(1.6.9)--3.8材料力学材料力学.ppt

Chapter 3 Torsion Mechanics of Materials3.8 Strain Energy in Torsion and Pure ShearStrain Energy in Torsion and Pure ShearPure TorsionFor linearly elastic material with small angles of twistFrom the principle of conservation of energyStrain Energy Work done by the loadStrain Energy in Torsion and Pure ShearNonuniform TorsionFor linearly elastic material with small angles of twist Bar consisting of prismatic segments with constant torque throughout each segmentStrain Energy in Torsion and Pure ShearNonuniform TorsionFor linearly elastic material with small angles of twist Bar with continuously varying cross sections or continuously varying torqueStrain Energy in Torsion and Pure ShearStrain-Energy Density in Pure ShearConsider an element of height h and thickness t.The shear forces V acting on the side faces:Strain Energy in Torsion and Pure ShearStrain-Energy Density in Pure ShearThe work done by the forces V as the element deforms The relative distancesFor linearly elastic material with small angles of twist From the principle of conservation of energy,the strain energy stored in the elementStrain Energy in Torsion and Pure ShearStrain-Energy Density in Pure Shearstrain-energy densityFor linearly elastic material with small angles of twist The strain energy per unit volumeStrain Energy in Torsion and Pure ShearExample 1Prismatic bar loaded with three different conditions.For each case of loading,obtain a formula for the strain energy stored in the bar.Case aCase bCase cStrain Energy in Torsion and Pure ShearExample 2The diameter of the bar varies linearly from dA at the left-hand end to dB at the right-hand end.Determine the angle of rotation at end A of the bar by equating the strain energy to the work done by the load.Strain Energy in Torsion and Pure ShearExample 2SolutionThe work done by the applied torqueThe strain energy of the tapered barStrain Energy in Torsion and Pure ShearExample 2SolutionFrom the principle of conservation of energyOnly when the bar is subjected to a single load,and then only when the desired angle corresponds to that load.End