# Stability analysis of three-dimensional thick rectangular plate using direct variational energy method

• Festus Chukwudi Onyeka Edo State University Uzairue, Edo State, Nigeria
Keywords: 3-D plate theory, Deflection function, Energy variation method, Exact trigonometry, Polynomial shape function, Rectangular thick plate, Stability analysis of thick plates

### Abstract

This study investigated the elastic static stability analysis of homogeneous and isotropic thick rectangular plates with twelve boundary conditions and carrying uniformly distributed uniaxial compressive load using the direct variational method. In the analysis, a thick plate energy expression was developed from the three-dimensional (3-D) constitutive relations and kinematic deformation; thereafter the compatibility equations used to resolve the rotations and deflection relationship were obtained. Likewise, the governing equations were derived by minimizing the equation for the potential energy with respect to deflection. The governing equation is solved to obtain an exact deflection function which is produced by the trigonometric and polynomial displacement shape function. The degree of rotation was obtained from the equation of compatibility which when equated to the deflection function and put into the potential energy equation formulas for the analysis were obtained after differentiating the outcome with respect to the deflection coefficients. The result obtained shows that the non-dimensional values of critical buckling load decrease as the length-width ratio increases (square plate being the highest value), this continues until failure occurs. This implies that an increase in plate width increases the probability of failure in a plate. Hence, it can be deduced that as the in-plane load on the plate increase and approaches the critical buckling, the failure in a plate structure is abound to occur. Meanwhile, the values of critical buckling load increase as the span-thickness ratio increases for all aspect ratios. This means that, as the span-thickness ratio increases an increase in the thickness increases the safety in the plate. It also indicates that the capacity of the plate to resist buckling decreases as the span-depth ratio increases. To establish the credibility of the present study, classical plate theory (CPT), refined plate theory (RPT) and exact solution models from different studies were employed to validate the results. The present works critical buckling load varied with those of CPT and RPT with 7.70% signifying the coarseness of the classical and refined plate theories. This amount of difference cannot be overlooked. The average total percentage differences between the exact 3-D study (Moslemi et al., 2016), and the present model using polynomial and trigonometric displacement functions is less than 1.0%. These differences being so small and negligible indicates that the present model using trigonometric and polynomial produces an exact solution. Thus, confirming the efficacy and reliability of the model for the 3-D stability analysis of rectangular plates.

### References

F. C. 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.

F. C. Onyeka and 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.

F. C. Onyeka and O. M. Ibearugbulem, “Load analysis and bending solutions of rectangular thick plate,” International Journal of Emerging Technologies, vol. 11, no. 3, pp. 1030-1110, 2020.

C. H. Aginam, C. A. Chidolue and C. A. Ezeagu, “Application of direct variational method in the analysis of isotropic thin rectangular plates,” ARPN Journal of Engineering and Applied Sciences, vol. 7, no. 9, pp. 1128-1138, 2012.

F. C. Onyeka, “Critical lateral load analysis of rectangular plate considering shear deformation effect,” Global Journal of Civil Engineering, vol. 1, pp. 16-27, 2020.

I. C. Erdem and I. Gerdemeli, “The problem of isotropic rectangular plate with four clamped edge,” Journal of Sadhana, vol. 32, no. 3, pp. 181-186, 2006.

F. C. 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.

K. Chandrashekhara, “Theory of plates,” University Press (India) Limited, 2001.

F. C. Onyeka and 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.

E. Ventsel, T. Krauthammer, “Thin plates and shells theory, analysis and applications,” Maxwell Publishers Inc; New York, 2001.

J. N. Reddy, “Classical theory of plates,” In Theory and Analysis of Elastic Plates and Shells, CRC Press, 2006, DOI: 10.1201/9780849384165-7

F. C. Onyeka and D. Osegbowa, “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.

O. M. Ibearugbulem, O. A. Oguaghamba, K. O. Njoku, and M. Nwaokeorie, “Using line continuum to explain work principle method for structural continuum analysis,” International Journal of Innovative Research and Development, vol. 3, no. 9, pp. 365-370, 2014.

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, 2020.

O. M. Ibearugbulem, J. C. Ezeh, and L. O. Ettu, “Energy methods in theory of rectangular plates using polynomial shape functions,” LIU House of Excellence Ventures, 2014.

S. P. Timoshenko and J. M. Gere, “Theory of elastic stability,” 2nd Edition, McGraw-Hill Books Company, New York, 1963, DOI: 10.1115/1.3636481

F. C. Onyeka, B. O. Mama, T. E. 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.

M. E. Shufrin, “Stability and vibration of shear deformable plates - first order and higher order analyses,” International Journal of Solids and Structures, vol. 42, no. 3-4, pp. 1225-1251, 2005.

