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03-03 Chap Gere - 236 CHAPTER 3 Torsion Strain Energy in...

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Strain Energy in Torsion Problem 3.9-1 A solid circular bar of steel ( G 11.4 10 6 psi) with length L 30 in. and diameter d 1.75 in. is subjected to pure torsion by torques T acting at the ends (se e figure). (a) Calculate the amount of strain energy U stored in the bar when the maximum shear stress is 4500 psi. (b) From the strain energy, calculate the angle of twist (in degrees). Solution 3.9-1 Steel bar 236 CHAPTER 3 Torsion L d T T L d T T G 11.4 10 6 psi L 30 in. d 1.75 in. max 4500 psi (Eq. 1) I P d 4 32 t max 16 T d 3 T d 3 t max 16 (a) S TRAIN ENERGY (Eq. 2) Substitute numerical values: (b) A NGLE OF TWIST Substitute for T and U from Eqs. (1) and (2): (Eq. 3) Substitute numerical values: f 0.013534 rad 0.775 f 2 L t max Gd U T f 2 f 2 U T U 32.0 in.-lb d 2 L t max 2 16 G U T 2 L 2 GI P ¢ d 3 t max 16 2 ¢ L 2 G ¢ 32 d 4 Problem 3.9-2 A solid circular bar of copper ( G 45 GPa) with length L 0.75 m and diameter d 40 mm is subjected to pure torsion by torques T acting at the ends (see figure). (a) Calculate the amount of strain energy U stored in the bar when the maximum shear stress is 32 MPa. (b) From the strain energy, calculate the angle of twist (in degrees)
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Solution 3.9-2 Copper bar SECTION 3.9 Strain Energy in Torsion 237 L d T T G 45 GPa L 0.75 m d 40 mm max 32 MPa (Eq. 1) I P d 4 32 t max 16 T d 3 T d 3 t max 16 ( A ) S TRAIN ENERGY (Eq. 2) Substitute numerical values: (b) A NGLE OF TWIST Substitute for T and U from Eqs. (1) and (2): (Eq. 3) Substitute numerical values: f 0.026667 rad 1.53 f 2 L t max Gd U T f 2 f 2 U T U 5.36 J d 2 L t max 2 16 G U T 2 L 2 GI P ¢ d 3 t max 16 2 ¢ L 2 G ¢ 32 d 4 Problem 3.9-3 A stepped shaft of solid circular cross sections (see figure) has length L 45 in., diameter d 2 1.2 in., and diameter d 1 1.0 in. The material is brass with G 5.6 10 6 psi. Determine the strain energy U of the shaft if the angle of twist is 3.0°. Solution 3.9-3 Stepped shaft d 2 d 1 T T L 2 L 2 d 2 d 1 T T L 2 L 2 d 1 1.0 in. d 2 1.2 in. L 45 in. G 5.6 10 6 psi (brass) 3.0 0.0523599 rad S TRAIN ENERGY (Eq. 1) Also, (Eq. 2) Equate U from Eqs. (1) and (2) and solve for T : S UBSTITUTE NUMERICAL VALUES : U 22.6 in.-lb U T f 2 G f 2 32 L ¢ d 1 4 d 2 4 d 1 4 d 2 4 f radians T Gd 1 4 d 2 4 f 16 L ( d 1 4 d 2 4 ) U T f 2 8 T 2 L G ¢ 1 d 2 4 1 d 1 4 U a T 2 L 2 GI P 16 T 2 ( L 2) Gd 2 4 16 T 2 ( L 2) Gd 1 4
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Problem 3.9-4 A stepped shaft of solid circular cross sections (see figure) has length L 0.80 m, diameter d 2 40 mm, and diameter d 1 30 mm. The material is steel with G 80 GPa. Determine the strain energy U of the shaft if the angle of twist is 1.0°. Soluton 3.9-4 Stepped shaft 238 CHAPTER 3 Torsion d 2 d 1 T T L 2 L 2 d 1 30 mm d 2 40 mm L 0.80 m G 80 GPa (steel) 1.0 0.0174533 rad S TRAIN ENERGY (Eq. 1) Also, (Eq. 2) U T f 2 8 T 2 L G ¢ 1 d 2 4 1 d 1 4 U a T 2 L 2 GI P 16 T 2 ( L 2) Gd 2 4 16 T 2 ( L 2) Gd 1 4 Equate U from Eqs. (1) and (2) and solve for T : S UBSTITUTE NUMERICAL VALUES : U 1.84 J U T f 2 G f 2 32 L ¢ d 1 4 d 2 4 d 1 4 d 2 4 f radians T G d 1 4 d 2 4 f 16 L ( d 1 4 d 2 4 ) Problem 3.9-5 A cantilever bar of circular cross section and length L is fixed at one end and free at the other (see figure). The bar is loaded by a torque T
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