Issue 35

P. Bernardi et al, Frattura ed Integrità Strutturale, 35 (2016) 98-107; DOI: 10.3221/IGF-ESIS.35.12 104 Numerical results vs. experimental observations Numerical analyses have been carried out by implementing the above described 2D-PARC model into a commercial FE code (ABAQUS, [28]). Taking advantage of the symmetry of the problem, only one half of each beam has been simulated, by adopting a FE mesh constituted by quadratic, isoparametric 8-node membrane elements with reduced integration (4 Gauss integration points). Numerical analyses have been performed under displacement control, by applying an increasingly displacement at the loading point, in order to achieve a better numerical convergence and evaluate also the post-peak behavior. Numerical and experimental results have been first compared by considering the global response, in terms of applied load vs midspan deflection, as can be seen from Figs. 3, 4. In more details, Figs. 3 and 4a-b refer to the three specimens tested by Vecchio and Shim [17]. On the same graphs, also the results obtained by Bresler and Scordelis [20] on nominally identical beams are reported. Fig. 3 shows a comparison between the experimental data relative to specimen OA2 and the numerical results obtained by adopting the 2D-PARC model, respectively implementing the non-linear constitutive relation described above or a simple linear elastic matrix for describing concrete behavior. These results highlight that the assumption of a constant value for concrete Young modulus during the analysis – equal to its initial value ( E c = E ci ) – obviously provides a stiffer response, which is more pronounced for higher values of the applied load, where the effect of mechanical non-linearity is more significant. Moreover, the adoption of a linear elastic matrix for concrete leads in this case to a slight underestimation of the failure load. A more refined description of concrete behavior allows not only an improved modeling of the experimental response until failure, but also a much more stable solution, since the numerical analysis is characterized by a better convergence. On the contrary, both numerical analyses correctly predict the experimental shear failure mode and the corresponding final crack pattern. It should be here pointed out that the numerical response obtained through the original formulation of 2D-PARC model described in [7] is almost superimposed to that provided by adopting the concrete modeling proposed in this work. For this reason, the corresponding curve has not been plotted, so as to allow a better comprehension of the graphs of Fig. 3. In both cases, concrete is indeed treated as a non-linear elastic material subjected to a biaxial state of stress, but the here proposed approach is preferable since it requires a reduced computational effort and leads to an improved convergence of the algorithm. 0 50 100 150 200 250 300 350 400 0 3 6 9 12 15 18 Vecchio-Shim beam Bresler-Scordelis beam NLFEA P [kN] δ [mm] OA2 NLFEA, E c =E ci Figure 3 : Comparison between experimental [17] and numerical results in terms of applied load vs. midspan deflection for specimen OA2. Numerical analyses have been repeated twice, by differently modeling concrete behavior. Figs. 4a-b show similar comparisons for the other two specimens tested by Vecchio and Shim [17], namely OA1 and OA3 beams. In this case, experimental data have been only compared with the numerical response provided by the improved formulation of 2D-PARC model proposed herein, so to better appreciate its effectiveness. The graphs highlight the good capability of the model to represent the global behavior both at serviceability (cracking load) and at ultimate limit state. The peak load is accurately predicted, as well as the brittle shear failure characterized by no ductility. An accurate simulation of the experimental behavior, both in terms of stiffness and failure load, has been also obtained for specimens tested by Podgorniak-Stanik [18], as shown in Figs. 4c-d. The adopted FE model is also able to predict that beams BN25D and BN50D (containing an additional distributed reinforcement on element sides, in addition to the main flexural bottom rebars) fail for higher shear stresses than the corresponding specimens BN25 and BN50, as provided by test results. Therefore, the satisfactory agreement between experimental and numerical responses proves that the proposed model is able to correctly describe the experimental behavior for different specimen geometries, as well as different reinforcement arrangements.