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Instituto Tecnológico de Aeronáutica

MP-206 1

Class notes

Analysis and design of composite structures


MP-206 2

4. Laminates


MP-206 3

Introduction

Two or more laminae bonded together

How will the laminate respond to loads?

Laminate properties are obtained from the laminae properties

Arbitrary orientation angles must be considered


MP-206 4

Introduction

Laminates have high bending stiffnessP P

3

3

3

33

1 812

48248 Ebh

PL

bh

PL

EI

PL =×

==∆3

3

3

33

2 3212

)2(48

48 Ebh

PL

hb

PL

IE

PL ==′

=∆

L L

∆1=4∆2


MP-206 5

Introduction

∆xT

∆xB

∆yT

∆yB

x

yxBxT E

E σσ2

1=

No bonding ⇒ yByT E

E ∆=∆2

1

Bonding ⇒ σyT = −σyB

Conditions to be satisfied:

• deformation compatibility

• stress × strain relations

• equilibrium


MP-206 6

4.1 Bending of thin isotropic plates


MP-206 7

Small thickness compared to other dimensions

Resists bending and membrane loads

Aeronautical panels

Different loadings and boundary conditions

Equilibrium described by fourth order differential equation

Thin plates


MP-206 8

z

xy My

My

Mx

Mx

Pure bending of thin plates

Bending moments Mx and My

Positive bending when they compress bottom surface

Plane sections remain plane after deformation

The mid plane does not deform


MP-206 9

σxσy

zdz

t/2

t/2

dxdy

n

ρxρy

n


Curvature radii ρx and ρy on plane xz and yz

Positive bending moments ⇒ positive curvatures


MP-206 10

σxσy

zdz

t/2

t/2

dxdy

n

ρxρy

n

yy

xx

zz

ρε

ρε == ,

+

−=

+

−=

xyy

yxx

EzEz

ρν

ρνσ

ρν

ρνσ 1

1,

1

1 22


Stresses and strains


MP-206 11

+=

+

−=

+=

+

−=

∫

∫

−

−

xy

t

t xyy

yx

t

t yxx

DdzEz

M

DdzEz

M

ρν

ρρν

ρν

ρν

ρρν

ρν

11

1

11

1

2/

2/2

2

2/

2/2

2

)1(121 2

32/

2/2

2

νν −=

−= ∫

−

Etdz

EzD

t

t

plate bending stiffness

∂∂+

∂∂−=

∂∂+

∂∂−=

2

2

2

2

2

2

2

2

x

w

y

wDM

y

w

x

wDM

y

x

ν

ν

2

2

2

2

1

1

y

w

x

w

y

x

∂∂−=

∂∂−=

ρ

ρ


Moment and curvature: isotropic plate


MP-206 12

Mx

Mx

2

2

2

2

x

w

y

w

∂∂−=

∂∂ ν Curvatures of opposite sign


Anticlastic surface

Case when My = 0

ρρρ111 ==

yx)1(

1

νρ +=

D

M

Case when Mx = My = M


MP-206 13

xyMy

MyMx

Mx

Myx

Myx (= − Mxy)Mxy

Mxy

τxy

zdz

t/2

t/2

dxdy

n

τyx

MxyMxy

A

B

C

D

E

F

Plate subject to bending and torsion

In the most general case there are tangential components

Tangent components produce torsion

Mxy and Myx are consequence of τxy ⇒ Mxy = −Myx


MP-206 14

xy

My

My

Mx

Mx

My

Mx

MtMn

αA

B

C

(a) (b)

Mxy

MxyMyx

Myx

Myx

Mxy


Moments Mx, My and Mxy

Decomposition in Mt and Mn


MP-206 15

My

Mx

MtMn A

B

C

Myx

Mxy

αααα cossinsincos BCMABMBCMABMACM xyxyyxn −−+=

ααα 2sinsincos 22xyyxn MMMM −+=

αααα sincoscossin BCMABMBCMABMACM xyxyyxt −+−=

αα 2cos2sin2 xy

yxt M

MMM +

−=


Moment transformation


MP-206 16

τxy

zdz

t/2

t/2

dxdy

n

τyx

MxyMxy

A

B

C

D

E

F

∫∫−−

−=⇒−=2/

2/

2/

2/

t

t

xyxy

t

t

xyxy zdzMzdydzdyM ττ

∫−

−=2/

2/

t

t

xyxy zdzGM γ

How to relate Mxy and w?


