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Lecture4b(1)
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Transcript

  • 1. Today: • Centre of Mass • Centroids of sections • Distributed Loads Book: Chapter 9.1 - 9.2 and 4.9
  • 2. Centre of Gravity
  • 3. Centre of mass: • Equal to centre of gravity if g = constant • Independent of gravity
  • 4. In formulas: For bars: For surfaces: For volumes: ; ; ; xdm ydm zdm x y z dm dm dm = = = ∫ ∫ ∫ ∫ ∫ ∫
  • 5. Center of mass for n particles ; 1 1 ∑ ∑ = = = n i i n i ii m mx x ; 1 1 ∑ ∑ = = = n i i n i ii m my y ; 1 1 ∑ ∑ = = = n i i n i ii m mz z
  • 6. Centroid (= geometric centre): • Equal to centre of mass if ρ = constant • Property dependent on the shape of the body
  • 7. So for a given area A: ; ;A A A A xdA ydA x y dA dA = = ∫ ∫ ∫ ∫ % %
  • 8. First moment of area ; ; ; x y A A y x A A Q ydA Q xdA Q Q x y dA dA = = = = ∫ ∫ ∫ ∫ % %
  • 9. Special cases: - If a body has an axis of symmetry, C is on this axis - If a body has two or more axes of symmetry, C is where both axes cross - if a body is symmetric about a point, C is that point
  • 10. 1 axis of symmetry multiple axis of symmetry symmetry about a point Special cases:
  • 11. Basic shapes:
  • 12. Example: Calculate centroid coordinates
  • 13. For a given composite area A ; xA yA x y A A ∑ ∑ = = ∑ ∑ % %
  • 14. Example: composite cross-sections Locate the x- and y – coordinates of the centroid
  • 15. Example continued:
  • 16. F
  • 17. Distributed force: Area distribution Aerodynamic pressure
  • 18. Distributed force: Line distribution
  • 19. Equivalent force
  • 20. Equivalent force
  • 21. Equivalent force ( )( )R R w x x= ∆ = ∆∑ ∑
  • 22. Magnitude of the equivalent force of a line distributed load
  • 23. Position of the line of action of the equivalent force
  • 24. Uniform load distribution
  • 25. Linearly varying load distribution
  • 26. Superposition principle
  • 27. Determine the reactions at A and B for the beam subjected to the uniform load distribution
  • 28. Determine the reactions at A and B for the beam subjected to the two linearly varying load distributions
  • 29. Determine the equivalent force and its line of action that replaces the forces exerted by the masses.

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