An electronic device uses an aluminum plate of 1-in. thickness and a top cross–sectional area of 4 in. × 4 in. to take a pure bending moment. The designer wants to replace the aluminum plate with graphite/epoxy unidirectional laminate. The ply thickness of graphite/epoxy is 0.125 mm (0.0049213

Question

An electronic device uses an aluminum plate of 1-in. thickness and a top cross–sectional area of 4 in. × 4 in. to take a pure bending moment. The designer wants to replace the aluminum plate with graphite/epoxy unidirectional laminate. The ply thickness of graphite/epoxy is 0.125 mm (0.0049213 in.).

1. Use the properties of aluminum and unidirectional graphite/epoxy as given in Table 3.4 and Table 2.2, respectively, to design a plate of graphite/epoxy with the same bending stiffness in the needed direction of load as that of the aluminum beam.
2. Does the laminate design decrease the mass? If so, by how much?

TABLE 2.2
Typical Mechanical Properties of a Unidirectional Lamina (USCS System of Units)

Property	Symbol	Units	Glass/epoxy	Boron/epoxy	Graphite/epoxy
Fiber volume fraction	[latex]V_f[/latex]	—	0.45	0.50	0.70
Longitudinal elastic modulus	[latex]E_1[/latex]	Msi	5.60	29.59	26.25
Transverse elastic modulus	[latex]E_2[/latex]	Msi	1.20	2.683	1.49
Major Poisson’s ratio	[latex] u_{12}[/latex]		0.26	0.23	0.28
Shear modulus	[latex]G_{12}[/latex]	Msi	0.60	0.811	1.040
Ultimate longitudinal tensile strength	[latex](\sigma_1^T)_{ult}[/latex]	ksi	154.03	182.75	217.56
Ultimate longitudinal compressive strength	[latex](\sigma_1^C)_{ult}[/latex]	ksi	88.47	362.6	217.56
Ultimate transverse tensile strength	[latex](\sigma_2^T)_{ult}[/latex]	ksi	4.496	8.847	5.802
Ultimate transverse compressive strength	[latex](\sigma_2^C)_{ult}[/latex]	ksi	17.12	29.30	35.68
Ultimate in-plane shear strength	[latex]( au_{12})_{ult}[/latex]	ksi	10.44	9.718	9.863
Longitudinal coefficient of thermal expansion	[latex]\alpha_1[/latex]	μin./in./°F	4.778	3.389	0.0111
Transverse coefficient of thermal expansion	[latex]\alpha_2[/latex]	μin./in./°F	12.278	16.83	12.5
Longitudinal coefficient of moisture expansion	[latex]\beta_1[/latex]	in./in./lb/lb	0.00	0.00	0.00
Transverse coefficient of moisture expansion	[latex]\beta_2[/latex]	in./in./lb/lb	0.60	0.60	0.60

TABLE 3.4
Typical Properties of Matrices (USCS System of Units)

Property	Units	Epoxy	Aluminum	Polyamide
Axial modulus	Msi	0.493	10.30	0.5075
Transverse modulus	Msi	0.493	10.30	0.5075
Axial Poisson’s ratio	—	0.30	0.30	0.35
Transverse Poisson’s ratio	—	0.30	0.30	0.35
Axial shear modulus	Msi	0.1897	3.915	0.1885
coefficient of thermal expansion	μin./in./°F	35	12.78	50
Coefficient of moisture expansion	in./in./lb/lb	0.33	0.00	0.33
Axial tensile strength	ksi	10.44	40.02	7.83
Axial compressive strength	ksi	14.79	40.02	15.66
Transverse tensile strength	ksi	10.44	40.02	7.83
Transverse compressive strength	ksi	14.79	40.02	15.66
Shear strength	ksi	4.93	20.01	7.83
Specific gravity	—	1.2	2.7	1.2

Accepted Answer

1. The bending stiffness, [latex]E_b[/latex], of the aluminum plate is given by:

[latex]E_b=EI[/latex] (5.23)

