Study of Numeric Convergence of the Method of R − functions in Problems of Constraint Torsion

This paper is devoted to the study of numeric convergence of Rvachev’s method of R − function. The method of R − function in this case is applied to the solution of the problem of constraint torsion of prismatic bodies of arbitrary section. First the problem is solved analytically when the section is rectangular and this solution is compared with results of the method of R − function. Then numeric convergence of R − function is studied, when the angles of rectangular section are rounded; it also is applied when the section has rectangular opening with rounded angles. Results compared show good agreement and convergence.


Introduction
Elements of any engineering structure independent of its purpose must be strong, rigid and light with the least consumption of material. So one of the basic problems of design is a study of stress − strain state in machine elements or structure elements of a given type and develop ment, on the basis of carried out investigations, of new, more rat ional constructive forms.
In this connection, optimal design of spatial pris matic elements of a structure of arbitrary section of different buildings and study of their strength properties present actual tasks and demand an application of modern methods of design, which allow to account real conditions of operation, configuration of a given element with consideration of material properties.
Elements of a structure in a nu mber o f cases demand maxi mal weight d ecrease du e to material consu mpt ion, caused by technology of fabrication (cavities, inclusions and depressions). These elements of a structure with geometrical featu res may p res ent p ris mat ic bod ies with d ifferent configuration of section, which are described by the system of differential equations in part ial derivatives of co mp lex type. To solve them approximat ion methods of Bubnov − Galerkin o r Rit z o r Vlas ov − Kan to rov ich typ e with boundary conditions are used. In their realization, one not always manages to build the system of coordinate functions,

Task Definition
To prove an algorith m of the procedure of R − function, we consider the problem of constraint torsion of prismat ic bodies of arbitrary section in Cartesian system of coordinates (x, y, z). We will assume that one of sections (z=0) is fixed (u=v=w=0), and to another section (z=c) torsion mo ment is applied (М ). Lateral sides are free fro m loads. If the body has cavities, its surface is also taken as free. An equation of balance for this body has the form: : ( ( , ) ) 0; : ( ( , ) ) 0 x y x y x y x a x y y where ) (z θ − is a torsion angle; θ I − relative angle of twisting; ϕ(x,y) − torsion function; r 2 (ϕ)=(I p +2I d +I k )/i ϕϕ ;  p   dxdy  x  y  x  y  y  x  G  I  I  I  I  or   dxdy  y  P  x  P  M   dxdy  G  I  dxdy  G  I   dxdy  y  x  G  I  dxdy  y  x  G  I   ;  ) ) , is checked in a given points х and у. If the condition is satisfied, the components of displacement are calculated: If the conditions are not satisfied, the steps are repeated beginning fro m the second one.
If the section of a given body is of co mplex configuration or has cavities or depressions, the procedure of R − function by Bubnov − Galerkin method is used. To build the function ϕ(х, у) for described algorith m, the following functional is used: Now we will consider for co mparison the results of procedure of R -function in design of prismatic body of solid section under follo wing conditions: а=1s m; b=1sm; с=4 sm; The problem was solved analytically (a.s.) and using the procedure of R − function (R s.) with solution structure , 0 ω ϕ ω ϕ where D= ; y y conjunction. C ij − are unknown coefficients to be defined; X i (x), Y j (y) − full system of basic polyno mes (order ones, trigonomet ric, Chebyshev's or other).

Experiment in Numbers
Results of design of torsion function and components of tensors of stresses are shown in Tables 1 and 2 for different section of body and for different values of х and у respectively.
In Tables 1 and 2 in each b lock the first line corresponds to zero solution in (5) and in R − function, and the following lines -to approximat ions.
As seen from the tables, in the third and fourth approximations the values of torsion function coincide by two signs; and in arbitrary on х and у by three signs in mean ing figures. When the solution ϕ is built analyt ically for rectangular solid section with procedure R − function, they coincide up to the third sign of mean ing figures. With decrease of the area of section of the body, its coincidence and convergence become better. So in approximation of the section to quadrate it is advisory to increase the number of terms in solutions. Table 2 shows the values of components of stresses and good agreement and convergence. Now consider design of the body with rectangular cavity with the same values (а 1 =a/10; b 1 =b/10), the surfaces of cavities are free fro m loads. The structure of solution has the form (12), and forms of boundaries of section of a given body are determined by the following way: In this case the following boundary conditions are added to (2): Here numeric convergence of torsion function and its derivatives are studied; results are given in Table 3 for different number of terms in (13), where good agreement and convergence are observed for different sections of prismat ic body with rectangular cavity. Decrease of the area o f section and occurrence of cavity improve convergence of results, because the number of nodes and systems of coordinate functions are increasing.
With an increase of the length of body and decrease of the section of body the values of tangential stresses at z=с are increasing up to reaching the size of large side of section and then they transfer to parallel state to z axis. On the basis of results we may assume that procedure of R-function may be substantially applied in design of pris matic bodies with complex configuration of section.
Results show that the physics of the process in these prismat ic bodies of rectangular section with rectangular cavity is correctly expressed. Because of the small value of cavity there is no great d ifference between the change of section in values of stresses for solid bodies and prismat ic bodies with cavity. The change of curves (Z z , Z y ,Z x ) along the length of body may be observed in Figures (1−3).   Figures 1, 2, 3 dotted, broken and solid lines correspond to the following dimensions of section: а=1s m, b =1s m; а=1s m, b=0.5s m; а=1sm, b=0.2sm. Curves are given for one and the same coordinate's х and у, and external load. All curves qualitatively coincide with curves of design of prismat ic body of solid section. The least value for normal stress is obtained for quadratic section, the greatest valuefor narro w rectangular section; an increase of the length of body does not lead to sharp rise of stress values, step − by −step they increase up to the size of large side of section, then they tend to approximate to "z" − axis (Fig.1.) in normal stresses, and in tangential stresses are parallel to "z" − axis (Fig. 2,3). Now consider the effect of angle points of quadratic section and quadratic cavity on numeric convergence of procedure of R − function.
Consider pris matic body of solid quadratic section with rounded angles with the following values of radiuses r 1 =0.
Results of design are given in Tab le 5.
As seen from the Table near the angle the values of torsion function coincide in one figure, so results obtained earlier are correct. If the radius is less and the number of terms in the structure of solution of R − function is greater, the results will be better.
The values of components of tensor of stresses are given below (Tables 6, 7, 8):    To study the effect of external and internal angle points on the behavior of stress state of prismatic bodies with arbitrary forms of cavity the body with rounded angles with different radiuses of external and internal quadrate was considered. Results of study of stresses Z z ⋅10 4 /G are given in Tab le 9 with init ial data obtained earlier.
As seen from the Table rounded character of external angle effects less than rounded character of internal angle, that is cavity.

Conclusions
Therefore, on the base of above given calculations, we can assert, that the R − function procedures can be used for calculations of prismat ic bodies of any section with any cavity configuration.
So we may state that worked out algorith m on the basis of the method of successive approximat ions and R-functions may be applied to the solution of p ractical problems of constraint torsion in prismat ic do mains of arb itrary section with d ifferent forms of cavities or inclusions, as well as side depressions; here methodology permits to use different hypothesis to determine the angle of torsion.
The paper presents the results of co mputational experiment and investigations, connected with design of practically necessary elements of structure of the type of prismat ic bodies of arbitrary section with d ifferent form of cavity.