The superposition method in the problem of bending of a thin isotropic plate clamped along the contour

G. O. Alcybeev ; D. P. Goloskokov ; A. V. Matrosov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 18 (2022) no. 3, pp. 347-364

Résumé

In this work, the general solution of the differential equation for the bending of a thin isotropic plate under the action of a normal load applied to its plane is constructed by the superposition method. The solutions obtained by the method of initial functions in the form of trigonometric series are taken as two solutions, each of which allows satisfying the boundary conditions on two opposite sides of the plate. Two ways of satisfying the boundary conditions of a clamped plate are studied: the method of expansion into trigonometric Fourier series and the collocation method. It is shown that both methods give the same results and sufficiently fast convergence of the solution at all points of the plate except for small neighborhoods of the corner points. The constructed solution made it possible to study the behavior of the shear force in the corner points. Computational experiments have shown that when keeping 390 terms in the trigonometric series of the solution, the shear force is close to zero, but not identically equal.

Détail

Keywords: isotropic plate, bending of a thin plate, clamped plate, method of initial functions, MIF, computer algebra, Maple.

Bibliographie

[1] Timoshenko S. P., Woinowsky-Krieger S., Theory of plates and shells, 2$^\text{nd}$ ed., McGraw-Hill Publ, New York, 1987, 580 pp. | MR

[2] Meleshko V. V., “Selected topics in the history of the two-dimensional biharmonic problem”, Appl. Mech. Rev., 56:1 (2003), 33–85 | DOI | MR

[3] Meleshko V. V., “Bending of an elastic rectangular clamped plate: exact versus 'engineering' solutions”, Journal of Elasticity, 48:1 (1997), 1–50 | DOI | MR | Zbl

[4] Ceribasi S., Altay G., Cengiz Dokmeci M., “Static analysis of superelliptical clamped plates by Galerkin's method”, Thin-Walled Structures, 46:2 (2008), 122–127 | DOI

[5] Khan Y., Tiwari P., Ali R., “Application of variational methods to a rectangular clamped plate problem”, Computers and Mathematics with Applications, 63:4 (2012), 862–869 | DOI | MR | Zbl

[6] Altunsaray E., Bayer I., “Deflection and free vibration of symmetrically laminated quasi-isotropic thin rectangular plates for different boundary conditions”, Ocean Eng., 57 (2013), 197–222 | DOI

[7] Ceribasi S., Altay G., “Free vibration of super elliptical plates with constant and variable thickness by Ritz method”, Journal of Sound and Vibration, 319:1–2 (2009), 668–680 | DOI

[8] Altunsaray E., “Static deflections of symmetrically laminated quasi-isotropic super-elliptical thin plates”, Ocean Eng, 141 (2017), 337–350 | DOI

[9] Nwoji C. U., Onah H. N., Mama B. O., Ike C. C., “Ritz variational method for bending of rectangular kirchhoff plate under transverse hydrostatic load distribution”, Mathematical Modelling of Engineering Problems, 5:1 (2018), 1–10 | DOI

[10] Aginam C. H., Chidolue C. A., Ezeagu C. A., “Application of direct variational method in the analysis of isotropic thin rectangular plates”, ARPN Journal of Engineering and Applied Sciences, 7:9 (2012), 1128–1138

[11] Festus O., Okeke E. T., John W., “Strain-displacement expressions and their effect on the deflection and strength of plate”, Advances in Science, Technology and Engineering Systems, 5:5 (2020), 401–413 | DOI

[12] Wang X., Yuan Z., “Buckling analysis of isotropic skew plates under general in-plane loads by the modified differential quadrature method”, Applied Mathematical Modelling, 56 (2018), 83–95 | DOI | MR | Zbl

[13] Ike C. C., Onyia M. E., Rowland-Lato E. O., “Generalized integral transform method for bending and buckling analysis of rectangular thin plate with two opposite edges simply supported and other edges clamped”, Advances in Science, Technology and Engineering Systems, 6:1 (2021), 283–296 | DOI

[14] Imrak E., Fetvaci C., “The deflection solution of a clamped rectangular thin plate carrying uniformly load”, Mechanics Based Design of Structures and Machines, 37:4 (2009), 462–474 | DOI

[15] Meleshko V. V., Gomilko A. M., Gourjii A. A., “Normal reactions in a clamped elastic rectangular plate”, Journal of Engineering Mathematics, 40 (2001), 377–398 | DOI | MR | Zbl

[16] Matrosov A. V., Suratov V. A., “Stress-strain state in the corner points of a clamped plate under uniformly distributed normal load”, Materials Physics and Mechanics, 36:1 (2018), 124–146

[17] Matrosov A. V., “An exact analytical solution for a free-supported micropolar rectangle by the method of initial functions”, Zeitschrift für Angewandte Mathematik und Physik, 73:2 (2022), 74 | DOI | MR | Zbl

[18] Goloskokov D. P., Matrosov A. V., “A superposition method in the analysis of an isotropic rectangle”, Applied Mathematical Sciences, 10:54 (2016), 2647–2660 | DOI

[19] Goloskokov D. P., Matrosov A. V., “Method of initial functions in analyses of the bending of a thin orthotropic plate clamped along the contour”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 17:4 (2021), 330–344 (In Russian) | DOI | MR

[20] Lur'e A. I., Three dimensional problems of theory of elasticity, Willey Publ, New York, 1964, 495 pp. | MR | MR | Zbl

