Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity

  • Akgoz, Bekir (Civil Engineering Department, Division of Mechanics, Akdeniz University) ;
  • Civalek, Omer (Civil Engineering Department, Division of Mechanics, Akdeniz University)
  • Received : 2012.05.07
  • Accepted : 2013.10.02
  • Published : 2013.10.25
⟨ Previous Next ⟩

Abstract

The buckling problem of linearly tapered micro-columns is investigated on the basis of modified strain gradient elasticity theory. Bernoulli-Euler beam theory is used to model the non-uniform micro column. Rayleigh-Ritz solution method is utilized to obtain the critical buckling loads of the tapered cantilever micro-columns for different taper ratios. Some comparative results for the cases of rectangular and circular cross-sections are presented in graphical and tabular form to show the differences between the results obtained by modified strain gradient elasticity theory and those achieved by modified couple stress and classical theories. From the results, it is observed that the differences between critical buckling loads achieved by classical and those predicted by non-classical theories are considerable for smaller values of the ratio of the micro-column thickness (or diameter) at its bottom end to the additional material length scale parameters and the differences also increase due to increasing of the taper ratio.

Keywords

References

  1. Akgoz, B. and Civalek, O. (2011a), "Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams", Int. J. Eng. Sci., 49(11), 1268-1280. https://doi.org/10.1016/j.ijengsci.2010.12.009
  2. Akgoz, B. and Civalek, O. (2011b), "Application of strain gradient elasticity theory for buckling analysis of protein microtubules", Curr. Appl. Phys., 11(5), 1133-1138. https://doi.org/10.1016/j.cap.2011.02.006
  3. Akgoz, B. and Civalek, O. (2012), "Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory", Arch. Appl. Mech., 82(3), 423-443. https://doi.org/10.1007/s00419-011-0565-5
  4. Akgoz, B. and Civalek, O. (2013a), "Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory", Compos. Struct., 98, 314-322. https://doi.org/10.1016/j.compstruct.2012.11.020
  5. Akgoz, B. and Civalek, O. (2013b), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004
  6. Alizada, A.N. and Sofiyev, A.H. (2011), "Modified Young's moduli of nano-materials taking into account the scale effects and vacancies", Meccanica, 46, 915-920. https://doi.org/10.1007/s11012-010-9349-1
  7. Civalek, O. and Akgoz, B. (2010), "Free vibration analysis of microtubules as cytoskeleton components: Nonlocal Euler-Bernoulli beam modeling", Scientia Iranica, Trans. B-Mech. Eng., 17(5), 367-375.
  8. Civalek, O., Demir, C. and Akgoz, B. (2010), "Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model", Math. Comput. Appl., 15(2), 289-298.
  9. Civalek, O. and Demir, C. (2011), "Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory", Appl. Math. Model., 35, 2053-2067. https://doi.org/10.1016/j.apm.2010.11.004
  10. Darbandi, S.M., Firouz-Abadi, R.D. and Haddadpour, H. (2010), "Buckling of variable section columns under axial loading", J. Eng. Mech., 136(4), 472-476. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000096
  11. Demir, C., Civalek, O. and Akgoz, B. (2010), "Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique", Math. Comput. Appl., 15(1), 57-65.
  12. Eisenberger, M. (1991), "Buckling loads for variable cross-section members with variable axial forces", Int. J. Solids Struct., 27(2), 135-143. https://doi.org/10.1016/0020-7683(91)90224-4
  13. Elishakoff, I. and Bert, C.W. (1988), "Comparison of Rayleigh's noninteger-power method with Rayleigh-Ritz method", Comput. Methods Appl. Mech. Eng., 67(3), 297-309. https://doi.org/10.1016/0045-7825(88)90050-3
  14. 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. https://doi.org/10.1063/1.332803
  15. Fleck, N.A., Muller, G.M., Asbhy, M.F. and Hutchinson, J.W. (1994), "Strain gradient plasticity: theory and experiment", Acta Metall. Mater., 42(2), 475-487. https://doi.org/10.1016/0956-7151(94)90502-9
