Numerical Lower Bound Limit Analysis Using Second-order Cone Programming And Its Applications

Stability analyses or collapse calculations, which include bearing capacity of foundations, lateral earth pressures, and stability of natural and man made slopes and cuts, are important in geoteehnieal engineering to ensure that foundations and earthworks have adequate factors of safety against all possible modes of failure. In practical applications, limit analysis is recognized as the most powerful procedure for estimating the collapse of stability problems in a direct manner. Recently, more reliable stability calculations for complex situations can be achieved by optimizing limit analysis solution using mathematical programming methods, while spatial discretization of the filed variables can be accomplished using numerical methods. The main objective of this thesis is to develop and provide a novel approach to the lower bound limit analysis using conic programming.An analytical method is applied to stability problem of sheet pile type retaining walls under suitable idealization from the point of limit analysis. Both upper and lower bound analyses are carried out to estimate the range of the exact solutions. The efficiency and validity of limit analysis to deep foundation is investigated in this thesis. Moreover, Based on the fundamental of variational principles in conjunction with the limit theorems of classical plasticity, the functional form of the upper bound analysis is derived.In this study, this thesis describes a new formulation, based on linear finite elements and non-linear programming, for computing rigorous lower bounds in 1, 2 and 3 dimensions. By assuming linear variation of nodal variables, the formulation of limit analysis on collapse loads and factor of safety leads to an optimization problem with a large number of cone-shaped yield restrictions. Being analogous formulation in native form, the predictor-corrector method, an algorithm for second order cone programming (SOCP), is used to tackle the resultant nonlinear programming problem. From the point of nonlinear optimization theory, there exists strong duality between static and kinematic principles of limit analysis. The primal-dual interior point method is recognized as the mighty and fast numerical algorithm to solve large scaled convex programming problems. In SOCP algorithm based on the interior-point method, which derived from the work of the Nesterov and Todd (1997) on self-scaled cones and employs a Mehrotra (1992) type predictor-corrector extension and sparse linear algebra to improve the computational efficiency, it's no need to smooth the yield surface in the vicinity of the apex. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. Computational results are also presented to document that the implementation can solve very large problems robustly and efficiently.The numerical lower bound limit analysis is then applied in predictions of collapse loads for foundations on soil and masonry structures. The first analysis focus on shallow and embedded rigid footings subjected to vertical and inclined, concentric loading. These analyses required several modifications of the existing programs to generalize soil-footing interface properties and to select optimization functions. Also three different approaches have been used in the present research to perform lower bound analysis of embedded foundations. Closed-form and exact solutions for these classic bearing capacity problems, provide a useful basis for evaluating the accuracy of the numerical procedures. Especially, the results of the analysis are used to propose definitive values of depth factors for embedded strip footing. A comparison is made between slip-line, empirical and semi-empirical solutions. These results are helpful in reducing the uncertainties related to the method of analysis in bearing capacity calculations, paving the way for more cost-effective foundation design.In addition, the calculation of the collapse load on masonry structures is the second part of the applications. To achieve the complete description on cracking, slip and crushing of unit-joint interface material, a novel, three-surface elasto-plastic cap model in which the three surfaces intersect smoothly is introduced and developed here. Sample computations demonstrating the very good performance of the algorithm without any modifications are also presented.The major part of this thesis described the formulation of numerical lower bound limit analysis for structural element. The proposed formulation is evaluated by:â…°) comparison with analytical closed-form solutions for a series of 2-D structural frame problems published in the literature; andâ…±) analysis of laterally loaded sheet pile embedded in clay. The formulation of structural elements represents the first step toward the use of limit analysis for stability problems of soil-structure interaction such as deep foundation and excavation.The efficiency of the methodology proposed is illustrated with several applications in plane strain, demonstrating that it can be used in complex, realistic problems as a supplement to other methods.