Effect of Bed and Crack on the Natural Frequency for the Timoshenko Beam Using Finite Element Method

Document Type : Persian


1 Assistant Prof., Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran

2 MSc Student, Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran


In this study, the natural frequencies and mode shapes of beams without cracks and cracked Timoshenko beams is calculated with different boundary conditions using finite element method. The energy method is used to solve the equations. Hardness and softness matrices for Timoshenko beam without crack are obtained by solving the potential and kinetic energy equations. Then for investigation of cracked condition, the cracked element stiffness matrix is used and the beam natural frequencies are obtained by entering the boundary conditions of the beam. After that the effect of bed is investigated by addition of it to the equations.


[1] Gudmundsun, P., The Dynamic Behavior of Slender Structures with Cross-Sectional Cracks, Journal of Mechanics and Physics of Solids, Vol. 1, No. 4, 1983, pp. 329-345.
[2] Silva, J.M., Gomes, A.J., Experimental Dynamic Analysis of Cracked Free-free Beams, Journal of Experimental Mechanics, Vol. 30, No. 1, 1990, pp. 20-25.
[3] Qian, G.L., Gu, S.N., Jiang, J.S., The Dynamic Behavior and Crack Detection of a Beam with a Crack, Journal of Sound and Vibration, Vol. 138, No. 2, 1990, pp. 233-243.
[4] Pandey A.K., Biswas M., Damage Detection in Structures Using Changes in Flexibility, Journal of Sound and Vibration, Vol. 169, No. 1, 1994, pp. 3–17.
[5] Narkis Y., Identification of Crack Location in Vibrating Simply supported Beams, Journal of Sound and Vibration, Vol. 172, No. 4, 1994, pp. 549-558.
[6] Lele S.P., Maiti S.K., Modeling of Transverse Vibration of Short Beams for Crack Detection and Measurement of Crack Extension, Journal of Sound and Vibration, Vol. 257, No. 3, 2002, pp. 559-583.
[7] Kim J.T., Stubbs N., Crack Detection in Beam-Type Structures Using Frequency Data, Journal of Sound and Vibration, Vol. 259, No. 1, 2003, pp. 145-160.
[8] Lin, H. P., ‘‘Direct and Inverse Methods on Free Vibration Analysis of Simply Supported Beams with a Crack’’, Journal of Engineering Structures, Vol. 26, 2004, pp. 427-436.
 [9] Swamidas A.S.J., Yang X.F., Seshadri R., Identification of Cracking in Beam Structures Using Timoshenko and Euler Formulations, Journal of Engineering Mechanics, Vol. 130, No. 11, 2004, pp. 1297-1308.
[10] Nahvi H., Jabbari M., Crack Detection in Beams Using Experimental Modal Data and Finite Element Method, Journal of Mechanical Science, Vol. 47, 2005, pp. 1477-1497.
[11] Vakili-Baghmisheh M., Peimani M., Homayoun Sadeghi M., Ettefagh M., Crack Detection in Beam Like Structures Using Genetic Algorithms, Journal of Applied Soft Computing, Vol. 8, 2008, pp. 1150-1160.
[12] Ariaei A., Ziaei-Rad S., Ghayour M., Vibration analysis of beams with open and breathing cracks subjected to moving masse, journal of sound and vibration, Vol. 326, 2009, pp. 709-724.
[13] Dimaragonas A.D., Vibration of cracked structures-A state of the art review, Engineering fracture mechanics, Vol. 55, 1996, pp.831-857.
[14] Hetenyi M., Beams on elastic foundation. The university of Michigan press, Ann Arbor, U.S.A, 1946.
[15] Chen Y., Huang Y., Shin E., Response of an in finite Timoshenko beam on a viscoelastic foundation to a harmonic moving loads, Journal of sound and vibration, Vol. 241, 2001, pp. 809-842.
]16[ چهارمحالی­پور، ا.، نحوی، ح.، بررسی رفتار ارتعاشی تیرهای دارای ترک، دانشکده مکانیک، دانشگاه آزاد خمینی‌شهر، 1387.