1. Moments of Inertia

2. Chapter Objectives
Method for determining the moment of inertia for an area

3. Chapter Outline Definitions of Moments of Inertia for Areas
Parallel-Axis Theorem for an Area
Moments of Inertia for Composite Areas

4. 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 beam and depends on a reference axis.
The smallest M.I about any axis passes through the centroid.

5. Definition of Moments of Inertia for Areas
Moment of Inertia
Consider area A lying in the x-y plane
Be definition, moments of inertia of the differential plane area dA about the x and y axes
For entire area, moments of
inertia are given by

6. Parallel Axis Theorem for an Area
For moment of inertia of an area known about an axis passing through its centroid, determine the moment of inertia of area about a corresponding parallel axis using the parallel axis theorem
Consider moment of inertia of the shaded area
A differential element dA is located at an arbitrary distance y’ from the centroidal x’ axis

7. Parallel Axis Theorem for an Area
The fixed distance between the parallel x and x’ axes is defined as dy
For moment of inertia of dA about x axis
For entire area
First integral represent the moment of inertia of the area about the centroidal axis

8. Parallel Axis Theorem for an Area
Second integral = 0 since x’ passes through the area’s centroid C
Third integral represents the total area A
Similarly

10. Moment of inertia of a rectangle

11. Moment of inertia of a triangle

12. Moment of inertia of a parabolic spandrel

13. Moment of inertia of a circle

14. 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

15. 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

17. Example 1
Compute the moment of inertia of the composite area about the x axis.

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

19. Solution

20. Example 2
Compute the moment of inertia of the composite area about the x axis & y-axia.

21. Solution

22. Example 3
Compute the moment of inertia of the composite area about the y-axis.

23. Solution

24. Example 4
Compute the moment of inertia of the composite area about the x/-axia.

25. Solution

26. Example 5
Determine the moment of inertia of the beam’s cross-sectional area about the x and Y axes.

28. Example 6
Determine the moment of inertia of the beam’s cross-sectional area about the x/ axis.

30. Example 7
Determine the moment of inertia of the composite area about the x &y-axes.
7.5 mm

32. Determine the H-beam,s moment of inertia about the x &y-axes.
Example 8
Determine the H-beam,s moment of inertia about the x &y-axes.