  16. Gere, J.M. and Carter, W.O. (1962), "Critical buckling loads for tapered columns", J. Struct. Eng., 88(1), 1-11.
  17. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  18. Liew, K.M., Hu, Y.A. and He, X.Q. (2008), "Flexural wave propagation in single-walled carbon nanotubes", J. Comput. Theor. Nanosci., 5(4), 581-586. https://doi.org/10.1166/jctn.2008.019
  19. Lim, C.W. (2009), "Equilibrium and static deflection for bending of a nonlocal nanobeam", Adv. Vib. Eng., 8(4), 277-300.
  20. Lim, C.W. (2010), "On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: Equilibrium, governing equation and static deflection", Appl. Math. Mech., 31, 37-54. https://doi.org/10.1007/s10483-010-0105-7
  21. Lim, C.W., Niu, J.C. and Yu, Y.M. (2010), "Nonlocal stress theory for buckling instability of nanotubes: New predictions on stiffness strengthening effects of nanoscales", J. Comput. Theor. Nanosci., 7, 2104-2111. https://doi.org/10.1166/jctn.2010.1591
  22. Lim, C.W. and Wang, C.M. (2007), "Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams", J. Appl. Phys., 101, 54312-316. https://doi.org/10.1063/1.2435878
  23. Liu, G.R., Cheng, Y., Mi, D. and Li, Z.R. (2005), "A study on self-insertion of peptides into single-walled carbon nanotubes based on molecular dynamics simulation", Int. J. Modern Phys. C, 16, 1239-1250. https://doi.org/10.1142/S0129183105007856
  24. Shen, H.S. (2010), "Nonlocal shear deformable shell model for bending buckling of micro tubules embedded in an elastic medium'', Phys. Lett. A, 374, 4030-4039. https://doi.org/10.1016/j.physleta.2010.08.006
  25. Shen, L., Shen, H.S. and Zhang, C.L. (2010a), "Temperature-dependent elastic properties of single layer graphene sheets", Mater. Design, 31, 4445-4449. https://doi.org/10.1016/j.matdes.2010.04.016
  26. Shen, L., Shen, H.S. and Zhang, C.L. (2010b), "Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments", Comput. Mater. Sci., 48, 680-685. https://doi.org/10.1016/j.commatsci.2010.03.006
  27. Sudak, L.J. (2003), "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94(11), 7281-7287. https://doi.org/10.1063/1.1625437
  28. Şimsek, M. (2010). "Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory", Int. J. Eng. Sci., 48(12), 1721-1732. https://doi.org/10.1016/j.ijengsci.2010.09.027
  29. Şimsek, M. (2011), "Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059
  30. Timoshenko, S.P. and Gere, J.M. (1961), Theory of Elastic Stability, McGraw-Hill, New York.
  31. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  32. Wang, Q. and Liew, K.M. (2007), "Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures", Phys. Lett. A, 363(3), 236-242. https://doi.org/10.1016/j.physleta.2006.10.093
  33. Wang, Q., Liew, K.M. and Duan, W.H. (2008), "Modeling of the mechanical instability of carbon nanotubes", Carbon, 46(2), 285-290. https://doi.org/10.1016/j.carbon.2007.11.022
  34. Wang, C.M., Wang, C.Y. and Reddy, J.N. (2005). Exact Solutions for Buckling of Structural Members, Chap. 2, CRC Press, Boca Raton, Florida.
