The effective shell material constants after stress relaxations are defined in
Eqs. (5.59), (5.60), and (5.50). In summary, with successive substitutions,
Eqs. (5.128), (5.129), (5.130), (5.131), (5.132), (5.133), (5.134), (5.135), (5.136),
(5.137), and (5.138) can be written as 11 equations for u
0
ð Þ
i , u
1
ð Þ
a , φ
(0) , φ
(1) , p
(0) , p
(1) ,
n
(0) , and n
(1) . At the boundary of a shell with an in-plane unit exterior normal n and
an in-plane unit tangent s, we may prescribe
N nn or u
0
ð Þ
n , N ns or u
0
ð Þ
s , Q
0
ð Þ
n3 or u
0
ð Þ
3 ,
M nn or u
1
ð Þ
n , M ns or u
1
ð Þ
s ,
D
0
ð Þ
n
or φ
0
ð Þ , D
1
ð Þ
n
or φ
1
ð Þ ,
ð5:147Þ
J
p 0
ð Þ
n
or p
0
ð Þ , J
p 1
ð Þ
n
or p
1
ð Þ ,
J
n 0
ð Þ
n
or n
0
ð Þ , J
n 1
ð Þ
n
or n
1
ð Þ
:
ð5:148Þ
For the classical theory of a shell in coupled extension and bending without shear
deformations, we set the relevant shear strains to zero:
2S
0
ð Þ
31 ¼ u
1
ð Þ
1 À
u
0
ð Þ
1
R 1
þ
1
A 1
∂u
0
ð Þ
3
∂α 1
¼ 0,
2S
0
ð Þ
23 ¼ u
1
ð Þ
2 À
u
0
ð Þ
2
R 2
þ
1
A 2
∂u
0
ð Þ
3
∂α 2
¼ 0:
ð5:149Þ
This allows us to express u
1
ð Þ
1 and u
1
ð Þ
2 in terms of the zero-order displacements for
extension and bending. Furthermore, we ignore the rotatory inertia 2ρh
3 /3 in
Eqs. (5.131) and (5.132) to obtain
∂ 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 0, ð5:150Þ
∂ 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 0, ð5:151Þ
which yields expressions for the transverse shear forces Q 31 and Q 32 in terms of the
bending and twisting moments, or effectively the constitutive relations for Q 31 and
Q 32 . The mechanical equations of motion for the classical theory are left to be
Eqs. (5.128), (5.129), and (5.130) for u
0
ð Þ
i :
∂ 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 ,
ð5:152Þ
5.11 Equations for Shells
139