Skip to main content
SpringerLink
Log in

Stability buckling and bending of nanobeams including cutouts

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This manuscript developed a comprehensive model and numerical studies to illustrate the effect of perforation parameters on critical buckling loads and static bending of thin and thick nanobeams for all boundary conditions, for the first time. Analytical closed-form solutions are presented for buckling loads and static deflections, respectively. Euler–Bernoulli beam theory is exploited for thin beam analysis, and Timoshenko beam theory is proposed to consider a shear effect in case of thick beam analysis. Nonlocal differential form of elasticity theory is included to consider a size scale effect that is missing in case of classical theory and macro-analysis. Geometrical adaptations for perforated beam structures are illustrated in simplest form. Equilibrium equations for local and nonlocal beam are derived in detail. Numerical studies are illustrated to demonstrate influences of long-range atomic interaction, hole perforation size, number of rows of holes and boundary conditions on buckling loads and deflection of perforated nanobeams. The recommended model is helpful in designing nanoresonators and nanoactuators used in NEMS structures and nanotechnology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Abdelrahmaan AA, Eltaher MA, Kabeel AM, Abdraboh AM, Hendi AA (2019) Free and forced analysis of perforated beams. Steel Compos Struct 31(5):489–502. https://doi.org/10.12989/scs.2019.31.5.489

    Article  Google Scholar 

  2. Almitani KH, Abdelrahman AA, Eltaher MA (2019) On forcd and free vibrations of cutout squared beams. Steel Compos Struct 32(5):643–655. https://doi.org/10.12989/scs.2019.32.5.643

    Article  Google Scholar 

  3. Almitani KH, Abdelrahman AA, Eltaher MA (2020) Stability of perforated nanobeams incorporating surface energy effects. Steel Compos Struct. https://doi.org/10.12989/scs.2020.35.4.000

    Article  Google Scholar 

  4. Apuzzo A, Barretta R, Faghidian SA, Luciano R, De Sciarra FM (2019) Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams. Compos B Eng 164:667–674. https://doi.org/10.1016/j.compositesb.2018.12.112

    Article  Google Scholar 

  5. Aria AI, Friswell MI (2019) A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos B Eng 166:233–246. https://doi.org/10.1016/j.compositesb.2018.11.071

    Article  Google Scholar 

  6. Azelmad E, Salmi A, El Kennassi E, Bousshine L (2018) Elastoplastic behavioranalysis of clamped circular perforated thin plates. IOSR J Mech Civ Eng 15(2):23–37. https://doi.org/10.9790/1684-1502022337

    Article  Google Scholar 

  7. Bessaim A, Ahmed Houari MS, Abdelmoumen Anis B, Kaci A, Tounsi A, Adda Bedia EA (2018) Buckling analysis of embedded nanosize FG beams based on a refined hyperbolic shear deformation theory. J Appl Comput Mech 4(3):140–146. https://doi.org/10.22055/jacm.2017.22996.1146

    Article  Google Scholar 

  8. Bohlooly M, Malekzadeh Fard K (2019) Buckling and postbuckling of concentrically stiffened piezo-composite plates on elastic foundations. J Appl Comput Mech 5(1):128–140. https://doi.org/10.22055/jacm.2018.25539.1277

    Article  Google Scholar 

  9. Bourouina H, Yahiaoui R, Sahar A, Benamar MEA (2016) Analytical modeling for the determination of nonlocal resonance frequencies of perforated nanobeams subjected to temperature-induced loads. Physica E 75:163–168. https://doi.org/10.1016/j.physe.2015.09.014

    Article  Google Scholar 

  10. Bourouina H, Yahiaoui R, Kerid R, Ghoumid K, Lajoie I, Picaud F, Herlem G (2020) The influence of hole networks on the adsorption-induced frequency shift of a perforated nanobeam using non-local elasticity theory. J Phys Chem Solids 136:109201. https://doi.org/10.1016/j.jpcs.2019.109201

    Article  Google Scholar 

  11. Davey K, Darvizeh R, Sedqi Z (2020) A tessellated continuum approach for the static analysis of perforated structures. Comput Struct 227:106140. https://doi.org/10.1016/j.compstruc.2019.106140

    Article  Google Scholar 

  12. De Pasquale G, Veijola T, Somà A (2010) Modelling and validation of air damping in perforated gold and silicon MEMS plates. J Micromech Microeng 20(1):015010. https://doi.org/10.1088/0960-1317/20/1/015010

    Article  Google Scholar 

  13. Duncan JP, Upfold RW (1963) Equivalent elastic properties of perforated bars and plates. J Mech Eng Sci 5(1):53–65

    Article  Google Scholar 

  14. Ebrahimi F, Farazmandnia N, Kokaba MR, Mahesh V (2019) Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory. Eng Comput. https://doi.org/10.1007/s00366-019-00864-4

