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Summary of Classical Lamination Theory (CLT) Calculations
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Summary of Classical Lamination Theory (CLT) Calculations
Numerical examples illustrating discussion in:
![Page 3: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/3.jpg)
Summary of Classical Lamination Theory (CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads Section 6.8.1: A CLT Analysis When Loads Are Known
![Page 4: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/4.jpg)
Summary of Classical Lamination Theory (CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads Section 6.8.1: A CLT Analysis When Loads Are Known
Section 6.8.2: A CLT Analysis When MidplaneStrains and Curvatures are Known
![Page 5: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/5.jpg)
Summary of Classical Lamination Theory (CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads Section 6.8.1: A CLT Analysis When Loads Are Known
Section 6.8.2: A CLT Analysis When MidplaneStrains and Curvatures are Known
(Sections 6.8.1 and 6.8.2 are nearly identical…)
![Page 6: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/6.jpg)
Summary of Classical Lamination Theory (CLT) Calculations
Numerical examples illustrating discussion in:
Section 6.8.1: A CLT Analysis When Loads Section 6.8.1: A CLT Analysis When Loads Are Known
Section 6.8.2: A CLT Analysis When MidplaneStrains and Curvatures are Known
(Sections 6.8.1 and 6.8.2 are nearly identical…)
![Page 7: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/7.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each material
![Page 8: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/8.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) 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
Mat’l Number
E11 (psi) E22 (psi) ν12 G12 (psi) α11 (in/in-ºF)
α22 (in/in-ºF)
Ply thick (in)
Gr/Ep 1 25 x 106 1.5 x 106 0.30 1.9 x 106 -0.5x10-6 15 x10-6 0.005 Gl/Ep 2 8.0 x106 2.3 x 106 0.28 1.1 x106 3.7x10-6 14 x10-6 0.005
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Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate description
![Page 10: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/10.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate description
T]30/90/30/0[ − A 4-ply laminate. This description is adequate if same material is used for all plies.
Example:
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Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate description
T]30/90/30/0[ − 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
Graphite/Epoxy
Glass/Epoxy
Example:
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Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate description
Example:
inintzz
inintzz
inintzz
inmtzz
inintz
inin
010.0005.0005.0
005.0005.0000.0
000.0005.0005.0
005.0005.001.0
010.02/)020(.2/
020.0)4(0.005 thicknesslaminate Total
434
323
212
101
0
=+=+==+−=+==+−=+=−=+−=+=
−=−=−===
![Page 13: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/13.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate descriptiond) Specify mechanical and thermal loads
![Page 14: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/14.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate descriptiond) Specify mechanical and thermal loads
Example:
FTFTFT
MN
MN
MN
servicecure
xyxy
yyyy
xxxx
°−=∆⇒°=°=
−−==
−==−−==
275 75 350
in/inlbf 0854.0 lbf/in 4.64
in/inlbf 22.0 lbf/in 377
in/inlbf 0.4 lbf/in 520
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Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:
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Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each material
![Page 17: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/17.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each material
( )
( )
( ) ( ) ( )
−
−
−
−
=
=
12
2221211
2211
2221211
221112
2221211
221112
2221211
211
66
2212
1211
00
0
0
00
0
0
][
G
EE
EE
EE
EE
EE
EE
EE
E
Q
Qνν
νν
νν
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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: xx 66 010452.01014.25Example:
psi
x
xx
xx
Q
psi
x
xx
xx
Q
EpGl
EpGr
=
=
6
66
66
/
6
66
66
/
1010.100
010353.210659.0
010659.010184.8
][
1090.100
010508.110452.0
010452.01014.25
][
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Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply][Q
![Page 20: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/20.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply][Q
[ ]
=
662616
262212
161211
QQQ
QQQ
QQQ
Q
![Page 21: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/21.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply][Q
