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 [ , ] = − $ $ ( $ − $ ) $ [1 + 3 ! 3 " ( 1 # !" + 1)] 0. ! 0. " + 2 3 ( " ( 1 $ !" + 1) 0. ! 0. " + 2 # 0. ! 0. " $

1360

5

(14)

so that, definitely [ , ] = − with the position: =

$ $ G 5 4 − $ H $

(15)

$ $ ; $ = @1 + 3 ! 3 " (2 + #$ ) + 2 3 ( " (1 + #$ )C ; = $ $ − 3 ( " # 2 ) " #$

(16)

In the motion equation, the incremental work of the inertial loads is: [− ̈, ] = − 6∗ ̈ $ $

(17)

where :

6∗ = 6 (8 $ ) (1 + #$ ## $ ) ; #$ = 8 # (% " % ! ) 8 # (% ! )

(18)

with 6 (8 $ ) and 6 (8 # 8) the moment of inertia of the segments I and II around C 1 e C 2 . The horizontal forces G 1 A/g, G 2 A/g work for the additional horizontal displacements of the centers of the segments I and II: [ , ] = 3 " 9 $ ( $ − $ ) $ + 3 ! 9 # ( # − # ) #$ $ (19) Taking into account that the angles q 1 e q 2 are small quantities we have [ , ] = $ : ' 9 $ [ :; + :; $ ] $ (20) with :; = $ + 3 ! 3 " ) ! ) " #$ # ; :; = $ + 3 ! 3 " ) ! ) " ## $ # (21) Substitution of (15), (17) e (20) into (5) gives the equation of motion: − 6∗ ̈ $ + $ $ $ G $ $ − 9 : :; H − $ $ G $ $ + 3 ( " # 2 ) " − 9 : :; H = 0 (22) Further, neglecting, as in the Housner column, the quantity / ⋅ :; compared to $ $ we obtain ̈ $ − # $ + # G 5 4 − 9 : 5 () 5 H = 0 (23) where the quantity = P 3 " ) " 8 #∗ (24)