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.     

Bearing capacity of two interfering strip footings from lower bound finite elements limit analysis

A rigorous lower bound solution, with the usage of the finite elements limit analysis, has been obtained for finding the ultimate bearing capacity of two interfering strip footings placed on a sandy medium. Smooth as well as rough footing–soil interfaces are considered in the analysis. The failure load for an interfering footing becomes always greater than that for a single isolated footing. The effect of the interference on the failure load (i) for rough footings becomes greater than that for smooth footings, (ii) increases with an increase in ?, and (iii) becomes almost negligible beyond S/B > 3. Compared with various theoretical and experimental results reported in literature, the present analysis generally provides the lowest magnitude of the collapse load. Copyright © 2011 John Wiley & Sons, Ltd.     

Vertical uplift capacity of two interfering horizontal anchors in sand using an upper bound limit analysis

The vertical uplift resistance of two interfering rigid rough strip anchors embedded horizontally in sand at shallow depths has been examined. The analysis is performed by using an upper bound theorem of limit analysis in combination with finite elements and linear programming. It is specified that both the anchors are loaded to failure simultaneously at the same magnitude of the failure load. For different clear spacing (S) between the anchors, the magnitude of the efficiency factor (ξ_γ) is determined. On account of interference, the magnitude of ξ_γ is found to reduce continuously with a decrease in the spacing between the anchors. The results from the numerical analysis were found to compare reasonably well with the available theoretical data from the literature.     

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.     

Bearing capacity of non-associative coaxial granular materials by upper bound limit analysis and finite elements

In this paper, an upper bound estimate of the limit load on non-associative coaxial granular materials is presented. The kinematic approach of the upper bound limit analysis has been utilised. The failure mechanism is assumed to coincide with the direction of the shear bands at every point throughout the body. The shear band orientation in non-associative coaxial materials, i.e. those with the same major principal stress and major principal strain increment directions, can be found based on the angle of dilation and the major principal stress direction. Therefore, having known the stress field at limiting equilibrium, the orientation of the shear bands and hence, the failure mechanism can be obtained. In this study, the stress field is first determined by the method of stress characteristics. Then, the finite element interpolation technique is used to interpolate the stress field and to find the orientation of the shear bands at every point within the field. Once the failure mechanism and the stress state at every point along velocity discontinuities have been found, the upper bound limit analysis has been performed to estimate the limit load.     

Bearing capacity of two interfering footings

By using an upper bound limit analysis in conjunction with finite elements and linear programming, the ultimate bearing capacity of two interfering rough strip footings, resting on a cohesionless medium, was computed. Along all the interfaces of the chosen triangular elements, velocity discontinuities were employed. The plastic strains were incorporated using an associated flow rule. For different clear spacing (S) between the two footings, the efficiency factor (ξ_γ) was determined, where ξ_γ is defined as the ratio of the failure load for a strip footing of given width in the presence of the other footing to that of a single isolated strip footing having the same width. The value of ξ_γ at S/B = 0 becomes equal to 2.0, and the maximum ξ_γ occurs at S/B = S_cr/B. For S/B?S_cr/B, the ultimate failure load for a footing becomes almost half that of an isolated footing having width (2B + S), and the soil mass below and in between the two footings deforms mainly in the downward direction. In contrast, for S/B>S_cr/B, ground heave was noticed along both the sides of the footing. As compared to the available theories, the analysis provides generally lower values of ξ_γ for S/B>S_cr/B. Copyright © 2007 John Wiley & Sons, Ltd.     

