Civil Engineering Reference
In-Depth Information
cNN
0
0
0
0
0
⎡
⎤
x
17
⎢
⎥
0
cNN
0
0
0
cNN
⎢
y
28
y
2 2
⎥
⎢
⎥
L
0
0
cNN
0
cNN
0
z
39
z
3 1
∫
c
=
⎢
⎢
⎥
dx
12
0
0
0
cNN
0
0
⎥
θ
4 0
0
⎢
⎥
0
0
cNN
0
cNN
0
⎢
z
59
z
5 1
⎥
⎢
⎥
0
cNN
0
0
0
cNN
⎣
⎦
y
68
y
6 2
(9.31)
70
c
0
0
0
0
0
⎡
⎤
x
⎢
⎥
0 4
c
0
0
0
3
c L
−
⎢
y
y
⎥
⎢
⎥
0
0
54
c
0
13
c L
0
L
z
z
⎢
⎥
=
0
0
0
70
c
0
0
⎢
⎥
420
θ
⎢
⎥
2
0
0
13
cL
0
3
cL
0
−
−
⎢
⎥
z
z
⎢
⎥
2
0 3
cL
0
0
0
−
3
cL
⎣
⎦
y
y
2
7
⎡
⎤
cN
0
0
0
0
0
x
⎢
⎥
2
8
0
cN
0
0
0
cNN
⎢
⎥
y
y
8
12
⎢
⎥
L
2
9
0
0
cN
0
cNN
0
⎢
⎥
z
z
9
11
∫
c
=
dx
⎢
⎥
22
2
0
0
0
cN
0
0
⎢
⎥
0
θ
10
⎢
⎥
2
0
0
cN N
0
cN
0
⎢
⎥
z
11
9
z
11
⎢
⎥
2
0
cN N
0
0
0
cN
⎣
⎦
y
12
8
y
12
(9.32)
140
c
0
0
0
0
0
⎡
⎤
x
⎢
⎥
0
156
c
0
0
0
22
c L
−
⎢
y
y
⎥
⎢
⎥
0
0
156
c
0
22
c L
0
L
z
z
⎢
⎥
=
0
0
0
140
c
0
0
⎢
⎥
420
θ
⎢
⎥
2
0
0
22
cL
0
4
cL
0
⎢
⎥
z
z
⎢
⎥
2
0
22
cL
0
0
0
4
cL
−
⎣
⎦
y
y
Similarly, the element mass matrix is given by:
mm
⎡
⎤
11
12
T
mm
m
=
mm
where
(9.33)
⎢
⎥
21
12
⎣
⎦
21
22
The necessary integrations will involve the same combinations of shape functions as
those that have been applied to obtain the damping matrix, and thus
