Civil Engineering Reference
In-Depth Information
where
σ
x
,
σ
y
and
τ
xy
are the only non-zero stress components and
F
x
,
F
y
are “body forces”
(units of force/length
3
).
2. Constitutive (plane stress)
=
1
ν
0
ν
10
00
1
σ
x
σ
y
τ
xy
x
y
γ
xy
E
(2.51)
2
1
−
ν
−
ν
2
where
E
is Young's modulus,
ν
is Poisson's ratio, and
x
,
y
and
γ
xy
are the independent
small strain components.
3. Strain-displacement
∂
∂x
0
=
u
v
x
y
γ
xy
∂
∂y
0
(2.52)
∂
∂y
∂
∂x
where
u
and
v
are the components of displacement in the
x
and
y
directions.
Equations (2.50) to (2.52) can be written in the form
[
A
]
T
{
σ
} =−{
f
}
{
σ
} =
[
D
]
{
}
(2.53)
{
} =
[
A
]
{
e
}
where
u
v
F
x
F
y
σ
x
σ
y
τ
xy
x
y
γ
xy
{
σ
} =
,
{
} =
,
{
e
} =
,
{
f
} =
(2.54)
∂
∂x
0
1
ν
0
ν
10
00
1
E
∂
∂y
[
A
]
=
0
,
[
D
]
=
(2.55)
2
1
−
ν
−
ν
2
∂
∂y
∂
∂x
We shall only be concerned in this topic with “displacement” formulations in which
{
σ
}
and
{
}
are eliminated from (2.53) as follows:
[
A
]
T
{
σ
} =−{
f
}
[
A
]
T
[
D
]
{
} =−{
f
}
(2.56)
[
A
]
T
[
D
][
A
]
{
e
} =−{
f
}
