03-01ChapGere.0015

03-01ChapGere.0015 - For the tube T 5 2 I P t max d 2 f 5...

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Problem 3.3-16 A hollow aluminum tube used in a roof structure has an outside diameter d 2 5 100 mm and an inside diameter d 1 5 80 mm (see figure). The tube is 2.5 m long, and the aluminum has shear modulus G 5 28 GPa. (a) If the tube is twisted in pure torsion by torques acting at the ends, what is the angle of twist f (in degrees) when the maximum shear stress is 50 MPa? (b) What diameter d is required for a solid shaft (see figure) to resist the same torque with the same maximum stress? (c) What is the ratio of the weight of the hollow tube to the weight of the solid shaft? Solution 3.3-16 Hollow aluminum tube SECTION 3.3 Circular Bars and Tubes 195 d 2 d 1 d d 2 5 100 mm d 1 5 80 mm L 5 2.5 m G 5 28 GPa t max 5 50 MPa (a) A NGLE OF TWIST FOR THE TUBE (b) D IAMETER OF A SOLID SHAFT t max is the same as for tube. Torque is the same. T 5 2 t max d 2 ¢ p 32 ( d 2 4 2 d 1 4
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Unformatted text preview: ) For the tube: T 5 2 I P t max d 2 f 5 5.12 8 Ê f 5 2(50 MPa)(2.5 m) (28 GPa)(100 mm) 5 0.08929 rad f 5 2 t max L Gd 2 f 5 TL GI p 5 ¢ 2 I p t max d 2 ≤¢ L GI p ≤ t max 5 Tr I p 5 Td 2 2 I p , Ê T 5 2 I p t max d 2 F OR THE SOLID SHAFT : (c) R ATIO OF WEIGHTS The weight of the tube is 51% of the weight of the solid shaft, but they resist the same torque. W tube W solid 5 (100 mm) 2 2 (80 mm) 2 (83.9 mm) 2 5 0.51 W tube W solid 5 A tube A solid 5 d 2 2 2 d 1 2 d 2 d 5 83.9 mm Ê d 3 5 (100 mm) 4 2 (80 mm) 4 100 mm 5 590,400 mm 3 Solve for d 3 : d 3 5 d 2 4 2 d 1 4 d 2 t max 5 16 T p d 3 5 16 p d 3 ¢ 2 t max d 2 ≤ ¢ p 32 ≤ ( d 2 4 2 d 1 4 ) d 2 d 1 d A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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