Objective: to investigate the stress-strain state of a thin, homogeneous, isotropic, semi-infinite rectangular plate that is clamped along two parallel edges, with the third edge considered either fixed or free. The study also aims to demonstrate that the derived semi-infinite plate solution can be applied to assess stress-strain fields in finite-size rectangular plates. Methods: the issue is tackled using L. V. Kantorovich's method with trial (basis) functions constructed from Jacobi polynomials that satisfy the boundary conditions on the parallel clamped edges of the semi-infinite plate. These basis functions exhibit quasi-orthogonality properties for their first and second derivatives, which permits separation of the coupled ordinary differential equations produced by Kantorovich's procedure into independent ordinary differential equations that admit analytical solutions. Results: an approximate analytical solution for the bending of the semi-infinite rectangular plate has been obtained various clamping configurations of the plate. The derived solution demonstrates rapid convergence for both deflections and bending moments. It is shown that the stress-strain state of finite-size rectangular plates can be effectively investigated by applying the solution of the semi-infinite plate bending. To facilitate this approach, solutions for several boundary-condition configurations of the finite-size rectangular platebending issues have been developed. Namely, the Bubnov-Galerkin solution was used for a plate clamped along its contour, and M. Levy's solution was employed for a plate clamped along two parallel edges and pinned along the remaining two. Practical importance: the algorithm proposed herein for solving semiinfinite plate bending issues is recommended for practical application in determining the stress-strain state of finite-size rectangular plates.
Semi-infinite rectangular plate, plate bending, L. V. Kantorovich method, Bubnov — Galerkin method, M. Levy solution, Jacobi polynomials
1. Timoshenko S. P., Voynovskiy-Kriger S. Plastinki i obolochki. M.: Nauka, 1966. 636 s.
2. Gopalachariulu. Zaschemlennye polubeskonechnye plastiny // Prikladnaya mehanika. 1966. № 1. S. 195–197.
3. Goloskokov P. G. Izgib pryamougol'noy plity, zhestko zadelannoy po dvum protivopolozhnym storonam // Izv. vuzov. Ser: Stroitel'stvo i arhitektura. 1959. № 11–12. S. 25–34.
4. Kantorovich L. V. Ispol'zovanie idei metoda Galerkina v metode privedeniya k obyknovennym differencial'nym uravneniyam // PMM. 1942. Vyp. 6. S. 31–40.
5. Aver'yanova G. V., Goloskokov D. P. Raschet tonkih plastin pri pomoschi polinomov special'nogo vida // Exponenta Pro. Matematika v prilozheniyah. 2004. № 1. S. 70–76.
6. Goloskokov D. P., Goloskokov P. G. Metod polinomov v zadachah teorii tonkih plit. SPb.: SPGUVK, 2008. 254 s.
7. Goloskokov D. P., Matrosov A. V., Olemskoy I. V. Izgib zaschemlennoy tonkoy izotropnoy plastiny metodom Kantorovicha s ispol'zovaniem special'nyh polinomov // Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Processy upravleniya. 2023. T. 19, vyp. 4. S. 423–442.
8. Goloskokov D. P., Matrosov A. V. Bending of clamped orthotropic thin plates: polynomial solution // Mathematics and Mechanics of Solids. 2022. Vol. 27, no 11. Pp. 2498–2509.
9. Sege G. Ortogonal'nye mnogochleny. M.: Fiz-matlit, 1962.
