Paper Submission: 27 November 2020
Author Notification: 7 to 10 days
Journal Publication: November 2020
Igor Bokov , Elena Strelnikova
A fundamental solution of the elasticity theory equations for transversely isotropic plates was obtained. To construct the twodimensional elasticity theory equations, the approximation method of displacements, stresses and strains using Fourier series by Legendre polynomials on the transverse coordinate was used. This approach has allowed to take into account the transverse shear and normal stresses. Since the classical Kirchhoff-Love theory does not consider these stresses, the research based on the refined theories of the stress-strain state of transversely isotropic plates under concentrated force actions is an urgent scientific and technical problem. The fundamental solution of these equations was found using the two-dimensional Fourier integral transform and the generalization method, built with a special G-function. The method allows to reduce the system of resolvent differential static equations of flat plates and shells to a system of algebraic equations. Then the inverse Fourier transform restores fundamental solution. Numerical studies that demonstrate behavior patterns of the stress-strain state components depending on the elastic constants of transversely isotropic material were performed. The approach demonstrates the development of the refined theory of plates and shells based on the three-dimensional elasticity theory.
 Ambartsumyan, S. A. 1974. The general theory of anisotropic shells. M.: Science, 446.
 Goldenveizer, A. L. 1944. Research of the stress condition of the spherical shell. Applied Mathematics and Mechanics, Vol. 6, 441–467.
 Flügge, W. 1966. Concentrated forces on shells. Proceedings of Xlth Internat. Congress of Applied Mechanics. Munich: Springer Verlag, 270–276.
 Łukasiewicz, S. A. 1976. Introduction of concentrated loads in plates and shells. Progress in Aerospace Sciences, Vol. 17, 109–146.
 Darevskii, V. M. 1966. Contact problems of the theory of shells (the action of local loads on the shell). Proceedings of VI All-Union Conference on the theory of shells and plates. M.: Nauka, 927–933.
 Zhigalko, Y. P. 1966. Calculation of thin elastic cylindrical shells at local loads (literature review, methods and results). Investigations in the theory of plates and shells, Vol. 4, 3–41.
 Goldenveizer, A. L. 1954. On the question of the calculation of shells on concentrated forces. Applied Mathematics and Mechanics, Vol. 8, № 2, 181–186.
 Vekua, I. N. 1970. Variational principles of the theory of shells. Publisher of Tbilisi, 300.
 Reissner, E. 1985. Reflections on the theory of elastic plates. Applied Mechanics Reviews, Vol. 38, № 11, 1453–1464.
 Kiel, N. A. 1973. On the action of local loads on the shell. Proceedings of the universities. Construction and architecture, № 3, 43–46.
 Ganowicz, R. 1973. On fundamental singularity in the theory of shallow cylindrical shells. Arch. mech. Stosowanej, Vol. 25, № 6, 985–992.
 Jahanshahi, A. 1963. Force Singularities of Shallow Cylindrical Shells. Journal of Applied Mechanics. ASME International, Vol. 30, № 3, 342–345.
 Velichko, P. M., Khizhnyak, V. K., Shevchenko, V. P. 1971. Deformation of shells of positive curvature in the concentrated actions. Stress concentration, Vol. 3, 31.
 Velichko, P. M., Khizhnyak, V. K., Shevchenko, V. P. 1968. Research local stresses in plates and shells under concentrated load. Proceedings of III All-Union Congress on theor. mechanics. M.: Publishing House of the USSR Academy of Sciences, 67.
 Velichko, P. M., Khizhnyak, V. K., Shevchenko, V. P. 1975. Local stresses in the shells of positive, neutral and negative curvature. Proceedings of X All-Union Conf. on the theory of shells and plates. Vol. 1. Tbilisi: Publishing house Mitsniereba, 31–41.
 Lukasiewicz, S. 1979. Local loads in plates and shells. Alpen aan den Rijn, Sijthoff and Noordhoff. Warszawa: PWM, 569.
 Sanders, J. L. 1970. Singular Solutions to the Shallow Shell Equations. Journal of Applied Mechanics, Vol. 37, № 2, 361–366.
 Simmonds, J. G., Bradley, M. R. 1976. The Fundamental Solution for a Shallow Shell With an Arbitrary Quadratic Midsurface. Journal of Applied Mechanics, Vol. 43, № 2, 286–290.
 Vladimirov, V. S. 1976. Generalized functions in mathematical physics. M.: Science, 280.
 Schwartz, L. 1965. Mathematical Methods for the Physical Sciences. M.: World, 412.
 Edwards, R. 1969. Function analysis. M.: World, 1071.
 Vekua, I. N. 1955. A method for calculating prismatic shells. Pr. Tbilis. Mat. Inst., № 21, 191–253.
 Vekua, I. N. 1965. The theory of thin shallow shells of variable thickness. Tbilisi: Metsniereba, 103.
 Cicala, P. 1959. Sulla teoria elastica della plate sottile. Gorn. Genio civile, Vol. 97, № 4, 238–256.
 Poniatowski, V. V. 1962. On the theory of plates of medium thickness. Applied Mathematics and Mechanics, Vol. 26, № 2, 335–341.
 Poniatowski, V. V. 1964. On the theory of bending of anisotropic plates. Applied Mathematics and Mechanics, Vol. 28, № 6, 1033–1039.
 Bokov, I. P., Strelnikova, E. A. 2015. Construction of fundamental solutions static equation isotropic plates of medium thickness. Eastern-European Journal of Enterprise Technologies, Vol. 4, № 76, 27–34. DOI: 10.15587/1729- 4061.2015.47232
 Han, H. 1988. Elasticity Theory: Fundamentals of linear theory and its application. M.: World, 344.
 Pelekh, B. L., Lazko, V. A. 1982. Laminated anisotropic plates and shells with stress concentrators. K.: Science thought, 296.
 Sneddon, I. 1955. Fourier transform. M.: Foreign literature publishing house, 668.
 Khizhnyak, V. K., Shevchenko, V. P. 1980. Mixed problem in the theory of plates and shells. DonGU: Donetsk, 128.
 Dementyev, A. D., Nazarov, L. A., Nazarova, L. A. 2002. Applications of the theory of elasticity. Novosibirsk, 224.
A. N. Podgorny Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Kharkov, Ukraine,
No. of Downloads: 124 | No. of Views: 211