  35. Zhang, C.L. and Shen, H.S. (2007), "Thermal buckling of initially compressed single-walled carbon nanotubes by molecular dynamics simulation", Carbon, 45(13), 2614-2620. https://doi.org/10.1016/j.carbon.2007.08.007
  36. Zhang, Y.Q., Liu, G.R. and Han, X. (2004), "Analysis of strain localization for ductile materials with effect of void growth", Int. J. Mech. Sci., 46(7), 1021-1034. https://doi.org/10.1016/j.ijmecsci.2004.07.011
  37. Zhang, Y.Q., Liu, G.R. and Han, X. (2006), "Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure", Phys. Lett. A, 349(5), 370-376. https://doi.org/10.1016/j.physleta.2005.09.036
  38. Zhang, Y.Q., Liu, G.R., Qiang, H.F. and Li, G.Y. (2006), "Investigation of buckling of double-walled carbon nanotubes embedded in an elastic medium using the energy method", Int. J. Mech. Sci., 48(1), 53-61. https://doi.org/10.1016/j.ijmecsci.2005.09.010
  39. Zhang, Y.Q., Liu, G.R. and Wang, J.S. (2004), "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression", Phys. Rev. B, 70(20), 205430. https://doi.org/10.1103/PhysRevB.70.205430

Cited by

  1. Size-dependent piezoelectric energy-harvesting analysis of micro/nano bridges subjected to random ambient excitations vol.27, pp.2, 2018, https://doi.org/10.1088/1361-665X/aaa1a9
  2. On size-dependent vibration of rotary axially functionally graded microbeam vol.101, 2016, https://doi.org/10.1016/j.ijengsci.2015.12.008
  3. Experimental and mathematical analysis of a piezoelectrically actuated multilayered imperfect microbeam subjected to applied electric potential vol.184, 2018, https://doi.org/10.1016/j.compstruct.2017.10.062
  4. Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams vol.129, 2017, https://doi.org/10.1016/j.compositesb.2017.07.024
  5. Buckling analysis of tapered nanobeams using nonlocal strain gradient theory and a generalized differential quadrature method vol.4, pp.6, 2017, https://doi.org/10.1088/2053-1591/aa7111
  6. Nonlinear size-dependent dynamic buckling analysis of embedded micro cylindrical shells reinforced with agglomerated CNTs using strain gradient theory vol.23, pp.12, 2017, https://doi.org/10.1007/s00542-017-3407-8
  7. Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method vol.132, pp.11, 2017, https://doi.org/10.1140/epjp/i2017-11757-4
  8. Vibrational behavior of rotating pre-twisted functionally graded microbeams in thermal environment vol.157, 2016, https://doi.org/10.1016/j.compstruct.2016.08.031
  9. Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM vol.149, 2016, https://doi.org/10.1016/j.compstruct.2016.04.024
  10. A unified formulation for static behavior of nonlocal curved beams vol.59, pp.3, 2016, https://doi.org/10.12989/sem.2016.59.3.475
  11. Bending, buckling and vibration of small-scale tapered beams vol.120, 2017, https://doi.org/10.1016/j.ijengsci.2017.08.005
  12. Nonlinear static and transient isogeometric analysis of functionally graded microplates based on the modified strain gradient theory vol.153, 2017, https://doi.org/10.1016/j.engstruct.2017.10.002
  13. Torsional vibration analysis of nanorods with elastic torsional restraints using non-local elasticity theory 2018, https://doi.org/10.1049/mnl.2017.0751
  14. Analytical solutions for static bending of edge cracked micro beams vol.59, pp.3, 2016, https://doi.org/10.12989/sem.2016.59.3.579
  15. A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation vol.176, 2017, https://doi.org/10.1016/j.compstruct.2017.06.039
  16. Predicting dynamic response of large amplitude free vibrations of cantilever tapered beams on a nonlinear elastic foundation vol.87, pp.4, 2017, https://doi.org/10.1007/s00419-016-1221-x
  17. Geometrically nonlinear free vibration analysis of axially functionally graded taper beams vol.18, pp.4, 2015, https://doi.org/10.1016/j.jestch.2015.04.003
