Civil Engineering Reference
In-Depth Information
The strains in the element are
.
/
!
ε
x
ε
y
γ
xy
{
ε
}=
(17.74)
0
"
Direct and shear strains may be defined in the form
∂
w
∂
x
∂v
∂
y
∂
w
∂
y
∂v
∂
x
ε
x
=
ε
y
=
γ
xy
=
+
(17.75)
Substituting for
w
and
v
in Eq. (17.75) from Eq. (17.69) gives
ε
x
=
α
2
ε
y
=
α
6
γ
xy
=
α
3
+
α
5
or in matrix form
.
/
0
!
"
α
1
α
2
α
3
α
4
α
5
α
6
010000
000001
001010
{
ε
}=
(17.76)
which is of the form
{
ε
}=
[
C
]
{
α
}
(see Eqs (17.51) and (17.52))
[
A
−
1
]
δ
e
Substituting for
{
α
}
(
=
{
}
) we obtain
[
C
][
A
−
1
]
{
δ
e
{
ε
}=
}
(compare with Eq. (17.53))
or
δ
e
{
ε
}=
[
B
]
{
}
(see Eq. (17.63))
where [
C
] is defined in Eq. (17.76).
In step five we relate the internal stresses
{
σ
}
to the strain
{
ε
}
and hence, using step
δ
e
four, to the nodal displacements
{
}
. For plane stress problems
.
/
!
σ
x
σ
y
τ
xy
{
σ
}=
(17.77)
0
"
and
!
σ
x
E
−
νσ
y
E
ε
x
=
σ
y
E
−
νσ
x
E
ε
y
=
(see Chapter 7)
"
τ
xy
G
=
2(1
+
ν
)
γ
xy
=
τ
xy
E
