Analytical Elasticity Solution for Accurate Prediction of Stresses in a Rectangular Plate Bending Analysis Using Exact 3-D Theory

Document Type : Research Paper


1 Department of Civil Engineering, Edo State University Uzairue, Edo State, 312102, Nigeria.

2 Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, 410101, Nigeria.

3 Department of Civil Engineering, Michael Okpara University of Agriculture, Umudike, Abia State, 440109, Nigeria.


The bending attributes of a uniformly-loaded thick plate was modelled with three-dimensional (3-D) elasticity plate theory using exact polynomial displacement function. Plates with free support at its third edge and simply supported at other edges (SSFS), were covered in this study. The effect of shear deformation along with the transverse normal strain stress was considered in this model obviating the coefficients of shear correction. The total potential energy expression was formulated from 3-D kinematic and constitutive relations. The slope and deflection relationship was obtained from the equilibrium equation developed from the energy functional transformation. The solution of the equilibrium equation produced an exact polynomial deflection function while the coefficient of deflection of the plate was formed from the governing equation employing a direct variation approach. The formular for computing the displacement-stress components of the plate was established from these solutions in order to evaluate the bending properties of the plate. The solutions realized herein certifies that the 3D model is exact and consistent compared to refined plate theories applied by previous authors in the available literature. The total average percentage variation of the center deflection values obtained by Onyeka and Okeke, (2020) and Gwarah (2019), is 3.28%. This showed that at 97% confidence level, 3D model is most suitable and safe for analyzing the bending characteristics of thick plates unlike the 2-D RPTs.


