The natural frequencies and mode shapes of an Euler-Bernoulli beam with a rectangular cross- section, which has a surface crack, is investigated. The crack is modeled as a change (sudden or gradual) in the cross-section of the beam, and a modified perturbation approach is used assuming that the crack geometry is much smaller than the beam cross section. Computations of natural frequencies and mode shapes were carried out for various crack shapes and compared with a range of experiments and finite element analyses. It is concluded that the suggested modified perturbation approach gives reliable results with minimal effort for eigenfrequencies of cracked beams. Furthermore, as a new feature, the present perturbation method includes the shape of the crack in eigenfrequency computations and in principle, can work for any type of disturbance on the surface including a small bump for example.

Keywords:
## Perturbation, Surface Crack Euler-Bernoulli Beam,

- [1] R. D. Adams, P. Cawley, C. J. Pye, B. J. Stone, “A vibration technique for non-destructively assessing the integrity of structures,” Journal of Mechanical Engineering Science, vol. 20, no. 2, pp. 93–100, 1978.
- [2] L. Rubio, “An efficient method for crack identification in simply supported Euler – Bernoulli Beams,” Journal of Vibration and Acoustics, vol. 131, no. 5, pp. 1-6, 2009.
- [3] A. Morassi, “Crack-induced changes in eigenparameters of beam structures,” Asce - Journal Engineering Mechanics, vol. 119, no. 9, pp. 1798–1803, 1993.
- [4] M. H. H. Shen, C. Pierre, “Natural modes of Bernoulli-Euler Beams with symmetric cracks,” Journal of Sound and Vibration, vol. 138, no. 1, pp. 115–134, 1990.
- [5] N. Papaeconomou, A. Dimarogonas, “Vibration of cracked beams,” Computational Mechanics, vol. 5, no. 2, pp. 88–94, 1989.
- [6] T. G. Chondros, A. D. Dimarogonas, “Vibration of a cracked cantilever beam,” Journal of Vibration and Acoustics, vol. 120, no. 3, pp. 742–746, 1998.
- [7] N. T. Khiem, L. K. Toan, “A novel method for crack detection in beam-like structures by measurements of natural frequencies,” Journal of Sound and Vibration, vol. 333, no. 18, pp. 4084–4103, 2014.
- [8] J. F. Sáez, A. Morassi, M. Pressacco, L. Rubio, “Unique determination of a single crack in a uniform simply supported beam in bending vibration,” Journal of Sound and Vibration, vol. 371, no. 9, pp. 94–109, 2016.
- [9] T. D. Chaudhari, S. K. Maiti, “Modelling of transverse vibration of beam of linearly variable depth with edge crack,” Engineering Fracture Mechanics, vol. 63, no. 4, pp. 425–445, 1999.
- [10] Q. He, Y. Lin, “Assessing the severity of fatigue crack using acoustics modulated by hysteretic vibration for a cantilever beam,” Journal of Sound and Vibration, vol. 370, no. 26, pp. 306–318, 2016.
- [11] F. B. Nejad, A. Khorram, M. Rezaeian, “Analytical estimation of natural frequencies and mode shapes of a beam having two cracks,” International Journal of Mechanical Sciences, vol. 78, no. 1, pp. 193–202, 2014.
- [12] K. Mazanoglu, I. Yesilyurt, M. Sabuncu, “Vibration analysis of multiple-cracked non-uniform beams,” Journal of Sound and Vibration, vol. 320, no. 4-5, pp. 977–989, 2009.
- [13] X. F. Yang, A. S. J. Swamidas, R. Seshadri, “Crack identification in vibrating beams using the energy method,” Journal of Sound and Vibration, vol. 244, no. 2, pp. 339–357, 2001.
- [14] K. Aydin. “Free vibration of functionally graded beams with arbitrary number of surface cracks,” European Journal of Mechanics - A/Solids, vol. 42, no. 11-12, pp. 112–124, 2013.
- [15] S. Caddemi, A. Morassi, “Multi-cracked Euler–Bernoulli Beams: Mathematical modeling and exact solutions,” International Journal of Solids and Structures, vol. 50, no. 6, pp. 944–956, 2013.
- [16] M. Kisa, J. Brandon, M. Topcu, “Free vibration analysis of cracked beams by a combination of finite elements and component mode synthesis methods,” Computers and Structures, vol. 67, no. 4, pp. 215–223, 1998.
