On the generalized Clausen problem

Abstract

We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load 𝐹 of the column and the length 𝐿, find crosssectional area 𝐴, such that the volume 𝑊 of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.