8 PAX Short Course Laminat-Analysis

8/12/2019 8 PAX Short Course Laminat-Analysis

1/29

2003, P. Joyce


2/29

MacromechanicalMacromechanical Analysis ofAnalysis ofLaminatesLaminates


3/29

2003, P. Joyce

StressStressStrain Relations for anStrain Relations for an

Isotropic BeamIsotropic Beam

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

Px

z

EA

P

A

P

xxxx == and

Assumes that the normal stress and strain are uniform and constant in the

beam and are dependent on the load P being applied at the centroid of

the cross-section.One dimensional analysis


4/29

2003, P. Joyce


Isotropic BeamIsotropic Beam

Consider the same prismatic beam in a pure bending momentM.The beam is assumed to initially straight and the applied loads pass through

a plane of symmetry to avoid twisting.

x

z

andI

Mzzxxxx ==

Neglects transverse shear.

Assumes plane sections remain plane.

Is the second moment of area (often mistakenly referred to as the moment of inertia.)

M


5/29

2003, P. Joyce


Isotropic BeamIsotropic BeamFinally consider the beam under combined loading.

Px

z

z

z

EI

Mz

EA

P

xx

xx

xx

+=

+=

+=

0

0

1

Where 0 is the strain aty = 0 (through the centroid),and = the curvature of the beam.

M


6/29

2003, P. Joyce

StrainStrain--Displacement EquationsDisplacement Equations

for an Anisotropic Laminatefor an Anisotropic Laminate Use Classical Lamination Theory (CLT) to develop similar

relationships in 3D for a laminate (plate) under combinedshear and axial forces and bending and twisting moments.

The following assumptions are made to develop the

relationships: Each lamina is homogeneous and orthotropic

The laminate is thin and is loaded in plane only (plane stress)

Displacements are continuous and small throughout the laminate

Each lamina is elastic (stress-strain relations are linear)

No slip occurs between the lamina interfaces

Transverse shear strains are negligible

The transverse normal strain is negligible


7/29

2003, P. Joyce

StrainStrain--Displacement EquationsDisplacement Equations

for an Anisotropic Laminatefor an Anisotropic Laminate

+

=

xy

y

x

xy

y

x

xy

y

x

z

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

curvaturesmidplanetheareand

strainsmidplanethearewhere0

0

0

xy

y

x

xy

y

x

Can derive the following

Strain-displacement equation:


8/29

2003, P. Joyce

Strain and Stress in a LaminateStrain and Stress in a Laminate

=

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQ

QQQ

QQQ

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,previousthengSubstituti

0

0

0

+

=

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQ

QQQ

QQQ

z

QQQ

QQQ

QQQ

The reduced transformed stiffness matrix, Qxy corresponds to that of the ply

located at the point along the thickness of the laminate.


9/29

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.


10/29

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?


11/29

2003, P. Joyce

Force and Moment ResultantsForce and Moment Resultants The stresses in each lamina can be integrated to give resultant forces andmoments (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 nplies 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

h0h/2

mid-planeh

1

surface)(bottom2

andsurface)(top2

110 th

hh

h ==


12/29

2003, P. Joyce

Force and Moment ResultantsForce and Moment Resultants

Integrating the global stresses in each lamina gives the resultant

forces per unit length in thex-yplane through the laminate

thickness as

dzN

dzN

dzN

h

h

xyxy

h

h

yy

h

h

xx

=

=

=

2/

2/

2/

2/

2/

2/

Similarly, integrating the stresses in each lamina gives he resulting

moments per unit length in thex-yplane through the thickness of

the laminate.

