1. An electronic device uses an aluminum plate of cross-section 4 in. × 4 in. to take a pure bending moment of 13,000 lb-in. The factor of safety is 2. Using the properties of aluminum given in Table 3.4, find the thickness of the plate. 2. The designer wants at least to halve the thickness of the

Question

1. An electronic device uses an aluminum plate of cross-section 4 in. × 4 in. to take a pure bending moment of 13,000 lb-in. The factor of safety is 2. Using the properties of aluminum given in Table 3.4, find the thickness of the plate.

2. The designer wants at least to halve the thickness of the plate to make room for additional hardware on the electronic device. The choices include unidirectional laminates of graphite/epoxy, glass/epoxy, or their combination (hybrid laminates). The ply thickness is 0.125 mm (0.0049213 in.). Design a plate with the lowest cost if the manufacturing cost per ply of graphite/epoxy and glass/epoxy is ten and four units, respectively. Use the properties of unidirectional graphite/epoxy and glass/epoxy laminae from Table 2.2.

3. Did your choice of the laminate composite 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 maximum normal stress in a plate under bending is given by

[latex] \sigma=\pm \frac{M \frac{t}{2}}{I} [/latex], (5.20)

where
M = bending moment (lb-in.)
t = thickness of plate (in.)
I = second moment of area (in.[latex]^4[/latex])

For a rectangular cross-section, the second moment of area is

[latex] I=\frac{b t^{3}}{12}, [/latex] (5.21)

where b = width of plate (in.).
Using the given factor of safety, [latex]F_s = 2[/latex], and given b = 4 in., the thickness of the plate using the maximum stress criterion is

[latex] t=\sqrt{\frac{6 M F_{s}}{b \sigma_{u l t}}} [/latex] (5.22)

where [latex]σ_{ult} = 40.02 Ksi[/latex] from Table 3.4

[latex] t=\sqrt{\frac{6(13000) 2}{4(40.02) 10^{3}}} \ \space \ =0.9872 in.[/latex]

2. Now the designer wants to replace the 0.9872 in. thick aluminum plate by a plate of maximum thickness of 0.4936 in. (half that of aluminum) made of laminated composites. The bending moment per unit width is

[latex] M_{x x}=\frac{13,000}{4} \ \space \ = 3,250 lb-in./in.[/latex]

Using the factor of safety of two, the plate is designed to take a bending moment per unit width of

[latex]M_{xx}=3,250 imes 2 \ \space \ =6,500 lb-in./in.[/latex]

The simplest choices are to replace the aluminum plate by an all graphite/epoxy laminate or an all glass/epoxy laminate. Using the procedure described in Example 5.3 or using the PROMAL[latex]^2[/latex] program, the strength ratio for using a single 0° ply for the previous load for glass/epoxy ply is

[latex]SR=5.494 imes 10^{-5}[/latex].

The bending moment per unit width is inversely proportional to the square of the thickness of the plate, so the minimum number of plies required would be

[latex] N_{G l / E p}=\sqrt{\frac{1}{5.494 imes 10^{-5}}} \ \space \ =135 plies.[/latex]

This gives the thickness of the all-glass/epoxy laminate as

[latex]t_{Gl/Ep}[/latex] = 135 × 0.0049213 in.
= 0.6643 in.

The thickness of an all-glass/epoxy laminate is more than 0.4935 in. and is thus not acceptable.
Similarly, for an all graphite/epoxy laminate made of only 0° plies, the minimum number of plies required is

[latex]N_{Gr/Ep}[/latex] = 87 plies.

This gives the thickness of the plate as

[latex]t_{Gr/Ep}[/latex] = 87 × 0.0049213
= 0.4282 in.

The thickness of an all-graphite/epoxy laminate is less than 0.4936 in. and is acceptable.

Even if an all-graphite/epoxy laminate is acceptable, because graphite/epoxy is 2.5 times more costly than glass/epoxy, one would suggest the use of a hybrid laminate. The question that arises now concerns the sequence in which the unidirectional plies should be stacked. In a plate under a bending moment, the magnitude of ply stresses is maximum on the top and bottom face. Because the longitudinal tensile and compressive strengths are larger in the graphite/epoxy lamina than in a glass/epoxy lamina, one would put the former as the facing material and the latter in the core.
The maximum number of plies allowed in the hybrid laminate is

[latex] N_{\max }=\frac{ ext { Maximum Allowable Thickness }}{ ext { Thickness of each ply }} \ \space \ =\frac{0.4936}{0.0049213} \ \space \ =100 plies.[/latex]

Several combinations of 100-ply symmetric hybrid laminates of the form [latex][0_n^{Gr} / 0_m^{Gl} / 0_n^{Gr}][/latex] are now subjected to the applied bending moment.
Minimum strength ratios in each laminate stacking sequence are found. Only if the strength ratios are greater than one — that is, the laminate is safe — is the cost of the stacking sequence determined. A summary of these results is given in Table 5.8.
From Table 5.8, an acceptable hybrid laminate with the lowest cost is case VI, [latex][0_{16}^{Gr} / 0_{68}^{Gl} / 0_{16}^{Gr}][/latex].

