On the Analysis of FGM Beams: FEM with Innovative Element

Document Type: Research Paper

Authors

1 School of Civil Engineering, College of Engineering, University of Tehran

2 School of Civil Engineering, College of Engineering, University of Tehran----Centre of Numerical Methods in Engineering, University of Tehran

Abstract

This paper aims at presenting a new efficient element for free vibration and instability analysis of Axially Functionally Graded Materials (FGMs) non-prismatic beams using Finite Element Method (FEM). Using concept of Basic Displacement Functions (BDFs), two- node element extends  to three-node element for obtaining much more exact results using FEM. First, BDFs are introduced and computed using energy method such as unit-dummy load method. Afterward, new efficient shape functions are developed in terms of BDFs during the procedure based on the mechanical behavior of the element in which presented shape functions benefit generality and accuracy from stiffness and force method, respectively. Finally, deriving structural matrices of the beam with respect to new shape functions; free vibration and instability analysis of the FGM beam are studied using finite element method for all types of AFGM beams and the convergence of FEM has been studied. The results from both free vibration and instability analysis are in perfect agreement with those of previously published.

Keywords


[1] Chakraborty A., Gopalakrishnan S., Reddy J.N., 2003, A new beam finite element for the analysis of functionally graded materials, International Journal of Mechanical Sciences 45(3): 519-539.
[2] Carrera E., Brischetto S., Robaldo A., 2008, Variable kinematic model for the analysis of functionally graded material plates, AIAA Journal 46(1): 194-203.
[3] Giunta G., Belouettar S., Carrera E., 2010, Analysis of FGM beams by means of classical and advanced theories, Mechanics of Advanced Materials and Structures 17(8): 622-635.
[4] Şimşek M., 2010, Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures 92(4): 904-917.
[5] Elishakoff I., Becquet R., 2000, Closed-form solutions for natural frequencies for inhomogeneous beams with one sliding support and the other clamped, Journal of Sound and Vibration 238: 540-546.
[6] Calio I., Elishakoff I., 2005, Closed-form solutions for axially graded beam-columns, Journal of Sound and Vibration 280: 1083-1094.
[7] Bequet R., Elishakoff I., 2001, Class of analytical closed-form polynomial solutions for clamped-guided inhomogeneous beams, Chaos, Solitons & Fractals 12: 1657-1678.
[8] Calio I., Elishakoff I., 2004, Closed-form trigonometric solutions for inhomogeneous beam-columns on elastic foundation, International Journal of Structural Stability and Dynamics 4(1):139-146.
[9] Elishakoff I., Candan S., 2001, Apparently first closed-form solution for vibrating inhomogeneous beams, International Journal of Solids and Structures 38(19): 3411-3441.
[10] Calio I., Elishakoff I., 2004, Can a trigonometric function serve both as the vibration and the buckling mode of an axially graded structure, Mechanics Based Design of Structures and Machines 32(4): 401-421.
[11] Elishakoff I., 2001, Inverse buckling problem for inhomogeneous columns, International Journal of Solids and Structures 38(3):457-464.
[12] Elishakoff I., Becquet R., 2000, Closed-form solutions for natural frequencies for inhomogeneous beams with one sliding support and the other pinned, Journal of Sound and Vibration 238: 529-553.
[13] Becquet R., Elishakoff I., 2001, Class of analytical closed-form polynomial solutions for guided-pinned inhomogeneous beams, Chaos, Solitons & Fractals 12(8): 1509-1534.
[14] Guede Z., Elishakoff I., 2001, Apparently the first closed-form solution for inhomogeneous vibrating beams under axial loading, Proceedings of the Royal Society A 457(2007): 623-649.
[15] Elishakoff I., Guede Z., 2001, A remarkable nature of the effect of boundary conditions on closed-form solutions for vibrating inhomogeneous Euler-Bernoulli beams, Chaos, Solitons & Fractals 12: 659-704.
[16] Elishakoff I., 2001, Euler’s problem revisited: 222 years later, Meccanica 36: 265-272.
[17] Elishakoff I., Guede Z., 2001, Novel closed-form solutions in buckling of inhomogeneous columns under distributed variable loading, Chaos, Solitons & Fractals 12(6): 1075-1089.
[18] Elishakoff I., Johnson V., 2005, Apparently the first closed-form solution of vibrating inhomogeneous beam with a tip mass, Journal of Sound and Vibration 286: 1057-1066.
[19] Elishakoff I., Pentaras D., 2006, Apparently the first closed-form solution of inhomogeneous elastically restrained vibrating beams, Journal of Sound and Vibration 298(1-2): 439-445.
[20] Wu L., Wang Q.S., Elishakoff I., 2005, Semi-inverse for axially functionally graded beams with an anti-symmetric vibration mode, Journal of Sound and Vibration 284(3-5): 1190-1202.
[21] Elishakoff I., Perez A., 2005, Design of a polynomially inhomogeneous bar with a tip mass for specied mode shape and natural frequency, Journal of Sound and Vibration 287(4-5): 1004-1012.
[22] Elishakoff I., 2001, Some unexpected results in vibration of nonhomogeneous beams on elastic foundation, Chaos, Solitons & Fractals 12(12): 2177-2218.
[23] Guede Z., Elishakoff I., 2001, A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure, Chaos, Solitons & Fractals 12(7): 1267-1298.
[24] Elishakoff I., Guede Z., 2004, Analytical polynomial solutions for vibrating axially graded beams, Mechanics of Advanced Materials and Structures 11(6): 517-533.
[25] Huang Y., Li X.F.A., 2010, New approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration 329(11): 2291-2303.
[26] Alshorbagy A.E., Eltaher M.A., Mahmoud F.F., 2011, Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling 35(1): 412-425.
[27] Singh K.V., Li G., 2009, Buckling of functionally graded and elastically restrained non-uniform columns, Composites Part B 40(5): 393-403.
[28] Shahba A., Rajasekaran S., 2012, Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials, Applied Mathematical Modelling 36(7): 3094-3111.
[29] Shahba A., Attarnejad R., Hajilar S., 2013, A mechanical-based solution for axially functionally graded tapered euler-bernoulli beams, Mechanics of Advanced Materials and Structures 20: 696-707.
[30] Zarrinzadeh H., Attarnejad R., Shahba A., 2012, Free vibration of rotating axially functionally graded tapered beams, Proceedings of The Institution of Mechanical Engineers Part G-journal of Aerospace Engineering 226: 363-379.
[31] Shahba A., Attarnejad R., Zarrinzadeh H., 2013, Free Vibration Analysis of Centrifugally Stiffened Tapered Functionally Graded Beams, Mechanics of Advanced Materials and Structures 20(5): 331-338.
[32] Shahba A., Attarnejad R., TavanaieMarvi M., Hajilar S., 2011, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites Part B 42(4): 801-808.
[33] Shahba A., Attarnejad R., Hajilara S., 2011, Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams, Shock and Vibration 18(5): 683-696.
[34] Attarnejad R., 2010, Basic displacement functions in analysis of nonprismatic beams, Engineering with Computers 27(6): 733-745.