Stability and optimal shape of Pflüger micro/nano beam

Abstract This paper deals with optimal shapes against buckling of an elastic nonlocal small-scale Pfluger beams with Eringen’s model for constitutive bending curvature relationship. By use of the Pontryagin’s maximum principle the optimality condition in form of a depressed quartic equation is obtained. The shape of the lightest (having the smallest volume) simply supported beam that will support given uniformly distributed follower type of load and axial compressive force of constant intensity without buckling, is determined numerically. A special attention is paid to the influence of the characteristic small length scale parameter of the nonlocal constitutive law to both critical load and optimal shape of the analyzed beams. For the case when distributed follower type of load is zero, our results reduce to those obtained recently for compressed nonlocal beam. Also the post buckling shape of the optimally shaped rod is studied numerically.