# Theoretical Analysis of Five-Point Bending and Springback for Preforming Process of ERW Pipe FFX Forming

Authors: Lidong Ma; Haoxi Ma; Zijian Liu; Shuo Chen
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Flexible forming excellent (FFX) forming of electric resistance welding (ERW) pipe is an advanced roll forming method. In order to understand FFX forming mechanism and set reasonable roll parameters, five-point bending process and springback process for preforming section of ERW pipe FFX forming are analyzed. Under various assumptions, the bending process of sheet metal is divided into full elastic bending stage and elastic-plastic bending stage by analyzing the forming law and simplifying the contact state model of roll bending process. Based on the theory of plane bending theory, the five-point bending springback model is established. Mathematical expressions of bending curvature and rotation angle of arbitrary particle on plate under arbitrary state are derived. The curvature calculation formulas of each particle on plate after springback are obtained, and then the calculation formulas of bending angle and rebound angle after springback are obtained. The theoretical analysis results are verified by the wide plate five-point bending experiment. The theoretical value of springback angle agrees well with the experimental results. The maximum deviation of springback angle is 0.42 degrees. The results show that the five-point bending model is reasonable. It provides theoretical support for the preforming model of flexible roll forming.


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## Body

## 1. Introduction

In recent years, ERW pipe products have caused extensive concern [1, 2]. High-grade ERW pipes are mainly used for oil wells, submarine pipelines, oil and gas pipelines, some special purpose professional pipes, etc. The forming process for producing high-grade ERW pipes is mainly divided into two types: the first one is pressure forming, such as JCO forming and UOE forming. The second one is roll forming, including row roll forming, cage roll forming, and other flexible roll forming methods. Roll forming is cost-effective compared with other sheet metal forming processes. Flexible roll forming can use the same unit to form a certain range of products and become more and more flexible. The motion of the rolls can be controlled by many axes enabling a large variation in the profiles along the formed sheet, called 3D roll forming or flexible roll forming technology [3]. Nowadays, flexible roll forming technology has been applied in many fields. Zhong [4] researched the flexible roll forming process and established carbon fiber composite process control software. Joo [5] proposed a profile design method for manufacturing variable section parts by flexible forming process. Kari [6] proved that UHS steel can be bent by flexible roll forming process through experimental research. And flexible roll forming technology has also been widely used in the manufacture of automotive component, hull structure, and three-dimensional surface parts [7–9]. Nakata Flexible forming excellent (FFX) forming is an advanced flexible roll forming technology [10, 11].Multipoint bending forming is a popular forming method and a flexible manufacturing technique. It is especially suited to the three-dimensional forming of varied large sheet [12]. The five-point bending forming proposed in the paper is a multipoint bending forming technology, it is the forming method for the preforming section of the FFX forming process, and the middle of the sheet metal is in a state of reverse bending deformation called W-forming, which makes the edge bending deformation sufficient and the amount of springback small. In actual production, this forming process can reduce the defects such as bad edge bending caused by material properties fluctuation and roll wear.Springback is the elastically driven change during unloading and following forming [13]. In the past twenty years, more and more researchers began to pay attention to and study this topic, and this topic has developed rapidly. Carden et al. [14] measured the springback by stretching and bending experiments. Chen [15] and Wagoner [16] carried out finite element simulation of springback of sheet metal forming by different methods. Zhao et al. [17] established the complex equation of planar bending elastic and successfully applied it to the expanding and setting round process. On this basis, Zhao et al. also carried out some research on the prediction and control of the springback in setting round process for pipe-end of large pipe and the springback analysis of profile plane stretch–bending in the loading method of pretension and moment [18, 19]. Liu et al. [20] proposed an analytical method for predicting springback and residual stress distribution of thick plates after multipoint forming. Bending springback is one of the most important factors in ERW pipe forming process. At present, few experts research on bending springback of FFX forming technology, so it cannot provide theoretical guidance for engineering practice. The preforming section of FFX forming is a key process affecting the final product quality. Therefore, it is of great significance to carry out theoretical research on the bending springback process of the FFX forming preformed section.Based on the classical elastic-plastic bending theory and the analytical theory of three-point bending and four-point bending of JCO welded pipe forming [21, 22], the in-depth study of the bending and springback process of the preforming process is carried out, and the mechanical model of the five-point bending springback of the plate is established. It lays a theoretical foundation for realizing the prediction model of roll position parameters of the preformed section in FFX intelligent control system of ERW pipe.


