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Introduction to modelization of thick and heterogeneous plates
Arthur Lebée, Karam Sab
To cite this version:
Arthur Lebée, Karam Sab. Introduction to modelization of thick and heterogeneous plates. Rencontres
franciliennes de mécanique, Jun 2014, Paris, France. �hal-01134568�
’
Introduction to modelization of thick and heterogeneous plates
Arthur Leb´ ee, Karam Sab
Laboratoire Navier (UMR CNRS 8205)
Universit´e Paris-Est - ´Ecole des Ponts ParisTech - IFSTTAR
11/06/2014
The 3D Problem
σ
ij,jt= 0 on Ω.
σ
ijt= C
ijkl(x
3) ε
tklon Ω . σ
i3t= ± T
i±on ω
±. ε
tij= u
(i,j)ton Ω .
u
it= 0 on ∂ω × ] − t /2, t /2[
I monoclinic C
∼∼: C αβγ3 = C 333α = 0
I symmetrically laminated plate
I symmetric transverse load T
−± = p 2
3e
−3
C
∼∼(x
3): even
Ω T
−+T
−−
ω
+ω ω
−∂ω e
−3
e
−2
e
−1
⇒ pure bending:
I u 3 and σ α3 even / x 3
I u α , σ αβ and σ 33 odd / x 3
Introduction
Building a plate model?
For typical width L let t → 0
I Solve a 2D problem
I “fair” 3D displacement localization
I “fair” 3D stress localization
Some energy principles...
I Statically admissible fields:
SA 3 D = n
σ ij / σ ij ,j = 0 and σ
∼± t 2
· ±e
−3 = P 2
3e
−3
o
SA 3 D,0 ⇔ P 3 = 0
I Kinematically admissible fields:
KA 3 D =
u i / u i = 0 on ∂ω ×
− 2 t , t 2
KA 3 D,0 = KA 3 D
I Orthogonality between KA 3D,0 and SA 3 D,0 :
∀u ∈ KA 3 D,0 , σ ∈ SA 3 D,0 , Z
σ : ε ( u )d Ω = 0
Introduction
Some energy principles...
I Potential energy:
u
−
3 D = argmin
−
u ∈KA
3DW 3D ε
∼( u
−)
− Z
ω
P 3 u + 3 + u 3 − 2 d ω
where W 3D = 1 2 R
Ω ε
∼( u
−) : C
∼∼: ε
∼( u
−)d Ω
I Complementary energy:
σ
∼3 D = argmin
σ ∈SA
3Dn
W ∗3 D σ
∼o
where W ∗3 D = 1 2
Z
Ω
σ
∼: S
∼∼: σ
∼d Ω
The 2-energy principle
∀ u ˆ
−∈ KA 3D , ∀ σ ˆ
∼∈ SA 3 D : W 3D
ε
∼u
−ˆ
− S
∼∼: σ ˆ
∼= W ∗3D
ˆ
σ
∼− C
∼∼: ε
∼u
−ˆ
= W ∗3 D
ˆ σ
∼− σ
∼3 D
+ W 3 D
ε
∼u
−ˆ − u
−3 D
⇒ W
∗3Dσ ˆ
∼− C
∼∼: ε
∼u
−ˆ
provides an error estimate in terms of constitutive equation.
Prager and Synge (1947); Morgenstern (1959); Braess et al. (2010)
Introduction
Contents
The case of homogeneous and isotropic plates The case of laminated plates
Applications
Cylindrical bending of laminates Extension to periodic plates
The case of cellular sandwich panels
Why all periodic plates are not “Reissner” like...
Contents
The case of homogeneous and isotropic plates The case of laminated plates
Applications
The case of homogeneous and isotropic plates
Natural scaling of the stress
SA 3 D
σ αβ,β + σ α3,3 = 0 σ α3,α + σ 33,3 = 0 σ 33 ( ± t/2) = ± P 3 /2 σ α3 ( ± t/2) = 0
⇒
σ α3 = − Z x
3−t/2
σ αβ,β dz σ 33 = −
Z x
3t/2
σ α3,α dz − P 3 /2
σ αβ ∼ t 0 ⇒ σ α3 ∼ t 1 , σ 33 ∼ t 2 and P 3 ∼ t 2
From 3D equilibrium to 2D
Plate generalized stresses:
( M αβ (x 1 , x 2 ) = h x 3 σ αβ i ∼ t 2
Q α (x 1 , x 2 ) = h σ α3 i ∼ t 2 h•i = Z
t2
−
2t• dx 3
2D equilibrium equations:
h σ α3,α + σ 33,3 i = 0 h x 3 (σ αβ,β + σ α3,3 ) i = 0 ⇒
Q α,α + P 3 = 0 M αβ,β − Q α = 0
Boussinesq (1871); Mindlin (1951)...
