## 1 Introduction  
Thin plates are widely used as elements of building and engineering structures. The Kirchhoff plate theory is the basis of engineering design and calculation of such elements. It is an approximate zeroth-order shear deformation plate theory. Discrepancy between order of basic differential equation and number of boundary conditions is the main disadvantage of the Kirchhoff theory. There are considered only three from six physical equations. Shearing and normal transverse stresses are not taken into account. They are determined from equation of equilibrium. The higher-order shear deformation plate theories are free from this discrepancy. According to the Kirchhoff plate theory solution of thin isotropic plates is equivalent to solution of fourth-order partial differential equation [1, 2]  
\[\
abla^{2}\
abla^{2}w=\frac{q}{D}, \tag{1}\]  
for given boundary conditions. In the above equation \(w\) is the deflection of the plate, and \(D\) is its flexural rigidity. Function \(q(x_{1},x_{2})\) describes distribution of load on upper surface of the plate. Uniform equation corresponds to (1) has the form  
\[\
abla^{2}\
abla^{2}w=0. \tag{2}\]  
In many cases, Kirchhoff theory gives satisfactory results and sufficient accuracy for practical purpose and correctly describes behavior of the structure. It is based on the linear distribution of the displacements through the plate thickness. As a result, parabolic distributions of the shearing stresses are obtained. For comparison [PERSON] plate theory [3] also takes the same distribution of displacements but gives incorrect uniform distribution of the shearing stresses through the plate thickness. Higher-order theories like [PERSON], [PERSON], [PERSON] can represent the kinematics better. However, they involve higher-order stress resultants that are difficult to interpret physically. Therefore, such theories should be used only when necessary [4]. In addition to its inherent simplicity and low computational cost, the first-order plate theory often provides a sufficiently accurate description of the global response for thin to moderately thick plates. Higher-order theories provide a small increase in accuracy relative to the first-order plate theory solution, at the expense of a significant increase in computational effort [5]. Therefore, the choice of the Kirchhoff theory to analysis of thin-walled engineering structures is still justified. Let us note that exact solution of boundary problems of theory of the plates is possible only for simplest cases. At present, there are many various methods to solve thin plates, which can be divided into two basic groups: analytical [2; 6; 7; 8] and numerical ones [6; 9; 10; 11]. Analytical methods are more exact than numerical methods but they allow the solving of boundary problems of plates with a small number of unknown parameters and constrained by canonical contour as rectangle, square, triangle, circle or ellipse. Numerical methods are less exact but they allow the solving of arbitrary configuration plates [12] at the expense of a significant increasing of unknown parameters. It increases the time for calculations and leads to the accumulation of computational errors. Numerical methods are currently reaching their limits. Among numerical methods, the Finite Element Method (FEM) [13; 14] and Finite Boundary Method (FBM) [15; 16; 17; 18] are the most common. The finite element method allows us to approximately solve plate of arbitrary configuration, where the analytical solution is impossible. Research on this method can be conventionally divided into: theoretical (related the construction of finite element model) and applied (related to the use of the method to solve complicated technical problems). In recent years, many methods which are free from disadvantages of standard numerical ones have been suggested. They create separate group called analytical-numerical methods. According to these methods part of equations are performed in analytical manner and others with help of numerical procedures.