Some Conventional and Unconventional Educational Column Stability Problems

Two interesting problems are considered for enriching the curriculum of the Strength of Materials course, in the light of recently developed functionally graded materials (FGMs), characterized with the smooth variation of the elastic modulus. These are problems associated with buckling of columns with variable flexural rigidity along the axis of the column. A simple semi-inverse method is proposed for determining closed-form solutions of axially inhomogeneous columns. In order for the presentation to be given in one package, the conventional problems are also recapitulated along with the novel ones. The main approach adopted here is the use of the second-order differential equation, instead of the fourth-order one, obtained by integrating twice and having a physical meaning, since it could be derived based on moment equilibrium. This is essentially the same idea as utilized in the textbook by Timoshenko and Gere;¹the difference is that this paper develops a semi-inverse formulation.