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# Lecture4c(1)

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### Transcript of "Lecture4c(1)"

1. 1. Today: Area moments of Inertia Steiner theorem Mass moments of inertia Book: Chapter 10.1,10.2, 10.4 & 10.8 + hand-out on BB
2. 2. Area Moments of Inertia: - Cross-sectional property - Used to indicate resistance of shape against bending
3. 3. Cross-sectional properties - Dimensions h, b (height, width) - Area A = h x b (height x width) - First moment of Area: yQ xdA= ∫ D D B B C y dy b/2 b/2 h/2 h/2 ∫= A x ydAQ
4. 4. Moments of inertia: Second moment of Area - Defined as: - Unit: [mm4 ] 2 2 x A y A I y dA I x dA = = ∫ ∫
5. 5. Rectangle Calculate Ix and Iy D D B B C y dy b/2 b/2 h/2 h/2
6. 6. Polar moment of Inertia: - Area Moment about z- axis - Signifies resistance against torsion - Denoted by J0 or Ip: 2 2 2 0 x y dA dA dA J r dA I I y dA x dA= = + = +∫ ∫ ∫
7. 7. Example: Circle Calculate Ix, Iy and J0
8. 8. Steiner theorem: Allows you to calculate moment of inertia about another axis than through the centroid 2 2 2 0 0 x x y y y x I I Ad I I Ad J J Ad = + = + = +
9. 9. Rectangle Calculate Ix and Iy about B-B and D-D and B-D D D B B C y dy b/2 b/2 h/2 h/2
10. 10. Example: I-beam Calculate Ix & Iy about C.G. b t t h
11. 11. Thin-walled structures: - t << h, b - all higher order terms of t MUST be neglected
12. 12. Example: T-profile (thin-walled) 2t t b b
13. 13. Example: Ring (thin-walled) Calculate J0 R t
14. 14. Mass moment of Inertia Defined as the integral of the second moment about an axis of all elements of mass dm which compose the body Resistance against acceleration
15. 15. Mass moment of Inertia Mass moment of inertia about the z- axis is: Unit: [kgm2 ] Steiner theorem: 2 m I r dm= ∫ 2 GI I md= +
16. 16. Example: Disc ρ = 8000kg/m3 t = 10 mm Calculate IO
17. 17. Example: Pendulum
18. 18. Note: At the exam the standard area and mass moments of inertia for circles, cylinders, rings, rectangles and triangles (area only) as well as the location of their centroids are given on a formula sheet. See copy of formula sheet on Blackboard
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