Investigation of buckling of rectilinear beams with additional constraint at an arbitrary internal point

Abstract

Summary The problems of stability and instability (buckling) of compressed rectilinear beams are considered. The beams are treated as one-dimensional elastic bodies possessing stiffnesses of two kinds: the stiffness with respect to extension–compression and the stiffness with respect to bending. The ends of beams are hinged, but along with traditional setting of a problem, characterized by one movable hinge and a given compressive load, the problem with both immovable end hinges fixing the ends of a beam in a compressed state, is considered. In addition to the ends, the beams are assumed to be constrained in a certain way at some internal point located at an arbitrary specified distance from one of the ends. These constraints are supposed to prohibit either lateral displacements with free rotation (a hinge), or rotation with free lateral displacements. Analytical solutions for all four types of problems regarding the onset of instability are obtained and examined with respect to the position of additional constraint. It turned out that results for the cases of one movable end hinge with given compressive load and of fixed end hinges, coincide. In the case of additional hinge, its middle position provides maximal value of the critical force, whereas its limiting end position corresponds to its minimal value. As for the case of additional constraint prohibiting rotation, its middle position corresponds to minimal value of the critical force, whereas its maximal value is attained at a certain internal point of a beam located approximately at the distance of one-sixth of its length from one of the ends.