The Finite Element modelling of structural failure using the standard displacement-based formulation of solid mechanics has demonstrated to result in spuriously mesh dependent calculations in terms of crack trajectory. Ensuingly, failure mechanisms obtained with this method present the critical issue of spurious mesh bias dependency depending on the FE mesh orientation adopted to perform the computation. The authors addressed this matter in the past by adopting a mixed strain/displacement finite element formulation to approach the nonlinear solid mechanics problem. The local convergence in terms of strains and stresses that is guaranteed with this approach permits to produce mesh bias objective calculations [1-2]. The present work presents the Adaptive Formulation Refinement (AFR) strategy [3] which is adopted for an efficient analysis of failure in quasi-brittle materials. With this methodology, computations are performed starting from a standard displacement-based FE approach, and the mixed FE formulation is adaptively activated only in the regions where the fracture develops. The standard FE is preserved in the rest of the structural domain, enabling important savings in computational cost while maintaining the mesh objectivity and enhanced accuracy of the results obtained by the mixed FE. This AFR is combined in the localized structural failure simulations with an Adaptive Mesh Refinement (AMR) approach introduced to further increase the cost-effectiveness of the analyses [3]. The octree-based AMR scheme allows to adaptively refine the mesh only in specific areas of the domain. The proposed AMR strategy permits to start the calculations with an initially relatively coarse mesh and perform refinement operations only where fracture onset and evolution occurs. This allows to simulate the phenomenon of structural failure with the necessary mesh resolution and in a more cost-effective manner. The cost-effectiveness and accuracy of the methodology put forward is investigated through a comprehensive set of numerical simulations including both benchmark problems and laboratory tests reported in the literature. The analyses demonstrate that the approach combining the AFR and AMR methods results in mesh objective computations in terms of collapse mechanisms, fracture paths, bearing capacities and force-displacement curves. Experiments are reproduced in 2D and 3D calculations with precision.