336_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

336_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Unformatted text preview: Sec_3.9-3.11.qxd 330 9/27/08 1:14 PM Page 330 CHAPTER 3 Torsion Problem 3.10-4 A thin-walled steel tube of rectangular cross section (see figure) has centerline dimensions b and h 150 mm 100 mm. The wall thickness t is constant and equal to 6.0 mm. (a) Determine the shear stress in the tube due to a torque T 1650 N m. (b) Determine the angle of twist (in degrees) if the length L of the tube is 1.2 m and the shear modulus G is 75 GPa. Solution 3.10-4 Thin-walled tube b h 10.8 10 75 GPa T 2tAm t t2 t: ; 9.17 MPa (b) ANGLE OF TWIST (Eq. 3-72) TL GJ f 0.002444 rad 0.140° 0.015 m2 bh 6 1.2 m G J 1650 N m L Eq. (3-71) with t1 6.0 mm T Eq. (3-64): Am 100 mm t (a) SHEAR STRESS (Eq. 3-61) 150 mm ; 2b2h2t b+h J m4 Tube (1) Problem 3.10-5 A thin-walled circular tube and a solid circular bar of Bar (2) 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 U1 in the tube to the strain energy U2 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 THIN-WALLED TUBE (1) r2 Am tmax T U1 T 2L 2GJ 2G(2pr3t) prtt2 L max G A But rt 2p ‹ U1 T 2tAm At2 L max 2G 2 rt SOLID BAR (2) T A 2 pr2 IP tmax Tr2 IP 2T 2 2pr t 2 2 rt 12pr2ttmax22L 2 r3t A J max U2 RATIO U1 2 U2 T 3 pr2 tmax 2 3 pr2 3 22 (pr2 tmax)2L pr2 tmaxL 4G p4 8G a r2 b 2 T2L 2GIP 2 But pr2 p4 r 22 A ; ‹ U2 2 Atmax L 4G ...
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