539.3

loaded plate, three-dimensional stressed state, stress tensor, Lamé equations.

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

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    2018. P. 33–39. URL: http://elartu.tntu.edu.ua/handle/lib/24882.
12. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity. Int. Appl. Mech. Vol. 45. No. 7. 2009. P. 730–741. URL: https://doi.org/10.1007/s10778-009-0225-4.

1. Sadd M. H. Elasticity. Theory, applications, and numerics, Amsterdam: Academic Press: 2014. 600 p.
2. Donell L. H. Beams, plates and shells. New York: McGraw-Hill: 1976. 568 p.
3. Timoshenko S. P., Woinowsky-Krieger S. Theory of Plates and Shells, second edition, New York: McGraw-Hill, 1959.595 p.
4. Shaldyrvan V. A. Some results and problems in the three-dimensional theory of plates (Review), Int. Appl. Mech. Vol. 43. No. 2. 2007. P. 160–181.
5. Meleshko V. V. Selected topics in the history of the twodimensional biharmonic problem. Applied Mechanics Reviews. 56 (1). 2003. P. 33–85.
6. Grinchenko V. T. The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems, The Journal of Engineering Mathematics, 46, No. 3. 2003. P. 281–297.
7. Wang W., Shi M. X. Thick plate theory based on general solutions of elasticity, Acta Mechanica. 123. 1997. P. 27–36.
8. Kobayashi H. A. Survey of Books and Monographs on Plates, Mem. Fac. Eng., Osaka City Univ. 38. 1997. P. 73–98.
9. Revenko V. P., Revenko A. V. Determination of Plane Stress-Strain States of the Plates on the Basis of the Three-Dimensional Theory of Elasticity, Materials Science. Vol. 52. No. 6. 2017. P. 811–818. URL: https://doi.org/10.1007/s11003-017-0025-7.
10. Revenko V .P. Reduction of a three-dimensional problem of the theory of bending of thick plates to the solution of two two-dimensional problems. Materials Science. 51. No. 6. 2015. P. 785–792. URL: https://doi.org/10.1007/s11003-016-9903-7.
11. Revenko V. P. Development of two-dimensional theory of thick plates bending on the basis of general solution of Lamé equations, Scientific journal of the Ternopil National Technical University. No. 1 (89).
    2018. P. 33–39. URL: http://elartu.tntu.edu.ua/handle/lib/24882.
12. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity. Int. Appl. Mech. Vol. 45. No. 7. 2009. P. 730–741. URL: https://doi.org/10.1007/s10778-009-0225-4.