Reinforced concrete slabs supported on columns fail by punching when a conical plug of concrete perforates the slab above the column. This failure has been mostly investigated experimentally, so what can numerical simulation contribute to understand this phenomenon? To answer this question, a computational simulation tool based on the finite element method is developed. This numerical model is implemented into a computer code in applying the object-oriented programming concept. The requirements that the numerical model should fulfill are derived from a review of the experimental results published in the literature and obtained in our laboratory. The numerical model reproduces the non-linear material behavior characterizing reinforced concrete structures by decoupling the actions of steel reinforcement and concrete. The steel reinforcement is represented by uniaxial truss elements which follow a bi-linear stress-strain response. The concrete is modeled with continuum elements which are described at the constitutive level within the framework of the incremental flow theory of plasticity. The concrete triaxial strength is delimitated with a new failure criterion. A plastic flow rule is derived so as to reproduce the evolution of the plastic strain observed experimentally. Cracking induces strain-softening which refers to a gradual decrease in tensile strength with increasing deformations. The reduction of the tensile strength is controlled by an isotropic decohesion process which is monitored by constant fracture energy. The stiffness degradation due to cracking is reproduced with an isotropic elastic damage model. A perfect bond between concrete and steel is assumed. The capabilities of the numerical model are illustrated by simulating the localized shear failure in soil under a footing, the uniaxial tensile, and the confined compressive tests in plain and reinforced concrete specimens. The simulation of punching failure is investigated for three circular reinforced concrete slabs. The comparison with experimental results indicates that: (1) the punching failure mechanism –characterized by a localized inclined punching crack– is generated, (2) the value of the punching load is predicted, (3) the cracking sequence is reproduced, (4) the global response is slightly too stiff. For slabs with orthogonal reinforcement, the perfect bond hypothesis does not allow to capture the punching failure mechanism and the numerical model is improved by relaxing this hypothesis. The developed computational simulation tool reproduces the punching failure observed experimentally and is consequently used to investigate the failure mechanism. First, it is shown that the punching crack results from a crack coalescence phenomenon at the top of the slab followed by a crack propagation in the direction of the corner slab-column. Second, a condition for punching failure to occur is determined. Third, it is illustrated that the size-effect observed experimentally is reproduced and is reflected by a modification of the tensile stress distribution along the punching crack. Finally, a parametric study of the punching failure is performed revealing that punching failure is governed by the tensile strength of concrete. It is also shown that increasing the percentage of reinforcement raises the punching load and reduces the ductility of failure. Lastly, the computational simulation tool allows to demonstrate that in a circular slab, the applied bending moment is not determinant for the value of the punching load.