What this calculator does
Section modulus is a single number that describes how well a cross-section resists bending, combining its second moment of area and its size in one figure. For a rectangle it is S = bh² / 6, where b is the width and h is the height measured in the direction of bending. A bigger section modulus means the same bending moment produces less stress.
The reason section modulus gets looked up on its own, separate from a full bending stress calculation, is that it simplifies the stress formula to σ = M / S once you have it, and it is the figure engineers compare directly against a material's allowable stress when sizing a beam. Timber tables, steel section handbooks and structural codes all publish section modulus values for exactly this reason.
The formula
For a rectangular cross-section, section modulus is width times height squared, divided by 6. For any other shape, if you already know the second moment of area (I) and the distance from the neutral axis to the outer fibre (y), section modulus is simply I divided by y. Both routes describe the same property, just starting from different known quantities.
| Term | Meaning |
|---|---|
| S | Section modulus, in mm³ or cm³. |
| b, h | Width and height of a rectangular cross-section, in the same units. |
| I | Second moment of area of the cross-section. |
| y | Distance from the neutral (centroidal) axis to the extreme fibre, the farthest point from the axis. |
The inputs explained
| Field | What to enter |
|---|---|
| Cross-section | Choose the rectangular calculation (from width and height) or the custom calculation (from a known second moment of area and distance to the extreme fibre). |
| Width (b) (mm) | The width of the rectangular section, measured perpendicular to the direction of bending. |
| Height (h) (mm) | The height of the rectangular section, measured in the direction of bending. |
| Second moment of area (I) (cm⁴) | The second moment of area of the cross-section, for the custom calculation. |
| Distance from neutral axis to extreme fibre (y) (mm) | The distance from the neutral axis to the outer edge of the cross-section, for the custom calculation. |
When to use it
Sizing a timber or steel beam
Once the maximum bending moment on a beam is known, dividing it by the material's allowable stress gives the minimum section modulus required, which is then matched against a catalogue of standard beam or joist sizes.
Comparing two cross-sections of the same area
Section modulus shows why a beam turned on edge is so much stiffer than one laid flat: the h² term means height contributes far more to bending resistance than width does, for the same amount of material.
Feeding into a bending stress check
Once section modulus is known, the maximum bending stress in a beam simplifies to σ = M / S, a much quicker check than working with the bending moment, distance to the extreme fibre and second moment of area separately each time.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How section modulus changes with beam height at a fixed 100 mm width
A fixed 100 mm width, across a range of section heights.
| Height | Section modulus (S) | Second moment of area (I = bh³/12) |
|---|---|---|
| 50 mm | 41,667 mm³ | 104.17 cm⁴ |
| 100 mm | 166,667 mm³ | 833.33 cm⁴ |
| 150 mm | 375,000 mm³ | 2,812.50 cm⁴ |
| 200 mm | 666,667 mm³ | 6,666.67 cm⁴ |
| 300 mm | 1,500,000 mm³ | 22,500.00 cm⁴ |
| 400 mm | 2,666,667 mm³ | 53,333.33 cm⁴ |
Questions
What is the difference between section modulus and second moment of area?
Second moment of area (I) describes how a cross-section's material is distributed relative to its neutral axis, independent of size. Section modulus (S = I / y) divides that by the distance to the outer fibre, turning it into a figure that plugs directly into the bending stress formula σ = M / S.
Does section modulus already account for material strength?
No. Section modulus is purely a property of the cross-section's shape and size. It is combined with the material's allowable stress separately, when checking whether a given bending moment is safe for a particular beam.
Why does a rectangle turned on edge have a much higher section modulus?
Because section modulus depends on height squared but only on width to the first power. Rotating the same rectangular timber from lying flat to standing on edge swaps which dimension is h, which is why joists and floor beams are always installed on edge rather than flat.
Which section modulus should I use for an asymmetric cross-section?
An asymmetric section (such as a T-shape) has two different distances from the neutral axis to its top and bottom fibres, so it has two section moduli, one for each side. Use whichever distance corresponds to the fibre under the greater stress for that loading case.
To take section modulus the rest of the way to a bending stress figure, see the bending stress calculator.