[latex] =E\left(\frac{1}{12} b h^{3} ight), [/latex]

where
E = Young’s modulus of aluminum
b = width of beam
h = thickness of beam

[latex] E_{b}=10.3 imes 10^{6}\left(\frac{1}{2}(4)(1)^{3} ight) \ \space \ =3.433 imes 10^6 lb-in.^2 [/latex]

To find the thickness of a graphite/epoxy laminate with unidirectional plies and the same flexural rigidity, let us look at the bending stiffness of a laminate of thickness, h:

[latex]E_b=E_xI \ \space \ =E_x\frac{1}{12}bh^3[/latex],

where [latex]E_x[/latex] = Young’s modulus in direction of fibers.
Because [latex]E_x = E_1 = 26.25 Msi[/latex] for a 0° ply from Table 2.2,

[latex] 3.433 imes 10^{6}=26.25 imes 10^{6}\left(\frac{1}{2} 4 h^{3} ight) [/latex],

giving

h = 0.732 in.

Thus, a 1-in. thick aluminum beam can be replaced with a graphite/epoxy laminate of 0.732 in. thickness. Note that, although the Young’s modulus of graphite /epoxy is approximately 2.5 times that of aluminum, the thickness of aluminum plate is approximately only 1.4 times that of the graphite /epoxy of laminate because the bending stiffness of a beam is proportional to the cube of the thickness. Thus, the lightest beam for such bending would be influenced by the cube root of the Young’s moduli. From the thickness of 0.732 in. of the laminate and a thickness of 0.0049312 in. of the lamina, the number of 0° graphite/epoxy plies needed is

[latex] n=\frac{0.732}{0.0049213}=149 . [/latex]

The resulting graphite/epoxy laminate then is [latex][0_{149}][/latex].

2. The volume of the aluminum plate [latex]V_{Al}[/latex] is

[latex]V_{Al}[/latex] = 4 × 4 × 1.0
= 16 in.³

The mass of the aluminum plate is (specific gravity = 2.7 from Table 3.2; density of water is [latex]3.6127 × 10^{–2} lbm/in.^3[/latex]):

[latex]M_{Al} = V_{Al} ρ_{Al} \ = 16 × (2.7 × 3.6127 × 10^{–2}) \ = 1.561 lbm.[/latex]

The volume of a [latex][0_{149}][/latex] graphite/epoxy laminate is

[latex]V_{Gr/Ep}[/latex] = 4 × 4 × 0.0049213 × 149
= 11.73 in.³

The density of a graphite/epoxy from Example 5.5 is

[latex] ho_{G r / E p}=0.5853 imes 10^{-1} \frac{ lbm }{ in ^{3}} . [/latex]

The mass of the graphite/epoxy laminate beam is

[latex]M_{Gr/Ep}[/latex] = (0.5853 × 10–1) (11.73)
= 0.6866 lbm.

Therefore, the percentage saving in using graphite/epoxy composite laminate over aluminum is

[latex] \frac{M_{A l}-M_{G r / E p}}{M_{A l}} \ \space \ =\frac{1.561-0.6866}{1.561} imes 100 \ \space \ = 56\%.[/latex]

TABLE 3.2
Typical Properties of Matrices (SI System of Units)

Property	Units	Epoxy	Aluminum	Polyamide
Axial modulus	GPa	3.4	71	3.5
Transverse modulus	GPa	3.4	71	3.5
Axial Poisson’s ratio	—	0.30	0.30	0.35
Transverse Poisson’s ratio	—	0.30	0.30	0.35
Axial shear modulus	GPa	1.308	27	1.3
coefficient of thermal expansion	μm/m/°C	63	23	90
Coefficient of moisture expansion	m/m/kg/kg	0.33	0.00	0.33
Axial tensile strength	MPa	72	276	54
Axial compressive strength	MPa	102	276	108
Transverse tensile strength	MPa	72	276	54
Transverse compressive strength	MPa	102	276	108
Shear strength	MPa	34	138	54
Specific gravity	—	1.2	3	1.2

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