[1]          F. Onyeka, C. Nwa-David, E. Arinze, Structural imposed load analysis of isotropic rectangular plate carrying a uniformly distributed load using refined shear plate theory, FUOYE Journal of Engineering and Technology, Vol. 6, No. 4, pp. 414-419, 2021.
[2]          F. Onyeka, B. O. Mama, C. Nwa-David, Application of variation method in three dimensional stability analysis of rectangular plate using various exact shape functions, Nigerian Journal of Technology, Vol. 41, No. 1, pp. 8–20-8–20, 2022.
[3]          F. Onyeka, Direct analysis of critical lateral load in a thick rectangular plate using refined plate theory, International Journal of Civil Engineering and Technology, Vol. 10, No. 5, pp. 492-505, 2019.
[4]          F. Onyeka, F. Okafor, H. Onah, Application of a new trigonometric theory in the buckling analysis of three-dimensional thick plate, International Journal of Emerging Technologies, Vol. 12, No. 1, pp. 228-240, 2021.
[5]          E. S. Ventsel, T. Krauthammer, 2001, Thin plates and shells: theory, analysis, and applications, Marcel Dekker,
[6]          K. Chandrashekhara, 2001, Theory of plates, Universities press,
[7]          F. Onyeka, Effect of stress and load distribution analysis on an isotropic rectangular plate, Arid Zone Journal of Engineering, Technology and Environment, Vol. 17, No. 1, pp. 9-26, 2021.
[8]          F. Onyeka, T. Okeke, C. Nwa-David, Static and buckling analysis of a three-dimensional (3-D) rectangular thick plates using exact polynomial displacement function, European Journal of Engineering and Technology Research, Vol. 7, No. 2, pp. 29-35, 2022.
[9]          F. C. Onyeka, Stability analysis of three-dimensional thick rectangular plate using direct variational energy method, Journal of Advances in Science and Engineering, Vol. 6, No. 2, pp. 1-78, 2022.
[10]        J. Reddy, Theory and Analysis of Elastic Plates and Shells, in Proceeding of.
[11]        O. Festus, E. T. Okeke, W. John, Strain–Displacement expressions and their effect on the deflection and strength of plate, Advances in Science, Technology and Engineering Systems, Vol. 5, No. 5, pp. 401-413, 2020.
[12]        P. Gujar, K. Ladhane, Bending analysis of simply supported and clamped circular plate, International Journal of Civil Engineering, Vol. 2, No. 5, pp. 69-75, 2015.
[13]        F. Onyeka, F. Okafor, Buckling solution of a three-dimensional clamped rectangular thick plate using direct variational method, building structure, Vol. 1, pp. 3, 2021.
[14]        J. Mantari, A. Oktem, C. G. Soares, A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates, International Journal of Solids and Structures, Vol. 49, No. 1, pp. 43-53, 2012.
[15]        F. Onyeka, D. Osegbowa, Stress analysis of thick rectangular plate using higher order polynomial shear deformation theory, FUTO Journal Series–FUTOJNLS, Vol. 6, No. 2, pp. 142-161, 2020.
[16]        O. Ibearugbulem, F. C. Onyeka, Moment and stress analysis solutions of clamped rectangular thick plate, European Journal of Engineering and Technology Research, Vol. 5, No. 4, pp. 531-534, 2020.
[17]        F. Onyeka, D. Osegbowa, E. Arinze, Application of a new refined shear deformation theory for the analysis of thick rectangular plates, Nigerian Research Journal of Engineering and Environmental Sciences, Vol. 5, No. 2, pp. 901-917, 2020.
[18]        R. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, 1951.
[19]        E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, 1945.
[20]        I. Sayyad, S. Chikalthankar, V. Nandedkar, Bending and free vibration analysis of isotropic plate using refined plate theory, Bonfring International Journal of Industrial Engineering and Management Science, Vol. 3, No. 2, pp. 40-46, 2013.
[21]        F. C. Onyeka, T. E. Okeke, Analysis of critical imposed load of plate using variational calculus, Journal of Advances in Science and Engineering, Vol. 4, No. 1, pp. 13-23, 2021.
[22]        C. Nwoji, H. Onah, B. Mama, C. Ike, M. Abd El Hady, A. Youssef, A. Bayoumy, Y. Elhalwagy, X. Wang, G. Ren, Ritz variational method for bending of rectangular Kirchhoff plate under transverse hydrostatic load distribution, Mathematical Modelling of Engineering Problems, Vol. 5, No. 1, pp. 1-10, 2018.
[23]        R. Szilard, Theories and applications of plate analysis: classical, numerical and engineering methods, Appl. Mech. Rev., Vol. 57, No. 6, pp. B32-B33, 2004.
[24]        S. Timoshenko, S. Woinowsky-Krieger, 1959, Theory of plates and shells, McGraw-hill New York,
[25]        C. Nwoji, B. Mama, C. Ike, H. Onah, Galerkin-Vlasov method for the flexural analysis of rectangular Kirchhoff plates with clamped and simply supported edges, IOSR Journal of Mechanical and Civil Engineering, Vol. 14, No. 2, pp. 61-74, 2017.
[26]        C. Ike, Equilibrium approach in the derivation of differential equations for homogeneous isotropic Mindlin plates, Nigerian Journal of Technology, Vol. 36, No. 2, pp. 346-350, 2017.
