The standard displacement-based finite element formulation has shown to produce spurious mesh dependent results when approaching the computation of quasi-brittle failure problems. Failure mechanisms calculated with this approach are spuriously dependent on the FE orientation of the mesh employed. The authors have addressed this critical issue in the past via the adoption of a mixed strain/displacement finite element formulation to calculate the localized structural failure problem. This method guarantees the local convergence of the solution in terms of stresses and strains, which represents a crucial feature in order to obtain mesh bias objective results. In the present work, an Adaptive Formulation Refinement (AFR) strategy is put forward to perform a computationally efficient calculation of the nonlinear solid mechanics problem. The computations are conducted starting with a standard displacement-based FE approach, and the mixed formulation is adaptively activated only in the areas of the domain where damage develops. The standard FE is maintained elsewhere, allowing for very significant saving in computational cost while preserving the quality of the results obtained with the mixed formulation in terms of mesh objectivity of the computed cracks. This AFR strategy is combined with an octree-based Adaptive Mesh Refinement (AMR) strategy to additionally enhance cost-effectiveness. This allows to initiate the calculations with an initially relatively coarse mesh and to adaptively refine the FE mesh only in the regions of the domain where cracks develop. The proposed combination of the AFR and AMR approaches allows the analysis of quasi-brittle structural failure with the required mesh resolution and in a more cost-efficient way. A comprehensive set of numerical simulations of both benchmark problems and laboratory tests available in the literature are reproduced in 2D and 3D. Computations show the ability of the methodology to reproduce experimental results in terms of collapse mechanisms, crack trajectories and force-displacement curves.