The behaviour of an elastic non-homogeneous half-space. Part I-line and point loads

In the first part of this paper solutions are developed for the response of a non-homogeneous half-space subjected to either a surface point load or a surface line load. The non-homogeneity considered is a variation in Young's modulus (E) with depth (z) which takes the form E=m_EZ^α where m_E is a constant and α is referred to as the non-homogeneity parameter. The variation of these solutions as the non-homogeneity parameter α varies between the limits of zero (homogeneous soil) to unity (Gibson soil) gives some fresh insight into both these limiting cases.     

Response of circular footings and anchor plates in non-homogeneous elastic soils

The axisymmetric elastic response of circular footings and anchor plates in a linearly non-homogeneous elastic soil is analysed. It is assumed that footings/anchors are flexible and subjected to axisymmetric vertical loads. The response of the footing/anchor is modelled by using the classical Poisson–Kirchhoff thin plate theory. A variational technique is used to analyse the interaction problem. A representation for the contact stress is established by using a fundamental solution corresponding to a unit vertical pressure acting over an annular region in the interior of the non-homogeneous soil. The fundamental solution can be derived by using rigorous analytical procedures. The influence of the footing flexibility and the degree of soil non-homogeneity on the displacements, bending moments and contact stresses of a surface footing is examined over a wide range of governing parameters. In the case of anchor plates the influence of depth of embedment, degree of soil non-homogeneity and anchor flexibility on the anchor displacement is investigated.     

On the dynamic stiffness of circular ring footings on an elastic stratum

Circular ring footings on an elastic stratum are considered. The static and dynamic stiffnesses are calculated using an efficient numerical technique. The results indicate that the static torsional and rocking stiffnesses of a ring footing do not deviate significantly from the corresponding stiffness of the circular footing for values of the inner-radius-to-outer-radius ratio up to about 0.75. The static horizontal and vertical stiffnesses change considerably only for values of this ratio greater than about 0.60. The change in the stiffness and damping coefficients is small for values of the ratio between 0 and 0.5.     

The interaction between a circular elastic plate and a transversely isotropic elastic half-space

The axisymmetric problem of flexure of a centrally loaded or uniformly loaded thin-elastic plate resting on or buried in a transversely isotropic elastic half-space is solved using a constraint variational scheme. A fundamental solution corresponding to a unit vertical pressure acting over an annular region in the interior of the half-space which is derived through the application of Hankel integral transforms is used to evaluate the strain energy of the elastic half-space. The deflected shape of the plate is represented by a power series of radial coordinate together with a term corresponding to a concentrated load derived from classical plate theory. The coefficients of the power series representation, which could be considered as generalized coordinates corresponding to the interaction problem, are determined through the minimization of a constraint energy functional. The energy functional consists of the strain energies of the plate and half-space, the potential of the applied loads and a term corresponding to plate edge boundary conditions incorporated with a penalty number. The entire formulation is developed as a convenient and flexible matrix solution scheme. The numerical stability and convergence of the solution scheme are established, and the response of the plate is investigated over a range of relative rigidities of practical interest.     

Bearing capacity of strip and circular footings in sand using finite elements

Design of shallow foundations relies on bearing capacity values calculated using procedures that are based in part on solutions obtained using the method of characteristics, which assumes a soil following an associated flow rule. In this paper, we use the finite element method to determine the vertical bearing capacity of strip and circular footings resting on a sand layer. Analyses were performed using an elastic–perfectly plastic Mohr–Coulomb constitutive model. To investigate the effect of dilatancy angle on the footing bearing capacity, two series of analyses were performed, one using an associated flow rule and one using a non-associated flow rule. The study focuses on the values of the bearing capacity factors N_q and N_γ and of the shape factors s_q and s_γ for circular footings. Relationships for these factors that are valid for realistic pairs of friction angle and dilatancy angle values are also proposed.     

The axial loading of a rigid circular anchor plate embedded in an elastic half-space

The present paper examines the axisymmetric problem related to the loading of a rigid circular anchor plate which is embedded in bonded contact with an isotropic elastic half-space. A Hankel transform development of the governing equations is used to reduce the associated mixed boundary value problem to a set of coupled Fredholm integral equations of the second kind. These equatons are solved in a numerical fashion to generate results of engineering interest. In particular, the results indicate the influence of the depth of embedment on the axial stiffness of the rigid anchor plate.     

