Bending of a ribbed plate under complex loading

D. P. Goloskokov ; A. V. Matrosov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 17 (2021) no. 2, pp. 120-130

Résumé

The article considers the problem of a rectangular plate, supported by a cross system of stiffening ribs, bending. In addition to the transverse load, the plate is subjected to forces in its plane, transmitted through the ribs. An analytical solution to the boundary value problem for the resolving differential equation with respect to the normal deflection of the plate, describing the deformation of a rectangular plate supported by stiffeners, is obtained. The solution is presented in the form of series in combinations of regular and special discontinuous functions, which converge quickly and lead to a simple computational algorithm. The influence of the ribs is taken into account in the equation in the form of additional terms containing factors with a delta function. This approach allows us to get rid of a number of assumptions regarding the interaction of the plate with its reinforcing elements. The use of the apparatus of generalized functions when modeling objects of this type simplifies the boundary conditions (there are no conditions for conjugation of various structural elements), but at the same time the differential equations become more complicated. The problem is reduced to the so-called partially degenerate equations. Development of analytical methods that allow obtaining exact solutions of differential equations of this type, and their introduction into computational practice, is one of the urgent tasks of the mechanics of objects with disturbed regularity.

Détail

Keywords: plate, stiffeners, mathematical model, numerical-analytical methods, special discontinuous functions, Dirac function, Heaviside function, Fourier series, orthogonal series.

Bibliographie

[1] Goloskokov D. P., “Calculation of the ribbed plate in the mixed form “deflection-function of efforts””, 2015 International Conference “Stability and Control Processes” in memory of V. I. Zubov, SCP-2015, Proceedings, St. Petersburg State University, St. Petersburg, 2015, 7342170, 386–388 | MR

[2] Goloskokov D. P., Matrosov A. V., “Analysis of elastic systems with discontinuous parameters”, 2017 International Conference “Constructive Nonsmooth Analysis and Related Topics” dedicated to the memory of V. F. Demyanov, CNSA-2017, Proceedings, St. Petersburg State University, St. Petersburg, 2017, 7973959

[3] Qin X., Liu S., Shen Y., Peng L., “Rib meshless optimization of stiffened plates resting on elastic foundation based on genetic algorithm”, Chinese Journal of Theoretical and Applied Mechanics, 52:1 (2020), 93–110

[4] Konev S. V., Fainshtein A. S., Teftelev I. E., “Application of Kantorovich — Vlasov method for shaped plate bending problem”, 5th International Conference on Industrial Engineering (ICIE-2019) (Sochi, Russian Federation, March 25–29, 2019), Lecture Notes in Mechanical Engineering, Sochi, 2020, 91–100 | DOI | MR

[5] Qin X., Liu S. S., Peng L. X., “Bending analysis of skew ribbed plates with a meshfree method”, 2nd International Conference on Modeling in Mechanics and Materials (Suzhou, China, March 29–31, 2019), IOP Conference Series Materials Science and Engineering, 531, no. 1, 2019

[6] Sapountzakis E. J., “An improved model for the analysis of plates stiffened by parallel beams including creep and shrinkage effects: Application to concrete or to composite steel-concrete structures”, International Journal on Engineering Applications, 6:2 (2018), 57–70

[7] Meng X., Sha L., Tong L., Yang X., “Experimental research on flexural performance of prestressed ribbed slab and composite slab”, Journal of Shenyang Jianzhu University (Natural Science), 33:1 (2017), 77–85

[8] Peng L. X., “Bending analysis of rectangular ribbed plates by the moving-least square meshfree method”, Chinese Journal of Computational Mechanics, 29:2 (2012), 210–216

[9] Sapountzakis E. J., Mokos V. G., “Shear deformation effect in plates stiffened by parallel beams”, Archive of Applied Mechanics, 79:10 (2009), 893–915 | DOI | Zbl

[10] Sapountzakis E. J., Mokos V. G., “Analysis of plates stiffened by parallel beams”, International Journal for Numerical Methods in Engineering, 70:10 (2007), 1209–1240 | DOI | Zbl

[11] Sapountzakis E. J., Katsikadelis J. T., “A new model for slab and beam structures — comparison with other models”, Computers and Structures, 80:5–6 (2002), 459–470 | DOI

[12] Goloskokov D. P., Matrosov A. V., “Comparison of two analytical approaches to the analysis of grillages”, 2015 International Conference “Stability and Control Processes” in memory of V. I. Zubov, SCP-2015, St. Petersburg State University, St. Petersburg, 2015, 7342169, 382–385

[13] Goloskokov D. P., Matrosov A. V., “Approximate analytical solutions in the analysis of thin elastic plates” (St. Petersburg, 1959), AIP Conference Proceedings, 2018, 070012 | DOI

[14] Matrosov A. V., Goloskokov D. P., “Analysis of elastic systems with nonsmooth boundaries”, 2017 International Conference “Constructive Nonsmooth Analysis and Related Topics” dedicated to the memory of V. F. Demyanov, CNSA-2017, Proceedings, St. Petersburg State University, St. Petersburg, 2017, 7973987

[15] Donnel L. G., Beams, plates and shells, McGraw-Hill Press, New York, 1976, 453 pp. | Zbl

[16] Timoshenko S. P., Woinowsky-Krieger S., Plates and shells, Nauka Publ, M., 1966, 636 pp. (In Russian)

[17] Goloskokov D. P., Numerical and analytical methods of calculation for elastic slimness-walling constructions for non-regular structure, SPbGUVK Publ, St. Petersburg, 2006, 258 pp. (In Russian)