SavedAdd your journal

Journals / European Journal of Engineering and Applied Sciences / 2026 / Cilt: 9 Sayı: 1

Natural-Frequency Uncertainty of Cracked Microbeams with Random Crack Flexibility

Rastgele Çatlak Esnekliğine Sahip Çatlaklı Mikro Kirişlerin Doğal Frekans Belirsizliği

Serpil Yılmaz Kutluay(IZMIR KATIP CELEBI UNIVERSITY)0000-0002-6276-6058
Duygu Atcı(IZMIR KATIP CELEBI UNIVERSITY)0000-0002-1998-5394
Pages
41–48
Publication Date
2026-07-31
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The results show that the coefficient of variation increases as the crack location moves toward the midspan, and the simply supported microbeam exhibits larger uncertainty than the clamped-clamped microbeam.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Uncertainty in crack flexibility can significantly affect the vibration characteristics of cracked microbeams, particularly in microscale structures.
📊
Key Findings & Data:Deterministic frequency maps are first generated in the crack parameter plane to identify regions where the modal response is most sensitive to crack location and crack flexibility.
💡
Conclusion & Impact:The results show that the coefficient of variation increases as the crack location moves toward the midspan, and the simply supported microbeam exhibits larger uncertainty than the clamped-clamped microbeam.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Çatlak esnekliği belirsizliği · çatlaklı mikro kirişler · doğal frekans

Abstract

Uncertainty in crack flexibility can significantly affect the vibration characteristics of cracked microbeams, particularly in microscale structures. This study investigates the stochastic free vibration behavior of cracked microbeams with random crack flexibility within the framework of modified couple stress theory. The crack is modeled as a torsional spring; its position defines the crack location, while its rotational flexibility characterizes the crack severity. In the stochastic formulation, the crack location is kept deterministic, and the crack flexibility is modeled as a random variable. Natural frequencies are obtained for simply supported and clamped-clamped microbeams using a numerical procedure based on the characteristic equation. Deterministic frequency maps are first generated in the crack parameter plane to identify regions where the modal response is most sensitive to crack location and crack flexibility. Monte Carlo simulation is then used to propagate the uncertainty in crack flexibility, and the resulting samples of the first mode natural frequency are evaluated using the mean, standard deviation, coefficient of variation, and percentile intervals. The results show that the coefficient of variation increases as the crack location moves toward the midspan, and the simply supported microbeam exhibits larger uncertainty than the clamped-clamped microbeam. A sensitivity analysis of the uncertainty level further confirms that greater dispersion in crack flexibility leads to greater uncertainty in the natural frequency.

Özet

Çatlak esnekliğindeki belirsizlik, özellikle mikro ölçekli yapılarda, titreşim karakteristiklerini önemli ölçüde etkileyen bir unsurdur. Bu çalışmada, rastgele çatlak esnekliğine sahip çatlaklı mikro kirişlerin stokastik serbest titreşim davranışı, değiştirilmiş çift gerilme teorisi çerçevesinde incelenmektedir. Çatlak, burulma yayı olarak modellenmiştir; yayın konumu çatlağın yerini tanımlarken, yay direngenliği, çatlak şiddetini karakterize etmektedir. Stokastik formülasyonda çatlak konumu sabit alınmış, çatlak esnekliği ise rastgele değişken olarak modellenmiştir. Basit mesnetli ve ankastre-ankastre mikro kirişler için doğal frekanslar, karakteristik denkleme dayalı sayısal bir yöntemle elde edilmiştir. Öncelikle, modal yanıtın çatlak konumu ve çatlak esnekliğine en duyarlı olduğu bölgeleri belirlemek amacıyla çatlak parametre düzleminde deterministik frekans haritaları oluşturulmuştur. Daha sonra, çatlak esnekliğindeki belirsizliği yaymak için Monte Carlo simülasyonu kullanılmış ve elde edilen birinci mod frekans örnekleri ortalama, standart sapma, değişim katsayısı ve yüzdelik aralıklar kullanılarak değerlendirilmiştir. Sonuçlar, değişim katsayısının, çatlak konumu kiriş ortasına yaklaştıkça arttığını ve basit mesnetli mikro kirişte, ankastre-ankastre mikro kirişe göre daha yüksek belirsizliğin ortaya çıktığını göstermektedir. Belirsizlik seviyesi duyarlılık analizi sonuçları ise, çatlak esnekliğindeki daha geniş dağılımın daha güçlü frekans belirsizliğine yol açtığını doğrulamaktadır.

