Technological parameters optimization and numerical simulation of hot pushing pipe bending process

Elbow pipes are widely used in many industries such as petroleum, chemical, and metallurgy. Of various elbow pipe forming technology, hot pushing pipe bending process is an efficient and economical one for processing elbow pipes. This paper introduced the technological process and plastic deformation mechanism of hot pushing pipe bending. Four deformation assumptions were proposed to describe the main deformation characteristics in the bending process. The proportional relationship between the bending deformation and the diameter expansion deformation was derived and the horn mandrel was designed. The forming process of the elbow was simulated and the metal flow in the forming process of the elbow was obtained. The results showed that controlling the axial compression and circumferential expansion in proportion is the key to getting constant wall thickness. By using the horn mandrel with variable curvature axis, it benefits the metal flowing uniformity from the concave side to the convexity, avoiding the increase of the inner wall thickness and the decrease of the outer wall thickness during pipe bending. Taking the minimum change rate and good uniformity of the concave side wall thickness as goals, the optimal process parameters were obtained, that is, the heating temperature is 750° Celsius, the pushing speed is 3 mm/s, the friction factor is 0.15, the bending angle is 45° and initial bending radius is 2.0 mm. It will play a positive role in revealing the pipe forming mechanism and have a guiding effect on production practice.


Introduction
Elbow pipes with uniform wall thickness, small curvature, thin wall and high intensity are used more and more in petroleum, chemical, metallurgy, boiler and other industrial production unit [1][2][3]. When using traditional bending process for forming, the wall thickness easily appears uneven because the convex side of the elbow pipe is under tensile forces and the concave side is under compression. In recent years, many scholars have researched and explored the pipe bending process [4][5][6][7][8][9][10].
The main forming methods of thin-walled elbow pipe with a small bending radius are hot pushing pipe bending method, molded bending method, cold pushing bending method, coreless winding method and so on. Of the various elbow pipe forming methods, hot pushing pipe bending process is an efficient and economical one for processing elbow pipes. In this process, the straight pipe billets are set on the horn mandrel and pushed by a push plate to obtain elbow pipes with required dimensions. It is inductively heated locally in the expanding and bending section of the horn mandrel. Theoretically, the hot pushing pipe bending process can acquire high quality, even wall thickness and small radius elbow pipe. At present, this process has been widely used in the production of transportation pipelines and metal structure parts. Plentiful studies are reported on this process by theoretical analysis, experimental research and numerical simulation [11][12][13][14].
There are many factors influencing the forming quality of elbow pipes, including geometric parameters of the core mandrel, pushing speed of the push plate, the friction factor between the pipe billet and the core mandrel, heating temperature of the pipe billet, etc. The pipe shape after forming, the wall thickness and the wall thickness uniformity are different with different technological parameters in * Lili Huang huang0539@sdjzu.edu.cn 1 the pushing process. The geometrical shape and size of the horn mandrel is the main technological parameter that affects the elbow pipe forming quality, and it also directly affects the rule of metal flowing and the plastic deformation mechanism. This paper induces the polar equation of horn mandrel axis with continuous curvature instead of the past design method of mandrel axis with single radius or double radii. The hot pushing pipe bending process is simulated by using the finite element software MSC.Marc. The key parameters of the forming process, i.e., bending angle of the core mandrel, heating temperature, initial bending radius, pushing speed and the friction factor, are optimized by the orthogonal test. Taking the minimum change rate and good uniformity of the concave side wall thickness of the pipe as goal, the optimized technological parameters are obtained. It has a certain guiding significance for designing geometrical shape of the core mandrel and defining reasonably technological parameters.
2 Technological process of hot pushing pipe bending

Technological characteristics of hot pushing pipe bending process
The technological process of hot pushing pipe bending is shown in Fig. 1. A number of pipe billets with a certain size are selected and set on a straight rod (Part I as shown in Fig. 1). The pipe billets are pushed forward along the rod by a push plate. The push plate presses against the last pipe billet and pushes all the pipe billets into guiding region of the horn mandrel (Part II). When the pipe billet enters into the bending and expanding region (Part III), it is heated to 700-900℃ by medium frequency induction heating coil.
The bending deformation and expanding deformation occur simultaneously during the pushing process. After passing through the deformation region of the horn mandrel, the pipe enters into the finishing region for reshaping (Part IV). The elbow pipes are pushed out from the big end of the horn mandrel one by one after reshaping. It can be seen from the hot pushing pipe bending process that, the high temperature deformation region is narrow and the cross section shape of the deformation region is maintained by the rigid section shape of the adjacent end. The pipe billet in this process deforms in the heated region and transfers force in the unheated region, which can avoid the defects such as local flattening and surface scratch appear in other bending methods. It is convenient to control the cross section distortion of bending parts.
The bending deformation and expanding deformation occur simultaneously when the straight pipe is pushed through the horn mandrel at high temperature. From a mechanical perspective, the forming process of hot pushing pipe bending is a three-dimensional dynamic contact problem with constant changing boundary conditions of force and displacement and dual nonlinearities of geometric and material. There are many factors that affect the forming quality of bending pipe, such as the shape size of the core mandrel, heating temperature, pushing speed, pipe billet dimensions and so on. These factors mentioned above are interrelated and changing constantly in the process of pipe bending forming, which brings great difficulties to the experimental research and theoretical analysis of this process.

Plastic deformation mechanisms
On the surface of the straight pipe billet, 5 × 5 mm square coordinate grids were printed. The pipe billet was pushed to different deformation stages and unloaded. Then the deformation of the element grids on various parts of the pipe at different deformation stages was measured with calipers [15]. Analysis of experimental results shows that: (1) The vertical and horizontal grid lines that are orthogonal on the surface of the billet before deformation remain substantially orthogonal during and after deformation (see Fig. 2a, b). (2) Before deformation, the equidistant parallel curve family (spacing is 5 mm) evenly distributed on the surface of the billet along circumferential direction. After deformation, the curves on the surface of the billet along circumferential direction no longer maintain parallel to each other. However, the spacing of the grid lines along the circumferential direction at the convex edge of the pipe is still 5 mm. At the concave edge of the pipe, the spacing of grid lines along the circumferential direction is compressed to 2.5 mm. After defor- Fig. 1 Schematic diagram of the hot pushing pipe bending process mation, the grid lines along the circumferential direction have no warp and distribute in the shape of fan on the surface of the pipe (see Fig. 2c). It can be assumed that: (i) The cross section of the pipe billet is always kept as a plane during the bending process; (ii) During the bending process, the neutral surface will gradually move to the outer arc of the pipe due to the action of the heterogeneous extrusion pressure on the cross section, and finally move to the convex edge of the pipe. (3) Before deformation, the equidistant parallel line family (spacing is 5 mm) evenly distributed along the axial direction on the surface of the pipe billet, and which becomes not equidistant parallel arc family on the pipe surface after deformation. The spacing of each arc on the convex side of the pipe remains basically unchanged. And the spacing of each arc on the concave side of the pipe is widened to a maximum of 10 mm. Therefore, it can be concluded that: (i) During the bending process, the circumferential expansion deformation is not uniform. The expanding deformation occurs only on the concave side of the pipe, and the convex side of the pipe does not produce the expanding deformation. (ii) In the bending process, the axial compression deformation is also not uniform. The con-cave side of the pipe has the largest axial compression deformation, and the convex side of the pipe has neither circumferential expanding deformation nor axial compression deformation.
Given the above, the deformation mainly occurs on the concave side of the pipe billet in the bending process. As the metal in the concave side of the pipe occurs axial compression and circumferential tensile deformation, the metal in the convex side only produces elastic deformation in order to ensure the deformation continuity and coordination. Therefore, the following assumptions are made: (1) The hot pushing pipe bending process is a combined deformation of eccentric compression and eccentric diameter expansion of the straight pipe billet (as shown in Fig. 2d, e). (2) Eccentric compression and eccentric diameter expansion occur simultaneously on the concave side of the elbow pipe and thus the metal on the concave side flows symmetrically along both sides to the convex side under the integrated effects of axial compressive stress and circumferential tensile stress. (3) The eccentric compression makes the wall thickness on the concave side of the elbow pipe thickened, and the eccentric diameter expansion makes that thinned. Therefore, as long as the eccentric compression and eccentric diameter expansion keep a certain proportion in the bending process, the pipe with uniform wall thickness can be guaranteed to be produced. (4) The bending plastic deformation mechanism requires that the bending forming process must meet the simple loading proportional deformation condition, and the metal deformation at all parts must always be in the state of pure shear strain. (5) Eccentric diameter expansion is the essential feature of this process which is different from other pipe bending processes. It ensures that when the concave pipe wall is bent and pressed, the thickness increase of the pipe wall can be pulled out smoothly along the ring direction, so as to ensure that the wall thickness of the bending pipe in the forming process is always the same.
In order to describe the main deformation characteristics in the bending process, four deformation assumptions are proposed as follows: (1) Planar section assumption The cross sections of the billet are always kept in plane and no warping occurs during the forming process.
(2) Longitudinal symmetry plane assumption The force and deformation of the bending pipe during forming is symmetrical to the longitudinal geometric symmetry plane divided along the convex and concave edges of the elbow pipe. The circular sliding friction between the inner surface of the pipe billet and the outer surface of the mandrel is also symmetrical to the longitudinal symmetry plane because of the symmetry of the plastic flow process of the metal from the concave edge to the convex edge of the elbow.

(3) Neutral line assumption
The axial coordinate grid line of the convex edge is the neutral line which always keeps the length unchanged in the bending forming process. If the thickness of the pipe wall is much less than the diameter of the elbow, it can be considered that the axial metal fibers on the longitudinal symmetry surface of the convex elbow edge keep the length unchanged during the forming process, and thus a neutral surface along the thickness direction of the elbow edge is forming.

(4) Wall thickness invariance assumption
The axial compression and circumferential expansion are strictly in proportion to ensure that the metal in each part of the elbow is always in the state of pure shear strain during the deformation process. According to the constant volume condition in plastic deformation, there is no strain in the direction of pipe wall thickness, which ensures that the pipe wall thickness remains constant during the bending process.

Horn mandrel design
Because the pipe billet is pushed out close to the horn mandrel, the geometric shape and size of the horn mandrel directly affect the metal flow law and the plastic deformation mechanism in the process of bending, and it is the main technological parameters that determine the quality of pipe bending. The type of horn mandrel is generally classified according to its axis form. There are two types of common horn mandrel: (1) Double radius horn mandrel: the axis is composed of two arcs of different curvature with smooth connection, and the curvature of each arc is constant. (2) Single radius horn mandrel: the axis is composed of an arc with a certain curvature. The curvature radius of the arc is the curvature radius of the elbow pipe.
Theoretically, neither of these two types of horn mandrel can guarantee that the bending deformation and the expanding deformation always maintain a strict proportional relationship during the forming process. Sometimes, it is even hard to guarantee the continuous smoothness of the deformation. The axis of the horn mandrel should be an arc with variable curvature in order to keep a certain proportion during the forming process. That is, the bending deformation and the expanding deformation should be gradually increased and continuous smooth.

Size design of the expanding and bending section
The bending deformation and expanding deformation should always be strictly proportional during the forming process. This proportional relationship can be described by a simple function. The amount of deformation of an elbow pipe can be expressed by the curvature and the expanding rate. The functional relationship between these two parameters will be deduced below. Figure 3 is the diagram of the straight pipe billet unit with the length of dL and the elbow pipe unit after deformation. According to the planar section assumption, the volume of the pipe unit dV 0 before deformation and the volume of elbow pipe unit dV w after deformation can be obtained as follows: where r 0 is the radius of pipe billet, t 0 is the wall thickness of pipe billet, r w is the radius of elbow pipe, t w is the wall thickness of elbow pipe, and R w is the curvature radius of the elbow pipe.
According to the volume invariance assumption, the wall thickness invariance assumption and neutral line hypothesis in the forming process, the following equation can be deduced from Eq. (1) and Eq. (2).
According to the geometric relationship in Fig. 3b, Eq. (4) can be obtained.
Based on Eq. (3) and Eq. (4), the curvature radius of the elbow can be deduced as follows.
In order to describe the forming process, the bending deformation angle θ is introduced as a measure of time. In this way, the section radius r i in the bending process can be expressed as a function of the time parameter θ.
refers to total expanding ratio, α is the maximum bending angle. The instantaneous curvature radius in deformation process is expressed as: By substituting Eq. (6) into Eq. (7), the polar coordinates equation of variable curvature of horn mandrel axis can be obtained: where R j ( ) is the polar radius of horn mandrel axis.
In a rectangular coordinate system, the axis equation of horn mandrel can be expressed as:

Integral mandrel design
The mandrel is mainly divided into three parts: the guiding section, the expanding and bending section and the shaping section. The horn mandrel used in this paper adds a straight conical expanding section, as shown in Fig. 4. The role of the guiding section is to transport and guide the billet. The purpose of the straight conical expanding section is to accurately position the pipe billet before bending and expanding, and prevent the subsequent pipe billet from having bad guides. Hence, the pipe billet will be more and more tightly clamped on the mandrel before plastic deformation occurs in the bending deformation stage. The size design of the straight conical expanding section requires that the deformation of the pipe billet in this stage is elastic deformation. The expanding and bending section is the main part of the horn mandrel, where the permanent plastic deformation is produced. The axis of this section can be obtained by Eq. (9). The role of the shaping section is to level the pipe so that the outer diameter and curvature of the elbow meet the requirements.

Geometric dimension design of pipe billet
The dimension of the pipe billet is one of the main technological parameters of this forming process. The billet is determined by the elbow pipe to be produced. The basic parameters of the billet include the outer diameter, length and thickness. According to the empirical equation for pipe billet calculation in the mechanical engineering manual, the outer diameter of the pipe billet can be calculated by the following equation: According to Eq. (3), the calculation equation of pipe billet length can be written as: The machining allowance of both ends of the elbow pipe should be considered when pipe billet is blanked. According to the experimental experience, the machining allowance is taken 0.3 times of the nominal diameter of the elbow in this paper. Thus, the calculation equation for the actual blanking length of the pipe billet is as follows: 4 Finite element numerical simulation of bending process

Establishment of the finite element model
Known geometrical dimensions of the elbow: nominal diameter D g is 150 mm, outer diameter is 168 mm, bending radius R w is 225 mm, wall thickness t is 6 mm, inner diameter is 156 mm, bending angle θ is 90°.
According to the calculation of Eq. (10), the outer diameter of the billet can be obtained d e = 124 mm. There is no corresponding specification in the table of hot rolled seamless steel pipe, a similar slightly larger pipe billet is chosen. Thus, the outer diameter of pipe billet is taken as 127 mm and the wall thickness is taken as 6 mm. The length of the pipe billet is 518 mm calculated by Eq. (12).
The finite element software MSC. MARC is used for simulation. The horn mandrel is designed and generated by 3D modeling software and then imported into simulation software. The deformation of the mandrel is very small in the forming process, so it can be considered to be a rigid body. In order to further improve the calculation efficiency, the reduced integral heat conduction element ELEMENT 123 is selected. The pipe billet is divided into 936 hexahedral grid elements. The plastic deformation and heat transfer of the pipe billet occur in the forming process. Thermodynamic coupling analysis of the deformation and heat transfer is needed, so the ELEMENT 7 full integral thermodynamic coupling element is selected for pipe billet. Fig. 4 Three-dimensional modeling of horn mandrel Figure 5 shows the finite element model of hot pushing pipe bending process. Due to the symmetry of the model, half of the model is selected for simulation to improve the calculation efficiency.
According to the experimental results and production experience, the heating temperature of the billet is set as 750 °C, and the pushing speed is set as 4 mm/s. Under ideal deformation conditions, the material constitutive equation of the low carbon steel elbow is ideal elastoplastic, so the yield stress and the maximum elastic strain expressed by logarithm are σ s = 90 MPa, ε e max = 0.25 respectively. The constitutive equation of ideal elastoplastic material is expressed by hyperbolic tangent function i = s ⋅ th [10 i ] . The friction factor between the mandrel and the billet is set as 0.2. The bending angle of the mandrel is 50°, and the initial bending radius is 1.4 mm.

Finite element simulation process and result analysis
Under the action of the push plate, the pipe billet passes through the guide section and gradually enters into the expanding and bending deformation section of the mandrel, where the diameter expanding and bending deformation occurs, and then it is pushed out through the shaping section after shaping. In this simulation model, three straight pipe billets are added. The first pipe billet has no obstruction force at the front end and is easy to be pushed out. Starting from the second pipe billet, it is hindered by the front pipe billet. Therefore, the simulation process can be considered successful only if the second pipe billet is pushed out normally and has a good quality. The finite element simulation of hot pushing pipe bending process is shown in Fig. 6. As can be seen, the second pipe can be pushed out smoothly, and then the third pipe will follow along the path of the second pipe. This process shows the rationality of the simulation, which has a guiding effect on the actual production.
In order to facilitate the analysis of deformation in the bending process, axial and circumferential coordinate systems of 90° elbow pipe are established, as shown in Fig. 7. The α angle is calibrated along each cross section of the elbow axis, and the value of α is 0-90° from the front end to the back end of the elbow. The β angle is calibrated as the circumferential position of the points on each section perpendicular to the elbow axis, and the β value from concave edge to convex edge of the elbow is 0-180°. In the axial direction, three characteristic positions of concave edge AB, middle edge EF and convex edge GH are selected. In the circumferential direction, three characteristic positions of front end AG, middle MN and back end BH are selected to analyze the bending process. Figure 8 shows the axial thickness distribution of convex and concave edges of the first elbow after it is pushed out. As can be seen from the figure, the axial thickness distribution of the concave edge is not uniform. The wall thickness decreases seriously especially in the range of 0-20° at the The hot pushing pipe bending process front end of the outlet. The wall thickness of the concave edge fluctuates around 6.2 mm in the rage of 20-90°. The wall thickness of the convex edge is uniformly distributed along the axial, which basically keeps at about 5.7 mm. The wall thickness of the first elbow pipe can fluctuate up to 20% and thus the first pipe does not meet the production needs. Figure 9 shows the axial thickness distribution of convex and concave edges of the second elbow pipe after it is pushed out. As can be seen from the figure, the axial thickness distribution of the concave edge fluctuates greatly at both ends of the elbow. This is because the front and rear ends of the second elbow are squeezed by the other elbows, where the compressive stress is large, leading to the increase of compression strain and the increase of thickness. Except for the two ends, the pipe wall thickness fluctuates between 5.8 and 6.2 mm within the range of 12-85°, and the fluctuation is less than 10%. The convex edge wall thickness is uniformly distributed along the axial direction with an average value of 5.8 mm, which basically meets the requirements of uniform wall thickness.
It can be seen from the simulation analysis that the first elbow pipe has thinning scrap section in the forming process. This is because the inside front end of the pipe billet has no metal constraints, which makes the material easy to flow freely forward, resulting in a sharp thinning of the inside. At the front end, the constraint of the pipe billet is the least and the thinning is the most severe. The further back, the constraint from the front metal is greater and the thinning  The axial thickness distribution of convex and concave edges of the second elbow is less. Excluding the first elbow pipe, the other pipes are constrained by both the front end and the back end, so the change of wall thickness tends to be stable gradually. It can also be seen from the simulation that the wall thickness of the concave edge fluctuates greatly, so it is necessary to optimize the forming process parameters to obtain the elbow pipe with uniform wall thickness.

Optimization of technological parameters
The technological parameters of hot pushing pipe bending mainly include heating temperature, pushing speed, friction factor, bending angle and initial bending radius of horn mandrel. The simulation experiment will be optimized for these parameters.
According to the production experience and the previous experiment results, the reasonable ranges of parameter values are selected to analyze the influence rules of the parameters on the bending forming. The specific values are as follows: heating temperatures of 650, 700, 750 and 800℃ are selected respectively. The pushing speeds are selected as 3, The change rate and uniformity of pipe wall thickness are the main factors affecting the forming quality of pipe bending. Due to the characteristics of hot-pushing forming technology of elbow, the thickness of convex edge of elbow is basically unchanged, so the change rate and uniformity of concave edge of elbow have a decisive effect on the forming quality of elbows. In this paper, the mean value and variance of the concave side wall thickness in the simulated experimental results are used to determine the variation and uniformity of the wall thickness, analyze the deformation rule and optimize the process parameters.

Orthogonal table design
Heating temperature A (℃), pushing speed B(mm/s), friction factor C, bending angle D(°) and initial bending radius E (mm) are taken as simulation test factors. Four levels of each of the five factors are selected. L 16 (4 5 ) orthogonal table (see Table 1) is used for simulation optimization design, and a total of 16 tests are needed. The test scheme is shown in Table 2. Table 3 shows the orthogonal test results. The mean value and variance of the wall thickness in the table are obtained by measuring the wall thickness at each position of the concave edge of the elbow pipe. The calculation equations are as follows:

Result analysis
Mean wall thickness of concave edges: Wall thickness variance of concave edges: where n = 54. The K value in Table 3 is the mean value of each factor at four levels, reflecting the influence of the corresponding factor on the experimental results. R is the range ( R = K max − K min ), reflecting the significance of the corresponding factor.
The average wall thickness of scheme 11 ( Table 3 is 6.1160 mm, which is the closest to the initial wall thickness of 6 mm. Scheme 6 (A 2 B 2 C 1 D 4 E 3 ) has the minimum variance of 0.0577, indicating that this group of schemes has the best wall thickness uniformity. According to the R value in the mean value analysis and variance analysis of wall thickness in the table, it can be seen that the influence of various factors on wall thickness is ABCED in descending order. The influence of each factor on the uniformity of wall thickness is CADBE from large to small. According to the value of K, the optimal parameter combination of mean wall thickness is A 3 B 1 C 1 D 2 E 4 , and the optimal parameter combination of wall thickness uniformity is Figure 10 shows the relationship between the level of each factor and the mean value of wall thickness, and Fig. 11 shows the relationship between the level of each factor and the variance of wall thickness, which reflects the change rule of the thinning rate and uniformity of wall thickness caused by the changes of each parameter. Combined with Table 3, the influence rule of each factor on the forming process can be analyzed and obtained: (1) The influence of factor A of heating temperature The heating temperature affects the plasticity and yield point of the metal. When the temperature is too high, the pipe billet will pile up and wrinkle, and when the temperature is too low, the pipe will crack. In this paper, the experimental material is low carbon steel. At 750℃, the material is in an ideal elastoplastic state under the appropriate pushing speed, which is conducive to the metal flow from concave edge to convex edge during the forming process, thereby ensuring that the wall thickness of the bending pipe is basically unchanged. According to Table 3 and Fig. 10, the mean wall thickness of the pipe bending at 750℃ is closest to the initial wall thickness.
(2) The influence of factor B of pushing speed  It can be seen from Table 3 that the pushing speed has little influence on the uniformity of wall thickness, and the main consideration should be the influence on the thinning rate of wall thickness. When other factors are fixed, the smaller the speed is, the smaller the thinning rate of wall thickness will be. Because the pushing speed is slow and the strain rate is small, it can provide enough time for the metal on concave edge flow to the convex edge. Therefore, the speed should be selected as a smaller value during optimizations. In consideration of the reasonable matching between heating temperature and pushing speed, a comprehensive and reasonable selection should be made.
(3) The influence of factor C of friction factor The process mostly adopts graphite lubrication, which can ensure the stability of friction coefficient under high temperature. As can be seen from the simulation results in Table 3, the increase of friction factor causes the increase of elbow wall thickness, and the uniformity tends to get better. However, the friction factor is proportional to the pushing resistance. The larger the friction factor is, the greater the friction resistance between the billet and the mandrel is, which will easily lead to metal accumulation or wall thickness thickening in the concave edges of the elbow. In other words, the increase of the friction factor restricts the toroidal flow of the metal.
(4) The influence of factor D of bending angle The thinning rate and uniformity of the wall thickness are the best at a certain bending angle. This is because the pushing resistance is small when the bending angle is small, which makes the expanding effect weak, the bending effect obvious, and the inner pipe wall thickened. With the increase of bending angle, the expansion effect is gradually increased. The eccentric compression and eccentric expansion keep a certain proportion during the forming process, which makes the wall thickness and uniformity of the elbow become better. When the bending angle is too large, the effect of diameter expansion increases and the uniformity of wall thickness becomes worse.
(5) The influence of factor E of initial bending radius The larger the initial bending radius is, the more drastic the change of the axis radius will be, which makes the metal flow of the pipe billet more intense. When the initial bending radius  The axial thickness distribution curves of the elbow pipe after optimization is 1.1R, the metal flow is slow, causing the inner wall thickness to increase and the poor uniformity. With the increase of the initial bending radius, the inner metal flows to the outer side more rapidly, and the inner wall thickness tends to decrease.
According to the comprehensive equilibrium analysis of the two indexes of wall thickness variation and uniformity, factor A has a significant influence on the two indexes. The effect of level 2 on the uniformity of wall thickness is better than that of level 3, and the effect of level 3 on the variation of wall thickness is better than that of level 2. After comprehensive consideration, level 3 is selected. The effect of factor B on the thickness variation is greater than that on the thickness uniformity, and level 1 is selected. The effect of factor C on the uniformity of wall thickness is greater than that on the change rate of wall thickness, and level 2 is selected. Factor D also has a greater effect on the uniformity of wall thickness than on the change rate of wall thickness. The effect of level 2 on the change rate of wall thickness is better than that of level 4, and the effect of level 4 on the uniformity of wall thickness is better than that of level 2. Level 3 is selected after comprehensive consideration. Factor E has a weak influence on the two indexes, the effect of level 3 on the uniformity of wall thickness is better than that of level 4, and the effect of level 4 on the change rate of wall thickness is better than that of level 3. Considering comprehensively, level 4 is selected. Therefore, the optimal solution is finally determined as Since the best scheme obtained in this test is not in the 16 groups of tests, additional test is needed to simulate the best scheme. Figure 12 shows the axial thickness distribution curves of the elbow pipe after optimization. For the first elbow pipe, the axial thickness distribution of the convex edge is decreasing slightly as can be seen in Fig. 12a. The thickness mean value is 5.89 mm and the wall thickness reduction ratio is about 1.83%.
The axial thickness distribution of the concave edge is increasing. The thickness mean value is 6.0987 mm and the variance is 0.10518. The second elbow pipe is better than the first one as can be seen in Fig. 12b. For the second elbow pipe, the thickness mean value of the convex edge is 5.93 mm and the wall thickness reduction ratio is about 1.17%. The thickness mean value of the concave edge is 6.0744 mm and the variance is 0.0106. Compared with the test results in Table 3, the additional test obtains better results and the wall thickness of the elbow pipe has little change in the bending process.
By using the optimal process parameters, the elbow pipes are obtained with smooth surface, no wrinkle or tear, as shown in Fig. 13a. The comparison between the simulation and experimental results of the concave edge wall thickness is shown in Fig. 13b. The wall thickness distribution of the experimental pipe is basically consistent with the simulation results, which shows the validity of the optimization methods.

