Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

p6.52 A stepped shaft has a major diameter D = 100 mm and a
minor diameter d = 75 mm. A fillet with a 10 mm radius is used to
transition between the two shaft segments. The maximum shear
stress in the shaft must be limited to 60 MPa. If the shaft rotates at
a constant angular speed of 500 rpm, determine the maximum
power that may be delivered by the shaft.
p6.53 A semicircular groove with a 6 mm radius is required in a
50 mm diameter shaft. If the maximum allowable shear stress in the
shaft must be limited to 40 MPa, determine the maximum torque
that can be transmitted by the shaft.
p6.54 A 1.25 in. diameter shaft contains a 0.25 in. deep
U-shaped groove that has a 1/8 in. radius at the bottom of the
groove. The maximum shear stress in the shaft must be limited
to 12,000 psi. If the shaft rotates at a constant angular speed of
6 Hz, determine the maximum power that may be delivered by
the shaft.
6.11 Torsion of Noncircular Sections
Prior to 1820, the year that A. Duleau published experimental results to the contrary, it was
thought that the shear stresses in any torsionally loaded member were proportional to the
distance from the longitudinal axis of the member. Duleau proved experimentally that
relationship does not hold for rectangular cross sections. An examination of Figure 6.19
will verify Duleau’s conclusion. If the stresses in the rectangular bar were proportional to
the distance from its axis, the maximum stress would occur at the corners. However, if
there is a stress of any magnitude at a corner, as indicated in Figure 6.19a, it could be
resolved into the components indicated in Figure 6.19b. If these components existed, the
two components shown by the blue arrows would also exist. But these last components
cannot exist, since the surfaces on which they are shown are free boundaries. Therefore,
the shear stresses at the corners of the rectangular bar must be zero.
The first correct analysis of the torsion of a prismatic bar of noncircular cross section
was published by Saint-Venant in 1855; however, the scope of this analysis is beyond the
elementary discussions of this book. 5 The results of Saint-Venant’s analysis indicate that,
in general, every section will warp (i.e., not remain plane) when twisted, except for members
with circular cross sections.
Free boundaries
T
τ = 0
τ = 0
Shear stress τ
resolved into
components
τ dA
(b)
T
(a)
FIGURE 6.19 Torsional shear stresses in a rectangular bar.
5
A complete discussion of the theory is presented in various books, such as I. S. Sokolnikoff, Mathematical
Theory of Elasticity, 2nd. ed. (New York: McGraw-Hill, 1956), pp. 109–134.
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