3D ultimate limit state analysis using discontinuity layout optimization

The recently developed discontinuity layout optimization (DLO) procedure uses limit analysis theory to directly obtain upper bounds on plane strain collapse loads of bodies and has successfully been applied to geotechnical problems. In this thesis, a new three-dimensional formulation of DLO is described. The new formulation is capable of directly estimating the collapse load of bodies involving Tresca and Mohr-Coulomb yield criteria, using efficient second order cone programming. The new formulation can be stated in kinematic, energy balance form or static, equilibrium form. The derivation from first principles of both kinematic and equilibrium forms is described, allowing full conceptualization of the DLO procedure. A number of simple benchmark problems are considered, demonstrating that good results can be obtained using the new formulation even when very coarse numerical discretisations are employed. The best reported upper bound for the compression of a purely cohesive block between two perfectly rough platens was improved upon. In DLO, the yield condition is only checked on predefined discontinuities, used to discretize the problem. Consequently, the estimated collapse loads are greater than the ‘exact’ collapse load ( i.e. they are ‘unsafe’). New methods generating continuous stress fields from discontinuous DLO solutions are developed based on the plane strain and three-dimensional equilibrium forms of DLO. These new fields are discretized in plane strain and three-dimensions using solid triangular and tetrahedral elements, respectively. The stress fields are explained in the context of determining alternative ‘lower bound’ forms of solution. An alternative method determining a continuum stress field directly ( i.e. not from a DLO solution) was also developed.