A tubular shaft similar to that shown in Figure 16.11 is to be designed that has an outsid...

A tubular shaft similar to that shown in Figure 16.11 is to be designed that has an outside diameter of 100 mm (4 in.) and a length of 1.25 m (4.1 ft). The mechanical characteristic of prime importance is bending stiffness in terms of the longitudinal modulus of elasticity. Stiffness is to be specified as maximum allowable deflection in bending; when subjected to three-point bending as in Figure 12.30, a load of 1700 N (380 lbf) is to produce an elastic deflection of no more than 0.20 mm (0.008 in.) at the midpoint position. Continuous fibers that are oriented parallel to the tube axis will be used; possible fiber materials are glass, and carbon in standard-, intermediate-, and high-modulus grades. The matrix material is to be an epoxy resin, and fiber volume fraction is 0.40.

(a) Decide which of the four fiber materials are possible candidates for this application, and for each candidate, determine the required inside diameter consistent with the preceding criteria.

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(b) For each candidate, determine the required cost and, on this basis, specify the fiber that would be the least expensive to use. Elastic modulus, density, and cost data for the fiber and matrix materials are given in Table 16.6.

Figure 16.11 Schematic representation of a tubular composite shaft, the subject of Design Example 16.1.

Figure 12.30 A three-point loading scheme for measuring the stress–strain behavior and flexural strength of brittle ceramics, including expressions for computing stress for rectangular and circular cross sections.

Table 16.6 Elastic Modulus, Density, and Cost Data for Glass and Various Carbon Fibers and Epoxy Resin

DESIGN EXAMPLE 16.1

Design of a Tubular Composite Shaft

A tubular composite shaft is to be designed that has an outside diameter of 70 mm (2.75 in.), an inside diameter of 50 mm (1.97 in.), and a length of 1.0 m (39.4 in.); such is represented schematically in Figure 16.11. The mechanical characteristic of prime importance is bending stiffness in terms of the longitudinal modulus of elasticity; strength and fatigue resistance are not significant parameters for this application when filament composites are used. Stiffness is to be specified as maximum allowable deflection in bending; when subjected to three-point bending as in Figure 12.30 (i.e., support points at both tube extremities and load application at the longitudinal midpoint), a load of 1000 N (225 lbf) is to produce an elastic deflection of no more than 0.35 mm (0.014 in.) at the midpoint position.

Continuous fibers that are oriented parallel to the tube axis will be used; possible fiber materials are glass, and carbon in standard-, intermediate-, and high-modulus grades. The matrix material is to be an epoxy resin, and the maximum allowable fiber volume fraction is 0.60.

This design problem calls for us to do the following:

(a) Decide which of the four fiber materials, when embedded in the epoxy matrix, meet the stipulated criteria.

(b) Of these possibilities, select the one fiber material that will yield the lowest-cost composite material (assuming fabrication costs are the same for all fibers).

Elastic modulus, density, and cost data for the fiber and matrix materials are given in Table 16.6.