Size-Dependent Analysis of Perforated Timoshenko Microbeams with Graded Porosity Using the Chebyshev Collocation Method

Year 2025, Volume: 17 Issue: 4, 213 - 228, 31.12.2025

İbrahim Keles

https://doi.org/10.24107/ijeas.1806663

Abstract

Perforated microbeams are widely employed in micro- and nano-electromechanical systems due to their lightweight structures and tunable mechanical properties. At small scales, however, classical elasticity fails to capture size effects and porosity-induced stiffness variations. This study develops a comprehensive framework for the static, dynamic, and electrostatic behavior of perforated Timoshenko microbeams with graded porosity, based on the Chebyshev collocation method. The model integrates nonlocal elasticity to account for size effects, graded porosity distributions to represent realistic microstructural variations, and electrostatic actuation to analyze pull-in instability. Validation against published results confirms the accuracy and efficiency of the proposed approach. Parametric investigations reveal that porosity distribution strongly affects stiffness and deflection responses, while size-dependent effects enhance rigidity and increase natural frequencies. Electrostatic pull-in voltage is found to increase with both porosity ratio and length scale parameter, indicating improved stability for micro- and nano-electromechanical systems devices. The findings demonstrate that the Chebyshev collocation method framework provides a robust and efficient tool for the design and optimization of next-generation perforated microstructures.

Keywords

Perforated microbeam , Chebyshev Collocation Method Graded , Porosity , Size-dependent Elasticity , Electrostatic Pull-in Instability

Ethical Statement

This study does not involve human participants or animals. Therefore, ethical approval was not required.

Supporting Institution

No specific funding was received for this study.

Project Number

Not applicable

Thanks

The authors would like to thank the Mechanical Engineering Department of Samsun University for providing computational resources

