Civil Engineering Reference
In-Depth Information
(
1, 1)
(1, 1)
4
3
s
2
b
r
1
2
(
1,
1)
(1,
1)
2
a
Figure 9.5
A rectangular
element of width 2
a
and
height 2
b
.
with the natural coordinates defined as in Figure 9.5. To compute the strain com-
ponents in terms of the natural coordinates, the chain rule is applied to obtain
∂
∂
∂
∂
∂
r
1
a
∂
∂
x
=
x
=
r
∂
r
(9.61)
∂
∂
∂
∂
s
∂
s
1
b
∂
∂
y
=
y
=
∂
s
Performing the indicated differentiations, the strain components are found to be
ε
x
ε
y
xy
{
ε
} =
u
1
u
2
u
3
u
4
v
1
v
2
v
3
v
4
s
−
1
4
a
1
−
s
4
a
1
+
s
4
a
1
+
s
4
a
−
0
0
0
0
r
−
1
4
b
1
+
r
4
b
1
+
r
4
b
1
−
r
4
b
=
0
0
0
0
−
r
−
1
1
+
r
1
+
r
1
−
r
s
−
1
1
−
s
1
+
s
1
+
s
−
−
4
b
4
b
4
b
4
b
4
a
4
a
4
a
4
a
(9.62)
showing that the normal strain
ε
x
varies linearly in the
y
direction, normal strain
ε
y
varies linearly in the
x
direction, and shear strain
xy
varies linearly in both
coordinate directions (realizing that the natural coordinate
r
corresponds to the
x
axis and natural coordinate
s
corresponds to the
y
axis).
From Equation 9.62, the
[
B
]
matrix is readily identified as
s
−
1
1
−
s
1
+
s
1
+
s
−
0
0
0
0
4
a
4
a
4
a
4
a
r
−
1
1
+
r
1
+
r
1
−
r
[
B
]
=
0
0
0
0
−
4
b
4
b
4
b
4
b
r
−
1
1
+
r
1
+
r
1
−
r
s
−
1
1
−
s
1
+
s
1
+
s
−
−
4
b
4
b
4
b
4
b
4
a
4
a
4
a
4
a
(9.63)
