Issue 41

R. M. De Salvo, Frattura ed Integrità Strutturale, 41 (2017) 350-355; DOI: 10.3221/IGF-ESIS.41.46 353 Figure 3 : Hypothesis of plastic hinge at node C. The plastic moments of the elements are assumed equal to 2M p for beams and M p for pillars. The stiffness of the beams is fixed twice that of pillars. The stretched fibers of the elements are the internal ones except for the central pillar whose stretched fibers are indicated by a small rectangle. In figure the plastic hinges are localized together with the bending moment distribution at collapse. The multiplier at collapse is equal to  p =5.50M p /Pl. These results have been obtained making use of the fundamental theorems of the Limit analysis. It is worth to note that, being the structure six times hyper- static, at collapse the number of plastic hinges is seven. Hypothesis is made that the last formed plastic hinge is that at node C (Fig. 3). In the same figure the rotations at plastic hinges are reported. These were calculated through the solution of the system of equations obtained by applying the displacement method and taking into account the convention on signs at Section 4. The horizontal displacement at node D (assumed positive if rightward) and the vertical displacement at section E (assumed positive if downward) are also reported, both clearly deducted by solution of the above-mentioned system. For rotations, the factor M p l/EI is omitted, for displacements the factor M p l/EI. For a greater clarity, the diagram of moments has been superimposed. The analysis of these results shows that the choice of the last plastic hinge at node C is false. It is in fact sufficient to observe that the rotation at node E is negative while the plastic moment at the same section has a positive sign. On the basis of the criterion showed in Section 3, an articulated movement similar and concordant to the collapse motion has to be built. It is defined by a clockwise rotation  of the pillar AD (expressed for less than the factor M p l/EI). This rotation, accordingly with the convention on signs in Section 4, gives to each plastic hinge a rotation well defined in value and sign, as indicated in Fig. 4. Under the hypothesis that the last plastic hinge finds at node C (Fig. 3), the rotations of plastic hinges are now summed to the rotations obtained through the impressed articulated motion (Fig. 4). Figure 4 : Scheme with an imprinted articulated motion. In this way, the following sums are obtained: