A Proposed Approach to Simulate Thin Quadrilateral Plates Using Generalized Differential Quadrature Method Based on Kirchhoff–Love Theory

Document Type : Research Article


Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran


In this study, an approach to free vibration analysis of thin quadrilateral plates using the generalized differential quadrature method based on the strong version of the governing equation is proposed. Hence, the governing equation of a thin quadrilateral plate is firstly obtained using the Kirchhoff–Love theory of plates (classical theory) to achieve this aim. The well-known differential quadrature method is then utilized to obtain the discretized form of the equations of motion. However, simulation of any arbitrary geometry using conventional Generalized Differential Quadrature Method based on classical theory is impossible. This drawback can be removed by defining the additional degrees of freedom in boundaries. Moreover, the combination of the Refined Differential Quadrature Method with geometry mapping is developed to simulate thin quadrilateral plates. Also, the multi-block or elemental strategy is implemented for problems with more geometric complexities. For this aim, geometry can be divided into several subdomains. Continuity conditions make the connection between adjacent elements at each interface. By establishing the whole discretized governing equations, the free vibration analysis of a thin plate will be provided via the achieved eigenvalue problem. The obtained results are compared and validated with available results in the literature that show high accuracy and a fast rate of convergence.


Main Subjects

[1] R. Bellman, J. Casti, Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications, 34(2) (1971) 235-238.
[2] C. Shu, Differential quadrature and its application in engineering, Springer Science & Business Media, 2012.
[3] F. Tornabene, N. Fantuzzi, F. Ubertini, E. Viola, Strong formulation finite element method based on differential quadrature: a survey, Applied Mechanics Reviews, 67(2) (2015).
[4] C. Shu, B.E. Richards, Application of generalized differential quadrature to solve two‐dimensional incompressible Navier‐Stokes equations, International Journal for Numerical Methods in Fluids, 15(7) (1992) 791-798.
[5] M. Javadi, V. Khalafi, Flutter reliability analysis of laminated composite plates, Amirkabir Journal of Mechanical Engineering, 53(6 (Special Issue)) (2021) 10-10.
[6] V. Khalafi, H. Shahverdi, S. Noori, Nonlinear Aerothermoelastic Analysis of Functionally Graded Rectangular Plates Subjected to Hypersonic Airflow Loadings, AUT Journal of Mechanical Engineering, 2(2) (2018) 217-232.
[7] X. Wang, Z. Yuan, Buckling analysis of isotropic skew plates under general in-plane loads by the modified differential quadrature method, Applied Mathematical Modelling, 56 (2018) 83-95.
[8] A.G. Striz, C. Weilong, C.W. Bert, Static analysis of structures by the quadrature element method (QEM), International Journal of Solids and Structures, 31(20) (1994) 2807-2818.
[9] L. Ke, Y. Wang, J. Yang, S. Kitipornchai, F. Alam, Nonlinear vibration of edged cracked FGM beams using differential quadrature method, Science China Physics, Mechanics and Astronomy, 55(11) (2012) 2114-2121.
[10] K. Torabi, H. Afshari, F.H. Aboutalebi, A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions, Computers & Mathematics with Applications, 67(3) (2014) 527-541.
[11] F.-L. Liu, K. Liew, Vibration analysis of discontinuous Mindlin plates by differential quadrature element method,  (1999).
[12] F.-L. Liu, K. Liew, Differential quadrature element method: a new approach for free vibration analysis of polar Mindlin plates having discontinuities, Computer Methods in Applied Mechanics and Engineering, 179(3-4) (1999) 407-423.
[13] N. Fantuzzi, F. Tornabene, E. Viola, Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes, International Journal of Mechanical Sciences, 79 (2014) 216-251.
[14] S. Moradi, H. Makvandi, D. Poorveis, K.H. Shirazi, Free vibration analysis of cracked postbuckled plate, Applied Mathematical Modelling, 66 (2019) 611-627.
[15] M. Ishaquddin, S. Gopalakrishnan, A novel weak form quadrature element for gradient elastic beam theories, Applied Mathematical Modelling, 77 (2020) 1-16.
[16] M.M. Navardi, Supersonic flutter analysis of thin cracked plate by Differential Quadrature Method, Master of Scince thesis, Amirkabir University of Technology University, Tehran, Iran (2015).
[17] Y. Wang, X. Wang, Y. Zhou, Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method, International Journal for Numerical Methods in Engineering, 59(9) (2004) 1207-1226.
[18] G. Karami, P. Malekzadeh, Application of a new differential quadrature methodology for free vibration analysis of plates, International Journal for Numerical Methods in Engineering, 56(6) (2003) 847-868.
[19] H. Shahverdi, M.M. Navardi, Free vibration analysis of cracked thin plates using generalized differential quadrature element method, Structural engineering and mechanics: An international journal, 62(3) (2017) 345-355.
[20] J.N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, 2003.
[21] C.W. Bert, M. Malik, The differential quadrature method for irregular domains and application to plate vibration, International Journal of Mechanical Sciences, 38(6) (1996) 589-606.
[22] N. Bardell, The free vibration of skew plates using the hierarchical finite element method, Computers & structures, 45(5-6) (1992) 841-874.
[23] M. Zamani, A. Fallah, M. Aghdam, Free vibration analysis of moderately thick trapezoidal symmetrically laminated plates with various combinations of boundary conditions, European Journal of Mechanics-A/Solids, 36 (2012) 204-212.
[24] A.W. Leissa, The free vibration of rectangular plates, Journal of sound and vibration, 31(3) (1973) 257-293.
[25] R. Solecki, Vibration of a simply supported L-shaped plate,  (1997).
[26] R. Solecki, Free-vibration of an L-shaped plate: the general solution and an example of a simply-supported plate with a clamped cutout,  (1996).
[27] F.-L. Liu, K. Liew, Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method, Journal of Sound and Vibration, 225(5) (1999) 915-934.