© 2003, P. Joyce

MacromechanicalMacromechanical Analysis of Analysis of LaminatesLaminates

© 2003, P. Joyce

Stress Stress ––Strain Relations for an Strain Relations for an Isotropic BeamIsotropic Beam

Consider a prismatic beam of cross-section A under an applied axial load P.

Px

z

EAP

AP

xxxx == εσ and

Assumes that the normal stress and strain are uniform and constant in thebeam and are dependent on the load P being applied at the centroid ofthe cross-section.

One dimensional analysis

© 2003, P. Joyce


Consider the same prismatic beam in a pure bending moment M.The beam is assumed to initially straight and the applied loads pass througha plane of symmetry to avoid twisting.

x

z

and I

Mzzxxxx == σ

ρε

Neglects transverse shear.Assumes plane sections remain plane.I s the second moment of area (often mistakenly referred to as the moment of inertia.)

M

© 2003, P. Joyce


Finally consider the beam under combined loading.

Px

z

κεερ

εε

ε

z

z

EIMz

EAP

xx

xx

xx

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+=

0

01

Where ε0 is the strain at y = 0 (through the centroid), and κ = the curvature of the beam.

M

© 2003, P. Joyce

StrainStrain--Displacement Equations Displacement Equations for an Anisotropic Laminatefor an Anisotropic Laminate

Use Classical Lamination Theory (CLT) to develop similar relationships in 3D for a laminate (plate) under combined shear and axial forces and bending and twisting moments.The following assumptions are made to develop the relationships:

Each lamina is homogeneous and orthotropicThe laminate is thin and is loaded in plane only (plane stress)Displacements are continuous and small throughout the laminateEach lamina is elastic (stress-strain relations are linear)No slip occurs between the lamina interfacesTransverse shear strains are negligibleThe transverse normal strain is negligible

© 2003, P. Joyce

StrainStrain--Displacement Equations Displacement Equations for an Anisotropic Laminatefor an Anisotropic Laminate

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

xy

y

x

xy

y

x

zκκκ

γεε

γεε

0

0

0

Consider the general case of a plate under in-plane shear and axial loading, as well as bending and twisting moments.

Nx

Ny

Nxy

Nyx

Mxy

Mx

My

Myx

curvatures midplane theare and

strains midplane theare where0

0

0

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

xy

y

x

κκκ

γεε

Can derive the followingStrain-displacement equation:

© 2003, P. Joyce

Strain and Stress in a LaminateStrain and Stress in a Laminate

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQQQQQQQ

γεε

τσσ

If the strains are known at any point along the thickness of the laminate, the stress-strain equation calculates the global stresses in each lamina

result, previous thengSubstituti

0

0

0

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQQQQQQQ

zQQQQQQQQQ

κκκ

γεε

τσσ

The reduced transformed stiffness matrix, Qxy corresponds to that of the ply located at the point along the thickness of the laminate.

© 2003, P. Joyce


Laminate Strain Variation Stress Variation

The stresses vary linearly only through the thickness of each lamina.The stresses may jump from lamina to lamina since the transformed reduced stiffness matrix changes from ply to ply.

© 2003, P. Joyce


These global stresses can then be transformed to local stresses through the Transformation equation.Likewise, the local strains can be transformed to global strains.Can then be used in the Failure criteria discussed previously.All that remains is how to find the midplane strains and curvatures of a laminate if the applied loading is known?

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment ResultantsThe stresses in each lamina can be integrated to give resultant forces and moments (or applied forces and moments.)Since the forces and moments applied to a laminate will be known, the midplane strains and plate curvatures can then be found.Consider a laminate made of n plies as shown, each ply has a thickness tk.The location of the midplane is h/2 from the top or bottom surface.The z coordinate of each ply surface is given by

h0 h/2mid-planeh1

surface) (bottom 2

and surface) (top 2 110 thhhh −==

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment ResultantsIntegrating the global stresses in each lamina gives the resultant forces per unit length in the x-y plane through the laminate thickness as

dzN

dzN

dzN

h

hxyxy

h

hyy

h

hxx

∫

∫

∫

−

−

−

=

=

=

2/

2/

2/

2/

2/

2/

τ

σ

σ

Similarly, integrating the stresses in each lamina gives he resulting moments per unit length in the x-y plane through the thickness of the laminate.

