratum of loading situations in numerous areas of engineering. Copyright © 2007 John Wiley & Sons, Ltd.     

A stepwise damping‐solvent extraction method for large‐scale dynamic soil–structure interaction analysis in time domain

Time‐domain analysis of dynamic soil–structure interaction based on the substructure method plays an increasing role in practical applications as compared with the frequency‐domain analysis. Efficient and accurate modelling of the unbounded soil or rock medium has been a key issue in such an analysis. This paper presents a subregional stepwise damping‐solvent extraction formulation for solving large‐scale dynamic soil–structure problems in the time domain. Accuracy and efficiency of the formulation are evaluated in detail for a classical problem involving a rigid strip foundation embedded in a half‐space. A practical large‐scale soil–structure interaction problem, which represents a high concrete gravity dam subjected to seismic load, is then analysed using the proposed method. Various responses of the dam, including time histories of the crest displacement and acceleration and contours of the peak principal stresses within the dam body, are presented. Comparisons are also made between these results with those obtained using other models for the unbounded medium. Copyright © 2007 John Wiley & Sons, Ltd.     

半无限横观各向同性介质中多裂纹相互作用分析

为评估含矩形裂纹的半无限横观各向同性介质的局部强度和稳定性,采用基于双横观各向同性材料基本解的对偶边界元数值方法,分析了在沿裂纹面法向和切向均布力分别作用下矩形裂纹的应力强度因子及两个共面或平行的矩形裂纹的相互作用问题。通过数值计算考察了自由面对应力强度因子值的影响,以及裂纹间距、边长比及自由面对共面或平行双裂纹相互作用效应的影响。结果表明,自由面的存在引起该类裂纹应力强度因子值大于无限域情况;裂纹形状和裂纹间距对共面或平行双裂纹相互作用效应均有较明显的影响,但自由面对共面或平行双裂纹的相互作用效应均影响较小。     

Vertical vibrations of a rigid disk embedded in a poroelastic medium

This paper considers the steady-state vertical vibrations of a rigid circular disk embedded at a finite depth below the free surface of a poroelastic medium. Biot's elastodynamic theory for porous media is used in the analysis. General solutions for axisymmetric poroelastic fields are obtained by using Hankel integral transforms. Analytical solutions for influence functions corresponding to four types of buried axisymmetric excitations are derived. The embedded disk problem is fomulated in terms of a set of coupled integral equations for unknown traction and pore pressure jumps across the disk. The kernel functions of the integral equations are the influence functions corresponding to buried vertical, radial and pore pressure ring loads. The system of integral equations is solved numerically by discretizing the disk into several concentric annular rings. Selected numerical solutions for displacements, vertical stress and pore pressure due to a buried fully flexible disk (uniform pressure) are also presented. The vertical compliances of a rigid disk are examined for different depths of embedment, poroelastic materials and hydraulic boundary conditions. Solutions for traction and pore pressure jumps are also examined. The present results are useful in the study of dynamic response of embedded foundations and anchors in poroelastic soils. Copyright © 1999 John Wiley & Sons, Ltd.     

A macro‐element for a circular foundation to simulate 3D soil–structure interaction

This paper presents a non‐linear interface element to compute soil–structure interaction (SSI) based on the macro‐element concept. The particularity of this approach lies in the fact that the foundation is supposed to be infinitely rigid and its movement is entirely described by a system of global variables (forces and displacements) defined in the foundation's centre. The non‐linear behaviour of the soil is reproduced using the classical theory of plasticity. Failure is described by the interaction diagram of the ultimate bearing capacity of the foundation under combined loads. The macro‐element is appropriate for modelling the cyclic or dynamic response of structures subjected to seismic action. More specifically, the element is able to simulate the behaviour of a circular rigid shallow foundation considering the plasticity of the soil under monotonic static or cyclic loading applied in three directions. It is implemented into FedeasLab, a finite element Matlab toolbox. Comparisons with experimental monotonic static and cyclic results show the good performance of the approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of soil–pile–structure interaction in a two‐layer ground during earthquakes considering liquefaction

This study is conducted with a numerical method to investigate the seismic behaviour among certain soils, single piles, and a structure. A series of numerical simulations of the seismic behaviour of a single‐pile foundation constructed in a two‐layer ground is carried out. Various sandy soils, namely, dense sand, medium dense sand, reclaimed soil, and loose sand, are employed for the upper layer, while one type of clayey soil is used for the lower layer. The results reveal that when a structure is built in a non‐liquefiable ground, an amplification of the seismic waves is seen on the ground surface and in the upper structure, and large bending moments are generated at the pile heads. When a structure is built in a liquefiable ground, a de‐amplification of the seismic waves is seen on the ground surface and in the upper structure, and large bending moments are generated firstly at the pile heads and then in the lower segment at the boundary between the soil layers when liquefaction takes place. Copyright © 2007 John Wiley & Sons, Ltd.     

