Summary of Classical Lamination Theory
(CLT) Calculations
Summary of Classical Lamination Theory
(CLT) Calculations
Numerical examples illustrating discussion in:
Summary of Classical Lamination Theory
(CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads
Are Known
Summary of Classical Lamination Theory
(CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads
Are Known
Section 6.8.2: A CLT Analysis When Midplane
Strains and Curvatures are
Known
Summary of Classical Lamination Theory
(CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads
Are Known
Section 6.8.2: A CLT Analysis When Midplane
Strains and Curvatures are
Known
(Sections 6.8.1 and 6.8.2 are nearly identical…)
Summary of Classical Lamination Theory
(CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads
Are Known
Section 6.8.2: A CLT Analysis When Midplane
Strains and Curvatures are
Known
(Sections 6.8.1 and 6.8.2 are nearly identical…)
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
Example: Suppose two materials are used…graphite-epoxy
and glass epoxy. From Table 3.1, typical properties are
Mat’l
Name
Gr/Ep
Gl/Ep
Mat’l
Number
1
2
E11 (psi)
E22 (psi)
6
6
25 x 10
8.0 x106
1.5 x 10
2.3 x 106
ν12
0.30
0.28
G12 (psi)
6
1.9 x 10
1.1 x106
α11
α22
(in/in-ºF)
-0.5x10-6
3.7x10-6
(in/in-ºF)
15 x10-6
14 x10-6
Ply thick
(in)
0.005
0.005
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
Example:
[0 / 30 / 90 / − 30]T
A 4-ply laminate. This
description is adequate if same
material is used for all plies.
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
Example:
[0 / 30 / 90 / − 30]T
Graphite/Epoxy
Glass/Epoxy
A 4-ply laminate. This
description is adequate if same
material is used for all plies. For
illustrative purposes assume
Gr/Ep used for 0º and 90º plies
and Gl/Ep used for ±30º plies
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
Example:
Total laminate thickness = 4(0.005in) = 0.020in
z0 = −t / 2 = −(.020in) / 2 = −0.010in
z1 = z0 + t1 = −0.01 + 0.005m = −0.005in
z 2 = z1 + t 2 = −0.005 + 0.005in = 0.000in
z3 = z 2 + t3 = −0.000 + 0.005in = 0.005in
z 4 = z3 + t 4 = 0.005 + 0.005in = 0.010in
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
d) Specify mechanical and thermal loads
Section 6.8.1: A CLT Analysis When
Loads Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
d) Specify mechanical and thermal loads
Example: N xx = 520 lbf/in
N yy = 377 lbf/in
N xy = 64.4 lbf/in
Tcure = 350° F
M xx = −4.0 lbf − in/in
M yy = 0.22 lbf − in/in
M xy = −0.0854 lbf − in/in
Tservice = 75° F
⇒ ∆T = −275° F
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
Q11 Q12
[Q] = Q12 Q22
 0
0
2


E
11


2
 E11 − ν 12
E22 
0  
 ν 12 E11E22 



0 = 
 E − ν 2 E 
Q66   11 12 22 

(0)


 ν 12 E11E22 


 E −ν 2 E 
 11 12 22 
 E11E22 


 E −ν 2 E 
 11 12 22 
(0)

(0) 


(0) 

(G12 )

Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
Example:
25.14 x10 6 0.452 x106

0


6
6
0
[Q]Gr / Ep = 0.452 x10 1.508 x10
 psi
6

0
0
1
.
90
10
x


8.184 x106

[Q]Gl / Ep = 0.659 x10 6

0


0.659 x106
0

6
2.353 x10
0
 psi
0
1.10 x106 

Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Q11 Q12 Q16 


Q = Q12 Q 22 Q 26 
Q

Q
Q
26
66 
 16
[ ]
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Q11 = Q11 cos 4 θ + 2(Q12 + 2Q66 ) cos 2 θ sin 2 θ + Q22 sin 4 θ
Q12 = Q 21 = Q12 (cos 4 θ + sin 4 θ ) + (Q11 + Q22 − 4Q66 ) cos 2 θ sin 2 θ
Q16 = Q 61 = (Q11 − Q12 − 2Q66 ) cos 3 θ sin θ − (Q22 − Q12 − 2Q66 ) cos θ sin 3 θ
Q 22 = Q11 sin 4 θ + 2(Q12 + 2Q66 ) cos 2 θ sin 2 θ + Q22 cos 4 θ
Q 26 = Q 62 = (Q11 − Q12 − 2Q66 ) cos θ sin 3 θ − (Q22 − Q12 − 2Q66 ) cos 3 θ sin θ
Q 66 = (Q11 + Q22 − 2Q12 − 2Q66 ) cos 2 θ sin 2 θ + Q66 (cos 4 θ + sin 4 θ )
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Example: For ply 1,
25.14 x10 6 0.452 x106