🔖

Cite This Article

Kutluay, S. Y., Atcı, D. (2026). Natural-Frequency Uncertainty of Cracked Microbeams with Random Crack Flexibility. European Journal of Engineering and Applied Sciences, Cilt: 9 Sayı: 1, 41–48. https://doi.org/10.55581/ejeas.1967401

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (32)

  1. Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  2. Park, S. K., & Gao, X. L. (2006). Bernoulli–Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16(11), 2355-2359. https://doi.org/10.1088/0960-1317/16/11/015
  3. Ma, H. M., Gao, X. L., & Reddy, J. (2008). A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids, 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  4. Kong, S., Zhou, S., Nie, Z., & Wang, K. (2008). The size-dependent natural frequency of Bernoulli–Euler micro-beams. International Journal of Engineering Science, 46(5), 427-437.https://doi.org/10.1016/j.ijengsci.2007.10.002
  5. Reddy, J. (2011). Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59(11), 2382-2399. https://doi.org/10.1016/j.jmps.2011.06.008
  6. Ansari, R., Gholami, R., & Sahmani, S. (2011). Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures, 94(1), 221-228.https://doi.org/10.1016/j.compstruct.2011.06.024
  7. Akgöz, B., & Civalek, Ö. (2013). Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Composite Structures, 98, 314-322.https://doi.org/10.1016/j.compstruct.2012.11.020
  8. Şimşek, M., Aydın, M., Yurtcu, H. H., & Reddy, J. N. (2015). Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory. Acta Mechanica, 226(11), 3807-3822. https://doi.org/10.1007/s00707-015-1437-9
  9. Li, Y. S., & Xiao, T. (2021). Free vibration of the one-dimensional piezoelectric quasicrystal microbeams based on modified couple stress theory. Applied Mathematical Modelling, 96, 733-750. https://doi.org/10.1016/j.apm.2021.03.028
  10. Al-Shewailiah, D. M. R., & Al-Shujairi, M. A. (2022). Static bending of functionally graded single-walled carbon nanotube conjunction with modified couple stress theory. Materials Today: Proceedings, 61, 1023-1037.https://doi.org/10.1016/j.matpr.2021.10.295
  11. Baghani, M., MohammadSalehi, M. & Dabaghian, P. H. (2016). Analytical couple-stress solution for size-dependent large-amplitude vibrations of FG tapered-nanobeams. Latin American Journal of Solids and Structures, 13(1), 95-118. https://doi.org/10.1590/1679-78252175
  12. Khorshidi, M. A., Shariati, M., & Emam, S. A. (2016). Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory. International Journal of Mechanical Sciences, 110, 160-169.https://doi.org/10.1016/j.ijmecsci.2016.03.006
  13. Kural, S., & Özkaya, E. (2017). Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation. Journal of Vibration and Control, 23(7), 1106-1114.https://doi.org/10.1177/1077546315589666
  14. Atcı, D., & Bağdatlı, S. M. (2017). Free vibrations of fluid conveying microbeams under non-ideal boundary conditions. Steel and Composite Structures, 24(2), 141-149.https://doi.org/10.12989/scs.2017.24.2.141
  15. Abdelrahman, A. A., Abd-El-Mottaleb, H. E., & Eltaher, M. A. (2020). On bending analysis of perforated microbeams including the microstructure effects. Structural Engineering and Mechanics: An International Journal, 76(6), 765-779. https://doi.org/10.12989/sem.2020.76.6.765
  16. Atcı, D. (2021). Free vibrations of nanobeams under non-ideal supports based on modified couple stress theory. Zeitschrift für Naturforschung A, 76(5), 427-434. https://doi.org/10.1515/zna-2020-0335