References

• Abdelrahman, A. A., Esen, I. and Eltaher, M. A., Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment. Applied Mathematics and Computation, 407, 126307, 2021.
• Almitani, K. H., Abdelrahman, A. A. and Eltaher, M. A., Influence of the perforation configuration on dynamic behaviors of multilayered beam structure. Structures, 28, 1413-1426, 2020.
• Almitani, K. H., Abdelrahman, A. A. and Eltaher, M. A., Stability of perforated nanobeams incorporating surface energy effects. Steel and Composite Structures, 35(4), 555-566, 2020.
• D’Annibale, F., Ferretti, M., and Luongo, A., Static and dynamic responses of micro-structured beams. Applied Sciences, 10(19), 6836, 2020.
• Abdelrahman, A. A., Eltaher, M. A., Kabeel, A. M., Abdraboh, A. M. And Hendi, A. A., Free and forced analysis of perforated beams. Steel and Composite Structures, 489-502, 2022.
• Assie, A., Akbaş, Ş. D., Bashiri, A. H., Abdelrahman, A. A. and Eltaher, M. A., Vibration response of perforated thick beam under moving load. The European Physical Journal Plus, 136(3), 283, 2021.
• Almitani, K. H., On forced and free vibrations of cutout squared beams. Steel and Composite Structures, 32(5), 643-655, 2019.
• Eltaher, M. A., Abdelmoteleb, H. E., Daikh, A. A. and Abdelrahman, A. A., Vibrations and stress analysis of rotating perforated beams by using finite elements method. Steel and Composite Structures, 41, 505-520., 2022.
• Uzun, B. and Yaylı, M. O., Bending analysis of a perforated microbeam with laplace transform. Konya Journal of Engineering Sciences, 11, 23-31, 2023.
• Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves. Journal of Applied Physics, 54(9), 4703-4710, 1983.
• Bourouina, H., Yahiaoui, R., Kerid, R., Ghoumid, K., Lajoie, I., Picaud, F. and Herlem, G., The influence of hole networks on the adsorption-induced frequency shift of a perforated nanobeam using non-local elasticity theory. Journal of Physics and Chemistry of Solids, 136, 109201,2020
• Kerid, R. and Bounnah, Y., Effects of structure design on electrostatic pull-in voltage of perforated nanoswitch with intermolecular surface forces. Journal of Ultrafine Grained and Nanostructured Materials, 54(2), 219-227, 2021.
• Abdelrahman, A. A., Abd El Mottaleb, H. E. and Eltaher, M. A., On bending analysis of perforated microbeams including the microstructure effects. Structural Engineering and Mechanics, An Int'l Journal, 76(6), 765-779, 2020.
• Mercan, K., Numanoglu, H. M., Akgoz, B., Demir, C. and Civalek, O., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Archive of Applied Mechanics, 87(11), 1797-1814, 2017.
• Akgoz, B. and Civalek, O., Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity. International Journal of Computational Methods, 9(02), 1240032, 2012.
• Kafkas, U., Size-Dependent Bending Response of Perforated Nanobeams on Winkler-Pasternak Foundation. International Journal of Engineering and Applied Sciences, 17(1), 1-16, 2025.
• Akbas, S. D., Dastjerdi, S., Akgoz, B. and Civalek, O., Dynamic analysis of functionally graded porous microbeams under moving load. Transport in Porous Media, 142(1), 209-227, 2022.
• Mercan, K., Numanoglu, H. M., Akgoz, B., Demir, C. and Civalek, O., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Archive of Applied Mechanics, 87(11), 1797-1814, 2017.
• Akgoz, B. and Civalek, O., Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity. International Journal of Computational Methods, 9(02), 1240032, 2012.
• Abouelregal, A. E., Akgöz, B. and Civalek, O., Nonlocal thermoelastic vibration of a solid medium subjected to a pulsed heat flux via Caputo–Fabrizio fractional derivative heat conduction. Applied Physics A, 128(8), 660, 2022.
• Akgöz, B., Mercan, K., Demir, Ç. and Civalek, O., Static analysis of beams on elastic foundation by the method of discrete singular convolution. International Journal of Engineering and Applied Sciences, 8(3), 67-73, 2016.
• Alibakhshi, A., Dastjerdi, S., Akgöz, B. and Civalek, O., Parametric vibration of a dielectric elastomer microbeam resonator based on a hyperelastic cosserat continuum model. Composite Structures, 287, 115386, 2022.
• Abouelregal, A. E., Civalek, O. And Akgoz, B., A size-dependent non-fourier heat conduction model for magneto-thermoelastic vibration response of nanosystems. Journal of Applied and Computational Mechanics, 11(2), 344-357, 2025.
• Abdelrahman, A. A., Eltaher, M. A., Kabeel, A. M., Abdraboh, A. M. and Hendi, A. A., Free and forced analysis of perforated beams. Steel and Composite Structures, 489-502, 2022.
• Chen, W. R. and Chang, H., Vibration analysis of bidirectional functionally graded Timoshenko beams using Chebyshev collocation method. International Journal of Structural Stability and Dynamics, 21(01), 2150009, 2021.
• Wattanasakulpong, N and Chaikittiratana, A., Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica, 50(5), 1331-1342, 2015.
• Wattanasakulpong, N. and Mao, Q., Dynamic response of Timoshenko functionally graded beams with classical and non-classical boundary conditions using Chebyshev collocation method. Composite Structures, 119, 346-354, 2015.
• Faroughi, S., Sari, M. S. and Abdelkefi, A., Nonlocal Timoshenko representation and analysis of multi-layered functionally graded nanobeams. Microsystem Technologies, 27(3), 893-911, 2021.
• Shimpi, R. P., Pakhare, K. S., Punith, P. and Guruprasad, P. J., The static pull-in instability analysis of electrostatically actuated shear deformable microbeams using single variable refined beam theory variants. Archive of Applied Mechanics, 92(10), 2917-2950 2022.
• Abouelregal, A. E., Civalek, O., Akgoz, B., Foul, A. and Askar, S. S., Analysis of thermoelastic behavior of porous cylinders with voids via a nonlocal space-time elastic approach and Caputo-tempered fractional heat conduction. Mechanics of Time-Dependent Materials, 29(2), 1-32, 2025.
• Chen, D., Yang, J. and Kitipornchai, S., Elastic buckling and static bending of shear deformable functionally graded porous beam. Composite Structures, 133, 54-61, 2015.
• Lim, C. W., Zhang, G. and Reddy, J., A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298-313, 2015.
• Abdelrahman, A. A. and Eltaher, M. A., On bending and buckling responses of perforated nanobeams including surface energy for different beams theories. Engineering with Computers, 38(3), 2385-2411, 2022.
• Hieu, D. V., Hoa, N. T., Duy, L. Q. and Kim Thoa, N. T., Nonlinear vibration of an electrostatically actuated‎ functionally graded microbeam under longitudinal magnetic‎ field. Journal of Applied and Computational Mechanics, 7(3), 1537-1549, 2021.