||Nonlinear Free Vibration Analysis of Functionally Graded Beams Using a Mixed Finite Element Method and a Comparative Artificial Neural Network
||Department of Civil Engineering
Artificial neural networks
Finite element methods
Functionally graded beams
Mixed Timoshenko beam theory
本文基於Hamilton原理和Timoshenko梁理論發展一套混合有限元素法用以分析材料性梯度梁在簡支承和固定端兩種組合邊界條件束制下之非線性自由振動行為。功能性梯度梁是由兩相材料組成，其材料性質之變化根據組成材料體積分率的冪次函數分佈，為一材料性質沿厚度方向平滑變化之梁，且功能性梯度梁之有效材料性質是以複合材料力學中兩相材料混合原理評估。此混合有限元素法之弱形式數學方程式是從能量泛函透過變分法推導而來，其中能量泛函有將馮卡門幾何非線性效應(von Kármán geometrical nonlinearity, VKGN)考慮在內。文中數值範例的振幅-頻率關係是以前述之混合有限元素法經過迭代運算求得。本混合有限元素解與文獻中之精確解相比較，其彼此之結果相當吻合，且在迭代求解過程中此法可以快速地達到收斂。在此篇文章中，作者進一步使用了多層感知倒傳式類神經網路來預測功能性材料梁之幾何非線性振動行為。經過適當的訓練後，類神經網路所預測之振幅-頻率關係與混合有限元素法比較有相當不錯的精確度，而其運算時間比需要經過迭代的混合有限元素法大幅節省。
Based on the Hamilton principle combined with the Timoshenko beam theory, the authors develop a mixed finite element (FE) method for the nonlinear free vibration analysis of functionally graded (FG) beams with combinations of simply-supported and clamped edge conditions. The FG beam is composed of a two-phase material, the material properties of which gradually and smoothly vary through the thickness direction according to the power-law distributions of the volume fractions of the constituents, and the effective material properties of the FG beam are estimated using the rule of mixtures. A weak-form formulation for the mixed finite element analysis is derived using the variational approach, in which the von Kármán geometrical nonlinearity is considered. The finite element solutions of the amplitude-frequency relations of the FG beam are obtained using an iterative process. Implementation of the mixed finite element method shows its solutions converge rapidly and that the convergent solutions closely agree with the accurate solutions available in the literature. A multilayer perceptron (MP) back propagation neural network (BPNN) is also developed to predict the nonlinear free vibration behavior of the FG beam. After appropriate training, the prediction of the amplitude-frequency relations of the MP BPNN is quite accurate with those obtained using the mixed FE method, and its computer execution time is less time consuming than that of the mixed FE method.
 M. Koizumi, Recent progress of functionally graded materials in Japan, Ceram. Eng. Sci. Proc. 13 (1992) 333-347.
 M. Koizumi, FGM activities in Japan, Compos. Part B 28 (1997) 1-4.
 S. Brischetto, E. Carrera, Analysis of nano-reinforced layered plates via classical and refined two-dimensional theories, Multidisc. Model. Mater. Struct. 8 (2012) 4-31.
 S. Brischetto, E. Carrera, Classical and refined shell models for the analysis of nano-reinforced structures, Int. J. Mech. Sci. 55 (2012) 104-117.
 D.K. Jha, T. Kant, R.K. Singh, A critical review of recent research on functionally graded plates, Compos. Struct. 96 (2013) 833-849.
 K.M. Liew, Z.X. Lei, L.W. Zhang, Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review, Compos. Struct. 120 (2015) 90-97.
 K.M. Liew, P. Xiang, L.W. Zhang, Mechanical properties and characteristics of microtubules: a review, Compos. Struct. 123 (2015) 98-108.
 H.S. Shen, Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC Press, Boca Raton, 2009.
 C.P. Wu, Y.C. Liu, A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells, Compos. Struct. 147 (2016) 1-15.
 C.P. Wu, K.H. Chiu, Y.M. Wang, A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells, CMC-Comput. Mater. Continua 8 (2008) 93-132.
