# Calculate the Euler buckling load of a column using the finite element method

I am trying to find the buckling load of following column:

I started from the system of equations $P = (K + K_g(N))U$, where $K$ is the stiffness matrix of a single 2D frame element and $K_g$ is the geometric stiffness matrix of the same element.

The boundary conditions give zero displacements and rotation in the node at the bottom of the column. So I tried to solve the equations for the 2nd part of the column, where the only unknowns are the displacements and the rotation in node 2, with $N = P$. But this gives me the same result as for a standard cantilever beam.

Here's an example on how to calculate the buckling load for a cantilever beam:

I also tried solving the complete system of equations with unknown displacements in node 1 and 2, but without success. And since I need to be able to calculate it by hand, this doesn't seem to be the most efficient way to do it.

I don't know what I'm doing worng. Could someone help me to start in the right way?

• you need to work at clarifying what you are asking. I don't get it. Whats this to do with finite elements? Jan 20, 2018 at 17:46
• I'm sorry for the confusion in my question. I added an example of a simple cantilever beam and tried to explain my question more clearly. I need to solve it for my courses about the finite element method, that's why I added the tag.
– user14605
Jan 20, 2018 at 20:22