    Article  Google Scholar 

  15. Eltaher MA, Emam SA, Mahmoud FF (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88. https://doi.org/10.1016/j.compstruct.2012.09.030

    Article  Google Scholar 

  16. Eltaher MA, El-Borgi S, Reddy JN (2016) Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos Struct 153:902–913. https://doi.org/10.1016/j.compstruct.2016.07.013

    Article  Google Scholar 

  17. Eltaher MA, Khater ME, Emam SA (2016) A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl Math Model 40(5–6):4109–4128. https://doi.org/10.1016/j.apm.2015.11.026

    Article  MathSciNet  MATH  Google Scholar 

  18. Eltaher MA, Kabeel AM, Almitani KH, Abdraboh AM (2018) Static bending and buckling of perforated nonlocal size-dependent nanobeams. Microsyst Technol 24(12):4881–4893. https://doi.org/10.1007/s00542-018-3905-3

    Article  Google Scholar 

  19. Eltaher MA, Abdraboh AM, Almitani KH (2018) Resonance frequencies of size dependent perforated nonlocal nanobeam. Microsyst Technol 24(9):3925–3937. https://doi.org/10.1007/s00542-018-3910-6

    Article  Google Scholar 

  20. Eltaher MA, Omar FA, Abdalla WS, Gad EH (2019) Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity. Waves Random Complex Media 29(2):264–280. https://doi.org/10.1080/17455030.2018.1429693

    Article  MathSciNet  Google Scholar 

  21. Eltaher MA, Mohamed N, Mohamed S, Seddek LF (2019) Postbuckling of curved carbon nanotubes using energy equivalent model. J Nano Res 57:136–157. https://doi.org/10.4028/www.scientific.net/JNanoR.57.136

    Article  Google Scholar 

  22. Eltaher MA, Almalki TA, Almitani KH, Ahmed KIE (2019) Participation factor and vibration of carbon nanotube with vacancies. J Nano Res 57:158–174. https://doi.org/10.4028/www.scientific.net/JNanoR.57.158

    Article  Google Scholar 

  23. Eltaher MA, Omar FA, Abdraboh AM, Abdalla WS, Alshorbagy AE (2020) Mechanical behaviors of piezoelectric nonlocal nanobeam with cutouts. Smart Struct Syst 25(2):219. https://doi.org/10.12989/sss.2020.25.2.219

    Article  Google Scholar 

  24. Eltaher MA, Mohamed NA (2020) Vibration of nonlocal perforated nanobeams with general boundary conditions. Smart Struct Syst 25(4):501–514. https://doi.org/10.12989/sss.2020.25.4.501

    Article  Google Scholar 

  25. Emam S, Eltaher M, Khater M, Abdalla W (2018) Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load. Appl Sci 8(11):2238. https://doi.org/10.3390/app8112238

    Article  Google Scholar 

  26. Erfani S, Akrami V (2019) A nonlinear macro-model for numerical simulation of perforated steel beams. Int J Steel Struct 19(5):1605–1623. https://doi.org/10.1007/s13296-019-00239-x

    Article  Google Scholar 

  27. Eringen AC (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

    Article  Google Scholar 

  28. Esmaeili M, Tadi Beni Y (2019) Vibration and buckling analysis of functionally graded flexoelectric smart beam. J Appl Comput Mech 5(5):900–917. https://doi.org/10.22055/jacm.2019.27857.1439

    Article  Google Scholar 

  29. Gabalis M, Urbonas D, Petruskevicius R (2014) A perforated microring resonator for optical sensing applications. J Opt 16(10):105003. https://doi.org/10.1088/2040-8978/16/10/10500

    Article  Google Scholar 

  30. Gao P, Sánchez-Dehesa J, Wu L (2018) Poisson-like effect for flexural waves in periodically perforated thin plates. J Acoust Soc Am 144(2):1053–1058. https://doi.org/10.1121/1.5051648

    Article  Google Scholar 

  31. Graham TJ, Hibbins AP, Sambles JR, Starkey TA (2019) Underwater acoustic surface waves on a periodically perforated metal plate. J Acoust Soc Am 146(6):4569–4575. https://doi.org/10.1121/1.5139651

    Article  Google Scholar 

  32. Hamed MA, Sadoun AM, Eltaher MA (2019) Effects of porosity models on static behavior of size dependent functionally graded beam. Struct Eng Mech 71(1):89–98. https://doi.org/10.12989/sem.2019.71.1.089

    Article  Google Scholar 

  33. Hamed MA, Abo-bakr RM, Mohamed SA, Eltaher MA (2020) Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core. Eng Comput. https://doi.org/10.1007/s00366-020-01023-w

