Formula used
- Rectangle: Ix = b·h³/12, Iy = h·b³/12.
- Hollow rectangle: Ix = (B·H³ − b·h³)/12, Iy = (H·B³ − h·b³)/12.
- Circle: Ix = Iy = π·d⁴/64.
- Hollow circle: Ix = Iy = π·(D⁴ − d⁴)/64.
- Section modulus: Sx = Ix/cx and Sy = Iy/cy, using extreme-fiber distances.
- Radius of gyration: rx = √(Ix/A), ry = √(Iy/A).
- Optional stress: σ = M/S (reported in MPa and psi).
How to use this calculator
- Select the cross-section shape that matches your member.
- Choose a length unit, then enter all dimensions in that unit.
- For hollow shapes, fill both outer and inner dimensions.
- Enable stress only if you have bending moment values.
- Press Calculate to view results above the form.
- Use CSV or PDF export to store or share results.
Example data table
| Shape | Dimensions (mm) | Ix (mm⁴) | Iy (mm⁴) | Sx (mm³) | Sy (mm³) |
|---|---|---|---|---|---|
| Solid Rectangle | b=100, h=200 | 6.6667e+07 | 1.6667e+07 | 6.6667e+05 | 3.3333e+05 |
| Solid Circle | d=100 | 4.9087e+06 | 4.9087e+06 | 9.8175e+04 | 9.8175e+04 |
| Hollow Circle | D=120, d=80 | 8.1681e+06 | 8.1681e+06 | 1.3614e+05 | 1.3614e+05 |
| I-Section | B=150, H=300, tf=15, tw=10 | 1.0787e+08 | 8.4600e+06 | 7.1910e+05 | 1.1280e+05 |
Understanding the outputs
This calculator reports centroidal Ix and Iy plus elastic section moduli Sx and Sy. In bending, Ix controls stiffness about the x-axis and Sx controls stress through σ = M/S. Use the same length unit for every dimension to keep the output consistent.
Why inertia grows fast
Inertia scales with the fourth power of size. A solid circle uses I = π·d⁴/64, so increasing diameter from 50 mm to 100 mm raises I by (100/50)⁴ = 16×. For rectangles, Ix = b·h³/12, so doubling height increases Ix by 2³ = 8×.
Section modulus for quick stress checks
Section modulus is inertia divided by the extreme-fiber distance. For a 100×200 mm rectangle, Ix ≈ 6.67×10⁷ mm⁴ and c = 100 mm, giving Sx ≈ 6.67×10⁵ mm³. If M = 10 kN·m (that is 10,000,000 N·mm), stress is about 15 MPa.
Hollow sections save weight
Hollow shapes often keep stiffness while reducing area. A hollow circle uses I = π·(D⁴ − d⁴)/64. With D=120 mm and d=80 mm, the example table shows Ix ≈ 8.17×10⁶ mm⁴, which is higher than a 100 mm solid bar (Ix ≈ 4.91×10⁶ mm⁴) despite less material.
I-sections concentrate material efficiently
I-sections place most area in flanges, far from the neutral axis. The sample I-section (B=150, H=300, tf=15, tw=10 mm) yields Ix ≈ 1.08×10⁸ mm⁴ and a large Sx, which helps both stiffness and bending stress.
T-sections and centroid shift
A T-section’s centroid is not at mid-depth. This tool applies the parallel-axis theorem and reports an extreme-fiber distance based on the larger of top or bottom fibers. This matters when one flange is thicker, because c changes and so does S.
Units and quick conversions
The calculator converts internally using millimeters. Outputs are returned in your selected unit powers: area in unit², inertia in unit⁴, modulus in unit³. For stress inputs, 1 kN·m = 1,000,000 N·mm, and 1 MPa ≈ 145.038 psi.
Practical design workflow
Start with geometry and confirm Ix/Iy for the bending direction. Next compute stress using σ = M/S. Finally compare to your allowable stress and check deflection using beam formulas. If results look odd, re-check which axis your moment acts about and confirm dimensions are correct.
FAQs
1) What is the difference between Ix and Iy?
Ix is inertia about the horizontal centroidal axis, while Iy is about the vertical centroidal axis. Use the axis that matches the direction your section bends.
2) Why does section modulus matter?
Section modulus turns inertia into a stress-ready quantity. For linear elastic bending, maximum stress equals bending moment divided by section modulus, so larger S reduces stress.
3) Are these results about the centroid?
Yes. Ix and Iy are computed about centroidal axes. For asymmetric sections like T-shapes, the centroid is found first, then inertia is assembled using the parallel-axis theorem.
4) Can I use inches or feet?
Yes. Select in or ft, then enter all dimensions in that unit. The tool outputs inertia in in⁴ or ft⁴ and modulus in in³ or ft³ automatically.
5) What if my inner size is bigger than the outer size?
The calculator will flag it. Inner dimensions for hollow shapes must be smaller than outer dimensions; otherwise, area becomes negative and physical results are invalid.
6) How accurate is the optional stress result?
It is a basic bending estimate using σ = M/S at the extreme fiber. It does not include shear, torsion, local buckling, stress concentrations, or code-specific factors.
7) Which output helps deflection checks?
Use the inertia about the bending axis, typically Ix for strong-axis bending. Deflection formulas use E·I, so inertia directly influences stiffness and serviceability.