[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.12989/scs.2014.17.5.641

Analytical solutions for bending of transversely or axially FG nonlocal beams

Nguyen, Ngoc-Tuan (Department of Architectural Engineering, Sejong University)
Kim, Nam-Il (Department of Architectural Engineering, Sejong University)
Lee, Jaehong (Department of Architectural Engineering, Sejong University)

Publication Information

Steel and Composite Structures / v.17, no.5, 2014 , pp. 641-665 More about this Journal

Abstract

This paper presents the analytical solutions for the size-dependent static analysis of the functionally graded (FG) beams with various boundary conditions based on the nonlocal continuum model. The nonlocal behavior is described by the differential constitutive model of Eringen, which enables to this model to become effective in the analysis and design of nanostructures. The elastic modulus of beam is assumed to vary through the thickness or longitudinal directions according to the power law. The governing equations are derived by using the nonlocal continuum theory incorporated with Euler-Bernoulli beam theory. The explicit solutions are derived for the static behavior of the transversely or axially FG beams with various boundary conditions. The verification of the model is obtained by comparing the current results with previously published works and a good agreement is observed. Numerical results are presented to show the significance of the nonlocal effect, the material distribution profile, the boundary conditions, and the length of beams on the bending behavior of nonlocal FG beams.

Keywords

functionally graded beam; nonlocal theory; analytical solution; power series method;

Citations & Related Records

Times Cited By KSCI : 1 (Citation Analysis)

Reference
Cited By KSCI

1	Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2013), "Static analysis of nanobeams using nonlocal FEM", J. Mech. Sci. Technol., 27(7), 2035-2041. DOI
2	Challamel, N. and Wang, C.M. (2008), "The small length scale effect for a non-local cantilever beam- a paradox solved", Nanotechnology, 19(34), 345703. DOI
3	Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E., 41(9), 1651-1655. DOI ScienceOn
4	Budynas, R.G. and Nisbett, J.K. (2010), Shigley's Mechanical Engineering Design, McGraw-Hill, New York, NY, USA.
5	Chakraborty, A., Gopalakrishnan, S. and Reddy, J.N. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539. DOI ScienceOn
6	Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420. DOI ScienceOn
7	Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013a), "Determination of neutral axis position and its effect on natural frequencies of functionally graded macro or nanobeams", Compos. Struct., 99, 193-201. DOI ScienceOn
8	Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2013b), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. DOI ScienceOn
9	Eringen, A.C. (1972a), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. DOI ScienceOn
10	Eringen, A.C. (1972b), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. DOI ScienceOn
11	Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. DOI ScienceOn
12	Ghannadpour, S.A.M., Mohammadi, B. and Fazilati, J. (2013), "Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method", Compos. Struct., 96, 584-589. DOI
13	Kang, Y.A. and Li, X.F. (2009), "Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force", Int. J. Non-Linear. Mech., 44(6), 696-703. DOI
14	Li, X.F. (2008), "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound. Vib., 318(4-5), 1210-1229. DOI ScienceOn
15	Hein, H. and Feklistova, L. (2011), "Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets", Eng. Struct., 33(12), 3696-3701. DOI
16	Huang, Y. and Li, X.F. (2010), "A new approach for free vibration of axially functionally graded beams with non-uniform cross-section", J. Sound. Vib., 329(11), 2291-2303. DOI ScienceOn
17	Li, Y., Zhang, H. and Zhang, N. (2005), "Stress analysis of functionally graded beam using effective principle axes", Int. J. Mech. Mater. Des., 2(3-4), 157-164. DOI
18	Li, X.F., Wang, B.L. and Han, J.C. (2010), "A higher-order theory for static and dynamic analyses of functionally graded beams", Arch. Appl. Mech., 80(10), 1197-1212. DOI
19	Nakamura, T., Wang, T. and Sampath, S. (2000), "Determination of properties of graded materials by inverse analysis and instrumented indentation", Acta. Mater., 48(17), 4293-4306. DOI
20	Nie, G.J., Zhong, Z. and Chen, S. (2013), "Analytical solution for a functionally graded beam with arbitrary graded material properties", Compos. Pt. B-Eng., 44(1), 274-282. DOI
21	Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312. DOI ScienceOn
22	Phadikar, J.K. and Pradhan, S.C. (2010), "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Comput. Mater. Sci., 49(3), 492-499. DOI ScienceOn
23	Reddy, J.N. (2002), Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York, NY, USA.
24	Pradhan, S.C. and Murmu, T. (2010), "Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever", Physica E, 42(7), 1944-1949. DOI ScienceOn
25	Rajasekaran, S. (2013), "Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods", Appl. Math. Model., 37(6), 4440-4463. DOI
26	Reddy, J.N. (1999), Theory and Analysis of Elastic Plates, Taylor & Francis, Philadelphia, PA, USA
27	Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2-8), 288-307. DOI ScienceOn
28	Reddy, J.N. (2010), "Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates", Int. J. Eng. Sci., 48(11), 1507-1518. DOI ScienceOn
29	Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696. DOI ScienceOn
30	Shahba, A. and Rajasekaran, S. (2012), "Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials", Appl. Math. Model., 36(7), 3094-3111. DOI
31	Shahba, A., Attarnejad, R., Marvi, M.T. and Hajilar, S. (2011), "Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions", Compos. Pt. B-Eng., 42(4), 801-808.
32	Shahba, A., Attarnejad, R. and Hajilar, S. (2013), "A mechanical-based solution for axially functionally graded tapered Euler-Bernoulli beams", Mech. Adv. Mater. Struct., 20(8), 696-707. DOI
33	Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Comput. Mater. Sci., 61, 257-265. DOI
34	Wang, L. and Hu, H. (2005), "Flexural wave propagation in single-walled carbon nanotubes", Phys. Rev. B., 71, 195412. DOI
35	Simsek, M. and Yurtcu, H.H. (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Compos. Struct., 97, 378-386. DOI
36	Taeprasartsit, S. (2012), "A buckling analysis of perfect and imperfect dunctionally graded columns", J. Mater. Des. Appl., 226(1), 16-33.
37	Uymaz, B. (2013), "Forced vibration analysis of functionally graded beams using nonlocal elasticity", Compos. Struct., 105, 227-239. DOI
38	Wang, C.M., Zhang, Y.Y. and He, X.Q. (2007), "Vibration of nonlocal Timoshenko beams", Nanotechnology, 18(10), 105401. DOI ScienceOn
39	Zhang, Y.Y., Wang, C.M. and Challamel, N. (2010), "Bending, buckling, and vibration of micro nanobeams by hybrid nonlocal beam model", J. Eng. Mech., 136(5), 562-574. DOI ScienceOn
40	Ying, A. K., Li, X. F. (2009), "Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force", Int. J. Non-Linear. Mech., 44(6), 696-703. DOI
41	Zhong, Z. and Yu, T. (2007), "Analytical solution of a cantilever functionally graded beam", Compos. Sci. Technol., 67(3-4), 481-488. DOI ScienceOn
42	Zhu, H. and Sankar, B.V. (2004), "A combined Fourier series-Galerkin method for the analysis of functionally graded beams", J. Appl. Mech., 71(3), 421-424. DOI ScienceOn

