## 5 Discussion  
Solution of equilibrium equation consists of two parts: general and particular solution which are independent of each other. The general solution identically satisfies differential equation and, in contrast to FEM, satisfies kinematic, static and mixed boundary conditions. The general solution obtained for a rectangular plate is correct for any plate inscribed in a rectangular contour. Particular solution satisfies non-uniform equation of equilibrium using boundary collocation method. It also allows the consideration of plates with arbitrary configurations. Since symmetrical problems are considered, deflections and bending moments are symmetrical functions of variables \((x_{1},x_{2})\) but slopes are nonsymmetrical ones.  
Advantages of suggested method:  
* Simplicity of structure modeling,
* Possibility to define static, kinematic and mixed boundary conditions,
* High accuracy and efficiency of calculation,
* Meshless approach for solving the problem.  
Neither a surface mesh as in FEM nor a boundary mesh as in BEM are required. Introduced nodes are not vertex of finite elements. Surface and edge nodes are generated automatically. Contrary to FEM [78], the presented method is based on continuum model of material. For this reason, operations such as structure discretization and finite element aggregation are unnecessary. Singular plate can be considered to be one global or macro plate element.  
Disadvantages of presented method:  
* It cannot be used directly to solve fracture mechanics problems,
* It is worked for simply connected structures; for each additional contour, it is necessary to introduce the new block of basic functions,
* It gives good results for regular plates; plates with irregular geometry are solved with less accuracy,
* Unlike FEM, system matrix is consisted from zero and non-zero blocks.  
Tables 1-6 contain the minimal and maximal values of static and kinematic magnitudes of the plate obtained by author's program based on proposed method written in Python programming language, using finite element method in ABAQUS package, and based on the formulae for a simply supported plate according to [PERSON]'s method, included in [PERSON]'s monograph [7].  
It is seen that analytical and numerical results of kinematic magnitudes coincide. Maximal values of the bending moments are close but they do not coincide. Twisting moments are differ significantly. Numerical values of bending moments are not zero along the edges of the plate. It follows that static boundary conditions are not met in ABAQUS package.  
The reason for this discrepancy is as follows:  
* Presented method is based on Kirchhoff theory of the plates which is deformation theory of zeroth-order where deformations of a cross-section are omitted. Instead generalized shearing force as a sum of shearing forces and derivative of twisting moments is introduced.
* ABAQUS package is based on Mindlin plate theory which is deformation theory of first-order where deformations of a cross-section are taken into account. Twisting moments and shearing forces are independent ones. It is reason that obtained values of bending and twisting moments are different.
* Relatively small number of finite elements. The Student Edition of the ABAQUS package is restricted to 1000 nodes.  
In analytical approach kinematic and static boundary conditions are performed directly at the separate points at the plate contour. Numerical methods perfectly satisfy kinematic continuity equations and kinematic boundary conditions only at the separate nodes. Equations of equilibrium and static boundary conditions are fulfilled approximately in the whole area of the plate using Principle of Virtual Work. It can be reason that kinematic boundary conditions are satisfied very well but static ones are performed with less accuracy in Finite Element Method.