This study deals with the buckling of inhomogeneous columns whose flexural rigidity
is graded along the axial direction, being simply supported at one end and clamped
at the other. The buckling load, mode shape, and grading profile are jointly analyzed
under the requirement that the mode shape be of polynomial form. Satisfaction of the
governing differential equation leads to a set of polynomial equations in the coefficients
of the shape polynomial. This system of polynomial equations is amenable to solution
in terms of an appropriately chosen algebraic number. Thus we find new solutions in
analytic form associated to polynomials ranging from order four to seven. The method,
introduced by Eisenberger and Elishakoff (2017) for a column simply supported at both
ends, can in principle provide polynomial solutions or any order; in addition, solutions
associated to lower-degree polynomials reappear in higher degrees. It is remarkable
that multiple solutions can appear for the postulated polynomial of a certain degree.
