# Section Properties Calculator

Inertia governs deflection, section modulus governs strength — they are not interchangeable.

- Page: https://jianzhujisuan.com/en/section-modulus
- Category: Steel Design (https://jianzhujisuan.com/en/c/steel-design)
- Chinese version: https://jianzhujisuan.com/zh/section-modulus.md
- Data checked: 2026-10-25

## Quick result

With the default inputs (Section shape Rectangle b×h, Width b / flange bf 200 mm, Depth h 400 mm, Outside diameter D 219 mm, Wall thickness t 8 mm, Web thickness tw 8 mm, Flange thickness tf 13 mm): **Second moment I 106667 cm⁴**

## Inputs

| Parameter | Type | Default | Unit | Note |
|---|---|---|---|---|
| Section shape | select | Rectangle b×h | — |  |
| Width b / flange bf | number | 200 | mm |  |
| Depth h | number | 400 | mm |  |
| Outside diameter D | number | 219 | mm |  |
| Wall thickness t | number | 8 | mm |  |
| Web thickness tw | number | 8 | mm |  |
| Flange thickness tf | number | 13 | mm |  |

## Formula

- **Rectangle**: I = bh³/12, W = bh²/6
- **Circular hollow**: I = π(D⁴−d⁴)/64, d = D−2t
- **I-section**: I = [bf·h³ − (bf−tw)(h−2tf)³]/12
- **Section modulus**: W = I/(h/2)
- **Radius of gyration**: r = √(I/A)
- **Mass per metre**: m = A·7850 (A in m²)

## Worked example

A 200×400 rectangle: area 800.0 cm², 106667 cm⁴ of inertia, 5333 cm³ of section modulus, radius of gyration 115.5 mm, 628.0 kg/m and a shape factor of 1.50.

## Questions

### What is the difference between inertia and section modulus?

Inertia is the integral of y² over the area and governs stiffness; section modulus is inertia divided by the extreme fibre distance and governs strength. Deflection uses I, bending capacity uses W. For a 200×400 rectangle, I is 1.067×10⁹ mm⁴ and W is 5.33×10⁶ mm³ — 200 times apart because the units differ, mm⁴ against mm³, so neither can be substituted into the other formula.

### Why is an I-section so efficient in bending?

Because the material sits far from the neutral axis. Capacity scales with the integral of y squared, so the same area at 200 mm out contributes sixteen times as much as at 50 mm. Thin the web and widen the flanges and inertia can go up two to three times for the same area. The penalties are local buckling in a thin web, needing stiffeners, torsion in thin flanges, and a weak minor axis — often a tenth of the major — so lateral restraint is essential.

### Why is a hollow section the same in every direction?

Because a circle is symmetric about any axis through its centre. That is valuable in columns, where buckling can go any way, so a tube needs no preferred orientation. It makes spiral-welded tube and steel pipe columns economical under biaxial moment or uncertain direction. The section modulus is I/(D/2), slightly below a rectangle of the same area, because the material sits on a ring rather than at the extremes.

### What is the shape factor?

It is the plastic modulus over the elastic one: 1.5 for a rectangle, about 1.12 to 1.15 for an I-section and 1.27 for a tube. It measures the reserve after first yield — a rectangle carries 50% more moment than its elastic limit, an I-section only 12%, so once the flanges yield an I-beam loses capacity quickly while a rectangle has a long plastic plateau. That matters in seismic design, where members expected to form hinges benefit from a higher shape factor because rotation capacity follows it.

## Related tools

- [Steel Beam Span Calculator](https://jianzhujisuan.com/en/steel-beam-span): The depth is not looked up in a table — it falls out of the deflection limit.
- [Fillet Weld Strength Calculator](https://jianzhujisuan.com/en/fillet-weld): A 6 mm leg is not a 6 mm throat — capacity follows the 4.2 mm throat.
- [Base Plate Design Calculator](https://jianzhujisuan.com/en/base-plate): Thicker is not the answer — check the bearing area first, then the cantilever.
- [Column Buckling Calculator](https://jianzhujisuan.com/en/column-buckling): Half the effective length means four times the Euler load — but the design resistance differs by less, and how much depends on slenderness.

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