Rectangular plate calculation algorithm for static loads with geometric nonlinearity

. The aim of this study is to develop an algorithm for calculating rectangular plates for statistical loads taking into account geometrical nonlinearity on the basis of difference equations of the method of sequential approximations taking into account full and partial contact with elastic base. For our study, we used methods: finite elements method and method of sequential approximations, and theory of plate calculation considering large deflections. Results: a method, algorithm, and calculation algorithm for rectangular plates in the geometrically nonlinear formulation using difference equations of the method of sequential approximations (SEA) taking into account full and partial contact with the elastic base have been developed. Conclusion: An algorithm has been developed for the computation of rectangular plates in geometrically nonlinear formulation using difference equations of the method of sequential approximations. It is recommended to use generalized difference equations of finite difference method when calculating rectangular plates for the action of piecewise distributed transverse loads without taking into consideration interaction with elastic base. Having more compact form of recording, the solution obtained by using them has comparable accuracy as the version using SEA difference equations. Key


Introduction
In the vast majority of cases the solution of practical problems in nonlinear structural mechanics is carried out using computational complexes based on the finite element method (FEM). To evaluate the accuracy and reliability of the results obtained in this way, it is necessary to develop other calculation methods. In this paper to solve problems in the calculation of rectangular plates in the geometrically nonlinear formulation, the method of sequential approximations (SEA), which has proved its worth in the calculation of building structures, is used.
In the framework of the present study, we aim to develop a numerical algorithm for computing rectangular plates for static loads in a geometrically non-linear formulation, including consideration of contact with an elastic base, by using difference equations of SEA.

Methods
The finite-element method and the method of sequential approximations; the theory of plate calculations with large deflections taken into account.

Results and discussion
Let us write down the system of nonlinear differential equations of the theory of flexible plates, known as von Karman equations, according to [12]: Here Н determines plate thickness; is cylindrical rigidity; μ is Poisson's ratio; q defines shear load intensity distributed according to a given law; Еmaterial modulus of elasticity; х, у are the coordinates.
Ф is stress function; w is deflection (vertical movement along the axis z); We apply the system of differential equations (1) and (2) to the calculation of flexible plates. The components w and Ф represent the main unknowns of the system. Through w, we define the bending moments in the x, y direction; the torsional moment and the shear forces by the following formulas [1]: Bending moments in x-and y-axis directions: The torque can be defined as: Transverse forces in the x and y axes: Expressions for determining the shear forces at the free edge are written down as follows [2] Let us multiply (2) by a 2 and write it down with (7) and (8). Then (2) will take the form: Here k = Let us write the equation (9) as follows: Similarly, write down (10): Here: Let us introduce notations to simplify the notation of the right-hand side of equations (15) and (16): Now, the system of two fourth-order differential equations (9) and (10) will be presented as a system of four second-order differential equations. For this purpose, let us write equation (15) with consideration of (21) and (23), equation (16) with consideration of (22), and equations (11) and (12) will be given here without changes. We obtain: The algorithm for solving the system of equations (24)-(27) is as follows. At the first stage we solve the problem in linear formulation. From solution (24) at =0 and, as a consequence of (23), g = q ̅, we find the values of the function т. Substituting the found т in (2.5) we calculate the values w. Using the known deflection values, numerically determine the derivatives, i.e., using (13) find the values of l, п and t. We calculate α according to (22). Having determined the right-hand side, solve equation (26). From its solution we find the values of the function f. This will allow the solution to (27) and determine ψ. Calculating by (14) the second partial derivatives of ψ, we find b, с and d. By determining the parameter  from (21) according to (23) we can calculate a refined value g, which is the right-hand side of equation (24). Thus, the loop is closed. Starting from the second iteration, the problem is solved in a non-linear formulation. The search for a numerical solution continues until a predetermined accuracy is achieved. Figure 1 shows a block diagram illustrating the proposed calculation algorithm: A difference approximation of a second-order partial differential equation of the general form is given in [5]: Differential equations (24) -(27) are special cases of (28). Therefore, we use (2.5.13) of [3], which is a difference analogue of differential equation (28), for approximations of differential equations like (24) -(27).
Rectangular domain of integration Ω(ξ, η) we divide it into sub-areas (I-IV) (Figure 2), i.e. into separate elements that have a common point i,j. To approximate differential equations of type (24) by difference equations of the SEA at interior points i,j, should be taken as (28) α = γ = 1, δ = β = σ = 0, α i = γ i = δ i = β i = σ i = 0 substitutedω for т, р for g. We assume that the desired functions are continuous within an element. The desired functions themselves, their first derivatives and the right-hand sides of the original differential equations can have finite discontinuities at the boundaries of E3S Web of Conferences 389, 01005 (2023) https://doi.org/10.1051/e3sconf/202338901005 UESF-2023 the element. Let us assume a square grid τ 1 = τ 2 = h 1 = h 2 = h, and write an approximation of the differential equation (24) using the SEA difference equations: In this equation: It is not enough to approximate differential equations (24)-(27) at regular points in order to construct a computational algorithm for flexible plates. It is necessary to consider the boundary conditions. First of all, by the example of square plate ( Figure 3) let us introduce the notation used in this paper: edge OA is rigid embedment, edge AB is hinged fixed support, edge BC is hinged fixed support and edge CO is free from fixation.
Equations (46) and (47) for the edge η = nh are written in a mirror image, ( ) and ̅ are reversed by reversing the sign. For the upper and lower edges we write the equations with substitution η, i,j respectively by ξ, j, i. For the angle i,j with two intersecting free edges is written the one-dimensional difference equation of the SEA [3] in the case of = = 1, = = = 0, = = = = 0 and 1 = 2 = ℎ 1 = ℎ 2 = ℎ: Here with ̅ = 0. After calculating the values т, w ̅ at all points in the field and on the contour from the joint solution of the compiled equations to determine m (ξ) , m (η) it is necessary to determine w ̅ ij ξξ according to the following work formula [4]: Next, we determine the values w ̅ ηη by (11). For a plate with fixed supports, write such conditions only substitute Analyzing the graphs shown in Figure 4, we see that for a given stiffness characteristics of the plate, its bending operation is satisfactorily described by the linear theory up to reaching deflections of 0.005 of the side length. Within these limits the deflections according to the linear and geometrically nonlinear theories are almost the same.

Conclusions
The original system of two differential inhomogeneous nonlinear equations of the fourth order is reduced to four single-type differential equations of the second order.
To solve the resulting system of second order partial differential equations we employ the method of sequential approximations (SEA) that has proved to be good for solving Poisson type equations.
Solving problems of rectangular plates under static loads considering geometric nonlinearity on the base of SEA difference equations comes down to joint solution of systems of algebraic equations for each regular point of the grid considering boundary conditions according to solution algorithm ( Figure 1).
The difference equations of SEA allow calculating of flexible plates with different types of boundary conditions for different loads, including discontinuous ones.