F. C. Onyeka, F. O. Okafor, H. N. Onah, “Buckling solution of a three-dimensional clamped rectangular thick plate using direct variational method,” IOSR Journal of Mechanical and Civil Engineering, vol. 18, no. 3 Ser. III, pp. 10-22, 2021.

F. C. Onyeka, F. O. Okafor, H. N. 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.

A. S. Sayyad and Y. M. Ghugal, “Buckling analysis of thick isotropic plates using exponential shear deformation theory,” Applied and Computational Mechanics, vol. 6, pp. 185-196, 2012.

A. S. Sayyad and Y. M. Ghugal, “Buckling and free vibration analysis of orthotropic plates by using exponential shear deformation theory,” Latin American Journal of Solids and Structures, vol. 11, no. 8, 2014.

F. C. Onyeka, B. O. Mama and J. Wasiu, “An analytical 3-D modelling technique of non-linear buckling behaviour of an axially compressed rectangular plate,” International Research Journal of Innovations in Engineering and Technology, vol. 6, no. 1, pp. 91-101, 2022.

A. S. Sayyad, B. N. Shinde, B. M and Y. M. Ghugal, “Bending, vibration and buckling of laminated composite plates using a simple four variable plate theory,” Latin American Journal of Solids and Structures, vol. 13, no. 3, 2016.

F. C. Onyeka, T. E. Okeke, J. Wasiu, “Strain-displacement expressions and their effect on the deflection and strength of plate,” Advances in Science, Technology and Engineering Systems Journal, vol. 5, no. 5, pp. 401 – 413, 2020.

G. R. Kirchhoff, “U’’ber das gleichgewicht and die bewe gung einer elastschen scheibe,” Journal f’’ ur Die Reine und Angewandte Mathematik, vol. 40, pp. 51-88 (in German), 1850, DOI: 10.1515/crll.1850.40.51

G. R. Kirchhoff, “U’’ber die schwingungen einer kriesformigen elastischen scheibe,” Annalen der Physik und Chemie, vol. 81, pp. 258-264 (in German), 1850.

A. M. Zenkour, “Exact mixed-classical solutions for the bending analysis of shear deformable rectangular plates,” Applied Mathematical Modelling, vol. 27, no. 7, pp. 515-534, 2003.

O. M. Ibearugbulem, L. S. Gwarah, and C. N. Ibearugbulem, “Use of polynomial shape function in shear deformation theory for thick plate analysis,” Journal of Engineering (IOSRJEN), vol. no. 6, pp. 8-20, 2016.

R. D. Mindlin, “Influence of rotary inertia and shear on flexural motion of isotropic elastic plates,” ASME Journal of Applied Mechanics, vol. 18, pp. 31 – 38, 1951.

A. Pica and R. D. Wood, “Post buckling behavior of plates and shells using Mindlin shallow shell formulation,” Computers and Structures Journals, vol. 12, no. 5, pp. 759–768, 1980.

F. C. Onyeka, and T. E. 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.

K. H. Lo, R. M. Christensen, and E. M. Wu, “A high-order theory of plate deformation, part-1: homogeneous plates,” ASME Journal of Applied Mechanics, vol. 44, pp. 663–668, 1977.

F. C. Onyeka., C. D. Nwa-David, and E. 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.

Y. M. Ghugal, and M. D. Pawar, “Buckling and vibration of plates by hyperbolic shear deformation theory,” Journal of Aerospace Engineering and Technology, vol. 1, no. 1, pp. 1–12, 2011.

Y. M. Ghugal and A. S Sayyad. “Free vibration of thick orthotropic plates using trigonometric shear deformation theory,” Latin American Journal of Solids and Structures, vol. 8, no. 3 (September, 2011): 229–243.

B. S. Reddy, “Bending behaviour of exponentially graded material plates using new higher order shear deformation theory with stretching effect,” International Journal of Engineering Research, vol. 3, Special 1, pp. 124-131, 2014.

A. Mahi, E. A. Adda Bedia, and A. Tounsi, “A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates,” Applied Mathematical Modelling, Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 9, pp. 2489–2508, 2015.

S. M. Gunjal, R. B. Hajare, A. S. Sayyad, and M. D. Ghodle, “Buckling analysis of thick plates using refined trigonometric shear deformation theory,” Journal of Materials and Engineering Structures, vol. 2, pp. 159–167, 2015.