The case of homogeneous and isotropic plates
“Kirchhoff’s assumption”
At leading order in t:
ε αβ = x 3 K αβ where K αβ = − U 3,αβ σ i3 ' 0 + O (t 1 ) plane stress
ε
336 = 0!!
In-plane stress:
σ αβ = C αβγδ
σε δγ = x 3 C αβγδ
σK δγ
where C αβγδ
σ= C αβγδ − C
αβ33C
3333C
33γδ:
The Kirchhoff-Love plate problem
Bending constitutive equation:
M αβ = h x 3 σ αβ i =
x 3 2 C αβγδ
σK δγ = − D αβγδ U 3,δγ
D
∼∼= t 3 12 C
∼∼σ
Statically admissible fields:
SA KL : { M αβ /M αβ,αβ + P 3 = 0 } Kinematically compatible fields:
KA KL :
U 3 /U 3 = 0 and U 3,n = 0 on ∂ω, n
−outer normal to ω
The case of homogeneous and isotropic plates
Building SA 3 D fields
σ ˆ αβ = x 3 C αβγδ
σd δγζ M ζ = 12x 3
t 3 M αβ ∼ t 0 σ ˆ α3 = −
Z x
3−t/2
σ αβ,β dz = 3 2t
1 − 4x 3 2 t 2
M αβ,β ∼ t 1
σ ˆ 33 = − Z x
3−t/2
σ α3,α dz − P 3 /2 = 3x 3 2t
1 − 4x 3 2 3t 2
P 3 ∼ t 2
⇔ σ ˆ
∼= s
∼∼M (x 3 ) : M
∼+ s
∼−Q (x 3 ) · M
∼· ∇
−+ s
∼P
3(x 3 )P 3
Building KA 3 D fields
Strains
ˆ ε
∼=
ε ˆ αβ = x 3 d αβγδ M γδ = S αβγδ σ ˆ δγ +
S αβ33 σ ˆ 33
ε ˆ α3 = 3 4Gt
1 − 4x 3 2 t 2
M αβ,β = 2S α3β3 σ ˆ β3 ε ˆ 33 = − 12ν x 3
Et 3 M αα = S 33αβ σ ˆ αβ +
S 3333 σ ˆ 33
The case of homogeneous and isotropic plates
Building KA 3 D fields
Integration
ˆ u 3 =
Z x
3ε ˆ 33 (z )dz + U 3 = u M 3αβ (x 3 )M βα
| {z }
∼t
1+ U 3
|{z}
∼t
−1where: u M 3αβ (x 3 ) = ν 2Et
12x 3 2 t 2 − 1
δ αβ and D
u M 3αβ (x 3 ) E
= 0
Building KA 3 D fields
Integration
ˆ u 3 =
Z x
3ε ˆ 33 (z )dz + U 3 = u M 3αβ (x 3 )M βα
| {z }
∼t
1+ U 3
|{z}
∼t
−1ˆ u α =
Z x
32ˆ ε α3 (z ) − u ˆ 3,α dz = u M αβγδ
⊗∇ (x 3 )M δγ,β
| {z }
∼t
2− x 3 U 3,α
| {z }
∼t
0where:
u M αβγδ
⊗∇ (x 3 )M δγβ = x 3 2Et
6(1 + ν)
1 − 4x 3 2 3t 2
M αβ,β + ν
1 − 4x 3 2 t 2
M ββ,α
The case of homogeneous and isotropic plates
Building KA 3 D fields
Integration
ˆ u 3 =
Z x
3ε ˆ 33 (z )dz + U 3 = u M 3αβ (x 3 )M βα
| {z }
∼t
1+ U 3
|{z}
∼t
−1ˆ u α =
Z x
32ˆ ε α3 (z ) − u ˆ 3,α dz = u M αβγδ
⊗∇ (x 3 )M δγ,β
| {z }
∼t
2− x 3 U 3,α
| {z }
∼t
0⇒ u ˆ
−= U 3 e
−3 − x 3 U 3,α e
−α + u M 3αβ (x 3 )M βα e
