Support software for plastic bending of waveguides on a rotary draw bending machine

The paper addresses the problem of ensuring the bending quality of the waveguide on an automated pipe rotary draw bending machine by winding. This bending method is currently the most common, but has limitations on the allowable values of bend angles and radii, which ensure the accuracy and quality of the bend region geometry without losing the stability of the section shape in the form of wrinkles and ruptures. Modern requirements for minimizing dimensions and masses require the use of the minimum possible bending radii, which depend on the geometry of the section and the material of the work piece. Calculations to determine the limits of permissible bend parameters using general or special purpose programs are very time-consuming require high-performance computers and high qualification of the computing engineer. To solve this problem, it is proposed to create support software that, based on interpolation of results of more accurate calculations, allows you to determine the areas of permissible bend parameters for the selected shape and section dimensions, as well as the work piece material. This approach allows you to quickly obtain solutions for both the engineer in the design of the structure and the operator of the pipe bending machine and to obtain a product with guaranteed bending quality.


Introduction
When designing long thin-walled structures of waveguides, the engineer-designer must ensure its minimum weight and dimensional parameters, which requires the use of the minimum possible radii and the maximum possible bending angles.Plastic bending of thin-walled waveguides of rectangular cross-section is a complex technological problem, since it is often accompanied by such unacceptable defects as general loss of wall stability, local formation of convexities and wrinkles in the compression area, unacceptable thinning and ruptures in the stretching area of the work piece.
The most versatile and common method is to bend on the rotary draw bending machine.The downside of this versatility is the absence in the accompanying documentation of clear recommendations for the required adjustment of the bend process parameters to obtain a qualitative result for a particular bent sample.General information is usually provided that explains the effect of individual settings on the bend process without specific numerical values.As a result, determining the specific numerical parameters of a pipe bending machine becomes an inefficient empirical task, which is usually solved by conducting test bends of a very large number of work pieces with an assessment of the quality of the resulting products, making appropriate changes to its settings and repeating all actions until the desired result is obtained.Another drawback of the empirical approach is that after the desired result is achieved, all the settings of the pipe bending machine will be valid only for one product, since the settings are a multifactorial function of the whole set of blank data and the required results: shape and dimensions of the cross-section, wall thickness, material, bend angle and radius, etc.Therefore, this approach is inefficient, especially in medium and small-scale production of waveguides and leads to unreasonably high time and resource costs for the machine and tool.
The solution to the problem is to create methods for calculating the stress-strain state of work pieces during bending with an assessment of the obtained result depending on the settings of the pipe bending machine.These calculation methods should allow the justification of the required bend parameters and thereby reduce the number of test bends of the work pieces and, accordingly, the time and resources that in this case will be necessary only to confirm the correctness of the calculations.Many scientific works [1][2][3][4][5][6] are devoted to this area, in which classical and new hybrid calculation methods are proposed.However, these methods require high-performance computers, long calculation time and high qualification of the calculator, which is not applicable under usual conditions.
To solve this problem, we propose to create support software that, based on the interpolation of the database of finished calculation results, allows you to determine the areas of permissible bend parameters for the selected shape and section dimensions, as well as the work piece material.This approach allows you to quickly obtain solutions for both the engineer in the design of the structure and the operator of the pipe bending machine and to obtain a product with guaranteed bending quality.

Methods
Consider a diagram of the waveguide bending on a rotary draw bending machine (Fig. 1).Before bending, waveguide is rigidly clamped by pressure die and clamp die rotates together with bend die, simultaneously pulling and bending work piece.When bending thin-walled work pieces, especially into small bend radii, it is necessary to use a mandrel and a wiper die to avoid excessive deformation of the section shape and prevent wrinkles.In case of excessive thinning of the outer wall, a boost block is used to give axial force along the work piece.From the point of view of mechanics, the waveguide is experienced deformation loading with developed finite plastic deformation in the bending region.Finite deformations and angles of rotation formed during bending, and one-sided restrictions-contacts on deformation of walls in the form of a mandrel, a bend die and a wiper die determine a highly nonlinear nature of the problem and complexity in solving.The need to analyze the limits, for example, in the form of a minimum bend radius, implies the appearance of a complex stress state in the bend region, in which the effects of loss of wall stability arise and the result begins to be influenced by various geometric and physical imperfections of both the work piece and the tool used.
The difficulty of modeling and solving such a problem led to the development of various approaches that can be divided into analytical and numerical.Consider the existing methods of calculating the stress-state of the work piece during bending, their main features and disadvantages in order to choose the most rational approach in relation to the problem of bending thin-walled waveguides with a rectangular cross section.

