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.     

Stresses from linearly distributed pressures over rectangular areas

This paper presents a general algebraical formula to determine the vertical stresses resulting from a linearly distributed surface pressure resting on an elastic medium. This problem and its various derivations has received considerable attention because of its applications in foundation engineering [1–5]. The equation presented in this paper has been determined by the integration of the Boussinesq [6] equation with respect to the general surface function of a linearly distributed loading regime. The equation can be applied to the case of an optimally designed rectangular footing subjected to biaxial bending. Unlike the current design formulae [1–7], it provides the flexibility of having various contact pressures at each of the four corners of the rectangular area. This may enable more accurate design procedures for pad foundations in the future. This paper explains the validity of the new formula by comparing its results with those produced by classical methods. It shows how it can be applied to determine the vertical stress distributions resulting from all types of linearly distributed surface pressures which occupy a rectangular area on an elastic mass. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis of footing resting on non-homogeneous soil by double spline element

A numerical method is proposed for the analysis of rectangular footing resting on an elastic soil layer. The footing is represented by double spline elements and the elastic soil medium by finite layers. The effect of the rigidity of footing and the non-homogeneity of the soil on the behaviour of such foundation system is investigated, and the results are presented in form of design charts such that they may be used for hand calculation for the estimation of the settlement of footings for a wide range of practical cases.     

墙下条形基础与层状横观各向同性地基共同作用

运用对偶积分方程来求解层状横观各向同性地基与墙下条形基础的共同作用问题。从直角坐标平面应变问题控制方程出发,通过傅里叶（Fourier）变换和层间连续性条件,可以得到层状横观各向同性地基的传递矩阵解。基于该传递矩阵解,并利用条形基础与地基接触的混合边值条件,推导出一组关于基础挠度和地基反力的对偶积分方程。考虑墙下条形基础受到竖向集中荷载的情况,利用弹性薄板理论先求解出条形基础挠度;随后应用雅可比（Jacobi）正交多项式和级数展开的方法,将对偶积分方程转化为线性代数方程组进行求解。编制了相应的计算程序,其计算结果与有限元软件ABAQUS的结果基本吻合,从而验证了所提理论的正确性。算例分析表明,板土相对刚度与地基成层性对地基反力、地表沉降和沿z轴竖向正应力有很大的影响。     

Performance of surface footing on geocell-reinforced soft clay beds

This paper presents the results of laboratory model tests carried out to develop an understanding of the behaviour of geocell-reinforced soft clay foundations under circular loading. Natural silty clay was used in this study. The geocells were prepared using biaxial polymer grid. The performance of the reinforced bed is quantified using non-dimensional factors i.e., Bearing capacity improvement factor (I_f) and Percentage reduction in footing settlement (PRS). The test results demonstrate that the geocell mattress redistributes the footing load over a wider area thereby improving the performance of the footing. The load carrying capacity of the clay bed is increased by a factor of up to about 4.5 times that of unreinforced bed. From the pressure-settlement responses, it is observed that the geocell-reinforced foundation bed behaves as a much stiffer system compared to the unreinforced case indicating that a substantial reduction in footing settlement can be achieved by providing geocell reinforcement in the soft clay bed. The maximum reduction in footing settlement obtained with the provision of geocell mattress of optimum size placed close to the footing is around 90%. Further improvement in performance is obtained with provision of an additional planar geogrid layer at the base of the geocell mattress.     

Vertical deformation of rigid foundations of arbitrary shape on layered soil media

A numerical procedure is described for the analysis of vertical deformation of smooth, rigid foundations of arbitrary shape on homogeneous and layered soil media. The contact area at the interface of the foundation and soil medium is approximated by square subdivisions. The response of the system is then obtained from the superposition of the influence of the individual subdivisions. The flexibility influence coefficients are based on equivalent smooth, rigid circular areas with the same contact area as the square subdivisions. For foundations on a homogeneous, isotropic elastic half-space, the flexibility coefficients are given analytically by the integrated forms of the Boussinesq's solution. For a layered soil medium, the flexibility coefficients are determined from an axisymmetric finite element analysis which is essentially two dimensional. Thus, there is no necessity for a full three-dimensional finite element analysis. Comparison with solutions obtained using the integral transform technique for smooth, rigid rectangular foundations on a homogeneous, isotropic elastic half-space shows good agreement. Parametric solutions are presented for the response of rectangular foundations on some ‘typical’ soil profiles. The use of a simplified method to estimate the settlement of rectangular foundations on a layered soil medium by superposing solutions for homogeneous, elastic strata is discussed.     

