270
7 Mathematical Models of Functionally Graded Beams in Temperature Field
σ xx (x, z) =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
E 1
∂u
∂ x
+ c
∂α
∂ x
− z
∂
2 w
∂ x 2
,
(c ≤ z ≤ c + h 1 )
E 3
∂u
∂ x
+ z
∂α
∂ x
− z
∂
2 w
∂ x 2
,
(−c ≤ z ≤ c)
E 2
∂u
∂ x
− c
∂α
∂ x
− z
∂
2 w
∂ x 2
, (−c − h 2 ≤ z ≤ −c)
(7.114)
the tangential stresses in the middle layer
σ xz = τ =
⎧
⎨
⎩
G 1 α 1 = 0,
(c ≤ z ≤ c + h 1 )
G 3 α 3 (x) = G 3 α (x) , (−c ≤ z ≤ c)
G 2 α 2 = 0, (−c − h 2 ≤ z ≤ −c)
(7.115)
σ xy = σ yz = 0,
as well as the higher order moments
m xy =
⎧
⎪ ⎨
⎪ ⎩
−l
2
1 G 1
∂
2 w
∂ x 2 ,
(c ≤ z ≤ c + h 1 )
l
2
3 G 3
2
∂α
∂ x
− 2
∂
2 w
∂ x 2
, (−c ≤ z ≤ c)
−l
2
2 G 2
∂
2 w
∂ x 2 ,
(−c − h 2 ≤ z ≤ −c) .
(7.116)
Applying (7.114)–(7.116), we find the forces and moments in each beam layer.
In the case of the first carrying load layer c ≤ z ≤ c + h 1 , we have
N 1 = b
c+h 1
c
σ xx dz = Bγ 1
∂u
∂ x
+ K γ 1
t 3
∂α
∂ x
− (t 1 + t 3 )
∂
2 w
∂ x 2
,
(7.117)
M 1 = M
0
1 + M
h
1 = b
c+h 1
c
σ xx (z − c) + m
1
xy
dz = −cN 1 + K γ 1 t 1
∂u
∂ x
+
+ Dγ 1 t 1
−1
3t 3
∂α
∂ x
−
4t 1
1 +
3
2 (1 + ν 1 )
l 1
h 1
2
+ 3t 3
∂
2 w
∂ x 2
.
(7.118)
In the case of the middle layer −c ≤ z ≤ c, we obtain:
N 3 = b
c
−c
σ xx dz = Bγ 3
∂u
∂ x
,
(7.119)