Lecture6c

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Lecture6c

  1. 1. Today: Virtual Work Book: Chapter 11.1-11.3 + hand-out on Blackboard
  2. 2. a) Calculate the reactions in A and B b) Calculate the normal forces in DD’ and EE’ with the correct signs for tension and compression. c) Draw the N, V and M lines for CDSEGA B E’ ED D’ C S G
  3. 3. + + -
  4. 4. A few tips… • When drawing N,V and M lines, clearly denote the jumps and kinks in the lines. Give the values of ‘special’ points on the curves (maxima, transition points).
  5. 5. A few tips… • Use enough paper, work neatly. It will reduce the number of unnecessary mistakes. • Before filling in the answering sheet, draw the N,V and M diagrams on a piece of scrap paper.
  6. 6. Work of a force (infinitesimal movement) The work done by a force F acting on a particle with a differential displacement dr is defined as: dW d= ×F r W = [Nm] = kg m2 /s2 ] = [J]
  7. 7. Work of a force (infinitesimal movement) The work done by a force F acting on a particle with a differential displacement dr is defined as: cosdW d α= F r
  8. 8. Work of a force (finite movement) ( )x x y y z zW d F dr F dr F dr= = + +∫ ∫F r
  9. 9. Static equilibrium of a particle
  10. 10. The principle of virtual work (particle) A particle is in equilibrium when for any virtual displacement, the virtual work is equal to zero. 0x x y y z zW u F u F u Fδ δ δ δ= + + =∑ ∑ ∑
  11. 11. Work of a concentrated moment
  12. 12. Work of a concentrated moment A B
  13. 13. Small-Angle approximations
  14. 14. Work of a concentrated moment A B
  15. 15. The principle of virtual work (body in two dimensions) A particle is in equilibrium when for any variational displacement and/or rotation, the virtual work is equal to zero. 0i i i ix x y y i iW u F u F Mδ δ δ δθ= + + =∑ ∑ ∑
  16. 16. Calculate the reaction forces and moment in A using the principle of virtual work.A B
  17. 17. Calculate the reaction forces and moment in A. Vertical reaction A B
  18. 18. Calculate the reaction forces and moment in A. Horizontal reaction A B
  19. 19. Calculate the reaction forces and moment in A. Moment A B
  20. 20. Calculate the reaction in B A C B
  21. 21. Calculate the reaction in B Replace hinge B by a reaction force.A C B
  22. 22. Calculate the reaction in B Replace hinge B by a reaction force. Due to the absence of hinge B, the beam can rotate A C B
  23. 23. Calculate the reaction in B Replace hinge B by a reaction force. Due to the absence of hinge B, the beam can rotate Express all virtual rotations in terms of A C B
  24. 24. Calculate the reaction moment in A B A S
  25. 25. Calculate the internal moment in C B A C
  26. 26. Calculate the internal moment in CBA C BA
  27. 27. Internal moment • Hinge: only different rotation angles on either side of the cut. • Reaction moments M
  28. 28. Calculate the shear force in C B A C
  29. 29. Calculate the shear force in CBA BA
  30. 30. Shear force • Shear hinge: only different displacements perpendicular to the member on either side of the cut. • Reaction forces V
  31. 31. Calculate the normal force in member FG of this truss structure. B A C D E F G H
  32. 32. Calculate the normal force in member FG of this truss structure.B A C D E F G H
  33. 33. Normal force • Telescope hinge: only different displacements parallel to the member on either side of the cut • Reaction forces N

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