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

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

48248 Ebh

PL =×

==∆3

2 3212

48 Ebh

PL ==′

∆1=4∆2

MP-206 5

Introduction

yxBxT E

E σσ2

No bonding ⇒ yByT E

E ∆=∆2

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

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

ρxρy

Curvature radii ρx and ρy on plane xz and yz

Positive bending moments ⇒ positive curvatures

MP-206 10

σxσy

ρxρy

ρε == ,

ρνσ

ρνσ 1

Stresses and strains

MP-206 11

ρρν

)1(121 2

νν −=

−= ∫

plate bending stiffness

∂∂+

∂∂−=

∂∂+

∂∂−=

Moment and curvature: isotropic plate

MP-206 12

∂∂−=

∂∂ ν Curvatures of opposite sign

Anticlastic surface

Case when My = 0

ρρρ111 ==

νρ +=

Case when Mx = My = M

MP-206 13

Myx (= − Mxy)Mxy

MxyMxy

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

(a) (b)

MxyMyx

Moments Mx, My and Mxy

Decomposition in Mt and Mn

MP-206 15

MtMn A

αααα cossinsincos BCMABMBCMABMACM xyxyyxn −−+=

ααα 2sinsincos 22xyyxn MMMM −+=

αααα sincoscossin BCMABMBCMABMACM xyxyyxt −+−=

αα 2cos2sin2 xy

Moment transformation

MP-206 16

MxyMxy

∫∫−−

−=⇒−=2/

xyxy zdzMzdydzdyM ττ

∫−

−=2/

xyxy zdzGM γ

How to relate Mxy and w?

