Geometric Nonlinearity - Theory of Nonlinear Structural Analysis - page 274

K ðÞ= K o + K g ðÞ, K e = K o + K f

ð7 : 148Þ

F f ðÞ= K f x d ðÞ,

where K f is the d × d stiffness matrix that is a function of the gravity loads on the leaning col-
umn and the corresponding story height, K o is the d × d initial stiffness matrix of the frame
computed by using the gravity loads on the frame only, K g ðÞis the change in stiffness matrix
due to the change in axial load on members during the dynamic loading, and K e is the d × d
elastic stiffness matrix of the structure. Let

< =

, A = ,

g x ðÞ
g y ðÞ
g z ðÞ

x d ðÞ
x d ðÞ

0 I

z ðÞ= g ðÞ= ha ðÞ= h ð7 : 149aÞ

− M d −1

K e

− M d −1 C dd

: ;

, ,

0
− h

0
M d −1

f g ðÞ= K g ðÞ x d ðÞ, f m ðÞ= K ðÞ x 0 d H = B = ðÞ ð7 : 149bÞ

F d = e A Δ t , H d = e A Δ t H Δ t , B d = e A Δ t B Δ t

ð7 : 149cÞ

where Δ t is the time step size. Then, it follows from Eq. (7.119) that

z k +1 = F d z k + H d a k + B d f g , k + B d f m , k ð7 : 150Þ

where z k , a k , f g , k , and f m , k are the discretized forms of z ( t ), a ( t ), f g ( t ), and f m ( t ), respectively.
The discretized forms of

the governing equations of the FAM follow directly from

Eqs. (7.125) and (7.126), which are
m k +1 + K 0 k Δ Θ 00 = K 0 k x d , k +1 − K 0 k Θ 0 k ð7 : 151Þ

x 0 d , k +1 = K −1

K 0 k Θ 0 k +1

ð7 : 152Þ

k

where m k , Θ 0 k , x d , k , and x 0 d , k are the discretized forms of m ( t ), Θ 00 ( t ), x d ( t ), and x 0 d ðÞ, respec-
tively, Δ Θ 00 = Θ 0 k +1 − Θ 0 k , and K k , K 0 k , and K 0 k are the condensed geometric nonlinear stiffness
matrices at time step k . The displacement vector associated with the DOFs with zero mass
moment of inertia (i.e. x r ( t )) can be written using Eq. (3.130) as:

x r , k = − K −1

rr K rd x d , k + K −1 rr K 0 r Θ 0 k

ð7 : 153Þ

where x r , k is the discretized form of x r ( t ).
If updates to geometric nonlinearity are ignored, then the condensed geometric nonlinear
stiffness matrices become:

K k = K t ðÞ= K ðÞ= K o , K 0 k = K 0 t ðÞ= K 0 ðÞ= K 0 o

ð7 : 154aÞ

K 0 k = K 00 t ðÞ= K 00 ðÞ= K 0 o , K g ðÞ= 0

ð7 : 154bÞ

According to Eq. (7.149b), this gives

f m , k = K o x 0 k f g , k = ½ x d , k = 0 , ð7 : 155Þ