66 Torsion of shafts

(a) Isotropic solids

A torque, T, applied to the ends of an isotropic bar of uniform section, and acting in the plane normal to the axis of the bar, produces an angle of twist B. The twist is related to the torque by the first equation below, in which G is the shear modulus. For round bars and tubes of circular section, the factor K is equal to J, the polar moment of inertia of the section, defined in Section 6.2. For any other section shape K is less than J. Values of K are given in Section 6.2.

If the bar ceases to deform elastically, it is said to have failed. This will happen if the maximum surface stress exceeds either the yield strength, oy, of the material or the stress at which it fractures. For circular sections, the shear stress at any point a distance r from the axis of rotation is

The maximum shear stress, rmax, and the maximum tensile stress, amax, are at the surface and have the values

Td0 GBd0

2K 21

If xmax exceeds ay/2 (using a Tresca yield criterion), or if amax exceeds the MOR, the bar fails, as shown in Figure 6.5. The maximum surface stress for the solid ellipsoidal, square, rectangular and triangular sections is at the

Torsion of shafts



T'ef W


Elastic deflection




T = Torque (Nm) q = Angle of twist g = Shear modulus (N/m2) I = Length (m) d = Diameter (m) K = See Figure 6.1 (m4) sy= Yield strength (N/m2) sf = Modulus of rupture (N/m2)

Spring deflection and failure

Figure 6.5 Torsion of shafts

Spring deflection and failure

F = Force (N) u = Deflection (m) R= Coil radius (m) n = Number of turns

F = Force (N) u = Deflection (m) R= Coil radius (m) n = Number of turns

Figure 6.5 Torsion of shafts points on the surface closest to the centroid of the section (the mid-points of the longer sides). It can be estimated approximately by inscribing the largest circle which can be contained within the section and calculating the surface stress for a circular bar of that diameter. More complex section-shapes require special consideration, and, if thin, may additionally fail by buckling. Helical springs are a special case of torsional deformation. The extension of a helical spring of n turns of radius R, under a force F, and the failure force Fcrit, is given in Figure 6.5.

(b) Metal foams

The shear moduli of open-cell foams scales as (p/ps)2 and that of closed-cell foams has an additional linear term (Table 4.2). When seeking torsional stiffness at low weight, the material index characterizing performance (see Appendix) is G/p or G1/2/p (solid and hollow shafts). Used as shafts, foams have, at best, the same index value as the material of which they are made; usually it is less. Nothing is gained by using foams as torsion members.

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