Characteristic Trigonometric Function Application to the Direct Variation Method for Clamped, Simply and Freely Supported Plate Under Uniform Load Distributions

Okeke Thompson Edozie , Onyeka Festus Chukwudi,Nwa-David Chidobere

Abstract

In this paper, the bending attributes of a uniformly loaded three-dimensional (3-D) plate are modelled using exact trigonometric shape functions; the effects of shear correction factors associated with refined plate theory (RPT) are obviated. As an advancement of the RPT theory, the equations of equilibrium of the current model were obtained from fundamental principles of elasticity by applying the three-dimensional (3-D) kinematic deformation and constitutive relations which results of the six stress components. The formulated 3-D kinematics and constitutive relations was used to obtain the energy equation which was later transformed into compatibility equation to determine the function of deflection and rotation The governing equations were obtained and solved in terms of trigonometry to get the exact deflection function of the plate. The rotation and deflection function were substituted into the energy equation to get the coefficients of deflection and rotation. Thereafter, these coefficients were substituted into the obtained displacement and stress equations to get the model for evaluating the bending of thick plates that was clamped on the first edge, free at the third edge, with the second and fourth edges simply supported respectively (CSFS). The recorded outcome confirms that the values for stresses and displacement obtained from this 3-D theory is more accurate and reliable compared to refined plate theories applied in previous studies. The overall average percentage differences between the present study and the studies by Onyeka (2021) and Gwarah (2019) for center deflection of a square plate was 4.8%. This showed about 95% confidence level for adoption of 3-D plate analogy which is required as the only reliable modeling theory for an exact bending solution of thick plates as unrealistic solutions are obtained from 2D shear deformation theories.

Keywords: 3-D plate theory, Exact bending solution, CSFS thick Plate, Trigonometric Displacement Functions

How to cite this article: Okeke Thompson Edozie , Onyeka Festus Chukwudi,Nwa-David Chidobere, Characteristic Trigonometric Function Application to the Direct Variation Method for Clamped, Simply and Freely Supported Plate Under Uniform Load DistributionsJournal of Materials Engineering, Structures and Computation 1(1) 2022 pp. 40-58