竖向圆形荷载作用下弹性地基半空间问题的解析解

竖向圆形荷载作用下弹性半空间问题的位移和应力解是桩基分析的基础。利用Hankel积分变换,首先导出了弹性地基半空间位移与应力的积分形式的通解。通过适当地引入边界条件和界面位移和应力的连续条件,求得了内部作用竖向圆形荷载时弹性地基半空间位移与应力的积分形式解。在此基础上,给出了不同深度处荷载作用投影范围内竖向位移和竖向正应力的平均值。数值结果验证了解析解的正确性。     

非均质地基承载力及破坏模式的FLAC数值分析

利用基于Lagrangian显式差分的FLAC算法，通过数值计算，对黏结力随深度线性增长的非均质地基上条形基础和圆形基础的极限承载力及地基破坏模式进行了对比计算与系统分析。研究表明：（1）随着地基黏结力沿深度非均匀变化系数的增大，地基的破坏范围逐渐集中在地基表层和基础两侧：（2）即使地基的非均质程度较小，当将非均质地基近似地按均质地基考虑时，由此所估算的承载力可能过于保守；（3）地基承载力系数随黏结力沿深度非均匀变化系数的增大而非线性地增大。与数值解相比，skempton与Peck等近似公式均可能高估了非均质地基承载力。     

Vertical vibrations of a rigid circular body on a non-homogeneous half-space interrupted by a frictionless plane

An exact formulation is presented for the problem of a rigid circular body performing harmonic vibrations on an elastic half-space whose shear modulus increases linearly with depth and is interrupted at some finite depth by a frictionless horizontal plane. The static case is derived in the limit of zero frequency vibrations while the known result for the uninterrupted half-space is recovered in either extreme limit of the horizontal frictionless plane coinciding with the surface or when it is pushed down to an infinite depth. It is shown that the maximum effect of the interruption occurs when the frictionless plane is at a depth where the shear modulus is about 1·6 times the surface shear modulus. Furthermore, this maximum effect is equivalent to a reduction of about 5 per cent of the surface shear modulus or a reduction of about 2½ per cent in the natural frequency of the rigid body on an uninterrupted half-space. The important conclusion, therefore, is that irrespective of the depth at which a half-space isso interrupted, the surface shear modulus is still the dominant parameter and that both the increase in shear modulus with depth and the interruption are not only secondary but also opposing effects.     

Vertical and horizontal deformation of an inhomogeneous elastic half-space

Vertical and horizontal deformations of surface footings have been studied for an inhomogeneous elastic half-space in which the shear modulus increases with an arbitrary power of depth, n, and Poisson's ratio is constant. A general solution for displacements has been obtained first for point loads applied in vertical and horizontal directions. These are then used in obtaining closed-form solutions for displacements of uniformly loaded circular and rectangular footings. Finally, a numerical method is described that can be used to analyse a rigid footing of an arbitrary shape, and results for rigid rectangular footings are given.     

Influence of subsurface fractures on surface deformation of an elastic half-space

The paper presents a solution for calculation of the surface uplift field of a semi-infinite elastic solid embedded with a pressurized elliptical crack oriented at any angle, and at any depth. The results presented are those for a uniform pressure loading on the crack face, simulating a static pressurized hydraulic fracture, but the mathematical technique developed can be extended directly to non-uniform normal loads or general shear loads in the form of a polynomial up to the third order. The result of a special case of the new solution, where the crack is penny shaped and parallel to the free surface, is compared with results from Sun's solution and from experiments which were designed and performed by the authors. The experiments involved pressurization of a penny-shaped crack in a polyurethane material which is linear elastic under low stresses, and measurement of surface deformation by holographic interferometry and Fizeau interferometry. The correspondence between theory and experiment is excellent.     

Static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads

The transfer matrix approach is used to solve the problem of static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads. The general problem is decoupled into two independent problems. The antiplane strain problem and the plane strain problem are considered in detail. Integral expressions for displacements and stresses at any point of the medium due to a normal line load and a shear line load, acting parallel to a symmetry axis, are obtained. In the case of a uniform half-space, closed form analytic expressions for displacements and stresses are derived. The procedure developed is quite easy and convenient for numerical computations.     