0


0° ply
6
6
[Q ]Gr / Ep = 0.452 x10 1.508 x10
0
 psi
6

0
0
1
.
90
x
10


Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Example: For ply 2,
5.823 x10 6
30° ply 
[Q ]Gl / Ep = 1.563 x106
1.784 x106

1.563 x106 1.784 x106 
6
6
2.907 x10 0.741x10  psi
0.741x106 2.00 x10 6 

Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Example: For ply 3,
1.508 x106
90° ply 
[Q ]Gr / Ep = 0.452 x106

0


0.452 x10 6
0

6
25.14 x10
0
 psi
0
1.90 x106 

Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
Example: For ply 4,
 5.823 x10 6
−30° ply 
[Q ]Gl / Ep =  1.563x106
− 1.784 x106

1.563x106
2.907 x106
− 0.741x106
− 1.784 x106 
6
− 0.741x10  psi
2.00 x106 

Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
Example:
191.4 x103 20.15 x103
n

Aij = ∑ Q ij ( z k − z k −1 ) = 20.15 x103 162.3 x103
k

k =1
0
0

{ }

 lbf
0

3  in
39.04 x10 

0
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
Example:
 − 778 27.8 − 89.2
1
Bij = ∑ Q ij ( z k2 − z k2−1 ) =  27.8
330 − 37.0 lbf
k
2 k =1
− 89.2 − 37.0 2.59 
n
{ }
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
Example:
0.672 − 0.446
 9.34
1
Dij = ∑ Q ij ( z k3 − z k3−1 ) =  0.672
− 0.185 lbf − in
2.46
k
3 k =1
− 0.446 − 0.185
1.30 
n
{ }
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
d) Assemble the [ABD] matrix
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
d) Assemble the [ABD] matrix
 Aij
[ABD] =  B
 ij
 A11
A
 12
Bij   A16
=
Dij   B11
 B12

 B16
A12
A16
B11
B12
A22
A26
A26
A66
B12
B16
B22
B26
B12
B22
B16
B26
D11
D12
D12
D22
B26
B66
D16
D26
B16 
B26 
B66 

D16 
D26 

D66 
Section 6.8.1: A CLT Analysis When
Loads Are Known
2. Calculate the [ABD] matrix:
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
d) Assemble the [ABD] matrix
Example:
 191x103

3
20.1x10

0
[ ABD] = 
 − 778

 − 27.8
 − 89.2
20.1x103
0
162 x103
0
0
39.0 x103
− 27.8
− 89.2
330
− 37.0
− 37.0
2.59
− 89.2 

− 37.0 
27.8
330
− 89.2 − 37.0
2.59 

9.34
0.672 − 0.446

0.672
2.46
− 0.185
− 0.446 − 0.185 1.30 
− 778
27.8
Section 6.8.1: A CLT Analysis When
Loads Are Known
3. Calculate the [abd] = [ABD]-1 matrix:
Section 6.8.1: A CLT Analysis When
Loads Are Known
3. Calculate the [abd] = [ABD]-1 matrix:
 a11
a
 12
−1 a16
[abd ] = [ABD] = 
 b11
b12

b16
a12
a22
a26
b21
b22
b26
a16
a26
a66
b61
b62
b66
b11
b21
b61
d11
d12
d16
b12
b22
b62
d12
d 22
d 26
b16 
b26 
b66 

d16 
d 26 

d 66 
Section 6.8.1: A CLT Analysis When
Loads Are Known
3. Calculate the [abd] = [ABD]-1 matrix:
Example:
 9.09 x10 −6

