Araştırma Makalesi
BibTex RIS Kaynak Göster

Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series

Yıl 2017, Cilt: 9 Sayı: 4, 89 - 99, 27.12.2017

Hayrullah Kadıoğlu , Mustafa Özgür Yaylı

https://doi.org/10.24107/ijeas.362242
Cited By: 3

Öz

In this study, buckling
analysis of a nano sized beam has been performed by using Timoshenko beam
theory and Eringen’s nonlocal elasticity theory. Timoshenko beam theory takes
into account not only bending moment but also shear force. Therefore, it gives
more accurate outcomes than Euler Bernoulli beam theory. Moreover, Eringen’s
nonlocal elasticity theory takes into account the small scale effect. Thus,
these two theories are utilized in this study. The vertical displacement function
is chosen as a Fourier sine series. 
Similarly, the rotation function is chosen as a Fourier cosine series.
These functions are enforced by Stokes’ transformation, and higher order
derivatives of them are obtained. These derivatives are written in the
governing equations for the buckling of nonlocal Timoshenko beams. Hence
Fourier coefficients are acquired.  Subsequently
boundary condition of established beam model is identified with Timoshenko beam
and Eringen’s nonlocal elasticity theories, and the linear equations are
obtained.  A coefficients matrix is
created by utilizing these linear systems of equations. When determinant of
this coefficient matrix is calculated, the critical buckling loads are
acquired. Finally, achieved outcomes are compared with other studies in the
literature.  Calculated results are also
presented in a series of figures and tables

Anahtar Kelimeler

Timoshenko beam theory Eringen’s nonlocal elasticity theory elastic buckling analysis Stokes’ transformation

Kaynakça

  • [1] Eringen, A. C., Nonlocal polar elastic continua, International journal of engineering science. 10, 1-16, 1972.
  • [2] Eringen A. C. and Edelen D. G. B., On nonlocal elasticity. Int. J. Eng. Sci, 10, 233–48, 1972.
  • [3] Eringen A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54, 4703–4710, 1983.
  • [4] Li, C. Y., Chou T. W., A structural mechanics approach for the analysis of carbon nanotubes, Int. J. Solids Struct. 40, 2487-2499, 2003.
  • [5] Chowdhury, R., Adhikari, S., Wang, C. W., Scarpa, F., A molecular mechanics approach for the vibration of single walled carbon nanotubes. Comput. Mater. Sci., 48, 730-735, 2010.
  • [6] Poncharal, P., Wang, Z. L., Ugarte, D., Heer, W. A. D., Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science, 283, 1513-1516, 1999.
  • [7] Wang, C. M., Zhang, Y. Y., Ramesh, S. S., Kitipornchai, S., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 17, 3904-3909, 2006.
  • [8] Ghannadpour, S. A. M., and Mohammadi, B., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory using Chebyshev polynomials. Advanced Materials Research, 123, 619-622, 2010.
  • [9] Yayli M. Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube Embedded In An Elastic Medium Using Nonlocal Elasticity. Int J Eng Appl Sci, 8(2), 40-50, 2016. [10] Yayli M. Ö., An Analytical Solution for Free Vibrations of A Cantilever Nanobeam with A Spring Mass System. Int J Eng Appl Sci, 7(4), 10-18, 2016.
  • [11] Civalek, Ö., Akgöz, B., Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler–Bernoulli beam modeling. Sci. Iranica Trans. B: Mech. Eng., 17, 367-375, 2010.
  • [12] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Appl. Math. Model., 35, 2053-2067, 2011.
  • [13] Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M., Beam bending solutions based on nonlocal Timoshenko beam theory. J. Eng. Mech., 134, 475-481, 2008.
  • [14] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys., 99, 73510-73518, 2006.
  • [15] Murmu, T., Pradhan, S.C., Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory. Physica E, 41, 1451-1456, 2009.
  • [16] Rahmani, O., Pedram, O., Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci, 77, 55-70, 2014.
  • [17] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88, 2013
  • [18] Thai, H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64, 2012.
  • [19] Reddy J. N., Pang, S. D., Nonlocal continuum theories of beam for the analysis of carbon nanotubes. Journal of Applied Physics, 103, 1-16, 2008.
  • [20] Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P., Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737, 2011
  • [21] Yayli, M.Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube. Acta Physica Polonica A, 127, 3, 678-683, 2015.
  • [22] Yayli, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions. Journal of Mechanical Science and Technology, 29, 8, 3373-3380, 2015.
  • [23] Artan R., Tepe A., The initial values method for buckling of nonlocal bars with application in nanotechnology. European Journal of Mechanics-A/Solids, 27, (3), 469-477, 2008.
  • [24] Demir, Ç., Civalek, Ö., Nonlocal finite element formulation for vibration. International Journal of Engineering and Applied Sciences, 8(2), 109-117, 2016.
  • [25] Demir, Ç., Civalek, Ö., On the analysis of microbeams. International Journal of Engineering Science, 121(Supplement C), 14-33, 2017.
  • [26] Demir, Ç., Civalek, Ö., Nonlocal deflection of microtubules under point load. International Journal of Engineering and Applied Sciences, 7(3), 33-39, 2015
  • [27] Akgoz, B., Civalek, Ö., Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity. Structural Engineering and Mechanics, 48(2), 195-205, 2013.
  • [28] Akgöz, B., Civalek, Ö., Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica 119, 1-12, 2016.
  • [29] Civalek, Ö., Demir Ç., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation 289, 335-352, 2016.
  • [30] Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Composite Structures 143, 300-309, 2016.
  • [31] Civalek, Ö., Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory. Journal of Composite Materials 42(26), 2853-2867, 2008.
  • [32] Yayli, M.Ö., On the axial vibration of carbon nanotubes with different boundary conditions. Micro & Nano Letters 9(11), 807-811, 2014.
  • [33] Yayli, M.Ö., Torsion of nonlocal bars with equilateral triangle cross sections. Journal of Computational and Theoretical Nanoscience 10(2), 376-379, 2013.
  • [34] Yayli, M.Ö., Weak formulation of finite element method for nonlocal beams using additional boundary conditions. Journal of Computational and Theoretical Nanoscience 8(11), 2173-2180, 2011.
  • [35] Yayli, M.Ö., Çerçevik A. E., 1725. Axial vibration analysis of cracked nanorods with arbitrary boundary conditions. Journal of Vibroengineering 17(6), 2907-2921, 2015.
  • [36] Yayli, M.Ö., Airy Stress Functions for Transverse Sinusoidally Loaded Beam in Nonlocal Elasticity. Journal of Computational and Theoretical Nanoscience 8(10), 2006-2012, 2011.