  17. Polat, S. C., & Bağdatlı, S. M. (2023). Investigation of stepped microbeam vibration motions according to modified couple stress theory. Zeitschrift für Naturforschung Section A: A Journal of Physical Sciences, 78(5). https://doi.org/10.1515/zna-2022-0286
  18. He, G., Li, Z., Gui, H., & Cao, D. (2025). Dynamic modeling and nonlinear vibrations of a rotating poly-crystalline silicon microplate based on the modified couple stress theory. European Journal of Mechanics-A/Solids, 105833. https://doi.org/10.1016/j.euromechsol.2025.105833
  19. Loya, J., López-Puente, J., Zaera, R., & Fernández-Sáez, J. (2009). Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model. Journal of Applied Physics, 105(4). https://doi.org/10.1063/1.3068370
  20. Beni, Y. T., Jafaria, A., & Razavi, H. (2014). Size effect on free transverse vibration of cracked nano-beams using couple stress theory. International Journal of Engineering-Transactions B: Applications, 28(2), 296-304. https://doi.org/ 10.5829/idosi.ije.2015.28.02b.17
  21. Ha, L. T., & Van Lien, T. (2025). Effect of the damping model to dynamic responses of the cracked functionally graded microbeams on the elastic foundation subjected to moving load. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 47(7), 312. https://doi.org/10.1007/s40430-025-05605-x
  22. Akbaş, Ş. D. (2018). Forced vibration analysis of cracked nanobeams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(8), 392.https://doi.org/10.1007/s40430-018-1315-1
  23. Hsu, J. C., Lee, H. L., & Chang, W. J. (2011). Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory. Current Applied Physics, 11(6), 1384-1388.https://doi.org/10.1016/j.cap.2011.04.026
  24. Akbaş, Ş. D. (2017). Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory. International Journal of Structural Stability and Dynamics, 17(03), 1750033. https://doi.org/10.1142/S021945541750033X
  25. Rahi, A., & Petoft, H. (2018). Free vibration analysis of multi-cracked micro beams based on Modified Couple Stress Theory. Journal of Theoretical and Applied Vibration and Acoustics, 4(2), 205-222. http://dx.doi.org/10.22064/tava.2019.89997.1113
  26. Esen, I., Özarpa, C., & Eltaher, M. A. (2021). Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment. Composite Structures, 261, 113552.https://doi.org/10.1016/j.compstruct.2021.113552
  27. Van Lien, T., & Thanh, B. C. (2025). Vibrations of cracked FGM microbeams based on Modified Coupled Stress Theory. Vietnam Journal of Science and Technology, 63(5), 1003-1020. https://doi.org/10.15625/2525-2518/22502
  28. Atcı, D. (2024). Nonlinear vibrations of cracked microbeams based on modified couple stress theory. European Journal of Mechanics-A/Solids, 106, 105321.https://doi.org/10.1016/j.euromechsol.2024.105321
  29. Atcı, D. (2025). Investigation of Crack Effects on the Nonlinear Vibrations of Microbeams with a Tip Mass in a Magnetic Field. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi, 27(79), 54-61. https://doi.org/10.21205/deufmd.2025277908
  30. Al Hussein, D. (2026). Nonlinear Random Vibration of Nanobeam with Material Nonlinear Third-Order Elastic Modulus. International Journal of Structural Stability and Dynamics, 2750297. https://doi.org/10.1142/S021945542750297X
  31. Kutluay Yılmaz, S., & Atcı, D. (2026). Uncertainty analysis of the free vibration of nanobeams with random non-ideal supports. International Journal of Engineering Approaches, 3(1), 57-63. https://doi.org/10.66160/ijea.1932021
  32. Luo, Z., Shi, Q. & Wang, L. (2022). Size-dependent mechanical behaviors of defective FGM nanobeam subjected to random loading. Applied Sciences, 12(19), 9896. https://doi.org/10.1007/s42417-026-02336-8