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Chapter 5

Q. 5.5

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 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

Step-by-Step

Verified Solution

1. The maximum normal stress in a plate under bending is given by

\sigma=\pm \frac{M \frac{t}{2}}{I} ,                   (5.20)

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

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

I=\frac{b t^{3}}{12},                (5.21)

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

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

where σ_{ult} = 40.02  Ksi from Table 3.4

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

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

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

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

M_{xx}=3,250\times 2 \\ \space \\ =6,500  lb-in./in.

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^2 program, the strength ratio for using a single 0° ply for the previous load for glass/epoxy ply is

SR=5.494\times 10^{-5}.

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

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

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

t_{Gl/Ep} = 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

N_{Gr/Ep} = 87 plies.

This gives the thickness of the plate as

t_{Gr/Ep} = 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

N_{\max }=\frac{\text { Maximum Allowable Thickness }}{\text { Thickness of each ply }} \\ \space \\ =\frac{0.4936}{0.0049213} \\ \space \\ =100  plies.

Several combinations of 100-ply symmetric hybrid laminates of the form [0_n^{Gr} / 0_m^{Gl} / 0_n^{Gr}] 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, [0_{16}^{Gr} / 0_{68}^{Gl} / 0_{16}^{Gr}].

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

Case Number of plies Minimum SR Cost
Glass/epoxy (m) Graphite/epoxy (2n)
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

V_{Al} = 4 × 4 × 0.9871
= 15.7936 in.³

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

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

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

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

The volume of graphite/epoxy in the hybrid laminate is

V_{Gr/Ep} = 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 3.6127 × 10^{–2}  lbm/in.^3:

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

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

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

The mass of the hybrid laminate then is

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

The percentage savings using the composite laminate over aluminum is

=\frac{1.540-0.4928}{1.540} \times 100 \\ =68\%.

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):
\rho_{c}=\rho_{f} V_{f}+\rho_{m} V_{m}

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 V_f 0.45 0.50 0.70
Longitudinal elastic modulus E_1 GPa 38.6 204 181
Transverse elastic modulus E_2 GPa 8.27 18.50 10.30
Major Poisson’s ratio \nu_{12} 0.26 0.23 0.28
Shear modulus G_{12} GPa 4.14 5.59 7.17
Ultimate longitudinal tensile strength (\sigma_1^T)_{ult} MPa 1062 1260 1500
Ultimate longitudinal compressive strength (\sigma_1^C)_{ult} MPa 610 2500 1500
Ultimate transverse tensile strength (\sigma_2^T)_{ult} MPa 31 61 40
Ultimate transverse compressive strength (\sigma_2^C)_{ult} MPa 118 202 246
Ultimate in-plane shear strength (\tau_{12})_{ult} MPa 72 67 68
Longitudinal coefficient of thermal expansion \alpha_1 μm/m/°C 8.6 6.1 0.02
Transverse coefficient of thermal expansion \alpha_2 μm/m/°C 22.1 30.3 22.5
Longitudinal coefficient of moisture expansion \beta_1 m/m/kg/kg 0.00 0.00 0.00
Transverse coefficient of moisture expansion \beta_2 m/m/kg/kg 0.60 0.60 0.60