Research Article
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BENDING ANALYSIS OF A PERFORATED MICROBEAM WITH LAPLACE TRANSFORM

Year 2023, Volume: 11, 23 - 31, 29.12.2023

Büşra Uzun Mustafa Özgür Yaylı

https://doi.org/10.36306/konjes.1384835

Abstract

In recent years, the analysis of materials and elements with dimensions at nano/micro levels has gained momentum. While analyzing these small-scale materials and elements, higher-order elasticity theories have started to be used instead of classical elasticity theories (CETs). One of these theories, which includes a small-scale parameter in its constitutive equations, is the modified couple stress theory (MCST). In this study, the bending analysis of a cantilever perforated microbeam is investigated by MCST and Euler-Bernoulli (EB) beam theory. First, the perforation characteristics of the microbeam are described and incorporated into the equation governing the bending problem based on the modified couple stress theory found in the literature. Then, the Laplace transform is applied to the governing equation. The known boundary conditions of the cantilever microbeam are substituted into the equation and the inverse Laplace transform is applied to obtain the deflection equation.

Keywords

Laplace transform Perforated microbeam Modified couple stress theory Bending

References

  • A. A. Abdelrahman, İ. Esen, and M. A. Eltaher, “Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment,” Applied Mathematics and Computation, vol. 407, pp. 126307–126307, Oct. 2021, doi: https://doi.org/10.1016/j.amc.2021.126307.
  • K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “Influence of the perforation configuration on dynamic behaviors of multilayered beam structure,” Structures, vol. 28, pp. 1413–1426, Dec. 2020, doi: https://doi.org/10.1016/j.istruc.2020.09.055.
  • K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “Stability of perforated nanobeams incorporating surface energy effects,” Steel and Composite Structures, vol. 35, no. 4, pp. 555–566, Jan. 2020, doi: https://doi.org/10.12989/scs.2020.35.4.555.
  • L. Luschi and F. Pieri, “An analytical model for the determination of resonance frequencies of perforated beams,” Journal of Micromechanics and Microengineering, vol. 24, no. 5, p. 055004, Apr. 2014, doi: https://doi.org/10.1088/0960-1317/24/5/055004.
  • A. Abdelrahman, M. A. Eltaher, A.M. Kabeel, A. M. Abdraboh, and A. A. Hendi, “Free and forced analysis of perforated beams,” Steel and Composite Structures, vol. 31, no. 5, pp. 489–502, Jan. 2019, doi: https://doi.org/10.12989/scs.2019.31.5.489.
  • E. Assie, Ş. D. Akbaş, A. H. Bashiri, A. A. Abdelrahman, and M. A. Eltaher, “Vibration response of perforated thick beam under moving load,” The European Physical Journal Plus, vol. 136, no. 3, Mar. 2021, doi: https://doi.org/10.1140/epjp/s13360-021-01224-2.
  • K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “On forced and free vibrations of cutout squared beams,” Steel and Composite Structures, vol. 32, no. 5, pp. 643–655, Jan. 2019, doi: https://doi.org/10.12989/scs.2019.32.5.643.
  • M. A. Eltaher, H. E. Abdelmoteleb, A. A. Daikh, and A. A. Abdelrahman. “Vibrations and stress analysis of rotating perforated beams by using finite elements method”, Steel and Composite Structures, vol. 41 no. 4, pp. 505, 2021, doi: https://doi.org/10.12989/scs.2021.41.4.505