Torsion moment Mxy

MP-206 17

−udz

∂w/∂x

∂∂−=

uxy ∂∂

∂−=∂∂+

∂∂=

wzGzdzGM

xyxy ∂∂∂−=

∂∂∂

∂∂∂=

∂∂∂=−= ∫∫

−−

223232/

)1()1(126

2 νν

Determination of shear strain γxy

MP-206 18

4.2. Classical laminate theory

MP-206 19

Lamina stress × strain behavior

γεε

τσσ

γεε

τσσ

662616

262212

161211

For layer k: σ k = [Q]k ε k

MP-206 20

−∆udz

∂w/∂x

),(),,(

yxwzyxw

wzyxvzyxv

wzyxuzyxu

=∂∂−=

∂∂−=

Stress and strain variation in the laminate

xyxyxy

κγγ

κεε

+=∂∂

∂−∂∂+

∂∂=

∂∂+

∂∂=

+=∂∂−

∂∂=

+=∂∂−

∂∂=

κκκ

γεε

MP-206 21

Stress and strain variation

strain distribution moduli stress

MP-206 22

Resultant laminate forces and moments

∑ ∫∫=− −

2/ 1 τσσ

τσσ

∑ ∫∫=− −

zdzzdz

2/ 1 τσσ

τσσ

zk−1zk

zN−1

MP-206 23

∑ ∫∑ ∫==

−−

662616

262212

161211

111 κ

γεε

τσσ

∑ ∫∑ ∫==

−−

662616

262212

161211

111 κ

γεε

τσσ

MP-206 24

[ ]∑∑ ∫=

662616

262212

161211

662616

262212

161211

Laminate matrices

[ ]∑∑ ∫=

Qzzzdz

1662616

262212

161211

662616

262212

161211

[ ]∑∑ ∫=

Qzzdzz

662616

262212

161211

662616

262212

161211

MP-206 25

κκκ

γεε

κκκ

γεε

662616

262212

161211

662616

262212

161211

662616

262212

161211

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

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

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

γxzγxz = w,x + ψx

−ψx

Rotation of plane section

MP-206 31

)(2 ,,,,

xyyxxyxy

ψψγ

+++=∂∂+

∂∂=

+=∂∂=

+=∂∂+

∂∂=

+=∂∂+

∂∂=

==∂∂=

Strain × displacement relations

u, v in-plane displacements

w transverse displacement

ψα rotation angle

MP-206 32

σyσxz

τxz τyz

Plate infinitesimal element: stress distributions

MP-206 33

Moments Shear forces

Membrane forces

Plate resultant forces and moments

MP-206 34

xy ∂∂

y ∂∂

x ∂∂+

xyyxxy

xyNyxN

yx ∂∂

Plate infinitesimal element: internal membrane forces

MP-206 35

dydx q

yx ∂∂

xy ∂∂

y ∂∂

x ∂∂+

y ∂∂

x ∂∂+

xyyxxy

Plate infinitesimal element: internal moments and shear forces

MP-206 36

Force equilibrium along x

Force equilibrium along y

∂∂

∂∂+ dxNdxdy

NNdyNdydx

∂∂

+ dyNdydxx

NNdxNdxdy

∂∂

Force equilibrium equations

0=+−

∂∂

∂∂+ qdxdydxQdxdy

QQdyQdydx

0=+∂

∂∂

Force equilibrium along z

MP-206 37

Moment equilibrium about x

Moment equilibrium about y

)( 222

∂∂++

∂∂

QQdxdydy

MMdxMdydx

xyxyxy

0=−∂

∂∂

yyxy Q

0=−∂

∂+∂

xxy Qx

Moment equilibrium equations

MP-206 38

yyyxxx nQnQQ +=

QPrescribed boundary shear force:

Natural boundary conditions: shear forces

MP-206 39

Prescribed boundary membrane forces: yyxx NN ,

yyyyyxxy

xxyxyxxx

Natural boundary conditions: membrane forces

MP-206 40

yyxx MM ,Prescribed boundary moments:

yyyyyxxy

xxyxyxxx

Natural boundary conditions: moments

MP-206 41

)()( ,,,,

xyyxxyxy

ψψγψεψε

ψγψγ

+=+===

Strain × displacement relations

γεε

τσσ

Constitutive relations for lamina k

MP-206 42

boundary conditions

yyxxyyyy

xxyxyxxx

yyyxxx

yyyxxy

yxyxxx

yxWVU ΘΘ ,,,,

Prescribed displacements and rotations

yyxxyyxx MMQNN ,,,,

Prescribed forces and moments

Equilibrium equations

MP-206 43

∑ ∫∑ ∫==

−−

∑ ∫∑ ∫==

−−

662616

262212

161211

111 ψψ

τσσ

∑ ∫∑ ∫==

−−

662616

262212

161211

111 ψψ

τσσ

MP-206 44

[ ]∑∑ ∫=

662616

262212

161211

662616

262212

161211

Laminate matrices

[ ]∑∑ ∫=

Qzzzdz

1662616

262212

161211

662616

262212

161211

[ ]∑∑ ∫=

Qzzdzz

662616

262212

161211

662616

262212

161211

[ ]∑∑ ∫=

s QzzdzQQ

11 5545

4544 )(][1

MP-206 45

4.4. Special cases of laminates

MP-206 46

Single layered configurations

21 ν−= Et

Single isotropic layer

2/)1(00

νDD]0[][ =B

)1(12 2

ν−= Et

Single specially orthotropic layer

tA ]0[][ =B

Single orthotropic layer

662616

262212

161211

tA ]0[][ =B

662616

262212

1612113

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

πππ )1(/.../

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

∆−

γεε

τσσ

Transformation into structural coordinate system

∆−

−−−

γεε

τσσ

sccscs

∆−

−−−−

sccscs

ααα

γεε

τσσ

662616

262212

161211

662616

262212

161211

−−

)(2022 21

αααααα

ααα

sccscs

MP-206 53

Thermal forces and momentsIntegration through the thickness

∑ ∫

1662616

262212

161211

1662616

262212

161211

ααα

κκκ

γεε

κκκ

γεε

662616

262212

161211

662616

262212

161211

662616

262212

161211

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

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

10× 90o

MP-206 58

Pre-failure deformation

======

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

03954.0/024.01977.0/12.0

08819.0/75.04409.0/27.2

==∆+−=∆−=

∆−=∆+=

TtNTtN

ττσσ

MP-206 59

Tsai-Hill failure criterion

21 =++−

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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