Vibration characteristics of FGM beams under the action of initial axial load-based on a n-th generalized shear beam theory
PU Yu1,2,ZHOU Fengxi2
1.College of Civil Engineering, Lanzhou Institute of Technology, Lanzhou 730050, China;
2.School of Civil Engineering, Lanzhou University of Technology, Lanzhou 730050, China
Abstract:The vibration behavior of functional gradient material(FGM) beams on elastic foundation under the action of initial axial load was investigated.Based on an extended n-th generalized shear deformation beam theory and applying Hamilton’s principle, the free vibration governing equations of the mechanical model for the system was developed, in which the unknown basic functions are the axial displacement, shear and bending components of deflection.Introducing boundary condition coefficients and applying a modified generalized differential quadrature(MGDQ) method, the static and dynamic responses of FGM beams were obtained.The availability and accuracy of the extended theory and the method were verified through numerical examples and herein the ideal value of n was proposed.The study refines the beam theories and the result can be used as a benchmark to verify or modify other various shear deformation beam theories.The numerical solution by using MGDQ was validated and the process can expand its applicable scope.The effects of the initial axial load, boundary conditions, material graded index, elastic foundation stiffness, and length-to-thickness ratio on the vibration characteristics of FGM beams were mainly analyzed.
蒲育1,2,周凤玺2. 基于n阶GBT初始轴向载荷影响下FGM梁的振动特性[J]. 振动与冲击, 2020, 39(2): 100-106.
PU Yu1,2,ZHOU Fengxi2. Vibration characteristics of FGM beams under the action of initial axial load-based on a n-th generalized shear beam theory. JOURNAL OF VIBRATION AND SHOCK, 2020, 39(2): 100-106.
[1] Chen W Q, Lu C F, Bian Z G. A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation[J]. Applied Mathematical Modelling, 2004, 28(10): 877-890.
[2] Ying J, Lu C F, Chen W Q. Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations[J]. Composite Structures, 2008, 84(3):209-219.
[3] Aydogdu M, Taskin V. Free vibration analysis of functionally graded beams with simply supported edges[J]. Materials and Design, 2007, 28(5): 1651-1656.
[4] Simsek M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories[J]. Nuclear Engineering and Design, 2010, 240(4): 697-705.
[5] Thai H T, Vo T P. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories[J]. International Journal of Mechanical Sciences, 2012, 62(1) :57-66.
[6] Li Shirong, Wan Zeqing, Zhang Jinghua. Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories[J]. Applied Mathematics and Mechanics, 2014, 35(5): 591-606.
[7] Li Shirong, Wang Xuan, Wan Zeqing. Classical and homogenized expressions for buckling solutions of functionally graded material Levinson beams [J]. Acta Mechanica Solida Sinica, 2015, 28(5): 592—604.
[8] Pradhan K K, Chakraverty S. Effects of different shear deformation theories on free vibration of functionally graded beams [J]. International Journal of Mechanical Sciences, 2014, 82(5): 149—160.
[9] 蒲育, 滕兆春. Winkler-Pasternak弹性地基FGM梁自由振动二维弹性解[J]. 振动与冲击, 2015, 34(20): 74-79.
PU Yu, TENG Zhao-chun. Two-dimensional elasticity solutions for free vibration of FGM beams resting on Winkler-Pasternak elastic foundation[J]. Journal of Vibration and Shock, 2015, 34(20): 74-79.
[10] 蒲育, 滕兆春. 基于一阶剪切变形理论FGM梁自由振动的改进型GDQ法求解[J]. 振动与冲击, 2018, 37(16): 212-218.
PU Yu, TENG Zhao-chun. Free vibration of FGM beams based on the first-order shear deformation theory by a modified generalized differential quadrature method[J]. Journal of Vibration and Shock, 2018, 37(16): 212-218.
[11] Reddy J N. A simple hgher-order theory for laminated composite plates[J]. Journal of Applied Mechanics, 1984, 51(4):745-752.
[12] Touratier M. An efficient standard plate theory[J]. International Journal of Engineering Science, 1991, 29(8):901-916.
[13] Soldatos K P. A transverse shear deformation theory for homogeneous monoclinic plates[J]. Acta Mechanica, 1992, 94(3-4):195-220.
[14] Karama M, Afaq K S, Mistou S. Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity[J]. International Journal of Solids and Structures, 2003, 40(6):1525-1546.
[15] Aydogdu M. A new shear deformation theory for laminated composite plates[J]. Composite Structures, 2009, 89(1):94-101.
[16] Nguyen T K, Nguyen T T, et al. Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory[J]. Composites: Part B, 2015, 76:273-285.
[17] Xiang Song, Kang Guiwen. A nth-order shear deformation for the bending analysis on the functionally graded plates [J]. European Journal of Mechanics A/Solids, 2013, 37: 336—343.