A large hydraulic cylinder is to be treated using autofrettage, taking 80% of the wall thickness into the plastic range. The external diameter of the vessel is 200mm and the internal diameter is 100mm, so that it must be regarded as a thick cylinder. The yield stress of the material is approximately 450MPa.

(a) Calculate the internal pressure required

(i) to cause first yield

(ii) to cause yield half-way through the wall thickness

(iii) to cause yield all the way across the wall

(b) Using your answers from part (a), sketch a curve of internal pressure versus radius of yielded zone boundary for the cylinder. Comment on the factor of safety against plastic collapse associated with the target amount of yield, taking into account the assumptions inherent in the calculations.

(c) Write down the four equations which describe the hoop and radial stresses in the elastic and plastic zones of a pressurised thick cylinder. Hence, sketch the stress distribution across the wall of the cylinder for case (ii) in part (a), indicating values of stress in MPa. Using this sketch, briefly explain the process of yielding and plastic failure of a thick-walled cylinder as the pressure increases.