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03-03ChapGere.0007 - Substitute this expression for the...

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Problem 3.9-9 A thin-walled hollow tube AB of conical shape has constant thickness t and average diameters d A and d B at the ends (see figure). (a) Determine the strain energy U of the tube when it is subjected to pure torsion by torques T . (b) Determine the angle of twist of the tube. Note: Use the approximate formula I P d 3 t /4 for a thin circular ring; see Case 22 of Appendix D. Solution 3.9-9 Thin-walled, hollow tube 242 CHAPTER 3 Torsion L t t T T A d A d B B L T T A B d(x) x dx t thickness d A average diameter at end A d B average diameter at end B d ( x ) average diameter at distance x from end A P OLAR MOMENT OF INERTIA (a) S TRAIN ENERGY ( FROM E Q . 3-54) (Eq. 1) From Appendix C: dx ( a bx ) 3 1 2 b ( a bx ) 2 2 T 2 Gt L 0 dx B d A ¢ d B d A L x R 3 U L 0 T 2 dx 2 GI P ( x ) I P ( x ) [ d ( x ) ] 3 t 4 t 4 B d A ¢ d B d A L x R 3 I
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Unformatted text preview: Substitute this expression for the integral into the equation for U (Eq. 1): (b) A NGLE OF TWIST Solve for f : f 5 2 TL ( d A 1 d B ) p Gt d A 2 d B 2 W 5 U Ê T f 2 5 T 2 L ( d A 1 d B ) p Gt d A 2 d B 2 Work of the torque T : W 5 T f 2 U 5 2 T 2 p Gt ? L ( d A 1 d B ) 2 d A 2 d B 2 5 T 2 L p Gt ¢ d A 1 d B d A 2 d B 2 ≤ 5 L ( d A 1 d B ) 2 d A 2 d B 2 5 2 L 2( d B 2 d A )( d B ) 2 1 L 2( d B 2 d A )( d A ) 2 5 2 1 2( d B 2 d A ) L B d A 1 ¢ d B 2 d A L ≤ x R 2 4 L # L dx B d A 1 ¢ d B 2 d A L ≤ x R 3 A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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