Page 127 - Shigley's Mechanical Engineering Design

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102 Mechanical Engineering Design
The last assumption depends upon the axisymmetry of the member, so it does not
hold true for noncircular cross sections. Consequently, Eqs. (3–35) through (3–37)
apply only to circular sections. For a solid round section,
πd 4
J = (3–38)
32
where d is the diameter of the bar. For a hollow round section,

π 4 4
J = d − d i (3–39)
o
32
where the subscripts o and i refer to the outside and inside diameters, respectively.
There are some applications in machinery for noncircular cross section members
and shafts where a regular polygonal cross section is useful in transmitting torque to a
gear or pulley that can have an axial change in position. Because no key or keyway is
needed, the possibility of a lost key is avoided. The development of equations for stress
and deflection for torsional loading of noncircular cross sections can be obtained from
the mathematical theory of elasticity. In general, the shear stress does not vary linearly
with the distance from the axis, and depends on the specific cross section. In fact, for a
rectangular section bar the shear stress is zero at the corners where the distance from
the axis is the largest. The maximum shearing stress in a rectangular b × c section bar
occurs in the middle of the longest side b and is of the magnitude
T . T 1.8
τ max = = 3 + (3–40)
αbc 2 bc 2 b/c
where b is the width (longer side) and c is the thickness (shorter side). They can not be
interchanged. The parameter α is a factor that is a function of the ratio b/c as shown in
5
the following table. The angle of twist is given by
Tl
θ = (3–41)
3
βbc G
where β is a function of b/c, as shown in the table.
b/c 1.00 1.50 1.75 2.00 2.50 3.00 4.00 6.00 8.00 10 ∞

α 0.208 0.231 0.239 0.246 0.258 0.267 0.282 0.299 0.307 0.313 0.333
β 0.141 0.196 0.214 0.228 0.249 0.263 0.281 0.299 0.307 0.313 0.333

Equation (3–40) is also approximately valid for equal-sided angles; these can be con-
sidered as two rectangles, each of which is capable of carrying half the torque. 6
It is often necessary to obtain the torque T from a consideration of the power and
speed of a rotating shaft. For convenience when U. S. Customary units are used, three
forms of this relation are
FV 2πTn Tn
H = = = (3–42)
33 000 33 000(12) 63 025

5 S. Timoshenko, Strength of Materials, Part I, 3rd ed., D. Van Nostrand Company, NewYork, 1955, p. 290.
6 For other sections see W. C. Young and R. G. Budynas, Roark’s Formulas for Stress and Strain, 7th ed.,
McGraw-Hill, New York, 2002.