Large-Amplitude Frequency Analysis of Bi-directional Functionally Graded with Non-Uniform Porous Beams using a Higher Order Shear Deformation Theory

Document Type : Research Article

Authors

1 Department of Mechanical Engineering,Shiraz Branch, Islamic Azad University, Shiraz, Iran

2 Mechanical Engineering Department, Shiraz Branch, Islamic Azad University, Shiraz, Iran

3 Department of Mechanical Engineering, University of North Texas, North Texas, USA

4 Associate professor, Department of Mechanical Engineering, Shiraz Branch Islamic Azad University

Abstract

This Paper deals with the large amplitude frequency behavior of porous bi-directional functionally graded beams subjected to various boundary conditions which are simply supported, clamped-simply supported, clamped-clamped, and clamped-free utilizing Reddy third-order shear deformation theory and Green’s tensor together with the Von Karman geometric nonlinearity. The material properties of the beam change according to power and exponential law in both directions. The equations of motion and associated boundary conditions are derived by means of Hamilton’s principle. A generalized differential quadrature method in conjunction with a direct numerical iteration method is selected to solve the system of equations. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Timoshenko beam theory. The results of extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratio, porosity coefficient, Tapered ratio, shear and elastic foundation parameters, and boundary conditions on the Large amplitude vibration frequencies of the bi-directional functionally graded beams. The results reveal that non-linear frequencies increase with the rise of elastic foundation and tapered coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non-dimensional frequencies. The results of this study, while carefully examining the frequencies of variable cross-sectional functionally graded beams, are effective in the optimal design of bi-directional beams and are very effective in predicting and detecting failure modes of these beams.

