PSI - Issue 44

Simona Coccia et al. / Procedia Structural Integrity 44 (2023) 1356–1363 Simona Coccia et al. / Structural Integrity Procedia 00 (2022) 000–000

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the connection of the wall with the horizontal constraint. The position of the three hinges fits the position of the pressure line, as shown in Fig. 1(b). The static limit value A L can be evaluated by means the limit analysis approach (Como, 2013). The system has a single degree of freedom, represented by the virtual rotation dq 1 of the segment I . The virtual rotation dq 2 of the body II in the undeformed configuration is equal to: # = #$ $ (2) where: k 21 = & % '% (3) During the motion, the center C 12 changes position and the segment I gains inclination (Fig. 2(a) and (b)). The absolute center C 2 of the segment II approaches the wall (Fig. 2(a) and (b)). This nonlinear effect plays a relevant role in the formulation of the motion equations of the wall because enhanced by the wall slenderness. In the deformed configuration, assuming that the segment I and II rotate of the angle q 1 and q 2 , we can obtain the connection between these two rotations, taking into account that the horizontal distance from C 1 and the centers of the two segments are equal (Fig. 2(c)): $ sin( $ − $ ) = # sin( # − # ) (4) where R 1 , b 1 , R 2 and b 2 depend on the position of the hinge C 12 and are obtained in the undeformed position of the wall. Considering in the deformed configuration that small increments of dq 1 e dq 2 take place, we can obtained the motion equation writing the dynamical equilibrium of the wall: [ , ] + [− ̈, ] + [ , ] = 0 (5) where [ , ] , [− ̈, ] , [ , ] respectively are the incremental works of the dead loads g , the inertial forces − ̈ , and the forces induced by the acceleration A , made for the increment displacements d u of the deformed mechanism in the passage of configuration of the wall between the time t and t + dt .

(a)

(b)

(c)

Fig. 2. virtual displacement: (a) in the undeformed configuration; (b) in the undeformed configuration; (c) geometrical characteristics of the wall.

The incremental work of the weights is: [ , ] = −

$ $ $ − # # # − ( #

(6)