  • H. Bourouina, R. Yahiaoui, R. Kerid, K. Ghoumid, I. Lajoie, F. Picaud and G. Herlem “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, vol. 136, pp. 109201–109201, Jan. 2020, doi: https://doi.org/10.1016/j.jpcs.2019.109201.
  • Kerid, R., and Bounnah, Y. (2021). 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.
  • U. Kafkas, B. Uzun, M. Ö. Yaylı, and G. Güçlü, “Thermal vibration of perforated nanobeams with deformable boundary conditions via nonlocal strain gradient theory,” Zeitschrift für Naturforschung, vol. 78, no. 8, pp. 681–701, Jun. 2023, doi: https://doi.org/10.1515/zna-2023-0088.
  • A. A. Abdelrahman, H. E. Abd-El-Mottaleb, and M. A. Eltaher, “On bending analysis of perforated microbeams including the microstructure effects,” Structural Engineering and Mechanics, vol. 76, no. 6, pp. 765–779, Jan. 2020, doi: https://doi.org/10.12989/sem.2020.76.6.765.
  • S. K. Park and X. L. Gao, “Bernoulli–Euler beam model based on a modified couple stress theory”, Journal of Micromechanics and Microengineering, vol. 16, no. 11, 2006, doi: https://doi.org/10.1088/0960-1317/16/11/015
  • M. Sobhy, and A. M. Zenkour, “The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations,” Mechanics of Advanced Materials and Structures, vol. 27, no. 7, pp. 525–538, Jul. 2018, doi: https://doi.org/10.1080/15376494.2018.1482579.
  • J. Awrejcewicz, V.А. Кrysko, С. П. Павлов, M. V. Zhigalov, L. A. Kalutsky, and A. V. Krysko, “Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress theory,” Nonlinear Dynamics, vol. 99, no. 2, pp. 919–943, May 2019, doi: https://doi.org/10.1007/s11071-019-04976-w.
  • M. Ö. Yaylı, “Mikro Ölçekteki Bir Konsol Kirişin Değiştirilmiş Gerilme Çifti Teorisine Göre Laplace Dönüşümüyle Eğilme Analizi,” 20. ULUSAL MEKANİK KONGRESİ, 2017, pp. 626-629.
  • M. Nazmul and I. Devnath, “Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform,” International Journal of Computational Materials Science and Engineering, vol. 10, no. 02, pp. 2150012–2150012, Jun. 2021, doi: https://doi.org/10.1142/s2047684121500123.
  • P. Bian and H. Qing, “Elastic buckling and free vibration of nonlocal strain gradient Euler‐Bernoulli beams using Laplace transform,” ZAMM - Journal of Applied Mathematics and Mechanics, vol. 102, no. 1, Sep. 2021, doi: https://doi.org/10.1002/zamm.202100152.
  • C.C. Ike, C. U., Nwoji, B. O., Mama, H. N., Onah and M. E. Onyia, “Laplace transform method for the elastic buckling analysis of moderately thick beams”, International Journal of Engineering Research and Technology, vol. 12, no. 10, pp. 1626-1638, 2019.
  • C. Li, H. Qing, and C. Gao, “Theoretical analysis for static bending of Euler–Bernoulli using different nonlocal gradient models,” Mechanics of Advanced Materials and Structures, vol. 28, no. 19, pp. 1965–1977, Jan. 2020, doi: https://doi.org/10.1080/15376494.2020.1716121.
  • F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong, “Couple stress based strain gradient theory for elasticity,” International Journal of Solids and Structures, vol. 39, no. 10, pp. 2731–2743, May 2002, doi: https://doi.org/10.1016/s0020-7683(02)00152-x.

