Issue 41

R. M. De Salvo, Frattura ed Integrità Strutturale, 41 (2017) 350-355; DOI: 10.3221/IGF-ESIS.41.46 352 The rotations of the two above-mentioned configurations are now algebraically summed up. These are obviously functions of the parameter that originated the configuration 2. By making, for each plastic hinge, the product between the bending moment and the correspondent rotation (which is the sum of configurations 1 and 2), the result, for the verses concordance, must be necessarily negative. This leads to k+1 inequalities, with each one furnishing a lower limit for the parameter  from which the configuration 2 depends. Clearly, the highest value between these limits is that one which simultaneously verifies all the inequalities. The substitution of this value in the expressions for rotations, permits to come to the simultaneous identification of the plastic hinge and of the complete picture of deformations. C ONVENTION ON SIGNS s it is known, using the displacement method for applications to 2-D framed structures, the rotations at the extremes of a beam are assumed positive if clockwise. The sign convention, according to the Fracture Calculus, is different. There is therefore the problem to adequate one convention to the other one. To this end, if the beam is arranged in a horizontal way and with the stretched fibers pointed downwards (Fig. 1), the rotation at the right extreme of the beam is transferred to the plastic hinge with the same sign, the rotation to the left extreme of the beam is instead transferred with the opposite sign. Figure 1 : Sign convention. The rotation of the plastic hinge will result, in value and sign, equal to the algebraic sum of the two rotations. As a consequence, if the angle formed by the two beams adjacent to the plastic hinge increases, the rotation of the plastic hinge will be positive; negative on the contrary (Fig. 1). Figure 2 : Collapse scheme. N UMERICAL APPLICATION ith the aim to make the above theory more accessible, it is considered worthwhile to solve in detail the case of the frame reported in Fig. 2. A W