[27]        F. Onyeka, B. Mama, T. Okeke, Exact three-dimensional stability analysis of plate using a direct variational energy method, Civil Engineering Journal, Vol. 8, No. 1, pp. 60-80, 2022.
[28]        N. G. R. Iyengar, 1988, Structural stability of columns and plates / N.G.R. Iyengar, Ellis Horwood ; Halsted Press, Chichester [England] : New York
[29]        J. Mantari, C. G. Soares, Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory, Composite Structures, Vol. 94, No. 6, pp. 1991-2000, 2012.
[30]        D. BHASKAR, A. G. Thakur, I. I. Sayyad, S. V. Bhaskar, Numerical Analysis of Thick Isotropic and Transversely Isotropic Plates in Bending using FE Based New Inverse Shear Deformation Theory, International Journal of Automotive and Mechanical Engineering, Vol. 18, No. 3, pp. 8882-8894, 2021.
[31]        F. Y. Tash, B. N. Neya, An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness, Applied Mathematical Modelling, Vol. 77, pp. 1582-1602, 2020.
[32]        Y. M. Ghugal, P. D. Gajbhiye, Bending analysis of thick isotropic plates by using 5th order shear deformation theory, Journal of Applied and Computational Mechanics, Vol. 2, No. 2, pp. 80-95, 2016.
[33]        F. Onyeka, Critical lateral load analysis of rectangular plate considering shear deformation effect, Global Journal of Civil Engineering, Vol. 1, pp. 16-27, 2020.
[34]        F. Onyeka, B. Mama, C. Nwa-David, Analytical modelling of a three-dimensional (3D) rectangular plate using the exact solution approach, IOSR Journal of Mechanical and Civil Engineering, Vol. 11, No. 1, pp. 10-22, 2022.
[35]        F. Onyeka, E. T. Okeke, Analytical solution of thick rectangular plate with clamped and free support boundary condition using polynomial shear deformation theory, Advances in Science, Technology and Engineering Systems Journal, Vol. 6, No. 1, pp. 1427-1439, 2021.
[36]        F. Onyeka, T. Okeke, New refined shear deformation theory effect on non-linear analysis of a thick plate using energy method, Arid Zone Journal of Engineering, Technology and Environment, Vol. 17, No. 2, pp. 121-140, 2021.
[37]        A. Y. Grigorenko, A. Bergulev, S. Yaremchenko, Numerical solution of bending problems for rectangular plates, International Applied Mechanics, Vol. 49, pp. 81-94, 2013.
[38]        F. Onyeka, T. Okeke, Elastic bending analysis exact solution of plate using alternative i refined plate theory, Nigerian Journal of Technology, Vol. 40, No. 6, pp. 1018–1029-1018–1029, 2021.
[39]        F. Onyeka, B. Mama, Analytical study of bending characteristics of an elastic rectangular plate using direct variational energy approach with trigonometric function, Emerging Science Journal, Vol. 5, No. 6, pp. 916-928, 2021.
[40]        A. Hadi, A. Rastgoo, A. Daneshmehr, F. Ehsani, Stress and strain analysis of functionally graded rectangular plate with exponentially varying properties, Indian Journal of Materials Science, Vol. 2013, 2013.
[41]        O. Rahmani, S. Norouzi, H. Golmohammadi, S. Hosseini, Dynamic response of a double, single-walled carbon nanotube under a moving nanoparticle based on modified nonlocal elasticity theory considering surface effects, Mechanics of Advanced Materials and Structures, Vol. 24, No. 15, pp. 1274-1291, 2017.
[42]        F. Ebrahimi, P. Haghi, Elastic wave dispersion modelling within rotating functionally graded nanobeams in thermal environment, Advances in nano research, Vol. 6, No. 3, pp. 201, 2018.
[43]        M. Z. Nejad, A. Hadi, A. Rastgoo, Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, pp. 1-10, 2016/06/01/, 2016.
[44]        Y. Z. YÜKSEL, Ş. D. AKBAŞ, Free vibration analysis of a cross-ply laminated plate in thermal environment, International Journal of Engineering and Applied Sciences, Vol. 10, No. 3, pp. 176-189, 2018.
[45]        Y. Z. Yüksel, D. Akbaş, Hygrothermal stress analysis of laminated composite porous plates, Structural Engineering and Mechanics, Vol. 80, No. 1, pp. 1-13, 2021.
[46]        Y. Z. Yüksel, Ş. D. Akbaş, Buckling analysis of a fiber reinforced laminated composite plate with porosity, Journal of Computational Applied Mechanics, Vol. 50, No. 2, pp. 375-380, 2019.
[47]        Ş. D. AKBAŞ, Stability of a non-homogenous porous plate by using generalized differantial quadrature method, International Journal of Engineering and Applied Sciences, Vol. 9, No. 2, pp. 147-155, 2017.
[48]        Ş. AKBAŞ, Static analysis of a nano plate by using generalized differential quadrature method, International Journal of Engineering and Applied Sciences, Vol. 8, No. 2, pp. 30-39, 2016.
[49]      L. S. Gwarah, Application of shear deformation theory in the analysis of thick rectangular plates using polynomial displacement functions, PhD Thesis Presented to the School of Civil Engineering, Federal University of Technology, Owerri, Nigeria, 2019.
Volume 54, Issue 2
June 2023
Pages 167-185
  • Receive Date: 30 November 2022
  • Revise Date: 29 December 2022
  • Accept Date: 29 December 2022