The exact solutions for the stability analysis of a one-step bar of va
rying cross-section subjected to concentrated and distributed axial lo
ads are found first. Then the exact solution of that bar is used to de
rive the eigenvalue equation of a multi-step bar of varying cross-sect
ion subjected to more complicated loads by using transition matrices.
All of the exact solutions are expressed in terms of Bessel's function
s and super-geometric series.