Ike, C. (2023). Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method. Alustath, 41(11), 1403-1413. doi: 10.30684/etj.2023.142504.1537
Charles Chinwuba Ike. "Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method". Alustath, 41, 11, 2023, 1403-1413. doi: 10.30684/etj.2023.142504.1537
Ike, C. (2023). 'Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method', Alustath, 41(11), pp. 1403-1413. doi: 10.30684/etj.2023.142504.1537
Ike, C. Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method. Alustath, 2023; 41(11): 1403-1413. doi: 10.30684/etj.2023.142504.1537

Single Variable Thick Plate Buckling Problems Using Double Finite Sine Transform Method

Engineering and Technology Journal
Volume 41, Issue 11, November 2023, Pages 1403-1413    PDF (591.84 K)
Document Type: Research Paper
DOI: 10.30684/etj.2023.142504.1537
Author
Charles Chinwuba Ike*
Civil Engineering Dept., Enugu State University of Science and Technology, Agbani, Enugu State, Nigeria.
Abstract
This paper derives buckling solutions for single variable thick plate buckling problems using the double finite sine transform method (DFSTM). The problem governing partial differential equation (GPDE), originally formulated by Shimpi and others, uses a refined plate theory (RPT) and accounts for transverse shear deformations, rendering it applicable to thick plates. The thick plate is simply supported and subjected to (i) uniform uniaxial compressive load in the x direction. (ii) uniform biaxial compressive load in the x and y axes. The DFSTM was applied to the GPDE, and the problem transformed into an algebraic equation, which was simpler in this case due to the Dirichlet boundary conditions satisfied by the sinusoidal kernel of the DFSTM. Analytical buckling solutions were determined for the two considered cases of uniaxial and biaxial compressive loads in terms of the buckling modes, Poisson’s ratio (m), and thickness (h) to least dimension (a) ratio. Critical buckling loads (Pcr) determined at the first buckling modes agreed with previously obtained Navier solutions for Mindlin, Reddy, and Refined plates. Pcr calculated for ratios of h/a equal to 0.01 converged to the solutions obtained using Kirchhoff-Love plate theory (KLPT), illustrating the applicability of the GPDE to thin and thin plate buckling.

Highlights
  • The double finite sine transform method was utilized to obtain exact buckling solutions for thick plate problems
  • The method simplified the problem to algebraic ones since the sinusoidal function satisfies the boundary conditions
  • The buckling solutions aligned with exact solutions for thick plates
Keywords
Double finite sine transform method; Single variable thick plate; Bending problem; Biaxial buckling; Critical buckling load
References
  1. C. Ike, Exact analytical solutions to bending problems of SFrSFr plates using variational Kantorovich-Vlasov method, J. Compu. Appl. Mech., 54 (2023) 186 – 203. https://doi.org/10.22059/jcamech.2023.351563.776
  2. C. Ike, Variational Ritz-Kantorovich-Euler Lagrange method for the elastic buckling analysis of fully clamped Kirchhoff thin plate, ARPN J. Eng. Appl. Sci., 16 (2021) 224 – 241.
  3. C. Ike, Generalized integral transform method for bending analysis of clamped rectangular thin plate, J. Comput. Appl. Mech., 53 (2022) 599 – 635. https://doi.org/10.22059/jcamech.2022.350620.768
  4. D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates, ASME J. Appl. Mech., 18 (1951) 31 – 38. https://doi.org/10.1115/1.4010217
  5. N. Reddy, A simple higher order theory for laminated composite plates, ASME J. Appl. Mech., 51(1981) 745 – 752.
  6. N. Reddy, A refined non-linear theory of plates with transverse shear deformation., Int. J. Solids Struct., 20 (1984) 881 – 896. https://doi.org/10.1016/0020-7683(84)90056-8
  7. P. Shimpi, Refined plate theory and its variants, AIAA J., 40 (2002) 137 – 146. https://doi.org/10.2514/2.1622
  8. P. Shimpi, R.A. Shetty, A. Guha, A single variable refined theory for free vibrations of a plate using inertia related terms in displacements, Eur. J. Mech. A. Solids, 65 (2017) 136 – 148.https://doi.org/10.1016/j.euromechsol.2017.03.005
  9. U. Nwoji, B.O. Mama, H.N. Onah, C.C. Ike, Flexural analysis of simply supported rectangular Mindlin plates under bisinusoidal transverse load, ARPN J. Eng. Appl. Sci., 13 (2018) 4480 – 4488.
