Parametric investigations into dynamics of cracked thin rectangular plates excited by a moving mass

Document Type : Article


1 Departement of Civil Engineering, University of Science and Culture (USC), Tehran, P.O. Box 13145-871, Iran

2 Departement of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran


Dynamic analysis of cracked thin rectangular plates subjected to a moving mass first is investigated in this paper. To this end, the eigenfunction expansion method is utilized to solve the governing equation. For the first time, the intact plate orthogonal polynomials in combination with admissible crack functions as a composition, are employed in the eigenfunction expansion method formulation, required professional computer programming to solve the equation. The proposed approach guarantees upper bound of the true solution, which is the property of an appropriate numerical solution. Parametric investigation is performed to determine the effect of the moving mass weight, the moving mass velocity, the crack length, and the crack angular orientation as well as the plate’s aspect ratio, on the dynamic response of cracked plates. The results confirm, that the moving mass has a greater impact than the moving load on the dynamic responses of cracked plates. Furthermore, there is nonlinear relation among enhancing the dynamic responses of the cracked plates with various boundary conditions, and magnifying the moving mass weights, elevating the moving mass velocities, lengthening the crack length, raising the inclined crack angels as well as augmenting the plates aspect ratios.


1. Lynn, P.P. and Kumbasar, N. "Free vibration of thin rectangular plates having narrow cracks with simply supported edges", Developments in Mechanics. In. Proceeding of the 10th Midwestern Mechanics Conference, Colorado State University, Fort Collins, Colorado, 4, pp. 911-928 (1967).
2. Stahl, B. and Keer, L.M. "Vibration and stability of cracked rectangular plates", International Journal of Solids and Structures, 8, pp. 69-91 (1972).
3. Solecki, R. "Bending vibration of a simply supported rectangular plate with a crack parallel to one edge", Engineering Fracture Mechanics, 18, pp. 1111-1118 (1983).
4. Qian, G.L., Gu, S.N., and Jiang, J.S. "A finite element model of cracked plates and application to vibration problems", Computers and Structures, 39, pp. 483-487 (1991).
5. Krawczuk, M. "Natural vibrations of rectangular plates with a through crack", Archive of Applied Mechanics, 63, pp. 491-504 (1993).
6. Bachene, M., Tiberkak, R., and Rechak, S. "Vibration analysis of cracked plates using the extended finite element method", Archive of Applied Mechanics, 79, pp. 249-262 (2009).
7. Yuan, J. and Dickinson, S.M. "The  flexural vibration of rectangular plate systems approached by using artificial springs in the Rayleigh-Ritz method", Journal of Sound and Vibration, 159, pp. 39-55 (1992).
8. Liew, K.M., Hung, K.C., and Lim, M.K. "A solution method for analysis of cracked plates under vibration", Engineering Fracture Mechanics, 48, pp. 393- 404 (1994).
9. Lee, H.P. and Lim, S.P. "Vibration of cracked rectangular plates including transverse shear deformation and rotary inertia", Computers and Structures, 49, pp. 717-718 (1993).
10. Huang, C.S. and Leissa A.W. "Vibration analysis of rectangular plates with side cracks via the Ritz method", Journal of Sound and Vibration, 323, pp. 974-988 (2009).
11. Haung, C.S., Leissa, A.W., and Chan, C.W. "Vibrations of rectangular plates with internal cracks or slits", International Journal of Mechanical Sciences, 53, pp. 436-445 (2011).
12. Haung, C.S., Leissa, A.W., and Li, R.S. "Accurate vibration analysis of thick, cracked rectangular plates", Journal of Sound and Vibration, 330, pp. 2079-2093 (2011).
13. Riks, E., Bankin, C.C., and Brogen F.A. "The buckling of a central crack in a plate under tension", Engineering Fracture Mechanics, 43, pp. 529-548 (1992).
14. Barut, A., Madenci, E., and Britt, V.O. "Starnes, J.R.: Buckling of a thin tension loaded composite plate with an inclined crack", Engineering Fracture Mechanics, 58, pp. 233-248 (1997).
15. Brighenti, R. "Buckling sensitivity analysis of cracked thin plates under tension or compression", Thin Walled Structures, 43, pp. 209-224 (2005).
16. Brighenti, R. "Buckling sensitivity analysis of cracked thin plates under membrane tension or compression loading", Nuclear Engineering and Design, 239, pp. 965-980 (2009).
