Civil Engineering Reference
In-Depth Information
T
⎡
Cov
⎤
⎡
(
) (
)
⎤
Ar
⋅
Ar
⎡ ⎤
T
⎡ ⎤
T
d
nn
dd
⋅
rr
⋅
n
n
Cov
⎢
⎥
⎡
⎤
⎢
⎥
⎢ ⎥
⎢ ⎥
rr
Cov
T
⎢
⎥
⎢
(
) (
)
⎥
⎢
⎥
T
T
Ar
⋅
Ar
⎢ ⎥
dd
⋅
⎢ ⎥
r
⋅
r
d
nn
Cov
AA
n
n
⎢
⎥
⎢
⎥
⎢
⎥
rr
⎢ ⎥
⎢ ⎥
Cov
T
T
T
⎢
⎥
⎢
⎥
⎢
⎥
=
E
T
=
E
(
) (
)
=
A
⋅
E
T
⋅
=⋅
Cov
⋅
A
dd
⋅
Ar
⋅
Ar
rr
⋅
⎢ ⎥
⎢ ⎥
dd
n
n
n
rr
n
nn
n
n
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ⎥
⎢ ⎥
Cov
C
T
T
T
ov
⎢
(
) (
)
⎥
⎢
⎥
⎢
⎥
dd
⋅
⋅
⎢ ⎥
rr
Ar
⋅
Ar
rr
⎢ ⎥
d
nn
n
n
⎢
⎥
⎢
⎥
⎢
⎥
Cov
⎢ ⎥
⋅
⎢ ⎥
⎣
⎦
T
T
Cov
dd
T
rr
rr
⎢
(
) (
)
⎥
⋅
⎢
⎥ ⎣ ⎦
⎣ ⎦
Ar
⋅
Ar
⎣
dd
⎦
n
⎣
⎦
nn
n
n
(9.129)
By introducing the relevant expressions in Eq. 9.127 the following is obtained
⎡
Cov
Cov
⎤
1
⎛
⎞
d
nn
⎡ ⎤
⎢
⎥
⎜
⎟
⎢ ⎥
⎢
⎥
2
−
ω
d
nn
⎜
⎟
⎢ ⎥
⎢
⎥
∞
⎜
⎟
⎢ ⎥
Cov
2
()
T
∫
⎢
⎥
=⋅
A
⋅
S
ωω
d
⋅
A
(9.130)
−
ω
ω
ω
dd
⎜
⎟
n
⎢ ⎥
rr
n
nn
⎢
⎥
⎜
⎟
⎢ ⎥
0
2
Cov
⎢
⎥
d
nn
⎜
⎟
⎢ ⎥
⎢
⎥
⎜
⎟
⎢
⎣ ⎦
4
Cov
⎢
⎥
⎝
⎠
⎣
d
nn
⎦
The corresponding response force vector
⎧
[
]
T
n
F
=
FF F F F F
F
⎡⎤
⎪
1
1
2
3
4
5
6
()
1
n
F
t
=
where
(9.131)
⎨
⎢
⎣⎦
n
F
T
[
]
F
=
FFFF F F
⎪
⎩
2
n
2
7
8
9
10
11
12
n
n
associated with element number
n
is defined by the local element dynamic equilibrium
condition (see Eq. 9.25)
Fmdcdkd
=
+
+
(9.132)
n
n
n
n
n
n
n
and thus
T
⎡
(
) (
)
⎤
T
⎡
⎤
E
E
v
=
FF
⋅
=
mdcdkd mdcdkd
+
+
⋅
+
+
FF
⎣
n
n
⎦ ⎢
nn
nn
nn
nn
nn
nn
⎥
nn
⎣
⎦
T
T
T
T
T
T
E
⎡
⎤
E
⎡
⎤
E
⎡
⎤
=⋅
md
⋅
d
⋅
mc
+⋅
d
⋅
d
⋅
c
+⋅
k
d
⋅
d
⋅
k
n
⎣
n
n
⎦
n
n
⎣
n
n
⎦
n
n
⎣
n
n
⎦
n
T
T
T
T
⎡
⎤
⎡
⎤
E
E
+⋅
k
d
⋅
d
⋅
mmd
+ ⋅
⋅
d
⋅
k
n
⎣
n
n
⎦
n
n
⎣
n
n
⎦
n
(9.133)