−3 + u M αβγδ
⊗∇ (x 3 )M δγ,β e
−α
Application of the Two-Energy principle
Consider U 3 KL and M
∼KL the solution of the Kirchhoff-Love plate problem and define:
σ
∼KL = s
∼∼M (x 3 ) : M
∼KL + s
∼−Q (x 3 ) · M
∼KL · ∇
−+ s
∼P
3(x 3 )P 3 u
−KL = U 3 KL e
−3 − x 3 U 3,α KL e
−α + u M 3αβ (x 3 )M βα KL e
−3 + u M αβγδ
⊗∇ (x 3 )M δγ,β KL e
−α
We have:
ε
∼( u
−KL ) − S
∼∼: σ
∼KL = s P 33
3P 3
S αβ33 0 0 0 0 S 3333
∼ t 2
The case of homogeneous and isotropic plates
Application of the Two-Energy principle
ε
∼( u
−KL ) − S
∼∼: σ
∼KL ∼ t 2 ⇒ W 3D
ε
∼( u
−KL ) − S
∼∼: σ
∼KL
∼ t 5
Would lead to a relative error in t 2 ...
σ
∼KL ∈ SA 3 D but u
−KL ∈ / KA 3 D
At best: relative error in t 1/2 ...
Reissner’s original plate model (1945)
min of W
∗3DLet us consider:
ˆ σ
∼= s
∼∼M (x 3 ) : M
∼+ s
∼−Q (x 3 ) · Q
−+ s
∼P
3(x 3 )P 3 with
SA RM = n
( M
∼, Q
−) / Q α,α + P 3 = 0 and M αβ,β − Q α = 0 on ω o
W ∗3 D σ
∼3D
≤ min
(M
∼, Q
−)∈SA
RMW ∗3 D σ ˆ
∼≤ W ∗3 D σ
∼KL
is a better approximation of W ∗3 D σ 3 D
The case of homogeneous and isotropic plates
Reissner’s original plate model (1945)
Dualization
Q α,α + P 3 = 0 × U 3 M αβ,β − Q α = 0 × ϕ α ⇒
Q α ↔ γ α = ϕ α + U 3,α
M αβ ↔ χ αβ = ϕ (α,β)
KA RM = { (U 3 , ϕ α ) / U 3 = 0 and ϕ α = 0 on ∂ω }
Reissner’s original plate model (1945)
Constitutive equation
W ∗RM M
∼, Q
−= W ∗3 D σ ˆ
∼= 1 2
Z
ω
T
M
∼Q
−P 3
d
∼∼0 5Et 6ν δ
∼0 5Gt 6 δ
∼0
6ν
5Et δ
∼0 140E 17t
M
∼Q
−P 3
d ω
⇒
χ αβ
|{z}
∼t
−1= d αβγδ M δγ
| {z }
∼t
−1+ 6ν 5Et δ αβ P 3
| {z }
∼t
1γ α = 6
5Gt Q α ∼ t 1
The case of laminated plates
Contents
The case of homogeneous and isotropic plates The case of laminated plates
Applications
Field Localization
Following the same procedure leads to:
ˆ σ
∼= s
∼∼M (x 3 ) : M
∼+
_s
∼R (x 3 ) ··· M
∼⊗ ∇
−| {z }
∼t
1+
__s
T (x 3 ) ···· M
∼⊗ ∇
−2 + s
∼P
3(x 3 )P 3
| {z }
∼t
2u ˆ
−= U 3 e
−3 − x 3 U 3,α e
−α + u M 3αβ (x 3 )M βα e
−3 + u R αβγδ (x 3 )M δγ,β e
−α
where R
_= M
∼ ⊗∇
−and T
∼∼= R
_⊗∇
−... Kirchhoff-Love error estimates still hold.
