Vibration Analysis of Functionally Graded Euler-Bernoulli Beams under Uniform Thermal Loading by Using Differential Quadrature Method

Year 2025, Volume: 1 Issue: 2, 35 - 50, 28.11.2025

Mustafa Tolga Yavuz , Caglar Uyulan , Sercan Acarer , İbrahim Ozkol

Abstract

In the last few decades, the use of functionally graded materials has become increasingly popular in various engineering applications exposed to high temperatures due to their thermal resistance, smooth transition in their mechanical properties, and superior ability to minimize thermal stresses. In this study, the dynamic characteristics of beam structures with uniform cross sections are investigated for different boundary conditions, slenderness ratios, temperature-dependent material properties, composition of the structure, and temperatures. Applying Hamilton’s principle, the governing equation is derived for the Euler-Bernoulli beam structures made of functionally graded materials. Then, the governing differential equation is solved by employing the generalized differential quadrature method. In this numerical solution technique, classical boundary conditions and the equation of vibration motion are transformed into a set of linear algebraic equations stated in orthogonal matrix form. Ultimately, the obtained numerical results are presented in the relevant figures and tables to show the influence of operating environment and boundary conditions on the dynamic behavior of the beam, in addition to the influence of the composition of the ceramic-metal mixtures, and interactions of other structural design parameters with each other. The findings show that increasing temperature and gradiation index significantly reduce the natural frequencies due to thermal softening effects, while changes in boundary conditions and slenderness ratios strongly influence the overall vibration response. These results highlight the importance of incorporating temperature-dependent material behavior and gradient composition in the accurate prediction and optimization of the dynamic performance of functionally graded beam structures.

Keywords

Differential Quadrature Method , Euler-Bernoulli Beam , Free Vibration Analysis , Functionally Graded Beam , Thermoelastic Beam

Ethical Statement

This article has no conflicts of interest with any individual or institution. This article does not require ethics committee approval.

Thanks

This study was produced from the Doctorate’s Thesis conducted by Mustafa Tolga YAVUZ under the supervision of Prof. Dr. Ibrahim OZKOL. The authors would like to thank Prof. Dr. Ibrahim OZKOL from the Department of Aeronautical Engineering in Istanbul Technical University for suggestions and discussions.

Isıl Yükleme Altında Fonksiyonel Olarak Derecelendirilmiş Euler-Bernoulli Kirişlerinin Diferansiyel Kareleme Yöntemi Kullanılarak Dinamik Analizi

Abstract

Son yıllarda gerek çeşitli mühendislik dalları gerekse havacılık sektöründe yüksek sıcaklığa maruz kalan yapılarda fonksiyonel derecelendirilmiş malzeme ile kaplı yapıların kullanımında artış görülmektedir. Bu malzemelerin ısıl gerilmeleri azaltmadaki başarısı, mekanik özellikler arasında yumuşak geçiş sunması ve ısıl şoklara karşı direnç sağlaması gibi sebepler bunda etkili olmuştur. Bu amaçla bu çalışmada, düzgün kesit alanına sahip Euler-Bernoulli kiriş teorisine göre modellenen kiriş yapıların yüksek sıcaklığa maruz kalması halinde dinamik karakteristikleri incelenmiştir. Ayrıca, farklı geometrik sınır şartları, çeşitli malzeme bileşimleri ve narinlik oranı gibi kiriş yapının titreşim cevabını etkileyen yapısal etmenlerin de frekans ve mod şekillerini nasıl değiştirdiği incelenmektedir. Titreşim hareketini açıklayan yönetici denklem, Hamilton prensibinden faydalanarak çıkarılmış ve diferansiyel kareleme yönteminden faydalanarak çözülmüştür. Elde edilen sonuçlar, artan sıcaklık ve seramik malzeme oranın frekans değerlerini azalttığını göstermektedir. Bu sonuçlar yüksek sıcaklığa maruz kalan kiriş yapılarında doğru malzeme oranının, geometrik sınır şartının ve narinlik oranının seçilmesine yardımcı olacağı için önemlidir.

Keywords

Diferansiyel Kareleme Yöntemi , Euler-Bernoulli Kirişi , Serbest Titreşim Analizi , Fonksiyonel Derecelendirilmiş Kiriş , Termoelastik Kiriş

Thanks

Prof.Dr.Ibrahim Ozkol'a bu yayının hazırlanmasına sağladığı destekten dolayı teşekkür ederiz.

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