ANALYSIS OF STRUCTURES WITH ARBITRARY KINEMATIC BOUNDARY CONDITIONS BY THE SEMI-ANALYTICAL FINITE ELEMENT METHOD

The successful application of ITU to the analysis of structures is largely due to the efficiency of the use of modern software packages, in connection with which the role of program systems that implement the solution process increases. The correct organization of the computing complex, the choice of optimal algorithms for solving systems of linear and nonlinear equations largely determine the possibilities of the method in terms of the structural complexity of the objects under consideration, the accuracy of the results obtained, and the complexity of setting nonlinear problems. Therefore, there is an increased interest in the development of fairly universal computing complexes based on ITU. One of the effective software complexes is the "Strength" system, designed to conduct comprehensive research in the field of mechanics of a deformable solid, the basic principles of construction of which are used in this work in the implementation of a semi-analytical version of the finite element method (FEM). In this work, solutions of a significant number of control problems of deformation of massive and thin-walled prismatic bodies under different boundary conditions and loads are obtained. In the process of solving new problems, the estimation of the convergence of results was carried out on the basis of a sequential increase in finite elements and contained terms of decomposition, an increase in the accuracy of systems of linear and nonlinear equations, and the accuracy of satisfaction with natural boundary conditions was checked. The developed effective method for solving new complex problems of deformation of prismatic bodies is implemented in the form of complex programs and can be used in design and construction practice in construction, mechanical engineering and other fields of technology.

Entry.A wide range of building structures or machine-building equipment consists of elements that are prismatic bodies of complex shape.Modern methods that are focused on the calculation of a wide class of structures consisting of massive and thin-walled elements [3,7,14] should, along with high accuracy of the description of the stress-strain state of objects of complex shape and structure, provide a high rate of convergence to an exact solution with minimal numerical Cost.In this work, solutions of a significant number of .controlproblems of deformation of massive and thin-walled prismatic bodies under different boundary conditions and loads.
Bending a hinged square plate.We consider the elastic equilibrium of a thin square plate with a side under the action of a load normal to its plane, the intensity of which is described by the formula: where 0 q -is the value of the load in the center of the plate.Coordinate axis 1 Z  and 2 Z  are directed along the sides of the square.The plate is hinged on all side.
The solution of this problem, obtained in the work [9], is taken as a reference.Calculations performed according to the developed method have shown that the finite elemental approach provides a good convergence of the solution to the exact one.Similarly, when approximating a plate with a 2x5 FE grid, the values of the maximum deflection in the center of the plate obtained by the exact and approximate methods differ by less than 1%.
Cylindrical panel loaded with its own weight.One of the most common examples for estimating the convergence of numerical methods for calculating thinwalled objects is the problem of elastic equilibrium of a non-flat rectangular cylindrical panel, hinged on two curved sides and free on the other two, loaded with its own weight (Fig. 1) [6].( 0) ( ) Fig. 2 shows the graphs of the error change in determining the deflection of the center of the panel calculated in relation to the reference solution [9], in of the dependence on the order of the matrix of the system of allowed equations.The solutions are obtained on the basis of various variants of shell finite elements, FEM when discretized along the one that forms and directs the SAFEM.Their analysis demonstrates the high efficiency of the FEM in the calculation of thin-walled objects.
As can be seen from the figure, the traditional [5] and semi-analytical versions of the FEM, based on the relations of the moment scheme, have approximately the same rate of convergence in this problem.So over the entire range, the change in the number of unknowns is the difference The results obtained by different variants of the FEM are less than 1%, however, the width of the matrix of the system of allowed equations of the SAFEM is much smaller.
Elastic equilibrium of a prismatic bar.The possibility of applying the developed numerical analysis apparatus to the calculation of massive spatial structures is investigated on the example of a prismatic beam with a square prismatic (fig.3).Crosssection -b , the length of the beam - The solution of a similar problem is obtained in [6] on the basis of the generalized method of finite integral transformations.Table 1 shows the values of vertical displacements at different points of the section , obtained on the basis of SAFEM and using the approach [6].At all points of the section under consideration, the discrepancy in the results is about 1%. ( 0) ( ) Table 1 1   ) The thickness of the slab 0,02m h  , the length of the side 2 5 а h  , the intensity of the external load 50 q E  , where E -is the modulus of elasticity of the material, the Poisson's ratio 0, 25   . The plate is clamped on all four sides.Due to the symmetry of the object, only the fourth part is considered, the design model of which is shown in fig. 5.
A comparison of the results obtained on the basis of SAFEM and decomposition by orthogonal functions in all three directions is given in Table 2 and 3   Table 2 shows the convergence of dimensionless displacements 1 1 v u E qa  depending on the visas of the total number of unknowns М, received by НМСЕ in the center of the plate (upper, middle and lower surfaces), as well as their errors relative to the solutions given in the paper [1].
Conclusion.The obtained data indicate a high rate of convergence of the developed approach in the analysis of structures with arbitrary kinematic boundary conditions.

ANALYSIS OF STRUCTURES WITH ARBITRARY KINEMATIC BOUNDARY CONDITIONS BY THE SEMI-ANALYTICAL FINITE ELEMENT METHOD
The successful application of FEM to the analysis of structures is largely due to the efficiency of the use of modern software packages, in connection with which the role of program systems that implement the solution process increases.The correct organization of the computing complex, the choice of optimal algorithms for solving systems of linear and nonlinear equations largely determine the possibilities of the method in terms of the structural complexity of the objects under consideration, the accuracy of the results obtained, and the complexity of setting nonlinear problems.Therefore, there is an increased interest in the development of fairly universal computing complexes based on FEM.One of the effective software complexes is the "Strength" system, designed to conduct comprehensive research in the field of mechanics of a deformable solid, the basic principles of construction of which are used in this work in the implementation of a semi-analytical version of the FEM.
In this work, solutions of a significant number of control problems of deformation of massive and thin-walled prismatic bodies under different boundary conditions and loads are obtained.In the process of solving new problems, the estimation of the convergence of results was carried out on the basis of a sequential increase in finite elements and contained terms of decomposition, an increase in the accuracy of systems of linear and nonlinear equations, and the accuracy of satisfaction with natural boundary conditions was checked.The developed effective method for solving new complex problems of deformation of prismatic bodies is implemented in the form of complex programs and can be used in design and construction practice in construction, mechanical engineering and other fields of technology.
Keywords: finite element method (FEM), semi-analytical finite element method (SAMSE), stress-strain state, elastic deformation, bending of hinged square plate, cylindrical panel, elastic equilibrium of prismatic beam, thick square plate clamped along the contour.

Fig. 2 .
Fig. 2. Graphs of changes in the error in determining the deflection of the center of the panel the object is loaded with a vertical uniformly distributed load of unit intensity.The kinematic boundary conditions are assumed to be the absence of vertical displacements in the plane3

Fig. 4
Fig. 4 shows the nature of the change in vertical displacement along the face 1 Z b   , 2 Z b   .The solid line shows the reference results, the circles show the displacement values calculated by the SAFEM.A good alignment of solutions obtained by different methods is evident.A thick square slab is clamped along the contour.Approximation by means of functional series requires a justification of the possibility of satisfying different types of kinematic

Fig. 4 .
Fig. 4. Graphs of changes in vertical displacement along the face of the beam