u 2 x, t
ð Þ ffi v x 3 , t
ð
Þ,
u 3 x, t
ð Þ ffi x 2 ψ x 3 , t
ð
Þ,
φ x, t
ð Þ ffi x 2 ϕ
1
ð Þ x 3 , t
ð
Þ,
Δp x, t
ð Þ ffi x 2 p
1
ð Þ x 3 , t
ð
Þ,
Δn x, t
ð Þ ffi x 2 n
1
ð Þ x 3 , t
ð
Þ:
ð4:21Þ
The field equations are
Q ,3 ¼ ρA€ v,
M ,3 À Q ¼ ρI €
ψ,
D
1
ð Þ
3,3 À D
0
ð Þ
2 ¼ qI p
1
ð Þ
À n
1
ð Þ
,
J
p 1
ð Þ
3,3 À J
p 0
ð Þ
2
¼ ÀqI _
p
1
ð Þ ,
J
n 1
ð Þ
3,3 À J
n 0
ð Þ
2
¼ qI _
n
1
ð Þ
:
ð4:22Þ
The constitutive relations are
M ¼
Z
A
x 2 T 3 dA ¼ c 33 Iψ ,3 þ e 33 Iϕ
1
ð Þ
,3 ,
Q ¼
Z
A
T 4 dA ¼ c 44 A v ,3 þ ψ
ð
Þþe 15 Aϕ
1
ð Þ ,
ð4:23Þ
D
0
ð Þ
2 ¼
Z
A
D 2 dA ¼ e 15 A v ,3 þ ψ
ð
ÞÀε 11 Aϕ
1
ð Þ ,
D
1
ð Þ
3 ¼
Z
A
x 2 D 3 dA ¼ e 33 Iψ ,3 À ε 33 Iϕ
1
ð Þ
,3 ,
ð4:24Þ
J
p 0
ð Þ
2
¼
Z
A
J
p
2 dA ¼ Àqp 0 μ
p
11 Aϕ
1
ð Þ
À qD
p
11 Ap
1
ð Þ ,
J
p 1
ð Þ
3
¼
Z
A
x 2 J
p
3 dA ¼ Àqp 0 μ
p
33 Iϕ
1
ð Þ
,3 À qD
p
33 Ip
1
ð Þ
,3 ,
ð4:25Þ
J
n 0
ð Þ
2
¼
Z
A
J
n
2 dA ¼ Àqn 0 μ
n
11 Aϕ
1
ð Þ
þ qD
n
11 An
1
ð Þ ,
J
n 1
ð Þ
3
¼
Z
A
x 2 J
n
3 dA ¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 :
ð4:26Þ
The above equations are for bending with shear deformation. For the special case of
bending without shear deformation which is usually applicable to very thin beams,
we set the beam shear strain v ,3 + ψ to zero, which implies that
ψ ¼ Àv ,3 :
ð4:27Þ
94
4 Bending of Beams