A contact problem for a poroelastic halfspace containing an embedded inextensible membrane

The paper examines the axisymmetric problem of the indentation of a poroelastic halfspace that is reinforced with an inextensible permeable/impermeable membrane located at a finite depth by a rigid indenter. The constitutive behavior of the poroelastic halfspace is described by the three-dimensional theory of poroelasticity proposed by M.A. Biot. The contact conditions between the indenter and the poroelastic halfspace are varied to accommodate both adhesive/frictionless contact and impermeable/permeable conditions. The formulation of the mixed boundary value problems uses the stress function approaches applicable to semi-infinite domains. Successive applications of Laplace and Hankel integral transforms are used to reduce the mixed boundary value problems to sets of coupled Fredholm integral equations of the second kind. These integral equations are solved using numerical approaches, applicable both for the solution of the systems of coupled equations and for Laplace transform inversion, to examine the time-dependent displacement of the rigid indenter. The analytical-numerical estimates for the time-dependent displacements of the rigid indenter are compared with results obtained using a finite element approach.     

Fundamental results concerning the settlement of a rigid foundation on an elastic medium due to an adjacent surface load

This paper examines the interaction between a rigid circular foundation resting in smooth contact with an isotropic elastic halfspace and a concentrated surface load which acts at a finite distance from the foundation. Owing to the action of the external load the rigid foundation experiences an extra settlement and a tilt. The expressions for the extra settlement and the tilt are evaluated in exact closed form. It is also shown that these deformations due to the external load satisfy Betti's reciprocal theorem. The auxiliary solution required for the application of the reciprocal theorem is derived from the analysis of the problem of a rigid circular foundation resting in smooth contact with an elastic medium and subjected to an eccentric load. The, results developed for the interaction between the rigid circular foundation and the external concentrated load are utilized to generate, among others, solutions for the settlement and tilt induced at a rigid foundation due to its interaction with uniformly or non-uniformly distributed loads with circular and square plan shapes.     

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.     

A theoretical estimate of the tilt of a surface foundation due to an inclined anchor load applied at the interior of the soil medium

This paper examines the problem of the interaction between a loaded rigid circular foundation located at the surface of an isotropic elastic halfspace and an inclined concentrated anchor load which is located at a finite depth along the axis of symmetry. Such inclined loads can be induced by, for example, anchor regions supporting earth retaining structures. The loaded rigid circular foundation resting in smooth contact with the elastic soil mass experiences a displacement and a tilt due to the action of the inclined anchor load. The magnitude of the rotational settlement is evaluated in exact closed form.     

修正等应变假定下刚性桩复合地基固结特性分析

刚性桩复合地基固结特性是影响工程进度的重要因素。为研究外荷载作用下刚性桩复合地基固结特性，结合刚性桩和桩间土应力?应变特点对等应变假定进行修正。在此基础之上，考虑涂抹效应、桩间土和下卧层土体固结特性的影响，推导得到桩间土和下卧层土体固结微分方程。基于双层地基固结理论，考虑单级匀速施加荷载和附加应力沿深度变化等边界条件，推导得到其固结度解析解。最后，利用数值模拟方法对解析解进行验证，并对影响刚性桩复合地基固结特性的因素进行分析。结果表明：理论计算与数值模拟结果较为吻合；刚性桩复合地基固结速率受到贯入比影响较大，当贯入比较小时，与天然地基相比，其固结速率较小，贯入比较大且桩端阻力系数较小时，复合地基固结速率大于天然地基；随着贯入比、桩?土相对刚度和垫层?土体相对刚度的增大，复合地基整体固结速率加快；随着置换率的增大，复合地基整体固结速率降低。     

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.     

含饱和土的层状场地的动力响应

利用精确的土层和半空间动力刚度矩阵求解了含饱和土的层状场地的动力响应,给出了含饱和土层状场地的自由场（入射P波、SV波）、动力Green函数及刚性条形基础动力刚度的计算方法,由此建立的场地模型更接近实际情况。建议方法的退化结果与单相场地结果一致。研究了界面不同透水条件对基础动力刚度的影响,并与相应单相场地结果进行了对比。研究表明：界面透水条件对竖向弹簧系数影响最大,饱和土体中孔隙水对场地动力响应具有不可忽视的影响。     