 A. Chakraborty, D.R. Mahapatra, S. Gopalakrishnan, Finite element analysis of free vibration and wave propagation in asymmetric composite beams with structural discontinuities, Compos. Struct. 55 (2002) 23-36.
 A. Chakraborty, S. Gopalakrishnan, J.N. Reddy, A new beam finite element for the analysis of functionally graded materials, Int. J. Mech. Sci. 45 (2003) 519-539.
 M. Aydogdu, Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method, Int. J. Mech. Sci. 47 (2005) 1740-1755.
 M. Aydogdu, V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges, Mater. Des. 28 (2007) 1651-1656.
 M. Simsek, Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method, Int. J. Eng. Appl. Sci. 1 (2009) 1-11.
 M. Simsek, Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories, Nucl. Eng. Des. 240 (2010) 697-705.
 劉佳樺,Large-Amplitude free vibration analysis of functionally graded beams resting on an elastic medium, NCKU, Tainan, Taiwan (2018).
 L.S. Ma, D.W. Lee, Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading, Eur. J. Mech. A/Solids 31 (2012) 13-20.
 F. Alinaghizadeh, M. Kadkhodayan, Investigation of nonlinear bending analysis of moderately thick functionally graded material sector plates subjected to thermomechanical loads by the GDQ method, J. Eng. Mech. 140 (2014) 04014012 (11 pages).
 D.G. Zhang, Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory, Compos. Struct. 100 (2013) 121-126.
 S.J. Salami, Extended high order sandwich panel theory for bending analysis of sandwich beams with carbon nanotube reinforced face sheets, Physica E 76 (2016) 187-197.
 H.S. Shen, Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments, Compos. Struct. 91 (2009) 9-19.
 L.L. Ke, J. Yang, S. Kitipornchai, Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams, Compos. Struct. 9 (2010) 676-683.
 V.K. Chaudhari, A. Lal, Nonlinear free vibration analysis of elastically supported nanotube-reinforced composite beam in thermal environment, Procedia Eng. 144 (2016) 928-935.
 C.M. Bishop, Neural Networks for Pattern Recognition, Oxford Press, New York, 1995.
 M.T. Hagan, H.B. Demuth, M. Beale, Neural Network Design, PWS Pub. Comp., New York, 1995.
 R.J. Schalkoff, Artificial Neural Networks, McGraw-Hill, New York, 1997.
 G. Yagawa, H. Okuda, Neural networks in computational mechanics, Arch. Comput. Methods Eng. 3-4 (1996) 435-512.
 H.E. Kadi, Modeling the mechanical behavior of fiber-reinforced polymetric composite materials using artificial neural networks-a review, Compos. Struct. 73 (2006) 1-23.
 Z. Zhang, K. Friedrich, Artificial neural networks applied to polymer composites: a review, Compos. Sci. Technol. 63 (2003) 2029-2044.
 Z. Waszczyszyn, L. Ziemianski, Neural networks in mechanics of structures and materials-new results and prospects of applications, Comput. Struct. 79 (2001) 2261-2276.
 I.C. Yeh, Modeling of strength of high-performance concrete using artificial neural networks, Cement Concrete Res. 28 (1998) 1797-1808.
 K. Papik, B. Molnar, R. Schaefer, Z. Dombovari, Z., Tulassay, J. Feher, Application of neural networks in medicine-a review, Med. Sci. Monit. 4 (1998) 538-546.
 S.A. Kalogirou, Artificial neural networks in renewable energy systems applications: a review, Renewable Sustainable Energy Rev. 5 (2001) 373-401.
 Y. Ootao, R. Kawamura, Y. Tanigawa, T. Nakamura, Neural network optimization of material composition of a functionally graded material plate at arbitrary temperature range and temperature rise, Arch. Appl. Mech. 68 (1998) 662-676.
 Y. Ootao, R. Kawamura, Y. Tanigawa, R. Imamura, Optimization of material composition of nonhomogeneous hollow sphere for thermal stress relaxation making use of neural network, Comput. Methods Appl. Mech. Engrg. 180 (1999) 185-201.
