3. Fatigue of a pressure vessel (10 points)

A cylindrical steel pressure vessel of 7.5 m diameter and 40 mm wall thickness is to operate

at working pressure of 5.1 MPa. The fracture toughness for the steel is 200 MPa√m. The

growth of the crack by fatigue may be represented approximately by the equation

where A =2.44×10¹ (MPa) m¹. The design assumes that failure will take place by fast

fracture from a crack which has extended gradually along the length of the vessel by fatigue.

To prevent fast fracture, the total number of loading cycles from zero to full load and back

to zero again must not exceed 3000. Find the minimum pressure to which the vessel must be

tested before use to guarantee against failure in under the 3000 load cycles.

Hint: The stress intensity factor for a crack along the length of a thin-walled pressure vessel

can be estimated by the following expression (Tada, H., Paris, P. C., & Irwin, G. R. (1973). The

stress analysis of cracks. Handbook, Del Research Corporation, 34, 635):

K₁ = √na-F(2),