Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 192 Figure 10 : Plain concrete rectangular beam subjected to end torsion load: specimen configuration and FE discretization. In Fig. 11, the global response curves and the warping displacement distribution are plotted, comparing the results obtained with the proposed FE with the experimental and analytical outcomes presented in [14]. The curves show that the proposed FE is able to correctly predict the strain-softening behavior of the material, experimentally observed. Instead, the same accuracy can not be accomplished with standard FE formulations. In fact, the lower curve represents the response obtained under the hypothesis of rigid cross-sections, but correcting the section torsional inertia J with a torsion factor, that is J =  H B 3 with  = 0.229, determined by using the semi-analytical solution based on the Fourier series (according to the ratio H/B = 2.0). In this case, the peak load is significantly underestimated. This is due to the incorrect distribution of the damage variable D over the cross-section, that without warping is similar to that of a circular section, where no warping occurs [15]. Comparing the damage distribution in the two cases (Fig. 12), i.e., that obtained with the proposed FE and with rigid section hypothesis, it is evident that, for the same value of rotation x  , in the element without warping a larger part of the section is characterized by high values of damage, thus the torsional stiffness of the beam is lower than that of the element with warping. For higher values of the rotation x  , the distribution of the damage in the FE with warping tends to show a pseudo-circular shape, similar to that of the element without warping. Thus, it seems that the two responses tend to coincide. However, the experimental tests in [14] show that the post-peak part of the response is usually characterized by abrupt failure of the specimen, so high values of the rotation are not reached. Figure 11 : Response of the rectangular beam subjected to concentrated torque: moment vs rotation curve and warping profile.