)sin(cossincos)22(
sincos)2(sincos)2(
cossincos)2(2sin
sincos)2(sincos)2(
sincos)4()sin(cos
sinsincos)2(2cos
4466
226612221166
3661222
36612116226
422
226612
41122
3661222
36612116116
22662211
44122112
422
226612
41111
θθθθ
θθθθ
θθθθ
θθθθ
θθθθ
θθθθ
++−−+=
−−−−−==
+++=
−−−−−==
−+++==
+++=
QQQQQQ
QQQQQQQQ
QQQQQ
QQQQQQQQ
QQQQQQ
QQQQQ
![Page 22: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/22.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply
Example: For ply 1,
][Q
psi
x
xx
xx
Q plyEpGr
=°
6
66
66
0/
1090.100
010508.110452.0
010452.01014.25
][
![Page 23: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/23.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply
Example: For ply 2,
][Q
psi
xxx
xxx
xxx
Q plyEpGl
=°
666
666
666
30/
1000.210741.010784.1
10741.010907.210563.1
10784.110563.110823.5
][
![Page 24: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/24.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply
Example: For ply 3,
][Q
psi
x
xx
xx
Q plyEpGr
=°
6
66
66
90/
1090.100
01014.2510452.0
010452.010508.1
][
![Page 25: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/25.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each ply
Example: For ply 4,
][Q
psi
xxx
xxx
xxx
Q plyEpGl
−−−−
=°−
666
666
666
30/
1000.210741.010784.1
10741.010907.210563.1
10784.110563.110823.5
][
![Page 26: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/26.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matrices
][Q
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Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matrices
][Q
Example:
{ }in
lbf
x
xx
xx
zzQAn
kkkkijij
=−=∑=
−3
33
33
11
1004.3900
0103.1621015.20
01015.20104.191
)(
![Page 28: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/28.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matrices
][Q
Example:
{ } lbfzzQBn
kkkkijij
−−−−−
=−= ∑=
−59.20.372.89
0.373308.27
2.898.27778
)(2
1
1
21
2
![Page 29: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/29.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matrices
][Q
Example:
{ } inlbfzzQDn
kkkkijij −
−−−−
=−= ∑=
−30.1185.0446.0
185.046.2672.0
446.0672.034.9
)(3
1
1
31
3
![Page 30: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/30.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matricesd) Assemble the [ABD] matrix
][Q
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Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matricesd) Assemble the [ABD] matrix
][Q
[ ]
=
=
662616662616
262212262212
161211161211
662616662616
262212262212
161211161211
DDDBBB
DDDBBB
DDDBBB
BBBAAA
BBBAAA
BBBAAA
DB
BAABD
ijij
ijij
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Section 6.8.1: A CLT Analysis When Loads Are Known
2. Calculate the [ABD] matrix:a) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matricesd) Assemble the [ABD] matrix
][Q
Example:
−−−−−−−−−−−
−−−−−
=
30.1185.0446.059.20.372.89
185.046.2672.00.373308.27
446.0672.034.92.898.27778
59.20.372.89100.3900
0.373308.27010162101.20
2.898.277780101.2010191
][3
33
33
x
xx
xx
ABD
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Section 6.8.1: A CLT Analysis When Loads Are Known
3. Calculate the [abd] = [ABD]-1 matrix:
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Section 6.8.1: A CLT Analysis When Loads Are Known
3. Calculate the [abd] = [ABD]-1 matrix:
161211161211 bbbaaa
[ ] [ ]
== −
662616662616
262212622212
161211612111
666261662616
262221262212
161211161211
1
dddbbb
dddbbb
dddbbb
bbbaaa
bbbaaa
bbbaaa
ABDabd
![Page 35: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/35.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
3. Calculate the [abd] = [ABD]-1 matrix:
Example: Example:
−−−−−
−−−−−
=
−−−
−−−
−−−
−−−−
−−−−−
−−−−−
870.0025.0114.010226.010203.010862.0
025.0586.0044.010454.01016.110129.0
114.0044.0188.010381.010196.010827.0
10226.010454.010381.0109.2600
10203.01016.110196.001064.810811.0
10862.010129.010827.0010811.01009.9
][
343
333
343
3336
43466
33366
xxx
xxx
xxx
xxxx
xxxxx
xxxxx
abd
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Section 6.8.1: A CLT Analysis When Loads Are Known
4. Calculate thermal stress and moment resultants:
![Page 37: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/37.jpg)
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 plycoefficients for each ply
![Page 38: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/38.jpg)
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 plycoefficients for each ply
))(sin()cos(2
)(cos)(sin
)(sin)(cos
2211
222
211
222
211
ααθθαθαθαα
θαθαα
−=
+=
+=
xy
yy
xx
![Page 39: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/39.jpg)