References

1. C. Onyeka (2019). 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.
2. C. Onyeka, B. O. Mama, C. D. Nwa-David (2022). Application of Variation Method in Three-Dimensional Stability Analysis of Rectangular Plate Using Various Exact Shape Functions. Nigerian Journal of Technology (NIJOTECH), Vol. 41 No. 1, pp. 8-20, 2022. DOI: http://dx.doi.org/10.4314/njt.v41i1.2
3. C. Onyeka, B. O. Mama, C. D. Nwa-David (2022). 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. DOI: http://dx.doi.org/10.24018/ejeng.2022.7.2.2725
4. C. Onyeka and O. T. Edozie (2020). Application of Higher Order Shear Deformation Theory in the Analysis of thick Rectangular Plate. International Journal on Emerging Technologies, Vol. 11, No. 5, pp. 62–67.
5. C. Onyeka, C. D. Nwa-David, E. E. Arinze (2021). Structural Imposed Load Analysis of Isotropic Rectangular Plate Carrying a Uniformly Distributed Load Using Refined Shear Plate Theory, FUOYE Journal of Engineering and Technology (FUOYEJET), Vol. 6, No. 4, pp. 414-419. DOI: http://dx.doi.org/10.46792/fuoyejet.v6i4.719
6. H. Aginam, C. A. Chidolue and C. A. Ezeagu (2012). 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.
7. C. Onyeka, C. D. Nwa-David, T. E. Okeke (2022). 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. 4, pp. 28-37. DOI: https://dx.doi.org/10.55708/js0104004
8. C. Onyeka (2022). Stability Analysis of Three-Dimensional Thick Rectangular Plate Using Direct Variational Energy Method. PhD Thesis University of Nigeria, Nsukka, Journal of Advances in Science and Engineering, 6–t, pp.1–78. DOI: https://doi.org/10.37121/jase.v6i2.187
9. Ventsel, T. Krauthammer (2001). Thin plates and shells theory, analysis and applications. Maxwell Publishers Inc; New York.
10. N. Reddy (2006). Classical theory of plates,” In Theory and Analysis of Elastic Plates and Shells, CRC Press. DOI: 10.1201/9780849384165-7
11. Chandrashekhara (2001). Theory of plates. University Press (India) Limited.
12. C. Onyeka, B. O. Mama, C. D. Nwa-David (2022). Analytical Modelling of a Three-Dimensional (3D) Rectangular Plate Using the Exact Solution Approach. IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE), Vol. 19, No. 1, pp. 76-88. DOI: 10.9790/1684-1901017688
13. C. Onyeka, T. E Okeke, C. D Nwa-David (2022). 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, pp. 106-115. DOI: https://dx.doi.org/10.55708/js0103011
14. C. Onyeka, B. O. Mama and T. E. Okeke, C. D. (2022). Exact three-dimensional stability analysis of plate using a direct variational energy method. Civil Engineering Journal, Vol. 8, no. 1, pp. 60-80. DOI: http://dx.doi.org/10.28991/CEJ-2022-08-01-05
15. P. Timoshenko, and S. Woinowsky-krieger (1970). Theory of plates and shells, (2nd Ed.). Mc Graw-Hill Book Co. P.379, Singapore.
16. Kirchhoff (1859). U¨ ber das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math., Vol. 40, pp. 51-88.
17. D. Mindlin (1951). Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Trans. ASME J. Appl. Mech., Vol. 18, pp. 31-38.
18. Reissner (1945). The effect of transverse shear deformation on the bending of elastic plates. Trans ASME J. Appl. Mech., Vol. 12, no. 2, pp. 69-77.
19. N. Reddy (1984). A simple higher-order theory for laminated composite plates. Trans. ASME J. Appl. Mech., Vol. 51, pp. 745-752.
20. N. Reddy, N. D. Phan (1985). Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J. Sound Vib., Vol. 98, no. 2, pp. 157-170.
21. C. Onyeka and B. O. Mama (2021). 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. DOI: http://dx.doi.org/10.28991/esj-2021-01320
22. C. Onyeka, B.O. Mama, C.D Nwa-David, T.E. Okeke (2022). Exact Analytic Solution for Static Bending of 3-D Plate under Transverse Loading. Journal of Computational Applied Mechanics. DOI: 10.22059/JCAMECH.2022.342953.721
23. C. Onyeka and T. E. Okeke (2021). Elastic bending analysis exact solution of plate using alternative I refined plate theory. Nigerian Journal of Technology, Vol. 40, no. 6, pp. 1018 –1029. DOI: http://dx.doi.org/10.4314/njt.v40i6.4
24. C. Onyeka, T. E. Okeke, J. Wasiu (2020). 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.
25. C. Ike (2017). Equilibrium Approach in the Derivation of Differential Equation for Homogeneous Isotropic Mindlin Plates. Nigerian Journal of Technology (NIJOTECH), Vol. 36, no. 2, pp. 346-350. DOI: https://doi.org/10.4314/njt.v36i2.4
26. N. Osadebe, C. C. Ike, H. N. Onah, C. U. Nwoji and F. O. Okafor (2016). Application of Galerkin-Vlasov Method to the Flexural Analysis of Simply Supported Rectangular Kirchhoff Plates under Uniform Loads. Nigerian Journal of Technology (NIJOTECH), Vol. 35, no. 4, pp. 732-738. DOI: https://doi.org/10.4314/njt.v35i4.7
