Your SlideShare is downloading. ×
0
Today:
Area moments of Inertia
Steiner theorem
Mass moments of inertia
Book: Chapter 10.1,10.2, 10.4 & 10.8 + hand-out on ...
Area Moments of Inertia:
- Cross-sectional property
- Used to indicate resistance
of shape against bending
Cross-sectional properties
- Dimensions h, b (height, width)
- Area A = h x b (height x width)
- First moment of Area:
yQ ...
Moments of inertia: Second moment of Area
- Defined as:
- Unit: [mm4
]
2
2
x
A
y
A
I y dA
I x dA
=
=
∫
∫
Rectangle
Calculate Ix and Iy
D D
B B
C
y
dy
b/2 b/2
h/2
h/2
Polar moment of Inertia:
- Area Moment about z- axis
- Signifies resistance against torsion
- Denoted by J0 or Ip:
2 2 2
0...
Example: Circle
Calculate Ix, Iy and J0
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 ...
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
Example: I-beam
Calculate
Ix & Iy about C.G.
b
t
t
h
Thin-walled structures:
- t << h, b
- all higher order terms of t MUST be neglected
Example: T-profile
(thin-walled)
2t
t
b
b
Example: Ring (thin-walled)
Calculate J0 R
t
Mass moment of Inertia
Defined as the integral of the second
moment about an axis of all
elements of mass dm which
compose...
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...
Example: Disc
ρ = 8000kg/m3
t = 10 mm
Calculate IO
Example: Pendulum
Note:
At the exam the standard area and mass moments
of inertia for circles, cylinders, rings, rectangles
and triangles (a...
Upcoming SlideShare
Loading in...5
×

Lecture4c(1)

148

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
148
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

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
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×