Effect of spatial variability of cross‐correlated soil properties on bearing capacity of strip footing

Geotechnical engineering problems are characterized by many sources of uncertainty. Some of these sources are connected to the uncertainties of soil properties involved in the analysis. In this paper, a numerical procedure for a probabilistic analysis that considers the spatial variability of cross‐correlated soil properties is presented and applied to study the bearing capacity of spatially random soil with different autocorrelation distances in the vertical and horizontal directions. The approach integrates a commercial finite difference method and random field theory into the framework of a probabilistic analysis. Two‐dimensional cross‐correlated non‐Gaussian random fields are generated based on a Karhunen–Loève expansion in a manner consistent with a specified marginal distribution function, an autocorrelation function, and cross‐correlation coefficients. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses was performed to study the effects of uncertainty due to the spatial heterogeneity on the bearing capacity of a rough strip footing. The simulations provide insight into the application of uncertainty treatment to geotechnical problems and show the importance of the spatial variability of soil properties with regard to the outcome of a probabilistic assessment. Copyright © 2009 John Wiley & Sons, Ltd.     

Efficient sparse polynomial chaos expansion methodology for the probabilistic analysis of computationally‐expensive deterministic models

The sparse polynomial chaos expansion (SPCE) methodology is an efficient approach that deals with uncertainties propagation in case of high‐dimensional problems (i.e., when a large number of random variables is involved). This methodology significantly reduces the computational cost with respect to the classical full PCE methodology. Notice however that when dealing with computationally‐expensive deterministic models, the time cost remains important even with the use of the SPCE. In this paper, an efficient combined use of the SPCE methodology and the Global Sensitivity Analysis is proposed to solve such problem. The proposed methodology is firstly validated using a relatively non‐expensive deterministic model that involves the computation of the PDF of the ultimate bearing capacity of a strip footing resting on a weightless spatially varying soil where the soil cohesion and angle of internal friction are modeled by two anisotropic non‐Gaussian cross‐correlated random fields. This methodology is then applied to an expensive model that considers the case of a ponderable soil. A brief parametric study is presented in this case to show the efficiency of the proposed methodology. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

Seismic Bearing Capacity of Shallow Strip Footing with Coulomb Failure Mechanism using Limit Equilibrium Method

In this paper, an effort is made to evaluate the seismic bearing capacity of shallow strip footing resting on c–ф soil. The formulation is developed to get a single coefficient of bearing capacity for simultaneous resistance of weight, surcharge and cohesion. Limit equilibrium method in Pseudo-static approach with Coulomb mechanism is applied here to evaluate the seismic bearing capacity. The seismic bearing capacity of footing (q_uE) is expressed in terms of single coefficient N_γE. The effect of various parameters viz. angle of internal friction of soil (ф), angle of wall friction (δ), cohesion (c), ratio of depth to width of footing (d_f/B₀), seismic acceleration (k_h, k_v) are studied on the variation of seismic bearing capacity co-efficients.     

Response of Vertically Loaded Pile in Clay: A Probabilistic Study

This study presents the response of a vertically loaded pile in undrained clay considering spatially distributed undrained shear strength. The probabilistic study is performed considering undrained shear strength as random variable and the analysis is conducted using random field theory. The inherent soil variability is considered as source of variability and the field is modeled as two dimensional non-Gaussian homogeneous random field. Random field is simulated using Cholesky decomposition technique within the finite difference program and Monte Carlo simulation approach is considered for the probabilistic analysis. The influence of variance and spatial correlation of undrained shear strength on the ultimate capacity as summation of ultimate skin friction and end bearing resistance of pile are examined. It is observed that the coefficient of variation and spatial correlation distance are the most important parameters that affect the pile ultimate capacity.     

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.     

Probabilistic evaluation of tunnel face stability in spatially random soils using sparse polynomial chaos expansion with global sensitivity analysis

The sparse polynomial chaos expansion is employed to perform a probabilistic analysis of the tunnel face stability in the spatially random soils. A shield tunnel under compressed air is considered which implies that the applied pressure is uniformly distributed on the tunnel face. Two sets of failure mechanisms in the context of the limit analysis theory with respect to the frictional and the purely cohesive soils are used to calculate the required face pressure. In the case of the frictional soils, the cohesion and the friction angle are modeled as two anisotropic cross-correlated lognormal random fields; for the purely cohesive soils, the cohesion and the unit weight are modeled as two anisotropic independent lognormal random fields. The influences of the spatial variability and of the cross-correlation between the cohesion and the friction angle on the probability density function of the required face pressure, on the sensitivity index and on the failure probability are discussed. The obtained results show that the spatial variability has an important influence on the probability density function as well as the failure probability, but it has a negligible impact on the Sobol’s index.     

