BS02-13

Section 2-13 Torsion

Twisting of a member by an external twisting moment or torque, is called torsion. The angle of twist for a round bar is


     theta = T * l / (G * J)             [2-42]

where  T = torque
       l = length 
       G = modulus of rigidity
       J = polar area moment of inertia

For a solid round bar, the shear stress is zero at the center and maximum at the surface. The distribution is proportional to the radius rho and is


     tau = T * rho / J                   [2-43]

Designating r as the radius to the outer surface, we have


     tau_max = T * r / J                 [2-44]

The assumptions used in the analysis are:
1. The bar is acted upon by a pure torque,
and the sections under consideration are
remote from the point of application of
the load and from a change in diameter.
2. Adjacent cross sections originally plane 
and parallel remain plane and parallel
after twisting, and any radial line remains
straight.
3. The material obeys Hooke's law.

Equation [2-44] applies only to circular sections. For a solid round section,


     J = pi()*d^4/32                     [2-45]

where d is the diameter of the bar. For a hollow round section,


     J = pi()/32*(d^4-di^4)              [2-46]

where di is the inside diameter, often referred to as the ID.

In using equation [2-44] it is often necessary to obtain the torque T from a consideration of the horsepower and speed of a rotating shaft. For convenience, three forms of this relation are


     P = 2 * pi() * T * n / (33000 * (12))
        = F * V / 33000
         = T * n / 63000                   [2-47]

     T = 63000 * P / n                     [2-48]

where   P = horsepower
        T = torque, lb * in
        n = shaft speed, rpm
        F = force, lb
        V = velocity, fpm

If SI units are used, the applicable equation is


     P = T * omega                         [2-49]

where   P = power, W
        T = torque, N*m
        omega = angular velocity, radians/s

Determination of the torsional stresses in noncircular members is a difficult problem, generally handled experimentally using a soap-film or membrane analogy, which we shall not consider here. Timoshenko and MacCullough,* however, give the following approximate formula for the maximum torsional stress in a rectangular section:


     tau_max = T / (w*t^2)*(3 + 1.8*t/w)   [2-50]

In this equation w and t are the width and thickness of the bar, respectively; they cannot be interchanged since t must be the shortest dimension. For thin plates in torsion, t/w is small and the second term may be neglected. The equation is also approximately valid for equal-sided angles; these can be considered as two rectangles each of which is capable of carrying half the torque.

* S. Timoshenko and Gleason H. MacCullough, "Elements of Strength of Materials," 3d ed., p. 265, D. Van Nostrand Company, Inc., New York, 1949. See also F.R. Shanley, "Strength of Materials," p. 509, McGraw-Hill Book Company, New York, 1957.

Mechanical Engineering Design Section 2-13 Torsion