P
0
ð Þ
1 ¼ À
1
A 1
∂p
0
ð Þ
∂α 1
, P
0
ð Þ
2 ¼ À
1
A 2
∂p
0
ð Þ
∂α 2
, P
0
ð Þ
3 ¼ Àp
1
ð Þ ,
P
1
ð Þ
1 ¼ À
1
A 1
∂p
1
ð Þ
∂α 1
, P
1
ð Þ
2 ¼ À
1
A 2
∂p
1
ð Þ
∂α 2
, P
1
ð Þ
3 ¼ À2p
2
ð Þ
ffi 0,
ð5:126Þ
N
0
ð Þ
1 ¼ À
1
A 1
∂n
0
ð Þ
∂α 1
, N
0
ð Þ
2 ¼ À
1
A 2
∂n
0
ð Þ
∂α 2
, N
0
ð Þ
3 ¼ Àn
1
ð Þ ,
N
1
ð Þ
1 ¼ À
1
A 1
∂n
1
ð Þ
∂α 1
, N
1
ð Þ
2 ¼ À
1
A 2
∂n
1
ð Þ
∂α 2
, N
1
ð Þ
3 ¼ À2n
2
ð Þ
ffi 0:
ð5:127Þ
S
0
ð Þ
ab are the extensional and in-plane shear strains (in the tangent plane). S
0
ð Þ
a3 are the
transverse shear strains related to bending. S
1
ð Þ
ab are related to bending curvatures. The
two-dimensional equations of motion are [9, 10]
∂ N 11 A 2
ð
Þ
∂α 1
þ
∂ N 21 A 1
ð
Þ
∂α 2
þ N 12
∂A 1
∂α 2
À N 22
∂A 2
∂α 1
þ Q 13 A 1 A 2
1
R 1
þ F
0
ð Þ
1
¼ 2ρhA 1 A 2 € u
0
ð Þ
1 ,
ð5:128Þ
∂ N 12 A 2
ð
Þ
∂α 1
þ
∂ N 22 A 1
ð
Þ
∂α 2
þ N 21
∂A 2
∂α 1
À N 11
∂A 1
∂α 2
þ Q 23 A 1 A 2
1
R 2
þ F
0
ð Þ
2
¼ 2ρhA 1 A 2 € u
0
ð Þ
2 ,
ð5:129Þ
∂
∂α 1
Q 13 A 2
ð
Þþ
∂
∂α 2
Q 23 A 1
ð
ÞÀN 11
A 1 A 2
R 1
À N 22
A 1 A 2
R 2
þ F
0
ð Þ
3
¼ 2ρhA 1 A 2 € u
0
ð Þ
3 ,
ð5:130Þ
∂ M 11 A 2
ð
Þ
∂α 1
þ
∂ M 21 A 1
ð
Þ
∂α 2
þ M 12
∂A 1
∂α 2
À M 22
∂A 2
∂α 1
À Q 31 A 1 A 2 þ F
1
ð Þ
1
¼
2ρh
3
3
A 1 A 2 € u
1
ð Þ
1 ,
ð5:131Þ
∂ M 12 A 2
ð
Þ
∂α 1
þ
∂ M 22 A 1
ð
Þ
∂α 2
þ M 21
∂A 2
∂α 1
À M 11
∂A 1
∂α 2
À Q 32 A 1 A 2 þ F
1
ð Þ
2
¼
2ρh
3
3
A 1 A 2 € u
1
ð Þ
2 :
ð5:132Þ
N ab are the extensional and in-plane shear forces. Q 3c ¼ Q c3 are the transverse shear
forces. M ab are bending and twisting moments. Eqs. (5.128) and (5.129) are for
extension, Eq. (5.130) for flexure, and Eqs. (5.131) and (5.132) for shear deformations in the α 1 and α 2 directions accompanying flexure. The two-dimensional charge
equations are [6]
136
5 Extension and Bending of Plates