I think the root of this problem is that for Eulerian elements the stiffness is significantly changing during the analysis itself. And as the alpha parameter is multiplied by the stiffness matrix, it has no meaning to calculate it based on a certain stiffness that is changing and using it with the changing stiffness won't represent reality.

On the other hand, the beta is multiplied by the mass matrix which is not changing and could simply be multiplied by a constant.

Please be carefull as some references and some software packages mix between alpha and beta in terminology and some may call beta delta

Thanks a lot for your answer

 You are always welcomed.