Based on the theory of Euler-Bernoulli curved beam,the shifting of neutral layer was considered when materials were gradually distributed trapezoidally along the arch thickness,and functionally graded material (FGM) arches with variable curvature were discretized into a number of curved arch elements along the arc direction.Every curved arch element was considered as a circular arch element with a constant radius,according to Hamilton variational principle,the in-plane free vibration equation of a FGM circular arch element was derived,then the element transfer matrix was deduced.Furthmore,using TMM,the in-plane free vibration characteristic equation of the FGM arch with variable curvature was derived,the in-plane free vibration natural frequencies of the FGM arch with variable curvature under two-clamped end boundary condition were solved,the results were compared with those previously reported.It was shown that TMM is effective to solve this problem.The influences of curvature varying coefficient and material volumn fraction varying coefficient on the in-plane free vibration frequencies of the FGM arch with variable curvature were analyzed.
[1] Chakraborty A, Gopalakrishnan S, Reddy J N. A new beam finite element for the analysis of functionally graded materials[J]. International Journal of Mechanical Sciences. 2003, 45(3):519-539.
[2] Goupee A, Vel S. Optimization of natural frequencies of bidirectional functionally graded beams[J]. Structural and Multidisciplinary Optimization, 2006, 32(6):473-484.
[3] Malekzadeh P, Setoodeh A R, Brrmshouri E. A hybrid layerwise and differential quadrature method for in-plane free vibration of laminated thick circular arches[J]. Journal of Sound and Vibration, 2008, 315(1-2): 212–225.
[4] Lü Q, Lü C F. Exact two-dimensional solutions for in-plane natural frequencies of laminated circular arches[J]. Journal of Sound and Vibration, 2008, 318(4-5): 982–990.
[5] Tseng Y P,Huang C S,Kao M S. In-plane vibration of laminated curved beams with variable curvature by dynamic stiffness analysis[J]. Composite Structures, 2000, 50(2): 103-114.
[6] Lim C W, Yang Q, Lü C F. Two-dimensional elasticity solutions for temperatire-dependent in-plane vibration of FGM circular arches[J]. Composite Structures, 2009, 90(3): 323-329.
[7] Carlos P F, Marcelo T P. The dynamics of thick curved beams constructed with functionally graded materials[J]. Mechanics Research Communications, 2010,37(6):565-570.
[8] Zeng Q C, Lim C W, Lü C F, et al. Asymptotic two-dimensional elasticity approach for free vibration of FGM circular arches [J]. Mechanics of Advanced Materials and Structures, 2012, 19(1-3):29-38.
[9] Ugurcan E. In-plane free vibrations of circular beams made of functionally graded material in thermal environment: Beam theory approach[J]. Composite Structures, 2015, 122:217-228.
[10] 胡海昌. 弹性力学的变分原理及其应用[M]. 北京: 科学出版社, 1982.
Hu H C. The variational principle of elasticity and its application[M]. Beijing: Science Press, 1982. (in Chinese)
[11] 向天宇, 郑建军. 变截面圆拱的自由振动[J]. 振动与冲击, 2000, 19(2):59-63.
Xiang T Y, Zheng J J. Free vibration of circular arches with variable cross-section[J]. Journal of vibration and shock, 2000, 19(2):59-63. (in Chinese)
[12] Zhao Y Y, Kang H J. In-plane free vibration analysis of cable-arch structure[J]. Journal of Sound and Vibration, 2008, 312(3): 363-379.
[13] 孙训方, 方孝淑, 关来泰. 材料力学[M]. 北京: 高等教育出版社, 1982.
Sun X F, Fang X S, Guan L T. Mechanics of Materials[M]. Beijing: Higher Education Press, 1982. (in Chinese)
[14] Li S R, Su H D, Cheng C J. Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment[J]. Applied Mathematics and Mechanics (English Edition), 2009, 30(8): 969-982.
[15] Viola E, Artioli M, Dilena M. Analytical and differential quadrature results for vibration analysis of damaged circular arches[J]. Journal of Sound and Vibration, 2005, 288(4-5): 887-906.
[16] 同济大学数学教研室. 线性代数[M]. 北京: 高等教育出版社, 1999.
Department of Mathematics, Tongji University. Linear Algebra[M]. Beijing: Higher Education Press, 1999. (in Chinese)