Calculating the plastic section modulus for any shape

In structural engineering, the plastic section modulus is the property of a beam section. The value of which corresponds to the bending capacity of the section when the complete section of the beam is yielding. Yielding refers to the plateau where the elastic behaviour of the material no longer applies.


There are many formulas to calculate the plastic section modulus for standard structural sections. However for non-standard shapes, the task is rather difficult. One would usually have to assume the shape to confirm to a particular formula and make the assumption that the result will be reasonably correct.


The calculation is rather straight-forward. Because the sumation of forces for the given section has to be zero for pure bending, the section will behave similar to an elastic section when the zero-strain at the neutral axis. Next we divide the section to top and bottom halves and calculate their individual 1st moment of area independtly, about the neutral axis of the entire section. The sum of the two values is the plastic section modulus.

There is however a better approach. By creating the section as an image, one can use the computer to determine the plastic section modulus. There is still a cavaet – not all sections are stable enough to deform to their plastic limit, the beam might fail either through local buckling or lateral instability. Therefore adequate experience is required when designing a non-standard plastic section.

The solution

I have created a program that by iterating each pixel of the image, calculate the properties of the section. The program ignores pixels that are not completely black, so it’s possible to leave notes and scale dimensions on the image without interferring with the actual calculation.

Download Program

A screenshot of the print out from the program

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5 Responses to Calculating the plastic section modulus for any shape

  1. Pingback: Calculating section properties from an image |

  2. Your blog is a wee bit slow to show up when using Safari.

  3. Discounts says:

    What great article and i have downloaded the program. I have been searching for ages on how to calculate section modulus for non standard shapes.

  4. Props says:

    The program seems to work correctly, but if you copy and paste an image from Autocad to Powerpoint, then save the image as a png, the image gets distorted and the properties don’t come out right. Taking a screenshot and using Powerpoint to crop it down, then save as a png, the properties seem to come out pretty good.

  5. I agree with what you are saying here especially the formulae for the non standard shapes you have.

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