Civil Engineering Reference
In-Depth Information
where
Q
i
is the axial force due to gravity on the leaning column of the
i
th floor, and
h
i
is the
story height of the
i
th floor.
While the
K
f
matrix takes care of the geometric nonlinear effects from all the gravity col-
umns, the stiffness matrix
K
(
t
) in Eq. (7.100) must consider both large
P
-
Δ
and small
P
-
δ
effects for the moment-resisting frame itself. Let this time-dependent global stiffness matrix
K
(
t
) be represented in the form:
K
ðÞ=
K
o
+
K
g
ðÞ
ð7
:
103Þ
where
K
o
denotes the initial stiffness of the frame due to the gravity loads only, and
K
g
(
t
)
denotes the change in stiffness due to the change in axial load on members during the dynamic
loading. Note that the
K
o
matrix is a constant stiffness matrix computed by using only the grav-
ity loads on the columns (which means
K
o
=
K
(
t
0
)=
K
(0), the stiffness matrix computed at time
step 0) and is not a function of time.
First, solving for the elastic displacement
x
0
(
t
) in Eq. (7.94) and substituting the result into
Eq. (7.100) gives
ðÞ+
K
ðÞ
x
00
ðÞ
Mx
ðÞ+
Cx
ðÞ+
K
ðÞ
x
ðÞ= −
Mg
ðÞ−
F
f
ð7
:
104Þ
Then substituting Eqs. (7.101) and (7.103) into Eq. (7.104), the equation of motion after con-
sidering both
P
-
δ
and
P
-
Δ
effects of geometric nonlinearity of the entire structure becomes
Mx
ðÞ+
Cx
ðÞ+
K
o
x
ðÞ= −
Mg
ðÞ−
K
f
x
ðÞ−
K
g
ðÞ
x
ðÞ+
K
ðÞ
x
00
ðÞ
ð7
:
105Þ
Define
K
e
=
K
o
+
K
f
ð7
:
106Þ
where
K
e
represents the elastic stiffness of the structure. With a negative definite
K
f
matrix as
shown in Eq. (7.102), it can be seen from Eq. (7.106) that det(
K
e
) < det(
K
o
). Therefore, the
natural periods of vibration of the structure based on
K
e
(i.e. reduced stiffness) are always
longer than those calculated based on
K
o
(i.e. initial stiffness). Finally, on substituting
Eq. (7.106) into Eq. (7.105), it follows that
Mx
ðÞ+
Cx
ðÞ+
K
e
x
ðÞ= −
Mg
ðÞ−
K
g
ðÞ
x
ðÞ+
K
ðÞ
x
00
ðÞ
ð7
:
107Þ
An alternate equation of Eq. (7.107) can be written by first pre-multiplying Eq. (7.99) by the
stiffness matrix
K
(
t
) and then substituting the result into the last term of Eq. (7.107). This gives
Mx
ðÞ+
Cx
ðÞ+
K
e
x
ðÞ= −
Mg
ðÞ−
K
g
ðÞ
x
ðÞ+
K
0
ðÞ
Θ
00
ðÞ
ð7
:
108Þ
7.4.3 State Space Analysis with Geometric and Material Nonlinearities
Let the material nonlinearity term (i.e.
K
(
t
)
x
00
(
t
)) and geometric nonlinearity term
(i.e. −
K
g
(
t
)
x
(
t
)) shown on the right-hand side of Eq. (7.107) be treated as the equivalent