−6
 − 0.811x10

0
[abd ] = 
 0.827 x10 −3

−3
0
.
129
x
10
−

 0.862 x10 −3
− 0.811x10 −6
0
0.827 x10 −3
8.64 x10 −6
0
0
26.9 x10 −6
− 0.196 x10 − 4
− 0.196 x10 −4
0.381x10 −3
0.188
0.454 x10 −3
− 0.044
− 0.044
0.586
0.114
0.025
− 1.16 x10 −3
0.203x10 −4
0.226 x10 −3
0.381x10 −3
− 0.129 x10 −3
− 1.16 x10 −3
0.454 x10 −3
0.862 x10 −3 
−4 
0.203x10 
0.226 x10 −3 

0.114 

0.025 
0.870 
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
α xx = α11 cos 2 (θ ) + α 22 sin 2 (θ )
α yy = α11 sin 2 (θ ) + α 22 cos 2 (θ )
α xy = 2 cos(θ ) sin(θ )(α11 − α 22 )
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
Example:
Ply
Number
Mat’l
Number
1
2
3
4
1
2
1
2
Fiber
angle
(deg)
0
30
90
-30
αxx
αyy
αxy
(in/in- ºF)
(in/in- ºF)
(in/in- ºF)
-0.5x10-6
6.28 x10-6
15 x10-6
6.28 x10-6
15 x10-6
11.4 x 10-6
-0.5x10-6
11.4 x 10-6
0
-8.92 x 10-6
0
8.92 x 10-6
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
b) Calculate thermal stress & moment resultants
n
{[
]
}
{[
]
}
{[
]
}
N Txx ≡ ∆T ∑ Q11α xx + Q12α yy + Q16α xy [z k − z k −1 ]
k
k =1
n
N Tyy ≡ ∆T ∑ Q12α xx + Q 22α yy + Q 26α xy [z k − z k −1 ]
k
k =1
n
N Txy ≡ ∆T ∑ Q16α xx + Q 26α yy + Q 66α xy [z k − z k −1 ]
k
k =1
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
b) Calculate thermal stress & moment resultants
{[
][
]}
{[
][
]}
{[
][
]}
M Txx
∆T n
≡
Q11α xx + Q12α yy + Q16α xy z k2 − z k2−1
∑
k
2 k =1
M Tyy
∆T n
≡
Q12α xx + Q 22α yy + Q 26α xy z k2 − z k2−1
∑
k
2 k =1
M Txy
∆T n
≡
Q16α xx + Q 26α yy + Q 66α xy z k2 − z k2−1
∑
k
2 k =1
Section 6.8.1: A CLT Analysis When
Loads Are Known
4. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
b) Calculate thermal stress & moment resultants
Example:
N Txx = −129lbf / in
M Txx = −0.401 lbf − in / in
N Tyy = −123lbf / in
M Tyy = 0.0005 lbf − in / in
N Txy = 0
M Txy = −0.0246 lbf − in / in
Section 6.8.1: A CLT Analysis When
Loads Are Known
5. Calculate midplane strains and curvatures
Section 6.8.1: A CLT Analysis When
Loads Are Known
5. Calculate midplane strains and curvatures
o 
ε xx
 a11
 o  
ε yy  a12
 o  a
γ xy 
16
=



κ xx   b11
κ  b12
 yy  
κ xy  b16
a12
a22
a26
b21
b22
b26
a16
a26
a66
b61
b62
b66
b11
b21
b61
d11
d12
d16
b12
b22
b62
d12
d 22
d 26
b16   N xx

b26   N yy


b66  N xy

d16   M xx
d 26  M yy

d 66   M
 xy
+ N Txx 
T 
+ N yy 
T 
+ N xy 
T 
+ M xx 
+ M Tyy 

+ M Txy 
Section 6.8.1: A CLT Analysis When
Loads Are Known
5. Calculate midplane strains and curvatures
Example:
o 
ε xx
0


 o  

−6
ε yy  − 1300 x10 in / in 
 o   900 x10 −6 rad 

γ xy  


=
−1
− 0.50 in

κ xx  
−1
κ  

0
.
40
in

 yy  
−
1

− 0.20 in
κ xy  
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
a) Calculate ply strains in x-y coordinate system
(usually at ply interfaces)
o 
κ 
ε xx  ε xx
xx

