MSE 554 MathCAD Project

Input Section (Material Properties)

Epoxy Carbon-Fiber

Youngs Modulus 1-direction Em1 3.2 109Pa Ef1 400 10

9Pa

Youngs Modulus 2-direction Em2 4.0 109Pa Ef2 300 10

9Pa

Shear Modulus G12 Gm 1.5 109Pa Gf 30 10

9Pa

Poissons Ratio, υ12 ν12m 0.35 ν12f 0.25

Coefficient of Thermal Expansion 1-direction αm1 6.5 105

1

Δ°C αf1 5.0 10

6

1

Δ°C

Coefficient of Thermal Expansion 2-direction αm2 1.0 105

1

Δ°C αf2 5.0 10

6

1

Δ°C

Vm 0.40 Vf 0.60Volume Fraction

Input Section (Material Properties)

Ply Properties

E1 Ef1 Vf Em1 Vm 241.28 GPa

Halpin-Tsai for E2

ξ 2

χEf2 Em2

Ef2 ξ Em2 E2

Em2 1 ξ χ Vf( )

1 χ Vf20.344 GPa

ξ 1

χGf Gm

Gf ξ Gm G12

Gm 1 ξ χ Vf( )

1 χ Vf5.062 GPa

ν12 ν12f Vf ν12m Vm 0.29

ν21 ν12E2

E1 0.024

Ply Properties

Q and Qbar properties

Q11E1

1 ν12 ν21( ) Q12

ν12 E2

1 ν12 ν21

Q22E2

1 ν12 ν21( ) Q66 G12

Q

Q11

Q12

0

Q12

Q22

0

0

0

Q66

243.003

5.942

0

5.942

20.489

0

0

0

5.062

GPa Q Matrix

M θ( ) cos θ( ) n θ( ) sin θ( )

Θ θ( )

M θ( )2

n θ( )2

M θ( ) n θ( )

n θ( )2

M θ( )2

M θ( ) n θ( )( )

2 M θ( ) n θ( )

2 M θ( ) n θ( )

M θ( )2

n θ( )2

Ψ θ( )

M θ( )2

n θ( )2

2 M θ( ) n θ( )

n θ( )2

M θ( )2

2 M θ( ) n θ( )( )

M θ( ) n θ( )

M θ( ) n θ( )

M θ( )2

n θ( )2

Qbar Matrices Qbar θ( ) Θ θ( ) Q Ψ θ( )

1

Qbar 0deg( )

243.003

5.942

0

5.942

20.489

0

0

0

5.062

GPa Qbar 60 deg( )

32.738

49.321

23.13

49.321

143.995

73.221

23.13

73.221

48.442

GPa

Qbar 20deg( )

193.076

29.84

64.238

29.84

22.62

7.277

64.238

7.277

28.96

GPa Qbar 20 deg( )

193.076

29.84

64.238

29.84

22.62

7.277

64.238

7.277

28.96

GPa

Qbar 45deg( )

73.906

63.781

55.629

63.781

73.906

55.629

55.629

55.629

62.902

GPa Qbar 90deg( )

20.489

5.942

0

5.942

243.003

0

0

0

5.062

GPa

Qbar 30 deg( )

