Vibration of four-phase bidirectional functionally graded beams based on trigonometric enriched beam element
Email:
vuthianninh@utc.edu.vn
Từ khóa:
BFG beam, higher-order shear deformation beam theory, free vibration, enriched beam element, finite element formula, natural frequency
Tóm tắt
Improving the convergence rate in finite element formulation plays an important role in studying the behavior of structures. This paper presents an efficient beam element to investigate the free vibration of bidirectional functionally graded beam. The beam is composed of four materials whose properties vary along both the length and thickness directions according to the power function, and these properties are evaluated by Voigt model. The equations of motion are derived using Hamilton’s principle within the framework of the higher-order shear deformation beam theory. A two-node beam element is formulated by enriching the conventional Lagrange and Hermite interpolations with trigonometric functions, leading to rapid convergence. The finite element formulation has been validated through comparison with previously published results, and showing good agreement. The enriched beam element is employed to compute the natural frequencies of bidirectional functionally graded (BFG) beams under different boundary conditions. The influence of the grading indices, slenderness ratio and boundary conditions on the natural frequency is examined in detail and highlightedTài liệu tham khảo
[1]. S. A. Sina, H. M. Navazi, H. Haddadpour, An analytical method for free vibration analysis of functionally graded beams, Materials & Design, 30 (2009) 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
[2]. X. Wang, S. Li, Free vibration analysis of functionally graded material beams based on Levinson beam theory, Applied Mathematics and Mechanics, 37 (2016) 861-878. https://doi.org/10.1007/s10483-016-2094-9
[3]. V. Kahya, M. Turan, Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory, Composites Part B: Engineering, 109 (2017) 108-115. https://doi.org/10.1016/j.compositesb.2016.10.039
[4]. M. Avcar, W. K. M. Mohammed, Free vibration of functionally graded beams resting on Winkler-Pasternak foundation, Arabian Journal of Geosciences, 11 (2018) 232. https://doi.org/10.1007/s12517-018-3579-2
[5]. I. Katili, T. Syahril, A. M. Katili, Static and free vibration analysis of FGM beam based on unified and integrated of Timoshenko’s theory, Composite Structures, 242 (2020) 112130. https://doi.org/10.1016/j.compstruct.2020.112130
[6]. M. Şimşek, Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions, Composite Structures, 133 (2015) 968-978. https://doi.org/10.1016/j.compstruct.2015.08.021
[7]. T. A. Huynh, X. Q. Lieu, J. Lee, NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem, Composite Structures, 160 (2017) 1178-1190. https://doi.org/10.1016/j.compstruct.2016.10.076
[8]. D. K. Nguyen, T. T. Tran, Free vibration of tapered BFGM beams using an efficient shear deformable finite element model, Steel and Composite Structures, 29 (2018) 363-377. https://doi.org/10.12989/scs.2018.29.3.363
[9]. Vu Thi An Ninh, Fundamental frequencies of bidirectional functionally graded sandwich beams partially supported by foundation using different beam theories, Transport and Communications Science Journal, 72 (2021) 362-377. https://doi.org/10.47869/tcsj.72.4.5
[10]. D. K. Nguyen, Q. H. Nguyen, T. T. Tran, V. T. Bui, Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica, 228 (2017) 141-155. https://doi.org/10.1007/s00707-016-1705-3
[11]. P. Ribeiro, Hierarchical finite element analyses of geometrically non-linear vibration of beams and plane frames, Journal of Sound and Vibration, 246 (2001) 225-244. https://doi.org/10.1006/jsvi.2001.3634
[12]. H. Y. Shang, R. D. Machado, J. E. Abdalla Filho, Dynamic analysis of Euler–Bernoulli beam problems using the generalized finite element method, Computers & Structures, 173 (2016) 109-122. https://doi.org/10.1016/j.compstruc.2016.05.019
[13]. Y. S. Hsu, Enriched finite element methods for Timoshenko beam free vibration analysis, Applied Mathematical Modelling, 40 (2016) 7012-7033. https://doi.org/10.1016/j.apm.2016.02.042
[14]. C. I. Le, N. A. T. Le, D. K. Nguyen, Free vibration and buckling of bidirectional functionally graded sandwich beams using an enriched third-order shear deformation beam element, Composite Structures, 261 (2021), 113309. https://doi.org/10.1016/j.compstruct.2020.113309
[15]. H. T. Thai, S. E. Kim, A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates, Composite Structures, 96 (2013) 165-173. https://doi.org/10.1016/j.compstruct.2012.08.025
[16]. M. Géradin, D. J. Rixen, Mechanical vibrations: theory and application to structural dynamics. John Wiley & Sons (2015).
