03-03 Chap Gere - 236 CHAPTER 3 Torsion Strain Energy in Torsion Problem 3.9-1 A solid circular bar of steel(G 11.4 106 psi with length L 30 in and

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

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Strain Energy in Torsion Problem 3.9-1A solid circular bar of steel (G11.4 106psi)with length L30 in. and diameter d1.75 in. is subjected topure torsion by torques Tacting at the ends (see figure).(a)Calculate the amount of strain energy Ustored in the barwhen 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 3.9-2A solid circular bar of copper (G45 GPa) with lengthL0.75 m and diameter d40 mm is subjected to pure torsion bytorques Tacting at the ends (see figure). (a)Calculate the amount of strain energy Ustored in the bar when themaximum shear stress is 32 MPa. (b)From the strain energy, calculate the angle of twist (in degrees)
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
Problem 3.9-4A stepped shaft of solid circular cross sections (see figure)has length L0.80 m, diameter d240 mm, and diameter d130 mm.The material is steel with G80 GPa.Determine the strain energy Uof the shaft if the angle of twist is 1.0°.Soluton 3.9-4Stepped shaft1TTL2L2d130 mmd240 mmL0.80 mG80 GPa (steel)1.00.0174533 radSTRAIN ENERGY(Eq. 1)Also, (Eq. 2)UTf8T2LG¢1d241d14UaT2L2GIP16T2(L2)Gd2416T2(L2)Gd14Equate Ufrom Eqs. (1) and (2) and solve for TSUBSTITUTE NUMERICAL VALUES:U1.84JUTf2Gf232L¢d14d24d14d24fradiansTGd14d24f16L(d14d24)Problem 3.9-5A cantilever bar of circular cross section and length Lisfixed at one end and free at the other (see figure). The bar is loaded by atorque Tat the free end and by a distributed torque of constant intensity tper unit distance along the length of the bar. (a)What is the strain energy U1 238 CHAPTER 3 Torsion d 2 d 2 :

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