Behaviour of model footing on pond ash

This paper focuses on the effective utilization of pond ash, as foundation medium. A series of laboratory model tests have been carried out using square, rectangular and strip footings on pond ash. The effects of dry density, degree of saturation of pond ash, size and shape of footing on ultimate bearing capacity of shallow foundations are presented in this paper. Local shear failure of a square footing on pond ash at 37% moisture content (optimum moisture content) is observed up to the values of dry density 11.20 kN/m³ and general shear failure takes place at the values of dry density 11.48 kN/m³ and 11.70 kN/m³. Effects of degree of saturation on ultimate bearing capacity were studied. Experimental results show that degree of saturation significantly affects the ultimate bearing capacity of strip footing. The effect of footing length to width ratio (L/B), on increase in ultimate bearing capacity of pond ash, is insignificant for L/B ≥ 10 in case of rectangular footings. The effects of size of footing on ultimate bearing capacity for all shapes of footings viz., square, rectangular and strip footings are highlighted.     

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.     

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。     

Probabilistic analyses of three-dimensional circular footing resting on two-layer c–ϕ soil system considering soil spatial variability

Acta Geotechnica - The prime objective of this paper is to study the effect of soil spatial variability on the three-dimensional probabilistic bearing capacity of a circular footing resting on the...     

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.     

Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

This paper presents a probabilistic analysis to compute the probability density function of the bearing capacity of a strip footing resting on a spatially varying rock mass. The rock is assumed to follow the generalised Hoek–Brown failure criterion. The uniaxial compressive strength of the intact rock (σc) was considered as a random field and the geological strength index was modelled as a random variable. The uncertainty propagation methodology employed in the analysis is the sparse polynomial chaos expansion. A global sensitivity analysis based on Sobol indices was performed. Some numerical results were presented and discussed.     

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.     

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。     

Effect of spatial correlation structure and transformation model on the design of shallow strip footing

Directional variability of spatial correlation is observed in natural soils due to their depositional characteristics and it influences the response of structures founded on these deposits. Nonetheless, the results presented in most of the available literature are based on the assumption of either isotropic spatial correlation or perfect spatial correlation of soil properties in horizontal and vertical directions. It is also observed from past studies that the effect of transformation model on the total uncertainty is quite significant. Hence, an effort has been made in this paper to study the effect of anisotropy of autocorrelation characteristics of cone tip resistance (q_c) and the transformation model on the bearing capacity of a shallow strip footing, founding on the surface of a spatially varying soil mass. The statistics in the vertical direction of the soil mass are taken from 8 Cone Penetration Test (CPT) records and statistics in the horizontal direction are assumed. For the case considered, it is observed that the transformation model significantly influences the degree of variability of design parameter. The results also show that isotropic correlation structure based on the vertical autocorrelation distance underestimates the variability of design parameter. On the other hand, perfect correlation in horizontal or vertical, or both directions, overestimates the variability of design parameters, and produces conservative estimates of allowable bearing capacity.     

Bearing capacity of strip foundations reinforced with geogrid sheets by using upper bound finite‐element limit analysis

By using the upper bound finite‐elements limit analysis, with an inclusion of single and two horizontal layers of reinforcements, the ultimate bearing capacity has been computed for a rigid strip footing placed over (i) fully granular, (ii) cohesive‐frictional, and (iii) fully cohesive soils. It is assumed that (i) the reinforcements are structurally strong so that no axial tension failure can occur, (ii) the reinforcement sheets have negligible resistance to bending, and (iii) the shear failure can take place between the reinforcement and soil mass. It is expected that the different approximations on which the analysis has been based would generally remain applicable for reinforcements in the form of geogrid sheets. A method has been proposed to incorporate the effect of the reinforcement in the analysis. The efficiency factors, η_c and η_γ, to be multiplied with N_c and N_γ , for finding the bearing capacity of reinforced foundations, have been established. The results have been obtained (i) for different values of ? in case of fully granular and cohesive‐frictional soils, and (ii) for different rates at which the cohesion increases with depth for a fully cohesive soil. The optimum positions of the reinforcements' layers have also been determined. The effect of the reinforcements' length on the results has also been analyzed. As compared to cohesive soils, the granular soils, especially with higher values of ?, cause a much greater increase in the bearing capacity. The results compare reasonably well with the available theoretical and experimental data from literature. Copyright © 2013 John Wiley & Sons, Ltd.