Lower and upper bound limit analyses for stability problems in geotechnical engineering

Abstract

Stability problems are important issues in geotechnical engineering, and many methods have been developed over the past forty years. However, virtually all of the existing methods have either practical or theoretical limitations when applied under general conditions. With the advances in computer hardware and numerical algorithms for convex programming, there are growing interests in the application of numerical limit analysis for the stability problems for the Ultimate Limit State (ULS) design. Lower and upper bound limit analyses are becoming competitive due to its ability to provide rigorous lower and upper bounds to the exact solutions without arbitrary assumptions. In this research, the Finite-Element-based Limit Analysis (FELA) in which the stress and velocity field are discretised with finite dimensional element spaces is discussed. Stability problems are formulated as the solution of convex optimizations in the present study. The Mohr-Coulomb (MC) yield criterion for plane strain analysis and the Drucker Prager (DP) yield criterion for full three-dimensional analysis are transformed to second order conic constraints such that the resulting large scale optimization problem could be solved by the standard Second Order Cone Programming (SOCP) solvers with great efficiency. For MC materials under the full three-dimensional condition, hyperbolic smoothing in the meridional plane and round-off of corners in the octahedral plane are applied to render the yield function to be everywhere differentiable. The NLP optimization stemming from the formulation is then solved efficiently by the primal-dual interior point algorithms. This technique will ensure that the developed method can achieve high accuracy while maintaining a fast solution suitable for complicated practical problems. Slip bands for most of the geotechnical problems are highly localized and the quality of the FELA solution is considerably affected by the configuration of the mesh. A variety of strategies for error estimation and mesh adaptation are investigated in the current work. The residual error estimator and recovery-based error estimators that have been proposed for the FEM are tailored for the FELA. Comparisions on the performances of these error estimators are made to the yield criterion slackness and the deformation based method in this study. With an iterative local refinement technique, a coarse mesh without prior knowledge on the approximate solution can be used as the initial discretisation input. The optimal mesh distribution and the failure mechanism will be obtained as part of the solution, which enables solutions with satisfactory degree of accuracy to be obtained from a poor initial mesh and the need for the prior knowledge in generating high quality mesh is then removed, which is of great practical value for complicated problems. The method as developed in the present study can usually solve a problem within five to fifteen minutes (for several thousand elements), which is considered to be acceptable so that the present work can be useful to both theoretical study as well as practical application. To consider the effect of the nonlinearity of the yield envelope on the collapse load, a power-type nonlinear yield criterion has also been studied in a systematic manner under the framework of the FELA. Comparisons are made between the results from the present study and those from the literature obtained by either conventional upper bound methods or limit equilibrium method. To facilitate the extending of the FELA, a code library written in objective-oriented programming language C++ has been developed. Abstract classes such as OptSolver, YieldCriterion, and LAsystem etc., are developed such that integration of other yield criteria and more powerful solvers that will appear in later development can be greatly simplified.