Semi-analytical solutions for optimal design of columns based on Hencky bar-chain model

Abstract

This paper is concerned with the shape optimization problem of columns for a given volume and length against buckling by using the discrete link-spring model or the so-called Hencky bar-chain model (HBM). This discrete beam model comprises a finite number of rigid segments connected by frictionless hinges and rotational springs. In particular, the rotational spring stiffness of HBM is a function of the square of cross-sectional area of columns with regular polygonal or circular cross-sectional shape. Therefore, the design of optimal rotational spring stiffnesses of a HBM allows one to obtain the optimal shape of a column provided that the assumed number of springs is sufficiently large. The present formulation of HBM for column optimization is prompted by some discrepancies in the volume calculations and the specification of the spring stiffness at the clamped end in Krishna and Ram (2007) discrete link-spring model formulation. By using the correct formulation and the semi-analytical method proposed by Krishna and Ram (2007), we determine the optimal shape of clamped-free, pinned-pinned, clamped-spring-supported columns. In addition, we extend the semi-analytical method to optimize the shape of clamped-free columns under distributed loads. Also presented herein are exact buckling solutions for the uniform HBM under axial load and selfweight as well as the non-uniform HBM under axial load with a specific class of spring stiffnesses.