15.3 (B). Show that the strain energy per unit volume of a material under a single direct stress is given by (stress x strain). Hence show that for a material under the action of the principal stresses o1,02 and o3 the strain energy per unit volume becomes 1 2E A thin cylinder 1m diameter and 3 m long is filled with a liquid to a pressure of 2 MN/m². Assuming a yield stress for the material of 240 MN/m2 in simple tension and a safety factor of 4, determine the necessary wall thickness of the cylinder, taking the maximum shear strain energy as the criterion of failure. For the cylinder material, E = 207 GN/m² and v 0.286. [14.4 mm.]

[Ans. 1.2 MPa] PROBLEM 7.99 16 A spherical gas container made of steel has a diameter of 5.4 m and a wall thickness of 10 mm. Knowing that the internal pressure is 400 kPa, determine the membrane stress in the container. [Ans. 53.8 MPa] 17

4.3. Assessing Risk of Aneurysm Rupture: Studies suggest that the maximum allowable stress (failure strength) for the wall tissue in an aortic aneurysm is o = 220 kPa, after accounting for factor of safety. A physician wants your expert opinion on the maximum diameter that an aortic aneurysm in a hypertensive patient (BP 200/90 mmHg) could be allowed to grow to before rupture becomes imminent. The wall thickness may be assumed to remain unchanged at t = 1.2 mm. State your answer in millimeters of diameter. INDBER

A sheet metal cutting machine cuts the sheet metal via shear force, V as shown in Fig. 1. The crosssection of sheet metal is rectangular with width, b = 3m and thickness, d = 5mm. The sheet metal ishaving yield strength σy = 220 MPa; Young’s modules, E = 200 GPa and Poisson’s ratio, ν = 0.3.Conduct the following analyses at the state of stress when maximum stress reaches σy and beforeshearing commences.  (Need answer Question c) & d) c. Sketch the YIELD LOCI for both cases above. d. As a sheet-metal engineer, which failure criterion in (b) will you use to determine the cuttingshear force? Explain the reason for your answer.

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