Application of the isoperimetric condition in problems of buckling for a rod

When studying buckling of a rectilinear elastic rod, the so-called linearized differential equation in which the first derivative of deflection is neglected in the expression for the curvature is considered together with the exact differential equation. The linearized equation is a linear differential equation of the second order with constant coefficients; it can be reduced to a uniform differential equation of the fourth order. Conditions under which it is possible to solve the boundary value problem for this equation allow one to obtain an exact value of the critical force. In this case, the solution of the boundary value problem is defined only with an accuracy of up to an unknown coefficient which cannot be determined from the boundary conditions. This unknown coefficient can be found from an additional condition expressing the invariance of the rod length.
In this work, buckling of a rectilinear rod squeezed by a longitudinal force is investigated. The problem is solved in the Euler formulation, i.e., the possibility of the existence of curved balance forms adjacent to the rectilinear form is investigated. Solutions of the boundary value problem for the linearized differential equation of the curved rod axis are constructed using an additional condition of the invariance of the rod length. Various ways of fixing of the rod ends by means of a rigid and elastic supports are considered. The influence of the elasticity coefficient of the support on the buckling process and configuration of a curved axis of the rod is investigated.