Section 6.5
Solid Mechanics Part II
Kelly
149
6.5 Plate Problems in Rectangular Coordinates
In this section, a number of important plate problems will be examined using Cartesian
coordinates.
6.5.1
Uniform Pressure producing Bending in One Direction
Consider first the case of a plate which bends in one direction only.
From 6.3.11 the
deflection and moments are
2
2
2
2
)
(
,
)
(
),
(
dx
w
d
D
x
M
dx
w
d
D
x
M
x
f
w
y
x
ν
−
=
=
=
(6.5.1)
The differential equation 6.4.9 reads
D
x
q
dx
w
d
)
(
4
4
−
=
(6.5.2)
The corresponding equation for a beam is
EI
x
p
dx
w
d
/
)
(
/
4
4
=
.
If
)
(
/
)
(
x
q
b
x
p
−
=
,
with
b
the depth of the beam, with
12
/
3
bh
I
=
, the plate will respond more stiffly than
the beam by a factor of
)
1
/(
1
2
ν
−
, a factor of about 10% for
3
.
0
=
ν
, since
(
)
b
EI
Eh
D
2
2
3
1
1
1
12
ν
ν
−
=
−
=
(6.5.3)
The extra stiffness is due to the constraining effect of
y
M
, which is not present in the
beam.
6.5.2
Deflection of a Circular Plate by a Uniform Lateral Load
A solution for a circular plate problem is presented next.
This problem will be examined
again in the section which follows using the more natural polar coordinates.
Consider a circular plate with boundary
2
2
2
a
y
x
=
+
,
(6.5.4)
clamped at its edges and subjected to a uniform lateral load
q
, Fig. 6.5.1.

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