The constrained buckling problem of geometrically imperfect beams: a mathematical approach for the determination of the critical instability points

The constrained buckling problem of geometrically imperfect beams with intermediate unilateral supports is studied in the present paper. The proposed methodology offers the ability to calculate analytically the critical loads and the buckling shape of beams with arbitrary initial geometric imperfections, for a variety of different initial contact conditions in the framework of elastic stability theory. The proposed mathematical approach is based on the formulation of the equilibrium equations in the deformed position, in which the function of the unilateral constraints is appropriately taken into account. The analytical solution is obtained after the splitting of the initial constrained non-homogeneous boundary value problem (BVP) into constrained subproblems and the utilization of a classical mathematical theorem from the field of ordinary non-homogeneous BVPs. The implementation of the presented technique is demonstrated through characteristic examples. In order to validate the proposed mathematical method, the obtained results are compared with the respective numerical ones. The latter are obtained through the utilization of geometric nonlinear finite element analysis. The paper ends with the presentation of an investigation on the variation of the critical load with respect to different positions of the unilateral constraints.