Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion

A probabilistic model is presented to compute the probability density function (PDF) of the ultimate bearing capacity of a strip footing resting on a spatially varying soil. The soil cohesion and friction angle were considered as two anisotropic cross‐correlated non‐Gaussian random fields. The deterministic model was based on numerical simulations. An efficient uncertainty propagation methodology that makes use of a non‐intrusive approach to build up a sparse polynomial chaos expansion for the system response was employed. The probabilistic numerical results were presented in the case of a weightless soil. Sobol indices have shown that the variability of the ultimate bearing capacity is mainly due to the soil cohesion. An increase in the coefficient of variation of a soil parameter (c or φ) increases its Sobol index, this increase being more significant for the friction angle. The negative correlation between the soil shear strength parameters decreases the response variability. The variability of the ultimate bearing capacity increases with the increase in the coefficients of variation of the random fields, the increase being more significant for the cohesion parameter. The decrease in the autocorrelation distances may lead to a smaller variability of the ultimate bearing capacity. Finally, the probabilistic mean value of the ultimate bearing capacity presents a minimum. This minimum is obtained in the isotropic case when the autocorrelation distance is nearly equal to the footing breadth. However, for the anisotropic case, this minimum is obtained at a given value of the ratio between the horizontal and vertical autocorrelation distances. Copyright © 2012 John Wiley & Sons, Ltd.     

Upper bound solutions of bearing capacity of strip footing by pseudo-dynamic approach

This paper presents the pseudo-dynamic analysis of seismic bearing capacity of a strip footing using upper bound limit analysis. However, in the literature, the pseudo-static approach was frequently used by several researchers to compute the seismic bearing capacity factor theoretically, where the real dynamic nature of the earthquake accelerations cannot be considered. Under the seismic conditions, the values of the unit weight component of bearing capacity factor N _γE are determined for different magnitudes of soil friction angle, soil amplification and seismic acceleration coefficients both in the horizontal and vertical directions. The results obtained from the present study are shown both graphically as well as in the tabular form. It is observed that the bearing capacity factor N _γE decreases significantly with the increase in seismic accelerations and amplification. The results are thoroughly compared with the existing values in the literature and the significance of the present methodology for designing the shallow footing is discussed.     

Bearing Capacity of Strip Footings Near Slopes

In the last decades a great attention was given by many authors to the evaluation of the static and seismic bearing capacity of footings near slopes. In this paper a model has been developed based on the limit equilibrium method, considering a circular surface propagates towards the slope until the sloping ground is reached. The bearing capacity is investigated considering either the distance of the footing from the edge of the slope and/or the effect of the footing embedment. A validation of the proposed model was made by a comparison with solutions taken from literature regarding the evaluation of the bearing capacity for a footing adjacent to a slope and for an inclined load. The loading conditions consist in vertical and horizontal stress on the footing and on the soil below the footing. Both the inertial and kinematic effects of the seismic loading have been analyzed, and a simple equation has been derived for the evaluation of the seismic bearing capacity. The static and seismic bearing capacity has been investigated as a function of the soil friction angle, of the seismic coefficient, of the sloping ground. Finally, the influence of the distance of the footing from the edge of the slope was taken into consideration in the evaluation of the bearing capacity, and a threshold distance at which the reduction of the bearing capacity due to the sloping ground vanishes has been defined.     

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.     

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.     

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。     

Influence of random heterogeneity of the friction angle on bearing capacity factor Nγ

ABSTRACT

Probabilistic methods in geotechnical engineering have received a lot of attention during the last decade and different methodologies are used to capture the inherent variability of soil in different geotechnical engineering problems. In this paper, numerical simulations are conducted to obtain the bearing capacity factor, N_γ, for a purely frictional heterogenous soil where the friction angle is modelled as randomly distributed throughout the domain and the effect of its spatial variability on N_γ is investigated. A finite element method, based on the upper bound limit analysis was combined with random field theory and linear programming to develop a probabilistic analysis. Monte Carlo simulations were performed and the effect of the variability of the friction angle defined by statistical parameters on the bearing capacity factor was investigated. Results show that the mean bearing capacity factor N_γ of a footing on a spatially variable cohesionless soil is generally higher than the deterministic N_γ obtained from a constant mean value. Increasing the heterogeneity of the friction angle by an increase in the coefficient of variation generally increases this deviation. This can be explained by the nonlinearity of the relationship between N_γ and the friction angle.     

