Section Modulus Calculator

This repository contains a small calculator for the section modulus of simple beam cross-sections, including Z-shaped, G-shaped, and rotated Z-shaped profiles.

What the calculator does

Section modulus is a geometric property that describes how well a cross-section resists bending. For a given shape, the section modulus is found by:

1. Defining the cross-section geometry.
2. Choosing a coordinate origin.
3. Calculating the centroid position $y_0$ with respect to that origin.
4. Computing the second moment of area $I$ of the section about the centroid.
5. Dividing by the largest vertical distance $c$ from the centroid to the outermost edge of the section:

$$ S = \frac{I}{c} $$

How the calculation works

The implementation follows the standard procedure for composite sections:

• Break the shape into simple rectangular parts.
• Choose an origin for the coordinate system.
• Compute the centroid of each rectangle and combine them to get the overall centroid:

$$ y_0 = \frac{\int y\ dA}{\int dA} $$

where $dA$ is the differential area element and $y$ is its vertical coordinate relative to the chosen origin.

• Compute the second moment of area for each rectangle relative to the overall centroid. For a rectangle, this is obtained by integrating the square of its distance from the reference axis over the rectangle area.
• The final second moment of area is the integral over the entire section of the squared distance from the centroidal axis:

$$ I = \int (y - y_0)^2\ dA $$

This value is independent of the original coordinate choice, because shifting the origin only changes the intermediate terms and the centroid location consistently.

• Determine the maximum vertical distance from the centroid to the section boundary:

$$ c = \max(\text{distance to top}, \text{distance to bottom}) $$

• Finally, compute the section modulus:

$$ S = \frac{I}{c} $$

Note that the integrals for each component were already solved and directly expressed in the code. Please refer to the comments in section_modulus.py for its choice of coordinate system.

Files in this repo

• section_modulus.py: Python implementation of the section modulus calculations for the supported profiles.
• index.html: A simple web page that visualizes the shapes and displays the computed values.
• tests/: pytest tests, including the independent reference calculation.

Input rules

All lengths use the same unit (for example mm); the result is in that unit cubed.

• Thickness must be greater than 0.
• Lengths must be 0 or more (a 0 means that part is left out, for example a G section with no top flange).
• At least one length must be greater than 0.

The Python functions raise ValueError for inputs that break these rules; the web page shows a message instead of a result.

Run locally

Open index.html in your browser, or serve the folder with a simple web server.

You can also test the live version here: https://bpy13.github.io/section_modulus/

Run the tests

In PowerShell, from the repo folder:

conda env create -f environment.yml   # first time only
conda activate section_modulus
python -m pytest
ruff check .
ruff format --check .

The tests compare every shape with an independent calculation (rectangles plus the parallel-axis theorem) in tests/reference.py. The web page has its own JavaScript copy of the formulas; tests/test_js_parity.py runs that copy with Node.js and checks it gives the same results as Python. That test is skipped, with a message, if Node.js is not installed.

License

MIT; see LICENSE.