Civil Engineering Reference
In-Depth Information
y
y
F
2
x
z
F
1
f
(
x
,
y
)
t
Figure 9.1
An illustration of plane stress
conditions.
or
{}=
[
D
]
{
ε
}
(9.4)
where
x
y
xy
{} =
(9.5)
is the column matrix of stress components,
1
0
E
1
0
[
D
]
=
(9.6)
1
−
2
00
1
−
2
is the elastic material property matrix for plane stress, and
ε
x
ε
y
xy
{
ε
} =
(9.7)
is the column matrix of strain components.
For a state of plane stress, the strain energy per unit volume, Equation 2.43,
becomes
1
2
(
u
e
=
x
ε
x
+
y
ε
y
+
xy
xy
)
(9.8)
or, using the matrix notation,
1
2
{
ε
}
1
2
{
ε
}
T
T
[
D
]
u
e
=
{}=
{
ε
}
(9.9)
T
allows the matrix operation to reproduce the quadratic form of the
strain energy. Note that a quadratic relation in any variable
z
can be expressed
Use of
{
ε
}
