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HAL Id: hal-01134568

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Submitted on 23 Mar 2015

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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�

(2)

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

(3)

The 3D Problem

 

 

 

 

 

 

σ

ij,jt

= 0 on Ω.

σ

ijt

= C

ijkl

(x

3

) ε

tkl

on Ω . σ

i3t

= ± T

i±

on ω

±

. ε

tij

= u

(i,j)t

on 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

3

e

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

(4)

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

(5)

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

3

e

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

(6)

Introduction

Some energy principles...

I Potential energy:

u

3 D = argmin

u ∈KA

3D

W 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

3D

n

W ∗3 D σ

o

where W ∗3 D = 1 2

Z

σ

: S

∼∼

: σ

d Ω

(7)

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

∗3D

σ ˆ

− C

: ε

u

ˆ

provides an error estimate in terms of constitutive equation.

Prager and Synge (1947); Morgenstern (1959); Braess et al. (2010)

(8)

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...

(9)

Contents

The case of homogeneous and isotropic plates The case of laminated plates

Applications

(10)

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

3

t/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

(11)

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

t

2

2t

• dx 3

2D equilibrium equations:

h σ α3,α + σ 33,3 i = 0 h x 3αβ,β + σ α3,3 ) i = 0 ⇒

Q α,α + P 3 = 0 M αβ,β − Q α = 0

Boussinesq (1871); Mindlin (1951)...

(12)

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

33

6 = 0!!

In-plane stress:

σ αβ = C αβγδ

σ

ε δγ = x 3 C αβγδ

σ

K δγ

where C αβγδ

σ

= C αβγδC

αβ33

C

3333

C

33γδ

:

(13)

The Kirchhoff-Love plate problem

Bending constitutive equation:

M αβ = h x 3 σ αβ i =

x 3 2 C αβγδ

σ

K 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

(14)

The case of homogeneous and isotropic plates

Building SA 3 D fields

σ ˆ αβ = x 3 C αβγδ

σ

d 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

(15)

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

(16)

The case of homogeneous and isotropic plates

Building KA 3 D fields

Integration

ˆ u 3 =

Z x

3

ε ˆ 33 (z )dz + U 3 = u M 3αβ (x 3 )M βα

| {z }

∼t

1

+ U 3

|{z}

∼t

−1

where: u M 3αβ (x 3 ) = ν 2Et

12x 3 2 t 2 − 1

δ αβ and D

u M 3αβ (x 3 ) E

= 0

(17)

Building KA 3 D fields

Integration

ˆ u 3 =

Z x

3

ε ˆ 33 (z )dz + U 3 = u M 3αβ (x 3 )M βα

| {z }

∼t

1

+ U 3

|{z}

∼t

−1

ˆ u α =

Z x

3

2ˆ 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

where:

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

(18)

The case of homogeneous and isotropic plates

Building KA 3 D fields

Integration

ˆ u 3 =

Z x

3

ε ˆ 33 (z )dz + U 3 = u M 3αβ (x 3 )M βα

| {z }

∼t

1

+ U 3

|{z}

∼t

−1

ˆ u α =

Z x

3

2ˆ 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

(19)

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

3

P 3

S αβ33 0 0 0 0 S 3333

 ∼ t 2

(20)

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 ...

(21)

Reissner’s original plate model (1945)

min of W

∗3D

Let 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

RM

W ∗3 D σ ˆ

≤ W ∗3 D σ

KL

is a better approximation of W ∗3 D σ 3 D

(22)

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 }

(23)

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 δ

0 5Gt 6 δ

0

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

1

γ α = 6

5Gt Q α ∼ t 1

(24)

The case of laminated plates

Contents

The case of homogeneous and isotropic plates The case of laminated plates

Applications

(25)

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

2

u 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.

(26)

The case of laminated plates

Building SA 3 D fields

ˆ σ αβ = x 3 C αβγδ

σ

d 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

(27)

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

(28)

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

12,2

Q

2

= R

121

+ R

222

= M

21,1

+ M

22,2

(29)

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

(30)

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,γ

R

αββ,α

+ P

3

= 0

M

αβ

Φ

αβ,

+ R

αβγ

αβγ

+ I

αβγ

U

3,

)

M

αβ

Φ

αβγ

n

γ

+ R

αββ

n

α

U

3

RM:

Q

α

= M

αβ,β

M

αβ

ϕ

(α,β)

+

M

αβ

n

β

ϕ

α

+

(31)

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,γ

R

αββ,α

+ P

3

= 0

M

αβ

Φ

αβ,

+ R

αβγ

αβγ

+ I

αβγ

U

3,

)

M

αβ

Φ

αβγ

n

γ

+ R

αββ

n

α

U

3

RM:

Q

α

= M

αβ,β

Q

α,α

+ P

3

= 0

M

αβ

ϕ

(α,β)

+ Q

α

α

+ U

3,α

)

M

αβ

n

β

ϕ

α

+

Q

α

n

α

U

3

When the plate is homogeneous, BG is exactly turned into RM

(32)

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,γ

R

αββ,α

+ P

3

= 0

M

αβ

Φ

αβ,

+ R

αβγ

αβγ

+ I

αβγ

U

3,

)

M

αβ

Φ

αβγ

n

γ

+ R

αββ

n

α

U

3

RM:

Q

α

= M

αβ,β

M

αβ

ϕ

(α,β)

+

M

αβ

n

β

ϕ

α

+

(33)

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

(34)

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...

(35)

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...

(36)

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

(37)

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

(38)

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

(39)

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

(40)

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

(41)

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

(42)

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

(43)

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

(44)

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

(45)

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

(46)

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

(47)

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

(48)

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

(49)

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

(50)

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

(51)

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

(52)

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

(53)

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

(54)

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

(55)

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

(56)

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

(57)

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

(58)

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

(59)

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

(60)

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

(61)

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

(62)

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

(63)

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

(64)

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

(65)

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

(66)

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

(67)

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

(68)

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

(69)

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

(70)

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

(71)

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

(72)

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

(73)

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

(74)

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

(75)

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

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