Year 2023, Volume: 11, 23 - 31, 29.12.2023

Büşra Uzun Mustafa Özgür Yaylı

https://doi.org/10.36306/konjes.1384835

Abstract

References

  • A. A. Abdelrahman, İ. Esen, and M. A. Eltaher, “Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment,” Applied Mathematics and Computation, vol. 407, pp. 126307–126307, Oct. 2021, doi: https://doi.org/10.1016/j.amc.2021.126307.
  • K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “Influence of the perforation configuration on dynamic behaviors of multilayered beam structure,” Structures, vol. 28, pp. 1413–1426, Dec. 2020, doi: https://doi.org/10.1016/j.istruc.2020.09.055.
  • K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “Stability of perforated nanobeams incorporating surface energy effects,” Steel and Composite Structures, vol. 35, no. 4, pp. 555–566, Jan. 2020, doi: https://doi.org/10.12989/scs.2020.35.4.555.
  • L. Luschi and F. Pieri, “An analytical model for the determination of resonance frequencies of perforated beams,” Journal of Micromechanics and Microengineering, vol. 24, no. 5, p. 055004, Apr. 2014, doi: https://doi.org/10.1088/0960-1317/24/5/055004.
  • A. Abdelrahman, M. A. Eltaher, A.M. Kabeel, A. M. Abdraboh, and A. A. Hendi, “Free and forced analysis of perforated beams,” Steel and Composite Structures, vol. 31, no. 5, pp. 489–502, Jan. 2019, doi: https://doi.org/10.12989/scs.2019.31.5.489.
  • E. Assie, Ş. D. Akbaş, A. H. Bashiri, A. A. Abdelrahman, and M. A. Eltaher, “Vibration response of perforated thick beam under moving load,” The European Physical Journal Plus, vol. 136, no. 3, Mar. 2021, doi: https://doi.org/10.1140/epjp/s13360-021-01224-2.
  • K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “On forced and free vibrations of cutout squared beams,” Steel and Composite Structures, vol. 32, no. 5, pp. 643–655, Jan. 2019, doi: https://doi.org/10.12989/scs.2019.32.5.643.
  • M. A. Eltaher, H. E. Abdelmoteleb, A. A. Daikh, and A. A. Abdelrahman. “Vibrations and stress analysis of rotating perforated beams by using finite elements method”, Steel and Composite Structures, vol. 41 no. 4, pp. 505, 2021, doi: https://doi.org/10.12989/scs.2021.41.4.505
  • H. Bourouina, R. Yahiaoui, R. Kerid, K. Ghoumid, I. Lajoie, F. Picaud and G. Herlem “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, vol. 136, pp. 109201–109201, Jan. 2020, doi: https://doi.org/10.1016/j.jpcs.2019.109201.
  • Kerid, R., and Bounnah, Y. (2021). 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.
  • U. Kafkas, B. Uzun, M. Ö. Yaylı, and G. Güçlü, “Thermal vibration of perforated nanobeams with deformable boundary conditions via nonlocal strain gradient theory,” Zeitschrift für Naturforschung, vol. 78, no. 8, pp. 681–701, Jun. 2023, doi: https://doi.org/10.1515/zna-2023-0088.
  • A. A. Abdelrahman, H. E. Abd-El-Mottaleb, and M. A. Eltaher, “On bending analysis of perforated microbeams including the microstructure effects,” Structural Engineering and Mechanics, vol. 76, no. 6, pp. 765–779, Jan. 2020, doi: https://doi.org/10.12989/sem.2020.76.6.765.
  • S. K. Park and X. L. Gao, “Bernoulli–Euler beam model based on a modified couple stress theory”, Journal of Micromechanics and Microengineering, vol. 16, no. 11, 2006, doi: https://doi.org/10.1088/0960-1317/16/11/015
  • M. Sobhy, and A. M. Zenkour, “The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations,” Mechanics of Advanced Materials and Structures, vol. 27, no. 7, pp. 525–538, Jul. 2018, doi: https://doi.org/10.1080/15376494.2018.1482579.
  • J. Awrejcewicz, V.А. Кrysko, С. П. Павлов, M. V. Zhigalov, L. A. Kalutsky, and A. V. Krysko, “Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress theory,” Nonlinear Dynamics, vol. 99, no. 2, pp. 919–943, May 2019, doi: https://doi.org/10.1007/s11071-019-04976-w.
  • M. Ö. Yaylı, “Mikro Ölçekteki Bir Konsol Kirişin Değiştirilmiş Gerilme Çifti Teorisine Göre Laplace Dönüşümüyle Eğilme Analizi,” 20. ULUSAL MEKANİK KONGRESİ, 2017, pp. 626-629.
  • M. Nazmul and I. Devnath, “Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform,” International Journal of Computational Materials Science and Engineering, vol. 10, no. 02, pp. 2150012–2150012, Jun. 2021, doi: https://doi.org/10.1142/s2047684121500123.
  • P. Bian and H. Qing, “Elastic buckling and free vibration of nonlocal strain gradient Euler‐Bernoulli beams using Laplace transform,” ZAMM - Journal of Applied Mathematics and Mechanics, vol. 102, no. 1, Sep. 2021, doi: https://doi.org/10.1002/zamm.202100152.
  • C.C. Ike, C. U., Nwoji, B. O., Mama, H. N., Onah and M. E. Onyia, “Laplace transform method for the elastic buckling analysis of moderately thick beams”, International Journal of Engineering Research and Technology, vol. 12, no. 10, pp. 1626-1638, 2019.
  • C. Li, H. Qing, and C. Gao, “Theoretical analysis for static bending of Euler–Bernoulli using different nonlocal gradient models,” Mechanics of Advanced Materials and Structures, vol. 28, no. 19, pp. 1965–1977, Jan. 2020, doi: https://doi.org/10.1080/15376494.2020.1716121.
  • F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong, “Couple stress based strain gradient theory for elasticity,” International Journal of Solids and Structures, vol. 39, no. 10, pp. 2731–2743, May 2002, doi: https://doi.org/10.1016/s0020-7683(02)00152-x.
There are 21 citations in total.

Details

Primary Language English
Subjects Civil Engineering (Other)
Journal Section Research Article
Authors

Büşra Uzun 0000-0002-7636-7170

Mustafa Özgür Yaylı 0000-0003-2231-170X

Publication Date December 29, 2023
Submission Date November 1, 2023
Acceptance Date November 23, 2023
Published in Issue Year 2023 Volume: 11

Cite

IEEE B. Uzun and M. Ö. Yaylı, “BENDING ANALYSIS OF A PERFORATED MICROBEAM WITH LAPLACE TRANSFORM”, KONJES, vol. 11, pp. 23–31, 2023, doi: 10.36306/konjes.1384835.
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