[2]. X. Wang, S. Li, Free vibration analysis of functionally graded material beams based on Levinson beam theory, Applied Mathematics and Mechanics, 37 (2016) 861-878. https://doi.org/10.1007/s10483-016-2094-9
[3]. V. Kahya, M. Turan, Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory, Composites Part B: Engineering, 109 (2017) 108-115. https://doi.org/10.1016/j.compositesb.2016.10.039
[4]. M. Avcar, W. K. M. Mohammed, Free vibration of functionally graded beams resting on Winkler-Pasternak foundation, Arabian Journal of Geosciences, 11 (2018) 232. https://doi.org/10.1007/s12517-018-3579-2
[5]. I. Katili, T. Syahril, A. M. Katili, Static and free vibration analysis of FGM beam based on unified and integrated of Timoshenko’s theory, Composite Structures, 242 (2020) 112130. https://doi.org/10.1016/j.compstruct.2020.112130
[6]. M. Şimşek, Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions, Composite Structures, 133 (2015) 968-978. https://doi.org/10.1016/j.compstruct.2015.08.021
[7]. T. A. Huynh, X. Q. Lieu, J. Lee, NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem, Composite Structures, 160 (2017) 1178-1190. https://doi.org/10.1016/j.compstruct.2016.10.076
[8]. D. K. Nguyen, T. T. Tran, Free vibration of tapered BFGM beams using an efficient shear deformable finite element model, Steel and Composite Structures, 29 (2018) 363-377. https://doi.org/10.12989/scs.2018.29.3.363
[9]. Vu Thi An Ninh, Fundamental frequencies of bidirectional functionally graded sandwich beams partially supported by foundation using different beam theories, Transport and Communications Science Journal, 72 (2021) 362-377. https://doi.org/10.47869/tcsj.72.4.5
[10]. D. K. Nguyen, Q. H. Nguyen, T. T. Tran, V. T. Bui, Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica, 228 (2017) 141-155. https://doi.org/10.1007/s00707-016-1705-3
[11]. P. Ribeiro, Hierarchical finite element analyses of geometrically non-linear vibration of beams and plane frames, Journal of Sound and Vibration, 246 (2001) 225-244. https://doi.org/10.1006/jsvi.2001.3634
[12]. H. Y. Shang, R. D. Machado, J. E. Abdalla Filho, Dynamic analysis of Euler–Bernoulli beam problems using the generalized finite element method, Computers & Structures, 173 (2016) 109-122. https://doi.org/10.1016/j.compstruc.2016.05.019
[13]. Y. S. Hsu, Enriched finite element methods for Timoshenko beam free vibration analysis, Applied Mathematical Modelling, 40 (2016) 7012-7033. https://doi.org/10.1016/j.apm.2016.02.042
[14]. C. I. Le, N. A. T. Le, D. K. Nguyen, Free vibration and buckling of bidirectional functionally graded sandwich beams using an enriched third-order shear deformation beam element, Composite Structures, 261 (2021), 113309. https://doi.org/10.1016/j.compstruct.2020.113309
[15]. H. T. Thai, S. E. Kim, A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates, Composite Structures, 96 (2013) 165-173. https://doi.org/10.1016/j.compstruct.2012.08.025
[16]. M. Géradin, D. J. Rixen, Mechanical vibrations: theory and application to structural dynamics. John Wiley & Sons (2015).
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Nhận bài
12/02/2025
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04/06/2025
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08/09/2025
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15/09/2025
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Kiểu trích dẫn
Vu Thi An, N. (1757869200). Vibration of four-phase bidirectional functionally graded beams based on trigonometric enriched beam element. Tạp Chí Khoa Học Giao Thông Vận Tải, 76(7), 965-979. https://doi.org/10.47869/tcsj.76.7.4
