The thin-walled cylindrical pressure vessel shown in Figure 6 has an internal diameter (D) of 2.5 m and a wall thickness (t) of 10 mm. In service it is to be subjected to an internal operating pressure (pi) of 4.8 MNm². During a cycle the pressure varies between zero and the operating pressure. The required fatigue life of the vessel is 3000 cycles. The design recognises the possible presence of manufacturing defects and that failure could occur by fast fracture. -TANK Figure 6 The vessel material has a fracture toughness of 100 MNm-3/2 and the fatigue crack growth is represented by the Paris-Erdogan equation, da =2.5×10-14 (AK) m cycle where AK is in MNm-3/2 dN You may assume that K₁ = 1.20√πa for the vessel defect. For a thin-walled pressure vessel, the stresses are given by, συ pD 2t pD 4t (a) Determine the critical crack length at which failure will occur. (b) Estimate the initial crack length which will just give the required life. (c) What hydraulic test pressure must the vessel be subjected to before going into service in order to ensure that the required life can be achieved?