FORMULAS Emax = VƒEƒ + (1 - Vj) Em 1 = gel [r+rs(f-2)]¹/2/nQ6 A cylindrical pressure vessel with closed ends has a radius R = 1 m and thickness t =

40 mm and is subjected to internal pressure p. The vessel must be designed safely against failure by yielding (according to the von Mises yield criterion) and fracture. Three steels with the following values of yield stress oy and fracture toughness Kic are available for constructing the vessel. Steel Kic(MPa √/m A: 4340 100 B: 4335 70 C: 350 Maraging 55 Fracture of the vessel is caused by a long axial surface crack of depth a. The vessel should be designed with a factor of safety S = 2 against yielding and fracture. (a) By considering equilibrium along the longitudinal (axial) and circumferential (hoop) di- rections determine expressions for the hoop stress and axial stress in terms of the internal pressure, p, the radius, R and the thickness, t. dy (MPa) 860 1300 1550 (4 marks) (b) For the three steels, find the maximum pressure the vessel can withstand without failure by yielding. Note, your calculation should include the factor of safety, S. (4 marks) (c) The fracture toughness for a long axial surface crack of depth a is given by Kic 1.12000 √na. Hence determine an expression for the maximum pressure as a function of crack length a and fracture toughness. Note, your calculation should again include the factor of safety, S. (3 marks) (d) Plot the maximum permissable pressure pe versus crack depth a, for the three steels. (3 marks) (e) Calculate the maximum permissable crack depth a for an operating pressure p = 12 MPa. (3 marks) (f) Calculate the failure pressure p, for a maximum detectable crack depth a = 1 mm. (3 marks)