MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II EXAMPLE PROBLEM IN FRACTURE Distributed: Wednesday, April 7, 2004 A spherical thin-walled pressure vessel is to be fabricated from formed plates. The capacity of the vessel is to be 14 m3, and the fluid it contains will be under a peak internal pressure of 50 MP a. You may use the standard stress analysis of thin-walled spherical pressure vessels to relate stress to pressure and vessel geometry. Three grades of mar-ageing steel are under consideration for constructing the vessel. The table gives the tensile yield strength (σy) and fracture toughness (KIc) of each grade. Each of the steels has mass density ρ = 7.8Mg/m3 . Grade σy (MPa) KIc (MPa √m) 200 1482 111 250 1690 94 300 2000 66 You are to specify the material and corresponding wall thickness so that the structure will 1. be of least weight, while 2. retaining a factor of safety of at least 2 against both (a) yielding and (b) brittle fracture. 1AA'RtIn carrying out your calculations, you may assume that the critical defect is a small-scale circular surface crack of radius “a”. The calibration for maximum stress intensity factor along a shallow semi-circular surface crack front in a large plate of thickness t � a, subject to tensile stress “σ” perpendicular to the crack plane is: KI(max) = 0.617σ√πa Figure 1: Schematic of spherical pressure vessel with small, semi-circular surface crack on inner surface. Furthermore, assume that the non-destructive examination (NDE) technology available to inspect the fabricated vessel has a “reliable” resolution limit of 10 mm. (In this context, the resolution limit can be taken as the surface length of the assumed semi-circular surface crack.) Suppose instead that enhanced NDE methods are available, with a reliable detection resolu-tion of 5 mm. Now which material would you choose to optimize the vessel, and what wall thickness would you specify? Discuss.
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