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.