- [17] D. Y. Zheng, N. J. Kessissoglou, “Free vibration analysis of a cracked beam by finite element method,” Journal of Sound and Vibration, vol. 273, no. 3, pp. 457–475, 2004.
- [18] M. Kisa, M. A. Gurel, “Free vibration analysis of uniform and stepped cracked beams with circular cross sections,” International Journal of Engineering Science, vol. 45, no. 2-8, pp. 364–380, 2007.
- [19] J. P. Blasques, R. D. Bitsche, “An efficient and accurate method for computation of energy release rates in beam structures with longitudinal cracks,” Engineering Fracture Mechanics, vol. 133, no. 1, pp. 56–69, 2015.
- [20] C. Dundar, I. F. Kara, “Three dimensional analysis of reinforced concrete frames with cracked beam and column elements,” Engineering Structures, vol. 29, no. 9, pp. 2262–2273, 2007.
- [21] T. Xu, A. Castel, “Modeling the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration loads,” Journal of Sound and Vibration, vol. 368, no. 4, pp. 135–147, 2016.
- [22] M. Heydari, A. Ebrahimi, M. Behzad, “Forced vibration analysis of a Timoshenko cracked beam using a continuous model for the crack,” Engineering Science and Technology, an International Journal, vol. 17, no. 4, pp. 194–204, 2014.
- [23] B. Panigrahi, G. Pohit, “Nonlinear modelling and dynamic analysis of cracked Timoshenko functionally graded beams based on neutral surface approach,” Journal of Mechanical Engineering Science, 230, vol. 9, pp. 1486–1497, 2016.
- [24] P. Gudmundson, “Eigenfrequency changes of structures due to cracks, notches or other geometrical changes,” Journal of the Mechanics and Physics of Solids, vol. 30, no. 5, pp. 339–353, 1982.
- [25] W. M. Hasan, “Crack detection from the variation of the eigenfrequencies of a beam on elastic foundation,” Engineering Fracture Mechanics, vol. 52, no. 3, pp. 409–421, 1995.
- [26] P. E. Cooley, J.C. Slater, O.V. Shiryayev, “Fatigue crack modeling and analysis in beams,” 53rd IAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference AIAA Paper, pp. 2012-1874, 2012.
- [27] M. H. H. Shen, C. Pierre, “Natural modes of Bernoulli-Euler Beams with a single-edge crack,” 31st ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference AIAA Paper, pp. 1990-1124, 1990.
- [28] Y. C. Chu, M. H. H. Shen, “Analysis of forced bilinear oscillators and the application to cracked beam dynamics,” AIAA Journal, vol. 30, no. 10, pp. 2512–2519, 1992.
- [29] T. D. Dang, R. K. Kapania, M. J. Patil, “Analytical modeling of cracked thin-walled beams under torsion,” AIAA Journal, vol. 48, no. 3, pp. 664–675, 2010.
- [30] Y. S. Lee, M. J. Chung, “A study on crack detection using eigenfrequency test data,” Computers and Structures, vol. 77, no. 3, pp. 327–342, 2000.
- [31] S. Moradi, P. Razi, L. Fatahi, “On the application of bees algorithm to the problem of crack detection,” Computers and Structures, vol. 89, no. 23-24, pp. 2169–2175, 2011.
- [32] K. H. Barad, D. S. Sharma, V. Vyas, “Crack detection in cantilever beam by frequency based method,” Procedia Engineering, vol. 51, pp. 770–775, 2013.
- [33] N. Hu, X. Wang, H. Fukunaga, Z. H. Yao, H. X. Zhang, Z. S. Wu, “Damage assessment of structures using modal test data,” International Journal of Solids and Structures, vol. 38, no. 18, pp. 3111–3126, 2001.
- [34] N. T. Khiem, L. K. Toan, “A novel method for crack detection in beam-like structures by measurements of natural frequencies,” Journal of Sound and Vibration, vol. 333, no. 18, pp. 4084–4103, 2014.
- [35] J. K. Sinha, M. I. Friswell, S. Edwards, “Simplified models for the location of cracks in beam structures using measured vibration data,” Journal of Sound and Vibration, vol. 251, no. 1, pp. 13–38, 2002.
- [36] H. I. Yoon, I. S. Son, S. J. Ahn, “Free vibration analysis of Euler-Bernoulli Beam with double cracks,” Journal of Mechanical Science and Technology, vol. 21, no. 3, pp. 476–485, 2007.