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 factors of circular foundations for a general c–ϕ soil using lower bound finite elements limit analysis

By using the lower bound limit analysis in conjunction with finite elements and linear programming, the bearing capacity factors due to cohesion, surcharge and unit weight, respectively, have been computed for a circular footing with different values of ?. The recent axisymmetric formulation proposed by the authors under ?=0 condition, which is based on the concept that the magnitude of the hoop stress (σ_θ) remains closer to the least compressive normal stress (σ₃), is extended for a general c–? soil. The computational results are found to compare quite well with the available numerical results from literature. It is expected that the study will be useful for solving various axisymmetric geotechnical stability problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimation of bearing capacity for multiple footings in sand

In the present study, the effects of multiple-footing configurations in sand on bearing capacity were investigated using field plate load tests and finite element analyses. Both strip and spread footings were considered in the finite element analyses. In each case, different footing distances were applied for the purposes of comparison among all of the results. From these results, it was observed that the load responses of multiple footings are similar to those of the single footing at distances greater than three times the footing width. Design equation and correlation parameters, necessary for quantifying the values of the bearing capacity ratio for the different multiple-footing configuration, were derived. Experimental test results from the literature were selected and used in verifying the proposed method.     

Seismic uplift capacity of inclined strip anchors in sand using upper bound limit analysis

Pseudo-static approach is adopted in this paper to determine the seismic uplift capacity of an inclined strip anchor using upper bound limit analysis. Two different failure mechanisms are considered to obtain the magnitudes of unit weight component of uplift factor f_γE for different values of soil friction angle, interface friction of anchor plate, anchor inclination, embedment ratio and horizontal seismic acceleration coefficient. The failure mechanism 1 consists of a triangular and quadrilateral rigid blocks; whereas the failure mechanism 2 comprises a logarithmic spiral failure zone with varied focus, sandwiched between a triangular and quadrilateral rigid blocks. It is observed that the magnitude of uplift factor f_γE decreases significantly with the increase in seismic acceleration but increases with the increase in embedment ratio and roughness of the anchor surface. However, a mixed trend in the values of f_γE can be observed for different inclination of the anchor, which is clearly discussed in this paper. The results are compared with the existing values in the literature and the significance of the present methodology for designing the inclined strip anchor is discussed.     

The influence of load inclination on the undrained bearing capacity of strip footings on slopes

The effect of inclined loading on the bearing capacity of foundations on horizontal ground surface is well established and both the exact solution and simpler empirical equations are available for the calculation of the failure loads. However, for footings on or near slopes complete solutions are available only for vertical loading. This paper investigates the influence of inclined loading on the horizontal and vertical failure loads. The finite element, upper bound plasticity and stress field methods are used to examine a wide range of geometries and soil properties. The methods are first validated against known solutions for two special cases and are subsequently employed to investigate the effect of the geometrical and material properties on the failure loads and the bearing capacity load interaction diagram. Based on this investigation an empirical equation is proposed for the load interaction diagram for undrained inclined loading of footings on or near slopes.     

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.     

Bearing capacity of ring footings on cohesionless soil under eccentric load

This paper presents a study on the bearing capacity of eccentrically-loaded rough ring footings resting over cohesionless soil. To this aim, a series of 3D numerical simulations were performed using the finite difference method. In order to consider the effect of load eccentricity, reduction factor method is applied. In this method, the ratio of an eccentrically-loaded bearing capacity to the bearing capacity of the same footing under vertical load is defined. Comparison between the results of the numerical simulations with those of analytical solutions and experimental data indicates good agreement. A mathematical expression is also introduced for eccentrically-loaded ring footings.     

Determination of passive earth pressure with lower bound finite elements limit analysis and modified pseudo-dynamic method

ABSTRACT

The present study deals with the determination of passive earth pressure under seismic condition by following the lower bound finite elements limit analysis and modified pseudo-dynamic methodology. In accordance with the lower bound finite elements formulation, the stress field was modelled using a three-noded triangular elements, while the passive pressure was determined via linear optimisation. The parametric study was performed, for a vertical rigid retaining wall, by varying the magnitude of seismic acceleration in horizontal direction (k_h) between 0 and 0.3, while the vertical seismic acceleration (k_v) was kept equal to 0 or 0.5 k_h. Furthermore, the damping coefficient for dry cohesionless backfill was kept ξ = 10%. The obtained results in various cases were found to be in good agreement with those found in the literature. It is expected that this method can be further used for solving other important geotechnical stability problems.     