## 2. Basic Theory of Bending Springback in Plate Forming Process

### 2.1. Basic Assumptions

(1) Flat section assumption means that the cross section of the plate is always perpendicular to the axis of the plate during deformation before and after.(2) Tension-free assumption means that the sheet has no longitudinal stress during the bending process.(3) During the bending process, the stress neutral layer, the strain neutral layer, and the geometric center layer of the plate always coincide.(4) The deformation area conforms to the assumption of small deformation, which means the strain distribution on the section is linear distribution, and the expression is as follows:(1)ε=yρwhereε is the strain; ρ is the radius of curvature of neutral layer; y is the distance from the mass point to the neutral layer on the section.(5) The material conforms to the exponential hardening model, and the expression is as follows:(2)σ=Eεε≤σsEσ=kεnε>σsEwhereσ is the stress; k is the coefficient of strength; n is the hardening index; σs is the material yield limit; E is the modulus of elasticity.


### 2.2. Analysis of Preforming Process of FFX Forming

The springback process is analyzed, as shown in Figure1 (Figure 1 shows the forming part of the first rack of preforming section in FFX forming mode). The bending state of the plate is analyzed, and different roll gaps correspond to different bending directions.Figure 1
Analysis of contact state of roll bending deformation process.At present, the specific roll profile data of FFX forming mode is not considered, it is simplified appropriately in the derivation process, and its deformation state is similar to Figure2. The bending process is divided into elastic bending stage and elastic-plastic bending stage according to whether the upper roller has been coated. Figure 3 shows the bending moment under different coating states (the bending moment state in actual forming process can be adjusted based on this model). There are two quantities involved, ρlim and ρmin, where ρmin is the minimum bending radius in the deformation process and ρlim is the elastic limit radius. The calculation formula is as follows:(3)ρlim=Et2σsFigure 2
Broad plate five-point bending diagram.Figure 3
Bending process and the distribution of bending moment.
(a)
Upper roller uncoating stage (b)
Upper roller coating stageTaking the coating state of plate and upper roller as the boundary point, the bending process of plate is divided into two stages. The first is that the plate begins to contact the upper roller. The second is that the plate coats the upper roller. In the first stage, the bending starts and the contact state between plate and five rollers is point contact, then the bending state changes into a five-point bending state. All particles in the wide plate may be in the elastic-plastic stage, and the distribution of bending moment is shown in Figure3(a). In the second stage, the plate begins to coat the upper roller until it is completely coated. At this time, the plate has entered the stage of elastic-plastic deformation, and the bending moment in the coating area reaches the maximum value. The distribution of bending moment is shown in Figure 3(b).


### 2.3. Five-Point Bending Mechanical Model of the Wide Plate

According to Figure2, the following expressions can be obtained from the static equilibrium equation and the geometric relationship:(4)P=2N1cos⁡α1+μsin⁡α1+N2(5)l1=L1-rd+t2sin⁡α1(6)l2=L2-rf+t2sin⁡α2where P is the bending force; μ is the friction coefficient; t is the thickness; rd and rf are the rounded corner of the level lower roller and the bottom roller, respectively; N1 and N2 are the reaction forces at the fulcrum of the lower roller and the bottom roller respectively; α1 and α2 are the corner of the fulcrum of the lower roller and the bottom roller respectively; l1 and l2 are the distance from the origin of coordinates to the cutting point of the lower roller and the bottom roller; L1 and L2 are the distance from the origin of coordinates to the center of the lower roller corner and the center of the bottom roller corner, respectively.


## 2.1. Basic Assumptions

(1) Flat section assumption means that the cross section of the plate is always perpendicular to the axis of the plate during deformation before and after.(2) Tension-free assumption means that the sheet has no longitudinal stress during the bending process.(3) During the bending process, the stress neutral layer, the strain neutral layer, and the geometric center layer of the plate always coincide.(4) The deformation area conforms to the assumption of small deformation, which means the strain distribution on the section is linear distribution, and the expression is as follows:(1)ε=yρwhereε is the strain; ρ is the radius of curvature of neutral layer; y is the distance from the mass point to the neutral layer on the section.(5) The material conforms to the exponential hardening model, and the expression is as follows:(2)σ=Eεε≤σsEσ=kεnε>σsEwhereσ is the stress; k is the coefficient of strength; n is the hardening index; σs is the material yield limit; E is the modulus of elasticity.