Torsion moment Mxy


MP-206 17

x

z

t/2

t/2

z

−udz

dx

∂w/∂x

y

wzv

x

wzu

∂∂−=

∂∂−=

yx

wz

x

v

y

uxy ∂∂

∂−=∂∂+

∂∂=

2

2γ

yx

wD

yx

wEt

yx

wGtdz

yx

wzGzdzGM

t

t

t

t

xyxy ∂∂∂−=

∂∂∂

+=

∂∂∂=

∂∂∂=−= ∫∫

−−

223232/

2/

22

2/

2/

)1()1(126

2 νν

γ


Determination of shear strain γxy


MP-206 18

4.2. Classical laminate theory


MP-206 19

Lamina stress × strain behavior

=

12

2

1

66

2212

1211

12

2

1

00

0

0

γεε

τσσ

Q

QQ

QQ

=

xy

y

x

xy

y

x

QQQ

QQQ

QQQ

γεε

τσσ

662616

262212

161211

For layer k: σ k = [Q]k ε k


MP-206 20

x

z

t/2

t/2

z

−∆udz

dx

∂w/∂x

),(),,(

),(),,(

),(),,(

0

0

yxwzyxw

y

wzyxvzyxv

x

wzyxuzyxu

=∂∂−=

∂∂−=

u0

Stress and strain variation in the laminate

xyxyxy

yyy

xxx

zyx

wz

x

v

y

u

x

v

y

u

zy

wz

y

v

y

v

zx

wz

x

u

x

u

κγγ

κεε

κεε

+=∂∂

∂−∂∂+

∂∂=

∂∂+

∂∂=

+=∂∂−

∂∂=

∂∂=

+=∂∂−

∂∂=

∂∂=

02

00

02

20

02

20

2

+

=

xy

y

x

xy

y

x

xy

y

x

z

κκκ

γεε

γεε

0

0

0


MP-206 21

Stress and strain variation

1

2

3

4

strain distribution moduli stress


MP-206 22

Resultant laminate forces and moments

∑ ∫∫=− −

=

=

N

k

z

z

kxy

y

xt

txy

y

x

xy

y

x k

k

dzdz

N

N

N

1

2/

2/ 1 τσσ

τσσ

∑ ∫∫=− −

=

=

N

k

z

z

kxy

y

xt

txy

y

x

xy

y

x k

k

zdzzdz

M

M

M

1

2/

2/ 1 τσσ

τσσ

k = 3

k = 2

k = 1

k

k = N

z2

z3

z1

z0

zk−1zk

zN−1

zN


MP-206 23

∑ ∫∑ ∫==

−−

+

=

=

N

k

z

zxy

y

x

xy

y

x

k

N

k

z

z

kxy

y

x

xy

y

x k

k

k

k

dzz

QQQ

QQQ

QQQ

dz

N

N

N

1 0

0

0

662616

262212

161211

111 κ

κκ

γεε

τσσ


∑ ∫∑ ∫==

−−

+

=

=

N

k

z

zxy

y

x

xy

y

x

