A Unified Method for Modeling of Axially and/or Transversally Functionally Graded Beams with Variable Cross-Section Profile

REFERENCES

• 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 .
   Web of Science ®Google Scholar
• Atmane , H. A. , Abdelouahed , T. , Meftah , S. A. , Belhadj , H. A. ( 2011 ). Free vibration behavior of exponential functionally graded beams with varying cross-section . Journal of Vibration and Control 17 ( 2 ): 311 – 318 .
   Web of Science ®Google Scholar
• Boresi , A. P. , Schmidt , R. J. ( 2003 ). Advanced Mechanics of Materials. , 6th ed. Hoboken , USA : John Wiley & Sons .
   Google Scholar
• Caliò , I. , Elishakoff , I. ( 2004 ). Can a trigonometric function serve both as the vibration and the bucking mode of an axially graded structure? Mechanics Based Design of Structures and Machines 32 ( 4 ): 401 – 421 .
   Web of Science ®Google Scholar
• Carrera , E. , Giunta , G. , Nali , P. , Petrolo , M. ( 2010 ). Refined beam elements with arbitrary cross-section geometries . Computers and Structures 88 ( 5–6 ): 283 – 293 .
   Web of Science ®Google Scholar
• Firouz-Abadi , R. D. , Haddadpour , H. , Novinzadeh , A. B. ( 2007 ). An asymptotic solution to transverse free vibrations of variable-section beams . Journal of Sound and Vibration 304 ( 3–5 ): 530 – 540 .
   Google Scholar
• Franciosi , C. , Mecca , M. (1998). Some finite elements for the static analysis of beams with varying cross section. Computers and Structures 69(2):191–196.
   Google Scholar
• Huang , Y. , Li , X. F. ( 2010 ). A new approach for free vibration of axially functionally graded beams with non-uniform cross-section . Journal of Sound and Vibration 329 ( 11 ): 2291 – 2303 .
   Web of Science ®Google Scholar
• Huang , Y. , Luo , Q. Z. ( 2011 ). A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint . Computers and Mathematics with Applications 61 ( 9 ): 2510 – 2517 .
   Google Scholar
• Jaworski , J. W. , Dowell , E. H. ( 2008 ). Free vibration of a cantilevered beam with multiple steps: Comparison of several theoretical methods with experiment . Journal of Sound and Vibration 312 ( 4–5 ): 713 – 725 .
   Web of Science ®Google Scholar
• Kapuria , S. , Bhattacharyya , M. , Kumar , A. N. ( 2008 ). Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation . Computer Structures 82 ( 3 ): 390 – 402 .
   Google Scholar
• Kang , Y. A. , Li , X. F. ( 2009 ). Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force . International Journal of Non-Linear Mechanics 44 ( 6 ): 696 – 703 .
   Web of Science ®Google Scholar
• Kukla , S. , Zamojska , I. ( 2007 ). Frequency analysis of axially loaded stepped beams by Green's function method . Journal of Sound and Vibration 300 ( 3–5 ): 1034 – 1041 .
   Google Scholar
• Li , Q. S. ( 2000 ). A new exact approach for determining natural frequencies and mode shapes of non-uniform shear beams with arbitrary distribution of mass or stiffness . International Journal of Solids and Structures 37 ( 37 ): 5123 – 5141 .
   Google Scholar
• Li , Q. S. ( 2003 ). Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads . International Journal of Engineering Science 41 ( 6 ): 569 – 586 .
   Google Scholar
• Lu , Z. R. , Huang , M. , Liu , J. K. , Chen , W. H. , Liao , W. Y. ( 2009 ). Vibration analysis of multiple-stepped beams with the composite element model . Journal of Sound and Vibration 322 ( 4–5 ): 1070 – 1080 .
   Web of Science ®Google Scholar
• Murín , J. , Aminbaghai , M. , Kutiš , V. ( 2010 ). Exact solution of the bending vibration problem of FGM beams with variation of material properties . Engineering Structures 32 ( 6 ): 1631 – 1640 .
   Google Scholar
• Naguleswaran , S. ( 2002 ). Vibration of an Euler-Bernoulli beam on elastic end supports and with up to three step changes in cross-section . International Journal of Mechanical Sciences 44 ( 12 ): 2541 – 2555 .
   Google Scholar
• Sankar , B. V. ( 2001 ). An elasticity solution for functionally graded beams . Composites Science and Technology 61 ( 5 ): 689 – 696 .
   Web of Science ®Google Scholar
• Sapountzakis , E. J. ( 2005 ). Torsional vibrations of composite bars of variable cross-section by BEM . Computer Methods in Applied Mechanics and Engineering 194 ( 18–20 ): 2127 – 2145 .
   Web of Science ®Google Scholar
• Sapountzakis , E. J. , Mokos , V. G. ( 2004 ). Nonuniform torsion of bars of variable cross section . Computers and Structures 82 ( 9–10 ): 703 – 715 .
   Google Scholar
• Shahba , A. , Attarnejad , R. , Marvi , M. T. , 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 .
   Web of Science ®Google Scholar
• Şimşek , M. ( 2010 ). Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load . Composite Structures 92 ( 10 ): 2532 – 2546 .
   Web of Science ®Google Scholar
• Şimşek , M. , Kocatürk , T. , Akbaş , Ş. D. (2012). Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load. Composite Structures 94(8):2358–2364.
   Web of Science ®Google Scholar
• Yoo , W. S. , Park , S. J. , Park , J. Y. , Sohn , J. H. ( 2004 ). Physical experiments and computer simulations of a stepped cantilever beam with a hybrid coordinate formulation . Mechanics Based Design of Structures and Machines 32 ( 4 ): 515 – 532 .
   Web of Science ®Google Scholar
• Zheng , T. X. , Ji , T. J. ( 2011 ). Equivalent representations of beams with periodically variable cross-sections . Engineering Structures 33 ( 3 ): 706 – 719 .
   Google Scholar
• Zhong , Z. , Yu , T. ( 2007 ). Analytical solution of a cantilever functionally graded beam . Composites Science and Technology 67 ( 3–4 ): 481 – 488 .
   Web of Science ®Google Scholar
• Zienkiewicz , O. C. , Taylor , R. L. , Zhu , J. Z. ( 2005 ). The Finite Element Method: Its Basis and Fundamentals. , 6th ed. Oxford , UK : Elsevier .
   Google Scholar
• #Communicated by G. Hulbert.
   Google Scholar