• Edward Adah Department of Civil and Environmental Engineering, University of Calabar, Nigeria.
  • David Onwuka Department of Civil Engineering, Federal University of Technology Owerri, Nigeria
  • Owus Ibearugbulem Department of Civil Engineering, Federal University of Technology Owerri, Nigeria
  • Chinenye Okere Department of Civil Engineering, Federal University of Technology Owerri, Nigeria
Keywords: Membrane strain, total potential energy, linear, nonlinear free vibration, rectangular plates


The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.


Leissa , A. W. & Quta, M. S. (2011). Vibration of Continuous Systems. McGraw-Hill Company, USA.

Dash, A. K. (2010). Large Amplitude Free Vibration Analysis of Composite Plates by Finite Element Method. M.Sc Thesis, National Institute of Technology, Rourkela.

Ducceschi, M. (2014). Nonlinear Vibrations of Thin Rectangular Plates: A Numerical Investigations with Application to Wave Turbulence and Sound Synthesis. Vibrations (Physics.class-ph).ENSTA Panotech

Ibearugbulem, O. M, Ezeh, J. C. & Ettu, L. O. (2014). Energy Methods in Theory of Rectangular Plates: Use of Polynomial Shape Functions. Liu House of Excellence Ventures, Owerri.

Adah, E. I., Ibearugbulem, O. M., Onwuka, D. O. & Okoroafor, S. U. (2019). Determination of Resonating Frequency of Thin Rectangular Flat Plates. International Journal of Civil and Structural Engineering Research, 7 (1), 16-22,

Hashemi, S. & Jaberzadeh, E. (2012). A Finite Strip Formulation for Nonlinear Free Vibration of Plates, 15 WCEE, Lisboa.

Kumar, R. & Goytom, D. (2017). Postbuckling and Nonlinear Free Vibration Response of Elastically Supported Laminated Composite Plates with Uncertain System Properties in Thermal Environment. Frontiers in Aerospace Engineering, 6 (): 1-27.

Varzandian, G. A. & Ziaei, S. (2017). Analytical Solution of Nonlinear Free Vibration of Thin Rectangular Plates with Various Boundary Conditions based on Non-local Theory. Amir Kabir Journal of science and research mechanical engineering, 48 (4): 121-124.

Onodagu, P. D. (2018). Nonlinear Dynamic Analysis of Thin Rectangular Plates using Ritz Method. PhD Thesis, Federal University of Technology, Owerri, Nigeria.

Yosibash, Z. & Kirby, R. M. (2005). Dynamic Response of various von-Karman nonlinear plate models and their 3-D counterparts. International Journal of Solids & Structures, 42, 2517-2531.

Mattieu, G., Tyekolo, D. & Belay, S. (2017). The nonlinear bending of simply supported elastic plate. RUDN Journal of Engineering researches. 18 (1), 58-69.

Kucukrendeci, I. (2017). Nonlinear vibration analysis of composite plates on elastic foundations in thermal environments. AKU. J. Sci. Eng. 17, 790-796.

Zergoune, Z., Harras, B. & Benanar, R. (2015). Nonlinear Free Vibration vibration of C-C-SS-SS symmetrically laminated carbon fibre reinforced plastic (CFRP) rectangular composite plates. World Journal of mechanics. 5, 22-32

El Kaak, R. & Bikri, K (2016). Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of thin circular functionally graded plates using iterative and explicit analysis solution. International Journal of Acoustics and Vibration, 21(2), 209-221.

Levy, S. (1942). Bending of Rectangular Plates with Large Deflections. Technical notes: National Advisory Committee for Aeronautics (NACA), N0. 846.

Enem, J. I. (2018). Geometrically Nonlinear Analysis of Isotropic Rectangular Thin Plates Using Ritz Method. PhD Thesis, Federal University of Technology, Owerri, Nigeria.

Elsami, M. R. (2018). Buckling and Postbuckling of Beams, Plates, and Shells. Springer International Publishing AG.

Manuel, S. (1984). Analytical Results for Postbuckling Behavior of Plates in Compression and in Shear. National Aeronautics and Space Administration (NASA). NASA Technical Memorandum 85766.

Bloom, F. & Coffin, D. (2001). Thin Plate Buckling and Postbuckling. London: Chapman & Hall/CRC.

Byklum, E. & Amdahl, J. (2002). A Simplified Method for Elastic Large Deflection Analysis of Plates and Stiffened Panels due to Local Buckling. Thin-Walled Structures, 40 (): 925-953.

Tanriöver H. & Senocak, E. (2004). Large Deflection Analysis of Unsymmetrically Laminated Composite plates: Analytical-numerical type Approach. International Journal of Non-linear Mechanics, 39: 1385-1392.

GhannadPour, S. A. M. & Alinia, M. M. (2006). Large Deflection Behavior of Functionally Graded Plates under Pressure Loads. Composite Structures, 75: 67-71.

Shufrin, I., Rabinovitch, O. & Eisenberger, M. (2008). A Semi-analytical Approach for the Nonlinear Large Deflection Analysis of Laminated Rectangular Plates under General out-of-plane loading. International Journal of Non-Linear Mechanics, 43:328-340.

Ducceschi, M., Touze, C., Bilbao, S. & Webb, C. J. (2013). Nonlinear dynamics of rectangular plates: investigation of model interaction in free and forced vibrations. Acta Mech

Stoykov, S. & Margenov, S. (2016). Finite Element Method for Nonlinear Vibration Analysis of Plates. Springer International Publishing Switzerland, 17-27.

Ibearugbulem, O. M., Adah, E. I., Onwuka, D. O. & Okere, C. E. (2020). Simple and Exact Approach to Postbuckling Analysis of Rectangular Plate, SSRG International Journal of Civil Engineering, 7 (6): 54-64,

Deutsch, A., Tenenbaum, J., & Eisenberger, M. (2019). Benchmark Vibration Frequencies of Square Thin Plates with all Possible Combinations of Classical Boundary Conditions. International Journal of Structural Stability and Dynamics, 19(11), 1950131-1-1950131-16

How to Cite
Adah, E., Onwuka, D., Ibearugbulem, O., & Okere, C. (2021). LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE. Journal of Civil Engineering, Science and Technology, 12(1), 15-25.