东南大学：《弹性力学》课程教学课件（英文讲稿）14 Energy Method and Variational Principle

Energy Method and Variational Principle

Energy Method and Variational Principle

OutlineWork Done by External LoadStrain EnergyThe Delta OperatorPrinciple of Virtual WorkPrinciple of Minimum Potential EnergyCastigliano's First TheoremDisplacement Variation: Ritz MethodDisplacement Variation: Galerkin MethodComplimentary Strain EnergyPrinciple of Complimentary Virtual WorkPrinciple of Minimum Complimentary Potential EnergyCastigliano's Second TheoremStress VariationStress Variation: Application to Plane ElasticityStress Variation: Application to Torsion of Cylinders2

Outline • Work Done by External Load • Strain Energy • The Delta Operator • Principle of Virtual Work • Principle of Minimum Potential Energy • Castigliano’s First Theorem • Displacement Variation: Ritz Method • Displacement Variation: Galerkin Method • Complimentary Strain Energy • Principle of Complimentary Virtual Work • Principle of Minimum Complimentary Potential Energy • Castigliano’s Second Theorem • Stress Variation • Stress Variation: Application to Plane Elasticity • Stress Variation: Application to Torsion of Cylinders 2

Work Done by External LoadA uniform rod is subjected to a slowly increasing loadThe elementary work done by the load P as the rodelongatesbyasmalldxisdU=Pdx= elementaryworkwhich is egual to the area of width dx under the load-deformation diagram.U=AreaThe total work done by the load for a deformation Xi,U =[Pdx = total work = strain energyxx1fPwhich results in an increase of strain energy in the rodP=kxPIn the case of a linear elastic deformation,PixU =[kxdx=±kx, =PxSOxX3

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. 1 0 d x U P x total work 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 load￾deformation diagram. dU P x elementary work   d 1 1 1 2 2 2 1 1 1 0 d x U kx x kx P x     In the case of a linear elastic deformation, Work Done by External Load 3

Energy ConversionWork done by surface and body forces on elastic solidsis stored inside the body in the form of strain energy.UA

• Work done by surface and body forces on elastic solids is stored inside the body in the form of strain energy. F T n U Energy Conversion 4

Strain Energy DensityToeliminatetheeffectsof size,evaluatethestrain-energy per unit volume,o"PdxUV。ALU。=,d,=strainenergydensity0OWEpE1The total strain energy density resulting from thedeformation is equal to the area under the curve to gj.As the material is unloaded, the stress returns to zero butthere is a permanent deformation. Only the strainenergy represented by the triangular area is recovered.Remainder of the energy spent in deforming the material isdissipated as heat.5

To eliminate the effects of size, evaluate the strain￾energy per unit volume, 1 1 0 0 0 d d x x x U P x V A L U strain energy density         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 . Strain Energy Density 5

Strain Energy for Normal StressIn an element with a nonuniform stress distribution,AUdUU = [U,dV = total strain energyU。= limdvAV→0AVFor values of U.02EUnder axial loading, x = P/ AdV=AdxI.2dudxEAdx :2AEdx2For a rod of uniform cross-section,2AE6

In an element with a nonuniform stress distribution, 0 0 0 d lim d total strain energy V d U U U U U V   V V        For values of U0 < UY , i.e., below the proportional limit, 2 d elastic strain energy 2 x U V E     Under axial loading, d d  x   P A V A x 2 2 0 0 1 d d d 2 2 d L L P u U x EA x AE x           AE P L U 2 2  For a rod of uniform cross-section, Strain Energy for Normal Stress   E 0 6

Strain Energy for Normal StressFor a beam subjected to a bending load.M:xdVdv2E1?2EKSetting dV = dA dx,I1M1lAdx1My2E1?2E1201M2d"wEIdxdxdx?2EI2=E>0PFor an end-loaded cantilever beam.BM=-PxAp2x?p?L3dx2EI6EI67

I M y x   For a beam subjected to a bending load, 2 2 2 2 d d 2 2 x M y U V V E EI      Setting dV = dA dx, 2 2 2 2 2 2 0 0 2 2 2 2 0 0 d d d d 2 2 1 d d d 2 2 d L L A A L L M y M U A x y A x EI EI M w x EI x EI x                       For an end-loaded cantilever beam, 2 2 2 3 0 d 2 6 L M Px P x P L U x EI EI      Strain Energy for Normal Stress   E 0 7

Strain Energy for Shear StressFor a material subjected to plane shearingstresses,YsIt.dyx10号一xyFor values of trwithin the proportional limit.TMU=IGr,=yYXT2GThe total strain energy is found fromOxj(1+vU=dvU.dyxJE2C→G>0;V>-18

For a material subjected to plane shearing stresses, 0 0 d xy U xy xy      For values of  xy within the proportional limit, 2 1 1 2 0 2 2 2 x y U G xy xy xy G        The total strain energy is found from 2 2 0 (1 ) d d d 2 xy U U V V V xy G E           Strain Energy for Shear Stress     G 0; 1  8

Strain Energy for Shear StressFor a shaft subjected to a torsional load.d2GJ7Setting dV = dA dx,2OdAdx:dx2GJ22G.10TpTxyTdpG.Jdxdx=dx2GJ2In the case of a uniform shaft.T?LU:2GJ9

J T xy    2 2 2 2 d d 2 2 xy T U V V G GJ       For a shaft subjected to a torsional load, Setting dV = dA dx, 2 2 2 2 2 2 0 0 2 2 0 0 d d d d 2 2 1 d d d 2 2 d L L A A L L T T U A x A x GJ GJ T x GJ x GJ x                          In the case of a uniform shaft, GJ T L U 2 2  Strain Energy for Shear Stress 9

Strain Energy for Hydrostatic Stressp3(1- 2v)-3pkk=6 +62 +63=-pE31+2GE31+2G-pK=3△V3(1-2v)p13(1-12v0kkFnm722 K2E→K>0;v<0.510

Strain Energy for Hydrostatic Stress         1 2 3 2 2 0 3 1 2 3 3 2 3 2 3 1 2 3 1 1 1 3 1 2 2 2 2 2 k k kk m kk m m p p E G p E G K V U p K E                                     10    K 0; 0.5 