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This will lead to a system of six first order equations. Hence, three boundary conditions on
each edge may be prescribed.
13.4.1
Fundamental Local Relationships
The kinematical relations of Eq. (13.23) and the equilibrium conditions of Eq. (13.31) still
hold for a plate with shear deformation. The material law, however, requires special
attention.
The material law of Eq. (13.29) needs to be supplemented with relationships between
shear resultants
q
x
,q
y
and the strains
γ
γ
yz
referred to the middle surface. To find these
relationships, set the internal work due to the distributed stresses and strains equal to the
internal work of the stress resultants and the strains of the middle surface. For example, for
q
x
,
the work
xz
,
−
W
i
=
z
τ
xz
γ
xz
dz dA
A
should be equal to the work
q
x
γ
xz
dA
A
which is expressed in terms of middle surface variables. From the former expression, with
γ
xz
=
τ
xz
/
G,
z
τ
3
q
x
2
t
1
2
z
t
2
2
t
2
2
xz
G
1
G
z
τ
γ
xz
dz dA
=
dz dA
=
−
dz dA
xz
t
2
A
A
A
−
3
2
2
q
x
Gt
dA
8
15
q
x
t
t
G
dA
6
5
q
x
6
5
q
x
Gt
dA
=
=
=
(13.85)
A
A
A
where the shear stress distribution of Eq. (13.46a) was employed, i.e.,
τ
=
(
3
q
x
/
2
t
)
[1
−
Since, in terms of the middle surface, the work is
A
q
x
γ
xz
dA
, we conclude that by
comparison with the final integral of Eq. (13.85)
xz
(
2
z
/
t
)
2
]
.
q
x
Gt
6
5
(γ
)
=
.
Thus, the material
xz
middle surface
law for shear becomes
q
x
q
y
Gt
5
γ
xz
γ
yz
/
6
=
.
(13.86)
5
/
6
The complete material law then appears as
m
x
m
y
m
xy
q
x
q
y
κ
x
κ
y
2
1
ν
0
K
ν
10
00
1
−
ν
2
0
E
B
0
0
V
=
=
(13.87)
κ
xy
γ
xz
γ
yz
Gt
5
/
60
05
0
/
6
s
=
E
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