O. M. Ibearugbulem, “Note on rectangular plate analysis,” Lambert Academic Publishing, 2016.

F. C. Onyeka and B. O. 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. 916–926, 2021.

N. J. Pagano, “Exact solutions for bidirectional composites and sandwich plates,” Journal of Composite Materials, vol. 4, pp. 20–34, 1970.

F. C. Onyeka and T. E. 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, 2021.

O. M. Ibearugbulem, S. I. Ebirim, U. C. Anya, and L. O. Ettu, “Application of alternative II theory to vibration and stability analysis of thick rectangular plates (isotropic and orthotropic),” Nigerian Journal of Technology, vol. 39, no. 1, pp. 52 – 62, 2020.

O. M. Ibearugbulem, H. E. Opara, C. N. Ibearugbulem, and U. C. Nwachukwu, “Closed form buckling analysis of thin rectangular plates,” IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE), vol. 16, no. 1, pp. 83-90, 2019.

O. M. Ibearugbulem, I. C. Onyechere, J. C. Ezeh, and U. C. Anya, “Determination of exact displacement functions for rectangular thick plate analysis,” FUTO Journal Series -FUTOJNLS, vol. 5, no. 1, pp. 101–116, 2019.

J. C. Ezeh, I. C. Onyechere, O. M. Ibearugbulem, U. C. Anya and L. Anyaogu, “Buckling analysis of thick rectangular flat SSSS plates using polynomial displacement functions,” International Journal of Scientific & Engineering Research, vol. 9, no. 9, pp. 387- 392, 2018.

T. T. Behrang and Z. K. Mohammad, “Buckling of rectangular plates partially restrained along opposite edges,” 3rd National and First International Conference in applied research on Electrical, Mechanical and Mechatronics Engineering, 2016.

I. I., Sayyad, S. B. Chikalthankar and V. M. Nandedkar, “Trigonometric shear deformation theory for thick plate analysis,” International Conference on Recent Trends in Engineering & Technology - Organized By: SNJB's Late Sau. K. B. Jain College of Engineering, Chandwad, 2013.

H. O. Ozioko, O. M. Ibearugbulem, J. C. Ezeh, and U. C. Anya, “Algorithm for exact solution of thick anisotropic plates,” Scholar Journal of Applied Sciences and Research, vol. 2, no. 4, pp. 11-25, 2019.

J. C. Ezeh, O. M. Ibearugbulem and C. I. Onyechere, “Pure bending of thin rectangular flat plates using ordinary finite difference method,” International Journal of Emerging Technology and Advanced Engineering, vol. 3, no. 3, pp. 20-23, 2013.

J. C. Ezeh, O. M. Ibearugbulem, H. E. Opara and O. A. Oguaghamba, “Galerkin’s indirect variational method in elastic stability analysis of all edges clamped thin rectangular flat plates,” International Journal of Research in Engineering and Technology, vol. 3, no. 4, 2014.

J. C. Ezeh, O. M. Ibearugbulem, A. N. Nwadike and U. J. Maduh, “The use of polynomial shape function in the buckling analysis of CCFC rectangular plate,” International Journal of Research in Engineering and Technology, vol. 2, no. 12, 2013.

A. S. Sayyad, “Comparison of various shear deformation theories for the free vibration of thick isotropic beams,” International Journal of Civil and Structural Engineering, vol. 2, no. 1, pp. 85-97, 2011.

F. C. Onyeka, F. O. Okafor, H. N. Onah, “Application of exact solution approach in the analysis of thick rectangular plate,” International Journal of Applied Engineering Research, vol. 14, no. 8, pp. 2043-2057, 2019.

A. S. Sayyad and Y. M. Ghugal, “Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory,” Applied and Computational Mechanics, vol. 6, pp. 65-82, 2012.

S. A. Sadrnejad, A.S. Daryan and M. Ziaei, “Vibration equations of thick rectangular plates using Mindlin plate theory,” Journal of Computer Science, vol. 5, no. 11, pp. 838-842, 2009.

S. K. Mandal, and P. K. Mishra, “Buckling analysis of rectangular plate element subjected to in-plane loading using finite element method,” International Journal of Scientific & Engineering Research, vol. 9, no. 4, 2018.

J. N. Reddy and N. D. Phan, “Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory,” Journal of Sound and Vibration, vol. 98, no. 2, pp. 157–170, 1985.

E. Reissner, “A note on bending of plates including the effects of transverse shearing and normal strains,” ZAMP: Zeitschrift fur Angewandte Mathematik und Physik, vol. 32, 764–767, 1981.

