Numeric analysis of elastic plane body static problem by the method of matched sections

Authors

DOI:

https://doi.org/10.20535/2521-1943.2024.8.4(103).313412

Keywords:

Method of matched sections, transfer matrix method, elastic plane body, triangular element, boundary conditions, plate with a circular hole

Abstract

The paper continues the series of authors' works on the elaboration of a principally new variant of the finite element method, FEM, for the treatment of various problems of mathematical physics, namely the method of matched section, MMS. The elastic plane body under static loading is considered here. As in FEM, the whole body is meshed into the small elements of, preferably, rectangular form. The main peculiarity of the method consists in the introduction of a set of main parameters dependent only on one coordinate variable, i.e. either  or . So, any differential equilibrium equation with two partial derivatives concerning  or  is broken out into two relatively simple equations concerning only one independent variable. This leads to the introduction of one additional constant showing the interchange between these two equations. The introduced constants can be derived from the equation of continuity of kinematic parameters in the center of each element. The main, for example, -dependent parameters are:  and  displacements in vertical (-) and horizontal (-) directions, respectively; normal  and tangential (shear)  forces in  direction, and  direction, respectively; and bending moment  and angle of rotation . Similar parameters are established for -direction. Based on the methodology of the transfer matrix method the analytical matrix-form dependence between these parameters in any point  or  and those at the lower and/or left border of the element are established. For the treatment of oblique and curvilinear boundaries, the right triangular element as a special degenerate case of the rectangular element is derived. The resulting system of linear equations is formulated for unknownThe paper continues the series of authors' works on the elaboration of a principally new variant of the finite element method, FEM, for the treatment of various problems of mathematical physics, namely the method of matched section, MMS. The elastic plane body under static loading is considered here. As in FEM, the whole body is meshed into the small elements of, preferably, rectangular form. The main peculiarity of the method consists in the introduction of a set of main parameters dependent only on one coordinate variable, i.e. either  or . So, any differential equilibrium equation with two partial derivatives concerning  or  is broken out into two relatively simple equations concerning only one independent variable. This leads to the introduction of one additional constant showing the interchange between these two equations. The introduced constants can be derived from the equation of continuity of kinematic parameters in the center of each element. The main, for example, -dependent parameters are:  and  displacements in vertical (-) and horizontal (-) directions, respectively; normal  and tangential (shear)  forces in  direction, and  direction, respectively; and bending moment  and angle of rotation . Similar parameters are established for -direction. Based on the methodology of the transfer matrix method the analytical matrix-form dependence between these parameters in any point  or  and those at the lower and/or left border of the element are established. For the treatment of oblique and curvilinear boundaries, the right triangular element as a special degenerate case of the rectangular element is derived. The resulting system of linear equations is formulated for unknown values of all parameters specified at the border of all elements. The efficiency and the superb accuracy of the MMS are demonstrated in the classical examples of bending of a long rectangular body (beam-like geometry) and tension at infinity of a 2D body with a small circular hole values of all parameters specified at the border of all elements. The efficiency and the superb accuracy of the MMS are demonstrated in the classical examples of bending of a long rectangular body (beam-like geometry) and tension at infinity of a 2D body with a small circular hole.

