Moment of Inertia (Common Shapes)
Moment of inertia formulas for common shapes including rectangles, circles, I-beams, and hollow sections.
For structural engineering.
The Concept
The moment of inertia (second moment of area) measures a cross-section's resistance to bending. A larger moment of inertia means the beam is stiffer and deflects less under load.
This is not the same as mass moment of inertia (used in rotational dynamics), though they share the same name. The area moment of inertia depends only on the geometry of the cross-section, not the material.
Variables
| Symbol | Meaning |
|---|---|
| I | Moment of inertia (area moment, measured in mm⁴, cm⁴, or in⁴) |
| y | Distance from the neutral axis to the element dA |
| dA | Infinitesimal area element of the cross-section |
| b | Width of the shape |
| h | Height of the shape |
| r | Radius (for circular sections) |
| d | Diameter (for circular sections) |
Common Shape Formulas (about centroidal axis)
| Shape | Moment of Inertia (I) |
|---|---|
| Rectangle (about horizontal centroid) | I = bh³ / 12 |
| Rectangle (about base) | I = bh³ / 3 |
| Solid circle | I = πd⁴ / 64 = πr⁴ / 4 |
| Hollow circle (tube) | I = π(d₀⁴ − dᵢ⁴) / 64 |
| Triangle (about centroid) | I = bh³ / 36 |
| Triangle (about base) | I = bh³ / 12 |
| I-beam / H-section | Use composite method (flanges + web) |
Parallel Axis Theorem
To find the moment of inertia about any axis parallel to the centroidal axis, add the product of the area and the square of the distance between axes. This theorem is essential for calculating I of composite shapes.
Example 1
A rectangular beam is 50 mm wide and 200 mm tall. What is its moment of inertia about the horizontal centroidal axis?
Apply: I = bh³ / 12 = 50 × 200³ / 12
I = 50 × 8,000,000 / 12
I = 33,333,333 mm⁴ = 33.33 × 10⁶ mm⁴
Example 2
A solid circular shaft has a diameter of 80 mm. What is its moment of inertia?
Apply: I = πd⁴ / 64 = π × 80⁴ / 64
I = π × 40,960,000 / 64
I = 2,010,619 mm⁴ ≈ 2.01 × 10⁶ mm⁴
When to Use It
Moment of inertia is fundamental in structural and mechanical engineering.
- Calculating beam deflection: δ = FL³ / (48EI) for a simply supported beam with center load
- Calculating bending stress: σ = My / I (the flexure formula)
- Selecting beam sizes for structural applications
- Comparing the stiffness of different cross-section shapes
- Designing shafts, columns, and frames to meet load requirements