  10. U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike, Theory of elasticity formulation of the Mindlin plate equations, Int. J. Eng. Technol., 9 (2018) 4344 – 4352.
  11. C. Ike, Equilibrium approach in the derivation of differential equations of homogeneous isotropic Mindlin plates, Niger. J. Technol., 36 (2017) 346 – 350. https://doi.org/10.4314/njt.v36i2.4
  12. C. Ike, Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates, J. Math. Models Eng., 4 (2018) 50 – 72. https://doi.org/10.21595/mme.2018.19825
  13. Norouzzadeh, R. Ansari, H. Rouhi, Isogeometric analysis of Mindlin nanoplates based on the integral formulation of nonlocal elasticity Multidiscip, Model. Mater. Struct., 14 (2018) 810 – 827. https://doi.org/10.1108/mmms-09-2017-0109
  14. Bao, S. Cen, C. Li, Distortion-resistant and locking free eight node elements effectively capturing the edge effects of Mindlin-Reissner plates, Eng. Comput., 34 (2017) 548 – 586. https://doi.org/10.1108/EC-04-2016-0143
  15. Do V.T., Pham V.V., Nguyen H.N., On the development of refined plate theory for static bending behaviour of functionally graded plates, Math. Model. Eng. Probl., 2836763, 2020. https//doi.org/10.1155/2020/2836763
  16. Soltani, A. Bassaim, M.S.A. Houari, A. Kaci, M. Benguediab, A. Tounsi, M.Sh. Alhodaly, A novel hyperbolic shear deformation theory for the mechanical buckling analysis of advanced composite plates resting on elastic foundations, Steel Compos. Struct., 30 (2019) 13 – 29. https://doi.org/10.12989/scs.2019.30.1.013
  17. Kumar. J. Singh, Assessment of higher order transverse shear deformation theories for modelling and buckling analysis of FGM plates using RBF based meshless approach, Multidiscip Model. Mater. Struct., 14 (2018) 891 – 907. https://doi.org/10.1108/mmms-07-2017-0069.
  18. Hadji, M. Avcar, Free vibration analysis of FG porous sandwich plates under various boundary conditions, J. Appl. Comput. Mech., 7 (2021) 505 – 519. https://doi.org/10.22055/jacm.2020.35328.2628
  19. Mohseni, A. Naderi, Effect of higher order shear and normal deformations theory in buckling analysis of thick functionally graded plates, J. Comput. Appl. Mech., 54 (2023) 347 – 364. https://doi.org/10.22059/jcamech.2023.361677.852
  20. C. Onyeka, B.O. Mama, Analytical study of bending characteristics of an elastic rectangular plate using direct variational energy approach with trigonometric function, Emerg. Sci. J., 5 (2021) 916-928. https://doi.org/10.28991/esj-2021-01320
  21. C. Onyeka, E.T. Okeke, Analytical solution of thick rectangular plate with clamped and free support boundary condition using polynomial shear deformation theory, Adv. Sci. Technol. Eng. Syst. J., 6 (2021) 1427 – 1439.https://dx.doi.org/10.25046/aj0601162
  22. C. Onyeka, T.E. Okeke, New refined shear deformation theory effect on non-linear analysis of a thick plate using energy method, Arid Zone. J. Eng. Technol. Environ., 17 ( 2021) 121 – 140.