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 plycoefficients for each plyExample:
Ply Number
Mat’l Number
Fiber angle (deg)
αxx (in/in- ºF)
αyy (in/in- ºF)
αxy (in/in- ºF)
1 1 0 -0.5x10-6 15 x10-6 0 2 2 30 6.28 x10-6 11.4 x 10-6 -8.92 x 10-6 3 1 90 15 x10-6 -0.5x10-6 0 4 2 -30 6.28 x10-6 11.4 x 10-6 8.92 x 10-6
![Page 40: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/40.jpg)
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 plycoefficients for each plyb) Calculate thermal stress & moment resultants
[ ] [ ]{ }
[ ] [ ]{ }
[ ] [ ]{ }∑
∑
∑
=−
=−
=−
−++∆≡
−++∆≡
−++∆≡
n
kkkkxyyyxx
Txy
n
kkkkxyyyxx
Tyy
n
kkkkxyyyxx
Txx
zzQQQTN
zzQQQTN
zzQQQTN
11662616
11262212
11161211
ααα
ααα
ααα
![Page 41: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/41.jpg)
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 plycoefficients for each plyb) Calculate thermalstress & moment resultants
[ ] [ ]{ }
[ ] [ ]{ }
[ ] [ ]{ }∑
∑
∑
=−
=−
=−
−++∆≡
−++∆≡
−++∆≡
n
kkkkxyyyxx
Txy
n
kkkkxyyyxx
Tyy
n
kkkkxyyyxx
Txx
zzQQQT
M
zzQQQT
M
zzQQQT
M
1
21
2662616
1
21
2262212
1
21
2161211
2
2
2
ααα
ααα
ααα
![Page 42: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/42.jpg)
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 plycoefficients for each plyb) Calculate thermal stress & moment resultants
Example:
ininlbfMN
ininlbfMinlbfN
ininlbfMinlbfN
Txy
Txy
Tyy
Tyy
Txx
Txx
/ 0246.0 0
/ 0005.0 /123
/ 401.0 /129
−−==
−=−=
−−=−=
![Page 43: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/43.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
5. Calculate midplane strains and curvatures
![Page 44: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/44.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
5. Calculate midplane strains and curvatures
+ T
xxxxoxx NNbbbaaa 161211161211ε
+
++
+
++
=
Txyxy
Tyyyy
Txxxx
Txyxy
Tyyyy
xxxx
xy
yy
xx
oxy
oyy
oxx
MM
MM
MM
NN
NN
NN
dddbbb
dddbbb
dddbbb
bbbaaa
bbbaaa
bbbaaa
662616662616
262212622212
161211612111
666261662616
262221262212
161211161211
κκκγεε
![Page 45: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/45.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
5. Calculate midplane strains and curvatures
Example:
0o
xxε
−
−
−
=
−
−
−
−
−
1
1
1
6
6
20.0
40.0
50.0
rad 10900
/ 101300
0
in
in
in
x
ininx
xy
yy
xx
oxy
oyy
xx
κκκγεε
![Page 46: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/46.jpg)
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)
+
=
xy
yy
xx
oxy
oyy
oxx
xy
yy
xx
z
κκκ
γεε
γεε
![Page 47: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/47.jpg)
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:
PLY NO Z-COORD EPSxx EPSyy GAMxy
------- -0.100E-01 0.005000 -0.005300 0.0029001
------- -0.500E-02 0.002500 -0.003300 0.0019002
------- 0.000E+00 0.000000 -0.001300 0.0009003
------- 0.500E-02 -0.002500 0.000700 -0.0001004
------- 0.100E-01 -0.005000 0.002700 -0.001100
Example:
![Page 48: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/48.jpg)
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
[ ]z
xy
yy
xx
zT
=
2/2/12
22
11
γεε
γεε
![Page 49: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/49.jpg)
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 EPS11 EPS22 GAM12
Example:
PLY NO Z-COORD EPS11 EPS22 GAM12
--------------------------------------------------------------0.10000E-01 0.005000 -0.005300 0.002900
1-0.50000E-02 0.002500 -0.003300 0.001900
--------------------------------------------------------------0.50000E-02 0.001873 -0.002673 -0.004073
20.00000E+00 0.000065 -0.001365 -0.000676
-------------------------------------------------------------0.00000E+00 -0.001300 0.000000 -0.000900
30.50000E-02 0.000700 -0.002500 0.000100
-------------------------------------------------------------0.50000E-02 -0.001657 -0.000143 -0.002821
40.10000E-01 -0.002599 0.000299 -0.007218
--------------------------------------------------------------
![Page 50: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/50.jpg)
Section 6.8.1: A CLT Analysis When Loads Are Known
6. For each ply: b) Calculate ply stresses in x-y coordinate system
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Section 6.8.1: A CLT Analysis When Loads Are Known
6. For each ply: b) Calculate ply stresses in x-y coordinate system
ply)each (for
662616
262212
161211
∆−∆−∆−
=
xyxy
yyyy
xxxx
xy
yy
xx
T
T
T
QQQ
QQQ
QQQ
αγαεαε
τσσ
![Page 52: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/52.jpg)
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 SIGxx SIGyy TAUxy
Example:
PLY NO Z-COORD 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
20.00000E+00 0.10155E+05 0.69006E+04 0.13317E+04
--------------------------------------------------------------0.00000E+00 0.55707E+04 -0.34266E+05 0.17100E+04
