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where the appropriate entry of B T EB of Eq. (6.32 ) has been employed. Integration first
over d
ξ
yields
d
1
1
2
1
3
2
Eabt
1
2
η + η
1
ν
2
k i 12 =
+
η
(
1
ν
2
)
a 2
b 2
0
12 b 2 d
1
η + η
2
ν
Eabt
1
2
1
=
+
η
(
ν
2
)
a 2
1
0
Now integrate over d
η
, giving
12 b 2
1
3
1
1
+
Eabt
1
ν
12 b 2
Eabt
1
3 a 2 +
1
ν
k i 12 =
+
=
(
1
ν
2
)
a 2
(
1
ν
2
)
Finally, we find
Et
8 b
a
b
Et
k i 12 =
a +
2
(
1
ν)
=
8
α +
2
β(
1
ν)
(6.33)
[
]
24
(
1
ν
2
)
24
(
1
ν
2
)
where
b
The other stiffness coefficients are obtained in the same fashion.
For this case of rectangular elements with eight DOF, closed form integration can be
performed. In general, if more nodes are introduced to the rectangular elements or for
elements for other shapes with many DOF, numerical integration is usually employed.
Now, for the element of Fig. 6.10 put together all 64 coefficients using the symmetry of
the stiffness matrix.
α =
b
/
a,
β =
a
/
Et
k i v i
=
24
(
1
ν
2
)
C αβ
A αβ /
ν
ν
ν
2
ν
3
A αβ
2
B αβ
u x 1
u x 2
u x 3
u x 4
u y 1
u y 2
u y 3
u y 4
2
3
A αβ
B αβ
A αβ /
2
ν
ν
ν
3
ν
2
3
2
A
C
ν
ν
ν
ν
3
αβ
αβ
2
3
2
A
ν 3
ν 2
ν 3
ν 2
αβ
×
(6.34)
A
B
βα
A
βα /
2
C
βα
βα
Symmetric
A
C
A
βα /
2
βα
βα
A
B
βα
βα
A
βα
where
α =
b
/
a,
β =
a
/
b,
ν 1 =
1
ν
,
ν 2 =
3
(
1
+ ν)
,
ν 3 =
3
(
1
3
ν)
,A
αβ =
8
α +
4
βν 1 ,B
αβ =
4
α
4
βν 1 ,C
αβ =−
8
α +
2
βν 1 , and A
,B
,C
are obtained by interchanging
α
and
β
βα
βα
βα
in A
, respectively.
We have now completed the formation of the stiffness matrix for a single element.
The element stiffness matrix of Eq. (6.34) is arranged such that the first four displace-
ments correspond to x direction displacements, while the second four are in the y direction.
For computationally assembling the global stiffness matrix, it is more convenient to place
together the displacements for each node. Such a rearrangement will be employed for many
of the subsequent numerical calculations in this chapter.
,B
,C
αβ
αβ
αβ
Formation of the Element Loading Vector
The surface integral III of Eq. (6.21) remains to be evaluated. The integration is carried out
element by element. In terms of the assumed displacements of Eq. (6.19), integral III can be
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