1. Moments of Inertia

2. Definition of Moments of Inertia for Areas:
Measures the beam ability to resist bending.
The larger the M.I the less the beam will bend.
M.I is a geometrical property of a section and depends on a reference axis.
The smallest M.I about any axis passes through the centroid.

3. Moments of Inertia for Composite Areas
Composite area consist of a series of connected simpler parts or shapes
Moment of inertia of the composite area = algebraic sum of the moments of inertia of all its parts
Procedure for Analysis
Composite Parts
Divide area into its composite parts and indicate the centroid of each part to the reference axis
Parallel Axis Theorem
Moment of inertia of each part is determined about its centroidal axis

4. Moments of Inertia for Composite Areas
Procedure for Analysis
Parallel Axis Theorem
When centroidal axis does not coincide with the reference axis, the parallel axis theorem is used
Summation
Moment of inertia of the entire area about the reference axis is determined by summing the results of its composite parts

6. Example
Compute the moment of inertia of the composite area about the x axis.

7. Solution
Composite Parts Composite area obtained by subtracting the circle form the rectangle. Centroid of each area is located in the figure below.

8. Solution

9. Example
Compute the moment of inertia of the composite area about the x axis & y-axia.

10. Solution

11. Example
Compute the moment of inertia of the composite area about the y-axis.

12. Solution

13. Example
Compute the moment of inertia of the composite area about the x/-axia.

14. Solution

15. Example
Determine the moment of inertia of the beam’s cross-sectional area about the x axis.

16. Solution

17. Example
Determine the moment of inertia of the composite area about the x axis.

18. Solution

19. Example
Determine the moments of inertia IXo and IYo of the L-secion shown below with respect to the centroidal axes shown in the same figure. All dimensions are in mm.

20. Solution

21. Example
Determine the moment of inertia of the cross-sectional area about x & y/ axes.

22. Solution

23. Example
Determine the moment of inertia of the composite area about the x &y-axes.
7.5 mm

24. Solution

25. Determine the H-beam,s moment of inertia about the x &y-axes.
Example
Determine the H-beam,s moment of inertia about the x &y-axes.

26. Solution

27. 1)For the T-section below , determine the moment of inertia about centroidal axes.

28. 2) Determine the moment of Inertia about x/ & y-axis

29. Assignment
3) Determine the moment of Inertia about y-axis