The present work is concerned with the study of the geometrically non-linear steady state periodic forced response of a clamped–clamped beam containing an open crack. The model based on Hamiltonʼs principle and spectral analysis, previously used to investigate various non-linear vibration problems, is used here to determine the effect of the excitation frequency and level of the applied harmonic force, concentrated at the cracked beam middle span, on its dynamic response at large vibration amplitudes. The formulation uses the “cracked beam functions”, denoted as ‘CBF’, previously defined in recent works, obtained by combining the linear theory of vibration and the linear fracture mechanics theory. The crack has been modelled as a linear spring which, for a given depth, the spring constant remains the same for both directions. The results obtained may be used to detect cracks in vibrating structures, via examination of the qualitative and quantitative changes noticed in the non-linear dynamic behaviour, which is commented in the conclusion.

Accepted:

Published online:

El bekkaye Merrimi ^{1};
Khalid El bikri ^{1};
Rhali Benamar ^{2}

@article{CRMECA_2011__339_11_727_0, author = {El bekkaye Merrimi and Khalid El bikri and Rhali Benamar}, title = {Geometrically non-linear steady state periodic forced response of a clamped{\textendash}clamped beam with an edge open crack}, journal = {Comptes Rendus. M\'ecanique}, pages = {727--742}, publisher = {Elsevier}, volume = {339}, number = {11}, year = {2011}, doi = {10.1016/j.crme.2011.07.008}, language = {en}, }

TY - JOUR AU - El bekkaye Merrimi AU - Khalid El bikri AU - Rhali Benamar TI - Geometrically non-linear steady state periodic forced response of a clamped–clamped beam with an edge open crack JO - Comptes Rendus. Mécanique PY - 2011 SP - 727 EP - 742 VL - 339 IS - 11 PB - Elsevier DO - 10.1016/j.crme.2011.07.008 LA - en ID - CRMECA_2011__339_11_727_0 ER -

%0 Journal Article %A El bekkaye Merrimi %A Khalid El bikri %A Rhali Benamar %T Geometrically non-linear steady state periodic forced response of a clamped–clamped beam with an edge open crack %J Comptes Rendus. Mécanique %D 2011 %P 727-742 %V 339 %N 11 %I Elsevier %R 10.1016/j.crme.2011.07.008 %G en %F CRMECA_2011__339_11_727_0

El bekkaye Merrimi; Khalid El bikri; Rhali Benamar. Geometrically non-linear steady state periodic forced response of a clamped–clamped beam with an edge open crack. Comptes Rendus. Mécanique, Volume 339 (2011) no. 11, pp. 727-742. doi : 10.1016/j.crme.2011.07.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.07.008/

[1] Mechanical Behaviour of Materials, Engineering Methods for Deformation, Fracture, and Fatigue, Prentice–Hall Int. Ed., 1993

[2] A.D. Dimarogonas, Crack identification in aircraft structures, in: 1st National Aircraft Conf., Greece, 1982.

[3] Damage in steel bridges, J. Struct. Eng., Volume 119 (1993) no. 1, pp. 150-167

[4] T.G. Chondros, A.D. Dimarogonas, Identification of cracks in circular plates welded at the contour, in: A.S.M.E. Design Eng. Technical Conf., 79-DET-106, St Louis, 1979.

[5] Identification of cracks in welded joints of complex structures, Journal of Sound and Vibration, Volume 69 (1980), pp. 531-538

[6] J.Y. Guigne, A.S.J. Swamidas, J. Guzzwell, Modal information from acoustic measurements for fatigue crack detection applications, in: Proceedings 11th Int. Conf. on Offshore Technology and Arctic Eng., vol. I, Part B, 1992.

[7] Fatigue crack propagation in resonating structures, Eng. Fracture Mech., Volume 34 (1989) no. 3, pp. 721-728

[8] Etude de lʼamorçage des fissures et de la vitesse de fissuration par fatigue de quelques aciers inoxidables austénitiques, Revue de Métallurgie (1974), pp. 931-941

[9] Large-amplitude random response of angle-ply laminated composite plates, American Institute of Aeronautics and Astronautics, Journal, Volume 20 (1982), pp. 1450-1458

[10] Developments in the acoustic fatigue design process for composite aircraft structures, Composite Structures, Volume 16 (1990), pp. 171-192

[11] The effect of large vibration amplitudes on the dynamic strain response of a clamped–clamped beam with consideration of fatigue life, Journal of Sound and Vibration, Volume 96 (1984), pp. 281-308

[12] F. Pérignon, Vibration forcée des structures minces, élastiques, non linéaires, Thèse, université Aix-Marseille II, 2004.

