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.
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