Моделирование потери устойчивости тяжёлого стержня на жёстком основании

Abstract

Исследуется устойчивость упругого весомого стержня, лежащего на абсолютно жёстком основании. Найдена критическая сила, при которой горизонтальный стержень, лежащий на жёстком основании, теряет устойчивость прямолинейной формы равновесия и отрывается от жёсткого основания полностью или только частью свой длины. Предварительно рассмотрен невесомый стержень с грузом посредине. Задача решается в линеаризованной постановке. Для определения прогибов применяется изопериметрическое условие, выражающее неизменность длины стержня при изгибе. Stability of the ponderable elastic column lying on an absolutely rigid basis is investigated. The ends of the column are closed up and do not turn at a core bend. The column contracts by longitudinal force. Previously a weightless column with freight in the middle was considered. Research is conducted at rather small values of deflections of the column. Curvature of a curved axis of the column is accepted equal by the second derivative of function of deflections. The problem is solved in the linearized statement. For definition of deflections of a curved axis of the column the linear differential equation of the fourth order is used. The nontrivial private solution of the differential equation under the boundary conditions corresponding to fixing conditions is only with an accuracy of indefinite coefficient. The isoperimetric condition expressing the column length invariance at a bend is applied to determination of this coefficient. The critical force with which the column loses stability is found. Cases of a full and partial bend of the column are considered. For determination of critical force, besides the condition of the existence of the nontrivial solution of the differential equation of a bend, the law of conservation of mechanical energy is applied. It is established that the critical force and configuration of a curved axis of the column depends not only on length of the column and geometry of cross section, but also on column material density.