Geometric Nonlinearity - Theory of Nonlinear Structural Analysis - page 262

where Θ 00 ( t ) is the plastic rotation at the PHLs.
Substituting Eqs. (7.96) and (7.97) into Eq. (7.95) and rearranging the terms, the first
governing equation of the FAM for dynamic analysis is obtained as follows:

m ðÞ+ K 00 ðÞ Θ 00 ðÞ= K 0 ðÞ

T x ðÞ

ð7 : 98Þ

The second governing equation of the FAM relates the inelastic displacement x 00 ( t ) at the DOFs
and the plastic rotation Θ 00 ( t ) in the PHLs. This equation can be written as

x 00 ðÞ= K ðÞ −1 K 0 ðÞ Θ 00 ðÞ ð7 : 99Þ

7.4.2 Nonlinear Dynamic Analysis with the Force Analogy Method
When the FAM is used, the stiffness force in the equation of motion is calculated by multiply-
ing the stiffness matrix K ( t ) with the elastic displacement x 0 ( t ). For an n -DOF system subjected
to earthquake ground motions, this equation can be written as

Mx ðÞ+ Cx ðÞ+ K ðÞ x 0 ðÞ= − Mg ðÞ− F f

ðÞ ð7 : 100Þ

where M is the n × n invertible mass matrix, C is the n × n damping matrix, x ðÞis the n ×1
velocity vector, x ðÞis the n × 1 acceleration vector, g ðÞis the n × 1 earthquake ground accel-
eration vector corresponding to the effect of ground motion at each DOF, and F f ( t ) is the n ×1
vector of additional forces imposed on the frame due to geometric nonlinearity (mainly the P - Δ
effect) of the gravity columns. This effect can often be modeled using a leaning column (or a
P - Δ column) in a two-dimensional analysis but requires a detailed modeling of the gravity
columns in a three-dimensional analysis. The relationship between this lateral force F f ( t )
and the lateral displacement can be written in the form:

ð7 : 101Þ

F f ðÞ= K f x ðÞ

where K f is an n × n stiffness matrix that is a function of the gravity loads on the leaning column
and the corresponding story height, but it is not a function of time. In a two-dimensional
analysis, this K f matrix usually takes the form similar to assembling the second term of
Eq. (7.9) for each element stiffness matrix:

2
4

3
5

− Q 1 = h 1 − Q 2 = h 2 Q 2 = h 2 0 0

. .

. . . . .

Q 2 = h 2 − Q 2 = h 2 − Q 3 = h 3

. .

. . . .

K f =

0 Q n −1 = h n −1 0

.

.

. .

Q n −1 = h n −1 − Q n −1 = h n −1 − Q n = h n Q n = h n

0 0 Q n = h n − Q n = h n

ð7 : 102Þ