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.     

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.     

上拔扩底基础与地基土体承载特性差异性分析

工程中多采用基础上拔静载试验中基础顶部荷载-位移曲线获取基础的承载力,忽略了基础周围土体的变形破坏过程,而实际工程中均是基础周围地基土体发生破坏。为研究扩底基础与其周围土体在抗拔承载特性方面的差异,以黄土地基中的9个扩底基础为研究对象,通过现场全尺寸基础的上拔静载试验,分别获得基础顶部与地表的上拔荷载-位移曲线,并进一步对基顶与地表处的荷载-位移曲线变化特征、抗拔承载力取值进行对比分析。结果表明,两处的荷载-位移曲线变化特征相似,相同上拔荷载作用下地表处的位移量均小于基础处位移量,差异以初始弹性阶段变形最为突出;两者在弹性极限荷载QL1取值方面,相差较大,但随着地基基础由弹性向塑性发展,差异逐渐减小,两者塑性极限荷载QL2取值基本相同。结合上拔扩底基础的破坏模式,分析出上述差异主要由于基础与周围土体之间变形不协调所致,加载初期基础顶部的上拔位移包括基础拔出量和上部土体压缩量,当上部土体压密后压缩变形消失,地基基础成为一个整体,上拔基础与周围土体的变形趋于协调。     

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.     

A simple model for inelastic footing response to transient loading

This paper presents a mechanical analogue which models the response of a rigid circular footing on an ideal elastoplastic half-space to transient loads. In the rational analysis of pile-driving dynamics, the response of soil at the base of a pile is often approximated by a footing on a semi-infinite half-space. Most existing base models employ the well-known Lysmer analogue to model the elastic response of the soil at the pile base, and account for the inelastic soil behaviour through the inclusion of a plastic slider with a slip load equal to the ultimate failure load of the footing. The improved model provides a force response which is significantly closer to the ideal response than existing models. The paper commences with a review of analytical solutions for the dynamic response of a rigid circular footing on an elastic half-space. Existing mechanical analogs for the system are reviewed, and an automatic matching process proposed which improves the accuracy of the analogs under transient loading. The inelastic response is then studied using the finite element method, and the mechanical analogs are modified to allow representation of the observed inelastic behaviour. Examples are presented illustrating close agreement between the proposed models and finite element analyses for a range of Poisson's ratio. The improved models have direct application for one-dimensional models of pile driving, particularly in the back-analysis of data from dynamic testing of piles. They are also applicable to studies of dynamic compaction.     

Probabilistic Analysis and Design of a Strip Footing on Layered Soil Media

In the present paper, the analysis of a strip footing resting on a layered soil system has been carried out considering the elastic moduli of soil layers as random variables. Three layers of soil have been considered and the analysis employs Monte Carlo simulation. The modulus of elasticity has been considered as random variable having lognormal distribution. Factors of safety with respect to settlement of footing and the interfacial stresses have been determined and have been related to the associated risk factor and coefficient of variation of the random variable. A detailed parametric study revealed that for a given risk level, the factors of safety is strongly dependent on the coefficient of variation of elastic modulus and only mildly upon other parameters of the soil?Cfoundation system. This facilitated the development of closed form equations for the upper bounds on factors of safety only in terms of allowable risk of failure and the coefficient of variation of elastic modulus.     

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.     

Developing artificial neural network models to predict allowable bearing capacity and elastic settlement of shallow foundation in Sharjah,United Arab Emirates

This research proposes the use of artificial neural network to predict the allowable bearing capacity and elastic settlement of shallow foundation on granular soils in Sharjah, United Arab Emirates. Data obtained from existing soil reports of 600 boreholes were used to train and validate the model. Three parameters (footing width, effective unit weight, and SPT blow count) are considered to have the most significant impact on the magnitude of allowable bearing capacity and elastic settlement of shallow foundations, and thus were used as the model inputs. Throughout the study, depth of footing was limited to 1.5 m below existing ground level and water table depth taken at the level of the footing. Performance comparison of the developed models (in terms of coefficient of determination, root mean square error, and mean absolute error) revealed that the developed artificial neural network models could be effectively used for predicting the allowable bearing capacity and elastic settlement. As such, the developed models can be used at the preliminary stage of estimating the allowable bearing capacity and settlements of shallow foundations on granular soils, instead of the conventional methods.     

Modeling of Beams on Reinforced Granular Beds

The paper describes a mechanical model for estimating the flexural response of a strip footing, supporting a column (imposing a concentrated load), resting on a compacted granular bed overlying a reinforcement layer for example, geogrids, geomats etc. below which lies a loose soil deposit. The footing is idealized as a beam and the reinforcing element is assumed to have finite bending stiffness and negligible frictional resistance. The upper and lower soil layers are idealized by a series of linear and discrete springs (Winkler springs) of different stiffness values. To find the response of such a model the governing differential equations have been derived and expressed in a nondimensional form. A closed form analytical solution of the same has been obtained subjected to appropriate boundary conditions. Using the present approach the resulting solution for a degenerated case of a long beam is found to be identical to the same of Hetenyi (1946, Beams on elastic foundations, University of Michigan press, Ann Arbor, MI). Parametric studies reveal that the ratio of flexural rigidity of upper and lower beam and the ratio of stiffness of the upper and lower soil layers affect significantly the response of the foundation.     

