∂ D
0
ð Þ
1 A 2
∂α 1
þ
∂ D
0
ð Þ
2 A 1
∂α 2
þ
1
R 1
þ
1
R 2
A 1 A 2 D
0
ð Þ
3 þ D
0
ð Þ
¼ 2hA 1 A 2 q p
0
ð Þ
À n
0
ð Þ
,
ð5:133Þ
∂ D
1
ð Þ
1 A 2
∂α 1
þ
∂ D
1
ð Þ
2 A 1
∂α 2
þ
1
R 1
þ
1
R 2
A 1 A 2 D
1
ð Þ
3 À A 1 A 2 D
0
ð Þ
3 þ D
1
ð Þ
¼
2h
3
3
qA 1 A 2 p
1
ð Þ
À n
1
ð Þ
:
ð5:134Þ
The two-dimensional continuity equations are:
∂ J
p 0
ð Þ
1 A 2
∂α 1
þ
∂ J
p 0
ð Þ
2 A 1
∂α 2
þ
1
R 1
þ
1
R 2
A 1 A 2 J
p 0
ð Þ
3
þ J
p 0
ð Þ
¼ À2hA 1 A 2 _
p
0
ð Þ ,
ð5:135Þ
∂ J
p 1
ð Þ
1 A 2
∂α 1
þ
∂ J
p 1
ð Þ
2 A 1
∂α 2
þ
1
R 1
þ
1
R 2
A 1 A 2 J
p 1
ð Þ
3
À A 1 A 2 J
p 0
ð Þ
3
þ J
p 1
ð Þ
¼ À
2h
3
3
qA 1 A 2 _
p
1
ð Þ ,
ð5:136Þ
∂ J
n 0
ð Þ
1 A 2
∂α 1
þ
∂ J
n 0
ð Þ
2 A 1
∂α 2
þ
1
R 1
þ
1
R 2
A 1 A 2 J
n 0
ð Þ
3
þ J
n 0
ð Þ
¼ 2hA 1 A 2 _
n
0
ð Þ ,
ð5:137Þ
∂ J
n 1
ð Þ
1 A 2
∂α 1
þ
∂ J
n 1
ð Þ
2 A 1
∂α 2
þ
1
R 1
þ
1
R 2
A 1 A 2 J
n 1
ð Þ
3
À A 1 A 2 J
n 0
ð Þ
3
þ J
n 1
ð Þ
¼
2h
3
3
qA 1 A 2 _
n
1
ð Þ
:
ð5:138Þ
The mechanical resultants and surface loads are defined by
N ab , Q 3c , Q c3
f
g¼ T
0
ð Þ
ab , T
0
ð Þ
3c , T
0
ð Þ
c3
n
o
¼
Z h
Àh
T ab , T 3c , T c3
f
g dα 3 ,
M ab ¼ T
1
ð Þ
ab ¼
Z h
Àh
T ab α 3 dα 3 ,
F
n
ð Þ
j ¼ α
n
3 T 3j A 1 A 2
Â
à h
Àh
, n ¼ 0, 1, a, b, c ¼ 1, 2:
ð5:139Þ
5.11 Equations for Shells
137