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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