Finite layer analysis of the behaviour of a raft on a consolidating soil

A numerical method is developed in this paper for the analysis of the behaviour of a raft resting on a consolidating soil. The response of the raft under an applied loading is determined using the finite layer method for the soil and the finite element method for the raft. By considering deflection compatibility on the contact surface, the distribution of contact pressure is computed at various time steps. The settlement and bending moment in the raft is then evaluated by applying the calculated contact pressure back to the raft. It is shown that, in some cases, the maximum moment in the raft occurs during consolidation and that checking the final moment in the raft by use of elastic theory may not be sufficient.     

Semi‐analytical solution to one‐dimensional consolidation for unsaturated soils with semi‐permeable drainage boundary under time‐dependent loading

This paper presents semi‐analytical solutions to Fredlund and Hasan's one‐dimensional consolidation of unsaturated soils with semi‐permeable drainage boundary under time‐dependent loadings. Two variables are introduced to transform two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. The pore‐water pressure, pore‐air pressure and settlement are obtained in the Laplace domain. Crump's method is adopted to perform the inverse Laplace transform in order to obtain semi‐analytical solutions in time domain. It is shown that the present solutions are more general and have a good agreement with the existing solutions from literatures. Furthermore, the current solutions can also be degenerated into conventional solutions to one‐dimensional consolidation of unsaturated soils with homogeneous boundaries. Finally, several numerical examples are provided to illustrate consolidation behavior of unsaturated soils under four types of time‐dependent loadings, including instantaneous loading, ramp loading, exponential loading and sinusoidal loading. Parametric studies are illustrated by variations of pore‐air pressure, pore‐water pressure and settlement at different values of the ratio of air–water permeability coefficient, depth and loading parameters. Copyright © 2017 John Wiley & Sons, Ltd.     

Least‐square support vector machine applied to settlement of shallow foundations on cohesionless soils

This paper examines the potential of least‐square support vector machine (LSVVM) in the prediction of settlement of shallow foundation on cohesionless soil. In LSSVM, Vapnik's ε‐insensitive loss function has been replaced by a cost function that corresponds to a form of ridge regression. The LSSVM involves equality instead of inequality constraints and works with a least‐squares cost function. The five input variables used for the LSSVM for the prediction of settlement are footing width (B), footing length (L), footing net applied pressure (P), average standard penetration test value (N) and footing embedment depth (d). Comparison between LSSVM and some of the traditional interpretation methods are also presented. LSSVM has been used to compute error bar. The results presented in this paper clearly highlight that the LSSVM is a robust tool for prediction of settlement of shallow foundation on cohesionless soil. Copyright © 2008 John Wiley & Sons, Ltd.     

Settlement analysis of rectangular piles in nonhomogeneous soil using a Winkler model approach

An analytical approach using a Winkler model is investigated to provide analytical solutions of settlement of a rectangular pile subjected to vertical loads in nonhomogeneous soils. For a vertically loaded pile with a rectangular cross section, the settlement influence factor of a normal pile in nonhomogeneous soils is derived from Mindlin's solution for elastic continuum analysis. For short piles with rectangular and circular cross sections, the modified forms of settlement influence factors of normal piles are produced taking into account the load transfer parameter proposed by Randolph for short circular piles. The modulus of subgrade reaction along a rectangular pile in nonhomogeneous soils is expressed by using the settlement influence factor related to Mindlin's solution to combine the elastic continuum approach with the subgrade‐reaction approach. The relationship between settlement and vertical load for a rectangular pile in nonhomogeneous soils is available in the form of the recurrence equation. The formulation of settlement of soils surrounding a rectangular pile subjected to vertical loads in nonhomogeneous soils is proposed by taking into account Mindlin's solution and both the equivalent thickness and the equivalent elastic modulus for layers in the equivalent elastic method. The difference of settlement between square and circular piles is insignificant, and the settlement of a rectangular pile decreases as the aspect ratio of the rectangular pile cross section increases. The comparison of results calculated by the present method for a rectangular pile in nonhomogeneous soils has shown good agreement with those obtained from the analytical methods and the finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

Interference effect of two nearby strip footings on layered soil: theory of elasticity approach

