# Section Modulus Calculator

URL: https://speedycalculator.com/calculators/physics/section-modulus/
Last updated: 2026-10-30

> The section modulus calculator finds the elastic section modulus and area moment of inertia of common beam sections – rectangle, square, hollow rectangle, solid and hollow circle, I-beam, channel, tee and angle.

## Inputs

- Section
- Width b (flange / outer) (mm, cm, m, in, µm)
- Height h (outer) (mm, cm, m, in, µm)
- Wall / web thickness t (mm, cm, m, in, µm)
- Flange thickness t_f (mm, cm, m, in, µm)
- Outer diameter D (mm, cm, m, in, µm)
- Inner diameter d (mm, cm, m, in, µm)

## Results

- Elastic section modulus S
- Area moment of inertia I
- Cross-section area
- Distance to the extreme fibre c
- Radius of gyration

## Formula

```
S = I / c (c = distance from the neutral axis to the extreme fibre),  σ_max = M / S
```

## Explanation

The section modulus tells how well a beam shape resists bending: maximum bending stress equals bending moment divided by S. Deep sections with material far from the centre (like I-beams) give the most strength per kilogram.

### How to use the Section Modulus Calculator

- **Section**
- **Width b (flange / outer)**
- **Height h (outer)**
- **Wall / web thickness t**
- **Flange thickness t_f**
- **Outer diameter D**
- **Inner diameter d**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose the section shape and enter its dimensions. Results are for bending about the horizontal (x) axis; for tees and angles the smaller (critical) section modulus is given.

### Inputs explained

FieldWhat to enterSectionChoose one: Square, Rectangle, Hollow rectangle, Solid circle …Width b (flange / outer)Units: mm, cm, m, in, µmHeight h (outer)Units: mm, cm, m, in, µmWall / web thickness tUnits: mm, cm, m, in, µmFlange thickness t_fUnits: mm, cm, m, in, µmOuter diameter DUnits: mm, cm, m, in, µmInner diameter dUnits: mm, cm, m, in, µm

### What each result means

- **Elastic section modulus S** = [I]÷[c]
- **Area moment of inertia I** = [I]
- **Cross-section area** = [A]
- **Distance to the extreme fibre c** = [c]
- **Radius of gyration** = sqrt([I]÷[A])

### Example values

How the main result changes when **Width b** changes and everything else stays at the example values above:

Width b (flange / outer) (mm)Elastic section modulus S (mm³)50333,333.333375500,000100666,666.6667125833,333.33331501,000,0002001,333,333.3333

## Worked example

Inputs — Section: Rectangle; Width b (flange / outer): 100 mm; Height h (outer): 200 mm.

Results — **Elastic section modulus S = 666,666.66667 mm³**; Area moment of inertia I = 66,666,666.66667 mm⁴; Cross-section area = 20,000 mm²; Distance to the extreme fibre c = 100 mm; Radius of gyration = 57.735 mm.

## Key terms

- **Neutral axis**: Line through the centroid where bending stress is zero.

## FAQ

**How do I get bending stress from S?**
σ = M/S.

**Why are I-beams efficient?**
Most material is in the flanges, far from the neutral axis.

**What is the difference between S and I?**
I measures stiffness; S = I/c measures bending strength.

**What about the plastic section modulus?**
It is larger (by the shape factor, 1.5 for rectangles) and used in plastic design.


## Sources

- [MIT OpenCourseWare – Mechanics of Materials](https://ocw.mit.edu/courses/2-001-mechanics-materials-i-fall-2006/)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)