The critical load of a beam refers to the load at which the beam becomes unstable and buckles. It is an important parameter to determine the structural integrity and stability of the beam. The Euler-Bernoulli Beam equation is commonly used to analyze the behavior of beams under different loading conditions.

To compute the critical load of the Euler-Bernoulli Beam equation using the finite element method (FEM) and the eigenvalue method, there are several factors that could contribute to the discrepancy between your computed values and the analytical values. Here are a few possible reasons and suggestions to address them:

By carefully examining and addressing these factors, you should be able to improve the accuracy of your computed critical load values. It is also helpful to consult with experienced researchers or experts in the field of structural analysis to get additional insights and guidance.

I will suggest: 1) assign realistic values for the problem first, 2) make sure that it is a beam (cross-section much smaller than the length), 3) are you model the beam as a 3D solid using 3D FEA? If yes, please make sure that the dofs are fixed at the end surfaces, and the mesh is symmetric with respect to the neutral axis is your beam has a symmetric cross-section.

Here are a few potential reasons why you might not be obtaining an accurate value compared to the analytical solution:

Please also make sure that the beam behaves linearly before buckling. Otherwise, I suggest applying preload that is close to the critical load and then performing the BUCKLE analysis.