Page 115 - Mechanical Behavior of Materials

116 Chapter 3 A Survey of Engineering Materials

2
π EI
P cr =
L 2
where E is the elastic modulus of the column material and I is the moment of inertia of the
cross-sectional area. Assume that the cross section is a thin-walled tube of wall thickness t
and inner radius r 1 , with the proportions t = 0.2r 1 being maintained, as the section size is
allowed to vary with material choice. The column must have a particular length L and resist
a load P that is lower than P cr by a safety factor X. Noting that the relative importance of
light weight and cost may vary with the application, make a preliminary selection of column
materials from Table 3.13 for the following:
(a) A structural compression member in a space station.
(b) A support for the second floor above the garage in a private residence.
3.18 A spherical pressure vessel that must hold compressed air is to be designed with a given
inner radius r 1 , and the wall thickness t may vary with material choice. The vessel must resist
a pressure p such that there is a safety factor X against the material exceeding its failure
strength.
(a) Considering weight and cost, and any other factors that you believe to be important,
make a preliminary materials selection from Table 3.13 for this application.
(b) Calculate the vessel thickness required for each material. Assume a vessel inner
radius of r 1 = 2 m, a pressure of p = 0.7 MPa, and a safety factor on the material
strength of X = 3. Comment on the values obtained. Why are some much larger than
others?
3.19 A leaf spring in the suspension system of an experimental vehicle is a beam of length
L = 0.5 m, with a rectangular cross section, as shown in Fig. P3.19. This part, as currently
designed with a low-alloy steel, has a width t = 60 mm and a depth h = 5 mm. However,
if possible, it is desirable to replace this steel with another material to reduce the weight
of the component. To avoid redesigning other related parts, the t dimension should not be
changed, but h can be varied, as long as it does not exceed 12 mm. The spring stiffness must
be k = P/v = 50 kN/m. Also, the spring hits a limit to its motion at v max = 30 mm, at which
point the stress should not be so large that the safety factor against material failure is less than
X = 1.4.
(a) First, considering only the k = 50 kN/m requirement, determine which materials in
Table 3.13 would provide a lighter weight component.
(b) Next, for each material, calculate the h necessary to meet the k = 50 kN/m require-
ment, and also the safety factor relative to σ c at v max = 30 mm. Eliminate any materials
that do not meet h ≤ 12 mm and X ≥ 1.4.
(c) Finally, compare the alloy steel design with the use of each of the remaining candidates,
considering cost and any other factors that you believe to be important.
3.20 A beam is simply supported at its ends, has a length L = 1.50 m, and is subjected to a
uniformly distributed load w = 2.00 kN/m, as in Fig. A.4(b). The beam is a hollow box
section, as in Fig. A.2(d), with proportions b 2 = h 2 and b 1 = h 1 = 0.70h 2 . There are two
design requirements: The safety factor against yielding or other failure of the material must
be at least X = 3.0, and the midspan deflection must not exceed 25 mm. Using properties from