  23. C. Onyeka, B.O. Mama, T.E. Okeke, Exact three-dimensional stability analysis of plate using a direct variational energy method, Civ. Eng. J., 8 (2022) 60 – 80. https://doi.org/10.28991/CEJ-2022-08-01-05
  24. C. Onyeka, C.D. Nwa-David, T.E. Edozie, Analytical solution for the static bending elastic analysis of thick rectangular plate structures, Eng. Technol. J., 40 (2022)1548 – 1559. https://doi.org/10.30684/etj.2022.134687.1244
  25. C. Onyeka, T.E. Okeke, B.O. Mama, Static elastic bending analysis of a three-dimensional clamped thick rectangular plate using energy method, High Tech Innov. J., 3 (2022) 267 – 281. https://doi.org/10.28991/HIJ-2022-03-03-03
  26. C. Onyeka, E.D. Nwa-David, B.O. Mama, Static bending solutions for an isotropic rectangular clamped/simply supported plate using 3-D plate theory, J. Comput. Appl. Mech., 54 (2023) 1–18. https://doi.org/10.22059/JCAMECH.2022.349835.764
  27. C. Onyeka, T.E. Okeke, C. Nwa-David, B.O. Mama, Analytical elasticity solution for accurate prediction of stresses in a rectangular plate using exact 3-D theory, J. Comput. Appl. Mech., 54 (2023) 167-185.https://doi.org/10.22059/jcamech.2022.351892.781
  28. C. Onyeka, B.O. Mama, C.D. Nwa-David Stress analysis of transversely loaded isotropic three-dimensional plates using a polynomial shear deformation theory, Eng. Technol. J., 41 (2023) 603 – 618.https://doi.org/10.30684/etj.2023.137410.1345
  29. N. Onah, M.E. Onyia, B.O. Mama, C.U. Nwoji, C.C. Ike, First principles derivation of displacement and stress function for three-dimensional elasto static problems, and application to the flexural analysis of thick circular plates, J. Comput. Appl. Mech., 51 (2020) 184 – 198. https://doi.org/10.22059/JCAMECH.2020.295989.471
  30. Singh, A. Kumar, Vibration and the buckling response of functionally graded plates, according to a refined hyperbolic shear deformation theory, Mech. Compos. Mater., 59 (2023) 725 – 742. https://doi.org/10.1007/s11029-023-10127-5
  31. Nareen, R.P. Shimpi, Refined hyperbolic shear deformation plate theory, Proc. Inst. Mech. Eng. C., J. Mech. Eng. Sci. 229 (2015) 2675 – 2686. http://dx.doi.org/10.1177/0954406214563739
  32. J.M. Ferreira, C.M.C. Roque, Analysis of thick plates by radial basis functions, Acta Mech., 217 (2011) 177 – 190.https://doi.org/10.1007/s00707-010-0395-5
  33. T. Gomaa, M.H. Baluch, H.H. Abdel-Rahman, A.K. Mohammed, Finite element modelling of thick isotropic plates, Eng. Comput., 8 (1991) 361 – 378. https://doi.org/10.1108/eb023845
  34. Zargaripoor, A. Bahrami, M. Nikkah Bahrami, Free vibration and buckling analysis of third-order shear deformation plate theory using exact wave propagation approach, J. Comput. Appl. Mech., 49 (2018) 102 – 124. https://doi.org/10.22059/JCAMECH.2018.249468.227
  35. Malikan, V.C. Nguyen, A novel one-variable first order shear deformation theory for biaxial buckling of a size-dependent plate based on Eringen’s nonlocal differential law, World J. Eng., 15 (2018) 633 – 645. https://doi.org/10.1108/WJE-11-2017-0357
  36. Zhong Y., Xu Q., Analysis bending solutions of clamped rectangular thick plate, Math. Probl. Eng., 7539276, 2017. https://doi.org/10.1155/2017/7539276
  37. N. Reddy, N.D. Phan, Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory, J. Sound Vib., 98 (1985) 158 – 170. https://doi.org/10.1016/0022-460X(85)90383-9
  38. A. Deepak, R.A. Shetty, K.K. Sudheer, G.L. Dushyanthkumar, Buckling analysis of thick plates using a single variable simple plate theory, J. Mines. Metals Fuels, 2021 (2021) 67 – 72.