In recent times, rapid urbanisation coupled with scarcity of land forces several structures to come up ever closer to each other, which may sometime cause severe damage to the structures from both strength and serviceability point of view, and therefore, a need is felt to devise simplified methods to capture the effect of footing interference. In the present study, an attempt has been made to model the settlement behaviour of two strip footings placed in close spacing on layered soil deposit consisting of a strong top layer underlying a weak bottom layer. Theory of elasticity is employed to derive the governing differential equations and subsequently solved by the finite difference method. The perfectly rough strip footings are considered to be resting on the surface of two-layer soil system, and the soil is assumed to behave as linear elastic material under a range of static foundation load. The effect of various parameters such as the elastic moduli and thickness of two layers, clear spacing between the footings and footing load on the settlement behaviour of closely spaced footings has been determined. The variation of vertical normal stress at the interface of two different soil layers as well as at the base of the failure domain also forms an important part of this study. The results are presented in terms of settlement ratio (ξ_δ), and their variation is obtained with the change in clear spacing between two footings. The present theoretical investigation indicates that the settlement of closely spaced footings is found to be higher than that of single isolated footing, which further reduces with increase in the spacing between the footings.     

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.     

考虑二期荷载的高填明洞加筋减载土压力计算

随着城市建设的迅速发展,既有高填明洞填土顶面将不可避免地出现新增结构物等二期荷载的情况。为了明确明洞回填完成后新增二期荷载对其顶部垂直土压力的影响,在已有明洞减载结构土压力计算公式的基础上,利用土体沉降弹性理论计算公式建立了土工格栅变形与土体沉降变形的关系,并考虑二期荷载作用,进一步完善了高填明洞减载前后的洞顶垂直土压力计算公式。建立数值分析模型,将数值分析结果与文中公式计算结果进行了对比。结果表明：公式计算与数值分析结果较为接近,表现出相同的规律性;有无减载措施对明洞回填土压力影响较大,填土越高,减载越显著,实际作用在明洞顶土压力越小。新增二期荷载作用下的明洞顶垂直土压力计算,应考虑其对土体沉降变形的影响,而非简单叠加。     

路基沉降预测的Usher模型应用研究

基于线性或近似线性加载情况下路基沉降过程和Usher曲线,将广泛用于经济和资源预测的Usher模型应用于路基沉降预测。对Usher模型微分方程式的分析表明,目前用于沉降预测的Logistic模型和Gompertz模型是Usher模型的两种简化形式,但Usher模型对于实际情况具有更强的适应性。阐述了Usher模型参数的计算方法,并结合某工程实例进行了计算分析,对比结果说明Usher模型用于沉降预测的效果较好,且比Logistic模型和Gompertz模型具有更高的预测精度,可供工程应用参考。     

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。     

Analysis of laterally loaded piles with rectangular cross sections embedded in layered soil

An analysis is developed to determine the response of laterally loaded rectangular piles in layered elastic media. The differential equations governing the displacements of the pile–soil system are derived using variational principles. Closed‐form solutions of pile deflection, the slope of the deflected curve, the bending moment and the shear force profiles can be obtained by this method for the entire pile length. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. The new analysis allows insights into the lateral load response of square, rectangular and circular piles and how they compare. Copyright © 2007 John Wiley & Sons, Ltd.     

Visco‐elastic load transfer models for axially loaded piles

Viscoelastic or creep behaviour can have a significant influence on the load transfer (t–z) response at the pile–soil interface, and thus on the pile load settlement relationship. Many experimental and theoretical models for pile load transfer behaviour have been presented. However, none of these has led to a closed‐form expression which captures both non‐linearity and viscoelastic behaviour of the soil. In this paper, non‐linear viscoelastic shaft and base load transfer (t–z) models are presented, based on integration of a generalized viscoelastic stress–strain model for the soil. The resulting shaft model is verified through published field and laboratory test data. With these models, the previous closed‐form solutions evolved for a pile in a non‐homogeneous media have been readily extended to account for visco‐elastic response. For 1‐step loading case, the closed‐form predictions have been verified extensively with previous more rigorous numerical analysis, and with the new GASPILE program analysis. Parametric studies on two kinds of commonly encountered loading: step loading, ramp (linear increase followed by sustained) loading have been performed. Two examples of the prediction of the effects of creep on the load settlement relationship by the solutions and the program GASPILE, have been presented. Copyright © 2000 John Wiley & Sons, Ltd.     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.