  18. Investigating the non-classical boundary conditions relevant to strain gradient theories vol.86, 2017, https://doi.org/10.1016/j.physe.2016.09.012
  19. Free Vibration of Edge Cracked Functionally Graded Microscale Beams Based on the Modified Couple Stress Theory vol.17, pp.03, 2017, https://doi.org/10.1142/S021945541750033X
  20. Thermal buckling of rotating pre-twisted functionally graded microbeams with temperature-dependent material properties vol.228, pp.3, 2017, https://doi.org/10.1007/s00707-016-1759-2
  21. Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium vol.18, pp.6, 2016, https://doi.org/10.12989/sss.2016.18.6.1125
  22. A review of continuum mechanics models for size-dependent analysis of beams and plates vol.177, 2017, https://doi.org/10.1016/j.compstruct.2017.06.040
  23. On vibration behavior of rotating functionally graded double-tapered beam with the effect of porosities vol.230, pp.10, 2016, https://doi.org/10.1177/0954410015619647
  24. Nonlinear instability of hydrostatic pressurized microtubules surrounded by cytoplasm of a living cell including nonlocality and strain gradient microsize dependency vol.229, pp.1, 2018, https://doi.org/10.1007/s00707-017-1978-1
  25. Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method vol.24, pp.3, 2017, https://doi.org/10.1080/15376494.2015.1128025
  26. Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method 2018, https://doi.org/10.1016/j.aej.2017.06.001
  27. Buckling of bimorph functionally graded piezoelectric cylindrical nanoshell 2018, https://doi.org/10.1177/0954406217738033
  28. Postbuckling of magneto-electro-elastic CNT-MT composite nanotubes resting on a nonlinear elastic medium in a non-uniform thermal environment vol.133, pp.3, 2018, https://doi.org/10.1140/epjp/i2018-11942-y
  29. Comprehensive nonlocal analysis of piezoelectric nanobeams with surface effects in bending, buckling and vibrations under magneto-electro-thermo-mechanical loading vol.5, pp.3, 2018, https://doi.org/10.1088/2053-1591/aab46d
  30. Effects of rotational restraints on the thermal buckling of carbon nanotube pp.1750-0443, 2018, https://doi.org/10.1049/mnl.2018.5428
  31. Finite Element Formulation for Linear Stability Analysis of Axially Functionally Graded Nonprismatic Timoshenko Beam pp.1793-6764, 2018, https://doi.org/10.1142/S0219455419500020
  32. Higher order couple stress theory of plates and shells vol.98, pp.10, 2018, https://doi.org/10.1002/zamm.201800022
  33. Nonlocal Analysis of Natural Vibrations of Carbon Nanotubes pp.1544-1024, 2018, https://doi.org/10.1007/s11665-018-3673-3
  34. A modified couple stress theory for buckling analysis of higher order inhomogeneous microbeams with porosities pp.2041-2983, 2019, https://doi.org/10.1177/0954406218791642
  35. The comparison of strain gradient effects for each component in static and dynamic analyses of FGM micro-beams vol.229, pp.9, 2018, https://doi.org/10.1007/s00707-018-2192-5
  36. Thermal effect on bending, buckling and free vibration of functionally graded rectangular micro-plates possessing a variable length scale parameter vol.24, pp.8, 2018, https://doi.org/10.1007/s00542-018-3773-x
  37. Free longitudinal vibration of a nanorod with elastic spring boundary conditions made of functionally graded material vol.13, pp.7, 2018, https://doi.org/10.1049/mnl.2018.0181
  38. Size-dependent nonlinear pull-in instability of the clamped cylindrical thin micro-/nanoshell based on the non-classical theories pp.0974-9845, 2019, https://doi.org/10.1007/s12648-018-1332-z
  39. Free vibrations of fluid conveying microbeams under non-ideal boundary conditions vol.24, pp.2, 2013, https://doi.org/10.12989/scs.2017.24.2.141
