Engineering Mechanics ›› 2017, Vol. 34 ›› Issue (6): 1-8.doi: 10.6052/j.issn.1000-4750.2015.10.0868

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NONLINEAR STATIC RESPONSES OF FGM BEAMS UNDER NON-UNIFORM THERMAL LOADING

MAO Li-juan1,2, MA Lian-sheng1,2   

  1. 1. School of Science, Lanzhou University of Technology, Lanzhou 730050, China;
    2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, China
  • Received:2015-10-27 Revised:2017-05-13 Online:2017-06-25 Published:2017-06-25

Abstract: Due to the variation in material properties through the thickness of functionally graded material (FGM) structures, an FGM beam simply supported at both ends exhibits characteristics quite different from those of a FGM beam clamped at both ends. An exact, closed form solution is obtained for nonlinear static responses of FGM beams subjected to non-uniform in-plane thermal loadings. The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear classical beam theory and the physical neutral surface concept. The two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformation. For an FGM beam clamped at both ends, the equation and the corresponding boundary conditions lead to a differential eigenvalue problem, whereas for an FGM beam simply supported at both ends, an eigenvalue problem does not arise due to the inhomogeneous boundary conditions. This consequently results in quite different behavior between a clamped and a simply supported FGM beams. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for thermal bending deformation is obtained as a function of the applied thermal load. By using the exact solutions, the nonlinear deformation problems for buckling, postbuckling and nonlinear bending of the beam can be investigated. Finally, the numerical analyses are carried out to investigate the effects of material gradient properties and thermal loads on the nonlinear static responses of FGM beams.

Key words: exact solution, static response, thermal loading, functionally graded material beam, multi-solution branch

CLC Number: 

  • TB34
[1] Librescu L, Oh S Y, Song O. Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability [J]. Journal of Thermal Stresses, 2005, 28(6/7): 649-712.
[2] Bhangale R K, Ganesan N. Thermoelastic vibration and buckling analysis of functionally graded sandwich beam with constrained viscoelastic core [J]. Journal of Sound and Vibration, 2006, 295(1/2): 294-316.
[3] 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.
[4] Kapuria S, Bhattacharyya M, Kumar A N. Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation [J]. Composite Structures, 2008, 82(3): 390-402.
[5] Li X F. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams [J]. Journal of Sound and Vibration, 2008, 318(4/5): 1210-29.
[6] Zhong Z, Yu T. Analytical solution of a cantilever functionally graded beam [J]. Composite Science and Technology, 2007, 67(3/4): 481-488.
[7] 牛牧华, 马连生. 基于物理中面FGM梁的非线性力学行为[J]. 工程力学, 2011, 28(6): 219-225.Niu Muhua, Ma Liansheng. Nonlinear mechanical behaviors of FGM beams based on the physical neutral surface [J]. Engineering Mechanics, 2011, 28(6): 219-225. (in Chinese)
[8] Vo T P, Thai H T, Nguyen T K, Inam F. Static and vibration analysis of functionally graded beams using refined shear deformation theory [J]. Meccanica, 2014, 49(1): 155-168.
[9] 余莲英, 张亮亮, 尚兰歌, 孙振冬, 高恩来, 井文奇, 高阳. 功能梯度曲梁弯曲问题的解析解[J]. 工程力学, 2014, 31(12): 4-10.Yu Lianying, Zhang Liangliang, Shang Lan'ge, Sun Zhendong, Gao Enlai, Jing Wenqi, Gao Yang. Bending solutions of functionally graded curved-beam [J]. Engineering Mechanics, 2014, 31(12): 4-10. (in Chinese)
[10] 文颖, 李特, 曾庆元. 柔性梁几何非线性/后屈曲分析的改进势能列式方法研究[J]. 工程力学, 2015, 32(11): 18-26.Wen Ying, Li Te, Zeng Qingyuan. Improved potential energy formulation for geometrically nonlinear/ post-buckling analysis of flexible beam structures [J]. Engineering Mechanics, 2015, 32(11): 18-26. (in Chinese)
[11] Grygorowicz M, Magnucki K, Malinowski M. Elastic buckling of a sandwich beam with variable mechanical properties of the core [J]. Thin-Walled Structures, 2015, 87: 127-132.
[12] Chen D, Yang J, Kitipornchai S. Elastic buckling and static bending of shear deformable functionally graded porous beam [J]. Composite Structures, 2015, 133: 54-61.
[13] Emama S A, Nayfeh A H. Postbuckling and free vibrations of composite beams [J]. Composite Structures, 2009, 88(4): 636-642.
[14] 马连生, 顾春龙. 剪切可变形梁热过屈曲解析解[J]. 工程力学, 2012, 29(2): 172-176.Ma Liansheng, Gu Chunlong. Exact solutions for thermal post-buckling of shear deformable beams [J]. Engineering Mechanics, 2012, 29(2): 172-176. (in Chinese)
[15] Ma L S, Lee D W. Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading [J]. European Journal of Mechanics A/Solids, 2011, 31(1): 13-20.
[16] 马连生, 张璐. 面内热载荷作用下功能梯度梁热过屈曲精确解[J]. 兰州理工大学学报, 2015, 41(1): 164-167.Ma Liansheng, Zhang Lu. Exact solutions for thermo-post-buckling of functionally graded material beams under in-plane thermal loading [J]. Journal of Lanzhou University of Technology, 2015, 41(1): 164-167. (in Chinese)
[17] Ma L S, Wang T J. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings [J]. International Journal of Solids and Structures, 2003, 40(13/14): 3311-3330.
[18] Ma L S, Wang T J. Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression [J]. Material Science Forum, 2003, 423/424: 719-724.
[19] Leissa A W. Conditions for laminated plates to remain flat under inplane loading [J]. Composite Structures, 1986, 6(4): 262-270.
[20] Leissa A W. A review of laminated composite plate buckling [J]. Applied Mechanics Review, 1987, 40(5): 575-591.
[21] Qatu M S, Leissa A W. Buckling or transverse deflections of unsymmetrically laminated plates subjected to in-plane loads [J]. AIAA Journal, 1993, 31(1): 189-194.
[22] 沈惠申. 功能梯度复合材料板壳结构的弯曲、屈曲和振动[J]. 力学进展, 2004, 34(1): 53-60.Shen Huishen. Bending, buckling and vibration of functionally graded plates and shells [J]. Advance in Mechanics, 2004, 34(1): 53-60. (in Chinese)
[23] Aydogdu M. Conditions for functionally graded plates to remain flat under in-plane loads by classical plate theory [J]. Composite Structures, 2008, 82(1): 155-157.
[24] Praveen G N, Reddy J N. Nonlinear transient thermo-elastic analysis of functionally graded ceramic–metal plates [J]. International Journal of Solids and Structures, 1998, 35(33): 4457-4476.
[25] Zhang D G, Zhou Y H. A theoretical analysis of FGM thin plates based on physical neutral surface [J]. Computational Materials Science, 2008, 44: 716-720.
[26] Wu L H. Thermal buckling of a simply supported moderately thick rectangular FGM plate [J]. Composite Structures, 2004, 64(2): 211-218.
[27] Popov A A. Parametric resonance in cylindrical shells: a case study in the nonlinear vibration of structural shells [J]. Engineering Structures, 2003, 25(6): 789-799.
[28] Looss G, Joseph D D. Elementary stability and bifurcation theory [M]. 2nd ed. Springer, New York, 1990: 75-79.
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