A perturbation method for optimization of rigid block mechanisms in the kinematic method of limit analysis

Collapse mechanisms consisting of sliding rigid blocks are used widely as the basis for computing bounds on limit loads in geotechnical and structural engineering problems. While these mechanisms are conceptually straightforward to analyze, evaluating kinematically admissible velocities for a particular arrangement of blocks can be a tedious process, and optimizing the geometry of the mechanism is often prohibitively cumbersome for more than a few blocks. In this paper, we present a numerical technique for evaluating and optimizing mechanisms composed of an arbitrary number of sliding triangular blocks, assuming plane strain and homogenous, ponderable material obeying the Mohr–Coulomb yield condition. In the proposed method, coordinates defining the vertices of the blocks are treated as unknowns, and the optimal geometry is found by successively perturbing the vertex coordinates and block velocities, starting initially from a user-specified arrangement of blocks. The method is applied to three different examples related to geotechnical engineering, each of which illustrate that the approach is an efficient way to evaluate bounds that are often close to the true limit load.