Civil Engineering Reference
In-Depth Information
§ 4. Membranes
In solving three-dimensional problems, it is often possible to work in two (or even
one) dimensions since the length of the domain in one of more space directions
is very small. In such cases it is useful to consider the problem for the lower-
dimensional continuum, and then discretize afterwards. Some typical examples
are bars, beams, membranes, plates, and shells. The simplest two-dimensional ex-
ample is the membrane. However, this example already shows that the reduction in
dimension cannot be accomplished by simply eliminating one coordinate. More-
over, the boundary conditions have some influence on the reduction process. There
are two cases depending on the boundary condition.
Plane Stress States
2
be a domain, and suppose
t>
0 is a number which is significantly
smaller than the diameter of
ω
. We suppose that there are only external forces
operating on the body
=
ω
×
(
−
Let
ω
⊂ R
t
t
2
)
, and that their
z
-components vanish so
that they depend only on
x
and
y
.
20
If the membrane is thin and a deformation in
the
z
-direction is possible, we have the so-called
plane stress state
, i.e.,
2
,
+
σ
ij
(x,y,z)
=
σ
ij
(x, y),
i, j
=
1
,
2
,
(
4
.
1
)
σ
i
3
=
σ
3
i
=
=
0
,
i
1
,
2
,
3
.
Then in particular
ε
i
3
=
ε
3
i
=
0
for
i
=
1
,
2. In order for
σ
33
=
0, by the
constitutive equations (3.6) we have
ν
ε
33
=−
−
ν
(ε
11
+
ε
22
).
(
4
.
2
)
1
If we now eliminate the strain
ε
33
, we get the constitutive equations for the plane
stress state:
σ
11
σ
22
σ
12
1
ν
0
ν
10
001
ε
11
ε
22
ε
12
E
=
−
ν
2
1
−
ν
or
ε
+
−
ν
(ε
11
+
ε
22
)I
.
ν
E
σ
=
(
4
.
3
)
1
+
ν
1
20
It is easier to visualize the situation if we exchange the
y
- and
z
-coordinates. Suppose
the middle surface of a thin wall lies in the
(x, y)
-plane, and that the displacement in the
y
-direction is very small. Now suppose the wall is subject to a load in the vertical direction
so that the external forces operate in a direction parallel to the (middle surface of the) wall.
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