The case of laminated plates
Building SA 3 D fields
ˆ σ αβ = x 3 C αβγδ
σd δγζ M ζ = s M αβγδ (x 3 )M δγ ˆ σ α3 = −
Z x
3−t/2
σ αβ,β dz = s R α3βγδ (x 3 )M δγ,β
ˆ σ 33 = − Z x
3−t/2
σ α3,α dz − P 3 /2 = s T 33αβγδ (x 3 )M δγ,βα + s P 33
3(x 3 )P 3
The Bending-Gradient constitutive equation
Extending Reissner’s approach?:
ˆ
σ
∼= s
∼∼M (x 3 ) : M
∼+
_∼s R (x 3 ) ··· R
_+
__s T (x 3 ) ···· T
∼∼+
s
∼P
3(x 3 )P 3
Let us define:
ˆ σ
∼∗ = s
∼∼M (x 3 ) : M
∼+
_s
∼R (x 3 ) ··· R
_with: ˆ σ
∼∗ · ∇
−= 0 + O(t 1 ) only.
W ∗BG M
∼, R
_= W ∗3 D σ ˆ
∼∗
= 1 2
Z
ω
M
∼: d
∼∼: M
∼+ R
_···
__f ··· R
_d ω
The case of laminated plates
The Bending-Gradient theory for thick plates
I Equilibrium equations:
BG:
R αβγ = M αβ,γ
× Φ αβγ
R αββ,α + P 3 = 0
× U 3
RM:
Q α = M αβ,β
× ϕ α
Q α,α + P 3 = 0
× U 3
I Mechanical meaning of R
_Q
α= R
αββ⇔
Q
1= R
111+ R
122= M
11,1+ M
12,2Q
2= R
121+ R
222= M
21,1+ M
22,2The Bending-Gradient theory for thick plates
I Equilibrium equations:
BG:
R αβγ = M αβ,γ × Φ αβγ
R αββ,α + P 3 = 0 × U 3 RM:
Q α = M αβ,β × ϕ α
Q α,α + P 3 = 0 × U 3
The case of laminated plates
The Bending-Gradient theory for thick plates
I Equilibrium equations:
BG:
R αβγ = M αβ,γ × Φ αβγ
R αββ,α + P 3 = 0 × U 3 RM:
Q α = M αβ,β × ϕ α
Q α,α + P 3 = 0 × U 3
Equilibrium Work of internal forces Work on Boundary BG:
( R
αβγ= M
αβ,γR
αββ,α+ P
3= 0
M
αβΦ
αβ,+ R
αβγ(Φ
αβγ+ I
αβγU
3,)
M
αβΦ
αβγn
γ+ R
αββn
αU
3RM:
Q
α= M
αβ,βM
αβϕ
(α,β)+
(ϕ
M
αβn
βϕ
α+
The Bending-Gradient theory for thick plates
I Equilibrium equations:
BG:
R αβγ = M αβ,γ × Φ αβγ
R αββ,α + P 3 = 0 × U 3 RM:
Q α = M αβ,β × ϕ α
Q α,α + P 3 = 0 × U 3
Equilibrium Work of internal forces Work on Boundary BG:
( R
αβγ= M
αβ,γR
αββ,α+ P
3= 0
M
αβΦ
αβ,+ R
αβγ(Φ
αβγ+ I
αβγU
3,)
M
αβΦ
αβγn
γ+ R
αββn
αU
3RM:
Q
α= M
αβ,βQ
α,α+ P
3= 0
M
αβϕ
(α,β)+ Q
α(ϕ
α+ U
3,α)
M
αβn
βϕ
α+
Q
αn
αU
3When the plate is homogeneous, BG is exactly turned into RM
The case of laminated plates
The Bending-Gradient theory for thick plates
I Equilibrium equations:
BG:
R αβγ = M αβ,γ × Φ αβγ
R αββ,α + P 3 = 0 × U 3 RM:
Q α = M αβ,β × ϕ α
Q α,α + P 3 = 0 × U 3
Equilibrium Work of internal forces Work on Boundary BG:
( R
αβγ= M
αβ,γR
αββ,α+ P
3= 0
M
αβΦ
αβ,+ R
αβγ(Φ
αβγ+ I
αβγU
3,)
M
αβΦ
αβγn
γ+ R
αββn
αU
3RM:
Q
α= M
αβ,βM
αβϕ
(α,β)+
(ϕ
M
αβn
βϕ
α+
Local Fields reconstruction
Once the plate problem is solved (U 3 BG , Φ
_BG , M
∼BG , R
_BG known), we suggest the following field reconstruction:
I σ
∼BG = s
∼∼M (x 3 ) : M
∼+
_s
∼R (x 3 ) ··· R
_+
__s T (x 3 ) ···· R
_⊗ ∇
−+ s
∼P
3(x 3 )P 3
I u
−BG = U 3 e
−3 − x 3 U 3,α e
−α + u M 3αβ (x 3 )M βα e
−3 + u R αβγδ (x 3 )R δγβ e
−α
Applications
Contents
The case of homogeneous and isotropic plates The case of laminated plates
Applications
Cylindrical bending of laminates Extension to periodic plates
The case of cellular sandwich panels
Why all periodic plates are not “Reissner” like...
