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The fully plastic torque for a solid shaft is therefore 33% greater than the maximum elastic torque. As in the case of beams this can be taken account of in design procedures to increase the allowable torque which can be carried by the shaft or it may be treated as an additional safety factor. In any event it must be remembered that should stresses in the shaft at any time exceed the yield point for the material, then some permanent deformation will occur.

3.11. Angles of twist of shafts strained beyond the elastic limit

Angles of twist of shafts in the partially plastic condition are calculated on the basis of the elastic core only, thus assuming that once the outer regions have yielded they no longer offer any resistance to torque. This is in agreement with the basic assumption listed earlier that radial lines remain straight throughout plastic torsion, i.e. 9PP = 0E for the core. For the elastic core, therefore,

3.12. Plastic torsion of hollow tubes

Consider the hollow tube of Fig. 3.19 with internal radius Rt and external radius R subjected to a torque sufficient to produce yielding to a radius R2. The torque carried by the

Stress distribution

Fig. 3.19. Plastic torsion of a hollow shaft.

Stress distribution

Fig. 3.19. Plastic torsion of a hollow shaft.

equivalent partially plastic solid shaft, i.e. ignoring the central hole, is given by eqn. (3.12) with Ri replacing R\ as

The torque carried by the hollow tube can then be determined by subtracting from the above the torque which would be carried by a solid shaft of diameter equal to the central hole and subjected to a shear stress at its outside fibre equal to r. i.e. from eqn. (3.11) torque on imaginary shaft

_ jtR\ ~ ~2~r but by proportions of the stress distribution diagram

Ri y

Therefore torque on imaginary shaft equal in diameter to the hollow core

Therefore, partially plastic torque for the hollow tube

7rXt

The fully plastic torque is then obtained when Ri = R\, i-e. TFP = gW/?, - 4R*] = R3 - R>] (3.16)

This equation could also have been obtained by adaptation of eqn. (3.13), subtracting a fully plastic core of diameter equal to the central hole.

As an aid in visualising the stresses and torque capacities of members loaded to the fully plastic condition an analogy known as the sand-heap analogy has been introduced. Whilst full details have been given by NadaP it is sufficient for the purpose of this text to note that

^ A. Nadai, Theory of Flow and Fracture of Solids, Vol. 1, 2nd edn., McGraw-Hill, New York, 1950.

if dry sand is poured on to a raised flat surface having the same shape as the cross-section of the member under consideration, the sand heap will assume a constant slope, e.g. a cone on a circular disc and a pyramid on a square base. The volume of the sand heap, and hence its weight, is then found to be directly proportional to the fully plastic torque which would be carried by that particular shape of cross-section. Thus by calibration, i.e. with a knowledge of the fully plastic torque for a circular shaft, direct comparison of the weight of appropriate sand heaps yields an immediate indication of the fully plastic torque of some other more complicated section.

### 3.13. Plastic torsion of case-hardened shafts

Consider now the case-hardened shaft shown in Fig. 3.20. Whilst it is often assumed in such cases that the shear-modulus is the same for the material of the case and core, this is certainly not the case for the yield stresses; indeed, there is often a considerable difference, the value for the case being generally much larger than that for the core. Thus, when the shaft is subjected to a torque sufficient to initiate yielding at the outside fibres, the normal triangular elastic stress distribution required to maintain straight radii must be modified, since this would imply that some of the core material is stressed beyond its yield stress. Since the basic assumption used throughout this treatment is that stress remains constant at the yield stress for any increase in strain, it follows that the stress distribution must be as indicated in Fig. 3.20. The shaft thus contains at this stage a plastic region sandwiched between two elastic layers. Torques for each portion must be calculated separately, therefore, and combined to yield the partially plastic torque for the case-hardened shaft. (Example 3.5.)

Case-hardened shaft cross-section

Fig. 3.20. Plastic torsion of a case-hardened shaft.

Case-hardened shaft cross-section

Fig. 3.20. Plastic torsion of a case-hardened shaft.

### 3.14. Residual stresses after yield in torsion

If shafts are stressed at any time beyond their elastic limit to a partially plastic state as described previously, a permanent deformation will remain when torque is removed. Associated with this plastic deformation will be a system of residual stresses which will affect the strength of the shaft in subsequent loading cycles. The magnitudes of the residual stresses are determined using the method described in detail for beams strained beyond the elastic limit on page 73, i.e. the removal of torque is assumed to be a completely elastic process so that the associated stress distribution is linear. The residual stresses are thus obtained by subtracting the elastic unloading stress distribution from that of the partially plastic loading condition. Now, from eqn. (3.12), partially plastic torque = TPP.

Therefore elastic torque to be applied during unloading — TPP.

The stress r' at the outer fibre of the shaft which would be achieved by this torque, assuming elastic material, is given by the torsion theory

Thus, for a solid shaft the residual stress distribution is obtained as shown in Fig. 3.21.

Solid shaft cross-section (partially plastic)

Fig. 3.21. Residual stresses produced in a solid shaft after unloading from a partially plastic state. Similarly, for hollow shafts, the residual stress distribution will be as shown in Fig. 3.22.

Solid shaft cross-section (partially plastic)

Hollow shaft cross-section (partially plastic)

Fig. 3.21. Residual stresses produced in a solid shaft after unloading from a partially plastic state. Similarly, for hollow shafts, the residual stress distribution will be as shown in Fig. 3.22.

(c) Residual stresses

(c) Residual stresses

Hollow shaft cross-section (partially plastic)

Fig. 3.22. Residual stresses produced in a hollow shaft after unloading from a partially plastic state.

3.15. Plastic bending and torsion of strain-hardening materials

(a) Inelastic bending

Whilst the material in this case no longer follows Hookes' law it is necessary to assume that cross-sections of the beam remain plane during bending so that strains remain proportional to distance from the neutral axis.

Consider, therefore, the rectangular section beam shown in Fig. 3.23(b) with its neutral axis positioned at a distance hi from the lower surface and h2 from the upper surface. Bearing in mind the assumption made in the preceding paragraph we can now locate the neutral axis position by the usual equilibrium conditions.

Fig. 3.23(a). Stress-strain curve for a beam in bending constructed from a strain-hardening material.

Fig. 3.23(a). Stress-strain curve for a beam in bending constructed from a strain-hardening material.

i.e. Since the sum of forces normal to any cross-section must always be zero then:

0 0