Upper bound limit analysis for finding interference effect of two nearby strip footings on sand

The ultimate bearing capacity of two closely spaced strip footings, placed on a cohesionless medium and loaded simultaneously to failure at the same magnitude of failure load, was determined by using an upper bound limit analysis. A logarithmic spiral radial shear zone, comprising of a number of triangular rigid blocks, was assumed to exist around each footing edge. The equations of the logarithmic spiral arcs were based on angles φ_L and φ_R rather than soil friction angle φ; the values of φ_L and φ_R were gradually varied in between 0 and φ. The ultimate bearing capacity was found to become maximum corresponding to a certain critical spacing between the footings. For spacing greater than the critical, the bearing capacity was found to decrease continuously with increase in the spacing. The extent of the spacing corresponding to which the ultimate bearing capacity becomes either maximum or equal to that of a single isolated footing increases with increase in φ. The results compare reasonably well with the available theoretical and experimental data.     

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。     

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.     

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.     

Bearing Capacity of Shallow Foundation on Two Clay Layers by Numerical Approach

Natural soils are often deposited in layers. The estimation of the bearing capacity of the soil, using conventional bearing capacity theory based on the properties of the top layer, introduces significant inaccuracies if the thickness of the top layer is comparable to the width of the rigid footing placed on the soil surface. Saturated normally consolidated and lightly overconsolidated clays indicate that under undrained condition the cohesion of soil mass increases almost linearly with depth. A few theoretical studies have been proposed in the literature to incorporate the variation of cohesion with depth in the computation of the ultimate bearing capacity of strip and circular footings. In this paper, after reviewing previous works, numerical computations using the FLAC code (Fast Lagrangian Analyses of Continua) are reported to evaluate the two layered clays effect on the bearing capacity beneath rigid strip footing subject to axial static load. The results of the bearing capacity relating to the relative thickness of the top layer, the strength ratio of the soil two-layered clays and the rates of the increase of soil cohesion with depth are presented in Tables and graphs. The obtained results are compared with previous published results available in the literature.     

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.     

Stability of a long unsupported circular tunnel in soils with seismic forces

By using the lower-bound finite element limit analysis, the stability of a long unsupported circular tunnel has been examined with an inclusion of seismic body forces. The numerical results have been presented in terms of a non-dimensional stability number (γH/c) which is plotted as a function of horizontal seismic earth pressure coefficient (k _h) for different combinations of H/D and ?; where (1) H is the depth of the crest of the tunnel from ground surface, (2) D is the diameter of the tunnel, (3) k _h is the earthquake acceleration coefficient and (4) γ, c and ? define unit weight, cohesion and internal friction angle of soil mass, respectively. The stability numbers have been found to decrease continuously with an increase in k _h. With an inclusion of k _h, the plastic zone around the periphery of the tunnel becomes asymmetric. As compared to the results reported in the literature, the present analysis provides a little lower estimate of the stability numbers. The numerical results obtained would be useful for examining the stability of unsupported tunnel under seismic forces.     

Load‐displacement and bearing capacity of foundations on granular soils using a multi‐surface kinematic constitutive soil model

A finite element approach based on an advanced multi‐surface kinematic constitutive model is used to evaluate the bearing capacity of footings resting on granular soils. Unlike simple elastic‐perfectly plastic models, often applied to granular foundation problems, the present model realistically accounts for stress dependency of the friction angle, strain softening–hardening and non‐associativity. After the model and its implementation into a finite element code are briefly discussed, the numerical difficulty due to the singularity at the footing edge is addressed. The bearing capacity factor N_γ is then calculated for different granular materials. The effect of footing size, shape, relative density and roughness on the ultimate bearing capacity are studied and the computed results compare very favourably with the general experimental trends. In addition, it is shown that the finite element solution can clearly represent counteracting mechanisms of progressive failure which have an important effect on the bearing capacity of granular foundations. Copyright © 2006 John Wiley & Sons, Ltd.     

