Issue 41

R. M. De Salvo, Frattura ed Integrità Strutturale, 41 (2017) 350-355; DOI: 10.3221/IGF-ESIS.41.46 351 I NTRODUCTION t is made reference to the previously wrote monograph titled “Un procedimento originale per l’individuazione della cerniera ultima e per la determinazione della deformazione completa allo stato di collasso”, published in the journal LaborEst n.11 [1]. The topic is taken up in relation to the fact that, thanks to the introduction of a fundamental concept, it has been possible to arrive to the solution of the same problem through direct calculation and therefore not through an iterative procedure. C ONCISE CONTENTS OF THE ABOVE - MENTIONED MONOGRAPH he main aspects that led to the solution of the problem can be subdivided in the following points: a. demonstration that it is necessary to individualize the last plastic hinge with the aim to know the deformation state at collapse; b. formulation, by means suitable analyses, of the Last Hinge Theorem, that can be stated as follows: given a framed structure, k-times hyper-static, whose collapse mechanism is known, if one of the k+1 plastic hinges is chosen as the last one, therefore the concordance of verses between plastic moments and rotations at all the k plastic hinges is necessary and sufficient condition to assert that this configuration is the real one; c. choice of any of the k+1 plastic hinges and assumption that it is the last one to be formed, and solution of the structure through the rotation method. Knowing the deformed state of the system, this method allows to proceed to the comparison of verses of plastic moments and rotations of the correspondent hinges. If, according to what already said, the comparison between verses concordance returns a positive result, then the selected hinge is the last one to be formed. On the contrary, the evaluation must be repeated considering a different hinge as the last one. The procedure will end once the above concordance will be satisfied for all the k residual plastic hinges. In this case, the last plastic hinge will be known as well as the complete deformation of the system (rotations at all the plastic hinges and displacements). The procedure illustrated above inevitably needs, for each iteration, the compilation and solution of a system of equations with the aim to obtain the deformative state of the structure in relation to the position of the plastic hinge assumed as the last one to be formed. T HE NEW PROCEDURE n this work an innovative criterion is presented. It moves from the consideration that, if the structure at its collapse condition is subjected to an articulated movement, similar and concordant to the crisis motion, this will not change the stress state of the system because at this stage all the bending moments already reached their pick values. The imposed articulated movement produces the kinematical effect to modify the deformative state of the system, determining the passage from one configuration to another one. This motion is known once a single parameter is fixed, namely the displacement of a point or the rotation of a beam. It can be therefore inferred that a bijective correspondence exists between this parameter and the resulting deformed configuration: the one is known once the other is fixed and vice-versa. On the basis of what above said and under the hypothesis that the kinematics at collapse is known by applying any suitable procedure (in particular the first or second theorem of the Limit Analysis [2, 3]), one of the k+1 plastic hinges can be arbitrarily fixed and, under the hypothesis that this is the last one to be formed, the obtained scheme can be studied through the rotation method [4]. The comparison, in verses, between the plastic moments (known) and rotations of the correspondent hinges (obtained by applying the above method) can be now carried out. It is useful to denote the set of these deformations as “configuration 1”. If the comparison returns a positive result in terms of verses concordance, then the selected hinge is that one really formed as the last one and the search for it ends here. But the most interesting case (and that is clearly the most frequent) is when the comparison reveals at least one discordance. In the second case, an articulated movement in the sense of the collapse motion is imposed through the choice of a unique parameter, so that a deformed configuration is obtained where each plastic hinge has a rotation whose value is dependent by this parameter. It is useful to denote the set of these deformations as “configuration 2”. I T I