143.995

49.321

73.221

49.321

32.738

23.13

73.221

23.13

48.442

GPa

Q and Qbar properties

Input Layer Details

z0

1.225 mm z1

0.875 mm z2

0.525 mm z3

0.175 mm z4

0.175mm z5

0.525mm

z6

0.875mm z7

1.225mm

θ1

0deg θ2

20deg θ3

45deg θ4

30 deg θ5

60 deg θ6

20 deg θ7

90deg

i 1 7 j 0 7

z

1.225

0.875

0.525

0.175

0.175

0.525

0.875

1.225

mm θ

0

0

0.349

0.785

0.524

1.047

0.349

1.571

θi

020

45

-30

-60

-20

90

deg

zj

-1.225-0.875

-0.525

-0.175

0.175

0.525

0.875

1.225

mm

Input Layer Details

ABD Matrix

A

i

Qbar θi z

izi 1

A

3.151 108

8.19 107

1.425 107

8.19 107

1.958 108

1.425 107

1.425 107

1.425 107

7.974 107

Pa m

B1

2i

Qbar θi z

i 2 zi 1 2

B

8.682 104

1.771 103

4.112 104

1.771 103

9.036 104

1.935 104

4.112 104

1.935 104

1.771 103

N

D1

3i

Qbar θi z

i 3 zi 1 3

D

175.689

20.506

1.248

20.506

120.775

0.9

1.248

0.9

19.428

Pa m3

Row1 augment A1

Pa m B

1

Pa m2

Row2 augment B1

Pa m2

D

1

Pa m3

ABD stack Row1 Row2( ) ABD Matrix

ABD

3.151 108

8.19 107

1.425 107

8.682 104

1.771 103

4.112 104

8.19 107

1.958 108

1.425 107

1.771 103

9.036 104

1.935 104

1.425 107

1.425 107

7.974 107

4.112 104

1.935 104

1.771 103

8.682 104

1.771 103

4.112 104

175.689

20.506

1.248

1.771 103

9.036 104

1.935 104

20.506

120.775

0.9

4.112 104

1.935 104

1.771 103

1.248

0.9

19.428

ABD Matrix

Coefficient of Thermal Expansion

α1αf1 Ef1 Vf αm1 Em1 Vm( )

Ef1 Vf Em1 Vm5.318 10

6

1

Δ°C

α2 αf2 Vf αm2 VmEf1 ν12m Em2 ν12f( )

Ef1 Vf Em2 Vmαm2 αf1( ) Vf Vm 7.7 10

6

1

Δ°C

αx θ( ) M θ( )2

α1 n θ( )2

α2

αy θ( ) n θ( )2

α1 M θ( )2

α2

αxy θ( ) 2 M θ( ) n θ( ) α1 α2( )

αmatrix θ( )

αx θ( )

αy θ( )

αxy θ( )

Coefficient of Thermal Expansion

Input Load

X 10000

Nload

1 X

5 X

2 X

Mload

0

0

0

NM stack Nload Mload( )

1 104

5 104

2 104

0

0

0

ΔT 80 Δ°C

Nt

i

Qbar θi αmatrix θ

i zi

zi 1 ΔT

1.743 105

1.251 105

1.177 104

N

m

Mt1

2i

Qbar θi αmatrix θ

i zi 2 z

i 1 2

ΔT

36.588

36.588

24.976

Nm

m

NMthermal stack Ntm

N Mt

m

N m

1.743 105

1.251 105

1.177 104

36.588

36.588

24.976

NMtotal NM NMthermal

1.643 105

7.505 104

3.177 104

36.588

36.588

24.976

Input Load

Calculate Strains and Curvatures

εκ ABD1

NMtotal

εκ

4.268 104

2.946 105

3.143 104

0.097

0.251

0.364

ε0 submatrix εκ 0 2 0 0( )ε0

4.268 104

2.946 105

3.143 104

MidplaneStrainsκ submatrix εκ 3 5 0 0( )

1

m

εx z( ) ε0 κ z

κ

0.097

0.251

0.364

1

m Midplane

Curvaturesεx 0( )

4.268 104

2.946 105

3.143 104

εyy strains through laminate thicknessεxx strains through laminate thickness

1 103 0 1 10

36 10

4

5 104

4 104

3 104

εx t( )0

t1 10

3 0 1 103

4 104

2 104

0

2 104

4 104

εx t( )1

t

This is εxxThis is εyy

γxy shear strains through laminate thickness

1 103 0 1 10

32 10

4

0

2 104

4 104

6 104

8 104

εx t( )2

t

Calculate Strains and Curvatures

Stresses and Strains in material direction (1,2,3)

