1

Chapter 2 - Macromechanical Analysis of a Lamina Exercise Set 2.1 The number of independent elastic constants in three dimensions are:

Anisotropic 21 Monoclinic 13 Orthotropic 9 Transversely Orthotropic 5 Isotropic 2

2.2

=C

13.675

6.39

10.846

0

0

0

6.39

6.58

6.553

0

0

0

10.846

6.553

12.316

0

0

0

0

0

0

7

0

0

0

0

0

0

2

0

0

0

0

0

0

6

Msi

=S

0.25

0.05

0.1935

0

0

0

0.05

0.3333

0.1333

0

0

0

0.1935

0.1333

0.3226

0

0

0

0

0

0

0.1429

0

0

0

0

0

0

0.5

0

0

0

0

0

0

0.1667

1Msi

2.3

a)

=

σ 1

σ 2

σ 3

τ 23

τ 31

τ 12

8.269

6.731

4.423

0

10

9

kPa

b) Compliance matrix -

1

=S

0.5

0.5

0.4

0

0

0

0.5

0.3

0.2

0

0

0

0.4

0.2

0.6

0

0

0

0

0

0

0.25

0

0

0

0

0

0

0.5

0

0

0

0

0

0

0.667

1GPa

c) =E 1 2 GPa

=E 2 3.333 GPa

=E 3 1.667 GPa

=ν 12 1

=ν 23 0.6667

=ν 31 0.6667

=G 12 4 GPa

=G 23 2 GPa

=G 31 1.5 GPa

d)

=W 3.335 10 2 Pa 2.10

=S

4.902 10 3

1.127 10 3

0

1.127 10 3

5.405 10 2

0

0

0

1.789 10 1

1GPa

=Q

204.98

4.28

0

4.28

18.59

0

0

0

5.59

GPa

2.11

=

ε 1

ε 2

γ 12

17.35

103.59

536.70

µmm

1

2.12 Modifying Equation (2.17) for an isotropic lamina under plane stress -

S

1E

ν

E

0

ν

E

1E

0

0

0

1G

Inverting the compliance matrix gives the reduced stiffness matrix -

Q

E

1 ν2

.E ν

1 ν2

0

.E ν

1 ν2

E

1 ν2

0

0

0

G

2.13 Compliance matrix for a two dimensional orthotropic material per Equations (2.87) -

S

1E 1

ν 21E 2

0

ν 12E 1

1E 2

0

0

0

1G 12

Matrix inversion yields -

Q

.11 .ν 21 ν 12

E 1

.ν 21

1 .ν 21 ν 12E 1

0

.ν 12

1 .ν 21 ν 12E 2

.11 .ν 21 ν 12

E 2

0

0

0

G 12

Compliance matrix for three dimensional orthotropic material per Equation (2.70) -

1

S

1E 1

ν 21E 2

ν 31E 3

0

0

0

ν 12E 1

1E 2

ν 32E 3

0

0

0

ν 13E 1

ν 23E 2

1E 3

0

0

0

0

0

0

1G 23

0

0

0

0

0

0

1G 31

0

0

0

0

0

0

1G 12

Matrix inversion yields -

C 11.

1 .ν 32 ν 231 .ν 32 ν 23

.ν 21 ν 12..ν 21 ν 32 ν 13

..ν 31 ν 12 ν 23.ν 31 ν 13

E 1

C 66 G 12

Proving Q11 ≠ C11 and Q66 = C66. 2.15

=E 1 5.599 Msi,

=E 2 1.199 Msi

=ν 12 0.2600

=G 12 0.6006 Msi 2.16

=

σ 1

σ 2

τ 12

9.019 10 1

5.098

2.366

MPa

2.17

=

ε 1

ε 2

γ 12

2.201

3.799

3.232

µinin

2.18

1

=Q

29.06

40.40

19.50

40.40

122.26

61.21

19.50

61.21

41.71

GPa

=S

6.383 10 2

2.319 10 2

4.195 10 3

2.319 10 2

3.925 10 2

4.676 10 2

4.195 10 3

4.676 10 2

9.063 10 2

1GPa

2.19 S11 S 22 S12 S 12 S22 S 11

S16 0 S26 0 S66 S 66

The values of c and s are interchanged for 0° and 90° laminas. The values of S11 and S22 are interchanged also since the local axes for the 0° lamina are rotated 90°.

2.20

A)

