Abstract
The paper considers the loss of structure stability as one of the most important marginal states. Though this type of limit conditions has been closely studied, there is still no agreement on their causes, hence, there is no uniform approach to defining the criteria that determine the critical state. The most popular criteria known in the structural analysis and the theory of structural stability are energy criteria of stability in the form of Timoshenko or Bryan. In the first case the analysis covers total work of all forces affecting the system at the moment of collapse. In the second the analysis deals with the system internal energy, which allows us to solve the problems considering thermal and similar effects. In spite of the simplicity of the first approach and general character of the second one, it is hardly possible to assert that they can be sufficient to cover the total scope of stability problems that may arise in engineering. The criterion of critical energy critical level permits us to formulate and solve stability problems without any limitations related with the smallness of displacements or the types of impacts affecting a system, so it can be applied to formulate boundary conditions. For better understanding of the said criteria and their illustration the paper presents some simple problems in the form of systems with lumped parameters. Structure stability criteria are studied as in case of the systems with one degree of freedom. The analysis covers the statements and solutions of stability problems in the form of Timoshenko, Bryan and the criterion of energy critical levels. These approaches are reviewed in terms of their advantages and disadvantages using the example of the model of a structure with one degree of freedom.
Highlights
Ключевые слова: устойчивость конструкций, системы с распределенными параметрами, критическая сила, критерий устойчивости в форме Тимошенко, критерий устойчивости в форме Брайана, критерий критических уровней энергии
The analysis covers the statements and solutions of stability problems in the form of Timoshenko, Bryan and the criterion of energy critical levels. These approaches are reviewed in terms of their advantages and disadvantages using the example of the model of a structure with one degree of freedom
Variational Criteria for Critical Levels of Internal Energy of a Deformable Solids//Applied Mechanics and Materials. – Vols. 578-579 (2014), pp. 15841587
Summary
Мы пренебрегаем рассмотрением стадии запасания энергии, а рассматриваем систему в момент потери устойчивости, когда сжимающая сила достигает критического значения. При этом внешняя сила Fc совершает возможную работу при переходе системы в новое равновесное состояние. Где перемещение силы можно представить через бесконечно малый угол поворота в виде:. При этом действительную работу внутренних сил, или потенциальную энергию, запасаемую упругими пружинами при переходе системы в новое положение, можно записать как. Полную энергию системы можно представить как сумму работы внешних сил и внутренних усилий. Откуда получим выражение критической силы для системы, представленной на рисунке 1: Fc rh . Величина критической силы полностью совпадает с выражениями, полученными в [1, 2] для классической модели. Если записать равенство работ внешних сил и внутренних усилий к моменту достижения критических продольных перемещений, получим
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