Yıl 2017, Cilt: 9 Sayı: 4, 89 - 99, 27.12.2017

Hayrullah Kadıoğlu , Mustafa Özgür Yaylı

https://doi.org/10.24107/ijeas.362242
Cited By: 3

Öz

Kaynakça

  • [1] Eringen, A. C., Nonlocal polar elastic continua, International journal of engineering science. 10, 1-16, 1972.
  • [2] Eringen A. C. and Edelen D. G. B., On nonlocal elasticity. Int. J. Eng. Sci, 10, 233–48, 1972.
  • [3] Eringen A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54, 4703–4710, 1983.
  • [4] Li, C. Y., Chou T. W., A structural mechanics approach for the analysis of carbon nanotubes, Int. J. Solids Struct. 40, 2487-2499, 2003.
  • [5] Chowdhury, R., Adhikari, S., Wang, C. W., Scarpa, F., A molecular mechanics approach for the vibration of single walled carbon nanotubes. Comput. Mater. Sci., 48, 730-735, 2010.
  • [6] Poncharal, P., Wang, Z. L., Ugarte, D., Heer, W. A. D., Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science, 283, 1513-1516, 1999.
  • [7] Wang, C. M., Zhang, Y. Y., Ramesh, S. S., Kitipornchai, S., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 17, 3904-3909, 2006.
  • [8] Ghannadpour, S. A. M., and Mohammadi, B., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory using Chebyshev polynomials. Advanced Materials Research, 123, 619-622, 2010.
  • [9] Yayli M. Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube Embedded In An Elastic Medium Using Nonlocal Elasticity. Int J Eng Appl Sci, 8(2), 40-50, 2016. [10] Yayli M. Ö., An Analytical Solution for Free Vibrations of A Cantilever Nanobeam with A Spring Mass System. Int J Eng Appl Sci, 7(4), 10-18, 2016.
  • [11] Civalek, Ö., Akgöz, B., Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler–Bernoulli beam modeling. Sci. Iranica Trans. B: Mech. Eng., 17, 367-375, 2010.
  • [12] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Appl. Math. Model., 35, 2053-2067, 2011.
  • [13] Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M., Beam bending solutions based on nonlocal Timoshenko beam theory. J. Eng. Mech., 134, 475-481, 2008.
  • [14] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys., 99, 73510-73518, 2006.
  • [15] Murmu, T., Pradhan, S.C., Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory. Physica E, 41, 1451-1456, 2009.
  • [16] Rahmani, O., Pedram, O., Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci, 77, 55-70, 2014.
  • [17] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88, 2013
  • [18] Thai, H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64, 2012.
  • [19] Reddy J. N., Pang, S. D., Nonlocal continuum theories of beam for the analysis of carbon nanotubes. Journal of Applied Physics, 103, 1-16, 2008.
  • [20] Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P., Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737, 2011
  • [21] Yayli, M.Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube. Acta Physica Polonica A, 127, 3, 678-683, 2015.
  • [22] Yayli, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions. Journal of Mechanical Science and Technology, 29, 8, 3373-3380, 2015.
  • [23] Artan R., Tepe A., The initial values method for buckling of nonlocal bars with application in nanotechnology. European Journal of Mechanics-A/Solids, 27, (3), 469-477, 2008.
  • [24] Demir, Ç., Civalek, Ö., Nonlocal finite element formulation for vibration. International Journal of Engineering and Applied Sciences, 8(2), 109-117, 2016.
  • [25] Demir, Ç., Civalek, Ö., On the analysis of microbeams. International Journal of Engineering Science, 121(Supplement C), 14-33, 2017.
  • [26] Demir, Ç., Civalek, Ö., Nonlocal deflection of microtubules under point load. International Journal of Engineering and Applied Sciences, 7(3), 33-39, 2015
  • [27] Akgoz, B., Civalek, Ö., Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity. Structural Engineering and Mechanics, 48(2), 195-205, 2013.
  • [28] Akgöz, B., Civalek, Ö., Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica 119, 1-12, 2016.
  • [29] Civalek, Ö., Demir Ç., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation 289, 335-352, 2016.
  • [30] Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Composite Structures 143, 300-309, 2016.
  • [31] Civalek, Ö., Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory. Journal of Composite Materials 42(26), 2853-2867, 2008.
  • [32] Yayli, M.Ö., On the axial vibration of carbon nanotubes with different boundary conditions. Micro & Nano Letters 9(11), 807-811, 2014.
  • [33] Yayli, M.Ö., Torsion of nonlocal bars with equilateral triangle cross sections. Journal of Computational and Theoretical Nanoscience 10(2), 376-379, 2013.
  • [34] Yayli, M.Ö., Weak formulation of finite element method for nonlocal beams using additional boundary conditions. Journal of Computational and Theoretical Nanoscience 8(11), 2173-2180, 2011.
  • [35] Yayli, M.Ö., Çerçevik A. E., 1725. Axial vibration analysis of cracked nanorods with arbitrary boundary conditions. Journal of Vibroengineering 17(6), 2907-2921, 2015.
  • [36] Yayli, M.Ö., Airy Stress Functions for Transverse Sinusoidally Loaded Beam in Nonlocal Elasticity. Journal of Computational and Theoretical Nanoscience 8(10), 2006-2012, 2011.
Toplam 35 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Hayrullah Kadıoğlu