闭合型地下连续墙基础竖向承载性能数值模拟

基于室内模型试验,就闭合型与单片地下连续墙基础在竖向承载性能上的差异进行了对比分析,并采用FLAC-3D软件进行数值分析来丰富室内模型试验,探讨了土体变形模量、密度、内聚力以及内摩擦角对闭合型地下连续墙竖向承载力的影响。结果表明：闭合型地下连续墙基础外侧摩阻力的发挥过程与单片地下连续墙基础大致相同,但由于土芯的存在,其内侧摩擦阻力发挥机理更复杂;闭合型与单片地下连续墙基础均可视为端承摩擦型基础;随着墙周土变形模量的增加,闭合型地下连续墙基础竖向位移显著减少,墙体轴力也减少;密度对闭合型地下连续墙基础沉降的影响不显著;内聚力对侧摩擦阻力的影响程度受地下连续墙和土体之间相对位移量的控制;只有闭合型地下连续墙基础的沉降量超过20 mm时,土体内摩擦角才对基础的竖向承载力有较大影响。     

土体参数对扩底墩竖向承载性状影响的数值分析

为了分析不同土体参数对扩底墩竖向承载性状的影响,利用有限单元法,建立了均匀地基中不同土体模量、内摩擦角、粘聚力的三维有限元模型,模拟分析了土体模量、内摩擦角和粘聚力变化时扩底墩的竖向承载性状。分析表明,扩底墩基础以端阻力为主,土体模量和内摩擦角对其竖向承载性状的影响较大,粘聚力影响较小。因此扩底墩持力层应选择土体模量和内摩擦角大的坚硬土层。     

基岩内抗拔桩极限承载力的计算方法

采用极限平衡法,利用幂函数形式的滑移面假设,考虑桩岩界面作用力影响,推导出等截面抗拔桩在单层地基中极限承载力的计算公式。以软岩抗拔桩侧摩阻力试验结果为依据,提出软岩抗拔桩幂函数滑移面参数 时破裂面接近实际形状,并将计算结果与试验结果对比,验证了理论计算模型与假设的准确性。以理论模型为基础,提出对于软岩抗拔桩,桩岩界面作用力参数 、岩石界面摩擦角 可分别取岩石黏聚力c和内摩擦角 参数的0.7～0.8倍折减,分析了软岩抗拔桩极限承载力与软岩 、c的关系,发现抗拔桩极限承载力随着软岩摩擦角 、软岩黏聚力c增加而增加,软岩黏聚力对抗拔桩极限承载力影响更大。     

The variation of Nγ with footing roughness using the method of characteristics

By using the method of characteristics, the effect of footing–soil interface friction angle (δ) on the bearing capacity factor N_γ was computed for a strip footing. The analysis was performed by employing a curved trapped wedge under the footing base; this wedge joins the footing base at a distance B_t from the footing edge. For a given footing width (B), the value of B_t increases continuously with a decrease in δ. For δ=0, no trapped wedge exists below the footing base, that is, B_t/B=0.5. On the contrary, with δ=?, the point of emergence of the trapped wedge approaches toward the footing edge with an increase in ?. The magnitude of N_γ increases substantially with an increase in δ/?. The maximum depth of the plastic zone becomes higher for greater values of δ/?. The results from the present analysis were found to compare well with those reported in the literature. Copyright © 2008 John Wiley & Sons, Ltd.     

Long-term dynamic bearing capacity of shallow foundations on a contractive cohesive soil

The calculation of the long-term dynamic bearing capacity arises from the need to consider the generation of maximum pore-water pressure developed from a cyclic load. Under suitable conditions, a long-term equilibrium situation would be reached, when pore-water pressures stabilized. However, excess pore-water pressure generation can lead to cyclic softening. Consequently, it is necessary to define both the cohesion and the internal friction angle to calculate the dynamic bearing capacity of a foundation in the long term, being necessary to incorporate the influence of the self-weight of soil and therefore the width of the foundation. The present work is based on an analysis of the results of cyclic simple shear tests on soil samples from the port of El Prat in Barcelona. From these experimental data, a pore-water pressure generation formulation was obtained that was implemented in FLAC2D finite difference software. A methodology was developed for the calculation of the maximum cyclic load that a footing can resist before the occurrence of the cyclic softening. The type of soil studied is a contractive cohesive soil, which generates positive pore-water pressures. As a numerical result, design charts have been developed for long-term dynamic bearing capacity calculation and the charts were validated with the application of a real case study.