1

Classical Laminated Plate Theory

2

CONSTITUENTS

STRUCTURE COMPOSITE

STRUCTURALELEMENT

ELEMENTARYSTRUCTURE

Micro Mechanics

Ex

Ey

G

Experimental Data

Finite Element Analysis

Combining Constituents

3

Material Forms

4

Plane-Stress Assumption

• Fiber-reinforced materials are utilized in beams, plates, cylinders and other structures

• Typically one characteristic geometric dimension is an order of magnitude less than the other two

• Three of the six components of stress are generally much smaller than the other three

5

Plane Stress Inaccuracies

• Errors in analysis near edges– The stresses 3, 23, 13 lead to delaminations

– Bonded joints can not be modeled– Adhesive or cocured interface can not be

evaluated

• The stress components equated to zero are forgotten and no attempt is made to estimate their magnitude

• erroneously assumed that 3 is zero

6

Stress Transformation

xy

y

x

22

22

22

12

2

1

sincoscossincossin

cossin2cossin

cossin2sincos

22

22

22

nmmnmn

mn2mn

mn2nm

T

7

Compliance Transformation Equations

xy

y

x

662612

262212

161211

xy

y

x

SSS

SSS

SSS

4466

226612221166

3661222

366121126

422

226612

41122

3661222

366121116

4412

2266121112

422

226612

41111

mnSmnSS4S2S22S

mnSS2S2mnSS2S2S

mSmnSS2nSS

mnSS2S2mnSS2S2S

mnSmnSSSS

nSmnSS2mSS

8

Reduced Stiffness Transformation

xy

y

x

662612

262212

161211

xy

y

x

QQQ

QQQ

QQQ

44

6622

6612221166

3662212

366121126

422

226612

41122

3662212

366121116

4412

2266221112

422

226612

41111

mnQmnQ2Q2QQQ

mnQ2QQmnQ2QQQ

mQmnQ2Q2nQQ

mnQ2QQmnQ2QQQ

mnQmnQ4QQQ

nQmnQ2Q2mQQ

9

Classical Lamination TheoryThe influence of fiber direction, stacking arrangements, material properties, and more on structural response

10

Laminate Coordinate System

• Laminate thickness H• Layer thickness h

– not all layers same h

– kth layer - hk

• z-axis downward from geometric midplane– can be between layers– can be within a layer

• Fiber angles identified relative to x axis

11

Laminate Nomenclature

T

T2

s

45/45/0/0/45/45

45/45/0/45/45

0/45

• Layer 1 is the most -z• Layer N is the most +z• To catergorize a

laminate as symmetric a mirror about the geometric midplane– material properties– fiber orientation– thickness of layer

12

The Kirchhoff Hypothesis

• Mid 1800’s, simplified analysis

• Beams, plates, shells• metal, wood,

concrete, and other materials

13

Initially Flat Laminated Plate Acted Upon by Various Loads

• Loads– applied moments, M

– distributed loads, q

– inplane loads, N

– point loads, P

• Multiple layers of fiber reinforced material

• Fibers parallel to the plane of the plate

• Layers are perfectly bonded

14

Deformation of Lines Normal to Geometric Midplane

• Before deformation are straight

• Despite the deformations caused by the applied loads, line AA’ remains straight and normal to the deformed geometric midplane and does not change length

15

Consequences of Kirchhoff Hypothesis

16

Implications of the Kichhoff Hypothesis in X-Z Plane

• No through-thickness strain• Small deformations• Two components of

translation– uo in x direction, horizontal

translation– wo in z direction, vertical

translation

• One component of rotation about y-axis

xwo

17

Resulting Displacement Fieldin XZ plane

y,xwz,y,xw

x

y,xwzy,xuz,y,xu

o

oo

18

Resulting Displacement Fieldin YZ plane

y,xwz,y,xw

y

y,xwzy,xvz,y,xv

o

oo

19

Strain-Displacement Relations from Theory of Elasticity

zu

xw

zw

yw

zv

yv

yu

xv

xu

zxz

yzy

xyx

20

Laminate Strains

yx

)y,x(wz2

y

y,xu

x

y,xv

y

z,y,xu

x

z,y,xvz,y,x

0x

y,xw

x

y,xw

z

z,y,xu

x

z,y,xwz,y,x

0y

y,xw

y

y,xw

z

z,y,xv

y

z,y,xwz,y,x

0z

y,xw

z

z,y,xwz,y,x

y

)y,x(wz

y

y,xv

y

z,y,xvz,y,x

x

)y,x(wz

x

y,xu

x

z,y,xuz,y,x

o2oo

xy

oo

xz

oo

yz

o

z

2

o2o

y

2

o2o

x

21

Laminate Strains Composed of Two Parts

• Extensional Stain of the Reference Surface

• Curvature of the Reference Surface– inverse of the radius of curvature– involves more than just second derivative