[13] A vibration technique for non-destructively assessing the integrity of structures, J. Mech. Eng. Sci., Volume 20 (1978), pp. 93-100

[14] Simple static and dynamic models for cracked elastic beams, International Journal of Fracture, Volume 17 (1972), pp. 71-76

[15] A numerical study of the eigenparameters of a damaged cantilever beam, Journal of Sound and Vibration, Volume 103 (1985) no. 3, pp. 301-310

[16] Analytical Methods in Rotor Dynamics, Appl. Sci. Publishers, London, 1983 (pp. 144–193)

[17] Coupling and bending of torsional vibration of cracked Timoshenko shaft, Ing. Arch., Volume 57 (1987), pp. 495-505

[18] I.W. Mayes, W.G.R. Davies, A method of calculating the vibrational behaviour of coupled rotating shafts containing a transverse cracks, Paper No. C254/80, in: I. Mech. E. Conf., 1980.

[19] P. Gudmundson, Changes in modal parameters resulting from small cracks, in: Proc. 2nd Int. Modal Analysis Conf., vol. 2, Union College, Orlando, NY, USA, 1984, pp. 690–697.

[20] Identification of crack location and magnitude in a cantilever beam from the vibration modes, Journal of Sound and Vibration, Volume 138 (1990), pp. 381-388

[21] Forced vibration behaviour and crack detection of cracked beams using instantaneous frequency, NDT&E International, Volume 38 (2005), pp. 411-419 | DOI

[22] Nonlinear vibration of edge cracked functionally graded Timoshenko beams, Journal of Sound and Vibration, Volume 324 (2009), pp. 962-982 | DOI

[23] Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response, Journal of Sound and Vibration, Volume 330 (2011), pp. 721-742 | DOI

[24] R. Benamar, Non-linear dynamic behaviour of fully clamped beams and rectangular isotropic and laminated plates, Ph.D. thesis, University of Southampton, 1990.

[25] The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part I: Simply supported and clamped–clamped beams, Journal of Sound and Vibration, Volume 149 (1991), pp. 179-195

[26] The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part II: Fully clamped rectangular isotropic plates, Journal of Sound and Vibration, Volume 164 (1991), pp. 399-424

[27] A semi-analytical approach to the non-linear dynamic response. Problem of S–S and C–C beams at large vibration amplitudes. Part I: General theory and application to the single mode approach to free and forced vibration analysis, Journal of Sound and Vibration, Volume 224 (1999), pp. 377-395

[28] A semi-analytical approach to the non-linear dynamic response. Problem of beams at large vibration amplitudes. Part II: Multimode approach to the forced vibration analysis, Journal of Sound and Vibration, Volume 255 (2002) no. 1, pp. 1-41 | DOI

[29] L. Azrar, R. Benamar, R.G. White, Non-linear free and forced response of beams at large vibration amplitudes by a semi-analytical method, in: Proceedings of the Seventh International Conference – Structural Dynamics, Southampton, England, 2000.

[30] Improvement of the semi-analytical method, based on Hamiltonʼs principle and spectral analysis, for determination of the geometrically non-linear free response of thin straight structures. Part I: Application to C–C and SS–C beams, Journal of Sound and Vibration, Volume 249 (2002) no. 2, pp. 263-305

[31] Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: The effects of large vibration amplitudes on the fundamental mode shape, Comput. Struct., Volume 81 (2003), pp. 2029-2043

[32] Geometrically non-linear free vibrations of clamped–clamped beams with an edge crack, Comput. Struct., Volume 84 (2006), pp. 485-502 | DOI

[33] Analyses of stresses and strains near the end of a crack transversing a plate, J. Appl. Mech., Volume 24 (1957), pp. 361-364

[34] H. Tada, The stress analysis of cracks handbook, in: 9th Cong. Appl. Mech., Brussels, 1957.

[35] ASTM Standarts, Part 31, 1968, pp. 1018–1030.

[36] Theory of Vibration with Applications, Prentice–Hall Inc., Englewood Cliffs, NJ, 1972

[37] Geometrically nonlinear transverse vibrations of discrete multi-degrees of freedom systems with a localised nonlinearity, International Journal of Mathematics and Statistics, Volume 4 ( Spring 2009 ) no. S09

*Cited by Sources: *

Comments - Policy