Volume 2, Issue 3 (3-2018)                   NMCE 2018, 2(3): 67-77 | Back to browse issues page

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Soltani M, Mohammadi M. Stability Analysis of Non-Local Euler-Bernoulli Beam with Exponentially Varying Cross-Section Resting on Winkler-Pasternak Foundation . NMCE 2018; 2 (3) :67-77
URL: http://nmce.kntu.ac.ir/article-1-149-en.html
1- Assistant Professor, Department of civil engineering, University of Kashan, Iran , msoltani@kashanu.ac.ir
2- MSc Student in Structural Engineering, University of Kashan, Kashan, Iran
Abstract:   (1239 Views)
In this paper, linear stability analysis of non-prismatic beam resting on uniform Winkler-Pasternak elastic foundation is carried out based on Eringen's non-local elasticity theory. In the context of small displacement, the governing differential equation and the related boundary conditions are obtained via the energy principle. It is also assumed that the width of rectangle cross-section varies exponentially through the beam’s length while its thickness remains constant. The differential quadrature method as a highly accurate mathematical methodology is employed for solving the equilibrium equation and obtaining the critical buckling load of simply supported beam. Several numerical results are finally provided to demonstrate the effects of different parameters such as elastic foundation modulus, nonlocal Eringen’s parameter and tapering ratio on the critical loads of an exponential tapered non-local beam lying on Winkler-Pasternak foundation. The numerical outcomes indicate that the critical loads of pinned-pinned beam decrease by increasing nonlocal parameter. Furthermore, results show that the elastic foundation enhances the stability characteristics of non-local Euler-Bernoulli beam with constant or variable cross-section. It is finally concluded that the effect of non-uniformity in the cross-section plays significant roles on linear stability behavior of non-local beam. 
Full-Text [PDF 991 kb]   (833 Downloads)    
Type of Study: Research | Subject: General
Received: 2017/06/7 | Revised: 2017/09/2 | Accepted: 2017/12/7 | ePublished ahead of print: 2017/12/19

