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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is the pulsation of the motion. At the dynamical collapse configuration the additional work of the vertical loads vanishes. So we have, from (15): [ , ] = − $ $ G 5 4 − $ H $ = 0 (25) The rotation $ is so the collapse rotation of the wall $ = 6;!! = 5 4 (26) When the motion of the wall is slow enough to cancel the acceleration ̈ $ , the additional deformation $ of the wall, at the limit, is null. The acceleration A , to the wall equilibrium must be such to give G 5 4 − 9 : 5 () 5 H = 0 (27) The acceleration A is now the acceleration of the static collapse A L : : + 9 = 5 4 () (28) According to (23) and (27) the equation of the motion of the wall takes the more significant form ̈ $ − # $ + # 6;!! G1 − : : + H = 0 (29) Let us assume now a seismic input characterized by 2 opposite sequence of magnitude A 0 and duration t 0 (Fig.4(a)). The first pulse starts when the wall is still in its vertical position. Thus, with the initial conditions 8 (0) = ̇ 8 (0) = 0 the motion of the wall is represented by: $ ( ) = 6;!! G : ' : + − 1H ( ℎ − 1) (30) So that the rotation and the rotation velocity at the base of the wall, at the end of the first pulse are $ ( < ) = 6;!! G : ' : + − 1H ( ℎ < − 1) ; ̇ $ ( < ) = 6;!! G : ' : + − 1H ℎ < (31) Assume that at the end of the first pulse the base rotation of the wall has not reached the value =>?? of the dynamical collapse. The equation of the motion of the wall under the second pulse has the acceleration A 0 of opposite sign. The new equation is: ̈ 88 − # 88 = − # 6;!! G : ' : + + 1H (32) whose solution, given by the function 88 ( ) is linked to the first motion rotation q I (t) with the conditions 88 (0) = 8 ( < ); ̇ 88 (0) = ̇ 8 ( < ) . The motion of the wall under the second pulse is specified as 88 ( ) = @ $ ̇ 8 ( < ) ℎ + ℎ @ 8 ( < ) − G : ' : + + 1H 6;!! C + G : ' : + + 1H 6;!! (33) This critical state is attained if: