© 2003, p. joyce
TRANSCRIPT
© 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
Stress Stress ––Strain Relations for an Strain Relations for an Isotropic BeamIsotropic Beam
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
Stress Stress ––Strain Relations for an Strain Relations for an Isotropic BeamIsotropic Beam
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
Strain and Stress in a LaminateStrain and Stress in a Laminate
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
Strain and Stress in a LaminateStrain and Stress in a Laminate
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
Laminated Composite AnalysisLaminated Composite Analysis