## 2.2. Analysis of Preforming Process of FFX Forming

The springback process is analyzed, as shown in Figure1 (Figure 1 shows the forming part of the first rack of preforming section in FFX forming mode). The bending state of the plate is analyzed, and different roll gaps correspond to different bending directions.Figure 1
Analysis of contact state of roll bending deformation process.At present, the specific roll profile data of FFX forming mode is not considered, it is simplified appropriately in the derivation process, and its deformation state is similar to Figure2. The bending process is divided into elastic bending stage and elastic-plastic bending stage according to whether the upper roller has been coated. Figure 3 shows the bending moment under different coating states (the bending moment state in actual forming process can be adjusted based on this model). There are two quantities involved, ρlim and ρmin, where ρmin is the minimum bending radius in the deformation process and ρlim is the elastic limit radius. The calculation formula is as follows:(3)ρlim=Et2σsFigure 2
Broad plate five-point bending diagram.Figure 3
Bending process and the distribution of bending moment.
(a)
Upper roller uncoating stage (b)
Upper roller coating stageTaking the coating state of plate and upper roller as the boundary point, the bending process of plate is divided into two stages. The first is that the plate begins to contact the upper roller. The second is that the plate coats the upper roller. In the first stage, the bending starts and the contact state between plate and five rollers is point contact, then the bending state changes into a five-point bending state. All particles in the wide plate may be in the elastic-plastic stage, and the distribution of bending moment is shown in Figure3(a). In the second stage, the plate begins to coat the upper roller until it is completely coated. At this time, the plate has entered the stage of elastic-plastic deformation, and the bending moment in the coating area reaches the maximum value. The distribution of bending moment is shown in Figure 3(b).


## 2.3. Five-Point Bending Mechanical Model of the Wide Plate

According to Figure2, the following expressions can be obtained from the static equilibrium equation and the geometric relationship:(4)P=2N1cos⁡α1+μsin⁡α1+N2(5)l1=L1-rd+t2sin⁡α1(6)l2=L2-rf+t2sin⁡α2where P is the bending force; μ is the friction coefficient; t is the thickness; rd and rf are the rounded corner of the level lower roller and the bottom roller, respectively; N1 and N2 are the reaction forces at the fulcrum of the lower roller and the bottom roller respectively; α1 and α2 are the corner of the fulcrum of the lower roller and the bottom roller respectively; l1 and l2 are the distance from the origin of coordinates to the cutting point of the lower roller and the bottom roller; L1 and L2 are the distance from the origin of coordinates to the center of the lower roller corner and the center of the bottom roller corner, respectively.


## 3. Full Elastic Bending Stage

When the upper roller is pressed down, the initial stage of contact between the strip and the upper roller is the full elastic bending stage. The full elastic bending stage refers to the fact that the minimum bending radiusρmin of each point on the neutral layer of the plate belt is always greater than the elastic limit bending radius ρlim. In this state, the strip will rebound after the external force is removed.At this stage, the bending moment acting on the strip presents a linear distribution. Define the bending moment of the strip section at the distance from origin 0 tox as Mx, then(7)Mx=N1l11-xl11cos⁡α1x∈0,l1N1l11+xl21cos⁡α1+N2x2x∈-l2,0According to the knowledge of material mechanics, the relation between bending moment and radius of curvature is(8)1ρx=MxEIwhere ρx is the radius of curvature at any point x on the neutral layer.From (7) and (8) we can obtain(9)1ρx=N1l1EI1-xl11cos⁡α1x∈0,l1N1l1EI1+xl21cos⁡α1+N2x2x∈-l2,0Define the curvature of the strip at any pointx as KxKx≥0, then(10)Kx=y′′x1+y′2x3/2Let dy/dx=tan⁡u, then d2y/dx2=(1/cos2⁡u)(du/dx). Because of Kx≥0, formula (10) can be simplified as(11)Kx=-cos⁡ududxFor Kx=1/ρx, we can get(12)-cos⁡ududx=N1l1EI1-xl11cos⁡α1x∈0,l1N1l1EI1+xl21cos⁡α1+N2x2x∈-l2,0Define the angle of rotation at the particle x as αx, then αx=u. Thus,(13)αx=sin-1⁡fx,pfx,p=-N1l1EIcos⁡α1x-x22l1,x∈0,l1-N1l1EIcos⁡α1x-x22l2-N2x4,x∈-l2,0According to the boundary conditionyx=l1=0, the deflection yx at any point in x∈0,l1 is(14)yx=∫0xfx,p1-f2x,pdx-∫0l1fx,p1-f2x,pdxThe displacementh of the upper roller is(15)h=ymax+rd+t21-cos⁡α1Since any particle in the steel plate is in an elastic deformation state, it rebound completely.