Bibtex | @araştırma makalesi { saufenbilder1170458, journal = {Sakarya University Journal of Science}, eissn = {2147-835X}, address = {}, publisher = {Sakarya Üniversitesi}, year = {2022}, volume = {26}, number = {6}, pages = {1233 - 1243}, doi = {10.16984/saufenbilder.1170458}, title = {Perturbatıon Solution for a Cracked Euler-Bernoulli Beam}, key = {cite}, author = {Sakman, Lütfi Emir} } |

APA | Sakman, L. E. (2022). Perturbatıon Solution for a Cracked Euler-Bernoulli Beam . Sakarya University Journal of Science , 26 (6) , 1233-1243 . DOI: 10.16984/saufenbilder.1170458 |

MLA | Sakman, L. E. "Perturbatıon Solution for a Cracked Euler-Bernoulli Beam" . Sakarya University Journal of Science 26 (2022 ): 1233-1243 < |

Chicago | Sakman, L. E. "Perturbatıon Solution for a Cracked Euler-Bernoulli Beam". Sakarya University Journal of Science 26 (2022 ): 1233-1243 |

RIS | TY - JOUR T1 - Perturbatıon Solution for a Cracked Euler-Bernoulli Beam AU - Lütfi EmirSakman Y1 - 2022 PY - 2022 N1 - doi: 10.16984/saufenbilder.1170458 DO - 10.16984/saufenbilder.1170458 T2 - Sakarya University Journal of Science JF - Journal JO - JOR SP - 1233 EP - 1243 VL - 26 IS - 6 SN - -2147-835X M3 - doi: 10.16984/saufenbilder.1170458 UR - Y2 - 2022 ER - |

EndNote | %0 Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi Perturbatıon Solution for a Cracked Euler-Bernoulli Beam %A Lütfi Emir Sakman %T Perturbatıon Solution for a Cracked Euler-Bernoulli Beam %D 2022 %J Sakarya University Journal of Science %P -2147-835X %V 26 %N 6 %R doi: 10.16984/saufenbilder.1170458 %U 10.16984/saufenbilder.1170458 |

ISNAD | Sakman, Lütfi Emir . "Perturbatıon Solution for a Cracked Euler-Bernoulli Beam". Sakarya University Journal of Science 26 / 6 (Aralık 2022): 1233-1243 . |

AMA | Sakman L. E. Perturbatıon Solution for a Cracked Euler-Bernoulli Beam. SAUJS. 2022; 26(6): 1233-1243. |

Vancouver | Sakman L. E. Perturbatıon Solution for a Cracked Euler-Bernoulli Beam. Sakarya University Journal of Science. 2022; 26(6): 1233-1243. |

IEEE | L. E. Sakman , "Perturbatıon Solution for a Cracked Euler-Bernoulli Beam", , c. 26, sayı. 6, ss. 1233-1243, Ara. 2022, doi:10.16984/saufenbilder.1170458 |

Sayıdaki Diğer Makaleler

Şevket Onur KALKAN, Lütfullah GÜNDÜZ

Synthesis, DPPH and ABTS Activity of Novel Furfuryl-Chalcone Derivatives

Fatih SÖNMEZ, Enes AKGÜN, Zuhal ŞAHİN

Traditional Medicinal Plants Used in Dermatologic Disorders in Ardahan, Iğdır and Kars

Songül KARAKAYA, Zehra KIMIŞOĞLU, Serdar Volkan YILMAZ, Özkan AKSAKAL, Yusuf Ziya SÜMBÜLLÜ, Ümit İNCEKARA, Ahmet POLAT

Lazim KAMBERİ, Senad ORHANİ, Mirlinda SHAQİRİ, Sejhan IDRİZİ