Civil Engineering Reference
In-Depth Information
forces applied to the structure, the equation can now be solved using the state space method.
To represent Eq. (7.107) in state space form, let the state vector
z
(
t
) be defined as
x
ðÞ
x
ðÞ
z
ðÞ=
ð7
:
109Þ
which is a 2
n
× 1 vector with a collection of states of the responses. It follows from
Eq. (7.107) that
=
x
ðÞ
x
t
+
a
ðÞ+
M
−1
x
ðÞ
x
t
0
I
0
−
h
0
−
K
g
ðÞ
x
ðÞ+
K
ðÞ
x
00
ðÞ
z
ðÞ=
−
M
−1
K
e
−
M
−1
C
ðÞ
ðÞ
ð7
:
110Þ
where
h
is an
n
× 3 matrix that relates the directions of each DOF with the global
x
-,
y
-, and
z
-directions (i.e. a collection of 0 s and 1 s in all entries), and
a
(
t
) is the 3 × 1 ground acceleration
vector in the three global directions of
g
x
ðÞ,
g
y
ðÞ, and
g
z
ðÞ. The relationship between the
ground acceleration vector
g
ðÞfor each DOF in Eq. (7.108) and the three-component ground
acceleration vector
a
(
t
) can be expressed as
<
=
g
x
ðÞ
g
y
ðÞ
g
z
ðÞ
g
ðÞ=
ha
ðÞ=
h
ð7
:
111Þ
:
;
To simplify Eq. (7.110), let
,
,
0
I
0
−
h
0
−
M
−1
ð7
:
112aÞ
A
=
H
=
B
=
−
M
−1
K
e
−
M
−1
C
f
m
ðÞ=
K
ðÞ
x
00
ðÞ
f
g
ðÞ= −
K
g
ðÞ
x
ðÞ,
ð7
:
112bÞ
where
A
is the 2
n
×2
n
state transition matrix in the continuous form,
H
is the 2
n
× 3 ground
motion transition matrix in the continuous form,
B
is the 2
n
×
n
geometric nonlinearity transi-
tion matrix in the continuous form,
f
g
(
t
) is the
n
× 1 equivalent force vector due to geometric
nonlinearity, and
f
m
(
t
) is the
n
× 1 equivalent force vector due to material nonlinearity. Then,
Eq. (7.110) becomes
z
ðÞ=
Az
ðÞ+
Ha
ðÞ+
Bf
g
ðÞ+
Bf
m
ðÞ
ð7
:
113Þ
Solving for the first-order linear differential equation in Eq. (7.113) gives
z
ðÞ=
e
A
t
−
t
ð Þ
z
t
ðÞ+
e
A
t
ð
t
t
o
ds
e
−
A
s
Ha
ðÞ+
Bf
g
ðÞ+
Bf
m
ðÞ
ð7
:
114Þ
where
t
o
is the time of reference when the integration begins, which is typically the time when
the states
z
(
t
o
) are known, such as the initial conditions.
