Energy Methodmi@sereo.cn1
1 Energy Method mi@seu.edu.cn
Contents·Work and Strain Energy（功与应变能）·StrainEnergyDensity（应变能密度）·Strain Energy due to Normal Stresses（正应力所致应变能)·StrainEnergydue to Shearing Stresses（切应力所致应变能)? Strain Energy due to Bending and Transverse Shear（弯矩和横力所致应变能对比）· Strain Energy due to a General State of Stress （一般应力状态所致应变能）·Work and Energy under a Single Load（单载下的功能互等原理）2
• Work and Strain Energy（功与应变能） • Strain Energy Density（应变能密度） • Strain Energy due to Normal Stresses（正应力所致应 变能） • Strain Energy due to Shearing Stresses（切应力所致应 变能） • Strain Energy due to Bending and Transverse Shear （弯矩和横力所致应变能对比） • Strain Energy due to a General State of Stress（一般应 力状态所致应变能） • Work and Energy under a Single Load（单载下的功能 互等原理） 2 Contents
Contents·Strain Energy cannot be Superposed（应变能的不可叠加性）·Work and Energy under Several Loads（多载下的功能原理）·Castigliano's Second Theorem（卡氏第二定理）·StaticallyIndeterminateTruss（超静定桁架）·Statically Indeterminate Shafts（超静定扭转轴）·Statically Indeterminate Beams（超静定梁）·Method of DummyLoad（虚力法）·Method of Unit Dummy Load（单位虚力法）3
• Strain Energy cannot be Superposed（应变能的不可 叠加性） • Work and Energy under Several Loads（多载下的功 能原理） • Castigliano’s Second Theorem（卡氏第二定理） • Statically Indeterminate Truss（超静定桁架） • Statically Indeterminate Shafts（超静定扭转轴） • Statically Indeterminate Beams（超静定梁） • Method of Dummy Load（虚力法） • Method of Unit Dummy Load （单位虚力法） Contents 3
Work done by External Loads and Strain Energy: A uniform rod is subjected to a slowly increasing load. The elementary work done by the load P as the rodelongates by a small dx isdW = Pdx =elementary workwhich is equal to the area of width dx under the load-deformation diagram: The total work done by the load for a deformation xi,=AreaW=Pdx =total work = strain energyxxiwhich results in an increase of strain energy in the rodP=kxDP. In the case of a linear elastic deformation.W=[Pdx=kxdx=k=Px=UCxxi4
• A uniform rod is subjected to a slowly increasing load • The total work done by the load for a deformation x1 , which results in an increase of strain energy in the rod. W P dx total work strain energy x 1 0 Work done by External Loads and Strain Energy • The elementary work done by the load P as the rod elongates by a small dx is which is equal to the area of width dx under the loaddeformation diagram. dW Pdx elementary work 1 1 1 1 2 2 2 1 1 1 0 0 x x W P dx kx dx kx Px U • In the case of a linear elastic deformation, 4
Strain Energy DensityToeliminatetheeffectsofsize.evaluatethestrainenergyperunitvolume,oXiP dxUVOAL81[ox dex = strainenergy densityu=0EEpEI. The total strain energy density resulting from thedeformation is equal to the area under the curve to j. As the material is unloaded, the stress returns to zerobut there is a permanent deformation. Only the strainenergy represented by the triangular area is recovered.: Remainder of the energy spent in deforming the materialis dissipated as heat.5
Strain Energy Density • To eliminate the effects of size, evaluate the strainenergy per unit volume, u d strain energy density L dx A P V U x x x 1 1 0 0 • As the material is unloaded, the stress returns to zero but there is a permanent deformation. Only the strain energy represented by the triangular area is recovered. • Remainder of the energy spent in deforming the material is dissipated as heat. • The total strain energy density resulting from the deformation is equal to the area under the curve to 1 . 5