Capacity analysis is a common preliminary step in the design of offshore foundations. Inaccuracies in traditional capacity analysis methods, and the advancement of numerical modelling capabilities, have increasingly led designers to optimise foundations using more complex methods. In this thesis, the ultimate limit state capacity of a range of foundation types is investigated using finite element limit analysis. Novel three-dimensional finite element limit analysis software is benchmarked against analytical solutions and conventional displacement finite element analysis. It is then used to find lower and upper bounds of foundation capacity, with adaptive mesh refinement used to reduce the bound gap over successive iterations of the solution. Rigid foundations subjected to short term loading on clay soil are analysed. The undrained soil is modelled as a rigid--plastic von Mises material, and attention is given to modelling any normal and/or shear stress limits at the foundation/soil interface. Shallow foundations, suction anchor foundations, and hybrid mudmat/pile foundations are considered. Realistic six degree-of-freedom load combinations are applied and results are reported in the form of normalised design charts, and tables, that are suitable for use in preliminary design. Relationships between loading combinations and failure mechanisms are also explored. A number of case studies based on authentic foundation designs are analysed. The results suggest that finite element limit analysis could provide an attractive alternative to displacement finite element analysis for preliminary foundation design calculations.

Limit analysis is a very powerful tool to find accurate solutions of several geotechnical stability problems. This analysis is based on the theory of the plasticity and it provides two limiting solutions within lower and upper bounds. With the advancement of the finite elements and different robust optimization techniques, the numerical limit analysis approach in association with finite elements is becoming quite popular to assess the stability of various complicated structures. The present thesis deals with the formulations and the implementation of the finite element limit analysis to obtain the solutions of different geotechnical axisymmetric stability problems. The objectives of the present thesis are twofold: (a) developing limit analysis formulations in conjunction with linear and nonlinear optimizations for solving axisymmetric stability problems related with soil and rock mechanics, and then (b) implementing these axisymmetric formulations for solving various important axisymmetric stability problems in geomechanics. Three noded linear triangular elements have been used throughout the thesis. In order to solve the different problems, the associated computer programs have been written in MATLAB. With reference to the first objective of the thesis, the existing finite element lower bound axisymmetric formulation with linear programming has been presented. A new technique has also been proposed for solving an axisymmetric geomechanics stability problem by employing an upper bound limit analysis in combination with finite elements and linear programming. The method is based on the application of the von-Karman hypothesis to fix the constraints associated with the magnitude of the circumferential stress (), and finally the method involves only the nodal velocities as the basic unknown variables. The required computational effort becomes only marginally greater than that needed for an equivalent plane strain problem. The proposed methodology has been found to be computationally quite efficient. A new lower bound axisymmetric limit analysis formulation, by using two dimensional finite elements, the three dimensional Mohr-Coulomb (MC) yield criterion, and nonlinear optimization has also been presented for solving different axisymmetric stability problems in geomechanics. The nonlinear optimization was carried out by employing an interior point method based on the logarithmic barrier function. The yield surface was smoothened (i) by removing the tip singularity at the apex of the pyramid in the meridian plane, and (ii) by eliminating the stress discontinuities at the corners of the yield hexagon in the plane. No inherent assumption concerning with the hoop stress needs to be made in this formulation. The Drucker-Prager (DP) yield criterion was also used for computing the lower bound axisymmetric collapse load. The advantage of using the DP yield criterion is that it does not exhibit any singularity in the plane. A new proposal has also been given to simulate the DP yield cone with the MC hexagonal yield pyramid. The generalized Hoek-Brown (HB) yield criterion has also been used. This criterion has been smoothened both in the meridian and  planes and a new formulation is prescribed for obtaining the lower bound axisymmetric problems in rock media in combination with finite elements and nonlinear optimization. With reference to the second objective, a few important axisymmetric stability problems in soil mechanics associated with footings and excavations have been solved in the present thesis. In all these problems, except that of a flat circular footing lying over either homogeneous soil or rock media, it is assumed that the medium is governed by the MC failure criterion and it follows an associated flow rule. For determining the collapse loads for a circular footing over homogenous soil and rock media, the problem has been solved with the usage of Drucker-Prager, Mohr-Coulomb and Hoek-Brown criteria. The bearing capacity of a circular footing lying over fully cohesive strata, with an inclusion of a sand layer is evaluated. The effects of the thickness and internal friction angle of the sand layer () on the bearing capacity have been examined for different combinations of cu/(b) and q; where (i) cu defines the undrained shear strength, (ii)  is the unit weight of sand, (iii) b corresponds to the footing radius, and (iv) q is the surcharge pressure. The results have been presented in the form of a ratio () of the bearing capacities with an insertion of the sand layer to that for a footing lying directly over clayey strata. It is noted that an introduction of a layer of medium dense to dense sand over soft clay improves considerably