Study on Free Vibration of Buckled Cracked Plates Using Differential Quadrature Element Method

Authors

Abstract

Vibration of postbuckled cracked plate has been investigated using the differential quadrature element method. The crack modeled as an open crack using a no-mass linear spring. The governing equations of vibration of a buckled cracked plate are derived using the Mindlin theory and considering the effect of initial imperfection. The answer consist of static and dynamic parts. First, differential equations are discretized using the differential quadrature element method and then the resulting nonlinear algebraic equations are solved using the arc-length strategy. Then, assuming small amplitude vibrations of the plate about its buckled state and exploiting the static solution in the linearized vibration equations, the dynamic equations are converted to a non-standard eigenvalue problem. Finally, natural frequencies and mode shapes of the buckled cracked plate are obtained solving the eigenvalue problem. The accuracy of the proposed approach is verified using the results obtained by an experimental setup and those obtained by the finite element method. Moreover, several case studies of buckled cracked plates have been solved and effects of selected parameters have been studied.

Keywords

Main Subjects

References

[1] Rice JR, Levy N (1972) The part-through surface crack in elastic plate. J Appl Mech: 185-194.
[2] Chen  YZ (2011) A numerical solution for a finite internally cracked plate using hybrid crack element method, Structural engineering and Mechanics. An Int j 40(6): 813-827.
[3] Brighenti R (2010) Influence of a central straight crack on the buckling behavior of thin plates under tension, compression or shear loading. Int J mech of materials 6: 73-87.
[4] Alinia MM, Hosseinzadeh SAA, Habashi HR (2007) Numerical modeling for buckling analysis of cracked shear panels. J thin-walled structures 45: 1058-1067.
[5] Moradi S, Alimouri P (2014) Crack detection of plate structures using differential quadrature method. J Mech Eng Sci 227(7): 1495-1504.
[6] Bose T, Mohanty AR (2013) Vibration analysis of a rectangular thin isotropic plate with a part-through surface crack of arbitrary orientation and position. J Sound Vib 332: 7123-7141.
[7] Huang CS, Leissa AW, Chan CW (2011) Vibrations of rectangular plates with internal cracks or slits. Int J Mech Sci 53: 436-445.
[8] Khadem SE, Rezaee M (2000) An analytical approach for obtaining the location and depth of an all-over part through crack on externally in-plane loaded rectangular plate using vibration analysis. J Sound Vib 230(2): 291-308.
[9] Ismal R, Cartmell MP (2012) an investigation into the vibration analysis of a plate with a surface crack of variable angular orientation. J Sound Vib 2929-2948.
[10] Israr A (2011) Model for vibration of crack plates for use with damage detection methodologies. Space Technol 1: 17-25.
[11] Israr A, Cartmell MP, Manoach E, Trendafilova I, Ostachowicz W, Krawczuk M, Zak A (2009) Analytical modeling and vibration analysis of partially cracked rectangular plates with different boundary conditions and loading. J Appl Mech-T ASME 76(1).
[12] Israr A, Atepor L (2009) Investigation of nonlinear dynamics of partially cracked plate. 7th International Conference on Modern Practice in Stress and Vibration Analysis, J Phys 181: 1-8.
[13] Bachene M, Tiberkak R, Rechak S, Maurice G, Hachi BK (2009) Enriched finite element for modal analysis of cracked plates. Struct Mat 463-471.
[14] Bachene M, Tiberkak R, Rechak S (2009) Vibration analysis of cracked plates using the extended finite element method. Comput Struct 249-262.
[15] Qian GL, Gu SN, Jiang JSh (1991) A finite element model of cracked plates and application to vibration problems. Comput Struct 483-487.
[16] Manoach E, Trendafilova I (2008) Large amplitude vibrations and damage detection of rectangular plates. J Sound Vib 315: 591-606.
[17] NG CF, White RG (1988) Dynamic behavior of postbuckled isotropic plates under in-plane compression. J Sound Vib 1-18.
[18] Osman T, Matbuly MS, Mohamed SA, Nassar M (2013) Analysis of cracked plates using localized multi-domain differential quadrature method. Appl Comput Math 109-114.
[19] Williams M, Griffin B, Homeijer B, Sankar B, Sheplak M (2014) Vibration of post-buckled homogeneous circular plates. IEEE Ultrasonics Symposium.
[20] Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Darabi MA (2014) Size-dependent vibrations of post-buckled functionally graded Mindlin rectangular microplates. Lat Am J Solids Stru 11: 2351-2378.
[21] Taczala M, Buczkowski R, Kleiber M (2016) Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment. Compos struct 137: 85-92.
[22] Shahverdi H, Navardi MM (2017) Free vibration analysis of cracked thin plates using generalized differential quadrature element method, Structural engineering and Mechanics. An Int J 62(3): 345-355.
[23] Bellman R, Kashef BG, Casti J (1972) Differential Quadrature: A technique for the rapid solution of  nonlinear partial differential equations. J Comput Phys 10: 40-45.
[24] Quan JR, Chang CT (1989) New Insights in Solving Distributed System of Equations by Quadrature-Method. J Compute Chem Eng 13: 1017-1024.
[25] Chen L, Zhang Z, Zhang W (2011) Inner boundary conditions of mindlin plate with a finite-length part-through crack, Second International Conference on Mechanic Automation and Control Engineering 1365-1368.
[26] Wempner GA (1971) Discrete approximation related to nonlinear theories of solids. Int J Solids Struct 7: 1581-1599.
[27] Riks E (1972) The application of Newton’s method to the problem of elastic stability. J Appl Mech 39: 1060-1065.
[28] Forde BDR, Stiemer SF (1987) Improved arc length orthogonality methods for nonlinear finite element analysis. Comput Struct 27: 625-630.
[29] Ilanko S, Dickinson SM (1987) The vibration and post-buckling of geometrically  imperfect, simply supported, rectangular plates under uni-axial loading, part I: theoretical approach. J Sound Vib 118(2): 313-336.
[30] Kapania RK, Yang TY (1987) Buckling, postbuckling, and nonlinear vibrations of imperfect plates. AIAA J  25: 1338-1346.
[31] Hosseini-Hashemi S, Heydar Roohi G, Hossein Rokni DT (2010) Exact free vibration study of rectangular Mindlin plates with all-over part-through open cracks. Comput Struct 88: 1015-1032.
Journal of Solid and Fluid Mechanics
Volume 8, Issue 4
January 2019
Pages 67-83

Files

Share

How to cite

Statistics