Civil Engineering Reference
In-Depth Information
To integrate Eq. (7.114) numerically, let
t
k
+1
=
t
,
t
k
=
t
o
, and
Δ
t
=
t
k
+1
−
t
k
, and the subscript
k
denotes the
k
th time step, then it follows from Eq. (7.114) that
z
k
+1
=
e
A
Δ
t
z
k
+
e
A
t
k
+1
ð
t
k
+1
t
k
ds
e
−
A
s
Ha
ðÞ+
Bf
g
ðÞ+
Bf
m
ðÞ
ð7
:
115Þ
Using the delta function approximation for the variables in the integral, where the ground accel-
eration vector
a
(
s
), equivalent geometric nonlinear force vector
f
g
(
s
), and equivalent material
nonlinear force vector
f
m
(
s
) take the form:
a
ðÞ=
a
k
δ
s
−
t
ð Þ
Δ
t
,
t
k
≤
s
<
t
k
+1
ð7
:
116aÞ
f
g
ðÞ=
f
g
,
k
δ
s
−
t
ð Þ
Δ
t
,
t
k
≤
s
<
t
k
+1
ð7
:
116bÞ
f
m
ðÞ=
f
m
,
k
δ
s
−
t
ð Þ
Δ
t
,
t
k
≤
s
<
t
k
+1
ð7
:
116cÞ
Substituting Eq. (7.116) into Eq. (7.115) and performing the integration gives
z
k
+1
=
e
A
Δ
t
z
k
+
Δ
t
e
A
Δ
t
Ha
k
+
Δ
t
e
A
Δ
t
Bf
g
,
k
+
Δ
t
e
A
Δ
t
Bf
m
,
k
ð7
:
117Þ
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. Let
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
:
118Þ
Then, Eq. (7.117) becomes
z
k
+1
=
F
d
z
k
+
H
d
a
k
+
B
d
f
g
,
k
+
B
d
f
m
,
k
ð7
:
119Þ
Equation (7.119) represents the recursive equation for calculating the dynamic response of
moment-resisting framed structures while considering updates on geometric nonlinearity as the
axial compressive force in columns changes with time.
In Examples 7.2 and 7.3, it has been demonstrated that ignoring updates on geometric non-
linearity results in minor differences in the responses. Therefore, for the case when updates
on geometric nonlinearity are ignored in the nonlinear dynamic analysis, the
K
g
(
t
) matrix as
given in Eq. (7.103) becomes
K
g
(
t
)=
0
. Therefore, from the same equation,
K
(
t
)=
K
o
. Then,
it follows from Eq. (7.112b) that
f
g
ðÞ= ½
x
ðÞ=
0
ð7
:
120aÞ
f
m
ðÞ=
K
o
x
00
ðÞ
ð7
:
120bÞ
and Eq. (7.119) becomes
z
k
+1
=
F
d
z
k
+
H
d
a
k
+
G
d
x
0
k
ð7
:
121Þ
