Civil Engineering Reference
In-Depth Information
3
(2.25, 1.5)
4
(1.25, 1)
y
1
(1, 0)
2
(2, 0)
x
Figure 9.7
Dimensions are in inches.
Axes are shown for orientation only.
■
Solution
The mapping functions are
1
4
[(1
−
r
)(1
−
s
)(1)
+
(1
+
r
)(1
−
s
)(2)
+
(1
+
r
)(1
+
s
)(2
.
25)
+
(1
−
r
)(1
+
s
)(1
.
25)]
x
(
r
,
s
)
=
1
4
[(1
−
r
)(1
−
s
)(0)
+
(1
+
r
)(1
−
s
)(0)
+
(1
+
r
)(1
+
s
)(1
.
5)
+
(1
−
r
)(1
+
s
)(1)]
y
(
r
,
s
)
=
and the terms of the Jacobian matrix are
∂
x
∂
r
=
1
2
=
J
11
∂
y
∂
r
=
1
4
(0
J
12
=
.
5
−
0
.
5
s
)
∂
x
∂
s
=
1
2
J
21
=
∂
y
∂
s
=
1
4
(2
.
5
−
0
.
5
r
)
J
22
=
and the determinant is
1
16
(4
−
r
+
s
)
|
J
| =
J
11
J
22
−
J
12
J
21
=
Therefore, the geometric matrix
[
G
]
of Equation 9.79 is known in terms of ratios of
monomials in
r
and
s
as
2
.
5
−
0
.
5
r
−
(0
.
5
−
0
.
5
s
)
0
0
4
4
−
r
+
s
[
G
]
=
0
0
−
2
2
−
2
2
2
.
5
−
0
.
5
r
−
(0
.
5
−
0
.
5
s
)
For plane stress with the values given, the material property matrix is
[
D
]
=
32
.
97(10)
6
psi
30
0
.
31 0
00
10
.
.
35