1. [1] Akgoz B., Civalek O., "Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity", Structural Engineering and Mechanics, vol. 48, 2013, p.195-205. [DOI:10.12989/sem.2013.48.2.195]
2. [2] Aydogdu M., "Axial vibration of the nanorods with the nonlocal continuum rod model", Physica E., vol. 41, 2009, p. 861-864. [DOI:10.1016/j.physe.2009.01.007]
3. [3] Bellman R.E., Casti J., "Differential quadrature and long-term integration", Journal of Mathematical Analysis and Applications, vol. 34, 1971, p. 235-238. [DOI:10.1016/0022-247X(71)90110-7]
4. [4] Civalek O., Demir C., Akgoz B., "Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model", Math. Comput. Appl, vol. 15, 2010, p. 289-298. [DOI:10.3390/mca15020289]
5. [5] Ebrahimi F., Salari E., Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method", Composite Part B., vol. 79, 2015, p. 156-169. [DOI:10.1016/j.compositesb.2015.04.010]
6. [6] Ebrahimi F., Mokhtari M., "Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method", Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 37(4), 2015, p. 1435-1444. [DOI:10.1007/s40430-014-0255-7]
7. [7] Eringen A.C., Suhubi E.S., "Nonlinear theory of simple micro-elastic solids-I", International Journal of Engineering Science, vol. 2, 1964, p. 189-203. [DOI:10.1016/0020-7225(64)90004-7]
8. [8] Eringen A.C., "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", Journal of Applied Physics, vol. 54, 1983, p. 4703-4710. [DOI:10.1063/1.332803]
9. [9] Fleck N.A., Muller G.M., Ashby M.F., Hutchinson, J.W., "Strain gradient plasticity: theory and experiment", Acta Metall. Mater., vol. 42, 1999, p. 475-487. [DOI:10.1016/0956-7151(94)90502-9]
10. [10] Ghannadpour S.A.M., Mohammadi B., Fazilati J., "Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method", Composite Structures, vol. 96, 2013, p. 584-589. [DOI:10.1016/j.compstruct.2012.08.024]
11. [11] Hosseini Hashemi S., Bakhshi Khaniki H., "Analytical solution for free vibration of a variable cross-section nonlocal nanobeam", IJE TRANSACTIONS B Applications, vol. 29(5), 2016, p. 688-696. [DOI:10.5829/idosi.ije.2016.29.05b.13]
12. [12] Khajeansari A., Baradaran G.H., Yvonnet J., "An explicit solution for bending of nanowires lying on Winkler-Pasternak elastic substrate medium based on the Euler-Bernoulli beam theory", International Journal of Engineering Science, vol. 52, 2012, p. 115-128. [DOI:10.1016/j.ijengsci.2011.11.004]
13. [13] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, vol. 51, 2003, p. 1477-1508. [DOI:10.1016/S0022-5096(03)00053-X]
14. [14] Liew K.M., He X.Q., Kitipornchai S., "Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix", Acta Materialia, vol. 54, 2006, p. 4229-4236. [DOI:10.1016/j.actamat.2006.05.016]
15. [15] Malekzadeh P., Shojaee M., "Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams", Composites Part B, vol. 52, 2013, p. 84-92. [DOI:10.1016/j.compositesb.2013.03.046]
16. [16] Mercan K., Civalek O., "DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix", Composite Structures, vol. 143, 2016, p. 300-309. [DOI:10.1016/j.compstruct.2016.02.040]
17. [17] Murmu T., Pradhan S.C., "Buckling analysis of a single-walled carbon nano-tube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM", Physica E., vol. 41, 2009, p. 1232-1239. [DOI:10.1016/j.physe.2009.02.004]
18. [18] Pandeya A., Singhb J., "A variational principle approach for vibration of non-uniform nanocantilever using nonlocal elasticity theory", Proced. Mater. Sci, vol. 10, 2015, p. 497-506. [DOI:10.1016/j.mspro.2015.06.087]
19. [19] Peddieson J., Buchanan G.R., McNitt R.P., "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., vol. 41(3-5), 2003, p. 305-312. [DOI:10.1016/S0020-7225(02)00210-0]
20. [20] Phadikar J., Pradhan S., "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Computational Mat Sci, vol. 49(3), 2010, p. 492-499. [DOI:10.1016/j.commatsci.2010.05.040]
21. [21] Reddy J.N., "Nonlocal theories for bending, buckling and vibration of beams", Int J Eng Sci, vol. 45, 2007, p. 288-307. [DOI:10.1016/j.ijengsci.2007.04.004]
22. [22] Rahmanian M., Torkaman-Asadi M.A., Firouz-Abadi R.D., Kouchakzadeh M.A., "Free vibrations analysis of carbon nanotubes resting on Winkler foundations based on nonlocal models", Physica B, vol. 484, 2016, p. 83-94. [DOI:10.1016/j.physb.2015.12.041]
23. [23] Refaeinejad V., Rahmani O., Hosseini S.A.H., "An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories", Scientia Iranica F, vol. 24(3), 2017, p. 1635-1653. [DOI:10.24200/sci.2017.4141]
24. [24] Stolken, J.S., Evans, A.G., "Microbend test method for measuring the plasticity length scale", Acta Materialia vol. 46, 1998, p. 5109-5115 [DOI:10.1016/S1359-6454(98)00153-0]
25. [25] Sudak L.J., "Column buckling of multi-walled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., vol. 94(11), 2003, p. 7281-7287. [DOI:10.1063/1.1625437]
26. [26] Torabi K., Rahi M., Afshari H., "Transverse Vibration for Non-uniform Timoshenko Nano-beams", Mechanics of advanced Composite Structures, Vol. 2 2015, p. 1-16.
27. [27] Tsiatas G.C., "A new efficient method to evaluate exact stiffness and mass matrices of non-uniform beams resting on an elastic foundation", Archive of Applied Mechanic, vol. 84, 2014, p. 615-623. [DOI:10.1007/s00419-014-0820-7]
28. [28] Wang CM, Zhang YY, He X.Q., "Vibration of Non-local Timoshenko Beams", Nanotechnology, vol. 18, 2007, p. 1-9. [DOI:10.1088/0957-4484/18/10/105401]
29. [29] Wang Q., Liew K.M., "Application of nonlocal continuum mechanics to static analysis of micro-and nano-structures", Physics Letters A, vol. 363, 2007, p. 236-242. [DOI:10.1016/j.physleta.2006.10.093]
30. [30] Wang B.L., Hoffman M., Yu A.B.M.," Buckling analysis of embedded nanotubes using gradient continuum theory", Mech Mater, vol. 45, 2012, p. 52-60. [DOI:10.1016/j.mechmat.2011.10.003]

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