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## Lecture 1 : Work and Energy methods

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**Lecture 1 : Work and Energymethods**Hans Welleman**Content**• Meeting 1 Work and Energy • Meeting 2 Castigliano • Meeting 3 Potential Energy Work and Energy methods**Lecture 1**• Essentials • Work, virtual work, theorem of Betti and Maxwell • Deformation or Strain Energy • Work methods and solving techniques • Virtual work • Strain Energy versus Work • Work method with unity load • Rayleigh Work and Energy methods**Work**uF F u Work and Energy methods**F=0**F u unloaded situation loaded situation Deformation or Strain Energy force u spring characteristics Work and Energy methods**For a kinematical admissible displacement Virtual Work is**generated by the forces y x z Virtual Work : Particle Particle Equilibrium conditions of a particle in 3D Equilibrium : Virtual Work is zero Work and Energy methods**VW : Rigid Body (in x-y plane)**• Same approach, with additional rotational degree of freedom (see CM1, chapter 15) In plane equilibirum conditions for a rigid body Equilibirum : Virtual Work is zero Work and Energy methods**MECHANISMS**Interaction Forces (at the interface) do not generate Work ! • Kinematically indeterminate • Possibilities for mechanisms ? Hinge, N, V no M Shear force hinge, N, M no V Telescope, V, M no N Work and Energy methods**RESULT**• For mechanisms holds: The total amount of virtual work is generated only by external forces Work and Energy methods**work = 0**= M only M generates work ! M MECHANISMS ????? • Not a sensible structure • Correct, but ……. Work and Energy methods**F**F M = M F M M With Loading ….... • Total (virtual) work is zero ! total work = 0 ! results in value of M Work and Energy methods**F**M M Example : M at the position of F F x-axis l z-axis a b Work and Energy methods**Standard Approach**• Generate Virtual Work for the chosen generalised force (forces or moments) • Only possible if the constrained degreeof freedom which belongs to the generalised force is released and is given a virtual displacement or virtual rotation • In case of a statically determinate structure this approach will result in a mechanism. Only the external load and the requested generalised force will generate Virtual Work (no structural deformation). • The total amount of Virtual Work is zero. Work and Energy methods**F**l AV z-as a b F AV Example : AV Work and Energy methods**“TASTE” FOR BEAMS**• Support Reactions - remove the support • Shear force - shear hinge • Moment - hinge • Normal force - telescope Work and Energy methods**Horizontal displacement =**Rotation Vertical Distance to Rotational Centre (RC) Compute the amount of Work… Example : Truss Force in bar DE ? Step 1: release the elongation degree of freedom of this bar with a telescope mechanism and generate virtual work with the normal force N Step 2: Determine the virtual Work Step 3 : Solve N Work and Energy methods**50 kN**5 kN/m x-axis 2,5 m 3,5 m z-axis Assignment : Virtual Workmoment at the support and support reaction at the roller Work and Energy methods**Fb**Fa uba uaa ubb uab Work and the reciprocal theorem 1 : first Fa than Fb A B 2 : first Fb than Fa Work and Energy methods**Work must be the same**• Order of loading is not important • This results in: theorem of BETTI Work and Energy methods**Rewrite BETTI in to:**Reciprocal theorem of Maxwell displacement = influencefactor x force Work and Energy methods**Result : Betti – Maxwell reciprocal theorem**Work and Energy methods**Strain Energy**• Extension (tension or compression) • Shear • Torsion • Bending • Normal- and shear stresses Work and Energy methods**force**dx N N N dx strain d Extension work oppervlak Work and Energy methods**Strain Energy**• In terms of the generalised stresses EC • In terms of the generalised displacements EV See lecture notes for standard cases Work and Energy methods**SUMMARY**Work and Energy methods**Work methods**• Work by external loads is stored in the deformable elements as strain energy (Clapeyron) • Aext = EV Work and Energy methods**F**EI A B x-axis wmax 0,5 l 0,5 l z-axis Example 2 :Work and Energy Work and Energy methods**Work = Energy ?**• Unknown is wmax • Determine the M-distribution and the strain energy (MAPLE) • Work = Strain Energy (Clapeyron) Work and Energy methods**Moment Distribution ?**• Basic mechanics (statics) ? • Take half of the model due to symmetry Work and Energy methods**Solution**Work and Energy methods**q**A B w(x) EI l Distributedload ? • Work = displacement x load (how?) • Strain Energy from M-line (ok) • Average displacement or something like that ???? Work and Energy methods**Alternative Approach:Work Method with Unity Load**• Add a unitiy load at the position for which the displacement is asked for. • Displacement w and M-line M(x) due to actual loading • Displacement w en M-line m(x) due to unity load Work and Energy methods**1,0 kN**F EI EI m(x) M(x) l l Approach • Add Unity Load (0 .. 1,0) • Add actual Load (0 .. F) • Total Work ? • Strain Energy ? Work and Energy methods**Result**Integral is product of well known functions. In the “good old times” a standard table was used. Now use MAPLE Work and Energy methods**Work Method with Unity Load**Work and Energy methods**1,0 kN**q EI wmax 0,5 l 0,5 l Example with distributed load Work and Energy methods**Approach**• Determine M(x) due to load q (see example 1) • Determine m(x) due to unity load (notes : example 2) Elaborate… Work and Energy methods**EI, EA**F l u F just before buckling only compression uF F after buckling compression and bending Application Work & EnergyBuckling CONCLUSION : Increase in Work during buckling is stored as strain energy by bending only. (Compression is the same) Work and Energy methods**Buckling (transition)**• (almost) Constant Normal Force • Deformation by compression remains constant T H U S • Work done by normal force and additional displacement is stored as strain energy by bending only Work and Energy methods**dx**x, u w w dw duF dx z, w Additional displacement Taylor approximation Work and Energy methods**Clapeyron : A = Ev**Work and Energy methods**F**f l Example • Assume a kinematically admissible displacement field • Elaborate the integrals in the expression and compute the Buckling Load … Kinematic boundary conditions are met Exact Buckling load is always smaller than the one found with Rayleigh (UNSAFE) Work and Energy methods