Method of finite bodies for determination of the plane stressed state of rectangular plates with a rectangular hole


  • V. P. Revenko Pistryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 3-b, Naukova str., L'viv, 79060



The paper is devoted to the determination of the stress-deformed state of structurally heterogeneous bearing rectangular plates with a rectangular hole. The new analytical-numerical method (finite bodies) was used, to find the stress state of the plate with a hole. The method of finite bodies uses the conditional partition of the doubly-connected surface of the plate into simpler connected rectangular parts. On the lines of conditional contact, the conditions of ideal contact are taken into account, which ensure the equality of stresses, deformations and displacements. The perturbed stressed state, which is presented in the form of a series of functions, which is rapidly intercepted at a distance from the outline of the hole, is considered. A finite sum of solutions of a plane problem is used and the stress state of a perturbed state is given as a sum of a series for nonorthogonal functions. The components of vector of displacements and stresses are written. The determination of the coefficients of the sum of a series is based on the proposed method of satisfying all boundary conditions and the conditions of ideal contact to find the minimum of a generalized quadratic form. The numerical criterion for the convergence of the method is theoretically established. It is shown that the accuracy of satisfaction of boundary conditions and conditions of ideal contact is estimated by one number – the minimum of a generalized quadratic form.

Key words: plate, rectangular hole, method finite bodies.

Pages of the article in the issue: 170-173

Language of the article: Ukrainian


MUSHELISHVILI, N. (1966) Nekotoryje osnovnyje zadachi matematizeskoj theorii uprygosti. Moskva: Nauka.

SAVY’N, G. (1968) Raspredelenie napryazhenij okolo otverstiy. Kiev: Naukova dumka.

REVENKO, V.P. and BAKULIN, V.N. (2016) Analytical and numerical method of finite bodies for calculation of cylindrical orthotropic shell with rectangular hole. Russian Mathematics. No 6. p. 1–11.

REVENKO, V.P. and REVENKO, A.V. (2014) Determination of Plane Stress-Strain States of the Plates on the Basis of the Three-Dimensional Theory of Elasticity. Materials Science. (52)6. p. 811-818.

REVENKO, V.P. (2008) Application of the least-squares method to the analysis of displacements and stresses in the twodimensional problem. Materials Science. (44)4. p. 482–488.

MELESHKO, V. V. (2003) Selected topics in history of the two-dimensional biharmonic problem. Appl. Mech. Rev. (56)1. p. 33–85.


How to Cite

Revenko, V. P. (2019). Method of finite bodies for determination of the plane stressed state of rectangular plates with a rectangular hole. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, (1), 170–173.



Differential equations, mathematical physics and mechanics