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

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Strain Energy in TorsionProblem 3.9-1A solid circular bar of steel (G11.4106psi)with lengthL30 in. and diameterd1.75 in. is subjected topure torsion by torquesTacting at the ends (see figure).(a)Calculate the amount of strain energyUstored in the barwhen the maximum shear stress is 4500 psi.(b)From the strain energy, calculate the angle of twist(in degrees).Solution 3.9-1Steel bar236CHAPTER 3TorsionLdTTLdTTG11.4106psiL30 in.d1.75 in.max4500 psi(Eq. 1)IPd432tmax16Td3Td3tmax16(a) STRAIN ENERGY(Eq. 2)Substitute numerical values:(b) ANGLE OF TWISTSubstitute forTandUfrom Eqs. (1) and (2):(Eq. 3)Substitute numerical values:f0.013534rad0.775f2LtmaxGdUTf2f2UTU32.0in.-lbd2Ltmax216GUT2L2GIP¢d3tmax162¢L2G¢32d4Problem 3.9-2A solid circular bar of copper (G45 GPa) with lengthL0.75 m and diameterd40 mm is subjected to pure torsion bytorquesTacting at the ends (see figure).(a)Calculate the amount of strain energyUstored in the bar when themaximum shear stress is 32 MPa.(b)From the strain energy, calculate the angle of twist(in degrees)
Solution 3.9-2Copper barSECTION 3.9Strain Energy in Torsion237LdTTG45 GPaL0.75 md40 mmmax32 MPa(Eq. 1)IPd432tmax16Td3Td3tmax16(A) STRAIN ENERGY(Eq. 2)Substitute numerical values:(b) ANGLE OF TWISTSubstitute forTandUfrom Eqs. (1) and (2):(Eq. 3)Substitute numerical values:f0.026667 rad1.53f2LtmaxGdUTf2f2UTU5.36Jd2Ltmax216GUT2L2GIP¢d3tmax162¢L2G¢32d4Problem 3.9-3A stepped shaft of solid circular cross sections(see figure) has lengthL45 in., diameterd21.2 in., anddiameterd11.0 in. The material is brass withG5.6106psi.Determine the strain energyUof the shaft if the angle of twistis 3.0°.Solution 3.9-3Stepped shaftd2d1TTL2L2d2d1TTL2L2d11.0 in.d21.2 in.L45 in.G5.6106psi (brass)3.00.0523599 radSTRAIN ENERGY(Eq. 1)Also,(Eq. 2)EquateUfrom Eqs. (1) and (2) and solve forT:SUBSTITUTE NUMERICAL VALUES:U22.6in.-lbUTf2Gf232L¢d14d24d14d24fradiansTGd14d24f16L(d14d24)UTf28T2LG¢1d241d14UaT2L2GIP16T2(L2)Gd2416T2(L2)Gd14
Problem 3.9-4A stepped shaft of solid circular cross sections (see figure)has lengthL0.80 m, diameterd240 mm, and diameterd130 mm.The material is steel withG80 GPa.Determine the strain energyUof the shaft if the angle of twist is 1.0°.d2dSoluton 3.9-4Stepped shaft238CHAPTER 3Torsion1TTL2L2d130 mmd240 mmL0.80 mG80 GPa (steel)1.00.0174533 radSTRAIN ENERGY(Eq. 1)Also,(Eq. 2)UTf28T2LG¢1d241d14UaT2L2GIP16T2(L2)Gd2416T2(L2)Gd14EquateUfrom Eqs. (1) and (2) and solve forT:SUBSTITUTE NUMERICAL VALUES:U1.84JUTf2Gf232L¢d14d24d14d24fradiansTGd14d24f16L(d14d24)Problem 3.9-5A cantilever bar of circular cross section and lengthLisfixed at one end and free at the other (see figure). The bar is loaded by atorqueTat the free end and by a distributed torque of constant intensitytper unit distance along the length of the bar.(a)What is the strain energyU1

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