Calculation of The Stability of the Form of Equilibrium of Discrete Systems = Розрахунок стійкості форми рівноваги дискретних систем

Abstract

The paper presents an algorithm for calculating the stability of the equilibrium form of the first kind of compressed discrete systems by the displacements method in combination with the methods of iterations and bisection. The use of the methods makes it possible to effectively determine the minimum critical stress or strain at the first bifurcation and their corresponding form of stability loss, both for statically determined and statically undetermined systems. This approach, using matrix forms, makes it possible to signifi-cantly simplify the calculations of the analytical condition for the stability loss of compressed discrete systems (the stability loss equation), which has high orders, as well as to construct the form of stability loss corresponding to a critical load, that is, to solve the problem of loss of equilibrium stability. The calculation actually leads to solving a nonlinear transcendental equation, which is the equation of stability loss. The difficulty lies in the absence of an analytical solution of such an equation due to the presence of com-plex of Zhukovsky functions, which have transcendental functions in their structure. Such solution can be performed only with the use of numerical methods. This algorithm for calculating the loss of equilibrium of the first kind of compressed discrete systems by displacement in combination with the methods of iteration and bisection is implemented in the software complex "Persist" for PC in Windows OS. The program was approbated and implemented in the educational process at the Department of Structural and Theoret-ical Mechanics of Poltava National Technical Yuri Kondratyuk University during the training of specialists in engineering specialties.

Наведено алгоритм обчислення стійкості рівноважної форми першого виду стиснутих дискретних систем методом переміщень у поєднанні з методами ітерації і бісекції. Використання методів дає можливість для ефективного визначення мінімальної критичної напруги або деформації при першій біфуркації та їх відповідної форми втрати стійкості, як для статично визначених, так і для статично невизначених систем.