  In view of the fundamental distinction between the force-controlled model and
the displacement-controlled model in buckling problems of structures and the
complexity of the asymptotic post-buckling analysis traditionally based on the
force-controlled model, alternatively, we provide a straightforward theoretical
procedure of the energy method for the buckling and post-buckling analysis
completely based on the displacement-controlled model. The Euler buckling
behavior is analytically tackled as a static displacement-controlled process as
an example of the theoretical procedure, where no force potential energy
component but compression and bending strain energy components are considered
precisely at the deformed configuration for the total potential energy of the
beams. Analytical solutions to the potential energy, structural deformation,
internal forces and their critical results are obtained in closed-form for the
beams with six typical boundary conditions. We find that these beams have only
one unique but universal normalized potential energy surface depending on two
dimensionless quantities. The valley bottom pathways on the potential energy
surface show that the critical buckling state is not only a bifurcation point
but also a valley-ridge inflection point, and the energy increases
quadratically before the point and increases linearly with a slope of 2 beyond
the point. The axial forces are gradually increasing during post-buckling, but
are almost constant, indicating a symmetric neutral bifurcation buckling. Our
theoretical expressions generally agree with the counterparts of the
force-controlled model but exactly indicate somewhat differences. The present
analytical procedure is basically useful to investigate any
displacement-controlled buckling problems of beams, plates and shells.