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.     

Reliability analysis for bearing capacity of surface strip footing using fuzzy finite element method

ABSTRACT

This paper presents the reliability analysis on the basis of the foundation failure against bearing capacity using the concept of fuzzy set theory. A surface strip footing is considered for the analysis and the bearing capacity is estimated using the conventional Finite Element Method (FEM). The spatial variability of the variables is taken into consideration to capture the physical randomness of the soil parameters for an isotropic field. A variation of the probability of failure (P_f) against a varying limiting applied pressure (q) is presented for different Coefficient of Variation (COV) of the variables and different scale of fluctuation (θ). The results reveal that the friction angle of soil (?) is the most influencing parameter among the other variables. Further, the influence of the scale of fluctuation (θ) on the probability of failure (P_f) is also examined. It is observed that for a particular COV of ?, higher value of θ predicts higher P_f whereas, P_f increases as COV of ? increases for a particular θ value. Later, a comparison study is accomplished to verify the viability of the present method and it can be noticed that the present method compares well with the other reliability method (First Order Reliability Method) to a reasonably good extent.     

The ultimate uplift capacity of multi-plate strip anchors in undrained clay

Soil anchors are commonly used as foundation systems for structures requiring uplift resistance such as transmission towers, or for structures requiring lateral resistance, such as sheet pile walls. Anchors commonly have more than one plate or bearing element and therefore there is a complex interaction between adjacent plates due to overlapping stress zones. This interaction will affect the failure mode and ultimate capacity. However, no thorough numerical analyses have been performed to determine the ultimate pullout loads of multi-plate anchors. By far the majority of the research has been directed toward the tensile uplift behaviour of single anchors (only one plate). The primary aim of this research paper is to use numerical modelling techniques to better understand plane strain multi-plate anchor foundation behaviour in clay soils. A practical design framework for multi-plate anchor foundations will be established to replace existing semi-empirical design methods that are inadequate and have been found to be excessively under or over conservative. This framework can then be used by design engineers to more confidently estimate the pullout capacity of multi-plate anchors under tension loading.     

基于非线性规划的有限元塑性极限分析下限法研究

在分析Sloan建立的有限元塑性极限分析线性规划数学模型存在的局限性基础上,提出了基于非线性规划的有限元塑性极限分析下限法数学模型。采用非线性屈服条件构建了下限法静力容许应力场,建立了求解超载系数、强度储备系数的下限法数学模型,并提出了针对塑性极限分析非线性规划数学模型的求解策略;最后对一个经典算例进行了深入分析,验证了方法的正确性。     

Bearing capacity of rectangular footings in uniform clay with deep embedment

This letter is concerned with the undrained bearing capacity of rectangular footings with various aspect ratios and embedment ratios in uniform clay. It covers thin plate foundations with low aspect ratios and high embedment depth with embedment ratio up to 150. The work is based on small strain finite element analysis (FEA). After verification of the FEA model against existing solutions of the bearing capacity factors of rectangular footings, a series of FEA results are obtained. Based on the FEA results, a simple formulation is proposed to calculate the bearing capacity factor for rectangle footing with different aspect ratio in any embedment depth, extending the existing solutions to cover a wider ranges of footing aspect ratios and embedment ratios.     

基于下限原理有限元的强度折减法

对于岩土工程中常用的强度折减系数,其规划问题是非线性的,不能直接利用线性规划进行求解。基于四边形单元的下限原理有限元法,根据强度折减系数与超载系数近似符合双曲函数的特点,通过调整强度参数使得超载系数逼近于1[1],可将边坡稳定性分析中常用的强度折减系数的非线性规划求解转化为线性规划求解问题。分析表明,采用拟合双曲线插值法求解强度折减系数的计算效率高于常规的二分法及割线法,且具有较好的收敛性;该方法能够充分利用当前高效的线性规划算法,便于工程应用。