## Wyn

R Re Rp

where RP is the radius of curvature in the plastic condition and RE is the elastic spring-back, calculated by applying the simple bending theory to the complete section with a moment of Mpp or Mfp as the case may be.

3.10. Torsion of shafts beyond the elastic limit - plastic torsion

The method of treatment of shafts subjected to torques sufficient to initiate yielding of the material is similar to that used for plastic bending of beams (ยง3.1), i.e. it is usual to assume a stress-strain curve for the shaft material of the form shown in Fig. 3.2, the stress being proportional to strain up to the elastic limit and constant thereafter. It is also assumed that plane cross-sections remain plane and that any radial line across the section remains straight.

Consider, therefore, the cross-section of the shaft shown in Fig. 3.18(a) with its associated shear stress distribution. Whilst the shaft remains elastic the latter remains linear, and as the torque increases the shear stress in the outer fibres will eventually reach the yield stress in shear of the material rv. The torque at this point will be the maximum that the shaft can withstand whilst it is completely elastic.

Shaft Stress cross-section distribution

Shaft Stress cross-section distribution

Yielded' area

(o) Maximum (b) Portially elastic plastic

(c) Fully plastic

Fig. 3.18. Plastic torsion of a circular shaft.

Yielded' area

(o) Maximum (b) Portially elastic plastic

(c) Fully plastic

Fig. 3.18. Plastic torsion of a circular shaft.

From the torsion theory

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