k

N

k

z

z

kxy

y

x

xy

y

x k

k

k

k

zdzz

QQQ

QQQ

QQQ

zdz

M

M

M

1 0

0

0

662616

262212

161211

111 κ

κκ

γεε

τσσ


MP-206 24

[ ]∑∑ ∫=

−=

−=

=

=−

N

kkkk

N

k

z

z

k

Qzzdz

QQQ

QQQ

QQQ

AAA

AAA

AAA

Ak

k1

11

662616

262212

161211

662616

262212

161211

)(][1

Laminate matrices

[ ]∑∑ ∫=

−=

−=

=

=−

N

kkkk

N

k

z

z

k

Qzzzdz

QQQ

QQQ

QQQ

BBB

BBB

BBB

Bk

k1

21

2

1662616

262212

161211

662616

262212

161211

)(2

1][

1

[ ]∑∑ ∫=

−=

−=

=

=−

N

kkkk

N

k

z

z

k

Qzzdzz

QQQ

QQQ

QQQ

DDD

DDD

DDD

Dk

k1

31

3

1

2

662616

262212

161211

662616

262212

161211

)(3

1][

1


MP-206 25

+

=

+

=

xy

y

x

xy

y

x

xy

y

x

xy

y

x

xy

y

x

xy

y

x

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

BBB

BBB

BBB

AAA

AAA

AAA

N

N

N

κκκ

γεε

κκκ

γεε

662616

262212

161211

0

0

0

662616

262212

161211

662616

262212

161211

0

0

0

662616

262212

161211


shear-extension coupling membrane-bending coupling

bend-twist coupling


MP-206 26

Laminate matrices: weights

91h3/35.5h2h6h5h12

61h3/34.5h2h5h4h11

37h3/33.5h2h4h3h10

19h3/32.5h2h3h2h9

7h3/31.5h2h2hh8

h3/30.5h2hh07

h3/3−0.5h2h0−h6

7h3/3−1.5h2h−h−2h5

19h3/3−2.5h2h−2h−3h4

37h3/3−3.5h2h−3h−4h3

61h3/3−4.5h2h−4h−5h2

91h3/3−5.5h2h−5h−6h1

[zk3 − (zk−1)

3]/3[zk2 − (zk−1)

2]/2(zk − zk−1)zkzk−1k

12

11

10

9

8

7

6

5

4

3

2

1

6h

5h

4h

3h

2h

h

0

−h

−2h

−3h

−4h

−5h

−6h


MP-206 27

4.3. Mindlinlaminate plate theory


MP-206 28

Assumptions

Domain: Ω = (x,y,z) ∈ ℜ3|−t/2 ≤ z ≤ t/2, (x,y) ∈ ℜ2

σz = 0

u(x,y,z) = u(x,y) + zψx(x,y)

v(x,y,z) = v(x,y) + zψy(x,y)

w(x,y,z) = w(x,y)


