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8/12/2019 8 PAX Short Course Laminat-Analysis
1/29
2003, P. Joyce
8/12/2019 8 PAX Short Course Laminat-Analysis
2/29
MacromechanicalMacromechanical Analysis ofAnalysis ofLaminatesLaminates
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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2003, P. Joyce
StressStressStrain Relations for anStrain Relations for an
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
8/12/2019 8 PAX Short Course Laminat-Analysis
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2003, P. Joyce
StressStressStrain Relations for anStrain Relations for an
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
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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:
8/12/2019 8 PAX Short Course Laminat-Analysis
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.
8/12/2019 8 PAX Short Course Laminat-Analysis
9/29
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.
8/12/2019 8 PAX Short Course Laminat-Analysis
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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?
8/12/2019 8 PAX Short Course Laminat-Analysis
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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 ==
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
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
+
=
+
=
==
==
8/12/2019 8 PAX Short Course Laminat-Analysis
15/29
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
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
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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.
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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.
8/12/2019 8 PAX Short Course Laminat-Analysis
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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 =
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
=
8/12/2019 8 PAX Short Course Laminat-Analysis
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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 =
8/12/2019 8 PAX Short Course Laminat-Analysis
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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.
8/12/2019 8 PAX Short Course Laminat-Analysis
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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.
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
8/12/2019 8 PAX Short Course Laminat-Analysis
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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.
8/12/2019 8 PAX Short Course Laminat-Analysis
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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
=== ,,
8/12/2019 8 PAX Short Course Laminat-Analysis
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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.
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2003, P. Joyce
Laminated Composite AnalysisLaminated Composite Analysis