zdzM

zdzM

dzzM

h

hxyxy

h

hyy

h

hxx

∫

∫

∫

−

−

−

=

=

=

2/

2/

2/

2/

2/

2/

τ

σ

σNx, Ny = normal force/unit length

Nxy = shear force/unit length

Mx, My = bending moment/unit length

Mxy = twisting moment/unit length

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment ResultantsIn matrix form

zdzMMM

dzNNN

h

hxy

y

x

xy

y

x

h

hxy

y

x

xy

y

x

∫

∫

−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

2/

2/

2/

2/

τσσ

τσσ

∑ ∫

∑ ∫

=

=

−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

n

k

h

hxy

y

x

xy

y

x

n

k

h

hxy

y

x

xy

y

x

zdzMMM

dzNNN

k

k

k

k

1

1

1

1

τσσ

τσσ

Substituting

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQQQQQQQ

γεε

τσσ

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment Resultants

The resultant forces and moments can be written in terms of the midplane strains and curvatures

dzzQQQQQQQQQ

zdzQQQQQQQQQ

MMM

zdzQQQQQQQQQ

dzQQQQQQQQQ

NNN

xy

y

xn

k

h

hssysxs

ysyyxy

xsxyxx

xy

y

xn

k

h

hssysxs

ysyyxy

xsxyxx

xy

y

x

xy

y

xn

k

h

hssysxs

ysyyxy

xsxyxx

xy

y

xn

k

h

hssysxs

ysyyxy

xsxyxx

xy

y

x

k

k

k

k

k

k

k

k

2

10

0

0

1

10

0

0

1

11

11

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∑ ∫∑ ∫

∑ ∫∑ ∫

==

==

−−

−−

κκκ

γεε

κκκ

γεε

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment Resultants

Since the midplane strains and plate curvatures are independent of the z coordinate and the transformed reduced stiffness matrix is a constant for each ply —

∑ ∫∫

∑ ∫∫

=

=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−

n

k

h

hxy

y

x

ssysxs

ysyyxy

xsxyxxh

hxy

y

x

kssysxs

ysyyxy

xsxyxx

xy

y

x

n

k

h

hxy

y

x

ssysxs

ysyyxy

xsxyxxh

hxy

y

x

kssysxs

ysyyxy

xsxyxx

xy

y

x

k

k

k

k

k

k

k

k

dzzQQQQQQQQQ

zdzQQQQQQQQQ

MMM

zdzQQQQQQQQQ

dzQQQQQQQQQ

NNN

1

2

0

0

0

1 0

0

0

11

11

κκκ

γεε

κκκ

γεε

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment ResultantsFrom the geometry (and a little calculus) we can solve the integrals

mid-planeh/2h0

( )

( )

( )31

32

21

2

1

31

21

1

1

1

−

−

−

−=

−=

−=

∫

∫

∫

−

−

−

kk

h

h

kk

h

h

kk

h

h

hhdzz

hhzdz

hhdz

k

k

k

k

k

k h1

© 2003, P. Joyce

Force and Moment ResultantsForce and Moment ResultantsFurthermore only the stiffnesses are unique for each layer, k.Thus, [ε0]x,y and [κ]x,y can be factored outside the summation sign

[ ] [ ] ( ) [ ] [ ] ( ) [ ]

[ ] [ ] ( ) [ ] [ ] ( ) [ ] yx

n

kkk

kyxyx

n

kkk

kyxyx

yx

n

kkk

kyxyx

n

kkk

kyxyx

hhQhhQM

hhQhhQN

,1

31

3,,

0

1

21

2,,

,1

21

2,,

0

11,,

31

21

21

κε

κε

⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡−=

⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡−=

∑∑

∑∑

=−

=−

=−

=−

Define —

[ ] ( ) [ ] ( ) [ ] ( )∑∑∑=

−=

−=

− −=−=−=n

kkk

kyxij

n

kkk

kyxij

n

kkk

kyxij hhQDhhQBhhQA

1

31

3,

1

21

2,

11, 3

1 ,21 ,

[A], [B], [D] are called the extensional, coupling, and bending stiffness matrices, respectively.

© 2003, P. Joyce

Laminated Composite AnalysisLaminated Composite Analysis[ ] [ ][ ] [ ][ ][ ] [ ][ ] [ ][ ] yxijyxijyx

yxijyxijyx

DBM

BAN

,,0

,

,,0

,

κε

κε

+=

+=

Combine into one general expression for laminate composite analysis relating the in-plane forces and moments to the midplane strains and curvatures —

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

y

x

xy

y

x

xy

y

x

xy

y

x

DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA

MMMNNN

κκκγεε

0

0

0

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

© 2003, P. Joyce

Laminated Composite AnalysisLaminated Composite Analysis

The extensional stiffness matrix [A] relates the resultant in-plane force to the in-plane strains.The bending stiffness matrix [D] relates the resultant bending moments to the plate curvatures.The coupling stiffness matrix [B] relates the force and moment terms to the midplane strains and midplane curvatures.

© 2003, P. Joyce

Laminate Special CasesLaminate Special CasesSymmetric: [B] = 0

Load-deformation equation and moment-curvature relation decoupled.

