a análise de placas laminadas compostas inteligentes

7.5 Solução de Navier 140
A11
+ L11
∂2u0 ∂
+ A66
∂x2 2u0 ∂y2 + (A12 + A66) ∂2v 0
+ B11
∂x∂y
∂ 2 ψx
∂2ψ3x ∂
+ L66
∂x2 2ψ3x ∂y2 + (L12 + L66) ∂2ψ3y ∂x∂y −
npiez
k=1
(A12 + A66) ∂2 u 0
+ A66
∂x∂y
+ (L12 + L66) ∂2 ψ3x
∂x∂y
B11
∂ψx
Ac55
+
∂x
+ Ac44
npiez
k=1
hk
2
+ L66
∂ψx
+ Ac45
∂ 2 w 0
+ B66
∂x2
e (k)
31
∂2ψx ∂y2 + (B12 + B66) ∂2ψy ∂x∂y
∂ϕk−1
∂x
∂2v 0 ∂
+ A22
∂x2 2v0 ∂y2 + (B12 + B66) ∂2ψx ∂x∂y
∂y
+ 3Dc55
∂y2
e (k)
15
∂ 2 ϕk−1
∂x 2
∂2ψ3y ∂
+ L22
∂x2 2 npiez
ψ3y
−
∂y2 k=1
∂ψy
+ Ac45
∂x
∂ψ3x
+
∂x
∂ψy
+ Ac44
∂y
∂ψ3x
+ 3Dc45
e (k)
14 + e (k)
25
∂2u0 ∂
+ B66
∂x2 2u0 ∂y2 + (B12 + B66) ∂2v 0
+ F11
k=1
∂ 2 ψ3x
+ F66
∂x2 ∂y
∂ 2 ϕk−1
∂x∂y
+ D11
∂x∂y
e (k)
36
+ Ac55
∂ϕk−1
∂x
∂ 2 w 0
∂x
∂ψ3y
+ 3Dc44
∂y
+ e(k)
24
+ e(k)
36
+ B66
+ e(k)
32
∂ϕk−1
∂y
+ qx = fx
∂2ψy ∂
+ B22
∂x2 2ψy ∂y2 ∂ϕk−1
∂y
+ 2Ac45
2 ∂x∂y
∂ 2 ϕk−1
∂y 2
(7.83)
+ qy = fy
∂ 2 w 0
∂ψ3y
+ 3Dc45
∂x
+ qz = fz
(7.84)
(7.85)
∂2ψx ∂
+ D66
∂x2 2ψx ∂y2 + (D12 + D66) ∂2ψy ∂x∂y
∂2ψ3x ∂y2 + (F12 + F66) ∂2ψ3y ∂x∂y
∂w
− Ac45ψy − Ac45
0
∂y − Ac55ψx
∂w
− Ac55
0
∂x − 3Dc45ψ3y − 3Dc55ψ3x
npiez 1
− (z
2hk k=1
2 k − z 2
k−1) e (k)
∂ϕk−1 ∂ϕk−1
31 + e(k) 36
∂x ∂y
npiez
hk
− e
2
(k)
∂ϕk−1 ∂ϕk−1
15 + e(k) 25 + my = fmx
∂x ∂y
(7.86)