the bearing capacity of the foundation. The improvement in the bearing capacity increases continuously (i) with decreases in cu/(b), and (ii) increases in  and q/(b). The bearing capacity factors, Nc, Nq and N, for a conical footing are obtained in a bound form for a wide range of the values of cone apex angle () and with  = 0, 0.5 and . The bearing capacity factors for a perfectly rough ( = conical footing generally increase with a decrease in . On contrary for  = 0, the factors Nc and Nq reduce gradually with a decrease in . For  = 0, the factor N for  ≥ 35o becomes minimum for  approximately equal to 90o. For  = 0, the factor N for  ≤ 30o, like in the case of  = , generally reduces with an increase in . It has also been intended to compute the bearing capacity factors Nc, Nq and N, for smooth and rough ring footing for different combinations of ri/ro and ; where ri and ro refer to inner and outer radii of the ring, respectively. It is observed that for a smooth footing, with a given value of ro, the magnitude of the collapse load decreases continuously with an increase in ri. On the other hand, for a rough base, for a given value of ro, hardly any reduction occurs in the magnitude of collapse load up to ri/ro ≈ 0.2, whereas beyond this ri/ro, the magnitude of the collapse load, similar to that of a smooth footing, decreases continuously with an increase in ri/ro. An attempt has also been made to determine the ultimate bearing capacity of a circular footing, placed over a soil mass which is reinforced with horizontal layers of circular reinforcement sheets. For performing the analysis, three different soil media have been separately considered, namely, (i) fully granular, (ii) cohesive frictional, and (iii) fully cohesive with an additional provision to account for an increase of cohesion with depth. The reinforcement sheets are assumed to be structurally strong to resist axial tension but without having any resistance to bending; such an approximation usually holds good for geogrid sheets. The shear failure between the reinforcement sheet and adjoining soil mass has been considered. The increase in the magnitudes of the bearing capacity factors (Nc and N) with an inclusion of the reinforcement has been computed in terms of the efficiency factors c and . The critical positions and corresponding optimum diameter of the reinforcement sheets, for achieving the maximum bearing capacity, have also been established. The increase in the bearing capacity with an employment of the reinforcement increases continuously with an increase in . The improvement in the bearing capacity becomes quite extensive for two layers of the reinforcements as compared to the single layer of the reinforcement. The stability of an unsupported vertical cylindrical excavation has been assessed. For the purpose of design, stability numbers (Sn) have been generated for both (i) cohesive frictional soils, and (ii) pure cohesive soils with an additional provision to account for linearly increasing cohesion with depth by using a non-dimensional factor m. The variation of Sn with H/b has been established for different values of m and ; where H and b refer to height and radius of the cylindrical excavation. A number of useful observations have been drawn about the variation of the stability number and nodal velocity patterns with changes in H/b,  and m. In the last, by using the smoothened generalized HB yield criterion, the ultimate bearing capacity of a circular footing placed over a rock mass is evaluated in a non-dimensional form for different values of GSI, mi, ci/(b) and q/ci. For validating the results, computations were exclusively performed for a strip footing as well. For the various problems selected in the present thesis, the failure and nodal velocity patterns have been examined. The results obtained from the analysis have been thoroughly compared with that reported from literature. It is expected that the various design charts presented here will be useful for the practicing engineers. The formulations given in the thesis can also be further used for solving various axisymmetric stability problems in geomechanics.

This thesis presents the implementation of the upper bound limit analysis in combination with finite elements and linear optimization for solving different stability problems in geomechanics under plane strain conditions. Although the nonlinear optimization techniques are becoming quite popular, the linear optimization has been adopted due to its simplicity in implementation and ease in attaining the convergence while performing the analysis. The objectives of the present research work are (i) to reduce the computational effort while using an upper bound finite element limit analysis with linear programming in dealing with geotechnical stability problems, and (ii) to obtain solutions for a few important geotechnical stability problems associated with reinforced earth, unsupported tunnels and a group of anchors. It is also intended to examine the developments of the failure patterns in all the cases. For carrying out the analysis for different stability problems, three noded triangular elements have been used throughout the thesis. The nodal velocities are treated as basic unknown variables and the velocity discontinuities are employed along the interfaces of all the elements. The soil mass is assumed to obey the Mohr-Coulomb’s failure criterion and an associated flow rule. The Mohr-Coulomb yield surface is linearized by means of an exterior regular polygon circumscribing the actual yield circle so that the finite element formulation leads to a linear programming problem. A simple technique has been proposed for reducing the computational effort while solving any geotechnical stability problem by using the upper bound finite element limit analysis and linear optimization. In the proposed method, the problem domain has been discretized into a number of different regions in which a particular order (number of sides) of the polygon has been specified to linearize the Mohr-Coulomb yield criterion. A greater order of the polygon needs to be chosen only in that part of the domain wherein the rate of the plastic strains becomes higher. The computational effort required to solve the problem with this implementation reduces considerably. By using the