The present study examines the stability of an embankment of cohesive-frictional material placed over a purely cohesive stratum which is reinforced by means of uniformly spaced vertical granular cylindrical columns. By using the kinematic yield homogenization concept, the macroscopic yield strength envelope associated with the reinforced clay layer has been obtained numerically for different volumetric fractions of the granular column to the composite unit cell by using the semidefinite programming (SDP) technique. An upper bound approximation of the macroscopic yield criterion, using the least square method, is expressed as a summation of a number of convex ellipsoidal envelopes so that the solution for computing the stability factors for the embankment on granular column (in the presence of surcharge pressure) can be obtained later with the usage of the second order cone programming (SOCP) while performing the upper bound finite elements limit analysis. The methodology avoids the usage of any stress variable and the complete procedure has been clearly described and the associated code has been written in MATLAB. The study will be not only useful for designing embankments constructed over soft clay reinforced with granular columns but also for solving different geomechanics stability problems by using the homogenization based kinematic limit analysis approach.

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The present study examines the stability of an embankment of cohesive-frictional material placed over a purely cohesive stratum which is reinforced by means of uniformly spaced vertical granular cylindrical columns. By using the kinematic yield homogenization concept, the macroscopic yield strength envelope associated with the reinforced clay layer has been obtained numerically for different volumetric fractions of the granular column to the composite unit cell by using the semidefinite programming (SDP) technique. An upper bound approximation of the macroscopic yield criterion, using the least square method, is expressed as a summation of a number of convex ellipsoidal envelopes so that the solution for computing the stability factors for the embankment on granular column (in the presence of surcharge pressure) can be obtained later with the usage of the second order cone programming (SOCP) while performing the upper bound finite elements limit analysis. The methodology avoids the usage of any stress variable and the complete procedure has been clearly described and the associated code has been written in MATLAB. The study will be not only useful for designing embankments constructed over soft clay reinforced with granular columns but also for solving different geomechanics stability problems by using the homogenization based kinematic limit analysis approach.

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By using the lower bound finite elements limit analysis, this paper presents a methodology to evaluate the bearing capacity of piles in cohesive soils. The non-homogeneous nature of soil has been assumed for which the cohesion increases linearly with depth. An axisymmetric formulation, comprising of a planar domain with a mesh of three noded triangular elements has been adopted. Here, Mohr–Coulomb (MC) yield criterion has been employed. The yield criterion is expressed in terms of three quadratic constraints so that the problem can be solved using second-order cone programming (SOCP). Bearing capacity factors have been generated as a function of (i) height to radius ratio of the pile, (ii) rate of increase of cohesion and (ii) roughness of base and shaft of pile. The results computed from the analysis have been compared thoroughly with that reported in literatures.@en

While performing the upper bound limit analysis, using an associated flow rule, the expressions for the power dissipation function corresponding to different forms of yield criteria for clays exhibiting shear strength anisotropy and transversely asymmetric properties (implying different strengths in compression and tension), have been derived in terms of velocities, strain rates and shear strength parameters without involving stress variables by using the concept of duality in an optimization theory. The optimization has been performed in an efficient manner by using the second order cone programming (SOCP). The problems of strip and square footings have been dealt with. The lower bound solutions have also been established for the plane strain problems, and the two bound solutions have been found to match very closely with each other. The results also match well with the different solutions reported in literature. The variations of the power dissipation function and nodal velocity patterns in the problem domain have also been studied. The proposed formulation(s) will be useful for solving different two and three-dimensional geomechanics problems in clays to account for shear strength anisotropy as well as transversely asymmetric properties.@en

Finite element-based upper bound and lower bound limit analysis are found to be an excellent tool for solving various stability structural problems with less computational cost. The accuracy and efficiency of the obtained solution is highly dependent on the intensity of elements in the high stress or stain gradient region. Therefore, an adaptive meshing procedure is required to reduce the gap between upper and lower bound solutions. The adaptive meshing procedure presented in this paper is based on two basic concepts (a) a posterior error indicator and (b) an unstructured mesh generator. The posterior error estimator is based on intensity of maximum shear strain rate (γmax) has been used. Open-source software TRIANGLE and TETGEN have been used to generate unstructured meshes for 2D and 3D problems, respectively. A plane strain and a 3D foundation problem have been analyzed and steps involve in calculation have been clearly mentioned in the paper. The present analysis method is found to be efficient and accurate.@en

Bearing capacity of rectangular foundations, with varying aspect ratios (1 ≤ L/B ≤ 5) and embedded at shallow to medium depths (D/B ≤ 5), has been evaluated by using the three-dimensional cell-based smoothed finite elements quasi-kinematic limit analysis. The results have also been obtained for circular and strip foundations by performing exclusively the axisymmetric and plane strain analyses. The associated optimization problem has been solved by employing the semidefinite programming (SDP). Shape factors (sc, sq and sγ) and depth factors (dc, dq and dγ) are given as a function of soil internal friction angle (ϕ). The shape factors have been expressed as a function of L/B, on the other hand, the depth factors have been given as a function of D/B and L/B. A thorough comparison of the results has been made with the different solutions available in literature. The variations of (i) power dissipation function, (ii) maximum plastic shear strain rates, and (iii) the nodal velocities patterns, have also been examined to interpret the failure mechanism.@en

