334
Jui-Liang Lin and Keh-Chyuan Tsai
Inelastic Properties of 3DOF Modal Systems In an inelastic state, Equation (23) becomes
+ C D MnD n n n + R n = −M n 1(Γxnu gx + Γzn u gz )
(28)
where the restoring force, Rn, is a 3×1 column vector. By idealizing the three n-th modal pushover curves of the original MDOF system as three bi-linear curves, the corresponding three post-yielding stiffness ratios (αx, αz and αθ) and three yielding forces (Axny, Azny and Aθny) can be obtained. The six unknown inelastic parameters of the three bi-linear rotational springs used in the 3DOF modal system are determined accordingly. These six unknown parameters include the post-yielding stiffness of the rotational springs, k’x, k’z and k’θ, and the yielding moments of the rotational springs, Myx and Myz and Myθ. ~ , From the derivation shown in the appendix, the mode shape of the active sub-mode, φ a is equal to [1 1 1]T. Thus, the modal inertia force distribution of the active sub-mode, sa, is
⎡m x ⎤ ~~ s a = Mφ a = ⎢ m z ⎥ ⎢ ⎥ ⎣ I ⎦ 3×1
(29)
The displacements and forces of the 3DOF modal system subjected to sa presented in ADRS format are
~ ~ Dx ~ Dz ~ = , D x ~ ~ = Dz , φ φ ax az ~ ~ ~ Vx Vz T ~ m =φ ~ m =φ ~ I φ ax x az z aθ ~ ~
~ Dθ ~ = Dθ ~ φ aθ ~ ~ ~ Vx V z T ~ = = = =A mx m z I
(30)
~
where Vx , Vz and T are the base shears and base torque of the 3DOF modal system
~
subjected to sa. It should be noted that the base torque T is calculated on the projection point of the lumped mass on the X-Z plane. When all the rotational springs are elastic, the incremental displacements of the 3DOF modal system subjected to incremental sa are
~ ⎡ ΔDx ⎤ ⎢ ~ ⎥ ⎢ ΔDz ⎥ ~ ⎥ ⎢ΔD ⎣ θ⎦
~ = K −1Δs a 3×1
(31)