DOI: https://doi.org/10.20535/2521-1943.2020.0.209618

Application of exponential functions in weighted residuals method in structural mechanics. Part 1: axisymmetrical shell problem.

Igor Orynyak, Yulia Bai

Abstract


Weighted residuals method gained a wide popularity during last years especially due to its application in finite element methods. Its goal is in approximate satisfaction of the governing differential equations while boundary conditions are to be fulfilled exactly. This goal is achieved by the proper choice of the sets of so-called trial (basic) functions which give the residuals. Residuals are multiplied by weight functions and minimized by integration over the whole area of task. In fact, they determine the peculiarity and advantages of each particular method. Most popular is the choice of trial and weight (test) function as the trigonometric and polynomial functions. In 2D applications so-called “beam functions” are often used, which are solutions of much simpler 1D problems for beam.

In this methodological paper we explore the possibility of using the sets of functions constructed on the consequent exponential functions, which satisfy boundary conditions. The method is investigated on example of very simple 1D axisymmetrical task for shell, where exact solution exists for any loading. For several examples of distributed or concentrated loading the proposed method is compared with similar Navier’s method, which is the expansion on trigonometric functions. Also the proper choice of weight functions is carefully investigated. It is noted, that proposed sets (symmetrical or asymmetrical) of exponential functions has a good perspective in application for more complicated problems in structural mechanics.

Keywords


axisymmetrical shell, distributed loading, concentrated force, Navier method, Galerkin method, sets of exponential functions.

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References


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GOST Style Citations


[1] K.-J. Bathe, Finite Element Procedures. 2nd ed. Klaus-Jurgen Bathe, Watertown, MA, 2016, 1065 p. https://doi.org/10.1201/9781315641645-2

[2] N.G. Afendikova, “Istoriya metoda Galerkina i ego rol' v tvorchestve M.V. Keldysha”, Preprinty IPM im. M.V. Keldysha, no. 77, pp. 3–17, 2014, https://doi.org/10.1007/s15015-014-1378-5

[3] S.P. Demidov, Teoriya uprugosti [Theory of elasticity]. Moscow, Russia: Vyssha shkola, 1979, 432 p.

[4] C.A.J. Fletcher, Computational Galerkin methods. Springer-Verlag, New-York, 1984, 320 p. https://doi.org/10.1007/978-3-642-85949-6

[5] S.P. Timoshenko and S. Woinowsky-Krieger, Theory of plates and shells. 2nd ed. McGraw-Hill Inc New York, 1959, 595 p.

[6] I. Elishakoff, A.P. Ankitha and A. Marzani, “Rigorous versus naïve implementation of the Galerkin method for stepped beams”, Acta Mechanica, vol. 230, pp. 3861–3873, 2019. https://doi.org/10.1007/s00707-019-02393-z

[7] G.B. Warburton, “The vibration of rectangular plates”, Proc. Inst. Mech. Engrs., vol. 168(1), pp. 371–384, 1954. https://doi:10.1243/PIME_PROC_1954_168_040_02

[8] P. Moreno-García, J.V. Araújo dos Santos and H. Lopes, “A review and study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates”, Archives of Computational Methods in Engineering, vol. 25, pp. 785–815, 2018. https://doi.org/10.1007/s11831-017-9214-7

[9] U. Lee, Spectral element method in structural dynamics. John Wiley and Sons (Asia), Singapore, 2009, 454 p. DOI:10.1002/9780470823767

[10] J.T.-S. Wang and C.-C. Lin, “Dynamic analysis of generally supported beams using Fourier series”, Journal of Sound and Vibration, vol. 196, no. 3, pp. 285–293, 1996. https://doi.org/10.1006/jsvi.1996.0484

[11] S. Zhang, L.Xu and R. Li, “New exact series solutions for transverse vibration of rotationally-restrained orthotropic plates”, Applied Mathematical Modeling, vol. 65, pp. 348–360, 2019. https://doi.org/10.1016/j.apm.2018.08.033





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