Estimation of the stress-strain state of orthotropic plates on elastic foundation using the bimoment theory

. The study is devoted to the development of the bimoment theory and the method for calculating thick plates in the framework of the three-dimensional theory of elasticity. The basic relations and equations of motion of the plate presented are constructed with respect to the forces, moments and bimoments arising from the nonlinearity of the law of distribution of displacements and stresses over the plate thickness. Bending and vibrations of isotropic and orthotropic plates are considered an example of calculations. The resulting solution showed the efficiency and accuracy of the proposed bimoment theory in assessing the stress-strain state of thick plates.


Introduction
The calculation of plates and shells occupies a special place in the study of structural elements. The main provisions of the general methodology for constructing the classical and refined theory of plates and shells are built on the basis of a number of simplifying hypotheses. The problem of bending, stability and oscillation of orthotropic cantilevered plates has not been sufficiently studied due to the complexity of the main differential equation of the problem and boundary conditions. It requires the development of reliable numerical-analytical methods for its solution. We consider the plate material to be ideally elastic, and then there is an infinite number of critical loads that change the shape of the plate equilibrium. In [1], we study the natural oscillations of a rectangular plate, two adjacent edges of which are fixed, and the other two are free; it is an element of many building structures. The deflection function is chosen as a sum of two hyperbolic trigonometric series. Both series obey the main equation of free vibration. This eigenvalue problem is similar to the problem of determining the frequency spectrum of free vibrations of a plate [2][3][4]. The study in [4] considered stability problems for anisotropic plates and shells and the methods for solution applicable to cantilevered plates. The dynamic stability of round three-layer plates made of viscoelastic composite materials was studied in [5,6].
In [7,8], an orthotropic viscoelastic plate of arbitrarily varying thickness is presented. The plate was subjected to dynamic periodic loading. Within the framework of the Kirchhoff-Love hypothesis, a mathematical model was constructed in a geometrically nonlinear statement, taking into account the shear forces of inertia. The Bubnov-Galerkin method based on polynomial approximation of deflection and displacement was used. The problem was reduced to solving systems of nonlinear integro-differential equations. The solution to the system was obtained for an arbitrarily varying plate thickness. For a weakly singular Koltunov-Rzhanitsyn kernel with variable coefficients, the resulting system was solved numerically based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The dynamic stability of the plate was studied depending on the geometric parameters of the plate, viscoelastic and non-homogeneous properties of the material. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel practically coincide with the results of the elastic problem. When using the Koltunov-Rzhanitsyn kernel, the differences between the problems of elasticity and viscoelasticity are significant and amount to more than 40%.
References [9][10][11][12] are devoted to the development of a mathematical model and a method for solving the problem of contact interaction of various multilayer structures with a combined base. As a result of solving specific problems, a number of new mechanical effects associated with the manifestation of internal force factors in three-layer beam systems interacting with combined bases were revealed.
The studies in [13,14] are devoted to solving problems of plate bending in the framework of the Reissner theory, in which the finite element method is used to calculate square fixed and hinged plates under a uniformly distributed load. In [15], analytical and numerical methods for calculating the edge elements of buildings in the form of axisymmetric thick plates for nuclear reactors were developed. The bending problem of a Reissner rectangular plate was solved by the iterative method [16].
Articles [17][18][19] are devoted to dynamic calculations of elements of the box-shaped structure of buildings for seismic resistance, taking into account the spatial work of boxshaped elements under the action of dynamic influences specified by the displacement of their lower part according to the sinusoidal law. The equations of motion for each of the plate and beam elements of the box structure of the building are given on the basis of the Kirchhoff-Love theory. Expressions are given for the forces, moments and stresses of plate elements that balance the movement of box elements, as well as boundary conditions and full contact conditions through displacements and force factors in the contact zones of plate and beam elements.
It should be noted that the theory of thick plates and methods for their calculation, taking into account forces, moments, and bimoments, were discussed in [20,21]. The study in [22] is devoted to the dynamic calculation of the box-shaped structure of buildings for seismic resistance, taking into account the spatial work of box-shaped elements under the action of dynamic action. The article develops a mathematical model and a numericalanalytical method for solving the problem of dynamics by the finite difference method and expanding the solution by the modes of natural vibrations in the spatial formulation of boxshaped elements under kinematic action.
In the spatial case of deformation of the plate along its thickness, nonlinear laws of distribution of displacements, deformations and stresses hold. Therefore, it is necessary to take into account all components of the stress and strain tensor: . In this case, in contrast to the conventional case of setting the problem for describing the field of spatial deformation of the plate, an account for tensile and shear forces, bending and torque E3S Web of Conferences 402, 07020 (2023) https://doi.org/10.1051/e3sconf/202340207020 TransSiberia 2023 moments is not enough; it is necessary to take into account bimoments. When constructing a general theory of plates within the framework of the three-dimensional theory of elasticity, researchers use various methods, for example, the method of hypotheses, the method of expansion of displacements in a series or the method of asymptotic solution, etc.
This article briefly presents the constitutive relations, the equations of motion, and the boundary conditions of the bimoment theory of plates developed in [22].

