МЕТОДИКА РАСЧЕТА СФЕРИЧЕСКИХ КУПОЛОВ НА ПРОЧНОСТЬ И УСТОЙЧИВОСТЬ
Сведения об авторах
- Санкт-Петербургский государственный архитектурно-строительный университет, Санкт-Петербург, Российская Федерация
- Санкт-Петербургский государственный архитектурно-строительный университет, Санкт-Петербург, Российская Федерация
Аннотация
Рассматривается расширение области применения разработанных авторами моделей, алгоритмов, программного обеспечения и методик исследования оболочечных конструкций на сферические оболочки (купола). Для данного вида конструкций предложен способ обхода особенности в вершине купола за счет выбора модифицированных аппроксимирующих функций. Математическая модель является геометрически нелинейной, учитывает поперечные сдвиги и представлена в виде функционала полной потенциальной энергии деформации. Для сведения вариационной задачи к решению системы алгебраических уравнений применялся метод Ритца. Полученная система решается методом продолжения решения по наилучшему параметру с адаптивным выбором сетки. Алгоритм реализован в среде аналитических вычислений Maple. Выполнен расчет стального купола при разных способах закрепления контура, получены значения критической нагрузки потери устойчивости и нагрузки потери прочности. Построены график зависимости «нагрузка – прогиб» и поля прогибов в докритической и закритической стадиях. Поля показаны в локальной и глобальной декартовой системах координат. Продемонстрирована сходимость метода Ритца по значению критической нагрузки. Выполнена верификация методики путем сравнения решений тестовой задачи с известным решением, полученным Э.И. Григолюком и Е.А. Лопанщиным. Результаты сравнения позволяют судить о достоверности получаемых данных. Выявлено, что для рассматриваемого купола потеря прочности наступает значительно раньше потери устойчивости, в связи с чем может быть рекомендовано для ее проектирования выбрать марку стали с более высоким значением предела текучести. Шарнирно-неподвижное закрепление в данном случае дает более высокое значение предельно допустимой нагрузки.
Ключевые слова
сферические оболочкикуполаустойчивостьпрочностьматематическая модельметод Ритцакритическая нагрузка
METHOD FOR CALCULATING SPHERICAL DOMES FOR STRENGTH AND BUCKLING
The paper considers new applications of the models, algorithms, software and methods developed by the authors to study shell structures of spherical shells (domes). For this type of structures, a method has been proposed to bypass the singularity at the top of the dome by choosing modified approximating functions. The mathematical model is geometrically nonlinear; it takes into account transverse shears, and is presented as a functional of the total potential strain energy. To reduce the variational problem to solving a system of algebraic equations, the Ritz method was used. The resulting system is solved by the method of continuing the solution using the best parameter with an adaptive mesh selection. The algorithm is implemented in the Maple analytical computing environment. A steel dome was estimated using different methods of border fixing, the values of the critical buckling load and the limit stress load were obtained. A graph of the load – deflection relationship and the deflection fields in the subcritical and supercritical stages were constructed. Fields are shown in the local and global Cartesian coordinate systems. The convergence of the Ritz method in terms of the critical load value is demonstrated. The methodology was verified by comparing the solution to the test problem with the known solution obtained by E.I. Grigolyuk and E.A. Lopanitsyn. The comparison results demonstrate the reliability of the data obtained. It was revealed that for the dome under consideration, the loss of strength occurs much earlier than the buckling, and therefore it can be recommended to select a steel grade with a higher yield strength for its design. A simply support border condition in this case gives a higher value of the maximum permissible load.
Keywords
spherical shellsdomesbuckling strengthmathematical modelRitz methodcritical load
Литература
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