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)5314110.22363/1815-5235-2026-22-4-317-330LKNCQZAnalytical and numerical methods of analysis of structuresАналитические и численные методы расчета конструкцийResearch ArticleFinite Element Modeling of Dynamic Deformation of Bidirectional Functionally Graded BeamsКонечноэлементное моделирование динамического деформирования двунаправенных функционально градиентных балокhttps://orcid.org/0000-0001-7799-75211961-6243VassermanIgor N.ВассерманИгорь Николаевич

Candidate of Physical and Mathematical Sciences, Researcher

кандидат физико-математических наук, научный сотрудник

igorw@icmm.ruhttps://orcid.org/0000-0002-2842-75113231-7996GlotIrina O.ГлотИрина Олеговна

Candidate of Physical and Mathematical Sciences, Senior Researcher

кандидат физико-математических наук, старший научный сотрудник

glot@icmm.ruhttps://orcid.org/0000-0001-8673-642X4809-4776ShardakovIgor N.ШардаковИгорь Николаевич

Doctor of Physical and Mathematical Sciences, Chief Researcher

доктор физико-математических наук, главный научный сотрудник

shardakov@icmm.ruInstitute of Continuous Media Mechanics, Ural Branch of the Russian Academy of Sciences (UB RAS)Институт механики сплошных сред УрО РАН29102026224VOL 22, NO4 (2025)ТОМ 22, №4 (2025)31733029102026Copyright ©; 2026, Vasserman I.N., Glot I.O., Shardakov I.N.Copyright ©; 2026, Вассерман И.Н., Глот И.О., Шардаков И.Н.2026Vasserman I.N., Glot I.O., Shardakov I.N.Вассерман И.Н., Глот И.О., Шардаков И.Н.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/structural-mechanics/article/view/53141

This article presents a finite element algorithm for the numerical solution of boundary value problems for the deformation state of elastic beams made of functionally graded materials. The presented algorithm assumes that the density and elastic properties of the beam depend on the coordinates of the cross-section and on the coordinate along the longitudinal axis. The presented formulation allows to solve static and dynamic problems. The Euler-Bernoulli model and the third-order Reddy model are used to describe the deformation state of the beam. The beam finite element utilizes a Hermitian approximation. Using Hamilton's variational principle, the beam equation of motion is obtained in matrix form. The numerical implementation of the resulting finite element algorithm is carried out in Python using the Numpy and Scipy libraries. As an example, a numerical analysis of the solution to the problem of the natural frequencies of vibrations of a bidirectional functionally graded cantilever beam is performed. The numerical convergence and reliability of the obtained solutions are demonstrated. The possibility of coupling of the longitudinal and transverse vibrations of a beam is demonstrated for certain distributions of the physical properties of the material in the beam’s cross-section. It is shown that distributing the material properties along the longitudinal axis improves the beam’s dynamic properties.

Представлен конечно-элементный алгоритм численного решения краевых задач о деформационном состоянии упругих балок из функционально градиентных материалов. Предполагается, что плотность и упругие свойства балки зависят от координат поперечного сечения и от координаты вдоль продольной оси. Данная постановка допускает возможность решения статических и динамических задач. Для описания деформированного состояния балки использована модель Эйлера - Бернулли и модель третьего порядка Редди. В балочном конечном элементе применена эрмитова аппроксимация. С помощью вариационного принципа Гамильтона получено уравнение движения балки в матричном виде. Численная реализация полученного конечно-элементного алгоритма осуществлена на языке Python с использованием библиотек Numpy и Scipy. В качестве примера осуществлен численный анализ решения задачи о частотах собственных колебаний двунаправленной функционально-градиентной консольной балки. Продемонстрирована численная сходимость полученных решений и их достоверность. Показана возможность взаимосвязи продольных колебаний балки с поперечными при некоторых распределениях физических свойств материала в поперечном сечении балки. Показано, что распределение свойств материала вдоль продольной оси позволяет улучшить динамические свойства балки.

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