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FIGURE 13.1
Coordinate system and middle surface of a plate.
are related by
x
=
∂
u
x
∂
y
=
∂
u
y
∂
γ
xy
=
∂
u
x
∂
y
+
∂
u
y
∂
x
y
x
or
u
x
u
y
∂
0
x
y
γ
xy
x
=
0
∂
y
(13.1c)
∂
y
∂
x
=
D
(
or
D
u
)
u
Material Law
For a thin structure, it is convenient to replace the stresses (
xy
) with
stress
resultants
in force per unit length. For a thin element of thickness
t
, these are defined as
σ
x
,
σ
y
, and
τ
t
/
2
t
/
2
t
/
2
=
2
σ
=
2
σ
=
2
τ
n
x
x
dz
n
y
y
dz
n
xy
xy
dz
(13.2)
−
t
/
−
t
/
−
t
/
These can be combined to form the vector
s
[
n
x
n
y
n
xy
]
T
As explained in Chapter 1, Section 1.3.1, these stress resultants can be related to the strains
=
xy
]
T
based on plane stress or plane strain assumptions. In the case of plane
stress, with the assumptions
=
[
γ
x
y
σ
=
τ
=
τ
=
0, it is found that
z
zx
zy
ν
n
x
n
y
n
xy
1
0
x
y
γ
xy
=
ν
1
0
D
00
1
−
ν
2
(13.3a)
s
=
E
2
where
D
=
Et
/(
1
−
ν
)
,or
1
−
ν
0
n
x
n
y
n
xy
x
1
Et
=
−
ν
1 0
002
y
(13.3b)
γ
(
1
+
ν)
xy
E
−
1
=
s
Conditions of Equilibrium
The equilibrium equations,
wh
ich
pr
ovi
d
e a relationship between the stress resultants and
the body forces (force/area)
p
V
=
p
V
y
]
T
[
p
V
x
are
∂
n
x
∂
x
+
∂
n
yx
∂
+
p
Vx
=
0
y
n
xy
=
n
yx
(13.4a)
∂
n
y
∂
y
+
∂
n
xy
∂
+
p
Vy
=
0
x
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