## 4. Elastic-Plastic Bending Stage

### 4.1. Upper Roller Uncoating Stage

At this stage, plastic deformation of a particle in the strip has occurred. The minimum bending radiusρmin is greater than the circular radius rp of the upper roller and less than the elastic limit bending radius, that is, rp<ρmin<ρlim.(  1) The Bending Process. According to the theory of wide plate free bending forming [23], another expression of the bending moment at the section x of the strip can be obtained.(16)Mx=∫AσwdA=kIpρxnwhere k is strengthening coefficient; n is hardening index; Ip is moment of inertia of plastic bending section of exponential hardening material, Ip=∫Awn+1dA.From (8), (10), and (11), we have(17)1ρx=MxkIp1/nDefine the angle of rotation at the particlex as αx, then αx=u; therefore, (18)αx=sin-1⁡fx,p(19)fx,p=-nn+1N1l1kIpcos⁡α11/nl11-1-xl11+n/nx∈0,l1(20)fx,p=nn+1N1l1kIpcos⁡α11+n/n-N1l1kIpcos⁡α11+xl2+N2x2kIp1+n/n·kIpcos⁡α1N1+N2/2cos⁡α1x∈-l2,0According to the boundary conditionyx=l1=0, the deflection yx of particle x is(21)yx=∫0xfx,p1-f2x,pdx-∫0l1fx,p1-f2x,pdxx∈0,l1The displacementh of the upper roller is(22)h=ymax+rd+t21-cos⁡α1(  2) The Springback Process. When x=0, the minimum bending radius is(23)ρmin=kIpcos⁡α1N1l11/nLetxs1 be the dividing point of the elastic-plastic deformation of the strip in the positive direction of x, then ρxs1=ρlim. Combining (3) and (23), we get(24)xs1=l1-2σsEtnkIpcos⁡α1N1Letxs2 be the dividing point of the elastic-plastic deformation of the strip in the negative direction of x, then ρxs2=ρlim. Combining (3) and (23), we get(25)xs2=l0·1-2σsIcos⁡α1N1l1where l0 is the distance from the point in x∈[-l2,-d2] whose bending moment is zero to the original point. And l0=(2N1l12-N2l22cos⁡α1)/(2N1l1-N2l2cos⁡α1).Define the radius of curvature at the pointx after springback as ρxu. According to unloading law, we have(26)1ρxu=1ρx-1ρeDefine the bending moment of elastic deformation asMe, then(27)Me=EIρeAccording to unloading law, the springback bending moment is equal to the bending moment at loading, that is,Mx=Me. The radius of curvature at particle x after rebound is(28)1ρxu=N1l1kIpcos⁡α11-xl11/n-N1l1EIcos⁡α11-xl1x∈0,l1(29)1ρxu=N1l1kIpcos⁡α11+xl2+N2x2kIp1/n-N1l1EIcos⁡α11+xl2-N2x2EIx∈-l2,0The rotation angleαxu at particle x after springback is(30)αxu=sin-1⁡gx,pwhere(31)gx,p=-nn+1N1l1kIpcos⁡α11/nl11-1-xl1n+1/n+N1l1EIcos⁡α1x-x22l1x∈0,l1(32)gx,p=nn+1N1l1kIpcos⁡α11+n/n-N1l1kIpcos⁡α11+xl2+N2x2kIp1+n/n·kIpcos⁡α1N1+Pcos⁡α1+N1l1x2/2l2+xEIpcos⁡α1+N2x24EIx∈-l2,0


