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

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)(27.14310)^2 Iy = 870476.1905 mm^4
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

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: 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.
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
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

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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