27. U. Nwoji, B. O. Mama, C. C. Ike and H. N Onah (2017). Galerkin-Vlasov method for the flexural analysis of rectangular Kirchhoff plates with clamped and simply supported edges. IOSR Journal of Mechanical and Civil Engineering (IOSR JMCE), Vol. 14, no. 2, pp. 61-74. DOI: https://doi.org/10.9790/1684-1402016174
28. Szilard (2004). Theories and Applications of Plates Analysis: Classical, Numerical and Engineering Methods. John Wiley and Sons Inc.
29. N. Kapadiya, A. D Patel (2015). Review of Bending Solutions of thin Plates”. International Journal of Scientific Research and Development (IJSRD), Vol. 3, no. 3, pp. 1709-1712.
30. O. Mama, C. U. Nwoji, C. C. Ike and H. N. Onah (2017). Analysis of Simply Supported Rectangular Kirchhoff Plates by the Finite Fourier Sine Transform Method. International Journal of Advanced Engineering Research and Science (IJAERS), Vol. 4, no. 3, pp. 285-291. DOI: https://doi.org/10.22161/ijaers.4.3.44
31. G. Iyengar (1988). Structural Stability of Columns and Plates. New York: Ellis Horwood Limited.
32. C. Onyeka and E. T. Okeke (2021). 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. doi: 10.25046/aj0601162
33. C. Onyeka, F. O. Okafor, H. N. Onah (2021). 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.
34. C. Onyeka, F. O. Okafor, H. N. Onah (2021). Buckling Solution of a Three-Dimensional Clamped Rectangular Thick Plate Using Direct Variational Method. IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE), vol. 18, no. 3 Ser. III, pp. 10-22. doi: 10.9790/1684-1803031022
35. A Shetty, S. A Deepak, K. K Sudheer, G. L Dushyanthkumar (2022). Thick Plate Bending Analysis using a Single Variable Simple Plate Theory, Materials Today: Proceedings, Vol. 54, no. 2, pp. 191-195. DOI: https://doi.org/10.1016/j.matpr.2021.08.289
36. P. Bhaskar, A. G. Thakur, I. I. Sayyad and S. V. Bhaskar (2021). 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 (IJAME), Vol. 18, no. 3, pp. 8882-8894. DOI: https://doi.org/10.15282/ijame.18.3.2021.04.0681
37. S. Sayyada, and Y. M. Ghugal (2012). Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory. Applied and Computational Mechanics Vol. 6, pp. 65–82.
38. L. Mantari and C. G Soares (2012). 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. DOI: https://doi.org/10.1016/j.compstruct.2012.01.005
39. Y. Tash and B. N. Neya (2020). An Analytical Solution for Bending of Transversely Isotropic Thick Rectangular Plates with Variable Thickness. Applied Mathematical Modelling, Vol. 77, no. 2, pp. 1582-1602. DOI: https://doi.org/10.1016/j.apm.2019.08.017
40. T. Thai, D. H. Choi (2013). Analytical Solutions of Refined Plate Theory for Bending, Buckling and Vibration Analyses of Thick Plates. Applied Mathematical Modelling, Vol. 37, pp. 8310–8323. DOI: http://dx.doi.org/10.1016/j.apm.2013.03.038
41. Zhong and Q. Xu (2017). Analysis Bending Solutions of Clamped Rectangular Thick Plate. Mathematical Problems in Engineering, Vol. 2017, pp. 1-6. DOI: https://doi.org/10.1155/2017/7539276
42. M. Ghugal, P. D. Gajbhiye (2016). 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. DOI: 10.22055/jacm.2016.12366
43. M. Ibearugbulem, J. C. Ezeh, L. O. Ettu, L. S. Gwarah (2018). Bending Analysis of Rectangular Thick Plate Using Polynomial Shear Deformation Theory, IOSR Journal of Engineering (IOSRJEN), Vol. 8, no. 9, pp. 53-61.
44. C. Onyeka (2021). Effect of Stress and Load Distribution Analysis on an Isotropic Rectangular Plate. Arid Zone Journal of Engineering, Technology & Environment, Vol. 17, no. 1, pp. 9-26.
45. C. Ezeh, O. M. Ibearugbulem, L. O. Ettu, L. S. Gwarah, I. C. Onyechere (2018). Application of shear deformation theory for analysis of CCCS and SSFS rectangular isotropic thick plates. Journal of Mechanical and Civil Engineering (IOSR-JMCE), Vol. 15, no. 5, pp. 33 – 42. DOI: 10.9790/1684-1505023342
46. M. Ibearugbulem, U. C Onwuegbuchulem, C. N. Ibearugbulem (2021). Analytical Three-Dimensional Bending Analyses of Simply Supported Thick Rectangular Plate. International Journal of Engineering Advanced Research (IJEAR), Vol. 3, no. 1, pp. 27–45.
47. Y. Grigorenko, A. S. Bergulev, S. N. Yaremchenko (2013). Numerical solution of bending problems for rectangular plates. International Applied Mechanics, Vol. 49, no. 1, pp. 81 – 94. DOI: https://doi.org/10.1007/s10778-013-0554-1
48. S. Gwarah (2019). 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.