Expanded superposition method for impedance functions of inclined‐pile groups

This paper presents a superposition method expanded for computing impedance functions (IFs) of inclined‐pile groups. Closed‐form solutions for obtaining horizontal, vertical, and rocking IFs, estimated by using pile‐to‐pile interaction factors, are proposed. IFs of solitary inclined piles, crossed IFs, and explicit incorporation of compatibility conditions for pile‐head movements are also appropriately taken into consideration. All of these factors should be known in advance and will be computed and shown for the most relevant cases. The accuracy of the proposed closed‐form solutions is verified for 2 × 2 and 3 × 3 square inclined‐pile groups embedded in an isotropic viscoelastic homogeneous half‐space soil medium, with hysteretic damping. The pile‐to‐pile interaction factors are computed by means of a three‐dimensional time‐harmonic boundary elements–finite elements coupling formulation. The results indicate that the IFs obtained from the proposed method are in good agreement with those obtained from the coupling formulation. Furthermore, crossed vertical‐rocking IFs of solitary piles need to be appropriately considered for obtaining rocking IFs when the number of piles is small. Copyright © 2015 John Wiley & Sons, Ltd.     

Source signature and elastic waves in a half‐space under a sustainable line‐concentrated impulsive normal force

First, the response of an ideal elastic half‐space to a line‐concentrated impulsive normal load applied to its surface is obtained by a computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. Then, the geometry is determined of the obtained waves and the source signature—the latter is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Behind the dilatational precursor wave, there exists a pencil of three plane waves extending from the vertex at the impingement point of the precursor wave on the stress‐free surface of the half‐space to three points located on the other two boundaries of the solution domain. These four wave‐arresting points (end points) of the three plane waves constitute the source signature. One wave is an inhibitor front in the behaviour of the normal stress components and the particle velocity, while in the behaviour of the shear stress component, it is a surface‐axis wave. The second is a surface wave in the behaviour of the horizontal components of the dependent variables, while the third is an inhibitor wave in the behaviour of the shear stress component. An inhibitor wave is so named, since beyond it, the material motion is dying or becomes uniform. A surface‐axis wave is so named, since upon its arrival, like a surface wave, the dependent variable in question features an extreme value, but unlike a surface wave, it exists in the entire depth of the solution domain. It is evident from this work that Saint‐Venant's principle for wave propagation problems cannot be formulated; therefore, the above results are a consequence of the particular model proposed here for the line‐concentrated normal load. Copyright © 2002 John Wiley & Sons, Ltd.     

Source signature and elastic waves in a half‐space under a momentary shear line impulse

The transient deformation of an elastic half‐space under a line‐concentrated impulsive vector shear load applied momentarily is disclosed in this paper. While in an earlier work, the author gave an analytical–numerical method for the solution to this transient boundary‐value problem, here, the resultant response of the half‐space is presented and interpreted. In particular, a probe is set up for the kinematics of the source signature and wave fronts, both explicitly revealed in the strained half‐space by the solution method. The source signature is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Fourteen wave fronts exist behind the precursor shear wave S: four concentric cylindrical, eight plane, and two relativistic cylindrical initiated at propagating centres that are located on the stationary boundaries of the solution domain. A snapshot of the stressed half‐space reveals that none of the 14 wave fronts fully extend laterally. Instead, each is enclosed within point bounds. These wave arresting points and the two propagating centres of the relativistic waves constitute the source signature. The obtained 14 wave fronts are further combined into 11 disparate wave fronts that are grouped into four categories: an axis of symmetry wave—so named here by reason of being a wave front that is contiguous to the axis of symmetry, three body waves, five surface waves and two inhibitor waves—so named here by reason that beyond them the material motion dies out. Of the three body waves, the first is an unloading shear wave, the second is a diffracted wave and the third is a reflected longitudinal two‐branch wave. Of the two inhibitor waves, the first is a two‐joint relativistic wave, while the second is a two‐branch wave. The wave system, however, is not the same for all the dependent variables; a wave front that appears in the behaviour of one dependent variable may not exist in the behaviour of another. It is evident from this work that Saint–Venant's principle for wave propagation problems cannot be formulated. Therefore, the above results are valid for the particular proposed model for the momentary line‐concentrated shear load. The formulation of the source signature, the wave system, and their role in the half‐space transient deformation are presented here. Copyright © 2003 John Wiley & Sons, Ltd.