Scaled boundary finite‐element analysis of a non‐homogeneous axisymmetric domain subjected to general loading

The scaled boundary finite‐element method is derived for elastostatic problems involving an axisymmetric domain subjected to a general load, using a Fourier series to model the variation of displacement in the circumferential direction of the cylindrical co‐ordinate system. The method is particularly well suited to modelling unbounded problems, and the formulation allows a power‐law variation of Young's modulus with depth. The efficiency and accuracy of the method is demonstrated through a study showing the convergence of the computed solutions to analytical solutions for the vertical, horizontal, moment and torsion loading of a rigid circular footing on the surface of a homogeneous elastic half‐space. Computed solutions for the vertical and moment loading of a smooth rigid circular footing on a non‐homogeneous half‐space are compared to analytical ones, demonstrating the method's ability to accurately model a variation of Young's modulus with depth. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Bearing Capacity of Embedded Strip Footings on Cohesionless Soil Under Vertical and Horizontal Loads

In the last decades a few attention was given to the evaluation of the bearing capacity of embedded footing under inclined loads on a frictional soil. This paper focuses on a numerical study using the finite-difference code Fast Lagrangian Analysis of Continua (FLAC), to evaluate the bearing capacity of embedded strip footing on a frictional soil. The soil is modeled by an elasto-plastic model with a Mohr–Coulomb yield criterion and associative flow rule; the effect of non-associativity of the soil on the bearing capacity is also investigated. The effect of the embedment is estimated though a depth factor, defined as a ratio of the bearing capacity of a strip footing at a depth D to that of a strip footing at the ground surface. The inclination effect is estimated by inclination factors, defined as the ratio of the limit vertical load for a footing under inclined loading to that of the vertically loaded footing. Both swipe and probe analyses were carried out to identify the vertical force–horizontal force (V–H) failure envelope. The results have been compared with those available in the literature.     

复合地基沉降计算的半解析元法

将复合地基视为横观各向同性体,利用半解析数值计算原理,又将复合地基位移竖向离散为多项式,在水平两个方向上选取解析函数模拟无限域地基进行计算,从而克服了有限元计算中人为设定计算边界的弊端,提高了计算精度,并将三维沉降问题转化为一维问题进行计算,收敛速度快,显著地减少了计算工作量,将计算结果与理论解进行对比,满足精度要求。通过改变复合地基置换率、加固深度和桩土模量比,采用半解析元法研究了复合地基的加固效果,计算结果表明,桩土模量比较大时,可忽略加固区压缩变形量;增大置换率对减小沉降作用很小,而增加加固深度能明显减小复合地基沉降,在以控制沉降量为主要目的复合地基优化设计中应首先考虑增加加固深度。     

Experimental and numerical analyses of circular footing on geogrid-reinforced granular fill underlain by soft clay

Experimental and numerical investigations into the bearing capacity of circular footing on geogrid-reinforced compacted granular fill layer overlying on natural clay deposit have been conducted in this study. A total of 8 field tests were carried out using circular model rigid footing with a diameter of 0.30 m. 3D numerical analyses were performed to simulate soil behavior using finite element program Plaxis 3D Foundation. The results from the FE analysis are in very good agreement with the experimental observations. It is shown that the degree of improvement depends on thickness of granular fill layer and properties and configuration of geogrid layers. Parameters of the experimental and numerical analyses include depth of first reinforcement, vertical spacing of reinforcement layers. The results indicate that the use of geogrid-reinforced granular fill layers over natural clay soils has considerable effects on the bearing capacity and significantly reduces the lateral displacement and vertical displacement of the footing.     

The influence of horizontal confinement on the bearing capacity factor Nγ of smooth strip footing

We studied the upper-bound ultimate bearing capacity of smooth strip shallow footings with symmetrical and asymmetrical horizontal confinements on purely frictional sand within the framework of upper-bound limit analysis. The subsoil follows the associated flow rule, and no surcharge on the soil surface is assumed. The contact between the soil and the horizontal confinement walls is assumed to be perfectly rough. The upper-bound solutions for the objective functions are obtained using nonlinear sequential quadratic programming. The results for the different internal friction angles φ are provided in terms of the variation of two parameters, namely, the bearing capacity factor N_γ and the correction factor of bearing capacity K_γ, with respect to the change in the clear spacing between the edge of smooth footing and the rigid vertical walls. The values of N_γ and K_γ increase with φ and decrease with the clear spacing between the edge of the smooth footing and the rigid vertical walls. N_γ and K_γ are more sensitive to this confining effect as φ increases. The numerical results, a comparative analysis with the results from previous studies, and design charts are also included.     