   o  
ε yy  = ε yy  + z κ yy 
γ   o  

 xy  γ xy  κ xy 
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
a) Calculate ply strains in x-y coordinate system
(usually at ply interfaces)
LAMINATE PLY STRAINS, x-y COORDINATE SYSTEM:
Example:
PLY NO Z-COORD
------1
------2
------3
------4
-------
EPSxx
EPSyy
GAMxy
-0.100E-01 0.005000 -0.005300 0.002900
-0.500E-02 0.002500 -0.003300 0.001900
0.000E+00 0.000000 -0.001300 0.000900
0.500E-02 -0.002500 0.000700 -0.000100
0.100E-01 -0.005000 0.002700 -0.001100
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
a) …usually also transform strains to 1-2
coordinate system
 ε xx 
 ε11 




 ε 22  = [T ] z  ε yy 
γ / 2
γ / 2
 12 
 xy  z
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
a) …usually also transform strains to 1-2
coordinate system
LAMINATE PLY STRAINS, 1-2 COORDINATE SYSTEM:
PLY NO Z-COORD
Example:
EPS11
EPS22
-------------------------------------------------------------0.10000E-01
0.005000 -0.005300
1
-0.50000E-02
0.002500 -0.003300
-------------------------------------------------------------0.50000E-02
0.001873 -0.002673
2
0.00000E+00
0.000065 -0.001365
------------------------------------------------------------0.00000E+00 -0.001300
0.000000
3
0.50000E-02
0.000700 -0.002500
------------------------------------------------------------0.50000E-02 -0.001657 -0.000143
4
0.10000E-01 -0.002599
0.000299
--------------------------------------------------------------
GAM12
0.002900
0.001900
-0.004073
-0.000676
-0.000900
0.000100
-0.002821
-0.007218
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
b) Calculate ply stresses in x-y coordinate system
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
b) Calculate ply stresses in x-y coordinate system
σ xx  Q11 Q12

 
σ yy  = Q12 Q 22
τ  Q
 xy   16 Q 26
Q16  ε xx − ∆Tα xx 


Q 26  ε yy − ∆Tα yy 
Q 66  γ xy − ∆Tα xy 
(for each ply)
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
b) Calculate ply stresses in x-y coordinate system
LAMINATE PLY STRESSES, x-y COORDINATE SYSTEM:
PLY NO Z-COORD
Example:
SIGxx
SIGyy
TAUxy
--------------------------------------------------------------0.10000E-01 0.12169E+06 0.42794E+03 0.55100E+04
1
-0.50000E-02 0.59756E+05 0.23131E+04 0.36100E+04
--------------------------------------------------------------0.50000E-02 0.23372E+05 0.57335E+04 0.63146E+04
2
0.00000E+00 0.10155E+05 0.69006E+04 0.13317E+04
-------------------------------------------------------------0.00000E+00 0.55707E+04 -0.34266E+05 0.17100E+04
3
0.50000E-02 0.27052E+04 0.14874E+05 -0.19000E+03
-------------------------------------------------------------0.50000E-02 -0.27044E+04 0.82159E+04 0.32505E+04
4
0.10000E-01 -0.12352E+05 0.10865E+05 0.42260E+04
--------------------------------------------------------------
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
b) usually also transform stresses to 1-2 coord
system
σ xx 
σ 11 