S. E. Kim, and H. T. Thai, “Buckling analysis of plates using the two variable refined plate theory,” Thin-Walled Structures, vol. 47, pp. 455-462, 2009.

H. T. Thai and S. E. Kim, “Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theory,” Composite Structures, vol. 93, pp. 1738-1746, 2011.

Y. O. Mahmoud, and E. S. Osama Mohammed, “Buckling analysis of thin laminated composite plates using finite element method,” International Journal of Engineering Research and Advanced Technology, vol. 3, no. 3, pp. 1-18, 2017.

F. C. Onyeka, B. O. Mama, C. D. 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, 2022.

F. C. Onyeka, B. O. Mama, C. D. 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.

A. V. Moslemi, N. N. Bahram, and V. A. Javad, “3-D elasticity buckling solution for simply supported thick rectangular plates using displacement potential functions,” Applied Mathematical Modelling, vol. 40, pp. 5717–5730, 2016.

F. C. Onyeka, B. O. Mama, C. D. 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.

F. C. Onyeka, T. E. Okeke and C. D. Nwa-David. "Buckling analysis of a three-dimensional rectangular plates material based on exact trigonometric plate theory,” Journal of Engineering Research and Sciences, vol. 1, no. 3, 106-115, 2022.

S. P. Timoshenko and S. W. Krieger, “Theory of plates and shells,” 2nd Ed. McGraw – Hill Inc, Auckland, 1959.

S. P. Timoshenko and S. W. Krieger, “Theory of plates and shells,” (2nd Ed.). McGraw – Hill Inc, Auckland, 1970.

N. G. Iyengar, “Structural stability of columns and plates,” Ellis Horwood Limited, New York, 1988.

A. C. Ugural, “Stresses in plates and shells (2nd Ed.),” McGraw – Hill Inc, Singapore, 1999.

R. Szilard, “Theories and application of plate analysis: classical, numerical and engineering methods,” John Wiley and Sons Inc, 2004.

E. H. Mansfield, “The bending and stretching of plates,” Macmillan, New York, 1964.

A. S. Zaki, “Reliability analysis of the free vibration of composite plates,” M.Sc. Thesis, Department of Aerospace Engineering, Cairo, 2006.

R. T. Fenner, “Engineering Elasticity,” An Application of Numerical and Analytical Techniques. Ellis Horwood Ltd, Chichester, 1986.

F. O. Okafor, and O. T. Udeh, “Direct method of analysis of an isotropic rectangular plate using characteristic orthogonal polynomials,” Nigerian Journal of Technology, vol. 34, no. 2, pp. 232 – 239, 2015.

H. R. Kabir, and R. A. Chaudhuri, “Boundary-continuous Fourier solution for clamping Mindlin plates,” Journal of Engineering Mechanics, vol. 118, no. 7, pp. 1457–1467, 1992.

R. Li, B. Wang, and P. Li, “Hamiltonian system-based benchmark bending solutions of rectangular thin plates with a corner point-supported,” International Journal of Mechanical Sciences, vol. 85, pp. 212–218, 2014.

Y. Zhong and R. Li, “Exact bending analysis of fully clamped rectangular thin plates subjected to arbitrary loads by new simplistic approach,” Mechanics Research Communications, vol. 36, no. 6, pp. 707–714, 2009.

Y. Zhong and J. H. Yin, “Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method,” Mechanics Research Communications, vol. 35, no. 4, pp. 268–275, 2008.

R. P. Shimpi and H. G. Patel, “A two variable refined plate theory for orthotropic plate analysis,” International Journal of Solids and Structures, vol. 43, pp. 6783–6799, 2006.

F. C. Onyeka, C. D. Nwa-David, T. E. Okeke, "Study on stability analysis of rectangular plates section using a three-dimensional plate theory with polynomial function,” Journal of Engineering Research and Sciences, vol. 1, no. 3, pp. 28-37, 2022.

F. C. Onyeka, T. E. Okeke, C. D. Nwa-David, "Stability Analysis of a Three-Dimensional Rectangular Isotropic Plates with Arbitrary Clamped and Simply Supported Boundary Conditions,” IOSR Journal of Mechanical and Civil Engineering, vol. 19, no. 1, Ser. IV, pp. 01-09, 2021.

Published
2022-04-30
How to Cite
Onyeka, F. C. (2022). Stability analysis of three-dimensional thick rectangular plate using direct variational energy method. Journal of Advances in Science and Engineering, 6(2), 1-78. https://doi.org/10.37121/jase.v6i2.187
Section
Thesis