Contents
The case of homogeneous and isotropic plates The case of laminated plates
Applications
Cylindrical bending of laminates Extension to periodic plates
The case of cellular sandwich panels
Why all periodic plates are not “Reissner” like...
Applications Cylindrical bending of laminates
Pagano’s boundary value problem (Pagano, 1969)
CFRP layers with different orientiations:
x 3 p 3 /2
p 3 /2
σ
11(x
3) = 0 σ
12(x
3) = 0 u
3(x
3) = 0 x 1
x 2
L
Practical Localization...
Kirchhoff-Love
I σ
∼KL = s
∼∼M (x 3 ) : M
∼KL +
(( (( (( (( ( ( s
∼−
Q (x 3 ) · M
∼KL · ∇
−+ s
∼P
3(x 3 )P 3
I u
−KL = U 3 KL e
−3 − x 3 U 3,α KL e
−α + u M 3αβ (x 3 )M βα KL e
−3 +
(( (( (( (( ( u M αβγδ
⊗∇ (x 3 )M δγ,β KL e
−α
Bending-Gradient
I σ
∼BG = s
∼∼M (x 3 ) : M
∼+
_s
∼R (x 3 ) ··· R
_+
_
s
_
T (x 3 ) ···· R
_⊗ ∇
−+
s
∼P
3(x 3 )P 3 I u
−BG = U 3 e
−3 − x 3 U 3,α e
−α + u M 3αβ (x 3 )M βα e
−3 + u R αβγδ (x 3 )R δγβ e
−α
Applications Cylindrical bending of laminates
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 1.00
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 1.39
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 1.95
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 2.71
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 3.79
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 5.28
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 7.37
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 10.28
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 14.34
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 20.00
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [30 ◦ , − 30 ◦ , 30 ◦ ] stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.6−0.4−0.2 0.0 0.2 0.4 0.6 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 1.00
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 1.39
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 1.95
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 2.71
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 3.79
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 5.28
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 7.37
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 10.28
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
0.0 0.2
0.4 KL
BG Pagano
0.0 0.2 0.4
L/t = 14.34
KL BG Pagano
0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Stress distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−10 −5 0 5 10 t2σ11(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ22(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
−0.4
−0.2 0.0 0.2 0.4
L/t = 20.00
KL BG Pagano
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−10 −5 0 5 10 t2σ12(a/2, b/2, x3)/(p3λ2)
−0.4
−0.2 0.0 0.2
0.4 KL
BG Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Applications Cylindrical bending of laminates
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano
Displacement distributions for a [45 ◦ , − 45 ◦ ] 4 , 45 ◦ stack
−2.0−1.5−1.0−0.50.0 0.5 1.0 1.5 2.0 u1(0, b/2, x3)/(p3λ3)
−0.4
−0.2 0.0 0.2 0.4
x3/t
KL BG Pagano
−0.4−0.3−0.2−0.10.0 0.1 0.2 0.3 0.4 u2(a/2,0, x3)/(p3λ)
−0.4
−0.2 0.0 0.2 0.4 KLBG
Pagano
0.0 0.2 0.4 0.6 0.8 1.0 1.2 u3(a/2, b/2, x3)/huP ag3 i
−0.4
−0.2 0.0 0.2
0.4 KLBG
Pagano