软土地基上高填方刚性涵洞地基承载力分析

山区沟谷软土地基上高填方刚性涵洞的应用较为广泛,然而,现有的计算理论对该类条件下涵洞地基承载力的认识还不够充分,对地基承载力提出过高的要求,反而为结构带来了不利影响。通过数值模拟和试验手段对涵洞的地基承载力进行深入分析。探讨基础埋深、宽度及软土固结对涵洞地基承载力的影响。研究表明,当基础埋深系数 5时,涵洞地基承载力特征值随着基础埋深的增大近似线性增加,当 5时,基础埋深对地基承载力特征值影响逐渐减小;但基础宽度对软土地基上刚性涵洞地基承载力特征值的影响甚小,实际工程中可不予考虑。此外,试验结果表明,固结度和固结压力对软土的黏聚力和内摩擦角有复合影响,固结度较大时,黏聚力和内摩擦角随固结压力的增大而明显增大。固结度和固结压力对内摩擦角的影响比对黏聚力的影响要大。高填方涵洞地基极限承载力随着软土固结度的增大而提高,当固结度达到90%时,地基极限承载力通常可提高36%以上;地基极限承载力随固结压力的增加呈非线性增大,其提高的幅度逐渐减小。     

The behaviour of an impermeable flexible raft on a deep layer of consolidating soil

Analytic solutions to the problem of the time-settlement behaviour of raft foundations have been limited in the past to flexible or rigid loadings, and have treated the foundation as being completely permeable. In this paper, solutions are presented for smooth circular rafts of any flexibility causing consolidation of a deep homogeneous clay layer, where the raft may be considered permeable or impermeable. Results for the time-dependent behaviour of contact stresses, pore pressures, raft displacements and moment in the raft are presented.     

考虑时间效应的刚性基础沉降分析

土体变形的时效性反映了土体固结与流变的统一,亦即地基刚度本身具有时效性,通过采用解析的方法建立了考虑时间效应的地基柔度矩阵,得出考虑时间效应的刚性基础计算方法,并编制相应程序对考虑时间效应的刚性基础进行计算,分析并得出其内力、变形等随时间发展的规律：加载初期沉降发展较快,后期虽然沉降速率有很大递减,但沉降大小仍将随着时间推移而增加;计算区域基底反力在加荷初期变化较大,角点反力增大,中心点反力减小。随着时间的增长,反力不断调整并逐渐趋于稳定,呈现边缘大、中间小的马鞍形特征。     

On the indentation of a poroelastic layer

The paper examines the axisymmetric contact problem related to the indentation of a fluid saturated poroelastic layer by a smooth rigid punch. The layer rests in bonded contact with a rigid impermeable base and the surface of the layer is considered to be either permeable or impermeable. The paper develops the integral equations governing the problem for the generalized case where the pore fluid exhibits compressibility. The numerical results presented in the paper illustrate the influence of the relative layer thickness, drainage conditions and the compressibility of the pore fluid on the degree of consolidation settlement of the indenting punch.     

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.     

Torsional stiffness of non-uniform and hollow rigid piers embedded in isotropic elastic media

This paper examines the torsional response of a rigid pier type foundation, with a non-uniform or hollow cross-section which is embedded in bonded contact with a layered elastic half space. The tractions which act at the axisymmetric boundary surface between the pier and the surrounding elastic medium are represented by discretized regions of uniform traction. The compatibility of deformation at the boundary is used to determine the interface stress distribution. The torque-rotation response for the, rigid pier foundation is obtained for different choices of the pier geometry and shear modulus mistnatch between the layer and the underlying half space.     

考虑桩体刺入变形的透水混凝土桩复合地基固结理论研究

现有的多元复合地基利用两种或者两种以上的桩型对地基土进行加固,以期提高地基承载力、加速地基固结。提出了一种兼具散体材料桩和刚性不排水桩自身优点的透水混凝土桩复合地基固结模型。工程中刚性桩与土体的压缩量并不相等,这与等应变条件存在着不一致的地方。为此,抛弃传统等应变假设,考虑桩体向垫层和下卧层的刺入变形,对传统等应变假定进行修正,同时考虑了扰动区土体水平渗透系数的抛物线变化和水在桩体内的径向、竖向渗流,给出了此类固结问题的控制方程和解答以及复合地基固结度的表达式,并对复合地基固结度的解进行了讨论。最后与其他复合地基解做了比较并对固结性状进行了分析,并针对透水混凝土桩孔隙引起的桩阻作用对地基固结速度的影响进行了验证。     

On the mechanics of contact between a flexible transducer diaphragm located at a rigid boundary and an elastic material

In this paper we examine the mechanics of contact between a flexible elastic diaphragm and an isotropic elastic halfspace region where contact is established through the application of a normal stress. It is shown that a convenient solution for the maximum displacement and maximum flexural moment in the flexible diaphragm can be obtained by employing a variational technique. Copyright © 2006 John Wiley & Sons, Ltd.     

