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

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)

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

TABLE 2.1
Typical Mechanical Properties of a Unidirectional Lamina (SI System of Units)