σ 22  = [T ] σ yy 
τ 
τ 
 12 
 xy 
Section 6.8.1: A CLT Analysis When
Loads Are Known
6. For each ply:
b) usually also transform stresses to 1-2 coord
system
LAMINATE PLY STRESSES, 1-2 COORDINATE SYSTEM:
PLY NO Z-COORD
Example:
SIG11
SIG22
TAU12
--------------------------------------------------------------0.10000E-01 0.12169E+06 0.42794E+03 0.55100E+04
1
-0.50000E-02 0.59756E+05 0.23131E+04 0.36100E+04
--------------------------------------------------------------0.50000E-02 0.24431E+05 0.46744E+04 -0.44802E+04
2
0.00000E+00 0.10495E+05 0.65610E+04 -0.74342E+03
-------------------------------------------------------------0.00000E+00 -0.34266E+05 0.55707E+04 -0.17100E+04
3
0.50000E-02 0.14874E+05 0.27052E+04 0.19000E+03
-------------------------------------------------------------0.50000E-02 -0.27893E+04 0.83009E+04 -0.31034E+04
4
0.10000E-01 -0.10208E+05 0.87202E+04 -0.79402E+04
--------------------------------------------------------------
Summary of Classical Lamination Theory
(CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads Are Known
Section 6.8.2: A CLT Analysis When Midplane Strains and
Curvatures are Known
(Sections 6.8.1 and 6.8.2 are nearly identical…)
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
d) Specify midplane strains/curv & therm loads
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
1. Define the problem:
a) Specify number of different materials used
b) Specify properties for each material
c) Specify laminate description
d) Specify midplane strains/curv & therm loads
κ xx = − 0 .50 in − 1
o
Example: ε xx
=0
ε oyy = − 1300 µ in / in
κ yy = 0 .40 in −1
o
γ xy
= 900 µ rad
κ xy = − 0 .20 in −1
Tcure = 350 ° F
Tservice = 75 ° F
⇒ ∆ T = − 275 ° F
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
2. Calculate the [ABD] matrix
a) Calculate the [Q] matrix for each material
b) Calculate the [Q ] matrix for each ply
c) Calculate the [Aij], [Bij], and [Dij] matrices
d) Assemble the [ABD] matrix
(in this case do not need the [abd] matrix)
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
3. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
b) Complete calculation of thermal stress and
moment resultants
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
3. Calculate thermal stress and moment resultants:
a) Calculate effective thermal expansion
coefficients for each ply
b) Complete calculation of thermal stress and
moment resultants
N Txx = −129lbf / in
M Txx = −0.401 lbf − in / in
N Tyy = −123lbf / in
M Tyy = 0.0005 lbf − in / in
N Txy = 0
M Txy = −0.0246 lbf − in / in
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
 N xx   A11
N  
 yy   A12
 N xy   A16
M  = 
 xx   B11
M yy   B12

 
 M xy   B16
A12
A16
B11
B12
A22
A26
A26
A66
B12
B16
B22
B26
B12
B16
D11
D12
B22
B26
D12
D22
B26
B66
D16
D26
o   NT 
B16  ε xx
xx

 o 
T 

B26  ε yy   N yy 
 T 


o

B66 γ xy   N xy 

− T 
D16  κ xx   M xx 
D26  κ yy  M Tyy 


 
T
D66  κ xy   M 

  xy 
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
 N xx   A11
N  
 yy   A12
 N xy   A16
M  = 
 xx   B11
M yy   B12

 
 M xy   B16
A12
A16
B11
B12
A22
A26
A26
A66
B12
B16
B22
B26
B12
B16
D11
D12
B22
B26
D12
D22
B26
B66
D16
D26
o   NT 
B16  ε xx
xx

 o 
T 

B26  ε yy   N yy 
 T 


o

B66 γ xy   N xy 

− T 
D16  κ xx   M xx 
D26  κ yy  M Tyy 


 
T
D66  κ xy   M 

  xy 
(this is the only difference in calculation….)
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
 N xx   191E 3 20.1E 3
0
− 778
27.8
− 89.2   0   − 129 
N  
 − 1300µ   − 123 
E
E
−
20
.
1
3
162
3
0
27
.
8
330
37
.
0
yy

 


 
 N xy   0

0
39.0 E 3 − 89.2 − 37.0
2.59   900µ   0
=
−

M  
 

−
778
−
27
.
8
−
89
.
2
9
.
34
0
.
672
−
0
.
446
−
0
.
50
−
0
.
401


 xx  
 
M yy   − 27.8
− 37.0 0.672
− 0.185  0.40   0.0005 
330
2.46



 
 
− 0.446 − 0.185 1.30   − 0.20  − 0.0246
2.59
 M xy   − 89.2 − 37.0
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
 N xx  
520 lbf/in

N  

377
lbf/in
 yy  

 N xy   64.4 lbf/in 
M  = 

 xx   − 4.0 lbf - in/in 
M yy   0.22 lbf - in/in 

 

 M xy  − 0.085 lbf - in/in 
Section 6.8.2: A CLT Analysis When
Midplane Strains and Curvatures Are Known
6. For each ply:
a) Calculate ply strains in x-y and 1-2 coordinate
systems (at ply interfaces)
b) Calculate ply stresses in x-y and 1-2 coordinate
systems (at ply interfaces)
Analysis complete!