proposed method, the bearing capacity has been computed for smooth as well as rough strip footings and the results obtained are found to be quite satisfactory. The ultimate bearing capacity of a rigid strip footing placed over granular, cohesive-frictional and purely cohesive soils, reinforced with single and a group of two horizontal layers of reinforcements has been determined. The necessary formulation has been introduced to incorporate the inclusion of reinforcement in the analysis. The efficiency factors, and , to be multiplied with Nc and Nγ for finding the bearing capacity of reinforced foundations, have been established. The results have been obtained (i) for different values of soil friction angles in case of granular and cohesive-frictional soils, and (ii) for different rates at which the cohesion increases with depth for purely cohesive soil under undrained condition. The optimum positions of the reinforcements' layers corresponding to which and becomes maximum, have been established. The effect of the length of the reinforcements on the results has also been analyzed. As compared to cohesive soil, the granular soils, especially with greater values of frictional angle, cause much more predominant increase in the bearing capacity. The stability of a long open vertical trench laid in a fully cohesive and cohesive-frictional soil has been determined with an inclusion of single and a group of two layers of horizontal reinforcements. For different positions of the reinforcement layers, the efficiency factor (ηs), has been determined for several combinations of H/B, m and where H and B refer to height and width of the trench, respectively, and m accounts for the rate at which the cohesion increases linearly with depth for a fully cohesive soil with = 0. The effect of height to width of the long vertical trench on the stability number has been examined for both unreinforced and reinforced soils. The optimal positions of the reinforcements layers, corresponding to which becomes maximum, have been established. The required length of reinforcements to achieve maximum efficiency factor corresponding to optimum depth of reinforcement has also been determined. The magnitude of the maximum efficiency factor increases continuously with an increase in both m and . The effect of pseudo-static horizontal earthquake body forces on the stability of a long unsupported circular tunnel (opening) formed in a cohesive frictional soil has been determined. The stability numbers have been obtained for various values of H/D (H = tunnel cover, D = diameter of the tunnel), internal friction angle of soil, and the horizontal earthquake acceleration coefficient The computations revealed that the values of the stability numbers (i) decreases quite significantly with an increase in , and (ii) become continuously higher for greater values of H/D and . The failure patterns have also been drawn for different combinations of H/D, and . The geometry of the failure zone around the periphery of the tunnel becomes always asymmetrical with an inclusion of horizontal seismic body forces. The interference effect on the stability of two closely spaced parallel (twin) long unsupported circular tunnels formed in fully cohesive and cohesive-frictional soils has been evaluated. The variation of the stability number with S/D has been established for different combinations of H/D, m and ; where D refers to the diameter of each tunnel, S is the clear spacing between the tunnels, and is the internal friction angle of soil and m accounts for the rate at which the cohesion increases linearly with depth for a soil with = 0. On account of the interference of two tunnels, the stability number reduces continuously with a decrease in the spacing between the tunnels. The minimum spacing between the two tunnels required to eliminate the interference effect increases with (i) an increase in H/D and (ii) a decrease in the values of both m and . The failure patterns have also been generated for a few cases with different values of S/D. The size of the failure zone is found to become smaller for greater values of m and . The horizontal pullout capacity of a group of two vertical strip anchors embedded, along the same vertical plane in sand, at shallow depths has been determined. At collapse, it is assumed that the anchor plates are subjected to the same uniform horizontal velocity without any bending or tilt. The pullout resistance increases invariably with increases in the values of embedment ratio, friction angle of the sand mass and anchor-soil interface friction angle. The effect of spacing (S) between the anchors on their group collapse load is examined in detail. For a given embedment ratio, the total group failure load becomes maximum corresponding to a certain optimal spacing (Sopt). The values of Sopt increases with an increase in the value of , but the changes in the value of H/B and do not have any significant effect on Sopt. The vertical uplift capacity of a group of two horizontal strip plate anchors with the common vertical axis buried in purely cohesive as well as in cohesive frictional soil has been computed. The variation of the uplift factors Fc, Fq and F , due to the contributions of soil cohesion, surcharge pressure and unit weight, respectively, has been evaluated for different combinations of S/B and H/B. As compared to a single isolated anchor, the group of two anchors generates significantly greater magnitude of Fc. On the other hand, the factors Fq and F , for a group of two anchors are found to become almost equal to that of a single isolated anchor as long as the levels of the lower plate in the group and the single isolated anchor are kept the same. For the group of two horizontal strip plate anchors in purely cohesive soil, an increase of cohesion of soil mass with depth and the effect of self weight of the soil have been incorporated. The uplift factor Fcy both due to cohesion and unit weight of the soil has also been computed for the anchors embedded in clay under undrained condition. For given embedment ratios, the factor Fcy increases linearly with an increase in the normalized unit weight of soil mass upto a certain value before attaining a certain maximum magnitude. The computational results obtained for different research problems would be useful for design.