• For small strains second derivative and curvature identical

22

Strain Notation

yx

y,xw2y,x

y

y,xu

x

y,xvy,x

y

y,xwy,x

y

y,xvy,x

x

y,xwy,x

x

y,xuy,x

o2oxy

oooxy

2

o2oy

ooy

2

o2ox

oox

23

Laminate Strains using Revised Notation

y,xzy,xz,y,x

0z,y,x

0z,y,x

0z,y,x

y,xzy,xz,y,x

y,xzy,xz,y,x

oxy

oxyxy

xz

yz

z

oy

oyy

ox

oxx

The yz, and xz are zero because the Kirchhoff hypothesisassumes that lines perpendicular to the reference surfacebefore deformation remain perpendicular after thedeformation; right angles in the thickness direction do notchange when the laminate deforms

24

Laminate Stresses

oxy

oxy

oy

oy

ox

ox

662612

262212

161211

xy

y

x

z

z

z

QQQ

QQQ

QQQ

25

[0/90]s Laminate, Axial 1000Laminate Stress & Strain

26

[0/90]s Laminate, Axial 1000Material Stress & Strain

27

Aluminum, Axial 1000

28

[0/90]s Laminate, xo 3.33 m-1

Laminate Stress & Strain

29

[0/90]s Laminate, xo 3.33 m-1

Material Stress & Strain

30

Aluminum,xo 3.33 m-1

31

Definitions of Stress Resultants

• Stress in each ply varies through the thickness

• It is convenient to define stresses in terms of equivalent forces acting at the middle surface

• Stresses at the edge can be broken into increments and summed

• The resulting integral is defined as the stress resultant, Ni [force per length]

32

Stress Resultant in X direction

2/h

2/h xx

2/h

2/h xx

x

dzN

dzyydz,0dzAs

ydzdirectionxinforceTotal

33

Stress and Moment Resultants

2/h

2/h xyxy

2/h

2/h yy

2/h

2/h xx

2/h

2/h xyxy

2/h

2/h yy

2/h

2/h xx

dzzM

dzzM

dzzM

dzN

dzN

dzN

bend

bend

twist

34

Putting the Resultants in Matrix Form and Summing

dzz

M

M

M

dz

N

N

N

k

n

0k

h

h

xy

y

x

xy

y

x

k

n

0k

h

h

xy

y

x

xy

y

x

k

1k

k

1k

35

Relating Stress to Strain

n

0k

h

h

h

h

2

oxy

oy

ox

k662616

262212

161211

oxy

oy

ox

k662616

262212

161211

xy

y

x

n

0k

h

h

h

hoxy

oy

ox

k662616

262212

161211

oxy

oy

ox

k662616

262212

161211

xy

y

x

k

1k

k

1k

k

1k

k

1k

dzz

QQQ

QQQ

QQQ

dzz

QQQ

QQQ

QQQ

M

M

M

dzz

QQQ

QQQ

QQQ

dz

QQQ

QQQ

QQQ

N

N

N

36

Performing the Integration

n

0k

31k

3k

oxy

oy

ox

k662616

262212

1612112

1k2k

oxy

oy

ox

k662616

262212

161211

xy

y

x

n

0k

21k

2k

oxy

oy

ox

k662616

262212

161211

1kkoxy

oy

ox

k662616

262212

161211

xy

y

x

hh3

1

QQQ

QQQ

QQQ

hh2

1

QQQ

QQQ

QQQ

M

M

M

hh2

1

QQQ

QQQ

QQQ

hh

QQQ

QQQ

QQQ

N

N

N

37

Defining Laminate Stiffness Terms

31k

3k

k

n

0kijij

21k

2k

k

n

0kijij

1kkk

n

0kijij

hhQ3

1D

hhQ2

1B

hhQA

38

Constitutive Equations in Matrix Form

xy

y

x

oxy

oy

ox

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

xy

y

x

xy

y

x

DDDBBB

DDDBBB

DDDBBB

BBBAAA

BBBAAA

BBBAAA

M

M

M

N

N

N

39

Symmetric Laminates

• For every layer to one side of the laminate reference surface with a specific thickness, material properties, and fiber orientation, there is another layer an identical distance on the opposite side

• All components of [B] are zero• 6x6 set of equations decouples into two 3x3

sets of equations

40

Balanced Laminates

• For every layer with a specified thickness, material properties, and fiber orientation, there is another layer with the identical thickness, material properties, but opposite fiber orientation somewhere in the laminate

• If a laminate is balanced, A16 and A26 are always zero– Q16 & Q26 from opposite orientation have

opposite signs

41

Effective Engineering Properties of a Laminate

11

12yx

22

12xy

66x

11

2122211

y

22

2122211

x

A

A

A

Ah

AG

hA

AAAE

hA

AAAE