30.50000E-02 0.27052E+04 0.14874E+05 -0.19000E+03
--------------------------------------------------------------0.50000E-02 -0.27044E+04 0.82159E+04 0.32505E+04
40.10000E-01 -0.12352E+05 0.10865E+05 0.42260E+04
--------------------------------------------------------------
![Page 53: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/53.jpg)
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
[ ]
=
xy
yy
xx
T
τσσ
τσσ
12
22
11
![Page 54: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/54.jpg)
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 SIG11 SIG22 TAU12
Example:
PLY NO Z-COORD 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
20.00000E+00 0.10495E+05 0.65610E+04 -0.74342E+03
--------------------------------------------------------------0.00000E+00 -0.34266E+05 0.55707E+04 -0.17100E+04
30.50000E-02 0.14874E+05 0.27052E+04 0.19000E+03
--------------------------------------------------------------0.50000E-02 -0.27893E+04 0.83009E+04 -0.31034E+04
40.10000E-01 -0.10208E+05 0.87202E+04 -0.79402E+04
--------------------------------------------------------------
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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…)
![Page 56: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/56.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate descriptiond) Specify midplane strains/curv & thermloads
![Page 57: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/57.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
1. Define the problem:a) Specify number of different materials usedb) Specify properties for each materialb) Specify properties for each materialc) Specify laminate descriptiond) Specify midplane strains/curv & thermloads
Example:
FTFTFT
in
ininin
in
servicecure
xyoxy
yyoyy
xxoxx
°−=∆⇒°=°=
−==
=−=
−==
−
−
−
275 75 350
20.0 rad 900
40.0 /1300
50.0 0
1
1
1
κµγ
κµε
κε
![Page 58: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/58.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
2. Calculate the [ABD] matrixa) Calculate the [Q] matrix for each materialb) Calculate the matrix for each ply][Qb) Calculate the matrix for each plyc) Calculate the [Aij], [Bij], and [Dij] matricesd) Assemble the [ABD] matrix
(in this case do not need the [abd] matrix)
][Q
![Page 59: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/59.jpg)
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 plycoefficients for each plyb) Complete calculation of thermal stress and
moment resultants
![Page 60: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/60.jpg)
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 plycoefficients for each plyb) Complete calculation of thermal stress and
moment resultants
ininlbfMN
ininlbfMinlbfN
ininlbfMinlbfN
Txy
Txy
Tyy
Tyy
Txx
Txx
/ 0246.0 0
/ 0005.0 /123
/ 401.0 /129
−−==
−=−=
−−=−=
![Page 61: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/61.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
T
Txx
o
oxxxx
N
NBBBAAAN ε161211161211
−
=
Txy
Tyy
Txx
Txy
Tyy
xy
yy
xx
oxy
oyy
xy
yy
xx
xy
yy
M
M
M
N
N
DDDBBB
DDDBBB
DDDBBB
BBBAAA
BBBAAA
M
M
M
N
N
κκκγε
662616662616
262212262212
161211161211
662616662616
262212262212
![Page 62: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/62.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
T
Txx
o
oxxxx
N
NBBBAAAN ε161211161211
−
=
Txy
Tyy
Txx
Txy
Tyy
xy
yy
xx
oxy
oyy
xy
yy
xx
xy
yy
M
M
M
N
N
DDDBBB
DDDBBB
DDDBBB
BBBAAA
BBBAAA
M
M
M
N
N
κκκγε
662616662616
262212262212
161211161211
662616662616
262212262212
(this is the only difference in calculation….)
![Page 63: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/63.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
−
−−
12902.898.27778031.203191 EENxx
−
−
−
−
−
−
−
−−−−−−−−−−−
−−−
=
0246.0
0005.0
401.0
0
123
20.0
40.0
50.0
900
1300
30.1185.0446.059.20.372.89
185.046.2672.00.373308.27
446.0672.034.92.898.27778
59.20.372.8930.3900
0.373308.270316231.20
µµ
E
EE
M
M
M
N
N
xy
yy
xx
xy
yy
![Page 64: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/64.jpg)
Section 6.8.2: A CLT Analysis When Midplane Strains and Curvatures Are Known
5. Calculate stress and moment resultants
lbf/in 520xxN
−
−=
in/in-lbf 085.0
in/in-lbf 22.0
in/in-lbf 0.4
lbf/in 4.64
lbf/in 377
xy
yy
xx
xy
yy
M
M
M
N
N
![Page 65: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/65.jpg)
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)
![Page 66: Summary of Classical Lamination Theory (CLT) Calculationscourses.washington.edu/mengr450/CLT_Summary.pdf · Summary of Classical Lamination Theory (CLT) Calculations Numerical examples](https://reader031.vdocuments.us/reader031/viewer/2022013102/5be4003709d3f2f4628c0b55/html5/thumbnails/66.jpg)
Analysis complete!