Postbuckling behavior of compressed bars with nonlinearly elastic supports

Supercritical bending of a rod compressed by a longitudinal force and having a non-linearly elastic resistance to rotation of its axis in hinged supports is considered. Hooke's law is adopted for the rod material. The deflection is presented as a trigonometric series, in which only the first three terms are left. Through this row, the angle of inclination of the axis of the rod is expressed. It is chosen as the desired function, with respect to which a resolving nonlinear integro-differential equation is obtained. To simplify it, the case is considered when the bent axis of the rod is a gentle curve. It is revealed that the usual approach, in which the compressive force is given, leads to a trivial solution with zero values for the coefficients of this series. Therefore, the first coefficient of the series, through which the deflection is represented, was chosen as the initial parameter. The equation containing the remaining desired coefficients and load was solved by the simple collocation method, since the use of the predetermined collocation method is impossible due to the nonlinearity of the resulting system of equations. The calculation results were analyzed depending on the choice of collocation points, which showed a small difference between the obtained solutions (the quadratic discrepancy between the calculated values did not exceed one percent, which indicates sufficient accuracy of the deflection approximation in the form proposed in the work). In contrast to the usual approach used in numerical methods, here it is not necessary to enter any parameter of the loading process (such as the length of the arc in the arc method) to pass the limit points. An approach is presented for an approximate calculation of the compliance of the considered rods in compression, taking into account their supercritical bending. The results of numerical studies of the supercritical behavior of rods for various geometric and mechanical characteristics of rods and supports are presented, and dependences between various parameters of the stress-strain state are obtained. Their analysis made it possible to reveal a number of features in the behavior of the considered rods depending on the properties of the nonlinear elastic resistance of the supports. In particular, it was found that under the considered parameters of the system under consideration, the compression force and the angle of inclination of the rod on the support are related by a relationship close to linear, the compliance of the compressible rod can increase by tens or more times depending on the stiffness of the springs, the rod can lose stability by snap at some values parameters of nonlinear elastic supports.