In the paper the size effect phenomenon in concrete is analysed. The results of numerical simulations of using FEM on geometrically similar un-notched and notched concrete beams under bending are presented. Concrete beams of four different sizes and five different notch heights under three-point bending test were simulated. In total 18 beams were analysed. Two approaches were used to describe cracks in concrete. First, eXtended Finite Element Method (XFEM) describing cracks as discrete cohesive ones with bilinear softening was chosen. Alternatively, an elasto-plastic constitutive law with Rankine criterion, associated flow rule and bilinear softening was defined. In order to ensure mesh-independent FE results, a non-local theory in an integral format as a regularisation technique was applied in the softening regime. In both approaches the influence of the decrease of the material parameters (mainly fracture energy) in the boundary layer on obtained maximum loads was studied. Additionally the influence of the averaging method in non-local plasticity was also examined. Obtained results were compared with experimental outcomes available in literature.