O

Residual rodiol stresses re-plotted on horizontal base line

Fig. 3.32. Determination of residual radial stresses by elastic unloading.

Working stress distributions

Finally, if the stress distributions due to an elastic internal working pressure Pw are superimposed on the residual stress state then the final working stress state is produced as in Figs. 3.33 and 3.34.

The elastic working stresses are given by eqns. (3.42) and (3.43) with PA replaced by Pw. Alternatively a Lamé line solution can be adopted. The final stress distributions show that

Elastic looding stresses under working pressure P.

Elastic looding stresses under working pressure P.

Hoop working stresses

(by addition of above diagrams taking oecount of sign)

Fig. 3.33. Evaluation of hoop working stresses.

Hoop working stresses

(by addition of above diagrams taking oecount of sign)

Fig. 3.33. Evaluation of hoop working stresses.

Fig. 3.34. Evaluation of radial working stresses.

the maximum tensile stress, instead of being at the bore as in the plain cylinder, is now at the elastic/plastic interface position. Application of the Tresca maximum shear stress failure criterion:

i.e. aH — or = (Ty/n also indicates the elastic/plastic interface as now more critical than the internal bore - see Fig. 3.35.

Fig. 3.35. Distribution of maximum shear stress = j (ay — or) by combination of Figs. 3.33 and 3.34.

Effect of axial stresses and end restraint

Depending on the end conditions which can be assumed for the cylinder during both the autofrettage process and its normal working condition a further complication can arise since the axial stresses oz which are produced can affect the application of the Tresca criterion.

Strictly, Tresca requires the use of the greatest difference in the principal stresses which, if oz is zero, = oH — oy. If, however, az has a value it must be used in conjunction with oH and oy to produce the greatest difference.

The procedure used above to determine residual hoop and radial stresses and subsequent working stresses should therefore be repeated for axial stresses with values in the plastic region being found as suggested by Franklin and Morrison^ from:

and axial stresses under elastic conditions being given by eqn. (10.7)$ with P2 = 0 and P\ — Pa or Pw as required.

(c) Rotating discs

It will be shown in Chapter 4 that the centrifugal forces which act on rotating discs produce two-dimensional tensile stress systems. At any given radius the hoop or circumferential stress is always greater than, or equal to, the radial stress, the maximum values occurring at the inside radius. It follows, therefore, that yielding will first occur at the inside surface when the speed of rotation has increased sufficiently to make the circumferential stress equal to the tensile yield stress. With further increase of speed, plastic penetration will gradually proceed towards the centre of the disc and eventually complete plastic collapse will occur.

t G.J. Franklin and J.L.M. Morrison, Autofrettage of cylinders: reduction of pressure/external expansion curves and calculation of residual stresses. Proc. J. Mech. E. 174 (35) 1960.

i E.J. Hearn, Mechanics of Materials I, Butterworth-Heinemann, 1997.

Now for a solid disc the equilibrium eqn. (4.1) derived on page 120 is, with oH = oy, dor 2 2 oy — or — r—~ = pr co dr dor 2 2

dr r3co2

Now since the stress cannot be infinite at the centre where r = 0, then A must be zero.

3 2 nco

0 0

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