zdzM

zdzM

dzzM

h

h

xyxy

h

h

yy

h

h

xx

=

=

=

2/

2/

2/

2/

2/

2/

Nx, Ny = normal force/unit length

Nxy = shear force/unit length

Mx, My = bending moment/unit length

Mxy = twisting moment/unit length


13/29

2003, P. Joyce

Force and Moment ResultantsForce and Moment Resultants In matrix form

zdzMM

M

dz

N

N

N

h

h

xy

y

x

xy

y

x

h

h

xy

y

x

xy

y

x

=

=

2/

2/

2/

2/

=

=

=

=

n

k

h

h

xy

y

x

xy

y

x

n

k

h

h

xy

y

x

xy

y

x

zdz

M

M

M

dz

N

N

N

k

k

k

k

1

1

1

1

Substituting

=

xy

y

x

ssysxs

ysyyxy

xsxyxx

xy

y

x

QQQ

QQQ

QQQ


14/29

2003, P. Joyce


The resultant forces and moments can be written in terms

of the midplane strains and curvatures

dzz

QQQ

QQQ

QQQ

zdz

QQQ

QQQ

QQQ

M

M

M

zdz

QQQ

QQQ

QQQ

dz

QQQ

QQQ

QQQ

N

N

N

xy

y

xn

k

h

h

ssysxs

ysyyxy

xsxyxx

xy

y

xn

k

h

h

ssysxs

ysyyxy

xsxyxx

xy

y

x

xy

y

xn

k

h

h

ssysxs

ysyyxy

xsxyxx

xy

y

xn

k

h

h

ssysxs

ysyyxy

xsxyxx

xy

y

x

k

k

k

k

k

k

k

k

2

10

0

0

1

10

0

0

1

11

11

+

=

+

=

==

==


15/29

2003, P. Joyce


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

h

xy

y

x

ssysxs

ysyyxy

xsxyxxh

h

xy

y

x

kssysxs

ysyyxy

xsxyxx

xy

y

x

n

k

h

h

xy

y

x

ssysxs

ysyyxy

xsxyxxh

h

xy

y

x

kssysxs

ysyyxy

xsxyxx

xy

y

x

k

k

k

k

k

k

k

k

dzz

QQQ

QQQ

QQQ

zdz

QQQ

QQQ

QQQ

M

M

M

zdz

QQQ

QQQQQQ

dz

QQQ

QQQQQQ

N

NN

1

2

0

0

0

1 0

0

0

11

11


16/29

2003, P. Joyce

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

mid-planeh/2h0

( )

( )

( )3132

2

1

2

1

3

1

2

1

1

1

1

=

=

=

kk

h

h

kk

h

h

kk

h

h

hhdzz

hhzdz

hhdz

k

k

k

k

k

k h1


17/29

2003, P. Joyce

Force and Moment ResultantsForce and Moment Resultants Furthermore only the stiffnesses are unique for each layer, k.

Thus, [0]x,y

and []x,y

can be factored outside the summation sign

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

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

k

kk

k

yxyx

n

k

kk

k

yxyx

yx

n

k

kk

k

yxyx

n

k

kk

k

yxyx

hhQhhQM

hhQhhQN

,

1

3

1

3

,,

0

1

2

1

2

,,

,

1

2

1

2

,,

0

1

1,,

31

21

2

1

+

=

+

=

=

=

=

=

Define

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

=

=

===n

k

kk

k

yxij

n

k

kk

k

yxij

n

k

kk

k

yxij hhQDhhQBhhQA

1

3

1

3

,

1

2

1

2

,

1

1,3

1,

2

1,

[A], [B], [D] are called the extensional, coupling, and bending

stiffness matrices, respectively.


18/29

2003, P. Joyce

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

[ ] [ ][ ] [ ][ ] yxijyxijyxyxijyxijyx

DBM

BAN

,,

0

,

,,

0

,

+=

+=

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

DDDBBB

DDDBBB

DDDBBB

BBBAAA

BBBAAA

BBBAAA

M

M

M

N

N

N

0

0

0

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211


19/29

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.


20/29

2003, P. Joyce

Laminate Special CasesLaminate Special Cases Symmetric: [B] = 0

Load-deformation equation and moment-curvaturerelation decoupled.

Balanced: A16 = A26 = 0.

Symmetric and Balanced:

Orthotropic with respect to inplane behavior.

011 12

0

11 22

x x

y y

N A A

N A A

= 0

66xy xyN A =


21/29

2003, P. Joyce

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

Some decoupling of the six equations.

Orthotropic with respect to both inplane and bending behavior.

0

11 12 11 12

0

12 22 12 22

0

11 12 11 12

0

12 22 12 22

x x

y y

x x

y y

N A A B B

N A A B B

M B B D D

M B B D D

=

0

66 66

0

66 66

xy xy

xy xy

N A B

M B D

=


22/29

2003, P. Joyce

Laminate Special CasesLaminate Special Cases Symmetric Cross-Ply:

[B] =0

A16= A26 = D16= D26=0.

Significant decoupling

Orthotropic with respect to both inplane and bending behavior.

0

11 120

11 22

x x

y y

N A AN A A

=

0

66xy xyN A =

0

11 120

11 22

x x

y y

M D DM D D

=

0

66xy xyM D =


23/29

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 thez 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.


24/29

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 asa function of applied loads and moments.


25/29

2003, P. Joyce

Laminate CompliancesLaminate Compliances Performing matrix inversions

=

=

M

N

dc

ba

M

M

M

N

N

N

dddccc

dddccc

dddcccbbbaaa

bbbaaa

bbbaaa

xy

y

x

xy

y

x

xy

y

x

xy

y

x

0

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

0

0

0

briefinor


26/29

2003, P. Joyce

Laminate CompliancesLaminate Compliances Where [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.


27/29

2003, P. Joyce

Engineering Constants for aEngineering Constants for a

MultiMulti--Axial LaminateAxial Laminate From the laminate compliances we can compute the

engineering constants

ss

ys

sy

yy

sy

ys

xx

sxxs

ss

xssx

yy

xy

yx

xx

yx

xy

ss

xy

yy

y

xx

x

a

a

a

a

a

aa

a

a

a

a

a

haG

haE

haE

===

===

===

111

As in UD lamina, symmetry implies

xy

sy

y

ys

xy

sx

x

xs

y

yx

x

xy

GEGEEE

=== ,,


28/29

2003, P. Joyce

Engineering Constants for aEngineering Constants for a

MultiMulti--Axial LaminateAxial Laminate Computational Procedure for Determination of Engineering Elastic

Properties1. Determine the engineering constants of UD layer,E

1,E2, 12, 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.


29/29

2003, P. Joyce

Laminated Composite AnalysisLaminated Composite Analysis

8 PAX Short Course Laminat-Analysis

Documents

Transcript of 8 PAX Short Course Laminat-Analysis