Eur. J. Mech. A/Solids 18 (1999) 903–913  1999 Èditions scientifiques et médicales Elsevier SAS. All rights reserved On the optimal shape of a Pflüger column Teodor M. Atanackovic, Srboljub S. Simic Faculty of Engineering, University of Novi Sad, Trg D. Obradovica 6, 21000 Novi Sad, Yugoslavia (Received 15 July 1998; revised and accepted 21 December 1998) Abstract – The optimal shape of a Pflüger column is determined by using Pontryagin’s maximum principle. It is shown that the boundary value problem relevant for determining the optimal distribution of material (i.e. cross-sectional area function) along the column axis has simple eigenvalue. Necessary conditions for local extremum of column volume are reduced to a boundary-value problem for a single second order nonlinear differential equation. We examined singular points of this equation and formulated extremal complementary variational principles for it. The optimal cross-sectional area function is obtained by numerical integration and by Ritz method. The error of the analytical approximate solution obtained by Ritz method is also estimated.  1999 Èditions scientifiques et médicales Elsevier SAS optimal shape / Pontryagin’s principle 1. Introduction The problem of determining the shape of the column of greatest efficiency is indeed an old one. For simply supported column loaded by concentrated forces at its ends it was formulated by Lagrange (see (Cox, 1992), for example). The solution of the problem was obtained by many authors. We mention the work of Keller (1960) that became classic. Later Tadjbakhsh and Keller (1962) treated other boundary conditions by the method developed in Keller (1960). It was, for the first time, pointed out by Olhoff and Rasmussen (1977) that the change of boundary conditions could lead to multiple eigenvalues of the equation relevant for determining the optimal cross-sectional area function. In this case one is faced with a so called multiple eigenvalue optimization problem (Seyranian et al., 1994; Seiranyan, 1995) and the optimization problem becomes more involved. In this work we propose to study the problem of determining the Pflüger column of greatest efficiency. A Pflüger column is a simply supported column loaded by uniformly distributed follower type of load (see Pflüger, 1975). The uniformly distributed follower load is a non-conservative load. It is interesting, however, that despite the non-conservative character of the load the stability analysis for Pflüger column could be based on static (Euler) method. In the first part of this paper we shall formulate the nonlinear system of equations describing the equilibrium configuration of a column. On the basis of these equations we shall derive the linear boundary value problem that determines the stability boundary. By analyzing the multiplicity of the lowest eigenvalue of this equation we shall confirm that it represents the bifurcation point of the nonlinear system. Then we shall formulate the optimization problem. The necessary conditions for the minimum of volume will be reduced to a single nonlinear differential equation. The rest of the paper is devoted to analysis of this equation, both numerical and variational. As results of this analysis we shall determine the optimal distribution of the material along the column axis and show that the optimal column has 19% smaller volume then the corresponding column with constant cross section.