Calculator Inputs
Example Data Table
These examples help verify expected input patterns and result scale.
| Section | Key inputs | Area | Centroidal Ix | Centroidal Iy |
|---|---|---|---|---|
| Rectangle | b = 100 mm, h = 200 mm | 20,000 mm² | 66,666,666.67 mm⁴ | 16,666,666.67 mm⁴ |
| Circle | d = 120 mm | 11,309.73 mm² | 10,178,760.20 mm⁴ | 10,178,760.20 mm⁴ |
| Symmetric I Section | B = 150 mm, H = 250 mm, tw = 10 mm, tf = 15 mm | 6,700 mm² | 67,584,666.67 mm⁴ | 8,441,666.67 mm⁴ |
Formula Used
The second moment of inertia measures area distribution around an axis. It is also called the area moment of inertia.
Core definitions
Ix = ∫ y² dA
Iy = ∫ x² dA
J = Ix + Iy
Parallel axis theorem
Ix' = Ixc + A × dy²
Iy' = Iyc + A × dx²
r = √(I / A)
Common shape formulas
- Rectangle:
Ix = b h³ / 12,Iy = h b³ / 12 - Circle:
Ix = Iy = π d⁴ / 64 - Hollow circle:
Ix = Iy = π (d⁴ - di⁴) / 64 - Triangle:
Ix = b h³ / 36,Iy = h b³ / 48 - Composite section:
I = Σ(Ic + A × distance²)
How to Use This Calculator
- Select a section type from the list.
- Enter dimensions using one consistent unit.
- Add dx and dy only when using a shifted reference axis.
- Use custom composite rows for built-up sections or holes.
- Enter optional c distances when you need section modulus values.
- Press calculate and review area, centroid, inertia, and checks.
- Download CSV or PDF for documentation.
Second Moment of Inertia Guide
What the Value Means
The second moment of inertia describes how an area is spread around a reference axis. A larger value usually means the section is harder to bend about that axis. Engineers use it when checking beams, shafts, frames, and machine parts. It does not measure mass by itself. It measures the shape strength created by geometry. That is why deep beams often resist bending better than shallow beams.
Why Axis Choice Matters
The same section can have very different Ix and Iy values. A tall rectangle has a large Ix and a smaller Iy. Rotating the section changes the bending response. The centroidal axis is often the starting point. A shifted axis needs the parallel axis theorem. This calculator applies that theorem with dx and dy inputs.
Composite Sections
Real sections are often built from many simple areas. A plate, web, flange, or cutout can be entered as a component. The calculator first finds the combined centroid. Then it shifts each local inertia to that centroid. Holes should use negative area. This approach is useful for channels, tees, built-up plates, and irregular sections.
Design Use
The output can support early design checks. Section modulus helps estimate bending stress. Radius of gyration helps compare stiffness distribution. Polar moment is useful for area-based torsion comparisons. Always keep units consistent. A mixed unit entry can create very large errors. For final structural design, compare results with accepted standards and project rules.
FAQs
1. What is second moment of inertia?
It is a geometric property that shows how area is distributed around an axis. It is used for bending stiffness and stress calculations.
2. Is it the same as mass moment of inertia?
No. Second moment of inertia uses area. Mass moment of inertia uses mass distribution and applies to rotational dynamics.
3. Which unit should I use?
Use one consistent length unit for every input. The area output uses that unit squared. Inertia uses that unit to the fourth power.
4. What are dx and dy?
They are distances from the centroidal axis to a shifted reference axis. dy shifts Ix. dx shifts Iy.
5. How do I enter a hole?
Use the custom composite option. Enter the hole with negative area. The calculator treats its local inertia as negative.
6. What is section modulus?
Section modulus equals inertia divided by extreme fiber distance. It is commonly used to estimate bending stress.
7. Why are Ix and Iy different?
They depend on the axis. A shape may spread more area vertically than horizontally, creating different resistance values.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet records. Use the PDF button for a clean calculation report.