Pw t( ) i 0

i j

break

zj 1 t t z

jif

j 1 length z( ) 1for

i

σx t( ) i Pw t( )

Qbar θi εx t( ) αmatrix θ

i ΔT i 0 i length z( )if

0MPa( ) otherwise

σL t( ) i Pw t( )

Θ θi 1

σx t( )

εL t( ) i Pw t( )

Ψ θi 1

εx t( )

t 1.225 mm 1.224 mm 1.225mm

Stresses in each ply in the1-direction

Stresses in each ply in the2-direction

1 0 150

0

50

100

σL t( )0

MPa

t

mm

1 0 15

10

15

20

σL t( )1

MPa

t

mmStresses in each ply in the3-direction

1 0 14

2

0

2

4

σL t( )2

MPa

t

mm

Strains in each ply in the1-direction

Stresses in each ply in the2-direction

1 0 10.06

0.05

0.04

0.03

0.02

0.01

0

εL t( )0

%

t

mm

1 0 10.04

0.02

0

0.02

0.04

εL t( )1

%

t

mmStresses in each ply in the3-direction

1 0 10.1

0.05

0

0.05

0.1

εL t( )2

%

t

mm

Stresses and Strains in material direction (1,2,3)

Warpage Surface Plot for Thermal Loads

εκ ABD1

NMthermal

εκ

4.587 104

4.947 104

3.082 105

0.036

0.06

0.176

κ submatrix εκ 3 5 0 0( )1

m

κ

0.036

0.06

0.176

1

m

ω x y( )1

20.036 x

2 0.06 y

2

0.176

2x y

Warpage Surface Plot for 1m X 1m laminate

ω

Warpage Surface Plot for Thermal Loads

Maximum Stress Failure Analysis

A transverse tensile failue occurs in the 0 degreeoriented ply when X = 29579 MPa. At this stress level, F2t > 50 MPa

F1t 2000MPa F2t 50Pa F6 70MPa F1c 800MPa F2c 150MPa

X 29579

Nload

1 X

5 X

2 X

Mload

0

0

0

NM stack Nload Mload( )

2.958 104

1.479 105

5.916 104

0

0

0

ΔT 80 Δ°C

Nt

i

Qbar θi αmatrix θ

i zi

zi 1 ΔT

1.743 105

1.251 105

1.177 104

N

m

Mt1

2i

Qbar θi αmatrix θ

i zi 2 z

i 1 2

ΔT

36.588

36.588

24.976

Nm

m

NMthermal stack Ntm

N Mt

m

N m

1.743 105

1.251 105

1.177 104

36.588

36.588

24.976

NMtotal NM NMthermal

1.448 105

2.284 104

7.093 104

36.588

36.588

24.976

εκ ABD1

NMtotal

εκ

3.643 104

8.814 104

9.901 104

0.359

0.859

1.42

ε0 submatrix εκ 0 2 0 0( )ε0

3.643 104

8.814 104

9.901 104

κ submatrix εκ 3 5 0 0( )1

m

εx z( ) ε0 κ z

κ

0.359

0.859

1.42

1

m

εx 0( )

3.643 104

8.814 104

9.901 104

Pw t( ) i 0

i j

break

zj 1 t t z

jif

j 1 length z( ) 1for

i

σx t( ) i Pw t( )

Qbar θi εx t( ) αmatrix θ

i ΔT i 0 i length z( )if

0MPa( ) otherwise

σL t( ) i Pw t( )

Θ θi 1

σx t( )

t 1.225 mm 1.224 mm 1.225mm

Stresses in each ply in the 1-direction Stresses in each ply in the 2-direction

1 0 1100

0

100

200

300

σL t( )0

MPa

t

mm

1 0 110

20

30

40

50

60

σL t( )1

MPa

t

mm

Stresses in each ply in the3-direction

1 0 120

10

0

10

σL t( )2

MPa

t

mm

Maximum Stress Failure Analysis