Structural Mechanics of Engineering Constructions and BuildingsStructural Mechanics of Engineering Constructions and BuildingsСтроительная механика инженерных конструкций и сооружений1815-52352587-8700Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)4522010.22363/1815-5235-2025-21-2-138-154NQBZIKAnalysis of thin elastic shellsРасчет тонких упругих оболочекResearch ArticleSystems of Approximating Functions when Using Variational Methods for Calculating Thin-Walled Building StructuresСистемы аппроксимирующих функций при использовании вариационных методов для расчета тонкостенных строительных конструкцийhttps://orcid.org/0000-0001-7911-40677406-9199KarpovVladimir V.КарповВладимир Васильевич

Doctor of Technical Sciences, Professor of the Department of Information Systems and Technologies

доктор технических наук, профессор кафедры информационных систем и технологий

vvkarpov@lan.spbgasu.ruSaint Petersburg State University of Architecture and Civil EngineeringСанкт-Петербургский государственный архитектурно-строительный университет1107202521213815423072025Copyright ©; 2025, Karpov V.V.Copyright ©; 2025, Карпов В.В.2025Karpov V.V.Карпов В.В.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/structural-mechanics/article/view/45220

The question of the use of approximating functions in the calculation of thinwalled building structures is investigated and the requirements that they must satisfy are analyzed. A rule is formulated that allows one to distinguish between the principal boundary conditions and natural ones. It is shown that the approximating functions must satisfy the principal boundary conditions, while the natural boundary conditions are included in the equilibrium equations and are satisfied automatically when solving a boundary value problem. The accuracy of their fulfillment depends on the accuracy of the solution of the problem itself. An example shows what errors can result from the use of approximating functions that satisfy the specified boundary conditions, but do not satisfy the completeness conditions. Some systems of functions for which the completeness condition in the energy space has been proven are considered. Using the example of Legendre orthogonal polynomials, a technique is given for forming approximating functions that satisfy the specified boundary conditions and the completeness conditions of a system of functions. The efficiency of using the obtained approximating functions in solving boundary value problems using the Galerkin method is shown.

Исследуется вопрос использования аппроксимирующих функций в задачах расчета тонкостенных строительных конструкций и анализируются требования, которым они должны удовлетворять. Сформулировано правило, позволяющее отличить главные краевые условия от естественных. Показано, что аппроксимирующие функции должны удовлетворять главным краевым условиям, а естественные краевые условия входят в уравнения равновесия и выполняются автоматически при решении краевой задачи. Точность их выполнения зависит от точности решения самой задачи. На примере показано, к каким ошибкам может приводить использование аппроксимирующих функций, удовлетворяющих заданным краевым условиям, но не удовлетворяющих условиям полноты. Рассмотрены некоторые системы функций, для которых доказано условие полноты в энергетическом пространстве. На примере ортогональных многочленов Лежандра приводится методика формирования аппроксимирующих функций, удовлетворяющих заданным краевым условиям и условиям полноты системы функций. Показана эффективность использования полученных аппроксимирующих функций при решении краевых задач методом Б. Г. Галеркина.

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