  40. Strain gradient theory for vibration analysis of embedded CNT-reinforced micro Mindlin cylindrical shells considering agglomeration effects vol.62, pp.5, 2013, https://doi.org/10.12989/sem.2017.62.5.551
  41. Static behavior of thermally loaded multilayered Magneto-Electro-Elastic beam vol.63, pp.4, 2013, https://doi.org/10.12989/sem.2017.63.4.481
  42. Surface effects on vibration and buckling behavior of embedded nanoarches vol.64, pp.1, 2017, https://doi.org/10.12989/sem.2017.64.1.001
  43. Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment vol.64, pp.1, 2013, https://doi.org/10.12989/sem.2017.64.1.121
  44. A nonlocal strain gradient theory for nonlinear free and forced vibration of embedded thick FG double layered nanoplates vol.68, pp.1, 2018, https://doi.org/10.12989/sem.2018.68.1.103
  45. Dynamic response of functionally graded annular/circular plate in contact with bounded fluid under harmonic load vol.65, pp.5, 2018, https://doi.org/10.12989/sem.2018.65.5.523
  46. Wave dispersion characteristics of nonlocal strain gradient double-layered graphene sheets in hygro-thermal environments vol.65, pp.6, 2018, https://doi.org/10.12989/sem.2018.65.6.645
  47. A unified formulation for modeling of inhomogeneous nonlocal beams vol.66, pp.3, 2013, https://doi.org/10.12989/sem.2018.66.3.369
  48. Two-dimensional thermo-elastic analysis of FG-CNTRC cylindrical pressure vessels vol.27, pp.4, 2013, https://doi.org/10.12989/scs.2018.27.4.525
  49. A new quasi-3D higher shear deformation theory for vibration of functionally graded carbon nanotube-reinforced composite beams resting on elastic foundation vol.66, pp.6, 2018, https://doi.org/10.12989/sem.2018.66.6.771
  50. Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter vol.28, pp.1, 2013, https://doi.org/10.12989/scs.2018.28.1.013
  51. Bending of a cracked functionally graded nanobeam vol.6, pp.3, 2013, https://doi.org/10.12989/anr.2018.6.3.219
  52. Free vibration of axially functionally graded beams using the asymptotic development method vol.173, pp.None, 2013, https://doi.org/10.1016/j.engstruct.2018.06.111
  53. On the Internal Resonances of Size-Dependent Clamped-Hinged Microbeams: Continuum Modeling and Numerical Simulations vol.11, pp.3, 2013, https://doi.org/10.1142/s1758825119500224
  54. Hygro-thermal wave propagation in functionally graded double-layered nanotubes systems vol.31, pp.6, 2013, https://doi.org/10.12989/scs.2019.31.6.641
  55. Thermal buckling analysis of embedded graphene-oxide powder-reinforced nanocomposite plates vol.7, pp.5, 2013, https://doi.org/10.12989/anr.2019.7.5.293
  56. Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions vol.33, pp.1, 2019, https://doi.org/10.12989/scs.2019.33.1.133
  57. Finite element analysis and axial bearing capacity of steel reinforced recycled concrete filled square steel tube columns vol.72, pp.1, 2019, https://doi.org/10.12989/sem.2019.72.1.043
  58. Flexoelectric effects on dynamic response characteristics of nonlocal piezoelectric material beam vol.8, pp.4, 2013, https://doi.org/10.12989/amr.2019.8.4.259
  59. Longitudinal vibration of carbon nanotubes with elastically restrained ends using doublet mechanics vol.26, pp.2, 2020, https://doi.org/10.1007/s00542-019-04512-1
  60. Size-dependent thermo-mechanical free vibration analysis of functionally graded porous microbeams based on modified strain gradient theory vol.42, pp.5, 2013, https://doi.org/10.1007/s40430-020-02340-3
  61. A comprehensive review on the modeling of smart piezoelectric nanostructures vol.74, pp.5, 2013, https://doi.org/10.12989/sem.2020.74.5.611
  62. On scale-dependent stability analysis of functionally graded magneto-electro-thermo-elastic cylindrical nanoshells vol.75, pp.6, 2013, https://doi.org/10.12989/sem.2020.75.6.659