## 2 Literature Review of the Methods for Solving Plate Bending Problems  
Exact solution of the boundary problems of the plate theory are possible only for simplest cases. So far, many methods of approximate solution of plates have been developed. All of them can be divided into two basic groups: analytical [1, 2, 6, 7] and numerical ones [2, 9, 10, 15]. Their analysis is given in fundamental monograph by editor [PERSON] [21]. Formulation of the problem of the plate theory usually is performed one of the three techniques: system of equilibrium differential equations, variation approach and method of integral equations. Solution of differential equations of equilibrium causes development of analytical methods. Numerical methods are less exact but they allow the solving of arbitrary configuration plates [2, 10] at the expense of a significant increasing the number of unknowns. It increases the time for calculations and leads to accumulation of computational errors. Analytical methods include: Ritz-Timoshenko [8], Bubnov-Galerkin [22] and other [23] variational methods, [PERSON]'s [24] method of single and double trigonometric series, [PERSON]'s method of functional series [1, 8, 25] and spline-function method [26, 27] which is widely used for solution of the plate with discontinuous configuration. [PERSON] method is based on the principle of minimum of the structure potential energy (Principle of Virtual Works). \[\delta\Pi=\delta(V-A)=0, \tag{3}\]  
Here \(\Pi\) is potential energy of the structure; \(V\)--specific potential energy; \(A\) is work of outer load applied to plate structure; \(\delta\)--Lagrange variation symbol. Minimalization of error \(R\) of approximate solution of Equation (4)  
\[D\
abla^{2}\
abla^{2}w-q=R\
eq 0, \tag{4}\]  
with respect to test functions is object of Bubnov-Galerkin method. [PERSON]'s method only deals with rectangular plate free supported at the contour and arbitrary loaded. [PERSON]'s method allows the solving of the plates with two opposite edges free supported. Other edges can be arbitrary fixed. Numerical methods includes: method of boundary collocation [9, 28] and Trefftz collocation method [13, 29], finite element method (FEM) [14, 30, 31, 32, 33, 34], finite boundary method (FBM) [15, 17, 18] and method of time-space elements (MTSE) [35]. Method of boundary collocation demands satisfaction of integral equations and Trefftz collocation method demands to perform equilibrium equation. In both methods, the boundary conditions are performed at separate points called collocation points at the edges of the plate. Trefftz method does not contain singular integrals and it is more exact in comparison with other methods. Variant of collocation method is worked in paper [36] for a plate inscribed in rectangular contour. Collocation points are Chebyshev nodes. The authors argue that with the number of computational nodes increasing, the accuracy is not improved because of the round-off error effect. Finite element method (FEM) follows from variation formulation and finite boundary method (FBM) does from boundary integral equation. In the paper [34] four nodal plate element based on Reissner-Mindlin plate theory is worked. Mesh refinement increase the accuracy of the solution. Finite element modeling of stiffened and unstiffened orthotropic plates is presented in the paper [37]. In recent years, besides numerical and analytical methods, many analytical-numerical ones appear. There are: boundary integral method (BIM) or boundary integral equation method [17], global element method (GEM) [38] and global-local finite element method (GLFEM) [11], spline finite element method [39], scaled boundary finite element method [40], least squares collocation method [41], symplectic superposition method [42] and matrix method [43]. BIM is part of general method BEM associated with calculation singular integrals containing in BEM. According to global element method (GEM) [38] considered area is divided into some simple subareas (greater than in FEM).