MP-206 29

x

y

z

ψx

ψyCarefully check sign convention

Assumptions

The thickness t may be a function of x, y

σz = 0 is the plate stress assumption

Plane sections remain plane but not normal to mid surface


MP-206 30

x

zw,x

γxzγxz = w,x + ψx

w

−ψx

Rotation of plane section


MP-206 31

)(2 ,,,,

,,

,,

xyyxxyxy

yyyyy

xxxxx

zvu

x

v

y

u

zvy

v

zux

u

ψψγ

ψε

ψε

+++=∂∂+

∂∂=

+=∂∂=

+=∂∂=

xxxz

yyyz

zzz

wx

w

z

u

wy

w

z

v

wz

w

ψγ

ψγ

ε

+=∂∂+

∂∂=

+=∂∂+

∂∂=

==∂∂=

,

,

, 0

Strain × displacement relations

u, v in-plane displacements

w transverse displacement

ψα rotation angle


MP-206 32

τxy

zdz

t/2

t/2

dydx

n

τyx

z

yx

σyσxz

dz

t/2

t/2

dydx

n

z

yx

τxz τyz

Plate infinitesimal element: stress distributions


MP-206 33

∫

∫

∫

−

−

−

=

=

=

2/

2/

2/

2/

2/

2/

t

t

xyxy

t

t

yy

t

t

xx

dzzM

dzzM

dzzM

τ

σ

σ

Moments Shear forces

∫

∫

−

−

=

=

2/

2/

2/

2/

t

t

yzy

t

t

xzx

dzQ

dzQ

τ

τ

∫

∫

∫

−

−

−

=

=

=

2/

2/

2/

2/

2/

2/

t

t

xyxy

t

t

yy

t

t

xx

dzN

dzN

dzN

τ

σ

σ

Membrane forces

Plate resultant forces and moments


MP-206 34

t/2

t/2

dydx

dxx

NN xy

xy ∂∂

+

dyy

NN y

y ∂∂

+dx

x

NN x

x ∂∂+

xNyN

∫

∫

∫

−

−

−

==

=

=

2/

2/

2/

2/

2/

2/

t

t

xyyxxy

t

t

yy

t

t

xx

dzNN

dzN

dzN

τ

σ

σ

z

yx

xyNyxN

dyy

NN yx

yx ∂∂

+

Plate infinitesimal element: internal membrane forces


MP-206 35

t/2

t/2

dydx q

dyy

MM yx

yx ∂∂

+dx

x

MM xy

xy ∂∂

+

dyy

MM y

y ∂∂

+dx

x

MM x

x ∂∂+

dyy

QQ y

y ∂∂

+dx

x

QQ x

x ∂∂+

xQ

xM

yQ

yM

xyM

yxM

∫

∫

∫

∫

∫

−

−

−

−

−

=

=

==

=

=

2/

2/

2/

2/

2/

2/

2/

2/

2/

2/

t

t

yzy

t

t

xzx

t

t

xyyxxy

t

t

yy

t

t

xx

dzQ

dzQ

zdzMM

zdzM

zdzM

τ

τ

τ

σ

σ

z

yx

Plate infinitesimal element: internal moments and shear forces


MP-206 36

Force equilibrium along x

Force equilibrium along y

0=−

∂∂

++−

∂∂+ dxNdxdy

y

NNdyNdydx

x

NN yx

yxyxx

xx 0=

∂∂

+∂

∂y

N

x

N yxx

0=−

∂∂

++−

∂∂

+ dyNdydxx

NNdxNdxdy

y

NN xy

xyxyy

yy

0=∂

∂+

∂∂

x

N

y

N xyy

Force equilibrium equations

0=+−

∂∂

++−

∂∂+ qdxdydxQdxdy

y

QQdyQdydx

x

QQ y

yyx

xx

0=+∂

∂+

∂∂

qy

Q

x

Q yx

Force equilibrium along z


MP-206 37

Moment equilibrium about x

Moment equilibrium about y

02

)(

2

)(

2

)( 222

=+−

∂∂++

∂∂

+

+

∂∂

+−+

∂∂

+−

dyqdx

dyQ

dydx

y

QQdxdydy

y

QQ

dxdyy

MMdxMdydx

x

MMdyM

xx

xy

y

yyy

xyxyxy

0=−∂

∂+

∂∂

yyxy Q

y

M

x

M

0=−∂

∂+∂

∂x

xxy Qx

M

y

M

Moment equilibrium equations


MP-206 38

x

yQ

Qyy

Qxxy

x

yyyxxx nQnQQ +=

n

QPrescribed boundary shear force:

Natural boundary conditions: shear forces


MP-206 39

Prescribed boundary membrane forces: yyxx NN ,

x

yy

x

Nxy

Nxy

Nxx

Nyy

n

Nxx

Nyy

yyyyyxxy

xxyxyxxx

NnNnN

NnNnN

=+

=+

Natural boundary conditions: membrane forces


MP-206 40

yyxx MM ,Prescribed boundary moments:

