Civil Engineering Reference
In-Depth Information
Finally, the kinematics are compatible with (4.1) and (4.2) provided
u
i
(x,y,z)
=
u
i
(x, y),
i
=
1
,
2
,
u
3
(x,y,z)
=
z
·
ε
33
(x, y)
O
(z)
are neglected in the strain term.
and terms of order
Plane Strain States
If boundary conditions are enforced at
z
=±
t/
2 which ensure that the
z
-com-
ponent of the displacement vanishes, then we have the
plane strain state
:
ε
ij
(x,y,z)
=
ε
ij
(x, y),
i, j
=
1
,
2
,
(
4
.
4
)
ε
i
3
=
ε
3
i
=
=
0
,
i
1
,
2
,
3
.
The associated displacements satisfy
u
i
(x,y,z)
=
u
i
(x, y)
for
i
=
1
,
2, and
u
3
=
0. It follows from
ε
33
=
0 along with (3.7) that
σ
33
=
ν(σ
11
+
σ
22
),
(
4
.
5
)
and
σ
33
can be eliminated. We obtain the constitutive equations for the plane strain
state if we restrict (3.6) to the remaining components:
σ
11
σ
22
σ
12
1
ε
11
ε
22
ε
12
.
−
ν
ν
0
E
=
ν
1
−
ν
0
(
4
.
6
)
(
1
+
ν)(
1
−
2
ν)
0
0
1
−
2
ν
Membrane Elements
Both plane elasticity problems lead to a two-dimensional problem with the same
structure as the full three-dimensional elasticity problem.
The displacement model thus involves the (two-dimensional and isopara-
metric versions of the) conforming elements which also play a role for scalar
elliptic problems of second order:
(a) bilinear quadrilateral elements,
(b) quadratic triangular elements,
(c) biquadratic quadrilateral elements,
(d) eight-node quadrilateral elements in the serendipity class.
On the other hand, the simplest linear triangular elements are frequently unsat-
isfactory. For practical problems, there are often preferred directions because of
certain geometric relationships. In this case, higher order elements or quadrilateral
elements prove to be more flexible.
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