How to Solve for the Moment of Inertia of Irregular or Compound Shapes
Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.
What Is Moment of Inertia?
Moment of Inertia also called as "Angular Mass or Rotational Inertia" and "Second Moment of Area" is the inertia of a rotating body with respect to its rotation. Moment of Inertia applied to areas has no real meaning when examined by itself. It is merely a mathematical expression usually denoted by symbol I. However, when used in applications like flexural stresses in beams, it begins to have significance. The mathematical definition moment of inertia indicates that an area is divided into small parts dA, and each area is multiplied by the square of its moment arm about the reference axis.
I = ∫ ρ^{2} dA
The notation ρ (rho) corresponds to the coordinates of the center of differential area dA.
StepByStep Procedure in Solving for the Moment of Inertia of Composite or Irregular Shapes
1. Identify the xaxis and yaxis of the complex figure. If not given, create your axes by drawing the xaxis and yaxis on the boundaries of the figure.
2. Identify and divide the complex shape into basic shapes for easier computation of moment of inertia. When solving for the moment of inertia of a composite area, divide the composite area into basic geometric elements (rectangle, circle, triangle, etc) for which the moments of inertia are known. You can show the division by drawing solid or broken lines across the irregular shape. Label each basic shape to prevent confusion and miscalculations. An example is shown below.
3. Solve for the area and centroid of each basic shape by creating a tabular form of the solution. Obtain the distances from the axes of the centroid of the whole irregular shape before continuing to the computation of the moment of inertia. Always remember to subtract areas corresponding to holes. Refer to the article below for the computation of centroid distances.
4. Once you obtained the location of the centroid from the axes, proceed to the calculation of the moment of inertia. Compute for the moment of inertia of each basic shape and refer for the formula for the basic shapes given below.
Below are the moment of inertia of basic shapes for its centroidal axis. To calculate the moment of inertia of a compound shape successfully, you must memorize the basic formula of the moment of inertia of basic geometric elements. These formulas are only applicable if the centroid of a basic shape coincides with the centroid of the irregular shape.
5. If the centroid of the basic shape does not coincide, it is necessary to transfer the moment of inertia from that axis to the axis where the centroid of the compound shape is located using the 'Transfer Formula for Moment of Inertia'.
The moment of inertia with respect to any axis in the plane of the area is equal to the moment of inertia with respect to a parallel centroidal axis plus a transfer term composed of the product of the area of a basic shape multiplied by the square of the distance between the axes. The Transfer formula for Moment of Inertia is given below.
6. Get the summation of the moment of inertia of all basic shapes using transfer formula.
Example 1: Square Hole Punch
Solution
a. Solve for the centroid of the whole compound shape. Since the figure is symmetrical in both directions, then its centroid is located on the middle of the complex figure.
b. Solve for the moment of inertia of the complex figure by subtracting the moment of inertia of area 2 (A2) from area 1 (A1). There is no need to use the transfer formula of moment of inertia since the centroid of all basic shapes coincide with the centroid of the compound shape.
Example 2: CShape
Solution
a. Solve for the centroid of the whole complex shape by tabulating the solution.
Label  Area (mm^4)  xbar (mm)  ybar (mm)  Ax  Ay 

A1  800  40  50  32000  40000 
A2  800  40  10  32000  8000 
A3  1200  10  30  12000  36000 
TOTAL  2800 

 76000  84000 
b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.
Example 3  Snake Shape
Solution
a. Solve for the centroid of the whole complex shape by tabulating the solution.
Label  Area  xbar (mm)  ybar (mm)  Ax  Ay 

A1  300  15  5  4500  1500 
A2  500  35  25  17500  12500 
A3  300  55  45  16500  13500 
TOTAL  1100  38500  27500 
b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.
Example 4: IShape
Solution
a. Solve for the centroid of the whole compound shape. Since the figure is symmetrical in both directions, then its centroid is located on the middle of the complex figure.
b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.
Example 5: Complex Figure
Solution
a. Solve for the centroid of the whole complex shape by tabulating the solution.
Label  Area  xbar (mm)  ybar (mm)  Ax  Ay 

A1  157.0796327  10  34.24413182  1570.796327  191.3237645 
A2  600  10  15  6000  9000 
A3  300  26.67  10  8001  3000 
TOTAL  1057.079633 

 15571.79633  12191.32376 
b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.
© 2019 Ray
Comments
Ray (author) from Philippines on January 02, 2021:
Hi, I am an engineer and these are applications of what I've learned from my college. Thanks!
Bareq on July 05, 2020:
What are the sources of this report?