[21] Vlasov V. Z., “Method of initial functions in problems of the theory of elasticity”, Izv. Akad. Nauk SSSR. OTN, 1955, no. 7, 49–69 (In Russian)

[22] Vlasov V. Z., “Method of initial functions in problems of theory of thick plates and shells”, Proceedings of 9$^\text{th}$ Intern. Congress Appl. Mech., 1957, no. 6, 321–330 | MR

[23] Maliev A. S., “On the choice of functions in general solutions of the problem of the equilibrium of an isotropic elastic body”, Sbornik nauch. trudov Leningrad Elektrotehnical Institute, 1952, no. 4, 180–244 (In Russian)

[24] Sundara Raja Iyengar K. T., Pandya S. K., “Analysis of orthotropic rectangular thick plates”, Fibre Sci. Technol., 18:1 (1983), 19–36 | DOI

[25] Galileev S. M., Matrosov A. V., “Method of initial functions in the computation of sandwich plates”, Intern. Appl. Mech., 31:6 (1995), 469–476 | DOI

[26] Kovalenko M. D., “The lagrange expansions and nontrivial null-representations in terms of homogeneous solutions”, Dokl. Physics, 42:2 (1997), 90–92 | MR

[27] Dubey S. K., “Analysis of homogeneous orthotropic deep beams”, Journal of Struct. Eng. (Madras), 32:2 (2005), 109–116

[28] Patel R., Dubey S. K., Pathak K. K., “Analysis of RC brick filled composite beams using MIF”, Procedia Eng., 51 (2013), 30–34 | DOI

[29] Galileev S. M., Tabakov P. Y., “Mathematical foundations of the method of initial functions for the application to anisotropic plates”, 2$^\text{nd}$ Intern. Conference on Mech. Nanotechnol. Cryog. Eng., 2013, 59–63

[30] Patel R., Dubey S. K., Pathak K. K., “Analysis of infilled beams using method of initial functions and comparison with FEM”, Intern. Journal of Eng. Sci. Technol., 17 (2014), 158–164

[31] Matrosov A. V., “A numerical-analytical decomposition method in analyses of complex structures”, 2014 Intern. Conference on Computer Technologies in Physical and Engineering Applications, ICCTPEA 2014. Proceedings (St Petersburg, 2014), 104–105 (No 6893305)

[32] Matrosov A. V., Shirunov G. N., “Numerical-analytical computer modeling of a clamped isotropic thick plate”, Intern. Conference on Computer Technologies in Physical and Engineering Applications, ICCTPEA 2014. Proceedings (St Petersburg, 2014), 2014, 96 (No 6893300)

[33] Goloskokov D. P., Matrosov A. V., “Comparison of two analytical approaches to the analysis of grillages”, 2015 Intern. Conference on “Stability and Control Processes” in memory of V. I. Zubov, SCP 2015. Proceedings, Izdatel'skii Dom Fedorovoi G. V., St Petersburg, 2015, 382–385 (No 7342169)

[34] Matrosov A. V., “A superposition method in analysis of plane construction”, 2015 Intern. Conference on “Stability and Control Processes” in memory of V. I. Zubov, SCP 2015. Proceedings, Izdatel'skii Dom Fedorovoi G. V., St Petersburg, 2015, 414–416 (No 7342156)

[35] Olodo E. T., Adanhounme V., Hounkonnou M. N., “Exact solution of the harmonic problem for a rectangular plate in flat deformation by the method of initial functions”, Intern. Journal of Appl. Mech. Eng., 22:2 (2017), 349–361 | DOI

[36] Matrosov A. V., Shirunov G. N., “Analyzing thick layered plates under their own weight by the method of initial functions”, Materials Physics and Mechanics, 31:1–2 (2017), 36–39

[37] Asutkar P., Shinde S. B., Patel R., “Study on the behaviour of rubber aggregates concrete beams using analytical approach”, Intern. Journal of Eng. Sci. Technol., 20 (2017), 151–159

[38] Goloskokov D. P., Matrosov A. V., “Approximate analytical approach in analyzing an orthotropic rectangular plate with a crack”, Materials Physics and Mechanics, 36:1 (2018), 137–141

[39] Kovalenko M. D., Menshova I. V., Kerzhaev A. P., Yu G., “On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip”, Zeitschrift für Angewandte Mathematik und Physik, 69 (2018), 121 | DOI | MR | Zbl

[40] Goloskokov D. P., Matrosov A. V., “Approximate analytical solutions in the analysis of thin elastic plates”, AIP Conference Proceedings, American Institute of Physics Publ, 2018 (No 070012)

[41] Owczarek S., Owczarek M., “Heat transport analysis in rectangular shields using the Laplace and Poisson equations”, Energies, 13 (2020), 1714 | DOI

[42] Matrosov A. V., Kovalenko M. D., Menshova I. V., Kerzhaev A. P., “Method of initial functions and integral Fourier transform in some problems of the theory of elasticity”, Zeitschrift für Angewandte Mathematik und Physik, 71:1 (2020), 24 | DOI | MR | Zbl

[43] Lamé G., Leçon sur la théorie mathémathique de l'élasticité des corps solids, Bachelier, Paris, 1852, 335 pp.

[44] Matrosov A. V., “Computational peculiarities of the method of initial functions”, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2019, 37–51 | MR