17. Zeng, H.C., Huang, C.S., Leissa, A.W., et al. "Vibration and stability of a loaded side- cracked rectangular plate via the MLS-Ritz method", Thin-Walled Structures, 106, pp. 459-470 (2016).
18. Xue, J. and Wang, Y. "Free vibration analysis of a  at stiffened plate with side crack through the Ritz method", Archive of Applied Mechanics, 89, pp. 2089-2102 (2019).
19. Xue, J., Wang, Y., and Chen, L. "Nonlinear vibration of cracked rectangular Mindlin plate with in- plane preload", Journal of Sound and Vibration, 481, Art. 115437 (2020).
20. Huang, C.S., Lee, M.C., and Chang, M.J. "Vibration and buckling analysis of internally cracked square plates by the MLS-Ritz approach", International Journal of Structural Stability and Dynamics, 18(9), Art. 1850105 (2018).
21. Xue, J., Wang, Y., and Chen, L. "Buckling and free vibration of a side- cracked Mindlin plate under axial in-plane load", Archive of Applied Mechanics, 90, pp. 1811-1827 (2020).
22. Huang, C.S., Lee, H.T., Li, P.Y., et al. "Threedimensional buckling analyses of cracked functionally graded material plates via the MLS-Ritz method", Thin-Walled Structures, 134, pp. 189-202 (2019).
23. Huang, C.S., Lee, H.T., Li P.Y., et al. "Three- dimensional free vibration analyses of preloaded cracked plates of functionally graded materials via the MLSRitz method", Materials, 14(24), Art. 7712 (2021).
24. Kiani, K. and Z_ ur, K.K. "Vibrations of doublenanorod-systems with defects using nonlocal-integralsurface energy-based formulations", Composit Structures, 256, pp. 113028-113043 (2021).
25. Mote, C.D. "Stability of circular plates subjected to moving load", Journal of the Franklin Institute, 290, pp. 329-344 (1970).
26. Fryba, L., Vibration of Solids and Structures Under Moving Loads, Thomas Telford, London (1999).
27. Cifuentes, A. and Lalapet, S. "A general method to determine the dynamic response of a plate to a moving mass", Computers and Structures, 42, pp. 31- 36 (1992).
28. Shadnam, M.R., Mofid, M., and Akin, J.E. "On the dynamic response of rectangular plate with moving mass", Thin Wall Structures, 39, pp. 797-806 (2001).
29. Rofooei, F.R. and Nikkhoo, A. "Application of active piezoelectric patches in controlling the dynamic response of a thin rectangular plate under a moving mass", International Journal of Solids and Structures, 46, pp. 2429-2443 (2009).
30. Nikkhoo, A. and Rofooei, F.R. "Parametric study of the dynamic response of thin rectangular plates traversed by moving mass", Acta Mechanica, 223, pp. 15-27 (2012).
31. Kiani, K. "Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations", Physica E., 44, pp. 229-248 (2011).
32. Kiani, K. "Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part II: Parametric studies", Physica E., 44, pp. 249-269 (2011).
33. Kiani, K. "Vibrations of biaxially tensioned- embedded nanoplates for nanoparticle delivery", Indian Journal of Science and Technology, 6, pp. 4894-4902 (2013).
34. Nikkhoo, A., Hassanabadi, M.E., Azam, S.E., et al. "Vibration of a thin rectangular plate subjected to series of moving inertial loads", Mechanics Research Communications, 55, pp. 105-113 (2014).
35. Rofooei, F.R, Enshaeian, A., and Nikkhoo, A. "Dynamic response of geometrically nonlinear, elastic rectangular plates under a moving mass loading by inclusion of all inertial components", Journal of Sound and Vibration, 394, pp. 497-514 (2017).
36. Nikkhoo, A., Tafakor, R., and Mofid, M. "An exact solution to the problems of  exo-poro-elastic structures rested on elastic beds acted upon by moving loads", Scientia Iranica, 27, pp. 2326-2341 (2019).
37. Chakraverty, S., Vibration of Plates, CRC Press, Boca Raton (2010).
38. Bhat, R.B. "Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method", Journal of Sound and Vibration, 102(4), pp. 493-499 (1985).
39. Huang, C.S. and Huang, S.H. "Analytical solutions based on fourier cosine series for the free vibrations of functionally graded material rectangular mindlin plates", Materials, 13(17), Art. 3820 (2020).
40. Timoshenko, S. and Winowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, New York (1959).
41. Fryba, L. "History of winkler foundation", Vehicle System Dynamics Supplement, 24, pp. 7-12 (1995).
42. Chopra, A.K., Dynamics of Structures Theory and Applications to Earthquake Engineering in SI, Pearson Prentice Hall, Upper Saddle River (2019).
43. Leissa, A.W., Vibration of Plates, US Government Printing Office, Washington, DC (1969).
44. Boresi, A.P. and Schmidt, R.J., Advanced Mechanics of Materials, John Wiley and Sons, Inc., New York (2002).