03-03ChapGere.0007 - 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

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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 f of the tube. Note: Use the approximate formula I P < 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 5 thickness d A 5 average diameter at end A d B 5 average diameter at end B d ( x ) 5 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 1 bx ) 3 52 1 2 b ( a 1 bx ) 2 5 2 T 2 p Gt # L 0 dx B d A 1 ¢ d B 2 d A L x R 3 U 5 # L 0 T 2 dx 2 GI P ( x ) I P ( x ) 5 p [ d ( x )] 3 t 4 5 p t 4 B d A 1 ¢ d B 2 d A L x R 3 I P 5 p d 3 t 4 d ( x ) 5 d A 1 ¢ d B 2 d A L x Therefore,
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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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This note was uploaded on 09/20/2009 for the course COE 3001 taught by Professor Armanios during the Spring '08 term at Georgia Institute of Technology.

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