    Article  Google Scholar 

  34. Hashemian M, Foroutan S, Toghraie D (2019) Comprehensive beam models for buckling and bending behavior of simple nanobeam based on nonlocal strain gradient theory and surface effects. Mech Mater 139:103209. https://doi.org/10.1016/j.mechmat.2019.103209

    Article  Google Scholar 

  35. Jena SK, Chakraverty S (2019) Dynamic behavior of an electromagnetic nanobeam using the Haar wavelet method and the higher-order Haar wavelet method. Eur Phys J Plus 134(10):538. https://doi.org/10.1140/epjp/i2019-12874-8

    Article  Google Scholar 

  36. Jena SK, Chakraverty S, Malikan M, Tornabene F (2019) Stability analysis of single-walled carbon nanotubes embedded in winkler foundation placed in a thermal environment considering the surface effect using a new refined beam theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2019.1698437

    Article  Google Scholar 

  37. Jena SK, Chakraverty S, Jena RM, Tornabene F (2019) A novel fractional nonlocal model and its application in buckling analysis of Euler–Bernoulli nanobeam. Mater Res Exp 6(5):055016. https://doi.org/10.1088/2053-1591/ab016b

    Article  Google Scholar 

  38. Jena SK, Chakraverty S, Malikan M (2019) Implementation of Haar wavelet, higher order Haar wavelet, and differential quadrature methods on buckling response of strain gradient nonlocal beam embedded in an elastic medium. Eng Comput. https://doi.org/10.1007/s00366-019-00883-1

    Article  Google Scholar 

  39. Jena SK, Chakraverty S, Tornabene F (2019) Buckling behavior of nanobeams placed in electromagnetic field using shifted Chebyshev polynomials-based rayleigh-ritz method. Nanomaterials 9(9):1326. https://doi.org/10.3390/nano9091326

    Article  Google Scholar 

  40. Jena SK, Chakraverty S, Malikan M, Tornabene F (2020) Effects of surface energy and surface residual stresses on vibro-thermal analysis of chiral, zigzag, and armchair types of SWCNTs using refined beam theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1754239

    Article  Google Scholar 

  41. Jena SK, Chakraverty S, Malikan M (2020) Implementation of non-probabilistic methods for stability analysis of nonlocal beam with structural uncertainties. Eng Comput. https://doi.org/10.1007/s00366-020-00987-z

    Article  Google Scholar 

  42. Jena SK, Chakraverty S, Malikan M (2020) Vibration and buckling characteristics of nonlocal beam placed in a magnetic field embedded in Winkler-Pasternak elastic foundation using a new refined beam theory: an analytical approach. Eur Phys J Plus 135(2):164. https://doi.org/10.1140/epjp/s13360-020-00176-3

    Article  Google Scholar 

  43. Jena SK, Chakraverty S, Malikan M (2020) Application of shifted Chebyshev polynomial-based Rayleigh-Ritz method and Navier’s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation. Eng Comput. https://doi.org/10.1007/s00366-020-01018-7

    Article  Google Scholar 

  44. Jeong KH, Amabili M (2006) Bending vibration of perforated beams in contact with a liquid. J Sound Vib 298(1):404–419. https://doi.org/10.1016/j.jsv.2006.05.029

    Article  Google Scholar 

  45. Kalaiselvi S, Sujatha L, Sundar R (2019) Analysis of damping optimization through perforations in proof-mass of SOI capacitive accelerometer. Analog Integr Circ Sig Process. https://doi.org/10.1007/s10470-019-01560-5

    Article  Google Scholar 

  46. Kerid R, Bourouina H, Yahiaoui R, Bounekhla M, Aissat A (2019) Magnetic field effect on nonlocal resonance frequencies of structure-based filter with periodic square holes network. Physica E 105:83–89. https://doi.org/10.1016/j.physe.2018.05.021

    Article  Google Scholar 

  47. Khiloun M, Bousahla AA, Kaci A, Bessaim A, Tounsi A, Mahmoud SR (2019) Analytical modeling of bending and vibration of thick advanced composite plates using a four-variable quasi 3D HSDT. Eng Comput. https://doi.org/10.1007/s00366-019-00732-1

    Article  Google Scholar 

  48. Kim JH, Jeon JH, Park JS, Seo HD, Ahn HJ, Lee JM (2015) Effect of reinforcement on buckling and ultimate strength of perforated plates. Int J Mech Sci 92:194–205. https://doi.org/10.1016/j.ijmecsci.2014.12.016

    Article  Google Scholar 

  49. Luschi L, Pieri F (2014) An analytical model for the determination of resonance frequencies of perforated beams. J Micromech Microeng 24(5):055004. https://doi.org/10.1088/0960-1317/24/5/055004