Rocking vibration of a rigid strip footing on saturated soil

The dynamic response of a rigid strip footing lying on saturated soil is greatly affected by the pore pressure induced by a rocking moment. To consider the complex behavior of the soil under the rocking load, an analytical solution for a rigid strip foundation on saturated soil under a rocking moment is developed under the framework of Biot’s coupling theory. The boundary-value problem for the governing coupling equations for saturated soil is solved using a Fourier transform to yield a pair of dual integral equations. These dual integral equations are transformed into a set of linear equations using an infinite series of orthogonal Jacobi polynomials to yield the compliance functions. In addition, a parametric study has been carried out to examine the influence of: (1) the dimensionless frequency, (2) the dynamic permeability and (3) the Poisson’s ratio on saturated soil under a rocking rigid strip footing.     

Bearing capacity of foundations on slopes

By applying the lower bound finite element limit analysis in conjunction with non-linear optimisation, the bearing capacity factors, N_c, N_q and N_γ, due to the components of cohesion, surcharge and unit weight, respectively, have been estimated for a horizontal strip footing placed along a sloping ground surface. The variation of N_c, N_q and N_γ with changes in slope angle (β) for different soil friction angle (φ) have been computed for smooth as well as rough strip footings. The analysis reveals that along a sloping ground surface, in addition to N_γ, the factors N_c and N_q also vary considerably with changes in footing roughness. Compared to the smooth footing, the extent of the plastic zone around the footing becomes greater for the rough footing. The results obtained from the analysis are found to compare well with those previously reported in literature.     

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.     

移动荷载作用下非均匀地基的动力响应分析

基于线弹性动力学理论,结合坐标变换,建立了移动荷载作用下非均匀弹性半平面地基的动力控制方程,利用半解析法研究了移动荷载作用下二维非均匀地基的动力响应问题。采用傅里叶（Fourier）级数展开,假设了响应函数的级数形式,通过理论推导获得了剪切模量随深度任意变化的非均匀地基在移动荷载作用下各物理量的解析表达式。考虑土体的剪切模量沿厚度方向按幂函数梯度变化,通过数值算例分析并讨论了地基非均匀参数、荷载移动速度以及地基表面的剪切模量等对地基力学响应的影响规律,并与均质地基的计算结果进行了比较。数值结果表明：地基中各点的竖向位移随着土体表面剪切模量和表征土体非均匀性的梯度因子的增大而减小,随着荷载移动速度的增大而增大。在移动荷载作用下,非均匀地基与均匀地基的动力响应有着显著的区别。     

混凝土桩-泥皮-砂土接触面力学特性试验研究

为研究泥皮、粗糙度对桩-土接触面力学特性的影响规律,根据灌注桩成孔后的孔径-深度曲线,应用统计分析法获得了桩侧凸出尺寸和粗糙度的分布频率规律,以此构建了表面光滑和梯形凹槽混凝土板来模拟实际桩侧表面粗糙度。在此基础上,开展了不同泥皮厚度、粗糙度条件下的混凝土-砂土接触面大型直剪试验。其研究结果表明：无泥皮条件下粗糙接触面,其剪切应力-切向位移关系曲线呈软化型;泥皮厚度为5、10 mm条件下,呈硬化型。剪切模量G0.02随泥皮厚度增加而衰减。对光滑混凝土板,其接触面峰值剪切强度和峰值摩擦角随泥皮厚度的增加呈指数关系衰减;对粗糙混凝土板,峰值剪切强度和峰值摩擦角随泥皮厚度的增加近似呈线性衰减。初始泥皮越厚,试验后的泥皮土和泥皮越厚,接触面剪切强度越低。无泥皮条件下粗糙度对接触面峰值剪切强度的影响规律：存在一个临界粗糙度Icr =10 mm,当混凝土板的粗糙度I< Icr时,接触面峰值剪切强度和峰值摩擦角随粗糙度的增大而增大;当I≥Icr时,二者随着接触面粗糙度的增大而减小,泥皮存在会影响改变这一规律。     

