Page 327 - Mechanical Behavior of Materials

328 Chapter 7 Yielding and Fracture under Combined Stresses

material is 18 Ni maraging steel (250). What is the safety factor against yielding? (Note:
Check inside, outside, and several intermediate values of radius, as the most severely stressed
location is not known.)
7.32 A thick-walled tube with closed ends has inner and outer radii of 25 and 35 mm, respectively.
It contains an internal pressure of 25 MPa and is also subjected to a torque of 8.0 kN·m. The
material is 7075-T6 aluminum and a safety factor of 2.0 against yielding is required.
(a) What is the safety factor against yielding? Does it meet the required value?
(b) Assume that the inner radius is fixed at its given value. What adjusted value of the outer
radius is required to meet the safety factor?
7.33 Consider a thick-walled tube with closed ends having inner radius r 1 , outer radius r 2 , and
loaded only with an internal pressure p. Assume that, in a design situation, values of r 1 and p
are fixed, as well as a safety factor X against yielding. Further, a candidate material with yield
strength σ o has been selected.
(a) Develop an equation for the outer radius r 2 that is required as a function of the other
variables involved, that is, find r 2 = f ( r 1 , p , X , σ o ).
(b) What r 2 is required for r 1 = 40 mm, p = 100 MPa, X = 4.0, and σ o = 1791 MPa, where
the latter corresponds to the 18 Ni maraging steel of Table 4.2.
7.34 A rotating annular disc, as in Fig. A.9, has inner radius r 1 = 50, outer radius r 2 = 200, and
thickness t = 30 mm. It is made of 2024-T4 aluminum and rotates at a frequency of f = 230
revolutions/second.
(a) What is the safety factor against yielding?
(b) If a safety factor of 2.0 against yielding is required, what is the highest permissible
rotational frequency?
7.35 A solid circular shaft 1.0 m long must support a bending moment M = 1.0kN·m and a torque
T = 1.5kN·m. A safety factor of X = 2.0 against yielding is required.
(a) What shaft diameter d is required if the material is AISI 1020 steel? What is the
resulting mass of the shaft?
(b) Also, consider the possibility of making the shaft out of 2024-T4 aluminum, 7075-T6
aluminum, or one of the tempers of AISI 4140 steel from Prob. 4.33. Calculate the
required diameter and mass for each.
(c) Select a material for the shaft from among those considered in (a) and (b). Assume that
the shaft must be both light in weight and inexpensive, and also not prone to sudden
fracture. See Table 3.13 for useful data.

Section 7.6
7.36 For the situation of Prob. 7.21, what outside diameter is needed if load factors Y M = 1.50 and
Y T = 1.80 are required for moment and torque, respectively?
7.37 For a shaft loaded in bending and torsion, as in Prob. 7.23, develop a design equation for the
diameter d as a function of yield strength, bending moment M, torque T , and load factors Y M
and Y T for moment and torque, respectively. Employ (a) the maximum shear stress criterion,
and (b) the octahedral shear stress criterion.
7.38 A shaft made of gray cast iron is loaded in torsion and contains a groove as in Fig. A.12(d).
Dimensions are d 2 = 52.5, d 1 = 50, and ρ = 3.75 mm. The material has a tensile strength of