Метод суперпозиции в задаче изгиба защемленной по контуру тонкой изотропной пластинки
DOI:
https://doi.org/10.21638/11701/spbu10.2022.305Аннотация
В работе методом суперпозиции построено общее решение дифференциального уравнения изгиба тонкой изотропной пластинки под действием нормальной нагрузки, приложенной к ее плоскости. В качестве двух решений, каждое из которых позволяет удовлетворить граничным условиям на двух противоположных сторонах пластины, взяты решения, полученные методом начальных функций в виде тригонометрических рядов. Исследованы два способа удовлетворения граничным условиям жестко защемленной пластины: метод разложения в тригонометрические ряды Фурье и метод коллокаций. Показано, что оба метода дают одинаковые результаты и достаточно быструю сходимость решения во всех точках пластины, кроме малых окрестностей угловых точек. Построенное решение позволило изучить поведение перерезывающей силы в окрестностях угловых точек. Вычислительные эксперименты показали, что при удержании 390 членов в тригонометрических рядах решения перерезывающая сила близка к нулю, но не равна тождественно.
Ключевые слова:
изотропная пластинка, изгиб тонкой пластинки, защемленная по контуру пластинка, метод начальных функций, компьютерная алгебра, Maple
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Литература
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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. P. 104-105. N 6893305.
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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. St Petersburg: Izdatel’skii Dom Fedorovoi G. V., 2015. P. 382-385. N 7342169.
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Matrosov A. V., Shirunov G. N. Analyzing thick layered plates under their own weight by the method of initial functions // Materials Physics and Mechanics. 2017. Vol. 31. N 1-2. P. 36-39.
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. 2017. N 20. P. 151-159.
Goloskokov D. P., Matrosov A. V. Approximate analytical approach in analyzing an orthotropic rectangular plate with a crack // Materials Physics and Mechanics. 2018. Vol. 36. N 1. P. 137-141.
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. 2018. Vol. 69. P. 121.
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. N 070012.
Owczarek S., Owczarek M. Heat transport analysis in rectangular shields using the Laplace and Poisson equations // Energies. 2020. Vol. 13. P. 1714.
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Ceribasi S., Altay G., Cengiz Dokmeci M. Static analysis of superelliptical clamped plates by Galerkin’s method. Thin-Walled Structures, 2008, vol. 46, no. 2, pp. 122-127.
Khan Y., Tiwari P., Ali R. Application of variational methods to a rectangular clamped plate problem. Computers and Mathematics with Applications, 2012, vol. 63, no. 4, pp. 862-869.
Altunsaray E., Bayer I. Deflection and free vibration of symmetrically laminated quasi-isotropic thin rectangular plates for different boundary conditions. Ocean Eng. , 2013, vol. 57, pp. 197-222.
Ceribasi S., Altay G. Free vibration of super elliptical plates with constant and variable thickness by Ritz method. Journal of Sound and Vibration, 2009, vol. 319, no. 1-2, pp. 668-680.
Altunsaray E. Static deflections of symmetrically laminated quasi-isotropic super-elliptical thin plates. Ocean Eng., 2017, vol. 141, pp. 337-350.
Nwoji C. U., Onah H. N., Mama B. O., Ike С. C. Ritz variational method for bending of rectangular kirchhoff plate under transverse hydrostatic load distribution. Mathematical Modelling of Engineering Problems, 2018, vol. 5, no. 1, pp. 1-10.
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, 2012, vol. 7, no. 9, pp. 1128-1138.
Festus О., 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, 2020, vol. 5, no. 5, pp. 401-413.
Wang X., Yuan Z. Buckling analysis of isotropic skew plates under general in-plane loads by the modified differential quadrature method. Applied Mathematical Modelling, 2018, vol. 56, pp. 83-95.
Ike С. 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, 2021, vol. 6, no. 1, pp. 283-296.
Imrak E., Fetvaci C. The deflection solution of a clamped rectangular thin plate carrying uniformly load. Mechanics Based Design of Structures and Machines, 2009, vol. 37, no. 4, pp. 462-474.
Meleshko V.V., Gomilko A.M., Gourjii A. A. Normal reactions in a clamped elastic rectangular plate. Journal of Engineering Mathematics, 2001, vol. 40, pp. 377-398.
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, 2018, vol. 36, no. 1, pp. 124-146.
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, 2022, vol. 73, no. 2, p. 74.
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Kovalenko M.D. The lagrange expansions and nontrivial null-representations in terms of homogeneous solutions. Dokl. Physics, 1997, vol. 42, no. 2, pp. 90-92.
Dubey S.K. Analysis of homogeneous orthotropic deep beams. Journal of Struct. Eng. (Madras), 2005, vol. 32, no. 2, pp. 109-116.
Patel R., Dubey S.K., Pathak К. K. Analysis of RC brick filled composite beams using MIF. Procedia Eng., 2013, no. 51, pp. 30-34.
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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., 2014, no. 17, pp. 158-164.
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, pp. 104-105, no. 6893305.
Matrosov A. V., Shirunov G. N. Numerical-analytical computer modeling of a clamped isotropic thick plate. 2014 Intern. Conference on Computer Technologies in Physical and Engineering Applications, ICCTPEA 20Ц, Proceedings . St Petersburg, 2014, p. 96, no. 6893300.
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. St Petersburg, ‘Dom Fedorovoi G. V.’ Press, 2015, pp. 382-385, no. 7342169.
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. St Petersburg, ‘Dom Fedorovoi G. V.’ Press, 2015, pp. 414-416, no. 7342156.
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., 2017, vol. 22, no. 2, pp. 349-361.
Matrosov A.V., Shirunov G.N. Analyzing thick layered plates under their own weight by the method of initial functions. Materials Physics and Mechanics, 2017, vol. 31, no. 1-2, pp. 36-39.
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., 2017, no. 20, pp. 151-159.
Goloskokov D.P., Matrosov A.V. Approximate analytical approach in analyzing an orthotropic rectangular plate with a crack. Materials Physics and Mechanics, 2018, vol. 36, no. 1, pp. 137-141.
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, 2018, vol. 69, p. 121.
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.
Owczarek S., Owczarek M. Heat transport analysis in rectangular shields using the Laplace and Poisson equations. Energies, 2020, vol. 13, p. 1714.
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, 2020, vol. 71, no. 1, p. 24.
Lamé G. Leçons sur la théorie mathématique de l'élasticité des corps solides [Lectures on the mathematical theory of elasticity of solids]. Paris, Bachelier Publ., 1852, 335 p.
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, pp. 37-51.
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Опубликован
29.09.2022
Как цитировать
Алцыбеев, Г. О., Голоскоков, Д. П., & Матросов, А. В. (2022). Метод суперпозиции в задаче изгиба защемленной по контуру тонкой изотропной пластинки. Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления, 18(3), 347–364. https://doi.org/10.21638/11701/spbu10.2022.305
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Информатика
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Статьи журнала «Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления» находятся в открытом доступе и распространяются в соответствии с условиями Лицензионного Договора с Санкт-Петербургским государственным университетом, который бесплатно предоставляет авторам неограниченное распространение и самостоятельное архивирование.
