Novel Numerical Procedures for Limit Analysis: Implementation to Planar, Axisymmetric and Three-Dimensional Geomechanics Stability Problems
The Current research in the field of computational limit analysis dwells upon the development of numerical tools which are sufficiently efficient and robust to be used in engineering practices. This places demands on the numerical discretization strategies adopted as well as on the mathematical programming tools used to solve the associated optimization problems, which are the key ingredients of a typical computational limit analysis procedure. The traditional FEM based discretization is subjected to various issues, like: (i) volumetric locking, (ii) high sensitivity to mesh geometry, and (iii) difficulty in mesh generation and remeshing of higher order and 3D elements. Moreover, the optimization problem associated with limit analysis is nonlinear, non-smooth and sparse in nature. The linear and nonlinear programming are not so efficient in solving these type of problems. The three major objectives of present research work are (i) to improve the accuracy and efficiency of limit analysis solutions by using advanced spatial discretization techniques, (ii) to express the optimization problems associated with computational limit analysis as conic programming problem, so that it can be solved efficiently using primal-dual interior point method, and (iii) to solve complex planar, axisymmetric and three-dimensional geomechanics problems using developed numerical techniques. With reference to the first objective of the thesis, the bearing capacity of a circular footing resting on c- soil is solved by applying proposed axisymmetric kinematic limit analysis formulation to a planar domain which is discretized by three-noded and six-noded triangular elements with and without the provision of velocity discontinuities. The proposed formulation is casted as a semidefinite programming (SDP) problem independent of static variables. It is found that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and employing the velocity discontinuities along all the elements’ interfaces. Further, the concept of different smoothed finite element approaches along with adaptive meshing have been used to improve the performance of constant strain three noded triangular elements in kinematic limit analysis approach to solve plane strain and plane stress stability problems on basis of the Mohr-Coulomb yield criterion. By using duality of optimization theory the associated limit analysis formulation is framed as a second order cone programming (SOCP) problem. 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. For 3D kinematic limit analysis, a formulation by using the radial point interpolation method (RPIM) and the cell based smoothed finite element method (CS-FEM) also has been proposed. The RPIM uses higher order interpolation function, while, the CS-FEM uses a smoothed strain rate field to deal with the volumetric locking problem. The solutions obtained by using these methodologies are found to be accurate and upper bound in nature. With reference to the second objective of the thesis, a lower bound finite elements limit analysis formulation in combination with SDP is proposed to solve stability problems involving modified Hoek-Brown 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. An iterative approach has also been proposed to account for the true value of the exponent α in the generalized Hoek-Brown yield criterion still by using the formulation given on the basis of the modified Hoek-Brown yield criterion. It is found that the proposed approach remains quite accurate and is highly efficient to deal with any large scale optimization problem. 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 asymmetry (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 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 asymmetry. With reference to the second objective of the thesis, the bearing capacity of rectangular foundations, with varying aspect ratios ( 1 ≤ L/B ≤ 5 ) and embedded at shallow to medium depths (D/B ≤ 5 ), is evaluated by employing three-dimensional kinematic limit analysis, the radial point interpolation method (RPIM) and the cell based smoothed finite element method (CS-FEM). The shape factors are expressed as function of L/B only, which needs defining the depth factor as function of both L/B and D/B. New and updated values of shape and depthfactors for different combination of are L/B, D/B and are obtained and presented in the form of charts, which can be directly used to obtain the bearing capacities of foundations. For the extreme cases, the results are found to be very close to the published most accurate solutions for strip and circular footings. Even for intermediate values of L/B , the results from the current analysis compare equally well with that published in literature on the basis of (i) the upper bound finite element limit analysis by using continuous quadratic velocity field, and (ii) FLAC3D.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. Finally, by using kinematic limit analysis approach, an embankment placed over a purely cohesive stratum which is reinforced by means of a group of uniformly spaced vertical granular circular columns is analyzed. The yield surface of soil-stone column composite is obtained numerically by solving an auxiliary limit analysis problem associated with selected representative volume element (RVE). The obtained yield surface is then approximated as the sum of convex ellipsoidal sets and the optimization problem is framed as SOCP problem. The approximated numerical yield criterion has been used as yield function of stone column reinforced soil, while solving the embankment resting on stone column reinforced soil problem. Stability number charts have been provided for different soil and reinforced parameters, geometry of embankment and different percentage of reinforcements.
- Civil Engineering (CiE)