 Y. Ootao, R. Kawamura, Y. Tanigawa, R. Imamura, Optimization of material composition of nonhomogeneous hollow circular cylinder for thermal stress relaxation making use of neural network, J. Therm. Stresses 22 (1999) 1-22.
 M.W. Gardner, S.R. Dorling, Artificial neural networks (the multilayer perceptron)-a review of applications in the atmospheric sciences, Atmospheric Environment 32 (1998) 2627-2636.
 Y. Zou, L. Tong, G.P. Steven, Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures, J. Sound Vib. 230 (2000) 357-378.
 S. Chakraverty, V.P. Singh, R.K. Sharma, Regression based weight generation algorithm in neural network for estimation of frequencies of vibrating plates, Comput. Methods Appl. Mech. Engrg. 195 (2006) 4194-4202.
 M.R.S. Reddy, B.S. Reddy, V.N. Reddy, S. Sreenivasulu, Prediction of natural frequency of laminated composite plates using artificial neural networks, Eng. 4 (2012) 329-337.
 J. Jodaei, M. Jalal, M.H. Yas, Free vibration analysis of functionally graded annular plates by state-space based differential quadrature method and comparative modeling by ANN, Compos. Part B 43 (2012) 340-353.
 T. Subramani, S. Sharmila, Prediction of deflection and stresses of laminated composite plate with artificial network aid, Int. J. Modern Eng. Res. 4 (2014) 51-58.
 X. Liu, D. Guojun, N. Xiaoxia, A neural network method applied in prediction eigenvalue buckling for sandwich plates, Inform. Technol. J. 12 (2013) 8129-8134.
 E. Reissner, On a certain mixed variational theorem and a proposed application, Int. J. Numer. Methods Eng. 20 (1984) 1366-1368.
 E. Reissner, On a mixed variational theorem and on shear deformable plate theory, Int. J. Numer. Methods Eng. 23 (1986) 193-198.
 E. Carrera, An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates, Compos. Struct. 50 (2000) 183-198.
 E. Carrera, Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking, Arch. Comput. Methods Eng. 10 (2003) 216-296.
 E. Carrera, Assessment of theories for free vibration analysis of homogeneous and multilayered plates, Shock Vib. 11 (2004) 261-270.
 C.P. Wu, W.W. Lai, Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions, Compos. Struct. 122 (2015) 390-404.
 C.P. Wu, W.W. Lai, Free vibration of an embedded single-walled carbon nanotube with various boundary conditions using the RMVT-based nonlocal Timoshenko beam theory and DQ method, Physica E 68 (2015) 8-21.
 C.P. Wu, Z.L. Hong, Y.M. Wang, Geometrically nonlinear static analysis of an embedded multiwalled carbon nanotube and the van der Waals interaction, J. Nanomech. Micromech. 7 (2017) 04017012.
 R. Hill, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids 13 (1965) 213-222.
 B.S. Sarma, T.K. Varadan, Lagrange-type formulation for finite element analysis of non-linear beam vibrations, J. Sound Vib. 86 (1983) 61-70.
 B.S. Sarma, T.K. Varadan, Ritz finite element approach to nonlinear vibrations of beams, Int. J. Numer. Methods Eng. 20 (1984) 353-367.
 G.R. Bhashyam, G. Prathap, Galerkin finite element method for nonlinear beam vibrations, J. Sound Vib. 72 (1980) 191-203.
 B.S. Sarma, T.K. Varadan, G. Prathap, On various formulations of large amplitude free vibrations of beams, Comput. Struct. 29 (1988) 959-966.
 L.L. Ke, J. Yang, S. Kitipornchai, An analytical study on the nonlinear vibration of functionally graded beams, Meccanica 45 (2010) 743-752.
 M.N., Elmaguiri, M. Haterbouch, A. Bouayad, O. Oussouaddi, Geometrically nonlinear free vibration of functionally graded beams, J. Mater. Environ. Sci. 6 (2015) 3620-3633.
 H.M.Yin, L.Z.Sun, G.H.Paulino, Micromechanics-based elastic model for functionally graded materials with particle interactions, Acta Materialia 52 (2004) 3535-3543.