Civil Engineering Reference
In-Depth Information
leading as usual to the formation of [
B
]
=
[
A
][
S
]. The elastic stress-strain matrix in three
dimensions is given by,
ν
ν
1
0
0
0
1
−
ν
1
−
ν
ν
ν
1
0
0
0
−
ν
−
ν
1
1
ν
ν
1
0
0
0
1
−
ν
1
−
ν
E(
1
−
ν)
[
D
]
=
−
1
2
ν
(
1
+
ν)(
1
−
2
ν)
0
0
0
0
0
2
(
1
−
ν)
1
−
2
ν
0
0
0
0
0
−
ν)
2
(
1
1
−
2
ν
0
0
0
0
0
2
(
1
−
ν)
(2.84)
2.14 Plate-bending element
The bending of a thin plate is governed by the equation,
4
D
∇
w
=
q
(2.85)
4
where
∇
is the bi-harmonic operator,
D
is the flexural rigidity of the plate, given by,
3
Eh
D
=
(2.86)
2
12
(
1
−
ν
)
w
is the deflection in the transverse
z
-direction,
q
is a applied transverse distributed load,
and
h
is the plate thickness.
Solution of (2.85) directly, for example by Galerkin's method, appears to imply that
for a fixed
D
, a thin plate's deflection is unaffected by the value of Poisson's ratio. This
is in fact only true for certain boundary conditions and, in general, the integration by parts
in the Galerkin process will supply extra terms that are dependent on
ν
.
This is a case in which the energy approach provides a simpler formulation. The
strain energy in a piece of bent plate is given by (Timoshenko and Woinowsky-Krieger,
1959),
∂
∂
2
d
x
d
y
2
∂
2
2
2
2
2
1
2
D
w
∂x
∂
w
∂y
w
∂x
2
∂
w
∂y
w
∂x∂y
U
=
+
−
2
(
1
−
ν)
−
(2.87)
2
2
2
or
∂
2
d
x
d
y
2
∂
2
∂
2
2
2
2
2
1
2
D
w
∂x
w
∂y
2
ν
∂
w
∂x
2
∂
w
∂y
w
∂x∂y
U
=
+
+
+
2
(
1
−
ν)
2
2
2
(2.88)
