6.3 Modified Couple Stress Theory of Thermoelastic Curvilinear Panels
145
α P +
Y 12 + 3αT 12
2
x=L
x=0
= α ¯
P or
∂w
∂ x
x=L
x=0
=
____
∂w
∂ x
.
(6.46)
In relations (6.22)–(6.25) and (6.29), (6.30), we describe stresses and moments
by displacements u, ϕ and w and we obtain
N =
A
(1 + kz)σ 11 d A ∼ =
A
E(x, z, T (x, z))ε 11 − γ E(x, z, T (x, z))T (x, z)
d A =
=
A
E
∂u
∂ x
+ kw +
1
2
∂w
∂ x
− βku
2
+ Ez
∂ϕ
∂ x
−
(6.47)
−α Ez
3
∂ϕ
∂ x
+
∂
2 w
∂ x 2
− Eγ T
d A.
Introducing the notation
(J 0 , J 1 , J 2 , J 3 , J 4 , J 6 ) =
A
E(x, z, T (x, z))(1, z, z
2
, z
3
, z
4
, z
6
)d A,
(6.48)
N
T
=
A
Eγ T d A, M
T
=
A
Eγ T zd A, P
T
=
A
Eγ T z
3 d A .
(6.49)
yields N = N
e
− N
T
, where
N
e
= J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
− βku
2
+ (J 1 − α J 3 )
∂ϕ
∂ x
− α J 3
∂
2 w
∂ x 2 . (6.50)
We find equations governing M and P in the way analogous to N , i.e. we have
M = M
e
− M
T , where
M
e
= J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
− βku
2
+ (J 2 − α J 4 )
∂ϕ
∂ x
− α J 4
∂
2 w
∂ x 2 (6.51)
and P = P
e
− P
T , where
P
e
= J 3
∂u
∂ x
+ kw +
1
2
∂w
∂ x
− βku
2
+ (J 4 − α J 6 )
∂ϕ
∂ x
− α J 6
∂
2 w
∂ x 2 . (6.52)