### 4.2. Upper Roller Coating Stage

As the upper roller continues to press down, the bending degree of the plate increases, and the plate begins to coat the bottom roller until it is completely coated. At this point, the bending moment distribution is shown in Figure3(b). The bending moment at the section of the strip is(33)Mx=Mmax⁡1,x∈-d2,d1Mmax⁡1·l1-xl1-d1,x∈d1,l1Mmax⁡1l2+x-Mmax⁡2d2+xl2-d2,x∈-l2,-d2Mmax⁡2,x∈-L2,-l2According to (16), the maximum bending moment is(34)Mmax⁡1=kIprp+t/2n(35)Mmax⁡2=-kIprf+t/2nAccording to the mechanical and geometric relations, the bending moment loaded on thex section is(36)
M
x
=
N
1
l
1
1
-
d
1
l
1
1
cos
⁡
α
1
,
x
∈
-
d
2
,
d
1
N
1
l
1
1
-
x
l
1
1
cos
⁡
α
1
,
x
∈
d
1
,
l
1
N
1
l
1
-
d
1
cos
⁡
α
1
·
l
2
+
x
l
2
-
d
2
+
N
2
x
+
d
2
2
,
x
∈
-
l
2
,
-
d
2From (33), (34), (35), and (36), we obtain(37)d1=l1-kIpcos⁡α1N1rp+t/2n(38)d2=l2+2kIpN2rf+t/2nThe following discussion is based on the distribution of bending moment.Case 1.
The bending moment is a fixed value in-d2,d1.
(  1) The Bending Process. According to the geometric relationship in Figure 2, the deflection of particle x in the bending process is(39)yx=ymax-rp+t2-rp+t22-x2
The angle of rotation at particlex is(40)αx=-sin-1⁡xrp+t/2
(  2) The Springback Process. According to the unloading law, the radius of curvature at the particle x after springback is(41)1ρxu=1rp+t/2-MmaxEI
The rotation angle of particlex after unloading and springback is (42)axu=sin-1⁡x1rp+t/2-MmaxEICase 2.
Bending moment is in a linear distribution aread1,l1.
(  1) The Bending Process. According to Figure 3, the particle in d1,xt1 is in an elastic-plastic state, and the particle in xt1,l1 is in a fully elastic state. Then the radius of curvature at point x is(43)1ρx=l1-xMmax⁡1kIpl1-d11/nx∈d1,l1
Whenx=xt2, ρxt2=ρlim, the boundary point xt1 of elastic-plastic deformation can be obtained from (43). (44)xt1=l1-2σsEtnkIpl1-d1Mmax⁡1
Then the angle of rotation at particlex is(45)αx=sin-1⁡φ1xx∈d1,l1(46)φ1x=nn+1Mmax⁡1kIpl1-d11/nl1-x1+n/n-l1-d11+n/n-d1rp+t/2
According to the boundary conditionsyx=l=0, the deflection at the particle x is(47)yx=∫d1xφ1x1-φ12xdx-∫d1l1φ1x1-φ12xdx
The upper roller stroke is(48)h=ymax+rd+t21-cos⁡α1+rd+t2-rd+t22-d12
(  2) The Springback Process. According to the unloading law, the radius of curvature at particle x after springback is(49)1ρxu=Mmax⁡1l1-xkIpl1-d11/n-Mmax⁡1EIl1-xl1-d1x∈d1,xt1
The rotation angle at particlex∈d1,xt1 after unloading and springback is(50)αxu=sin-1⁡ϕ1x(51)ϕ1x=nn+1Mmax⁡1kIpl1-d11/nl1-xn+1/n-l1-d1n+1/n-Mmax⁡12EIl1-d1l1-x2-l1-d12-d11rp+t/2-Mmax⁡1EICase 3.
Bending moment is in a linear distribution area-l2,-d2.
(  1) The Bending Process. According to Figure 3, the particle in -l2,xt3∪xt2,-d2 is in an elastic-plastic state, and the particle in xt2,xt3 is in a fully elastic state.
The radius of curvature at particlex is(52)1ρx=l0+xMmax⁡1kIpl0-d21/nx∈-xt2,-d2where l0 is the distance from the point in x∈-l2,-d2 whose bending moment is zero to the origin point. And l0=(Mmax⁡1l1-Mmax⁡2l2)/(Mmax⁡1-Mmax⁡2).
Whenx=xt2, ρxt2=ρlim, the boundary point xt2 of elastic-plastic deformation is(53)xt2=-l0+2σsEtnkIpl0-d2Mmax
Whenx=-d2, the rotation angle of this particle can be obtained from (47) (54)αd=sin-1⁡d2rp+t/2
(  2) The Springback Process. According to the unloading law, the radius of curvature at particle x after springback is(55)1ρxu=Mmaxl0+xkIpl0-d21/n-MmaxEIl0+xl0-d2x∈-xt2,-d2
Then, the rotation angle at particlex∈xt2,-d2 after unloading and springback is(56)αxu=sin-1⁡ϕ2x(57)ϕ2x=-nn+1MmaxkIpl0-d21/nl0-xn+1/n-l0-d2n+1/n+Mmax2EIl0-d2l0-x2-l0-d22+d21rp+t/2-MmaxEI


