The results of the analysis problem are obtained by Spectral method. Necessary conditions for the maximum value of the first eigenvalue corresponding to given column volume are established to determine the optimal distribution of cross-sectional area along the column axis. | TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ Tập 48, số 5, 2010 Tr. 1-12 ANALYZING AND OPTIMIZING OF A PFLUGER COLUMN TRAN DUC TRUNG, BUI HAI LE, CAO QUOC HUONG ABSTRACT The optimal shape of a Pfluger column is determined by using Pontryagin’s maximum principle (PMP). The governing equation of the problem is reduced to a boundary-value problem for a single second order nonlinear differential equation. The results of the analysis problem are obtained by Spectral method. Necessary conditions for the maximum value of the first eigenvalue corresponding to given column volume are established to determine the optimal distribution of cross-sectional area along the column axis. Keywords: optimal shape; Pontryagin’s maximum principle. 1. INTRODUCTION The problem of determining the shape of a column that is the strongest against buckling is an important engineering one. The PMP has been widely used in finding out the optimal shape of the above-mentioned problem. Tran and Nguyen [12] used the PMP to study the optimal shape of a column loaded by an axially concentrated force. Szymczak [11] considered the problem of extreme critical conservative loads of torsional buckling for axially compressed thin walled columns with variable, within given limits, bisymmetric I cross-section basing on the PMP. Atanackovic and Simic [4] determined the optimal shape of a Pfluger column using the PMP, numerical integration and Ritz method. Glavardanov and Atanackovic [9] formulated and solved the problem of determining the shape of an elastic rod stable against buckling and having minimal volume, the rod was loaded by a concentrated force and a couple at its ends, the PMP was used to determine the optimal shape of the rod. Atanackovic and Novakovic [3] used the PMP to determine the optimal shape of an elastic compressed column on elastic, Winkler type foundation. The optimality conditions for the case of bimodal optimization were derived. The optimal cross-sectional area function was determined from the .