Analytical methods for calculation of plastic bending
The thin-walled geometry of waveguides assumes the use of shell theory for its calculation; however, the peculiarity of the shape of their cross-section in the form of right angles does not allow describing the stress-strain state of one system of the equation of shell theory.Therefore, we will simulate the waveguide with a complex structure of four plates.In Figure 2,a the waveguide is shown in a bent state during bending in the B-plane.B H t z x y q y q y q x q x q xy q xy q xy q xy a) waveguide in bent state b) loading of inner wall The most problematic area for obtaining a high-quality product and at the same time the most difficult part of the work piece for modeling and solving is internal wall of its cross-section (Fig. 2, b), which is in a state of complex loading.In the analytical approach, we consider that the load qx, qy, qxy is parallel to the middle surface of the plate and before the loss of stress stability is constant in thickness.Assuming the presence of curvatures kx, ky of the element in the corresponding directions, the nonlinear differential equilibrium equation has the form: The advantages of this approach are the clarity of the analytical solution and the wide possibilities for analyzing the contribution of individual components and their impact on the result.However, the general analytic solution of equation ( 1) is very difficult due to nonlinear terms and is obtained only for simple cases of loading conditions (qx, qy, qxy) and geometry (kx, ky) of the element.More complex boundary conditions in the form of uneven distribution of forces, variable wall thickness, anisotropy, etc. lead to great difficulties in obtaining an analytical solution.

Numerical methods for calculation of plastic bending
A more accurate accounting of the working conditions of the plate in the process of bending and its imperfection can provide numerical calculation methods, the most universal and common of which today is the finite element method (FEM).The modern level of development of the FEA allows you to take into account as much as possible all possible factors that determine the setting of the task and its solution.The implementation of FEA in many computer programs made this approach the most common in engineering calculations [7][8][9][10][11][12].At the same time, taking into account in the design diagram all the features of geometry, material properties, loading conditions, etc., leads to increased requirements for memory size and computer performance, long calculation time, problem of solution convergence, etc.
Currently, the following approaches to FEM-solving plastic problems are widespread: calculation of stability loss based on eigenvalues, implicit static method, explicit dynamic method.Each of these approaches has its own purpose and allows for a limited solution.However, despite the advantages of FEM, there is no single solution algorithm that would allow you to reliably solve the problem of plastic bending of a thin-walled waveguide on a pipe bending machine.Therefore, new algorithms for this calculation are being developed, most of which are based on existing methods and use their basic advantages to obtain a correct solution.For example, in work [13], a hybrid method is proposed, which is based on a combination of an explicit dynamic method, accounting for initial imperfections and an energy method.The appearance of folds during plastic deformation takes into account complex boundary conditions and is based on the introduction of initial small imperfections for subsequent explicit dynamic analysis.This approach allows you to obtain a higher accuracy of the solution than the methods used in it individually.

Accompanying software of waveguide bending process
The general disadvantages of all calculation methods are high requirements for the designer's qualification, computer resources and calculation duration, which do not allow using these methods in production to quickly determine the required machine settings and the range of permissible values of bend parameters, at which the product quality will be ensured.To solve this problem, we propose an approach that consists in creating a database of results of numerical calculation of plastic bending of work pieces from different materials with different geometric dimensions and parameters of bending by a numerical hybrid method.Calculations will be made for a limited set of parameters of the work pieces and the results obtained will be interpolated to obtain approximating functions.This allows you to have solutions for any intermediate parameters of the work pieces, for example, its dimensions.
Mathematically, this approach is to create functions that bind the two most important variables for a given material, shape, and section dimensions: bend angle, and bend radius.Since, in most cases, developers will be interested in the minimum allowable bend radius for a given section and material in the design and manufacture of the design, the desired function will be: where d,h,b,t -geometric dimensions of the work piece; max -maximum allowable linear deformation of the material; k -factor taking into account the method of manufacturing and adjustment of the bending machine;  -required bend angle.
Each section type is calculated and the minimum bend radius Rmin depends on the bend angle .In this case, in the expression (2), all parameters of the function except the bend angle  are taken constant.The bend angle  varies discretely from 10 0 to 180 0 in increments of 10 0 and in each case there is a value of the minimum bend radius Rmin at which the loss of section stability begins.That is, for each i-th type of section and material, the desired constraint is simplified from function (2) to a limited set of pairs of numbers that determine for each bend angle the value of the minimum bend radius: By interpolating these values by least squares, it is possible to obtain two mutual dependencies, which will be useful both in the design and in the manufacture of curved waveguides of a given section: The functions (4) obtained for a number of waveguide cross-section sizes allow interpolation according to other bend parameters, primarily the cross-section dimensions hxb and wall thickness t.This allows you to use the previous results to evaluate the bending capabilities of new section types from the same material when bending on the same type of machine.Similarly, it is possible to estimate the correlation of other bend parameters, for example, machine settings, which are conditionally indicated here as one parameter k.The results database and its interpolation will be presented as support software.
Thus, the proposed approach allows for a limited number of calculations to obtain wide possibilities for assessing the interaction of the shape and size of the section, material, machine settings, bend parameters, etc., to obtain very universal solutions that can be transferred to new products.

Discussion
Quality of plastic bending on the machine depends on many factors: the shape and size of the section, the plastic properties of the material, the loading conditions in the form of the presence of the mandrel and its parameters, the geometry and location of the clamps, etc.Therefore, general recommendations on the limits of bend parameters, which can be found in the general literature, are insufficient.For example, in [4] it is indicated that bending of pipes of circular cross-section with diameter D without mandrel is possible provided: ( ) min 2...3 RD = (5) and with mandrel the limits expand to: Such simplified constraints do not take into account other features of the work piece, such as wall thickness and maximum possible plastic deformations of the material.The situation is aggravated for non-symmetrical rectangular thin-walled cross-sections, which are waveguides.
The proposed method of estimating the allowable bending parameters of work pieces can be very easily implemented, since at each particular production there is a very limited set of work piece materials, shapes and cross-sectional dimensions.This allows each work piece to define the range of allowable bend parameters and values for the required bending machine settings.Another advantage of the combined approach is the high speed of obtaining results, since there is no need to create calculation models, conduct long-term calculations and process results.Also, when using such software, high qualification not required, since only the initial parameters of the work piece, which are known in advance, are set.The development of the proposed support software will allow you to obtain the necessary data for each section type by interpolating or extrapolating the results of the calculation of existing section sizes.
The accuracy of the results obtained depends both on the accuracy of numerical solutions and on the actual parameters of the work piece.Numerical solutions are based on convergence to specific numerical values, which depend on the accuracy of the bending process simulation.Verification of the obtained solutions can be control bends for a limited number of section sizes in order to check the correctness of the design boundaries of bend parameters.The bend results are also affected by the actual parameters of the work piece, primarily such as the spread of material properties, the presence of initial geometric deviations from the calculated ideal shape and dimensions.
To solve problems with deviations, it is proposed to establish safety margins for the calculated theoretical limits of bend parameters (4), after that obtained results are possible, but not guaranteed.Such a boundary can be a protective range of bend parameters of 5-10%, which will allow the production to check the correctness of the limit bend parameters for specific products.

Conclusion
The paper proposes a solution to the problem of obtaining a qualitative bend of thin-walled waveguides of rectangular cross-section by numerically solving a number of problems and interpolating the obtained results.As a result, it is possible to quickly assess the bending capabilities of both calculated and new types of waveguides with the exception of the appearance of various manufacturing defects during plastic bending.

Figure 1 .
Figure 1.Diagram of the plastic bending process.