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)