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which is known as the plate
stiffness
. The inverted relationship gives the curvatures in terms
of the stress resultants
κ
1
−
ν
0
m
x
m
y
m
xy
x
12
Et
3
=
κ
−
ν
1 0
002
y
(13.36)
2
κ
(
1
+
ν)
xy
E
−
1
=
s
Note that from Eq. (13.34), the relations of Eq. (13.35) in terms of displacements appear as
K
∂
w
2
x
2
y
m
x
=−
w
+
ν∂
K
∂
w
2
y
2
x
m
y
=−
w
+
ν∂
(13.37)
m
xy
=−
K
(
1
−
ν)∂
∂
w
x
y
From the material law of Eq. (13.25), i.e.,
E
σ
=
2
(
+
ν
)
x
x
y
1
−
ν
E
σ
y
=
2
(
y
+
ν
x
)
1
−
ν
τ
=
G
γ
xy
xy
and Eqs. (13.33) and (13.37), the stresses can be expressed in terms of the stress resultants
by
m
x
z
t
3
σ
x
=
/
12
m
y
z
t
3
σ
=
(13.38)
y
/
12
m
xy
z
t
3
τ
=
12
=
τ
xy
yx
/
Since
z
is measured from the middle surface, the maximum normal stresses will occur at
z
=±
t
/
2 on the top and bottom surfaces (see Fig. 13.8).
Conditions of Equilibrium
Consider the equilibrium conditions for the plate element in Fig. 13.6 in light of the as-
sumptions of Kirchhoff plate theory. Recall that the Kirchhoff plate theory omits the effect
of shear strains
γ
=
τ
/
γ
=
τ
/
G
on
the bending of the plate. Vertical forces
q
x
and
q
y
are not negligible. The external load
p
z
is carried by these shear forces together
with the moments
m
x
,
m
y
, and
m
xy
. The equilibrium conditions (Eq. 13.31) of the plate still
apply.
Substitute the second and third relations of Eq. (13.31a) into the first relation of Eq. (13.31a).
This gives the higher order equilibrium relation
G
and
xz
xz
yz
yz
2
m
x
∂
2
m
xy
∂
2
m
y
∂
∂
2
∂
y
+
∂
x
2
+
y
2
+
p
z
=
0
(13.39a)
x
∂
or
m
x
m
y
m
xy
+
x
y
[
∂
∂
2
∂
x
∂
y
]
[
p
z
]
=
0
(13.39b)
D
s
s
+
p
=
0
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