# Determining the critical load of a fixed beam with non-uniform cross-section?

I am a computer scientist, not a mechanical engineer, so it is absolutely possible that I am missing some basic understanding here.

I have a beam that is fixed in both ends and my goal is to approximate the critical force at which said beam would buckle. The beam fulfills all assumptions of Euler's critical load model, except that the cross-section is non-uniform.

Now I understand that I could use FEM by breaking the beam down into multiple elements, finding stiffness matrices and a matrix for the boundary conditions etc. but I am wondering if there isn't a simpler approach to (numerically?) finding the critical force in this scenario. I am looking to implement this routine into software I am working on, so while I could use third-party FEM libraries, I could not use existing software to help me.

• No, there is no simple method to solve this type of problem other than the numerical (FEM) method.
– r13
Jul 31, 2023 at 16:05
• Simply put you need to find the smallest energy buckled shape that has the same potential energy as the compression of the member if it were to remain straight, with that same load. Nov 13, 2023 at 3:25

You don't need to do a full FEM implementation for this problem. There are two simpler numerical approximations you can use. They are somewhat long and involved to write here, but I'll give you the references and a bird's eye view.

1. Use the beam-column frame theory. This is described in section 2.3 of Timoshenko and Gere's Theory of Elastic Instability. It requires you to get the beam properties and integrate the deflections and internal stresses. It's cumbersome, but not impossible.

2. Use the energy method. This is described in section 2.8 (and 2.9) of Timoshenko and Gere's Theory of Elastic Instability. In this method you basically "assume" a deformed shape (with some considerations the book treats) and then calculate the bending energy and the axial deformation energy. Buckling occurs when these energies are equal.

The simplified energy method is very simple, gives surprisingly accurate results, but requires heuristics to choose the approximated deformed shape.

I have personally implemented a partial version of 2.9 using Hermite polynomials of degree 3 (basically a simplified FEM formulation). I say partially because I only treated the simply supported case. The advantage of this method is that, once implemented it will treat all frame cases and can be made as accurate as desired, by simply increasing the number of divisions of the element. The disadvantage is that it requires the calculation of the first eigenvalue of a matrix, which can be tricky if you don't have access to a linear algebra package.

In any case, read the 3 sections I mentioned (it's about 10-15 pages in total IIRC) and you can choose whatever method fits your needs best.

If you choose to go with the Hermite polynomial route let me know and I'll find some reference to that.