References

  1. I. Orynyak and K. Danylenko, “Method of matched sections as a beam-like approach for plate analysis”, Finite Elements in Analysis and Design, vol. 230, p. 104103, 2024. DOI: https://doi.org/10.1016/j.finel.2023.104103.
  2. K. Danylenko and I. Orynyak, “Method of matched sections in application to thin-walled and Mindlin rectangular plates”, Mechanics and Advanced Technologies, vol. 7, no. 2 (98), pp. 205-215, 2023. DOI: https://doi.org/10.20535/2521-1943.2023.7.2.277341.
  3. I. Orynyak, A. Tsybulnyk, K. Danylenko, A. Oryniak and S. Radchenko, “Timestep-dependent element interpolation functions in the method of matched sections on the example of heat conduction problem”, Journal of Computational and Applied Mathematics, vol. 456, p. 116222, 2025. DOI: https://doi.org/10.1016/j.cam.2024.116222.
  4. I. Orynyak, A. Tsybulnyk and K. Danylenko, “Spectral realization of the method of matched sections for thin-plate vibration”, Archive of Applied Mechanics, vol. 95, no. 2, p. 51, 2025. DOI: https://doi.org/10.1007/s00419-024-02755-7.
  5. A. Hrennikoff, “Solution of Problems of Elasticity by the Framework Method”, Journal of Applied Mechanics, vol. 8, no. 4, pp. A169-A175, Dec. 1941. DOI: https://doi.org/10.1115/1.4009129.
  6. M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp, “Stiffness and deflection analysis of complex structures”, Journal of the Aeronautical Sciences, vol. 23, no. 9, pp. 805–823, 1956. DOI: https://doi.org/10.2514/8.3664.
  7. O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, The finite element method: its basis and fundamentals. Elsevier Butterworth-Heinemann, 2005, 752 p.
  8. V. Haktanir and E. Kiral, “Statical analysis of elastically and continuously supported helicoidal structures by the transfer and stiffness matrix methods”, Computers & Structures, vol. 49, no. 4, pp. 663–677, 1993. DOI: https://doi.org/10.1016/0045-7949(93)90070-T.
  9. F. N. Gimena, P. Gonzaga and L. Gimena, “Stiffness and transfer matrices of a non-naturally curved 3D-beam element”, Engineering Structures, vol. 30, no. 6, pp. 1770–1781, 2008. DOI: https://doi.org/10.1016/j.engstruct.2007.10.012.
  10. H. Zhong, Z. Liu, H. Qin and Y. Liu, “Static analysis of thin-walled space frame structures with arbitrary closed cross-sections using transfer matrix method”, Thin-Walled Structures, vol. 123, pp. 255–269, 2018. DOI: https://doi.org/10.1016/j.tws.2017.11.018.
  11. Y.-h. Cao, G.-m. Liu and Z. Hu, “Vibration calculation of pipeline systems with arbitrary branches by the hybrid energy transfer matrix method”, Thin-Walled Structures, vol. 183, p. 110442, 2023. DOI: https://doi.org/10.1016/j.tws.2022.110442.
  12. Y. Goto, Y. Morikawa and S. Matsuura, “Direct Lagrangian nonlinear analysis of elastic space rods using transfer matrix technique”, Journal of the Japan Society of Civil Engineers, vol. 1988, no. 392, pp. 173-182, 1988. DOI: https://doi.org/10.2208/jscej.1988.392_173.
  13. I. V. Orynyak and S. A. Radchenko, “A mixed-approach analysis of deformations in pipe bends. Part 3. Calculation of bend axis displacements by the method of initial parameters”, Strength of materials, vol. 36, no. 5, pp. 463–472, 2004. DOI: https://doi.org/10.1023/B:STOM.0000048394.98411.4f.
  14. E. Tufekci, U. Eroglu and S. A. Aya, “A new two-noded curved beam finite element formulation based on exact solution”, Engineering with computers, vol. 33, no. 2, pp. 261–273, 2017. DOI: https://doi.org/10.1007/s00366-016-0470-1.
  15. Y. Yamada and Y. Ezawa, “On curved finite elements for the analysis of circular arches”, International Journal for Numerical Methods in Engineering, vol. 11, no. 11, pp. 1635–1651, 1977. DOI: https://doi.org/10.1002/nme.1620111102.
  16. K. Fumio, “On the validity of the finite element analysis of circular arches represented by an assemblage of beam elements”, Computer Methods in Applied Mechanics and Engineering, vol. 5, no. 3, pp. 253–276, 1975. DOI: https://doi.org/10.1016/0045-7825(75)90001-8.
  17. E. L. Wilson, Three-Dimensional Static and Dynamic Analysis of Structures: A Physical Approach With Emphasis on Earthquake Engineering. Berkeley, California, USA: Computers and Structures, Inc., 2000.
  18. J. P. M. de Almeida and E. A. W. Maunder, Equilibrium finite element formulations. John Wiley & Sons, 2017, 304 p. DOI: https://doi.org/10.1002/9781118925782.
  19. J. P. M. de Almeida and J. Reis, “An efficient methodology for stress‐based finite element approximations in two‐dimensional elasticity”, International Journal for Numerical Methods in Engineering, vol. 121, no. 20, pp. 4649–4673, 2020. DOI: https://doi.org/10.1002/nme.6458.
  20. K. Olesen, B. Gervang, J. N. Reddy and M. Gerritsma, “A higher‐order equilibrium finite element method”, International Journal for Numerical Methods in Engineering, vol. 114, no. 12, pp. 1262–1290, 2018. DOI: https://doi.org/10.1002/nme.5785.
  21. S. Ham and K.-J. Bathe, “A finite element method enriched for wave propagation problems”, Computers & structures, vol. 94-95, pp. 1–12, 2012. DOI: https://doi.org/10.1016/j.compstruc.2012.01.001.
  22. F. Parrinello and G. Borino, “Hybrid equilibrium element with high‐order stress fields for accurate elastic dynamic analysis”, International Journal for Numerical Methods in Engineering, vol. 122, no. 21, pp. 6308–6340, 2021. DOI: https://doi.org/10.1002/nme.6793.
  23. I. Orynyak, Calculation of complex systems by transfer matrix method. Kyiv: “Igor Sikorsky Kyiv Polytechnical Institute”, 2022, 324 p.
  24. R. H. Macneal and R. L. Harder, “A proposed standard set of problems to test finite element accuracy”, Finite Elements in Analysis and Design, vol. 1, no. 1, pp. 3–20, 1985. DOI: https://doi.org/10.1016/0168-874X(85)90003-4.
  25. K. M. Rao and U. Shrinivasa, "A set of pathological tests to validate new finite elements", Sadhana, vol. 26, no. 6, pp. 549-590, 2001. DOI: https://doi.org/10.1007/BF02703459.
  26. M. R. Hematiyan, M. Arezou, N. Koochak Dezfouli and M. Khoshroo, "Some remarks on the method of fundamental solutions for two dimensional elasticity", Computer Modeling in Engineering & Sciences, vol. 121, no. 2, pp. 661–686, 2019. DOI: https://doi.org/10.32604/cmes.2019.08275.

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Published

2024-12-26

How to Cite

[1]
K. Danylenko and I. Orynyak, “Numeric analysis of elastic plane body static problem by the method of matched sections”, Mech. Adv. Technol., vol. 8, no. 4(103), pp. 428–440, Dec. 2024.

Issue

Vol. 8 No. 4(103) (2024)

Section

Mechanics

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Copyright (c) 2024 Кірілл Даниленко, Ігор Ориняк

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