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1、Chapter 3 Torsion Mechanics of Materials3.4 Circular Bars of Linearly Elastic MaterialsCircular Bars of Linearly Elastic MaterialsShear stresses in a circular bar in torsionHookes law in shearG is the shear modulus of elasticity is the shear strain in radiansPure Shearthe element is subjected to she
2、ar strains but no normal strainsCircular Bars of Linearly Elastic MaterialsShear stresses in a circular bar in torsionThe shear stresses vary linearly with the distance from the center of the bar.-the rate of twist,has units of radians per unit of length.The max shear stress maxis at the outer surfa
3、ce of the bar.Circular Bars of Linearly Elastic MaterialsShear stresses in a circular bar in torsionThe shear stresses acting on a cross-sectional plane are accompanied by shear stresses of the same magnitude acting on longitudinal planes.Equal shear stresses always exist on mutually perpendicular p
4、lanes.Longitudinal shear stresses in a circular barTheorem of conjugate shearing stressesCircular Bars of Linearly Elastic MaterialsThe Torsion FormulaThe relationship between the shear stresses and the torque TPolar moment of inertiaRate of twist Circular Bars of Linearly Elastic MaterialsThe Torsi
5、on FormulaThe maximum shear stressSolid circular cross sectionHollow circular cross sectionsCircular Bars of Linearly Elastic MaterialsThe Torsion FormulaPolar moment of inertiaSolid circular cross sectionHollow circular cross sectionsCircular Bars of Linearly Elastic MaterialsAngle of TwistFor pure
6、 torsion,the total angle of twist Rate of twist rad/mGIPTorsional RigidityTorsional stiffnessTorsional flexibilityEquations derived above are only for bars of circular cross sections that behave in a linearly elastic manner.Circular Bars of Linearly Elastic MaterialsStrength conditionStiffness condi
7、tionCircular Bars of Linearly Elastic MaterialsExampleDesign a steel(G=78 GPa)shaft,either as a solid circular bar or as a circular tube,to transmit a torque of 1200 Nm.=40MPa,=0.75/m.In case of tube construction,(a)Determine the required diameter d0 of the solid shaft.(b)Determine the required oute
8、r diameter d2 of the hollow shaft.(c)Determine the ratio of diameters and the ratio of weights of the hollow and solid shafts.Circular Bars of Linearly Elastic MaterialsSolution:Part a:Solid shaft.Strength conditionStiffness conditionStiffness governs the designCircular Bars of Linearly Elastic Mate
9、rialsSolution:Part b:Hollow shaft.Strength conditionStiffness conditionStiffness governs the designCircular Bars of Linearly Elastic MaterialsSolution:Part c:Ratios of diameters and weights.Ratio of diametersRatio of weightsThe hollow shaft uses only 47%as much material as does the solid shaft,while its outer diameter is only 14%larger.Circular Bars of Linearly Elastic MaterialsQuestionWhy circular tubes are more efficient in the use of materials than are solid circular bars?TTEnd