PSI - Issue 44

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Mauro Mezzina et al. / Procedia Structural Integrity 44 (2023) 566–573 Mauro Mezzina, Alfredo Sollazzo, Giuseppina Uva/ Structural Integrity Procedia 00 (2022) 000–000

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Fig. 3.

a) Multi-connected section subjected to torque; b) c) d) Equilibrium at a node.

Let m be the number of closed meshes, n the number of nodes, and t the number of connecting segments: + = + 1 For the considered example: 2+6 = 7+1. Determining the values of the tangential stresses in the different sections involves solving a system of m+n equations in t+1 unknowns. In fact, the unknowns are represented by the tangential stress fluxes ϕ i = τ i b i in the different segments and by the specific torsion angle θ t . The equations are represented by the n nodal equilibrium equations (of which only n-1 are independent); by the equilibrium equation to the rotation about the z -axis of the entire section: = � ℎ = �� ℎ =1 (4) and by m congruence equations, derived by imposing the equality of the torsional rotation for each closed mesh. In Eq. (4), h i is the arm of the resultant of the elementary force with respect to an arbitrary pole. The m congruence equations, for each closed mesh, are: = 4 2 � = 4 2 � 2 = 2 1 � (5) Eq. (5) can be obtained by applying Bredt's first formula (2): = 2 ( A i is the area enclosed by the midline C i of the considered mesh). In the present case, we obtain the following system of 4 equations in 4 unknowns: � − 2 1 2 2 0 0 0 (2 1 + ℎ ) 0 0 (2 2 + ℎ ) −ℎ − 2 1 ℎ ℎ − 2 2 ℎ � = � 0 0 0 