Download Stress concentration analysis of thick-walled laminate

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ACCEPTED MANUSCRIPT
in their plate theories. Some examples for theories of even higher order –
including not only the shear deformation effects but also, e. g., warping of
crossections – can be found in [15].
With the kinematic assumption (2) the strain-displacement relationships for
small displacements can be written in the form
⎡
⎡
⎤
⎤
⎡
⎤
⎡
⎤
⎡
⎤
∂ψx
∂u0
0
⎢
⎥
⎢ εx ⎥
⎢
⎥
⎢ εx ⎥
⎢ κx ⎥
∂x
∂x
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
∂ψy
⎢
⎥
∂v
0
0
⎢ε ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
ε
κ
⎢ y⎥
⎢
⎥
⎢
⎥
⎢
⎥
y
y
∂y
⎢
⎥
∂y
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎥
∂ψ
∂ψ
∂u
∂v
y
⎢γ ⎥ = ⎢
⎢
⎥
⎢
⎥
⎢
⎥
0
x
0
0
⎢
⎥+z⎢
=:
+
z
+
γ
κ
+
⎢ xy ⎥
⎥
⎢
⎥
⎢
xy ⎥ .
xy
⎢ ∂y
⎥
∂x
∂y
∂x
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ∂w
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
0
0
⎢
⎥
γ
0
⎢γxz ⎥
⎢
⎥
⎢
⎥
⎢
⎥
+
ψ
0
x⎥
xz ⎥
⎢ ∂x
⎢
⎥
⎢
⎥
⎢
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
⎣ ∂w
⎦
0
0 +ψ
γyz
γ
0
0
y
yz
∂y
(3)
Based on the strain-displacement-relations (3) and Hooke’s deformation law
(1), finally the structural law for thick-walled multilayered laminates according
to the first-order shear deformation theory is derived
⎡
⎤
⎢ Nx ⎥
⎢
⎥
⎢
⎥
⎢N ⎥
⎢ y ⎥
⎢
⎥
⎢
⎥
⎢N ⎥
⎢ xy ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ Mx ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ My ⎥
⎢
⎥
⎣
⎦
⎡
=
⎢A11 A12 A16 B11 B12
⎢
⎢
⎢
A22 A26
B22
⎢
⎢
⎢
⎢
A66 symm.
⎢
⎢
⎢
⎢
D11 D12
⎢
⎢
⎢
⎢
⎢ symm.
D22
⎢
⎣
Mxy
⎡
Qy
B16 ⎥
⎡
=
⎤
2
⎢−k2 A55
⎢
⎣
symm.
−k1 k2 A45 ⎥
−k12 A44
⎥
⎦
ε0x
⎤
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ε0 ⎥
⎢ y⎥
⎢
⎥
⎢
⎥
⎢γ 0 ⎥
⎢ xy ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ κx ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ κy ⎥
⎢
⎥
⎣
⎦
⎥
⎥
B26 ⎥
⎥
⎥
⎥
B66 ⎥
⎥
⎥
⎥
D16 ⎥
⎥
⎥
⎥
⎥
D26 ⎥
⎥
⎦
D66
⎤
⎢Qx ⎥
⎢ ⎥
⎣ ⎦
⎡
⎤
,
(4)
κxy
⎤
⎡
0
⎢−γxz ⎥
⎢
⎣
0
−γyz
⎥
⎦
,
with the shear correction factors 2 k1 , k2 , the stress resultants and moment
2
For details on the shear correction factors see e. g. [16]
5
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