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κ
x
κ
y
2
m
x
m
y
m
xy
q
x
q
y
1
−
ν
0
12
Et
3
−
ν
1 0
002
0
(
1
+
ν)
κ
xy
γ
xz
γ
=
(13.88)
6
/
50
06
1
Gt
0
/
5
yz
E
−
1
=
s
13.4.2 Boundary Conditions
The surface stress resultants of Eq. (13.57) still apply. The displacement boundary conditions
will be
w
=
w
,
θ
a
=
θ
a
,
θ
t
=
θ
t
(13.89a)
and the force boundary conditions
m
a
=
m
a
,
=
m
t
,
q
a
=
q
a
.
(13.89b)
t
Thus, for this plate with shear deformation, the specification of boundary conditions is
straightforward. This is in contrast to the previous model without shear deformation,
where it was necessary to define an equivalent shear force.
13.4.3
Local Form of the Governing Equations
The relationships of this section can be rearranged in a variety of ways. One approach of
particular interest involves defining two invariant combinations
≡
∂
x
(13.90)
Substitution of the kinematic relation of Eq. (13.23) and Hooke's law of Eq. (13.29) into the
second and third of the equilibrium expressions of Eq. (13.31a) leads to
θ
+
∂
θ
y
,
≡
∂
θ
−
∂
θ
x
x
y
x
y
y
K
1
−
ν
2
q
x
=
∂
x
−
∂
y
K
1
−
ν
2
q
y
=
∂
+
∂
y
x
Substitute these relationships into the first equation of the equilibrium conditions of
Eq. (13.31a) to find
2
y
The final two equations of the kinematic relations of Eq. (13.22), together with the expres-
sions for
2
2
2
x
K
∇
=−
p
z
∇
=
∂
+
∂
γ
xy
and
γ
xz
of Eq. (13.86), yield
K
Gt
6
5
1
−
ν
2
∂
w
=−
θ
+
∂
−
∂
x
x
x
y
K
Gt
6
5
1
−
ν
2
∂
w
=−
θ
+
∂
+
∂
y
y
y
x
These equations lead to a viable form of the governing equations
K
Gt
6
5
∇
6
5
p
z
G
2
2
∇
w
=−
+
=−
−
(13.91)
t
3
10
∇
K
G
6
5
1
−
ν
2
2
2
−
∇
=
−
=
0
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