  39. Timoshenko, S.P., Gere, G.M., Theory of Elastic Stability 2nd Edition, McGraw Hill Book Company, New York, 1961.
  40. T. Thai, D.H. Choi, Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates, Appl. Math. Modell., 37 (2013) 8310 – 8323.https://doi.org/10.1016/j.apm.2013.03.038
  41. Srinivas, A.K. Rao, Buckling of thick rectangular plates, AIAA J., 7 (1969) 1645 – 1646.
  42. H. Hashemi, K. Khorshidi, M. Amabili, Exact solutions for linear buckling of rectangular Mindlin plates, J. Sound Vib., 315 (2008) 318 – 342. http://dx.doi.org/10.1016/j.jsv.2008.01.059
  43. E. Onyia, E.O. Rowland-Lato, C.C. Ike, Galerkin-Kantorovich method for the elastic buckling analysis of thin rectangular SCSC plates, Int. J. Eng. Res. Technol., 13 (2020) 613 – 619. https://dx.doi.org/10.37624/IJERT/13.4.2020.613-619
  44. E. Onyia, E.O. Rowland-Late, C.C. Ike, Galerkin-Vlasov variational method for the elastic buckling analysis of SSCF and SSSS rectangular plates, Int. J. Eng. Res. Technol.,13 (2020) 1137 – 1146.http://dx.doi.org/10.37624/IJERT/13.6.2020.1137-1146
  45. A. Oguaghamba, C.C. Ike, Galerkin-Vlasov method for the elastic buckling analysis of Kirchhoff plate with one free edge and three simply supported edges under uniform uniaxial compression, ARPN J. Eng. Appl. Sci., 15 (2020) 1574 – 1581.
  46. N. Onah, C.U. Nwoji, C.C. Ike, B.O. Mama, Elastic buckling analysis of uniaxially compressed CCSS Kirchhoff plate using single finite Fourier sine transform method, Modell. Meas. Control, B, 87 (2018) 107 – 111.https://doi.org/10.18280/mmc_b.870208
  47. E. Onyia, E.O. Rowland-Late, C.C. Ike, Elastic buckling analysis of SSCF and SSSS rectangular thin plates using the single finite Fourier sine integral transform method, Int. J. Eng. Res. Technol., 13 (2020) 1147 – 1158. https://dx.doi.org/10.37624/IJERT/13.6.2020.1147-1158
  48. C. Ike, M.E. Onyia, E.O. Rowland-Lato, Generalized integral transform method for bending and buckling analysis of rectangular thin plate with two opposite edges simply supported and other edges clamped, Adv. Sci. Technol. Eng. Sys. J., 6 (2021) 283 – 296. http://dx.doi.org/10.25046/aj060133
  49. U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike, E.U. Ikwueze, Elastic buckling analysis of simply supported thin plates using the double finite sine transform method, Explorematics J. Innovative Eng. Technol., 1 (2017) 37 –47.
  50. A. Oguaghamba, C.C. Ike, Single finite Fourier sine integral transform method for the determination of natural frequencies of flexural vibration of Kirchhoff plates, Int. J. Eng. Res. Technol., 13 (2020) 470 – 476.https://dx.doi.org/10.37624/IJERT/13.3.2020.470-476
  51. O. Mama, O.A. Oguaghamba, C.C. Ike, Single finite Fourier sine integral transform method for the flexural analysis of rectangular Kirchhoff plate with opposite edges simply supported, other edges clamped for the case of triangular load distribution, Int. J. Eng. Res. Technol., 13 (2020) 1802 – 1813. https://dx.doi.org/10.37624/IJERT/13.7.2020.1802-1813
  52. P. Shimpi, P.J. Guruprasad, K.S. Pakhare, Single variable new first-order shear deformation theory for isotropic plates, Lat. Am. J. Solids Struct., 15 (2018) e124, 1-25. https://doi.org/10.1590/1679-78254842
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