Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

doi:10.1007/s10483-016-2042-9

Abstract

Abstract:

Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam's slenderness ratio, the crack's depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam's deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam's slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.

Key words: parameter study, crack gap, generalized function, Timoshenko beam, switching crack

2010 MSC Number:

34A05
30G30

Xiao YANG, Jin HUANG, Yu OUYANG. Bending of Timoshenko beam with effect of crack gap based on equivalent spring model. Applied Mathematics and Mechanics (English Edition), 2016, 37(4): 513-528.

TrendMD

References

[1] Jassim, Z. A., Ali, N. N., Mustapha, F., and Abdul-Jalil, N. A. A review on the vibration analysis for a damage occurrence of a cantilever beam. Engineering Failure Analysis, 31(7), 442-461(2013)
[2] Banan, M. R. and Hjelmstad, K. D. Parameter estimation of structures from static response, I:computational aspects. Journal of Structural Engineering, 120(11), 3243-3258(1994)
[3] Sung, S. H., Koo, K. Y., and Jung, H. J. Modal flexibility-based damage detection of cantilever beam-type structures using baseline modification. Journal of Sound and Vibration, 333(18), 4123-4138(2014)
[4] Dimarogons, A. Vibration of cracked structures:a state of the art review. Engineering Fracture Mechanics, 55(5), 831-857(1996)
[5] Palmeri, A. and Cicirello, A. Physically-based Dira?s delta functions in the static analysis of multicracked Euler-Bernoulli and Timoshenko beams. International Journal of Solids and Structures, 48(14-15), 2184-2195(2011)
[6] Challamel, N. and Xiang, Y. On the influence of the unilateral damage behaviour in the stability of cracked beam columns. Engineering Fracture Mechanics, 77(9), 1467-1478(2010)
[7] Patel, T. H. and Darpe, A. K. Influence of crack breathing model on nonlinear dynamics of a cracked rotor. Journal of Sound and Vibration, 311(3-5), 953-972(2008)
[8] Rezaee, M. and Hassannejad, R. Free vibration analysis of simply supported beam with breathing crack using perturbation method. Acta Mechanica Solida Sinica, 23(5), 459-470(2010)
[9] Buda, G. and Caddemi, S. Identification of concentrated damages in Euler-Bernoulli beams under static loads. Journal of Engineering Mechanics, 133(8), 942-956(2007)
[10] Khaji, N., Shafiei, M., and Jalalpour, M. Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions. International Journal of Mechanical Sciences, 51(9-10), 667-681(2009)
[11] Feng, X., Liu, Y. H., and Zhou, J. A method for static deflection analysis for cracked Timoshenko beam (in Chinese). Journal of Disaster Prevention and Mitigation Engineering, 29(6), 652-657(2009)
[12] Caddemi, S. and Calió, I. Exact solution of the multi-cracked Euler-Bernoulli column. International Journal of Solids and Structures, 45(5), 1332-1351(2008)
[13] Li, Q. S. Buckling of an elastically restrained multi-step non-uniform beam with multiple cracks. Archive of Applied Mechanics, 72(6-7), 522-535(2002)
[14] Kisa, M. Vibration and stability of multi-cracked beams under compressive axial loading. International Journal of the Physical Sciences, 6(11), 2681-2696(2011)
[15] Caddemi, S. and Calió, I. Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks. Journal of Sound and Vibration, 327(3-5), 473-489(2009)
[16] Caddemi, S. and Calió, I. The influence of the axial force on the vibration of the Euler-Bernoulli beam with an arbitrary number of cracks. Archive of Applied Mechanics, 82(6), 1-13(2012)
[17] Jun, O. S., Eun, H. J., Earmme, Y. Y., and Lee, C. W. Modelling and vibration analysis of a simple rotor with breathing crack. Journal of Sound and Vibration, 155(2), 273-290(1992)
[18] Hu, J. S., Feng, X., and Zhou, J. Study on nonlinear dynamic response of a beam with a breathing crack (in Chinese). Journal of Vibration and Shock, 28(1), 76-80(2009)
[19] Cicirello, A. and Palmeri, A. Static analysis of Euler-Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads. International Journal of Solids and Structures, 51(5), 1020-1029(2014)
[20] Fu, C. Y. The effect of switching cracks on the vibration of a continuous beam bridge subjected to moving vehicles. Journal of Sound and Vibration, 339(3), 157-175(2015)
[21] Bakhtiari-Nejad, F., Khorram, A., and Rezaeian, M. Analytical estimation of natural frequencies and mode shapes of a beam having two cracks. International Journal of Mechanical Sciences, 78(1), 193-202(2014)
[22] Caddemi, S. and Calió, I. The exact explicit dynamic stiffness matrix of multi-cracked Euler-Bernoulli beam and applications to damaged frame structures. Journal of Sound and Vibration, 332(12), 3049-3063(2013)
[23] Caddemi, S., Calió, I., and Marletta, M. The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks. International Journal of Non-Linear Mechanics, 45(7), 714-726(2010)
[24] Shen, M. H. H. and Chu, Y. C. Vibration of beams with a fatigue crack. Computers and Structures, 45(1), 79-93(1992)
[25] Gere, J.M. andTimoshenko, S.P. Mechanics of Materials, 2nd SI Edition, van Nostrand Reinhold, New York (1984)
[26] Han, S. M., Benaroya, H., and Wei, T. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225(5), 935-988(1999)