2. Chapter Objectives Chapter Outline
Method for determining the moment of inertia for an area
Definition of Radius of Gyration
Chapter Outline
Definitions of Moments of Inertia for Areas
Parallel-Axis Theorem for an Area
Radius of Gyration of an Area
Moments of Inertia for Composite Areas

3. 10.1 Definition of Moments of Inertia for Areas
The centroid of area is first order of moment of an area,
Moment of inertial is the second order of moment of an area
Moment of inertia is used to calculate, for example, when calculating a moment about an axis for distributed load, where the force is proportion to distance
Mainly applied in Mechanics of Materials, Structural Mechanics, Fluid Mechanics, Mechanical Design

4. 10.1 Definition of Moments of Inertia for Areas
For example the pressure of fluid p is proportional to depth y,
where is Specific Weight of fluid
The force on a small area is dF = pdA =
Moment of a force about the x axis is
The value is called
the moment of the area
about x-axis

5. 10.1 Definition of Moments of Inertia for Areas
Moment of Inertia
Consider the area A on the play x-y
The moment of inertial of the small are about x and y is
The moment of inertia about x and y axis is

6. 10.1 Definition of Moments of Inertia for Areas
Polar Moment of Inertia
The second order of moment of the area dA about O or about the z axis is (this axis is called polar axis)
r is the perpendicular distance from pole (z axis) to the area dA
Polar moment of inertia

7. 10.2 Parallel Axis Theorem for an Area
The theorem is used to find the moment of inertia of an area about an arbitrary axis parallel to the axis passing the centroid of the area (the moment of inertia about the centroid must be known)
Consider the area, in which x’ is passing through the centroid
The area dA has the distance from x to x’ equal to y’
The moment of inertial of the area dA
about x with the distance from the centroid x’ axis x’ equal to dy

8. 10.2 Parallel Axis Theorem for an Area
The first term is the moment of inertial about the axis passing the centroid
The second term is zeros since x’ passes the centroid
The third term is
Hence and
And
where and

9. 10.3 Radius of Gyration of an Area
Radius of gyration is used in column design is
This is similar to an equivalent radius when finding the moment of inertia
The equation is similar to the moment of inertia

10. Procedure for Analysis
Consider Integration of dA
Example Find Ix
Case 1
Consider dA parallel the x axis Ix
Case 2
Consider dA perpendicular to the x axis Ix
Find find Ix of small area from parallel axis theorem
Case 1
Case 2

11. Example 10.1
Determine the moment of inertia for the rectangular area with respect to
the centroidal x’ axis,
(b) the axis xb passing through the base of the rectangle,
(c) the pole or z’ axis perpendicular
to the x’-y’ plane and passing
through the centroid C.

12. Solution
Part (a)
Differential element chosen, distance y’ from x’ axis.
Since dA = b dy’,
Part (b)
By applying parallel axis theorem,

13. Solution Part (c) For polar moment of inertia about point C,
First determine,
Then:

14. Example 10.2
Determine the moment of inertia for the area about x axis.
Case 1
Case 2

15. Solution Solution I (case 1) Differential element area;
Moment of Inertia

16. Solution Solution II (case 2)
To determine the moment of inertia of the differential element area dA, by parallel-axis theorem.
For a small rectangle element
Then;
by parallel-axis theorem

17. Example 10.3
Determine the moment of inertia for the area about x axis.
Case 1
Case 2

18. Solution Solution I (case 1) Differential element area;
Moment of Inertia

19. Solution Solution II (case 2)
To determine the moment of inertia of the differential element area dA,
For a small rectangle element
Then;
Integrating with respect to x

20. 10.4 Moments of Inertia for Composite Areas
The composite area is composed of areas with many simple geometries
The moment of inertia of composite areas can be found by the sum of individual moment of inertia with sign (holes are negative)
Procedure for Analysis
Composite areas
Divide a composite area into simple geometrical areas and locate the centroids of each of the areas and find the moment of inertias about each of their own centroids
Parallel Axis Theorem
Apply the parallel axis theorem to calculate the moment of inertia about the required axis
Summation
Sum all of the moment of inertia, beware of signs

21. Example 10.4
Compute the moment of inertia of the composite area about the x axis.

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

23. Solution Parallel Axis Theorem Circle Rectangle Summation
For moment of inertia for the composite area,