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## Q. 5.6

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

## Verified Solution

1. The bending stiffness, $E_b$, of the aluminum plate is given by:

$E_b=EI$                 (5.23)

$=E\left(\frac{1}{12} b h^{3}\right),$

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

$E_{b}=10.3 \times 10^{6}\left(\frac{1}{2}(4)(1)^{3}\right) \\ \space \\ =3.433 \times 10^6 lb-in.^2$

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:

$E_b=E_xI \\ \space \\ =E_x\frac{1}{12}bh^3$,

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

$3.433 \times 10^{6}=26.25 \times 10^{6}\left(\frac{1}{2} 4 h^{3}\right)$,

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

$n=\frac{0.732}{0.0049213}=149 .$

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

2. The volume of the aluminum plate $V_{Al}$ is

$V_{Al}$ = 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 $3.6127 × 10^{–2} lbm/in.^3$):

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

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

$V_{Gr/Ep}$ = 4 × 4 × 0.0049213 × 149
= 11.73 in.³

The density of a graphite/epoxy from Example 5.5 is

$\rho_{G r / E p}=0.5853 \times 10^{-1} \frac{ lbm }{ in ^{3}} .$

The mass of the graphite/epoxy laminate beam is

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

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

$\frac{M_{A l}-M_{G r / E p}}{M_{A l}} \\ \space \\ =\frac{1.561-0.6866}{1.561} \times 100 \\ \space \\ = 56\%.$

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