    Article  Google Scholar 

  50. Luschi L, Pieri F (2016) An analytical model for the resonance frequency of square perforated Lamé-mode resonators. Sensors Actuat B: Chem 222:1233–1239. https://doi.org/10.1016/j.snb.2015.07.085

    Article  Google Scholar 

  51. Mohamed N, Mohamed SA, Eltaher MA (2020) Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng Comput. https://doi.org/10.1007/s00366-020-00976-2

    Article  Google Scholar 

  52. Mohite SS, Sonti VR, Pratap R (2008) A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures. J Microelectromech Syst 17(3):709–723. https://doi.org/10.1109/JMEMS.2008.921675

    Article  Google Scholar 

  53. Ouakad HM, Sedighi HM, Younis MI (2017) One-to-one and three-to-one internal resonances in MEMS shallow arches. J Comput Nonlinear Dyn. https://doi.org/10.1115/1.4036815

    Article  Google Scholar 

  54. Pascon JP (2019) Finite element analysis of functionally graded hyperelastic beams under plane stress. Eng Comput. https://doi.org/10.1007/s00366-019-00761-w

    Article  Google Scholar 

  55. Rao KS, Chand CG, Sravani KG, Prathyusha D, Naveena P, Lakshmi GS, Narayana TL (2019) Design, modeling and analysis of perforated RF MEMS capacitive shunt switch. IEEE Access 7:74869–74878. https://doi.org/10.1109/ACCESS.2019.2914260

    Article  Google Scholar 

  56. Reddy JN, Pang SD (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103:023511. https://doi.org/10.1063/1.2833431

    Article  Google Scholar 

  57. Sahmani S, Fattahi AM, Ahmed NA (2019) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput 35(4):1173–1189. https://doi.org/10.1007/s00366-018-0657-8

    Article  Google Scholar 

  58. Sedighi HM, Bozorgmehri A (2016) Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory. Acta Mech 227(6):1575–1591. https://doi.org/10.1007/s00707-016-1562-0

    Article  MathSciNet  MATH  Google Scholar 

  59. Sedighi HM, Sheikhanzadeh ASHKAN (2017) Static and dynamic pull-in instability of nano-beams resting on elastic foundation based on the nonlocal elasticity theory. Chin J Mech Eng 30(2):385–397. https://doi.org/10.1007/s10033-017-0079-3

    Article  Google Scholar 

  60. Simsek M (2019) Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory. Compos Struct 10:1–12. https://doi.org/10.1016/j.compstruct.2019.111041

    Article  Google Scholar 

  61. Wu Q, Chen H, Gao W (2019) Nonlocal strain gradient forced vibrations of FG-GPLRC nanocomposite microbeams. Eng Comput. https://doi.org/10.1007/s00366-019-00794-1

    Article  Google Scholar 

  62. Zhang ZJ, Zhang QC, Li FC, Yang JW, Liu JW, Liu ZY, Jin F (2019) Modal characteristics of micro-perforated sandwich beams with square honeycomb-corrugation hybrid cores: a mixed experimental-numerical study. Thin-Walled Struct 137:185–196. https://doi.org/10.1016/j.tws.2019.01.004

    Article  Google Scholar 

  63. Zhou CW, Lainé JP, Ichchou MN, Zine AM (2016) Numerical and experimental investigation on broadband wave propagation features in perforated plates. Mech Syst Signal Process 75:556–575. https://doi.org/10.1016/j.ymssp.2015.12.006

    Article  Google Scholar 

  64. Zhou XQ, Wang L, Yu DY, Zhang CY (2019) Experimental investigation of the dissipation characteristic of sandwich structures with periodically perforated viscoelastic damping material core. J Vib Control 25(14):2008–2024. https://doi.org/10.1177/1077546319844545

    Article  Google Scholar 

  65. Zulkefli MA, Mohamed MA, Siow KS, Majlis BY, Kulothungan J, Muruganathan M, Mizuta H (2018) Stress analysis of perforated graphene nano-electro-mechanical (NEM) contact switches by 3D finite element simulation. Microsyst Technol 24(2):1179–1187. https://doi.org/10.3390/mi8080236

    Article  Google Scholar 

Download references

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant No. G-44-135-1441. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Mostafa A. Hamed & M. A. Eltaher

  2. Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    N. A. Mohamed

  3. Mechanical Design & Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    M. A. Eltaher

Authors
  1. Mostafa A. Hamed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Mohamed
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Eltaher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Eltaher.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hamed, M.A., Mohamed, N.A. & Eltaher, M.A. Stability buckling and bending of nanobeams including cutouts. Engineering with Computers 38, 209–230 (2022). https://doi.org/10.1007/s00366-020-01063-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01063-2

Keywords

Search

Navigation