Mustafa Özgür Yaylı

Yayımlanma Tarihi 27 Aralık 2017
Kabul Tarihi 20 Aralık 2017
Yayımlandığı Sayı Yıl 2017 Cilt: 9 Sayı: 4

Kaynak Göster

APA Kadıoğlu, H., & Yaylı, M. Ö. (2017). Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series. International Journal of Engineering and Applied Sciences, 9(4), 89-99. https://doi.org/10.24107/ijeas.362242
AMA Kadıoğlu H, Yaylı MÖ. Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series. IJEAS. Aralık 2017;9(4):89-99. doi:10.24107/ijeas.362242
Chicago Kadıoğlu, Hayrullah, ve Mustafa Özgür Yaylı. “Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series”. International Journal of Engineering and Applied Sciences 9, sy. 4 (Aralık 2017): 89-99. https://doi.org/10.24107/ijeas.362242.
EndNote Kadıoğlu H, Yaylı MÖ (01 Aralık 2017) Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series. International Journal of Engineering and Applied Sciences 9 4 89–99.
IEEE H. Kadıoğlu ve M. Ö. Yaylı, “Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series”, IJEAS, c. 9, sy. 4, ss. 89–99, 2017, doi: 10.24107/ijeas.362242.
ISNAD Kadıoğlu, Hayrullah - Yaylı, Mustafa Özgür. “Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series”. International Journal of Engineering and Applied Sciences 9/4 (Aralık 2017), 89-99. https://doi.org/10.24107/ijeas.362242.
JAMA Kadıoğlu H, Yaylı MÖ. Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series. IJEAS. 2017;9:89–99.
MLA Kadıoğlu, Hayrullah ve Mustafa Özgür Yaylı. “Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series”. International Journal of Engineering and Applied Sciences, c. 9, sy. 4, 2017, ss. 89-99, doi:10.24107/ijeas.362242.
Vancouver Kadıoğlu H, Yaylı MÖ. Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series. IJEAS. 2017;9(4):89-9.

Cited By

Weighted Residual Approach for Bending Analysis of Nanobeam Using by Modified Couple Stress Theory
International Journal of Engineering and Applied Sciences
Mustafa Özgür YAYLI
https://doi.org/10.24107/ijeas.932580
Buckling of elastically restrained nonlocal carbon nanotubes under concentrated and uniformly distributed axial loads
Mechanical Sciences
Mouafo Teifouet Armand Robinson
https://doi.org/10.5194/ms-10-145-2019
Finite Element Model of Functionally Graded Nanobeam for Free Vibration Analysis
International Journal of Engineering and Applied Sciences
Büşra Uzun
https://doi.org/10.24107/ijeas.569798
 Kapak Resmi İndir

21357download