Civil Engineering Reference
In-Depth Information
where
K
is the initial stiffness matrix of the MDOF system, and
K
0
means the factor matrix
relating rotations with horizontal forces;
K
0T
means the transposed matrix of
K
0
;
K
00
denotes
the coefficient matrix that reflects the relationship between moments and rotations;
m
s,i
represents plastic rotation at the
i
th PHL.
Define
X
ðÞ
=
Φ
Y
ðÞ
ð8
:
82Þ
X
00
ðÞ=
Φ
Y
00
ðÞ
ð8
:
83Þ
where
Φ
means the modal matrix of the MDOF system and
Y(t)
denotes a vector of normal
coordinate. Substitute Eqs. (8.82) and (8.83) into Eq. (8.78), and multiply
Φ
T
on the left hand of
each item of Eq. (8.78), then Eq. (8.78) can be decoupled to
n
different separated equations. The
j
th modal equation of motion can be written as
Y
j
ðÞ+2
ς
j
ω
j
Y
j
ðÞ+
ω
j
Y
j
00
2
2
j
Y
j
ðÞ= −
γ
j
x
g
ðÞ+
ω
ðÞ
ð8
:
84Þ
in which
j
M
Φ
j
;
K
j
=
Φ
j
M
Φ
j
;
C
j
=
Φ
j
C
Φ
j
M
j
=
Φ
j
=
K
j
M
j
C
j
2
ω
j
M
j
γ
j
=
Ml
M
j
2
ω
;
ς
j
=
;
Although Eq. (8.78) can be decoupled to
n
different equations, Eqs. (8.79) and Eq. (8.80) still
cannot be decoupled through the simple modal decomposition due to the presence of plastic
rotation. A suitable method for establishing an equivalent SDOF system that is corresponding
to the
j
th modal SDOF system can be established through the NSPA procedure presented in
section 8.1. Detailed process are addressed as follows:
Step 1
: Define that the lateral load pattern is associated with the
j
th modal shape, as shown in
Figure 8.9(a) and written as
h
i
ðÞ
M
dd
s
j
ðÞ
M
dd
s
j
F
j
=
ðÞ
M
dd
s
j
λ
λ
ð8
:
85Þ
λ
Step 2
: Perform the NSPA procedure through Eqs. (8.41) to (8.46) and obtain a curve reflect-
ing the relationship of the base shear and roof displacement, as shown in Figure 8.9(b).
Step 3
: Assume the
j
th modal SDOF with unite height and transfer the relation of base shear
and roof displacement to the moment and displacement of the
j
th SDOF system through
Ω
j
=
V
j
H
=
V
j
ð8
:
86Þ
Y
j
=
X
l
+1
ð
Þ
j
,
n
γ
j
s
n
,
j
ð8
:
87Þ
