Solutions For Plane Strain And Axisymmetric Geomechanics Problems With Lower Bound Finite Elements Limit Analysis

Abstract

The present thesis illustrates the application of the lower bound limit analysis in combination with finite elements and linear programming for obtaining the numerical solutions for various plane strain and axisymmetric stability problems in geomechanics. For the different plane strain problems dealt in the thesis, the existing formulation from the literature with suitable amendments, wherever required, was used. On the other hand for various axisymmetric problems, the available plane strain methodology was modified and a new formulation is proposed. In comparison to the plane strain analysis, the proposed axisymmetric formulation requires only three additional linear constraints to incorporate the presence of the hoop/circumferential stress (σθ). Several axisymmetric geotechnical stability problems are solved successfully to demonstrate the applicability of the proposed formulation. In the entire thesis, three noded triangular elements are used for carrying out the analysis. The nodal stresses are treated as basic unknowns and the stress discontinuities are employed along the interfaces of all the elements. To ensure that the finite element formulation leads to a linear programming problem, the Mohr-Coulomb yield surface is approximated by a polygon inscribed to the parent yield surface. For solving different problems, computer programs are developed in ‘MATLAB’. The variation of the bearing capacity factor Nγ with footing-soil interface roughness angle δ is obtained for different soil friction angles. The magnitude of Nγ is found to increase extensively with an increase in δ. With respect to variation in δ, the obtained values of Nγ were found to be generally smaller than the results available in literature. The effect of the footing width on the magnitude of Nγ has been examined for both smooth and rough strip footings. An iterative computational procedure is introduced to account for the dependency of φ on the mean normal stress ( σm). Two well defined φ- σm curves from literature, associated with two different relative densities, are being chosen for performing the computational analysis. The magnitude of Nγ is obtained for different footing widths, covering almost the entire range of model and field footing sizes. For a value of the footing width greater than approximately 0.2 m and 0.4 m, for a rough and smooth footing, respectively, the magnitude of Nγ varies almost linearly on a log-log scale. The bearing capacity factors Nc, Nq and Nγ are computed for a circular footing both with smooth and rough footing interface. The bearing capacity factors for a rough footing are found to be consistently greater than those with a smooth interface, especially with grater values of soil friction angle (φ). An encouraging comparison between the obtained results and those available from the literature is noted. Bearing capacity factor Nc for axially loaded piles in clays whose cohesion increases linearly with depth has been estimated numerically under undrained (φ = 0) condition. The variation of Nc with embedment ratio is obtained for several rates of the increase of soil cohesion with depth; a special case is also examined when the pile base was placed in the stiff clay stratum overlaid by a soft clay layer. It has been noticed that the magnitude of Nc reaches almost a constant value for embedment ratio approximately greater than unity. The bearing capacity factor Nγ has been computed for a rough conical footing placed over horizontal ground surface. The variation of Nγ with the cone apex (interior) angle (β), in a range of 30º - 180º, is obtained for different values of friction angle ( φ). For φ< 30º, the magnitude of Nγ is found to decrease continuously with an increase in β from 30º to 180º. On the other hand, for φ > 30º , the minimum magnitude of Nγ is found to occur generally between β = 120 and β = 150º. In all the cases, it has been noticed that the magnitude of Nγ becomes maximum for β = 30o. The vertical uplift resistance of circular plate anchors, embedded horizontally in a clayey stratum whose cohesion increases linearly with depth, has been obtained under undrained ( φ = 0) condition. The variation of the uplift factor (Fc) with changes in the embedment ratio (H/B) has been computed for several rates of the increase of soil cohesion with depth. It has been noted that in all the cases, the magnitude of Fc increases continuously with H/B up to a certain value of Hcr/B beyond which the uplift factor becomes essentially constant. The results obtained from the analysis are noted to compare quite well with those published in literature. From the investigation reported in this thesis, it is expected that the proposed axisymmetric formulation will be quite useful for solving various axisymmetric geotechnical stability problem in a rapid manner. The available plane strain formulation has also been found to yield quite satisfactory solutions even for a problem where the soil friction angle depends on the state of stress at a point.