Bending of flexible round plates

In this paper, schemes for constructing solutions to boundary value problems for static calculation of flexible circular plates with the nonlinear theory of Lyava and Volmyr are presented. From the equations of the equilibrium system of the plates, given in curvilinear coordinates, the system of equilibrium equations for flexible round plates is obtained. Substituting the expressions for the efforts and shearing forces and introducing dimensionless quantities, we obtain a system of quasilinear quantities in displacements. To develop an automated system for static calculation of flexible round plates, we use central finite-difference schemes that approximate derivatives with second-order accuracy, we obtain a system of quasilinear algebraic equations. To test the constructed automatic system for static calculation, the difference equations are reduced to vector form. An implicit iterative process combined with the Gaussian elimination method is applied to the solution of the system of equations. When calculating iterative processes, it continues until the above conditions are met. After determining the required functions by the finite difference method, we calculate the calculated values. Using the obtained numerical results, we will construct their graphs.


Introduction
From the literature reviewed, it can be seen that most of the problems on flexible circular plates are solved in the Fepple-Karmana formulation, which is a special case of Lyava [1]. The constructed algorithms are not economical about their implementation on the computer. Therefore, the construction of an automated system for the complete calculation of flexible round plates with a given degree of accuracy becomes an urgent issue.
The problem of creating an automated system was first posed in the monograph by V.K. Kabulov [2]. The algorithmization problem is solved in four stages. At the first stage, depending on the geometric characteristics of the object and the physical properties of the material, the design scheme of this model is selected. The second stage is associated with the derivation of the original differential equations in the corresponding boundary and initial conditions. The choice of a computational algorithm and the numerical solution of the obtained solutions constitutes the third stage of research. The fourth stage ends with the analysis of the obtained numerical results described by the stress-strain state of the structure under consideration.
In particular, in [9], the algorithms for calculating specific structures -beams, plates and conical shells -are considered.
This work in the formulation of Love formulated boundary value problems of flexible circular plates in displacements. The corresponding system of two nonlinear partial differential equations is reduced to a system of two quasilinear differential equations.
The solution of a system of difference equations with different boundary conditions is reduced to the solution of systems of quasilinear equations.

Methods
The choice of a computational algorithm and the numerical solution of the formulated boundary value problems constitute the main stage of algorithmization This paper addresses the following issues: 1) Construction of a unified computational scheme for solving boundary value problems of static calculation of flexible round plates using the nonlinear Lyava theory; 2) Development of an automated system for static calculation of flexible round plates; 3) Approbation of the built automated system; 4) Study of the nature of the convergence of the applied numerical methods. Let us use the well-known equations of equilibrium of the plate in an arbitrary curvilinear coordinate system [1, 3,18,19]. Equations of equilibrium of flexible circular plates under the action of axisymmetric loads are derived from this system.
we obtain the equilibrium equations in displacements where    (7) is solved at 0 1 r   -for solid and at 0 1 r r   -for an annular circular plate and with the following boundary conditions Equations of equilibrium of flexible round plates (7) for the given boundary conditions can also be solved using the method of meshes [4,5,14].
Using the central difference formulas that approximate the derivatives with a second order accuracy [4,6], instead of equations (7), we obtain the following system of quasilinear algebraic equations [7,9]: Let us consider some different boundary conditions for flexible circular plates under uniformly distributed loads.
For a solid round plate hinged on the contour We get from the first, fourth and fifth (9) (10) Applying the central difference formulas with the second order of approximation [5] to the second, third, and sixth conditions (9), we find [6,15,16,20].
In vector form, conditions (10) and (11) are written as follows: and 1 1 0, Substituting (12) and (13) To solve the system of quasilinear algebraic equations (14), an implicit iterative process is applied in combination with the Gaussian elimination method, whose equations have the following form [4,6]: where, The iterative process in calculating (19) continues until the condition where  is the accuracy of the solution.
Using the forward sweep formulas (16)     The difference in the values of the deflection and moment of the plates in the nonlinear formulation of the problem with respect to the linear and total stresses relative to the bending in the linear formulation in the center is, respectively, at 12 II w -according to Ueyav.