Limit analysis is relevant in many practical engineering areas such as the design of mechanical structures or the analysis of soil mechanics. Assuming a rigid, perfectly-plastic solid subject to a static load distribution, the problem of limit analysis consists of finding the minimum multiple of this load distribution that will cause the body to collapse. This collapse multiplier results from solving an infinite dimensional saddle point problem, where the internal work rate is maximized over an admissible set of stresses -defined by a yield condition- and minimized over the linear space of kinematically admissible velocities for which the external work rate equals the unity. When strong duality is applied to this saddle point problem, the well-known convex (and equivalent) static and kinematic principles of limit analysis arise. In this thesis, an efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle, is discretized twice (one per bound) by means of different combinations of finite element spaces for the stresses and velocities. For each discretization, the interpolation spaces are chosen so that the attainment of an upper or a lower bound is guaranteed. The second step consists of solving the resulting discrete nonlinear optimization problems. Towards this end, they are reformulated into the canonical form of Second-order Cone Programs, which allows for the use of primal-dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis
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