Keywords

Main Subjects

References

[1] P. Zahedinejad, Free vibration analysis of functionally graded beams resting on elastic foundation in thermal environment, International Journal of Structural Stability and Dynamics, 16(07) (2016) 1550029.
[2] Y. Wen, Q. Zeng, A HIGH-ORDER FINITE ELEMENT FORMULATION FOR VIBRATION ANALYSIS OF BEAM-TYPE STRUCTURES, International Journal of Structural Stability and Dynamics, 9(04) (2009) 649-660.
[3] M. Aydogdu, V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges, Materials & design, 28(5) (2007) 1651-1656.
[4] M. Şimşek, Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories, Nuclear Engineering and Design, 240(4) (2010) 697-705.
[5] H.-T. Thai, T.P. Vo, Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories, International Journal of Mechanical Sciences, 62(1) (2012) 57-66.
[6] S. Chakraverty, K. Pradhan, Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions, Aerospace Science and Technology, 36 (2014) 132-156.
[7] N. Wattanasakulpong, A. Chaikittiratana, S. Pornpeerakeat, Chebyshev collocation approach for vibration analysis of functionally graded porous beams based on third-order shear deformation theory, Acta Mechanica Sinica, 34(6) (2018) 1124-1135.
[8] D. Zhou, A general solution to vibrations of beams on variable Winkler elastic foundation, Computers & structures, 47(1) (1993) 83-90.
[9] M. Eisenberger, Vibration frequencies for beams on variable one-and two-parameter elastic foundations, Journal of Sound and Vibration, 176(5) (1994) 577-584.
[10] S. Pradhan, T. Murmu, Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method, Journal of Sound and Vibration, 321(1-2) (2009) 342-362.
[11] P. Malekzadeh, G. Karami, A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations, Applied Mathematical Modelling, 32(7) (2008) 1381-1394.
[12] Ş.D. Akbaş, Free vibration and bending of functionally graded beams resting on elastic foundation, Research on Engineering Structures and Materials, 1(1) (2015) 25-37.
[13] M. Yas, S. Kamarian, A. Pourasghar, Free vibration analysis of functionally graded beams resting on variable elastic foundations using a generalized power-law distribution and GDQ method, Annals of Solid and Structural Mechanics, 9(1) (2017) 1-11.
[14] P. Tossapanon, N. Wattanasakulpong, Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation, Composite Structures, 142 (2016) 215-225.
[15] L. Li, X. Li, Y. Hu, Nonlinear bending of a two-dimensionally functionally graded beam, Composite Structures, 184 (2018) 1049-1061.
[16] J. Lei, Y. He, Z. Li, S. Guo, D. Liu, Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory, Composite Structures, 209 (2019) 811-829.
[17] 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.
[18] Z.-h. Wang, X.-h. Wang, G.-d. Xu, S. Cheng, T. Zeng, Free vibration of two-directional functionally graded beams, Composite structures, 135 (2016) 191-198.
[19] A. Ghorbanpour Arani, S. Niknejad, Dynamic stability analysis of Euler-Bernoulli and Timoshenko beams composed of bi-directional functionally graded materials, AUT Journal of Mechanical Engineering, 4(2) (2020) 201-214.
[20] N. Wattanasakulpong, A. Chaikittiratana, Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method, Meccanica, 50(5) (2015) 1331-1342.
[21] M.-J. Kazemzadeh-Parsi, F. Chinesta, A. Ammar, Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution, Materials, 14(21) (2021) 6660.
[22] J. Zhu, Z. Lai, Z. Yin, J. Jeon, S. Lee, Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy, Materials chemistry and physics, 68(1-3) (2001) 130-135.
[23] D. Chen, J. Yang, S. Kitipornchai, Free and forced vibrations of shear deformable functionally graded porous beams, International journal of mechanical sciences, 108 (2016) 14-22.
[24] Y.S. Al Rjoub, A.G. Hamad, Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method, KSCE Journal of Civil Engineering, 21(3) (2017) 792-806.
[25] N. Shafiei, S.S. Mirjavadi, B. MohaselAfshari, S. Rabby, M. Kazemi, Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams, Computer Methods in Applied Mechanics and Engineering, 322 (2017) 615-632.
[26] F.   Ebrahimi,   A.   Jafari,   Thermo-mechanical   vibration   analysis   of temperature-dependent   porous   FG   beams   based   on   Timoshenko   beam theory, Structural Engineering Mechanics, 59(2) (2016) 343-371.
[27] M. Khakpour, Y. Bazargan-Lari, P. Zahedinejad, M-J. Kazemzadeh-parsi, Vibrations evaluation of functionally graded porous beams in thermal surroundings by generalized differential quadrature method, Shock and Vibration, 4 (2022) 1-15.
[28] F. Ebrahimi, M. Zia, Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities, Acta Astronautica, 116 (2015) 117-125.
[29] S. Kitipornchai, L.-L. Ke, J. Yang, Y. Xiang, Nonlinear vibration of edge cracked functionally graded Timoshenko beams, Journal of sound and vibration, 324(3-5) (2009) 962-982.
[30] P. Malekzadeh, M. Shojaee, Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams, Composites Part B: Engineering, 52 (2013) 84-92.
[31] T. Yang, Y. Tang, Q. Li, X.-D. Yang, Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams, Composite Structures, 204 (2018) 313-319.
[32] M.H. Ghayesh, Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams, Applied Mathematical Modelling, 59 (2018) 583-596.
[33] A. Karamanlı, Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory, Composite Structures, 189 (2018) 127-136.
[34] K. Xie, Y. Wang, X. Fan, T. Fu, Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories, Applied Mathematical Modelling, 77 (2020) 1860-1880.
[35] M. Forghani, Y. Bazarganlari, P. Zahedinejad, M.J. Kazemzadeh-Parsi, Nonlinear frequency behavior of cracked functionally graded porous beams resting on elastic foundation using Reddy shear deformation theory, Journal of Vibration and Control,  (2022) 10775463221080213.
[36] M. Şimşek, Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method, Composite Structures, 112 (2014) 264-272.
[37] S. Sahmani, M. Bahrami, R. Ansari, Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory, Composite Structures, 110 (2014) 219-230.
[38] R. Bellman, B. Kashef, J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of computational physics, 10(1) (1972) 40-52.
[39] C.W. Bert, M. Malik, Differential quadrature method in computational mechanics: a review,  (1996).
[40] W. Chen, T. Zhong, The study on the nonlinear computations of the DQ and DC methods, Numerical Methods for Partial Differential Equations: An International Journal, 13(1) (1997) 57-75.
Amirkabir Journal of Mechanical Engineering
Volume 54, Issue 8
November 2022
Pages 1761-1788

Files

Share

How to cite

Statistics