=

ε x

ε y

γ xy

196.4

126.0

348.6

µmm

=

σ 1

σ 2

τ 12

9.800 10 2

6.098

6.340 10 1

MPa

=

ε 1

ε 2

γ 12

7.36

329.73

113.40

µmm

c) Principal normal stresses produced by applied global stresses -

σ maxσ x σ y

2

σ x σ y2

2

τ xy2

=σ max 6.162 MPa

1

σ minσ x σ y

2

σ x σ y2

2

τ xy2

=σ min 0.1623 MPa

Orientation of maximum principal stress -

θ pσ..1

2atan

.2 τ xyσ x σ y

.180 °π

=θ pσ 35.78 °

Principal normal strains -

ε maxε x ε y

2

ε x ε y2

2γ xy

2

2

=ε max 339.0µmm

ε minε x ε y

2

ε x ε y2

2γ xy

2

2

=ε min 16.64µmm

Orientation of maximum principal strain -

θ pε..1

2atan

γ xyε x ε y

.180 °π

=θ pε 39.30 °

d) Maximum shear produced by applied global stresses -

τ maxσ x σ y

2

2

τ xy2

=τ max 3.162 MPa

Orientation of maximum shear -

θ sτ..1

2atan

σ x σ y.2 τ xy

.180 °π

=θ sτ 9.22 °

Maximum shear strain -

γ max.2

ε x ε y2

2γ xy

2

2

=γ max 355.7µmm

Orientation of maximum shear strain -

θ sγ..1

2atan

ε x ε yγ xy

.180 °π

=θ sγ 5.70 °

1

2.21 =θ 34.98 °

2.22 =E x 2.272 Msi

=ν xy 0.3632

=m x 0.8530

=E y 3.696 Msi

=m y 9.538

=G xy 1.6 Msi 2.23

The maximum value of Gxy occurs at θ = 45° =G xy θ 3 26.54 GPa

The minimum value of Gxy occurs at θ = 0° =G xy θ 1 20.00 GPa

Displayed graphically -

0 10 20 30 40 50 60 70 80 9020

22

24

26

28

Ply Angle (°)

Eng

inee

ring

She

ar M

odul

us (G

Pa)

2.24 a) Graphite/Epoxy lamina

=ε a 11.60 %

b) Aluminum plate

1

=ε b 0.36 %

2.25 Given

a) Elastic modulus in x direction per Equation (2.112) - =G 12 10.10 GPa

b) Elastic modulus - =E x 13.4 GPa

2.26

=E x( ).30 ° 4.173 Msi

b) Only 1

G 12

.2 ν 12E 1

can be determined, G 12 and ν 12 cannot be determined individually.

2.27 Yes, for some values of θ, Ex < E2 if

<E 2 E 1 and <G 12

E 1

.2E 1E 2

ν 12

2.28 Yes, for some values of θ, Ex > E1 if

>G 12E 1

.2 1 ν 12

2.29 Basing calculations on a Boron/Epoxy lamina

The value of νxy is maximum for a 26° lamina. Displayed graphically –

1

2.30 Given Tabulation:

LWR ζ E1x (Msi) 2 1.504E-1 2.629 8 1.803E-2 2.275 16 4.724E-3 2.245 64 2.998E-4 2.235

Graph of ζ as a function of Length-to-Width ratio -

0 10 20 30 40 50 60 70

0.1

0.2

0.3

Length-to-Width Ratio (L/W)

Zeta

Graph of Engineering modulus for Finite LWR as a function of ζ -

1

0 10 20 30 40 50 60 702

2.5

3

3.5

Length-to-Width Ratio (L/W)

You

ng's

mod

ulus

for F

inite

L/W

(Msi

)

E x

2.32

=G 12 0.0488 GPa

b) From 35° ply data, one can find S11 and S12 for the lamina. However, for Equations (2.101a) and (2.101b), there are four unknowns: S11, S12, S22, and S66. Therefore, one cannot find S66 nor, in turn, the shear modulus, G12. Although the same is true in the case of the 45° ply, no manipulation of Equations (2.101a) and (2.101b) will allow S66 to be expressed in terms of S11 and S12 for the 35° ply.

2.33 Elastic properties of Boron/Epoxy from Table 2.2 -

=U 1 12.72 Msi

=U 2 13.52 Msi

=U 3 3.493 Msi

=U 4 4.113 Msi

=V 1 3.046 10 1 1Msi

=V 2 1.695 10 1 1Msi

=V 3 1.014 10 1 1Msi

=V 4 1.091 10 1 1Msi

1

2.35 Given

Failure Criterion Magnitude of Maximum Positive Shear Stress

Magnitude of Maximum Negative Shear Stress

Off-axis Shear Strength

Maximum Stress 134 MPa 70.44 MPa 70.44 MPa Maximum Strain: 134 MPa 68.99 MPa 68.99 MPa Tsai-Hill: 62.24 MPa 62.24 MPa 62.24 MPa Tsai-Wu (Mises-Hencky criterion)

139 MPa 59.66 MPa 59.66 MPa

2.36 The maximum stress failure theory gives the mode of failure. The Tsai-Wu failure theory is a unified

theory and gives no indication of the failure mode. 2.37 The Tsai-Wu failure theory agrees closely with experimentally obtained results. The difference between

the maximum stress failure theory and experimental results are quite pronounced. 2.38 Given

Maximum Strain: σ Mε

.249.9 MPa

Tsai-Wu (Mises-Hencky criterion): σ TWMH.259.9 MPa

The maximum biaxial stress that can be applied to a 60° lamina of Graphite/Epoxy is conservatively estimated at -249.9 MPa.

2.40 Given the strength parameters for an isotropic material

<σ 1

2σ 2

2 .6.25 τ 122 ..1. σ 1 σ 2 σ T

2 2.41 Given the strength parameters for a unidirectional Boron/Epoxy system -

Since <H 122 .H 11 H 22 the stability criterion is satisfied.

2.42 The units for the coefficient of thermal expansion in the USCS system are in/in/°F. In the SI system the

units for the coefficient of thermal expansion are m/m/°C. 2.43

=

ε C1

ε C2

γ C12

0

1200

0

µmm

1

=∆T 2_Offset 54.30 °C 2.44

=

α x

α y

α xy

10.403

6.653

6.495

µinin°F

2.45 The units for the coefficient of moisture expansion in the USCS system are in/in/lbm/lbm. In the SI system

the units for the coefficient of moisture expansion are m/m/kg/kg. 2.46

=

β x

β y

β xy

0.4500

0.1500

0.5196

mmkgkg