Formulation of the problem
Consider an orthotropic thick plate of constant thickness plan. Let us introduce the notation: -elastic moduli and -shear moduli; -Poisson's ratios of the plate material. We introduce the Cartesian coordinate system The components of the strain tensor are determined by the Cauchy relations.
The technique for constructing the bimomental theory of plates is based on the Cauchy relations, the generalized Hooke law, three-dimensional equations of the theory of elasticity and boundary conditions on the front surfaces, as well as on the expansion of displacements in an infinite Maclaurin series.
It should be noted that the expressions for forces, moments and bimoments, as well as the equations for these force factors, are constructed within the framework of the method of integrating three-dimensional equations of motion of the theory of elasticity along one of the coordinates, directed along the small dimension (along the thickness) of the plate. The method of expansion of the displacement component in one of the coordinates is used only when approximating the boundary conditions of the problem, set on the front surfaces of the plate.
The bimoment theory of plates [22] is described by two independent problems, each of which is formulated on the basis of nine equations with the corresponding boundary conditions. The first problem describes the symmetric problem of longitudinal vibrations, taking into account the transverse compression, and the second describes the asymmetric bending problem, taking into account the transverse shear of the material of the thick plate.
In the article, we consider bending-shear vibrations of a thick plate on an elastic foundation. Let us consider the problem of bending vibrations of a thick plate within the framework of the bimoment theory of thick plates [22].

Bimoment theory of thick plates
The problem of bending vibrations of a thick plate in the framework of the bimoment theory is described by equations for moments, shear forces and bimoments, as well as three equations for generalized displacements of the front surfaces of the plate. Forces, moments and bimoments are determined with respect to nine unknown functions defined by the following relations: The equations of motion in moments and forces have the form: ( Bending and torque moments are defined as: where 33 12 11 ,..., , E E E are the elastic constants determined in terms of the coefficients. The shear forces are defined as In equations (2), (3), the load terms are determined by the formulas: The bimoments 12 22 11 , P P P , generated during bending and shearing of the plate, are determined by the following formulas: The intensities of the transverse shear and normal bimoments 23   The equations for the bimoments in bending and transverse shear are obtained in the form: Equations (2), (3), (8) and (9) form a consistent system of six equations for nine unknown functions: The constructed six equations (2), (3), (8) and (9) contain nine unknown functions determined by formulas (1). As seen, three more equations are missing here. To construct these missing equations, we used the method of expansion of displacements in the Maclaurin series.
For the problem of bending vibrations, these equations can be written in the following form: Here 22 12 11, ,    are determined from Hooke's law, taking into account the conditions on the front surfaces: here . , , The system of equations (2)  Note that when constructing the equations of motion (10) and (11) . This assumption ensures the independence of the second problem of bending and vibrations of plates of the bimoment theory from the first problem which describes tension-compression with allowance for the transverse compression of the plate, which facilitates the construction of a solution to the problem. The following expressions are constructed for the contact forces that appear between the plate and the elastic foundation: where k1, k2, k3 are the coefficients of bed of elastic foundation. It is assumed that the edge of the plate 0 2  x is rigidly fixed. The remaining edges of the plate are free from supports. On the fixed edge of the plate, the displacements are zero: . The problem posed is described by the equations of motion of the bimoment theory of plates (2), (3), (8) - (11) and boundary conditions (12) - (14) and is solved by the finite difference method.
For the numerical solution to the problem posed, the method of finite differences was applied. To approximate the derivatives of displacements with respect to spatial coordinates, we use the formulas of central difference schemes. In this case, is the calculation step, M N , are the numbers of partitions, and t  is the time step. The calculation steps in terms of spatial coordinates and time are chosen as follows: ) , min( , , The maximum stresses of the plate are determined by expressions (15).

Analysis of numerical results
Calculations are made for isotropic and orthotropic square plates on an elastic foundation. Dimensionless numerical results of calculation of displacements and stresses for a square orthotropic plate SVAM 15:1 with elastic characteristics are obtained. , * 56 , 0 , * 12 , 1 , * 6 , 1 , * 6 , 4 The following conditions must be met at the free edge of the plate The stresses 12 22 11 , ,    in the plate are calculated by formulas (15 11 M also reaches the maximum and its value is -according to the bimoment theory and according to the Timoshenko theory. Here, the error is more than 22% according to Timoshenko's theory. in plan according to the bimoment theory and the Timoshenko theory.   The calculations performed according to the Timoshenko theory and the bimoment theory showed that with an increase in the thickness of the plate on an elastic foundation, the difference in the numerical results obtained by these theories increases. In addition, it was found that the stress values for orthotropic plates are much higher than for isotropic plates.
In calculations, the number of partitions into steps of difference schemes in dimensionless coordinates is taken as follows . The calculations were performed in the Delphi environment.

1.
A bimoment theory and a method were developed for estimating the stress state of thick plates on an elastic foundation based on the equations of the three-dimensional theory of elasticity.
2. The basic relations and equations of motion of the plate were constructed with respect to the forces, moments and bimoments arising from the nonlinearity of the law of distribution of displacements and stresses over the plate thickness.
3. As an example of the calculation, the bending and vibrations of isotropic and orthotropic plates on an elastic foundation were considered. 4. The effectiveness and accuracy of the proposed bimoment theory in solving applied problems of vibrations of plates on an elastic foundation was shown.
5. The results obtained for plates on an elastic foundation showed that with an increase in the thickness of the plate, the difference in the numerical results obtained by the bimoment theory and the Timoshenko theory increases. The stress values for orthotropic plates are much higher than the values for isotropic plates.