Moment of Inertial Calculations

For an in-depth understanding of moments of inertia, including definitions, practical applications, and advanced calculations, visit our main article: Comprehensive Guide to Moments of Inertia in Engineering.

1. Chart of Moments of Inertia for Standard Shapes

2. Worked Examples

2.1 Rectangle (Horizontal Axis)

Find the moment of inertia of a rectangle with b = 200 mm and h = 400 mm.

Solution:

 Ix = (1/12) × b × h³

Ix = (1/12) × 200 × 400³ = 1,066,666,667 mm⁴

2.2 Rectangle (Vertical Axis)

Find the moment of inertia of the same rectangle (b = 200 mm, h = 400 mm) about its vertical axis.

Solution: 

Iy = (1/12) × h × b³

Iy = (1/12) × 400 × 200³ = 533,333,333 mm⁴

2.3 Circle

Find the moment of inertia of a circle with r = 100 mm.

Solution: 

I = (π × r⁴) / 4

I = (π × 100⁴) / 4 = 7,854,000 mm⁴

2.4 Hollow Circle

Find the moment of inertia of a hollow circle with R = 150 mm and r = 100 mm.

Solution:

 I = (π × (R⁴ − r⁴)) / 4

I = (π × (150⁴ − 100⁴)) / 4 = 31,403,000 mm⁴

2.5 Triangle

Find the moment of inertia of a triangle with b = 300 mm and h = 600 mm.

Solution: 

Ix = (1/36) × b × h³

Ix = (1/36) × 300 × 600³ = 18,000,000,000 mm⁴

2.6 Semicircle

Find the moment of inertia of a semicircle with r = 80 mm.

Solution: 

I = (π × r⁴) / 8

I = (π × 80⁴) / 8 = 2,514,000 mm⁴

2.7 Ellipse (Major Axis)

Find the moment of inertia of an ellipse with a = 200 mm and b = 100 mm.

Solution: 

Ix = (π × a × b³) / 4

Ix = (π × 200 × 100³) / 4 = 157,079,600 mm⁴

2.8 Ellipse (Minor Axis)

Find the moment of inertia of the same ellipse about its minor axis.

Solution: 

Iy = (π × b × a³) / 4

Iy = (π × 100 × 200³) / 4 = 628,318,530 mm⁴

2.9 Thin Rectangular Plate (Length)

Find the moment of inertia of a thin rectangular plate with w = 500 mm and l = 1000 mm.

Solution: 

I = (1/12) × w × l³

I = (1/12) × 500 × 1000³ = 41,666,666,667 mm⁴

2.10 Thin-Walled Circular Tube

Find the moment of inertia of a thin-walled circular tube with R = 150 mm and t = 10 mm.

Solution: 

I = 2 × π × R³ × t

I = 2 × π × 150³ × 10 = 21,204,000 mm⁴

2.11 I-Beam

Find the moment of inertia of an I-beam with the following properties:

• Flanges: b = 200 mm, t = 20 mm
• Web: h = 400 mm, t = 10 mm

Solution:

Using the formula: I = I_flanges + I_web

For the flanges:

I_flanges = 2 × [(1/12) × b × t³ + b × t × (d/2)²]

Substitute values: I_flanges = 2 × [(1/12) × 200 × (20)³ + 200 × 20 × (210)²]

Calculation: I_flanges = 2 × (1,333,333 + 882,000,000) = 1,766,666,666 mm⁴

For the web:

I_web = (1/12) × t × h³

Substitute values: I_web = (1/12) × 10 × (400)³

Calculation: I_web = 5,333,333 mm⁴

Total moment of inertia: I = I_flanges + I_web = 1,766,666,666 + 5,333,333 = 1,771,999,999 mm⁴

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