Lower Bound Finite Element Limit Analysis for Solving Different Planar and Axisymmetric Stability Problems in Soil and Rock Media
The finite element limit analysis (FELA), which utilizes the concept of finite elements and an optimization procedure, is a very powerful technique to find an accurate solution of different stability problems in geomechanics. This analysis is based on the theory of plasticity, and it can bracket very precisely the numerical solutions from both above and below, by employing the upper bound (UB) and the lower bound (LB) formulations. Two different optimization methods, namely, linear programming and nonlinear programming, have been mostly implemented by various researchers till now for solving different stability problems both in soil and rock media. The optimization problems associated with the limit analysis in geomechanics are generally nonlinear and sparse in nature. Linear and nonlinear programming techniques are, however, not so efficient in solving these limit analysis problems. The primary objectives of the present research work are: (i) to implement the advanced optimization techniques in the LB-FELA for solving different plane strain and axisymmetric stability problems in soil media, (ii) to develop new efficient LB-FELA formulations by using the conic optimization technique for performing the plane strain and axisymmetric stability analysis in rock media, and then to implement these formulations for various important plane strain and axisymmetric stability problems in rock media. For all the analyses, the associated computer programs have been written in MATLAB, and the toolbox MOSEK has been used to solve the associated optimization problem. With reference to the first objective, two important stability problems in soil mechanics associated with the determination of (i) the pullout capacity of horizontal strip anchor, and (ii) the stability of vertical cylindrical excavation, have been solved in the present thesis. It is assumed that, the medium is governed by the Mohr-Coulomb (MC) failure criterion and it follows an associated flow rule. For both these problems, the LB-FELA has been implemented, and the associated optimization problems have been solved with the help of second order cone programming (SOCP) technique. The pullout capacity factor (𝐹𝛾) has been evaluated for a strip anchor plate of width B and embedded horizontally at a depth 𝐻 from the ground surface in a cohessionless soil medium. The anchor is subjected to pullout load (i) with an eccentricity 𝑒 from the center of the plate, and (ii) having an inclination 𝛼 with the vertical axis. The variation of 𝐹𝛾 with e/B has been presented for different combinations of 𝛼⁄𝜙 and 𝜆 (= H/B). It is noted that the magnitude of 𝐹𝛾 decreases continuously with an increase in e/B and 𝛼⁄𝜙 and the rate of reduction of 𝐹𝛾 with e/B tends to become greater for smaller values of 𝜆. The stability of an unsupported vertical cylindrical excavation in cohesive-frictional soil media has been assessed. An axisymmetric formulation has been used and, accordingly, the planar domain has been discretized with the help of three noded triangular elements. The assumption associated with the circumferential stress has been avoided by representing the MC yield criterion with the usage of the three second order conic constraints. An extensive parametric study has been carried out, and the computational results are presented in terms of non-dimensional stability numbers. The obtained results have been compared with that available in literature. With reference to the second objective of the thesis, a lower bound finite element limit analysis formulation in combination with the power cone programming (PCP) has been proposed to solve the plane stability problems involving generalized Hoek and Brown (GHB) yield criterion. The formulation does not require any assumption associated with either the value of the exponent in the GHB yield criterion or any smoothing of the yield surface. In order to demonstrate the applicability of the proposed method, three different standard stability problems in rock mechanics have been solved. These involve (i) finding the bearing capacity of strip footings on horizontal rock media, (ii) assessing the stability of finite rock slopes, and (iii) determining the stability of rectangular unlined tunnels in rock mass. It has been found that the proposed approach remains quite accurate and is highly efficient to deal with any largescale optimization problem. A new lower bound axisymmetric limit analysis formulation, by using two dimensional finite elements, three dimensional GHB yield criterion and the power cone programming, has been presented for solving different axisymmetric stability problems in rock mechanics. The formulation does not require any assumption associated with the circumferential stress. For the purpose of checking the efficacy and accuracy of the proposed computational approach, bearing capacity factors of a circular footing and stability numbers of a vertical cylindrical excavation have been determined. The present approach has been found to be computationally very robust and it generates very accurate solutions. The proposed LB-FELA formulations for planar and axisymmetric stability problem have been employed to solve few important stability problems in rock mechanics associated with footings, tunnels, and anchors. The bearing capacity factors 𝑁𝜎 and 𝑁𝑞 of a rough strip footing lying over the rock strata, have been evaluated in the presence of pseudo static horizontal earthquake body forces. The variation of 𝑁𝜎 and 𝑁𝑞 with the changes in the horizontal earthquake acceleration coefficient (𝑘ℎ), for different values of unit weight of rock mass (γ), ground surcharge pressure (𝑞0) and the associated GHB material shear strength parameters (GSI, 𝑚𝑖, 𝜎𝑐𝑖), have been thoroughly investigated. The analysis clearly reveals that the bearing capacity factors decreases quite significantly with an increase in the magnitude of the earthquake acceleration coefficient. An attempt has been made to examine the stability of a single and twin horse-shoe shaped unlined tunnels laid in rock media and loaded with surcharge pressure on ground surface. For a given cover (H) to diameter (B) ratio of the tunnels, the maximum permissible value of the surcharge pressure (𝑞per) on ground has been computed for different values of GSI, 𝑚𝑖, 𝜎𝑐𝑖, γ, D and S; here, D represents the disturbance factor of the rock mass and S refers to the clear spacing between the twin tunnels. The magnitude of 𝑞per/𝜎𝑐𝑖 (i) increases continuously with an increase in the value of H/B, 𝑚𝑖 and GSI; (ii) decrease continuously with an increase in D and (iii) remains almost unaffected with the changes in 𝜎𝑐𝑖/γB. The variation of 𝑞per/𝜎𝑐𝑖 with S shows that the maximum permissible value of the surcharge pressure initially decreases up to a certain critical spacing between the two tunnels then increases continuously with an increase in the spacing between the tunnels and finally reaches a constant value corresponding to a single isolated tunnel. In the last, the vertical uplift capacity of a horizontal circular anchor embedded in rock media has been determined by using the proposed lower bound axisymmetric finite element limit analysis formulation. Design charts have been developed which provide the variation of the uplift capacity factors 𝐹𝜎 and 𝐹𝑞 for the value of the embedment ratios increasing from 1 to 10 and for different values of GSI, 𝑚𝑖, 𝜎𝑐𝑖/γB, and 𝑞s/𝜎𝑐𝑖. The improvement in the uplift capacity factors have been noticed with an (i) increase in GSI, 𝑚𝑖, and 𝑞s/𝜎𝑐𝑖 and (ii) decrease in 𝜎𝑐𝑖/γB. For the various stability problems selected in the present thesis, the failure patterns have also been drawn in order to understand the development of the plastic zones within the chosen domain. The results obtained from the analysis have also been thoroughly compared with those reported in literature. It is expected that that the solutions presented in this thesis will be useful for the practicing engineers and the proposed axisymmetric formulation will be quite useful for solving various stability problems in soil and rock media in an efficient manner.
- Civil Engineering (CiE)