Elastic lateral dynamic impedance functions for a rigid cylindrical shell type foundation

Elastic lateral dynamic impedance functions are defined as the ratio of the lateral dynamic force/moment to the corresponding lateral displacement/rotation at the top ending of a foundation at very small strains. Elastic lateral dynamic impedance functions have a defining influence on the natural frequencies of offshore wind turbines supported on cylindrical shell type foundations, such as suction caissons, bucket foundations, and monopiles. This paper considers the coupled horizontal and rocking vibration of a cylindrical shell type foundation embedded in a fully saturated poroelastic seabed in contact with a seawater half‐space. The formulation of the coupled seawater–shell–seabed vibration problem is simplified by treating the shell as a rigid one. The rigid shell vibration problem is approached by the integral equation method using ring‐load Green's functions for a layered seawater‐seabed half‐space. By considering the boundary conditions at the shell–soil interface, the shell vibration problem is reduced to Fredholm integral equations. Through an analysis of the corresponding Cauchy singular equations, the intrinsic singular characteristics of the problem are rendered explicit. With the singularities incorporated into the solution representation, an effective numerical method involving Gauss–Chebyshev method is developed for the governing Fredholm equations. Selected numerical results for the dynamic contact load distributions, displacements of the shell, and lateral dynamic impedance functions are examined for different shell length–radius ratio, poroelastic materials, and frequencies of excitation. Copyright © 2016 John Wiley & Sons, Ltd.     

Vertical vibration of a rigid strip footing on viscoelastic half-space

This paper presents an analytical method for modeling the dynamic response of a rigid strip footing subjected to vertical-only loads. The footing is assumed to rest on the surface of a viscoelastic half-space; therefore, effects of hysteretic soil damping on the impedance of the foundation and the generated ground vibrations are considered in the solution. To solve the mixed boundary value problem, we use the Fourier transform to cast a pair of dual integral equations providing contact stresses, which are solved by means of Jacobi orthogonal polynomials. The resulting soil and footing displacements and stresses are obtained by means of the Fourier inverse transform. The solution provides more realistic estimates of footing impedance, compared to existing solutions for elastic soil, as well as of the attenuation of ground vibrations with distance of the footing. The latter is important for the estimation of machine vibration effects on nearby structures and installations.     

非均匀地基与均匀地基一维固结解答之间的转换关系

基于一维固结理论,研究了土层物理力学性质沿深度非均匀连续变化的固结问题。首先,利用分离变量法,获得了渗透系数和压缩系数随深度按指数函数变化的非均匀地基一维固结方程的解析表达式; 然后与Terzaghi固结理论的经典解答进行了比较,获得了两种解答之间的相似转换关系。这样,非均匀地基固结问题就可以用相同荷载以及边界条件下的均匀地基固结的经典解线性表示。因此,非均匀地基固结问题的求解转化为对相似转换系数的计算。该系数集中反映土层非均匀性对地基固结的影响,从而为解决非均匀地基一维固结问题提供了便捷途径。     

The eccentric loading of a rigid circular foundation embedded in an isotropic elastic medium

This paper examines the problem of the eccentric loading of a rigid circular disc-shaped foundation embedded in bonded contact with an istropic elastic medium of infinite extent. The solution of this problem is achieved by using a Hankel integral transform technique which reduces the problem to two sets of dual integral equations. These systems of dual integral equations represent the equations which govern the axisymmetric and asymmetric states of deformation induced by the loaded foundation. Explicit results are derived both for the displacement and rotation of the circular foundation and for the contact stress at the interface.     

线性加载下含砂垫层地基固结分析

针对含水平排水砂垫层地基的固结问题,建立了线性加载下夹砂垫层地基的二维固结模型,并通过边界转换法、积分变换法给出了相应的半解析解。在通过对解答的退化以及数值法验证半解析解正确性的基础上,分析了砂垫层与地基土的几何及物理参数对地基固结的影响。研究表明：对于地基土,地基固结速率随土体水平渗透系数增大而提高,随地基宽度的增大而降低;对于砂垫层,固结速率随砂垫层厚度、水平渗透系数的增大而提高;对于外荷载,加荷速率越快,孔压达到峰值所需时间也越短,地基固结速率亦随之提高。此外,以地基顶面砂垫层为例,综合考虑砂垫层的渗透系数与厚度对砂垫层透水性的影响,当组合参数 时（ 、 、 、 分别为砂垫层厚度、水平渗透系数、土体宽度和竖向渗透系数）,砂垫层近似为完全排水边界。