x

yy

x

Mxx

Myy

Mxy

Mxy

n

Myy

Mxx

yyyyyxxy

xxyxyxxx

MnMnM

MnMnM

=+

=+

Natural boundary conditions: moments


MP-206 41

)()( ,,,,

,,

,,

xyyxxyxy

yyyyy

xxxxx

zvu

zv

zu

ψψγψεψε

+++=

+=+=

yyyz

xxxz

zzz

w

w

w

ψγψγ

ε

+=+===

,

,

, 0

Strain × displacement relations

kxy

y

x

k

kxy

y

x

Q

=

γεε

τσσ

][kxz

yzks

kxz

yz Q

=

γγ

ττ

][

Constitutive relations for lamina k


MP-206 42

boundary conditions

0

0

0

0

0

,,

,,

,,

,,

,,

=−+

=−+

=++

=+

=+

yyxxyyyy

xxyxyxxx

yyyxxx

yyyxxy

yxyxxx

QMM

QMM

qQQ

NN

NN

yxWVU ΘΘ ,,,,

Prescribed displacements and rotations

yyxxyyxx MMQNN ,,,,

Prescribed forces and moments

Equilibrium equations


MP-206 43

∑ ∫∑ ∫==

−−

++

=

=

N

k

z

z xx

yy

k

N

k

z

z kxz

yz

x

yk

k

k

k

dzw

w

QQ

QQdz

Q

Q

1 ,

,

5545

4544

111

ψψ

ττ


∑ ∫∑ ∫==

−−

++

+

=

=

N

k

z

zxyyx

yy

xx

xy

y

x

k

N

k

z

z

kxy

y

x

xy

y

x k

k

k

k

dzz

vu

v

u

QQQ

QQQ

QQQ

dz

N

N

N

1,,

,

,

,,

,

,

662616

262212

161211

111 ψψ

ψψ

τσσ

∑ ∫∑ ∫==

−−

++

+

=

=

N

k

z

zxyyx

yy

xx

xy

y

x

k

N

k

z

z

kxy

y

x

xy

y

x k

k

k

k

zdzz

vu

v

u

QQQ

QQQ

QQQ

zdz

M

M

M

1,,

,

,

,,

,

,

662616

262212

161211

111 ψψ

ψψ

τσσ


MP-206 44

[ ]∑∑ ∫=

−=

−=

=

=−

N

kkkk

N

k

z

z

k

Qzzdz

QQQ

QQQ

QQQ

AAA

AAA

AAA

Ak

k1

11

662616

262212

161211

662616

262212

161211

)(][1

Laminate matrices

[ ]∑∑ ∫=

−=

−=

=

=−

N

kkkk

N

k

z

z

k

Qzzzdz

QQQ

QQQ

QQQ

BBB

BBB

BBB

Bk

k1

21

2

1662616

262212

161211

662616

262212

161211

)(2

1][

1

[ ]∑∑ ∫=

−=

−=

=

=−

N

kkkk

N

k

z

z

k

Qzzdzz

QQQ

QQQ

QQQ

DDD

DDD

DDD

Dk

k1

31

3

1

2

662616

262212

161211

662616

262212

161211

)(3

1][

1

[ ]∑∑ ∫=

−=

−=

=

=

−

N

kkskk

N

k

z

z k

s QzzdzQQ

QQ

AA

AAA

k

k1

11 5545

4544

5545

4544 )(][1


MP-206 45

4.4. Special cases of laminates


MP-206 46

Single layered configurations

21 ν−= Et

A

Single isotropic layer

−=

2/)1(00

01

01

][

νν

νAA

−=

2/)1(00

01

01

][

νν

νDD]0[][ =B

)1(12 2

3

ν−= Et

D

Single specially orthotropic layer

=

66

2212

1211

00

0

0

][

Q

QQ

QQ

tA ]0[][ =B

=

66

2212

12113

00

0

0

12][

Q

QQ

QQt

D

Single orthotropic layer

=

662616

262212

161211

][

QQQ

QQQ

QQQ

tA ]0[][ =B

=

662616

262212

1612113

12][

QQQ

QQQ

QQQt

D


MP-206 47

Symmetric laminates

Symmetric laminates are symmetric with respect to both geometry and material

For every layer k there must be another layer k’ symmetrically located about the mid plane with the same material and fiber orientation angle. Notice that it is possible to have symmetric laminates with an odd number of layers

In this case it is easy to show that [B] = [0]. Therefore, there is no membrane-bending coupling

Unsymmetric laminates present strong curvatures after cure


MP-206 48

Quasi-isotropic laminates

Laminates that possess isotropic extensional stiffness

−n

n

nn

πππ )1(/.../

2//0

Balanced laminates

Pairs of layers ±θ with same thickness, not necessarily symmetric

Anti-symmetric laminates

Usually needed to avoid coupling between bending and extension

Special applications where coupling is required


MP-206 49

Cross-ply laminates

All layers at 0o or 90o

Angle-ply laminates

All layers at −α or +α

Hybrid laminates

Mixture layers of two or more different materials

Matrices must be cure compatible


MP-206 50

4.5. Hygrothermalstresses


MP-206 51

Hygrothermal effects

Purely mechanical analyses are insufficient to describe the behavior of laminates subject to temperature gradients

