Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section
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Applied Mathematics and Mechanics (English Edition) https://doi.org/10.1007/s10483-020-2670-6
Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section∗ Yong HUANG† School of Mathematics and Big Data, Foshan University, Foshan 528000, Guangdong Province, China (Received Mar. 30, 2020 / Revised Jul. 27, 2020)
Abstract The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified. Key words higher-order beam theory, circular cross-section, bi-directional functionally graded (FG) beam, bending analysis, free vibration Chinese Library Classification O321 2010 Mathematics Subject Classification
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70J10, 74K10, 74H45
Introduction
Compared with traditional laminated materials, functionally graded (FG) materials can significantly reduce the risk for delamination failures. As a result, many kinds of structures fabricated by FG materials, including beams, plates, and shells, have been used in many different fields of engineering, and have been widely studied by various theories[1–10] . If a Timoshenko or higher-order beam model is adopted to investigate the free vibrations of axially FG beams, two coupled governing differential equations with variable coefficients will be derived, which makes it difficult to obtain the exact solutions due to the arbitrary gradient changes. Many researchers have introduced different numerical methods to solve such ∗ Citation: HUANG, Y. Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section. Applied Mathematics and Mechanics (English Edition) (2020) https://doi.org/10.1007/s10483-020-2670-6 † Corresponding author, E-mail: [email protected] Project supported by the Natural Science Foundation of Guangdong Province of China (No. 2018A030313258) ©The Author(s) 2020
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Yong HUANG
problems. Based on the finite element method, Shahba et al.[11] discussed the free vibrations of arbitrarily tapered and axially FG Timoshenko beams under various boundary conditions. Huang et
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