Page 357 - Introduction to Continuum Mechanics

Problems 341

5.49. A circular steel shaft is subjected to twisting couples of 5000 ft-lb (6780 N • m). Determine
the shaft diameter if the maximum shear stress is not to exceed 10,000 psi(69 MPa) and the
6
angle of twist is not to exceed 1.5 ° in 20 diameters of length. /* = 12 x 10 psi (82.7 GPa).
5.50. Demonstrate that the elastic solution for the solid circular bar in torsion is also valid for
a circular cylindrical tube in torsion. If a is the outside radius and b is the inside radius, how
must Eq. (5.13.10) for the twist per unit length be altered?
5.51. In Example 5.13.2, if the radius of the left portion isaj and the radius of the right portion
i
is «2> wnat s tne twisting moment produced in each portion of the shaft? Both shafts are of
the same material.

Fig. P5.3

5.52. Solve the previous problem if a\ - 3.0 cm, a*i = 2.5 cm, l\ - /2 = 75 cm, and M t = 700 N -m
5.53. For the circular shaft shown in Fig.P5.3, determine the twisting moment produced in each
part of the shaft.
5.54. A circular bar of one-inch (2.54 cm) radius is under the action of an axial tensile load of
30,000 lb(133 kN) and a twisting couple of 25,000 in-lbs(2830 N • m).
(a) Determine the stress throughout the bar.

(b) Find the maximum normal and shearing stress that occurs over all locations and all
cross-sectional planes throughout the bar.
5.55. Show that for any cylindrical bar of non-circular cross-section in torsion that the stress
vector at all points along the lateral boundary acting on any of the normal cross-sectional planes
must be tangent to the boundary.
5.56. Demonstrate that the displacement and stress for the elliptic bar in torsion may also be
used for an elliptic tube, if the inside boundary is defined by

where k<l.