Balanced: A16 = A26 = 0.Symmetric and Balanced:

Orthotropic with respect to inplane behavior.0

11 120

11 22

x x

y y

N A AN A A

εε⎡ ⎤⎡ ⎤ ⎡ ⎤

= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦0

66xy xyN A γ=

© 2003, P. Joyce

Laminate Special CasesLaminate Special CasesCross-Ply: A16 = A26 = B16 = B26 = D16 = D26 =0.

Some decoupling of the six equations.

Orthotropic with respect to both inplane and bending behavior.

011 12 11 12

012 22 12 22

011 12 11 12

012 22 12 22

x x

y y

x x

y y

N A A B BN A A B BM B B D DM B B D D

εεκκ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦0

66 660

66 66

xy xy

xy xy

N A BM B D

γκ⎡ ⎤⎡ ⎤ ⎡ ⎤

= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

© 2003, P. Joyce

Laminate Special CasesLaminate Special CasesSymmetric Cross-Ply:

[B] =0A16 = A26 = D16 = D26 =0.Significant decoupling

Orthotropic with respect to both inplane and bending behavior.

011 12

011 22

x x

y y

N A AN A A

εε⎡ ⎤⎡ ⎤ ⎡ ⎤

= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

066xy xyN A γ=

011 12

011 22

x x

y y

M D DM D D

κκ⎡ ⎤⎡ ⎤ ⎡ ⎤

= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

066xy xyM D κ=

© 2003, P. Joyce

Laminated Composite AnalysisLaminated Composite AnalysisThe following are steps for analyzing a laminated composite subjected to the

applied forces and moments:1. Find the values of the reduced stiffness matrix [Qij] for each ply.2. Find the value of the transformed reduced stiffness matrix [Qxy].3. Find the coordinates of the top and bottom surfaces of each ply.4. Find the 3 stiffness matrices [A], [B], and [D].5. Calculate the midplane strains and curvatures using the 6 simultaneous equations

(substitute the stiffness matrix values and the applied forces and moments).6. Knowing the z location of each ply compute the global strains in each ply.7. Use the stress-strain equation to find the global stresses.8. Use the Transformation equation to find the local stresses and strains.

© 2003, P. Joyce

Laminate CompliancesLaminate Compliances

Since multidirectional laminates are characterized by stress discontinuities from ply to ply, it is preferable to work with strains which are continuous through the thickness.For this reason it is necessary to invert the load-deformation relations and express strains and curvatures as a function of applied loads and moments.

© 2003, P. Joyce

Laminate CompliancesLaminate CompliancesPerforming matrix inversions

⎥⎦⎤

⎢⎣⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

MN

dcba

MMMNNN

dddcccdddcccdddcccbbbaaabbbaaabbbaaa

xy

y

x

xy

y

x

xy

y

x

xy

y

x

κε

κκκγεε

0

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

0

0

0

briefin or

© 2003, P. Joyce

Laminate CompliancesLaminate CompliancesWhere [a], [b], [c], and [d] are the laminate extensional, coupling, and bending compliance matrices obtained as follows:

[ ] [ ] [ ][ ]{ }[ ][ ] [ ][ ][ ] [ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ][ ][ ] [ ] [ ][ ]{ }[ ]BABDD

ABC

BAB

Dd

bcCDc

DBb

CDBAa

T

1*

1*

1*

1*

*1*

1**

*1**1

and

also

−

−

−

−

−

−

−−

−=

=

−=

=

=−=

=

−=

NB: the compliances that relate midplane strains to applied moments are not identical to those that relate curvatures to in-plane loads.

© 2003, P. Joyce

Engineering Constants for a Engineering Constants for a MultiMulti--Axial LaminateAxial Laminate

From the laminate compliances we can compute the engineering constants —

ss

yssy

yy

syys

xx

sxxs

ss

xssx

yy

xyyx

xx

yxxy

ssxy

yyy

xxx

aa

aa

aa

aa

aa

aa

haG

haE

haE

===

=−=−=

===

ηηη

ηνν

111

As in UD lamina, symmetry implies —

xy

sy

y

ys

xy

sx

x

xs

y

yx

x

xy

GEGEEEηηηηνν

=== , ,

© 2003, P. Joyce

Engineering Constants for a Engineering Constants for a MultiMulti--Axial LaminateAxial Laminate

Computational Procedure for Determination of Engineering ElasticProperties

1. Determine the engineering constants of UD layer, E1, E2, ν12, and G12.2. Calculate the layer stiffnesses in the principal material axes, Q11, Q22, Q12,

and Q66.3. Enter the fiber orientation of each layer, k.4. Calculate the transformed stiffnesses [Q]x,y of each layer, k.5. Enter the through thickness coordinates of the layer surfaces.6. Calculate the laminate stiffness matrices [A], [B], and [D].7. Calculate the laminate compliance matrix [a].8. Enter total laminate thickness, h.9. Calculate the laminate engineering properties in global, x, y axes.

© 2003, P. Joyce

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Transcript of © 2003, P. Joyce