An original method of direct calculation for the identification of the last hinge and the definition of the deformative state at collapse

The subject matter of these notes refers to the Ultimate Strength Design of 2-D steel framed structures and in particular to the analysis of the deformation state at collapse. The idea is based on 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. This motion is known once a single parameter is fixed, namely the displacement of a point or the rotation of a beam. When the collapse mechanism of the structure is already determined through any instrument of the Limit Analysis, a subsequent (k+1) plastic hinge can be arbitrarily fixed and assumed as the last developed one. It is therefore possible to solve the modified scheme through the rotation method and make a comparison in the verses between the known plastic moments and the rotations at the corresponding hinges. If the comparison is successful, in the sense that the checked verses are concordant, the selected hinge is actually the one formed as the last. On the contrary, the rotations resulting from an imprinted motion in the verse of collapse movement are algebraically added. If, for each hinge, the product between the plastic moment and the correspondent algebraic sum is made, this product has to be surely positive, due to the verses concordance. This can be translated in k+1 inequalities, with each one furnishing a lower limit for the parameter from which the articulated motion depends. Among these, the highest value is that one which makes all the inequalities to be simultaneously verified. The substitution of this value into the expressions for rotations permits to arrive to the simultaneous identifying of the last hinge and of the complete picture of deformations.