Thermal expansion coefficients must be known

][ βασε cTS ∆+∆+=

total strain

thermal strainmechanical strain

)]([ βαεσ cTQ ∆−∆−=

hygroscopic strain


MP-206 52

Thermal effects

Orthotropic lamina in plane stress

∆−

=

000

0

0

2

1

12

2

1

66

2212

1211

12

2

1

αα

γεε

τσσ

T

Q

QQ

QQ

Transformation into structural coordinate system

∆−

−−−

=

−−−

02200

0

0

2

2

2

1

22

22

22

66

2212

1211

22

22

22

αα

γεε

τσσ

T

sccscs

cscs

cssc

Q

QQ

QQ

sccscs

cscs

cssc

xy

y

x

xy

y

x

∆−

=

∆

−−−−

=

−

xy

y

x

xy

y

x

xy

y

x

xy

y

x

T

QQQ

QQQ

QQQ

T

sccscs

cscs

cssc

QQQ

QQQ

QQQ

ααα

γεε

αα

γεε

τσσ

662616

262212

161211

2

1

1

22

22

22

662616

262212

161211

022

−++

∆=

∆

−−

−=

)(2022 21

22

21

22

21

2

1

22

22

22

αααααα

αα

ααα

cs

cs

sc

TT

sccscs

cscs

cssc

xy

y

x


MP-206 53

Thermal forces and momentsIntegration through the thickness

∑ ∫

∑ ∫

=

=

−

−

∆

=

∆

=

N

k

z

z

kxy

y

x

k

Txy

Ty

Tx

N

k

z

z

kxy

y

x

k

Txy

Ty

Tx

k

k

k

k

zdzT

QQQ

QQQ

QQQ

M

M

M

dzT

QQQ

QQQ

QQQ

N

N

N

1662616

262212

161211

1662616

262212

161211

1

1

ααα

ααα

−

+

=

−

+

=

Txy

Ty

Tx

xy

y

x

xy

y

x

xy

y

x

Txy

Ty

Tx

xy

y

x

xy

y

x

xy

y

x

M

M

M

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

N

N

N

BBB

BBB

BBB

AAA

AAA

AAA

N

N

N

κκκ

γεε

κκκ

γεε

662616

262212

161211

0

0

0

662616

262212

161211

662616

262212

161211

0

0

0

662616

262212

161211


MP-206 54

Thermal effects

In virtually all laminates thermal effects cause residual thermal stresses because of the mismatch in thermal expansion coefficients from one lamina to the others

If the laminate is completely free there are no thermal residualmembrane forces or moments, i.e., [A] ε + [B] κ − NT = 0 and [B] ε + [D] κ − MT = 0. This is usually the condition of the laminate right after curing

If the laminate is constrained thermal residual forces and moments might arise

∆T ≠ 0


MP-206 55

Thermal effects

Even in completely free heterogeneous laminates thermal residualstresses will arise


MP-206 56

Thermal effects


MP-206 57

Strength of a cross-play laminate

x

y

E1 = 53.78 GPa, E2 = 17.93 GPa

ν12 = 0.25, G12 = 8.62 GPa

α1 = 6.3×10−6 oC−1, α2 = 20.52×10−6 oC−1

Xt = Xc = 1035 MPa, Yt = 27.6 MPa

Yc = 138 MPa, S = 41.4 MPa

Two 0o layers and ten 90o layers

Layer thickness: 0.127 mm

0o

0o

10× 90o


MP-206 58

Pre-failure deformation

000

0

0

][

00

0

0

][900

2900

1900

66

1112

1222

90

66

2212

1211

0

======

=

=

xyxy

xy

yx

Q

QQ

QQ

Q

Q

QQ

QQ

Q

αααααααα

A11 = 0.037207 GN/m A22 = 0.074405 GN/m

A12 = 0.0069767 GN/m A66 = 0.013137 GN/m

NxT = 0.41049 t ∆T MPa/oC, Ny

T = 0.43407 t ∆T MPa/oC, NxyT = 0

MxT = My

T = MxyT = 0

00

03954.0/024.01977.0/12.0

08819.0/75.04409.0/27.2

900

900

900

==∆+−=∆−=

∆−=∆+=

xyxy

xyxy

xxxx

TtNTtN

TtNTtN

ττσσ

σσ


MP-206 59

Tsai-Hill failure criterion

12

212

2

22

221

2

21 =++−

SYXX

τσσσσ 2222 )/( XYXyyxx =+− σσσσ

22 )(4621.05.57365.1 TYTt

N x ∆−+∆= Y [MPa] and ∆T [oC]

A) Cure at 132oC and laminate used at 21oC ⇒ ∆T = −111oC

0o layer: Nx/t = 43.37 MPa


B) Cure at 21oC and laminate used at 21oC ⇒ ∆T = 0oC



εx = 0.098%

0o layer:

class notes - instituto tecnológico de aeronáuticaarfaria/mp206_04.pdf · instituto tecnológico...

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