A numerical-analytical approach to solving problems on the stress–strain state of quadrangular plates of complex shape is proposed. The governing system of equations is presented in new orthogonal coordinates using transformations that take into account the plate geometry. A two-dimensional boundary-value problem, which is described by a system of partial differential equations derived with the spline-collocation method, is reduced to a one-dimensional one that is solved by the stable numerical discrete-orthogonalization method. The numerical results obtained for plates in the form of a trapezium and parallelogram are compared with the data obtained by other methods. The approach makes it possible to calculate deflections of quadrangular plates of complex shape made of anisotropic materials
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References
Ya. M. Grigorenko and L. V. Mol’chenko, Fundamentals of the Theory of Plates and Shells: Tutorial [in Ukrainian], Lybed’, Kyiv (1993).
Ya. M. Grigorenko, V. D. Budak, and O. Ya. Grigorenko, Solution of the Problems of the Shell Theory with Discrete-Continuous Methods. Tutorial [in Ukraine], Ilion, Mykolaiv (2010).
V. Birman, Plate Structures, Springer, New York (2011).
S. K. Godunov, “Numerical solution of boundary-value problems for systems of linear ordinary differential equations,” Usp. Math. Nauk, 16, No. 3, 171–174 (1961).
Ya. M. Grigorenko, N. N. Kryukov, and N. S. Yakovenko, “Using spline-functions to solve boundary-value problems for laminated orthotropic trapezoidal plates of variable thickness,” Int. Appl. Mech., 41, No. 4, 413–420 (2005).
N. N. Kryukov, “Design of oblique and trapezoidal plates using spline functions,” Int. Appl. Mech., 33, No. 5, 114–117 (1997).
W. Y. Li, Y. K. Cheung, F. Asce, and L. G. Tham, “Spline finite strip analysis of general plates,” J. Eng. Mech., 112, No. 1, 43–54 (1986).
A. R. Shahidi, M. Mahzoon, M. M. Saadatpour, and M. Azhari, “Nonlinear static analysis of arbitrary quadrilateral plates in very large deflections,” Commun. Nonlin. Sci. Numer. Simul., 12, 832–848 (2007).
A. H. Sheikh and M. Mukhopadhyay, “Geometric nonlinear analysis of stiffened plates by the spline finite strip method,” Comp. Struct., 76, 765–785 (2000).
I. Shufrin, O. Rabinovich, and M. Eisenberger, “A semi-analytical approach for the geometrically nonlinear analysis of trapezoidal plates,” Int. J. Mech. Sci., 52, 1588–1596 (2010).
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Translated from Prikladnaya Mekhanika, Vol. 53, No. 3, pp. 104–112, May–June, 2017.
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Grigorenko, A.Y., Pankrat’ev, S.A. & Yaremchenko, S.N. Solution of Stress-Strain Problems for Complex-Shaped Plates in a Refined Formulation. Int Appl Mech 53, 326–333 (2017). https://doi.org/10.1007/s10778-017-0814-6
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DOI: https://doi.org/10.1007/s10778-017-0814-6