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Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform

Year 2017, Volume: 9 Issue: 2, 103 - 111, 08.06.2017
https://doi.org/10.24107/ijeas.314635

Abstract

In this study, the static behavior of nanobeams subjected to end concentrated
loads is theoretically investigated in the Laplace domain. A closed form of
solution for the title problem is presented using Euler-Bernoulli beam theory.  Nonlocal elasticity theory proposed by Eringen
is used to represent small scale effect. A systems of differential
equations containing a small scale parameter is derived for nanobeams. Laplace
transformation is applied to this systems of differential
equations containing a small scale parameter. The exact static response of
the nanobeam with end concentrated loads is obtained by applying inverse
Laplace transform. The calculate results are plotted in a series of figures for
various combinations of concentrated loads.

References

  • [1] Eringen, A. C., Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1-16, 1972. [2] Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 41, 1651-1655, 2009. [3] Liu, T., Hai, M., Zhao, M., Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system, Eng. Fract. Mech. 75, 4909-4919, 2008. [4] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45, 288-307, 2007. [5] Narendar, S., Buckling analysis of micro-/nano-scale plates based on two variable refined plate theory incorporating nonlocal scale effects, Compos. Struct., 93, 3093-3103, 2011 [6] Pradhan, S.C., Phadikar, J.K., Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib., 325, 206-223, (2009). [7] Shen, L., Shen, H.S., Zhang, C.L., Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput. Mater. Sci., 48, 680-685, 2010. [8] Mercan, K., Civalek, Ö., Buckling Analysis of Silicon Carbide Nanotubes (SiCNTs). Int J Eng Appl Sci, 8(2), 101-108, 2016. [9] Mercan, K., Demir, Ç., Akgöz, B., Civalek, Ö., Coordinate Transformation for Sector and Annular Sector Shaped Graphene Sheets on Silicone Matrix. Int J Eng Appl Sci, 7(2), 56-73, 2015. [10] Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Compos Struct, 143, 300-309, 2016. [11] Gürses, M., Akgöz, B., Civalek, Ö., Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl Math Comput, 219, 3226–3240, 2012. [12] 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. [13] 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. [14] 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. [15] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Appl. Math. Model., 35, 2053-2067, 2011. [16] 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. [17] 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. [18] 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. [19] 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. [20] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88, 2013. [21] Thai, H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64, 2012. [22] 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. [23] Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P., Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737, 2011. [24] Yayli, M.Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube, Acta Physica Polonica A, 127, 3, 678-683, 2015. [25] Yayli, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions, Journal of Mechanical Science and Technology, 29, 8, 3373-3380, 2015. [26] 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. [27] Peddieson, J., Buchanan, G. R., McNitt, R. P., Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci, 41, (3-5), 305-312, 2013.
Year 2017, Volume: 9 Issue: 2, 103 - 111, 08.06.2017
https://doi.org/10.24107/ijeas.314635

Abstract

References

  • [1] Eringen, A. C., Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1-16, 1972. [2] Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 41, 1651-1655, 2009. [3] Liu, T., Hai, M., Zhao, M., Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system, Eng. Fract. Mech. 75, 4909-4919, 2008. [4] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45, 288-307, 2007. [5] Narendar, S., Buckling analysis of micro-/nano-scale plates based on two variable refined plate theory incorporating nonlocal scale effects, Compos. Struct., 93, 3093-3103, 2011 [6] Pradhan, S.C., Phadikar, J.K., Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib., 325, 206-223, (2009). [7] Shen, L., Shen, H.S., Zhang, C.L., Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput. Mater. Sci., 48, 680-685, 2010. [8] Mercan, K., Civalek, Ö., Buckling Analysis of Silicon Carbide Nanotubes (SiCNTs). Int J Eng Appl Sci, 8(2), 101-108, 2016. [9] Mercan, K., Demir, Ç., Akgöz, B., Civalek, Ö., Coordinate Transformation for Sector and Annular Sector Shaped Graphene Sheets on Silicone Matrix. Int J Eng Appl Sci, 7(2), 56-73, 2015. [10] Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Compos Struct, 143, 300-309, 2016. [11] Gürses, M., Akgöz, B., Civalek, Ö., Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl Math Comput, 219, 3226–3240, 2012. [12] 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. [13] 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. [14] 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. [15] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Appl. Math. Model., 35, 2053-2067, 2011. [16] 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. [17] 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. [18] 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. [19] 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. [20] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88, 2013. [21] Thai, H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64, 2012. [22] 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. [23] Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P., Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737, 2011. [24] Yayli, M.Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube, Acta Physica Polonica A, 127, 3, 678-683, 2015. [25] Yayli, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions, Journal of Mechanical Science and Technology, 29, 8, 3373-3380, 2015. [26] 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. [27] Peddieson, J., Buchanan, G. R., McNitt, R. P., Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci, 41, (3-5), 305-312, 2013.
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Details

Subjects Engineering
Journal Section Articles
Authors

Mustafa Özgür Yaylı

Süheyla Yerel Kandemir This is me

Publication Date June 8, 2017
Published in Issue Year 2017 Volume: 9 Issue: 2

Cite

APA Yaylı, M. Ö., & Yerel Kandemir, S. (2017). Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. International Journal of Engineering and Applied Sciences, 9(2), 103-111. https://doi.org/10.24107/ijeas.314635
AMA Yaylı MÖ, Yerel Kandemir S. Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. IJEAS. July 2017;9(2):103-111. doi:10.24107/ijeas.314635
Chicago Yaylı, Mustafa Özgür, and Süheyla Yerel Kandemir. “Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform”. International Journal of Engineering and Applied Sciences 9, no. 2 (July 2017): 103-11. https://doi.org/10.24107/ijeas.314635.
EndNote Yaylı MÖ, Yerel Kandemir S (July 1, 2017) Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. International Journal of Engineering and Applied Sciences 9 2 103–111.
IEEE M. Ö. Yaylı and S. Yerel Kandemir, “Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform”, IJEAS, vol. 9, no. 2, pp. 103–111, 2017, doi: 10.24107/ijeas.314635.
ISNAD Yaylı, Mustafa Özgür - Yerel Kandemir, Süheyla. “Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform”. International Journal of Engineering and Applied Sciences 9/2 (July 2017), 103-111. https://doi.org/10.24107/ijeas.314635.
JAMA Yaylı MÖ, Yerel Kandemir S. Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. IJEAS. 2017;9:103–111.
MLA Yaylı, Mustafa Özgür and Süheyla Yerel Kandemir. “Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform”. International Journal of Engineering and Applied Sciences, vol. 9, no. 2, 2017, pp. 103-11, doi:10.24107/ijeas.314635.
Vancouver Yaylı MÖ, Yerel Kandemir S. Bending Analysis of A Cantilever Nanobeam With End Forces By Laplace Transform. IJEAS. 2017;9(2):103-11.

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