Vertical vibration of an elastic strip footing on saturated soil

Based on Biot's dynamic coupled equations, the vertical vibration of an elastic strip footing on the surface of saturated soil is studied. Utilizing the Fourier transform, the governing dynamic differential equations for saturated poroelastic medium are solved. Considering the mixed boundary value conditions at the bottom of the foundation, a pair of dual integral equations about the vertical vibration of an elastic strip footing is derived, which can be converted to a set of linear equations by means of infinite series of orthogonal functions. The relation between the dynamic compliance coefficients and the dimensionless frequency tends to be gentle with decreasing footing rigidity, while the dimensionless frequency has only small effect on the dynamic compliance coefficients. When the dynamic permeability is large, its effect on the dynamic compliance coefficients should be taken into consideration. Furthermore, the dynamic compliance coefficients are found to be not sensitive to Poisson's ratio of the soil for footing on saturated soil. Copyright © 2007 John Wiley & Sons, Ltd.     

饱和地基上条形弹性基础的摇摆振动

基于Biot动力方程,研究了饱和均质弹性半空间上弹性条形基础的摇摆振动问题。通过Fourier积分变换求解了饱和土的动力控制方程,然后结合基础底部为混合边界的条件得到了弹性条形基础的摇摆振动对偶积分方程,利用正交多项式将对偶积分方程转化为求解一组线性代数方程组,同时利用复合Simpson法则,得到了动力柔度系数的表达式,通过算例得出了不同参数时地基动力柔度系数随无量纲频率的关系曲线。     

The influence of load inclination on the undrained bearing capacity of strip footings on slopes

The effect of inclined loading on the bearing capacity of foundations on horizontal ground surface is well established and both the exact solution and simpler empirical equations are available for the calculation of the failure loads. However, for footings on or near slopes complete solutions are available only for vertical loading. This paper investigates the influence of inclined loading on the horizontal and vertical failure loads. The finite element, upper bound plasticity and stress field methods are used to examine a wide range of geometries and soil properties. The methods are first validated against known solutions for two special cases and are subsequently employed to investigate the effect of the geometrical and material properties on the failure loads and the bearing capacity load interaction diagram. Based on this investigation an empirical equation is proposed for the load interaction diagram for undrained inclined loading of footings on or near slopes.     

Effect of irregularity on torsional surface waves in an initially stressed anisotropic porous layer sandwiched between homogeneous and non-homogeneous half-space

The present paper is concerned with the propagation of torsional surface waves in an initially stressed anisotropic porous layer sandwiched between homogeneous and non-homogeneous half-space. We assume the quadratic inhomogeneity in rigidity and density in the lower half-space and irregularity is taken in the form of rectangle at the interface separating the layer from the lower half-space. The dispersion equation for torsional waves has been obtained in a closed form. Velocity equation is also obtained in the absence of irregularity. The study reveals that the presence of irregularity, initial stress, porosity, inhomogeneity and anisotropy factor in the dispersion equation approves the significant effect of these parameters in the propagation of torsional waves in porous medium. It has also been observed that for a uniform media, the velocity equation reduces to the classical result of Love wave.     

The flexure of an infinite strip of finite width embedded in an isotropic elastic medium of infinite extent

This paper examines the problem of the flexure of a strip of finite width which is embedded in bonded contact with an isotropic elastic medium of infinite extent. The analysis of the problem is applicable to flexible beams of narrow width which experience flexure only in the longitudinal direction. Numerical results are developed for the maximum flexural moment induced in the embedded beam due to the action of a concentrated force.     

Importance of footing–soil separation on dynamic stiffness of piled embedded footings

Different phenomena such as soil consolidation, erosion, and scour beneath an embedded footing supported on piles may lead to loss of contact between soil and the pile cap underside. The importance of this separation on the dynamic stiffness and damping of the foundation is assessed in this work. To this end, a numerical parametric analysis in the frequency domain is performed using a rigorous three‐dimensional elastodynamic boundary element–finite element coupling scheme. Dimensionless plots relating dynamic stiffness functions computed with and without separation effects are presented for different pile–soil configurations. Vertical, horizontal and rocking modes of oscillation are analyzed for a wide range of dimensionless frequencies. It is shown that the importance of separation is negligible for frequencies below those for which dynamic pile group effects start to become apparent. Redistribution of stiffness contributions between piles and footing is also addressed. Copyright © 2009 John Wiley & Sons, Ltd.     

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.