SHELL STRESS ANALYSIS USING A VARIATIONAL METHOD BASED ON THREE-DIMENSIONAL FUNCTIONS WITH FINITE CARRIERS

A modifi cation of the method is presented, which allows one to analyze a thin shell calculation by specifying the geometry only in the Cartesian coordinate system. A variational method is presented for determining the stressstrain state of three-dimensional elastic structures based on the use of approximating functions with fi nite carriers of an arbitrary degree of approximation. The presented method can be successfully used both for the calculation of three-dimensional composite structures and for the calculation of thin shells using curvilinear coordinate systems, a comparison between the modeling of equation programming with the Ansys software gives a good indication for corrections of the method modifi cation.


INTRODUCTION
The Global Cartesian coordinate system is introduced x, y, z, in which the investigated three-dimensional structure is divided into subdomains in the form of curved hexagons V k (Fig.1). Hex faces are assumed to be piecewise smooth surfaces (Ω i , i=1, 6) . A local Cartesian coordinate system is introduced x, y, z, , associated with a structural element. Note that instead of the local Cartesian coordinate system, x, y, z, a curvilinear coordinate system can be chosen, which is considered in the work [1]. Figure 1: three-dimensional structure The local coordinate system, x, y, z, is chosen so that the equations of the faces can be set relative to the corresponding coordinate planes in the following form: Where Fi, i=1,6 -unambiguous class functions C 1 . The fi gure shows the numbers of the sides in parentheses; in circles, the numbers of the corner points (nodes) of the sub-region; and in squares, the numbers of the faces Ω i . Within subarea V k , a curvilinear coordinate system is introduced, β 1 , β 2 , β 3 , which is related to the coordinate system, x, y, z, in the following way: (2) where the functions, y 5 =y 5 (β 2 ,β 3 ), z 5 =z 5 (β 2 ,β 3 ), d 1 etc., are selected in this way [2], what's on the verge Ω i k , i=1,6 sub-regions V k equations (2) go into the equations of these faces, on the boundary lines λ j k , j=1,12 -into the equations of these lines; 0≤β 1 , β 2 , β 3 ≤1. Moreover, the coordinate system on these lines, β 1 , β 2 , β 3 , forms a uniform coordinate grid. As a result, if two sub domains are joined along a certain face and if this face in these subdomains is defi ned by the same equation, then the local coordinate grid on it for these subdomains will be the same. This ensures the continuity of the desired functions during transition from one subdomain to another. It also makes it easy to fulfi ll the geometric boundary conditions and those meant for joining the desired functions at the boundaries of the subdomains V k . Displacements defi ned in the global coordinate system x, y, z, within the subdomain are represented as:

D k inml -Unknown constants; form functions
Note that only part of the unknown have physical meaning, namely they are nodal displacements of a subdomain V k . The triple sum (3) is written in matrix form: Displacements in a local curvilinear coordinate system, β 1 , β 2 , β 3 , are recorded through movements in the global Cartesian coordinate system, x, y, z.
Where -matrix of guide cosines of a local curv linear coordinate system, β 1 , β 2 , β 3 in the local Cartesian coordinate system, x, y, z, [C k ] -matrix of guide cosines of the local Cartesian coordinate system, x, y, z, in the global cartesian coordinate system x, y, z. A class of problems is considered in the displacements and strains that are small; further, Hooke's law is valid. Substituting the approximating functions (4) into relations (5), then into the formula to determine the strains in a curvilinear coordinate system, using Hooke's law for the bulk stress state and the well-known formulas for the potential strain energy [4], we obtain the formula for the potential strain energy of the sub-region V k : Where [E] -elasticity matrix defi ned by known relations [3], [A k ] -differential matrix, which has the following form: To determine the stress-strain state of the structure, the Lagrange variation principle [5] is used, on the basis of which the condition should be satisfi ed: Where E -total design energy, E k -total energy subdomain V k , Π k -potential strain energy, which is calculated in an orthogonal curvilinear coordinate system, β 1 , β 2 , β 3 , δW k -variation of the work of external forces of a subdomain V k , K-number of subdomains. From variation equation (7), we obtain a system of equations to determine unknown constants D k inml .

METHOD IMPLEMENTATION
The distinctive features of the presented methodology are as follows: arbitrary order of approximation of the desired functions; integration in a curved coordinate system, β 1 , β 2 , β 3 , which makes the integration procedure the same for any body shape; there is no need to describe the geometry of objects in a curvilinear system, β 1 , β 2 , β 3 , since this system is internal and is constructed algorithmically based on the Cartesian coordinate system, x, y, z. Using the presented method, a thin cylindrical shell clamped at the ends (Fig. 2) was calculated L = 50 cm, loaded with an internal pressure of intensity q = 24MPa. Cylinder inner radius R = 50 cm, cylinder wall thickness t = 5 cm, elastic modulus E = 200 GPa, Poisson's ratio ν =0.3. Due to the presence of symmetry planes, oneeighth of the shell was considered, which was limited to the coordinate planes, x, y, z. In the calculation, two subdomains (two fi nite elements) were used.

RESULT AND DISCUSSIONS
The calculation results are shown in Table 1, where stress values are given. σ zz and σ yy are at points with coordinates (x, 0, z). The last two columns present the results from [6], in which this construction was divided into 300 elements. For a fairly thin cylindrical shell with free ends under the action of a self-balanced system of two concentrated forces (Fig. 3), Table 2 compares the results obtained by this method with the results of other authors. The results are given for the following numerical parameters: L = 26.2 cm; R = 12.5 cm; R / t = 52.5; F = 453 N; E = 74 GPa; ν = 0.3125; R is the radius of the median surface. The calculation was carried out by two fi nite elements using symmetry (1/8 of the shell). Here are the values of the maximum defl ection wmax for various values of the orders of the approximating functions. In [7], a solution was obtained for an inextensible shell, on the basis of which w max = 0.275 cm.   Figure 4: fi xed-fi xed supported by ANSYS The using of the Ansys software for analysis and comparison for case 1 and 2, Fig. 4 shows the auto-meshing that gave deformations in coordinates 50,8.33 up to 0.5 cm in fi xed-fi xed in Fig. 5 and 0.7 cm in free-free in Fig. 7. For case 2, the deformation result shows in Fig. 9 up to 0.28 cm compatible with the deformation analysis in Table 2; fi nite elements mesh are more than 4×4 which indicates more accuracy when the fi ne mesh is done in the Ansys software.

CONCLUSIONS
As can be seen from the calculation results, the presented method agrees very well with the results of calculations by other authors, in which a signifi cantly larger number of elements were used. Thus, the considered method allows one, describing the geometry of shells only in the Cartesian coordinate system, to determine the stress-strain of thin shells. Moreover, to obtain acceptable results, it is not necessary to split the structure into a large number of elements.