How to Solve for the Moment of Inertia of Irregular or Compound Shapes

Updated on December 15, 2019
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Ray is a Licensed Civil Engineer and specializes in Structural Engineering. He loves to write anything about education.

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 = ∫ ρ2 dA

The notation ρ (rho) corresponds to the coordinates of the center of differential area dA.

Moment of Inertia of Compound or Irregular Shapes
Moment of Inertia of Compound or Irregular Shapes | Source

Step-By-Step Procedure in Solving for the Moment of Inertia of Composite or Irregular Shapes

1. Identify the x-axis and y-axis of the complex figure. If not given, create your axes by drawing the x-axis and y-axis 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.

Division of Basic Shapes in Solving for Moment of Inertia
Division of Basic Shapes in Solving for Moment of Inertia | Source

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.

Area and Centroid of Basic Shapes for the Computation of Moment of Inertia
Area and Centroid of Basic Shapes for the Computation of Moment of Inertia | Source
Area and Centroid of Basic Shapes for the Computation of Moment of Inertia
Area and Centroid of Basic Shapes for the Computation of Moment of Inertia | Source

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.

Moment of Inertia and Radius of Gyration of Basic Shapes
Moment of Inertia and Radius of Gyration of Basic Shapes | Source
Moment of Inertia and Radius of Gyration of Basic Shapes
Moment of Inertia and Radius of Gyration of Basic Shapes | Source

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.

Transfer Formula of Moment of Inertia
Transfer Formula of Moment of Inertia | Source
Transfer Formula of Moment of Inertia
Transfer Formula of Moment of Inertia | Source

Example 1: Square Hole Punch

Solving for the Moment of Inertia of Compound Shapes
Solving for the Moment of Inertia of Compound Shapes | Source

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.

Location of centroid of the compound shape from the axes

x = 25 mm

y = 25 mm

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.

I = MOI of A1 - MOI of A2

I = bh^3/12 - bh^3/12

I = (50)(50)^3/12 - (25)(25)^3/12

I = 488281.25 mm^4

Example 2: C-Shape

Solving for the Moment of Inertia of Compound Shapes
Solving for the Moment of Inertia of Compound Shapes | Source

Solution

a. Solve for the centroid of the whole complex shape by tabulating the solution.

Label
Area (mm^4)
x-bar (mm)
y-bar (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
Location of centroid of the compound shape from the axes

x = 76000 / 2800

x = 27.143 mm

y = 84000 / 2800

y = 30 mm

b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.

Ix = MOI of A1 + MOI of A2 + MOI of A3

Ix = bh^3/12 + Ad^2 +  bh^3/12 + Ad^2 +  bh^3/12

Ix = (40)(20)^3/12 + (800)(20)^2 
   + (40)(20)^3/12 + (800)(20)^2
   + (20)(60)^3/12

Ix = 1053333.333 mm^4
Iy = MOI of A1 + MOI of A2 + MOI of A3

Iy = bh^3/12 + Ad^2 +  bh^3/12 + Ad^2 +  bh^3/12 + Ad^2

Iy = (20)(40)^3/12 + (800)(40 - 27.143)^2 
   + (20)(40)^3/12 + (800)(40 - 27.143)^2
   + (60)(20)^3/12 + (1200)(27.143-10)^2

Iy = 870476.1905 mm^4

Example 3 - Snake Shape

Solving for the Moment of Inertia of Compound Shapes
Solving for the Moment of Inertia of Compound Shapes | Source

Solution

a. Solve for the centroid of the whole complex shape by tabulating the solution.

Label
Area
x-bar (mm)
y-bar (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
Location of centroid of the compound shape from the axes

x = 38500 / 1100

x = 35 mm

y = 27500 / 1100

y = 25 mm

b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.

Ix = MOI of A1 + MOI of A2 + MOI of A3

Ix = bh^3/12 + Ad^2 +  bh^3/12 +  bh^3/12 + Ad^2

Ix = (30)(10)^3/12 + (300)(20)^2 
   + (10)(50)^3/12
   + (30)(10)^3/12 + (300)(20)^2

Ix = 349166.6667 mm^4
Iy = MOI of A1 + MOI of A2 + MOI of A3

Iy = bh^3/12 + Ad^2 +  bh^3/12 +  bh^3/12 + Ad^2

Iy = (10)(30)^3/12 + (300)(20)^2 
   + (50)(10)^3/12
   + (10)(30)^3/12 + (300)(20)^2

Iy = 289166.6667 mm^4

Example 4: I-Shape

Solving for the Moment of Inertia of Compound Shapes
Solving for the Moment of Inertia of Compound Shapes | Source

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.

Location of centroid of the compound shape from the axes

x = 20 mm

y = 20 mm

b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.

Ix = MOI of A1 + MOI of A2 + MOI of A3

Ix = bh^3/12 + Ad^2 +  bh^3/12 +  bh^3/12 + Ad^2

Ix = (40)(10)^3/12 + (400)(15)^2 
   + (10)(20)^3/12
   + (40)(10)^3/12 + (400)(15)^2

Ix = 193333.3333 mm^4
Iy = MOI of A1 + MOI of A2 + MOI of A3

Iy = bh^3/12 +  bh^3/12 +  bh^3/12 

Iy = (10)(40)^3/12 + (20)(10)^3/12 + (10)(40)^3/12 

Iy = 108333.3333  mm^4

Example 5: Complex Figure

Solving for the Moment of Inertia of Complex Figures
Solving for the Moment of Inertia of Complex Figures | Source

Solution

a. Solve for the centroid of the whole complex shape by tabulating the solution.

Label
Area
x-bar (mm)
y-bar (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
Location of centroid of the compound shape from the axes

x = 15571.79633 / 1057.079633

x =  14.73095862 mm

y = 12191.32376 / 1057.079633

y = 11.53302304 mm

b. Solve for the moment of inertia using the transfer formula. The word "MOI" stands for Moment of Inertia.

Ix = MOI of A1 + MOI of A2 + MOI of A3

Ix = (pi)r^4/4 + Ad^2 +  bh^3/12 + Ad^2 + bh^3/36 + Ad^2

Ix = (pi)(10)^4/4 + (157.0796327)(34.24413182 - 11.533)^2
   + (20)(30)^3/12 + (600)(15 - 11.533)^2
   + (20)(30)^3/36 + (300)(11.533 - 10)^2

Ix =  156792.0308 mm^4
Iy = MOI of A1 + MOI of A2 + MOI of A3

Iy = (pi)r^4/4 + Ad^2 +  bh^3/12 + Ad^2 + bh^3/36 + Ad^2

Iy = (pi)(10)^4/4 + (157.0796327)(14.73 - 10)^2
   + (30)(20)^3/12 + (600)(14.73 - 10)^2
   + (30)(20)^3/36 + (300)(26.67 - 14.73)^2

Iy = 94227.79522 mm^4

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    © 2019 Ray

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