Simple superposition approach for dynamic analysis of piled embedded footings

The effectiveness and accuracy of the superposition method in assessing the dynamic stiffness and damping coefficients (impedance functions) of embedded footings supported by vertical piles in homogeneous viscoelastic soil is addressed. To this end, the impedances of piled embedded footings are compared to those obtained by superposing the impedance functions of the corresponding pile groups and embedded footings treated separately, with the magnitude of the relative average differences being around 10–30%. The results are presented in a set of dimensionless graphs and simple expressions that can be used to estimate the dynamic stiffness and damping of piled embedded footings, provided that the impedance functions of the two individual components are known. This is precisely the reason why the superposition approach studied here is appealing, because such impedance functions for both embedded footings and pile groups are available for a wide range of cases. How to estimate the kinematic response functions of the system when those of the individual components are known is also discussed. To address the problem, parametric analyses performed using a 3D frequency‐domain elastodynamic BEM‐FEM formulation are presented for different pile–soil stiffness contrasts, embedment depths, pile‐to‐pile separations and excitation frequencies. Vertical, horizontal, rocking, and cross‐coupled horizontal‐rocking impedance functions, together with translational and rotational kinematic response functions, are discussed. The results suggest that the superposition concept, in conjunction with a correction strategy as that presented herein, can be employed in geotechnical design. For kinematic effects, the response functions of the embedded footing are found to provide reasonable estimates of the system's behaviour. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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。     

Numerical Study of Behavior of Circular Footing on Geogrid-Reinforced Sand Under Static and Dynamic Loading

A series of axi-symmetry models using finite element analyses were performed to investigate the behavior of circular footings over reinforced sand under static and dynamic loading. Geogrid was modeled as an elastic element and the soil was modeled using hardening soil model which use an elasto-plastic hyperbolic stress–strain relation. Several parameters including number of geogrid layers, depth to the first geogrid layer, spacing between layers and load amplitude of dynamic loading are selected in this paper to investigate the influence of these parameters on the performance of reinforced systems under both static and dynamic loads. The numerical studies demonstrated that the presence of geogrid in sand makes the relationship between contact pressure and settlement of reinforced system nearly linear until reaching the failure stage. The rate of footing settlement decreases as the number of loading cycles increases and the optimum values of the depth of first geogrid layer and spacing between layers is found 20% of the footing diameter. Some significant observations on the performance of footing-geogrid systems with change of the values of parametric study are presented in this paper.     

A three‐dimensional displacement approach for analysis of laterally loaded piles in nonhomogeneous soil

An analytical approach using the three‐dimensional displacement of a soil is investigated to provide analytical solutions of the horizontal response of a circular pile subjected to lateral loads in nonhomogeneous soil. The rocking stiffness coefficient of the pile shaft in homogeneous soil is derived from the analytical solution taking into account the three‐dimensional displacement represented in terms of scalar potentials in the elastic three‐dimensional analysis. The lateral stiffness coefficient of the pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient taking into account the rocking rotation of a rigid pile shaft. The relationship between horizontal displacement, rotation, moment, and shear force of a pile subjected to horizontal loads in nonhomogeneous soil is obtainable in the form of the recurrence equation. The formulation of the lateral displacement and rotation of the pile base subjected to lateral loads in nonhomogeneous soils is presented by taking into account Mindlin's equation and the equivalent thickness for soil layers in the equivalent elastic method. There is little difference between lateral, rocking, and couple stiffness coefficients each obtained from both the two‐dimensional and three‐dimensional methods except for the case of Poisson's ratio near 0.5. The comparison of results calculated by the current method for a pile subjected to lateral loads in homogeneous and nonhomogeneous soils has shown good agreement with those obtained from analytical and numerical methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Bearing capacity computation for a ring foundation using the stress characteristics method

The stress characteristics method (SCM) has been used to compute the bearing capacity of smooth and rough ring foundations. Two different failure mechanisms for a smooth footing, and four different mechanisms for a rough footing have been considered. For a rough base, a curvilinear non-plastic wedge has been employed below the footing. The analysis incorporates the stress singularities at the inner as well as outer edges of the ring footing. Bearing capacity factors, N_c, N_q and N_γ are presented as a function of soil internal friction angle (ϕ) and the ratio (r_i/r_o) of inner to outer radii of the footing.     

Ultimate Bearing Capacity of a Footing Considering the Interference of an Existing Footing on Sand

The ultimate bearing capacity of a new strip footing placed on a cohesionless soil medium, in the presence of an existing strip footing, the load on which is assumed to be known, has been determined. Both the footings are assumed to be perfectly rigid and rough. The analysis is carried out by using an upper bound finite element limit analysis. For different clear spacing (S) between the footings, the values of the efficiency factor (ξ_γ) were determined; where ξ_γ is defined as the ratio of the failure load for an interfering new footing of a given width (B) to that for a single isolated footing having the same width. For ϕ < 30°, it is generally noted that the magnitude of ξ_γ increases continuously with a decrease in S/B. For ϕ > 30°, on the other hand if the applied load on the existing footing is approximately greater than half the failure load for a single isolated footing having the same width, the peak magnitude of ξ_γ was found to occur at around S/B ≈ 0.1 rather than at S/B = 0. The increase in ξ_γ becomes further significant with an increase in the magnitude of the load on the existing footing.