Conclusions
This paper presents the forming principle and technological process of hot pushing pipe bending. On the basis of theoretical analysis and orthogonal experiment, hot pushing pipe bending process is simulated. The conclusions are summarized as follows: (1) The hot pushing pipe bending process is a combined deformation of eccentric compression and eccentric diameter expansion of the pipe billet. As long as the eccentric compression and eccentric diameter expansion keep a certain proportion in the bending process, the pipe with uniform wall thickness can be guaranteed to be produced. Four deformation hypotheses are proposed to describe the main deformation characteristics in the bending process, that is, planar section assumption, Fig. 13 The comparison of simulation and experimental results longitudinal symmetry plane assumption, neutral line assumption and wall thickness invariance assumption. (2) According to the four deformation hypotheses, the function relationship between the curvature and the diameter expansion rate is derived. On this basis, the polar coordinate equations of the axis of horn mandrel with variable curvature are deduced. By using the horn mandrel with variable curvature axis, it benefits the metal flowing uniformity from the concave side to the convexity, avoiding the increase of the inner wall thickness and the decrease of the outer wall thickness during pipe bending. (3) The forming process of the elbow is simulated and the metal flow in the forming process of the elbow is obtained. The first elbow pipe has thinning scrap section and the wall thickness is not uniform in the forming process. Excluding the first elbow pipe, the other pipes are constrained by both the front end and the back end, so the change of wall thickness tends to be stable gradually. (4) To optimize the technological parameters in the bending process, L 16 (4 5 ) orthogonal table is used for simulation.
The results show that the influence of various factors on wall thickness in descending order is heating temperature, pushing speed, friction factor, initial bending radius and bending angle. The influence of each factor on the uniformity of wall thickness in descending order is friction factor, heating temperature, bending angle, pushing speed, initial bending radius. It provides the convenience for process design and improvement. Aiming at the minimum change rate of concave wall thickness and good uniformity, the optimal process parameters are obtained in the selected factors and levels, that is, the heating temperature is 750° Celsius, the pushing speed is 3 mm/s, the friction factor is 0.15, the bending angle is 45° and initial bending radius is 2.0 mm.
Author contribution All the authors have contributed to the research conception and design. Material preparation, data collection and analysis were carried out by Huang Lili, Zhang Xiangwei, Lu Xiaoyang and Zhong Qingyun. The first draft of the manuscript was written by Huang Lili, and all authors commented on the previous version of the manuscript. All authors read and approved the final manuscript.
Funding This work is supported by the National Natural Science Foundation of China (Grant No. [11272188]).

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