03-03ChapGere.0012

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Problem 3.10-5 A thin-walled circular tube and a solid circular bar of the same material (see figure) are subjected to torsion. The tube and bar have the same cross-sectional area and the same length. What is the ratio of the strain energy U 1 in the tube to the strain energy U 2 in the solid bar if the maximum shear stresses are the same in both cases? (For the tube, use the approximate theory for thin-walled bars.) Solution 3.10-5 T HIN - WALLED TUBE (1) SECTION 3.10 Thin-Walled Tubes 247 Tube (1) Bar (2) A m 5 p r 2 J 5 2 p r 3 tA 5 2 p rt T 5 2 p r 2 t t max [ U 1 5 A t max 2 L 2 G But rt 5 A 2 p 5 p rt t max 2 L G U 1 5 T 2 L 2 GJ 5 (2 p r 2 t t max ) 2 L 2 G p r 3 t ) t max 5 T 2 tA m 5 T 2 p r 2 t S OLID BAR (2) R ATIO U 1 U 2 5 2 But p r 2 2 5 A U 2 5 A t max 2 L 4 G U 2 5 T 2 L 2 GI P 5 ( p r 2 3 t max ) 2 L 8 G ¢ p 2 r 2 4 5 p r 2 2 t max 2 L 4 G t max 5 Tr 2 I P 5 2 T p r 2 3 T 5 p r 2 3 t max 2 A 5 p r 2 2 I P 5 p 2 r 2 4 t r r 2 Problem 3.10-6 Calculate the shear stress t and the angle of twist f (in degrees) for a steel tube ( G 5 76 GPa) having the cross section shown in the figure. The tube has length
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