美国技术标准用弦线模量对我国技术更新的启示

用弦线模量计算地基变形是中国的原始自主创新。它改正了压缩模量和各种计算模型存在的问题,所以对计算结果有很大的改进。它依据土性物理指标用软件计算,是各种计算方法中最简单的方法。用于地基设计规范能够实现以变形控制设计的目标;用于黄土地区建筑规范,能够依据基础面积、基底压力、相邻基础影响等条件,并考虑地区差异、土性的微结构差异等条件真实地计算不同类型黄土的湿陷量,实现定量评价黄土湿陷性的目标。在中国对弦线模量研究的同时,美国在金属材料非线弹性变形计算中也研究并已经在其技术标准中采用了弦线模量代替杨氏模量,中国的同类标准也引用了它。这对在我国地基规范中采用弦线模量法进行技术更新,应该具有重要的启示作用。在简要介绍弦线模量法主要情况的基础上,进一步阐明这一观点。     

Elasto-viscoplastic finite element study of the effect of degradation on bearing capacity of footing on clay ground

The bearing capacity of footing has been studied by both conventional and numerical methods by many researchers. However, degradation of the microstructure of material, that is, a change in the microstructure of the soil, has not been adequately taken into account. Degradation of microstructure causes strain softening of materials and it leads to strain localization such as shear bands and slip bands. From an engineering point of view the strain localization is crucial because it is a precursor of failure. In the present study, finite element analyses of the bearing capacity of a shallow foundation on homogeneous and inhomogeneous saturated clay strata have been conducted using an elasto-viscoplastic soil constitutive model of microstructure change. A series of analyses of footing on clay deposit with different microstructure parameters have been carried out. Numerical results show that strain localization can be predicted during the loading of rigid footing on highly structured soil and strain localization affects the footing–soil interaction. The effects of footing roughness on the failure mechanism are also discussed in the study.     

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.     

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。     

Effect of interface characteristics on the influence coefficients of an embedded circular footing under horizontal and moment loading

By means of a semi-analytical FE approach an embedded circular footing under monotonic horizontal and moment loading is studied. In a non-homogeneous soil whose shear modulus is characterized by a power law variation with depth, horizontal, rocking and coupled modes of displacement, expressed in terms of influence factors are thoroughly examined. The exponent α that controls the shape of the stiffness variation with depth is termed shear modulus factor. Surface influence coefficients are considered for the situations where the interface between the soil and the footing is either perfectly rough or perfectly smooth. First, results of semi-analytical FE analysis in the case of rough footing are established and compared with those of another numerical method. Results of comparison show good agreement. Then, for different values of α the surface influence coefficients are presented for an embedded footing in perfect smooth contact with soil. The metacentre is referred to as the depth at which there is no coupling between the sliding and the rocking modes of footing deformations. Expressions for location and horizontal influence coefficient corresponding to this particular depth are developed and their variations with α examined. The paper finishes by showing the effect of interface conditions on the soil normal stresses developed beneath the embedded circular footing for the case of loading applied at the footing top.     

Numerical study of the bearing capacity for two interfering strip footings on sands

Current studies of bearing capacity for shallow foundations tend to rely on the hypothesis of an isolated footing. In practice a footing is never isolated; it is mostly in interaction with other footings. This paper focuses on a numerical study using the finite-difference code Fast Lagrangian Analysis of Continua (FLAC), to evaluate the bearing capacity for two interfering strip footings, subjected to centered vertical loads with smooth and rough interfaces. The soil is modeled by an elasto-plastic model with a Mohr–Coulomb yield criterion and associative flow rule. The interference effect is estimated by efficiency factors, defined as the ratio of the bearing capacity for a single footing in the presence of the other footing to that of the single isolated footing. The efficiency factors have been computed individually to estimate the effects of cohesion, surcharge, and soil weight using Terzaghi’s equation, both in a frictional soil with surcharge pressures and in a cohesive-frictional soil with surcharge pressures. The results have been compared with those available in the literature.     

Stability of eccentrically loaded footings on slopes

The stability of eccentrically loaded strip footings on slopes was investigated using the method of finite element analysis based on the theory of elasto-plasticity. The analysis was done for two different soils involving three levels of slope angle, six footing locations, and two levels of load eccentricity plus central vertical loading. The strip footing analysed was a 3-ft (0.9 m) wide reinforced concrete footing embedded to a depth of 3 ft (0.9 m). The analysis focused on footing settlement, plastic yielding of soil, and ultimate bearing capacity. The results of analysis show that the influence of load eccentricity on footing pressure vs. footing centre settlement is negligibly small. However, the progressive soil yielding and ultimate bearing capacity are greatly affected by load eccentricity. Furthermore, the effect of load eccentricity differs considerably with the load location relative to the footing centre and slope crest. The ultimate bearing capacity for the eccentric load located on the slope side is significantly greater than that for the load located on the other side of the footing centre. For a 2(H): 1(V) slope in silty clay, the effect of slope on footing stability decreases with increasing footing location from slope crest as would be expected, and diminishes when the footing is located from the crest at about 5-times the footing width.     