TABLE 5.8
Cost of Various Glass/Epoxy–Graphite/Epoxy Hybrid Laminates

Case	Number of plies		Minimum SR	Cost
Case	Glass/epoxy (m)	Graphite/epoxy (2n)	Minimum SR	Cost
I	0	87	1.023	870
II	20	80	1.342	880
III	60	40	1.127	640
IV	80	20	0.8032	—
V	70	30	0.9836	—
VI	68	32	1.014	592
VII	66	34	1.043	604

3. The volume of the aluminum plate is

[latex]V_{Al}[/latex] = 4 × 4 × 0.9871
= 15.7936 in.³

The mass of the aluminum plate is (specific gravity = 2.7 from Table 3.2),

[latex]M_{Al} = V_{Al} ρ_{Al} \ = 15.793 × [(2.7) (3.6127 × 10^{–2})] \ = 1.540 lbm.[/latex]

The volume of the glass/epoxy in the hybrid laminate is

[latex]V_{Gl/Ep }[/latex]= 4 × 4 × 0.0049213 × 68
= 5.354 in.³

The volume of graphite/epoxy in the hybrid laminate is

[latex]V_{Gr/Ep}[/latex] = 4 × 4 × 0.0049213 × 32
= 2.520 in.³

Using the specific gravities of glass, graphite, and epoxy given in Table 3.1 and Table 3.2 and considering that the density of water is [latex]3.6127 × 10^{–2} lbm/in.^3[/latex]:

[latex]ρ_{Gl} = 2.5 × (3.6127 imes 10^{–2}) = 0.9032 × 10^{–1} lbm/in.^3 \ ρ_{Gr} = 1.8 × (3.6127 imes 10^{–2}) = 0.6503 × 10^{–1} lbm/in.^3 \ ρ_{Ep} = 1.2 × (3.6127 imes 10^{–2}) = 0.4335 × 10^{–1} lbm/in.^3[/latex]

The fiber volume fraction is given in Table 2.1 and, substituting in Equation (3.8), the density of glass/epoxy and graphite/epoxy laminae is

[latex]ρ_{Gl/Ep} = (0.9032 × 10^{–1}) (0.45) + (0.4335 × 10^{–1}) (0.55) \ = 0.6449 × 10^{–1 } lbm/in.^3 \ ρ_{Gr/Ep} = (0.6503 × 10^{–1}) (0.70) + (0.4335 × 10^{–1}) (0.30) \ = 0.5853 × 10^{–1} lbm/in.^3[/latex]

The mass of the hybrid laminate then is

[latex]M_h = (5.354) (0.6449 × 10^{–1}) + (2.520)(0.5853 × 10^{–1}) \ = 0.4928 lbm.[/latex]

The percentage savings using the composite laminate over aluminum is

[latex] =\frac{1.540-0.4928}{1.540} imes 100 \ =68\%.[/latex]

This example dictated the use of unidirectional laminates. How will the design change if multiple loads are present? Examples of multiple loads include a leaf spring subjected to bending moment as well as torsion or a thin pressure vessel subjected to an internal pressure to yield a biaxial state of stress. In such cases, one may have a choice not only of material systems and their combination, but also of orientation of plies. Combinations of angle plies can be infinite, so attention may be focused on angle plies of 0°, 90°, 45°, and –45° and their combinations. This reduces the possibilities to a finite number for a limited number of material systems; however, but the number of combinations can still be quite large to handle.

_______________________________

(3.8):
[latex] ho_{c}= ho_{f} V_{f}+ ho_{m} V_{m} [/latex]

TABLE 3.1
Typical Properties of Fibers (SI System of Units)

Property	Units	Graphite	Glass	Aramid
Axial modulus	Gpa	230	85	124
Transverse modulus	GPa	22	85	8
Axial Poisson’s ratio	—	0.30	0.20	0.36
Transverse Poisson’s ratio	—	0.35	0.20	0.37
Axial shear modulus	GPa	22	35.42	3
Axial coefficient of thermal expansion	μm/m/°C	-1.3	5	-5.0
Transverse coefficient of thermal expansion	μm/m/°C	7.0	5	4.1
Axial tensile strength	MPa	2067	1550	1379
Axial compressive strength	MPa	1999	1550	276
Transverse tensile strength	MPa	77	1550	7
Transverse compressive strength	MPa	42	1550	7
Shear strength	MPa	36	35	21
Specific gravity	—	1.8	2.5	1.4

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

TABLE 2.1
Typical Mechanical Properties of a Unidirectional Lamina (SI 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]	GPa	38.6	204	181
Transverse elastic modulus	[latex]E_2[/latex]	GPa	8.27	18.50	10.30
Major Poisson’s ratio	[latex] u_{12}[/latex]		0.26	0.23	0.28
Shear modulus	[latex]G_{12}[/latex]	GPa	4.14	5.59	7.17
Ultimate longitudinal tensile strength	[latex](\sigma_1^T)_{ult}[/latex]	MPa	1062	1260	1500
Ultimate longitudinal compressive strength	[latex](\sigma_1^C)_{ult}[/latex]	MPa	610	2500	1500
Ultimate transverse tensile strength	[latex](\sigma_2^T)_{ult}[/latex]	MPa	31	61	40
Ultimate transverse compressive strength	[latex](\sigma_2^C)_{ult}[/latex]	MPa	118	202	246
Ultimate in-plane shear strength	[latex]( au_{12})_{ult}[/latex]	MPa	72	67	68
Longitudinal coefficient of thermal expansion	[latex]\alpha_1[/latex]	μm/m/°C	8.6	6.1	0.02
Transverse coefficient of thermal expansion	[latex]\alpha_2[/latex]	μm/m/°C	22.1	30.3	22.5
Longitudinal coefficient of moisture expansion	[latex]\beta_1[/latex]	m/m/kg/kg	0.00	0.00	0.00
Transverse coefficient of moisture expansion	[latex]\beta_2[/latex]	m/m/kg/kg	0.60	0.60	0.60

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