ch3-3 - 236 CHAPTER 3 Torsion Strain Energy in Torsion...

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Strain Energy in Torsion Problem 3.9-1 A solid circular bar of steel ( G 5 11.4 3 10 6 psi) with length L 5 30 in. and diameter d 5 1.75 in. 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 4500 psi. (b) From the strain energy, calculate the angle of twist f (in degrees). Solution 3.9-1 Steel bar 236 CHAPTER 3 Torsion L d TT L d G 5 11.4 3 10 6 psi L 5 30 in. d 5 1.75 in. t max 5 4500 psi (Eq. 1) I P 5 p d 4 32 t max 5 16 T p d 3 Ê T 5 p 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 5 0.013534 Ê rad 5 0.775 8 f 5 2 L t max Gd U 5 T f 2 Ê f 5 2 U T U 5 32.0 in.-lb 5 p d 2 L t max 2 16 G U 5 T 2 L 2 GI P 5 ¢ p d 3 t max 16 2 ¢ L 2 G ¢ 32 p d 4 Problem 3.9-2 A solid circular bar of copper ( G 5 45 GPa) with length L 5 0.75 m and diameter d 5 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 f (in degrees)
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Solution 3.9-2 Copper bar SECTION 3.9 Strain Energy in Torsion 237 L d TT G 5 45 GPa L 5 0.75 m d 5 40 mm t max 5 32 MPa (Eq. 1) I P 5 p d 4 32 t max 5 16 T p d 3 Ê T 5 p 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 5 0.026667 rad 5 1.53 8 f 5 2 L t max Gd U 5 T f 2 Ê f 5 2 U T U 5 5.36 J 5 p d 2 L t max 2 16 G U 5 T 2 L 2 GI P 5 ¢ p d 3 t max 16 2 ¢ L 2 G ¢ 32 p d 4 Problem 3.9-3 A stepped shaft of solid circular cross sections (see figure) has length L 5 45 in., diameter d 2 5 1.2 in., and diameter d 1 5 1.0 in. The material is brass with G 5 5.6 3 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 5 1.0 in. d 2 5 1.2 in. L 5 45 in. G 5 5.6 3 10 6 psi (brass) f 5 3.0 85 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 5 22.6 in.-lb U 5 T f 2 5 p G f 2 32 L ¢ d 1 4 d 2 4 d 1 4 1 d 2 4 Ê f 5 radians T 5 p Gd 1 4 d 2 4 f 16 L ( d 1 4 1 d 2 4 ) U 5 T f 2 5 8 T 2 L p G ¢ 1 d 2 4 1 1 d 1 4 U 5 a T 2 L 2 GI P 5 16 T 2 ( L / 2) p Gd 2 4 1 16 T 2 ( L / p Gd 1 4
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Problem 3.9-4 A stepped shaft of solid circular cross sections (see figure) has length L 5 0.80 m, diameter d 2 5 40 mm, and diameter d 1 5 30 mm. The material is steel with G 5 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 5 30 mm d 2 5 40 mm L 5 0.80 m G 5 80 GPa (steel) f 5 1.0 85 0.0174533 rad S TRAIN ENERGY (Eq. 1) Also, (Eq. 2) U 5 T f 2 5 8 T 2 L p G ¢ 1 d 2 4 1 1 d 1 4 U 5 a T 2 L 2 GI P 5 16 T 2 ( L / 2) p Gd 2 4 1 16 T 2 ( L / p Gd 1 4 Equate U from Eqs. (1) and (2) and solve for T : S UBSTITUTE NUMERICAL VALUES : U 5 1.84 J U 5 T f 2 5 p G f 2 32 L ¢ d 1 4 d 2 4 d 1 4 1
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This note was uploaded on 03/30/2008 for the course CVEN 205 taught by Professor Muliana during the Spring '08 term at Texas A&M.

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ch3-3 - 236 CHAPTER 3 Torsion Strain Energy in Torsion...

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