	Farzad Ebrahimi. (2017) Mechanics of Advanced Materials and Structures Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory / , 1
	Tuan N. Nguyen. (2017) Computer Methods in Applied Mechanics and Engineering A novel three-variable shear deformation plate formulation: Theory and Isogeometric implementation / 326 , 376
	Farzad Ebrahimi. (2018) Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium / , 095440621771351
	Huu-Tai Thai. (2017) Composite Structures A review of continuum mechanics models for size-dependent analysis of beams and plates / 177 , 196
	Mohammad Rezaiee-Pajand. (2018) Mechanics of Advanced Materials and Structures Nonlocal static analysis of a functionally graded material curved nanobeam / , 1
	Xiao Liang Chen. (2017) Key Engineering Materials Measurement Model for Young's Modulus of Axially Functionally Graded Materials / 748 , 391
5	Farzad Ebrahimi. (2017) Journal of Thermal Stresses Through-the-length temperature distribution effects on thermal vibration analysis of nonlocal strain-gradient axially graded nanobeams subjected to nonuniform magnetic field / 40 (5) , 548
5	(2018) Chaos: An Interdisciplinary Journal of Nonlinear Science Nonlinear dynamics of contact interaction of a size-dependent plate supported by a size-dependent beam / 28 (5) , 053102
1	(2017) Steel & Composite structures : an international journal Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions / 25 (1) , 67
3	(2014) Steel & Composite structures : an international journal Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams / 23 (3) , 339
6	(2014) Steel & Composite structures : an international journal Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory / 26 (6) , 663
3	(2018) Steel & Composite structures : an international journal Size dependent bending analysis of micro/nano sandwich structures based on a nonlocal high order theory / 27 (3) , 371
1	(2014) Steel & Composite structures : an international journal Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter / 28 (1) , 13
6	(2018) Steel & Composite structures : an international journal Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory / 28 (6) , 749
2	(2020) Structural engineering and mechanics : An international journal Buckling and free vibration analyses of nanobeams with surface effects via various higher-order shear deformation theories / 74 (2) , 175
5	(2014) Steel & Composite structures : an international journal On the mechanics of nanocomposites reinforced by wavy/defected/aggregated nanotubes / 38 (5) , 533
None	(2022) Thin-walled structures A review of size-dependent continuum mechanics models for micro- and nano-structures / 170 (None) , 108562