By employing different smoothed finite element (S-FE) methods, the kinematic limit analysis approach has been presented by using three noded triangular elements to solve plane strain and plane stress stability problems on basis of the Mohr-Coulomb yield criterion. By using the concept of duality in conic programming, the expressions for the power dissipation function and the associated constraints have been written entirely in terms of kinematic variables. The optimization formulation is framed as a second order cone programming problem. Based on the absolute maximum magnitude of the shear strain rate, an algorithm to update the mesh has been provided. Amongst the different approaches, the node-based S-FE formulation has been found to be very accurate as well as efficient to solve large scale stability problems. The selective edge-based and the edge-based with a bubble node approaches also generate quite accurate solutions though require a little more computational time.@en

A three-dimensional kinematic limit analysis approach based on the radial point interpolation method (RPIM) has been used to compute collapse loads for rectangular foundations. The analysis is based on the Mohr-Coulomb yield criterion and the associated flow rule. It is understood that the internal plastic power dissipation function and flow rule constraints can be expressed entirely in terms of plastic strain rates without involving stresses. The optimization problem has been solved on basis of the semidefinite programming (SDP) by using highly efficient primal-dual interior point solver MOSEK in MATLAB. The results have been presented in terms of the variation of the shape factors with changes in the aspect ratio (L/B) of the footing for different values of soil internal friction angle (ϕ). Computations have revealed that the shape factors, sc and sq, due to effects of cohesion and surcharge increase continuously with (1) decrease in L/B and (2) increase in ϕ. On the other hand, the shape factor sγ, due to the effect of soil unit weight, increases very marginally with an increase in L/B up to (1) ϕ = 25° for a rough footing and (2) ϕ = 35° for a smooth footing. Thereafter, for greater values of ϕ, the variation of sγ with L/B has been found to be quite similar to that of the factors sc and sq. The variations of (1) nodal velocity patterns, (2) plastic power dissipation, and (3) maximum plastic shear strain rates have also been examined to interpret the associated failure mechanism.@en

The writers are thankful to the discussers for their constructive discussion on the work. Rather than keeping 𝛼=0.5 for all geological strength index (GSI) values, the discussers presented a method to account for the actual value of 𝛼 in the generalized Hoek-Brown (GHB) criterion by using the reduced values of the material input parameters to solve the problem by using the modified Hoek-Brown (MHB) criterion; for the definitions of GSI and 𝛼, refer to the original paper, and the MHB and GHB criteria are defined exactly the same way as introduced by the discussers. In the original paper, it was pointed out that due to an assumption of using 𝛼=0.5, a significant overestimation of the collapse load occurs for GSI≤30. The discussers provided a detailed investigation on exploring how the choice of 𝛼 affects the solutions for values of GSI≤30. Figs. 2–4 plotted by the discussers provide the difference between the MHB and GHB criteria for different values of GSI, stress range, and 𝑚𝑖. These figures explain the reason for the difference between the obtained solution and that reported in the literature by Merifield et al. (2006) and Clausen (2013). The main contribution of the discussers is that a new method has been proposed to take into account the true value of 𝛼 by choosing the reduced values of the input strength parameters (𝜎𝑐⁢𝑖, 𝑚𝑏, and 𝑠) to be used in the MHB criterion based on the least-square optimization so that the yield envelope of the MHB criterion fits closely with the GHB criterion. The method introduced by the discussers is simple and straightforward and can be easily implemented. By using two worked examples, the discussers indicated that the reduced material input parameters can significantly reduce the strength overestimation of the MHB criterion for lower values of GSI, although no comparison of the final solution with the GHB was provided in the discussion. The major disadvantage, however, associated with the recommended approach of the discussers is the choice of appropriate range of 𝜎1 in order to compute the reduced values of the input strength parameters to be used in the MHB criterion. It is, however, going to be a difficult exercise to select the correct range of 𝜎1. If the selected range of 𝜎1 is kept very large, it is not going to be an easy task to find the perfect match between the two yield envelopes using the GHB and the MHB criteria.@en

An upper-bound formulation for performing finite-element limit analysis by using semidefinite programming (SDP) for an axisymmetric stability problem involving the Mohr-Coulomb yield criterion has been presented. The SDP technique has an advantage in that it can deal with the yield criterion directly in its native form in terms of principal stresses and strains without any smoothing of the parent yield surface. The associated flow rule and plastic power dissipation are expressed entirely in terms of principal plastic strain rates. Nodal velocities and element plastic strain rates were framed as basic governing variables without involving stresses. The solution was obtained by using the SDP solver MOSEK in MATLAB. The problem of finding the bearing capacity of a circular footing was dealt with by choosing a planar domain that was discretized with (1) three-noded constant strain finite elements and (2) six-noded linear strain finite elements. The results were obtained with and without the provision of the velocity discontinuities, and were compared with those reported in literature. It was learned that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and using velocity discontinuities along all the elements’ interfaces.@en

The lower-bound finite elements limit analysis in combination with semidefinite programming (SDP) has been used to solve stability problems involving a modified Hoek-Brown (HB) yield criterion with the exponent 𝛼=0.5. In order to demonstrate the applicability of the proposed computational approach, bearing capacities of strip and circular foundations on rock mass have been determined. The results have been given in the form of nondimensional bearing capacity factors as a function of different material input parameters. By comparing the results obtained from the present analysis with that reported in literature, it is noted that the proposed approach remains quite accurate and is highly efficient to deal with any large-scale optimization problem. The method does not require any kind of the smoothing of the yield surface and it can even be implemented to deal with any three-dimensional problem as well.@en