Bearing Capacity Factors of Ring Footings by Using the Method of Characteristics

Ring footings can be more effective and economical than circular footings. In spite of similarities between circular and ring footings, their behaviors are different in some respects such as bearing pressure distribution under the footing and settlement. But no exclusive theoretical prediction of ultimate bearing capacity has been reported for ring footings. In the present study, stress characteristics method is employed for coding the bearing capacity of ring footing with horizontal ground surface. In the calculations, friction at the contact between the soil and foundation is considered. In this research, the soil obeys the Mohr–Coulomb yield criterion and that is cohesive–frictional-weighted with applied surcharge pressure. The bearing capacity factors Nγ, Nq and Nc for ring footings were calculated by a written code based on the method of characteristics. Bearing capacity was determined for different conditions of soil and different ratio of radii in comparison with the principle of superposition results. The findings show that the principle of superposition is effective for determining the bearing capacity of a ring footing.     

临近边坡的条形基础地基极限承载力数值分析

建筑物或构筑物基础临近边坡置放的情况在实际工程中十分普遍,但目前对于临近边坡基础的地基承载力及破坏模式尚缺乏深入研究。采用不连续布局优化（DLO）极限分析法建立数值模型,分析边坡几何尺寸、土体参数和基础位置对临坡条形基础的极限承载力和边坡破坏模式的影响,并对国内外现行规范推荐的计算方法进行评价。结果表明：极限承载力随边坡高度和边坡倾斜角的增大而减小,当坡高超过临界高度后,极限承载力将不受其影响;极限承载力随土体黏聚力和内摩擦角的增大而提高,滑动面随黏聚力的增大而变浅,随内摩擦角的增大而变深;极限承载力随基础与坡肩相对距离的增大而提高,当基础置放位置超过某临界距离后极限承载力不受边坡影响。在土体强度高、坡角较大时,《建筑地基基础设计规范》规定的临坡基础最小置放距离偏于危险,设计时仍需考虑边坡对承载力的减损作用;在土体强度较低、坡角较小时,规范规定值偏于保守。美国AASHTO规范对边坡地基极限承载力的取值在砂土边坡时较为可靠,但其仅适用于坡面破坏模式的情况;饱和黏土边坡的承载力曲线有悖于理论解,对临界距离的规定同样存在低估。     

Reliability bearing capacity analysis of footings on cohesive soil slopes using RFEM

Reliability analysis of bearing capacity of a strip footing at the crest of a simple slope with cohesive soil was carried out using the random finite element method (RFEM). Analyses showed that the coefficient of variation and the spatial correlation length of soil cohesion can have a large influence on footing bearing capacity, particularly for slopes with large height to footing width ratios. The paper demonstrates cases where a footing satisfies a deterministic design factor of safety of 3 but the probability of design failure is unacceptably high. Isotropic and anisotropic spatial variability of the soil strength was also considered.     

Non-linear analysis of geomechanical problems using coupled finite and infinite elements

In modeling of many geomechanics problems such as underground openings, soil-foundation structure interaction problems, and in wave propagation problems through semi-infinite soil medium the soil is represented as a region of either infinite or semi-infinite extent. Numerical modeling of such problems using conventional finite elements involves a truncation of the far field in which the infinite boundary is terminated at a finite distance. In these problems, appropriate boundary conditions are introduced to approximate the solution of the infinite or semi-infinite boundaries as closely as possible. However, the task of positioning the finite boundary in conventional finite element discretization and the definition of the boundary and its conditions is very delicate and depends on the modeller's skill and intuition. Moreover, such a choice is influenced by the size of the domain to be discretized. Consequently, the dimensions of the global matrices and the time required for solution of the problem will increase considerably and also selection of the arbitrary location of truncated boundary may lead to erroneous result. In order to over come these problems, mapped infinite elements have been developed by earlier researchers (Simoni and Schrefier, 1987). In the present work the applicability of infinite element technique is examined for different geomechanics problems. A computer program INFEMEP is developed based on the conventional finite element and mapped infinite element technique. It is then validated using selected problems such as strip footing and circular footing. CPU time taken to obtain solutions using finite element approach and infinite element approach was estimated and presented to show the capability of coupled modeling in improving the computational efficiency. Mesh configurations of different sizes were used to explore the enhancement of both computational economy and solution accuracy achieved by incorporation of infinite elements to solve elastic and elasto-plastic problems in semi-infinite/finite domain as applied to geotechnical engineering. © Rapid Science Ltd. 1998