Lecture 1 : Work and Energy methods

1. Lecture 1 : Work and Energymethods Hans Welleman

2. Content • Meeting 1 Work and Energy • Meeting 2 Castigliano • Meeting 3 Potential Energy Work and Energy methods

3. 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

4. Work uF F u Work and Energy methods

5. F=0 F u unloaded situation loaded situation Deformation or Strain Energy force u spring characteristics Work and Energy methods

6. 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

7. 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

8. 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

9. RESULT • For mechanisms holds: The total amount of virtual work is generated only by external forces Work and Energy methods

10. work = 0 = M only M generates work ! M MECHANISMS ????? • Not a sensible structure • Correct, but ……. Work and Energy methods

11. 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

12. F M M Example : M at the position of F F x-axis l z-axis a b Work and Energy methods

13. 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

14. F l AV z-as a b F AV Example : AV Work and Energy methods

15. “TASTE” FOR BEAMS • Support Reactions - remove the support • Shear force - shear hinge • Moment - hinge • Normal force - telescope Work and Energy methods

16. 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

17. 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

18. 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

19. Work must be the same • Order of loading is not important • This results in: theorem of BETTI Work and Energy methods

20. Rewrite BETTI in to: Reciprocal theorem of Maxwell displacement = influencefactor x force Work and Energy methods

21. Result : Betti – Maxwell reciprocal theorem Work and Energy methods

22. Strain Energy • Extension (tension or compression) • Shear • Torsion • Bending • Normal- and shear stresses Work and Energy methods

23. force dx N N N dx  strain d Extension work oppervlak Work and Energy methods

24. 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

25. SUMMARY Work and Energy methods

26. Work methods • Work by external loads is stored in the deformable elements as strain energy (Clapeyron) • Aext = EV Work and Energy methods

27. F EI A B x-axis wmax 0,5 l 0,5 l z-axis Example 2 :Work and Energy Work and Energy methods

28. Work = Energy ? • Unknown is wmax • Determine the M-distribution and the strain energy (MAPLE) • Work = Strain Energy (Clapeyron) Work and Energy methods

29. Moment Distribution ? • Basic mechanics (statics) ? • Take half of the model due to symmetry Work and Energy methods

30. Solution Work and Energy methods

31. 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

32. 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

33. 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

34. 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

35. Work Method with Unity Load Work and Energy methods

36. 1,0 kN q EI wmax 0,5 l 0,5 l Example with distributed load Work and Energy methods

37. 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

38. 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

39. 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

40. dx x, u w w dw duF dx z, w Additional displacement Taylor approximation Work and Energy methods

41. Clapeyron : A = Ev Work and Energy methods

42. 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