Issues with linear buckling analysis following ASME PTB-2022 example problem

Yeah, it tends to show something like this as the 0 buckling factor, and it can be ignored as long as you get correct/reasonable further buckling factors. Check the solution from the first (preload) step too.

You could also try changing the mesh. Not just refining it, but also using different element types. Currently, you have S3 and S4 elements. The Abaqus reference mentions S4R, this can be changed after right-clicking on each part and choosing Edit. You may try to generate a full quad mesh too (likely the best way is to add a Shell Gmsh item in the Geometry tab and set the algorithm to Quasi-structured quad).

I would also have a closer look at the connections - tie constraints. They look good at first glance, but it’s always recommended to run a frequency analysis first and see if connections work as expected everywhere. It’s better to use a continuous mesh instead to prevent stress concentrations and other undesirable behavior, but tie constraints often prove necessary.