## 4.1. Upper Roller Uncoating Stage

At this stage, plastic deformation of a particle in the strip has occurred. The minimum bending radiusρmin is greater than the circular radius rp of the upper roller and less than the elastic limit bending radius, that is, rp<ρmin<ρlim.(  1) The Bending Process. According to the theory of wide plate free bending forming [23], another expression of the bending moment at the section x of the strip can be obtained.(16)Mx=∫AσwdA=kIpρxnwhere k is strengthening coefficient; n is hardening index; Ip is moment of inertia of plastic bending section of exponential hardening material, Ip=∫Awn+1dA.From (8), (10), and (11), we have(17)1ρx=MxkIp1/nDefine the angle of rotation at the particlex as αx, then αx=u; therefore, (18)αx=sin-1⁡fx,p(19)fx,p=-nn+1N1l1kIpcos⁡α11/nl11-1-xl11+n/nx∈0,l1(20)fx,p=nn+1N1l1kIpcos⁡α11+n/n-N1l1kIpcos⁡α11+xl2+N2x2kIp1+n/n·kIpcos⁡α1N1+N2/2cos⁡α1x∈-l2,0According to the boundary conditionyx=l1=0, the deflection yx of particle x is(21)yx=∫0xfx,p1-f2x,pdx-∫0l1fx,p1-f2x,pdxx∈0,l1The displacementh of the upper roller is(22)h=ymax+rd+t21-cos⁡α1(  2) The Springback Process. When x=0, the minimum bending radius is(23)ρmin=kIpcos⁡α1N1l11/nLetxs1 be the dividing point of the elastic-plastic deformation of the strip in the positive direction of x, then ρxs1=ρlim. Combining (3) and (23), we get(24)xs1=l1-2σsEtnkIpcos⁡α1N1Letxs2 be the dividing point of the elastic-plastic deformation of the strip in the negative direction of x, then ρxs2=ρlim. Combining (3) and (23), we get(25)xs2=l0·1-2σsIcos⁡α1N1l1where l0 is the distance from the point in x∈[-l2,-d2] whose bending moment is zero to the original point. And l0=(2N1l12-N2l22cos⁡α1)/(2N1l1-N2l2cos⁡α1).Define the radius of curvature at the pointx after springback as ρxu. According to unloading law, we have(26)1ρxu=1ρx-1ρeDefine the bending moment of elastic deformation asMe, then(27)Me=EIρeAccording to unloading law, the springback bending moment is equal to the bending moment at loading, that is,Mx=Me. The radius of curvature at particle x after rebound is(28)1ρxu=N1l1kIpcos⁡α11-xl11/n-N1l1EIcos⁡α11-xl1x∈0,l1(29)1ρxu=N1l1kIpcos⁡α11+xl2+N2x2kIp1/n-N1l1EIcos⁡α11+xl2-N2x2EIx∈-l2,0The rotation angleαxu at particle x after springback is(30)αxu=sin-1⁡gx,pwhere(31)gx,p=-nn+1N1l1kIpcos⁡α11/nl11-1-xl1n+1/n+N1l1EIcos⁡α1x-x22l1x∈0,l1(32)gx,p=nn+1N1l1kIpcos⁡α11+n/n-N1l1kIpcos⁡α11+xl2+N2x2kIp1+n/n·kIpcos⁡α1N1+Pcos⁡α1+N1l1x2/2l2+xEIpcos⁡α1+N2x24EIx∈-l2,0


## 4.2. Upper Roller Coating Stage

As the upper roller continues to press down, the bending degree of the plate increases, and the plate begins to coat the bottom roller until it is completely coated. At this point, the bending moment distribution is shown in Figure3(b). The bending moment at the section of the strip is(33)Mx=Mmax⁡1,x∈-d2,d1Mmax⁡1·l1-xl1-d1,x∈d1,l1Mmax⁡1l2+x-Mmax⁡2d2+xl2-d2,x∈-l2,-d2Mmax⁡2,x∈-L2,-l2According to (16), the maximum bending moment is(34)Mmax⁡1=kIprp+t/2n(35)Mmax⁡2=-kIprf+t/2nAccording to the mechanical and geometric relations, the bending moment loaded on thex section is(36)
M
x
=
N
1
l
1
1
-
d
1
l
1
1
cos
⁡
α
1
,
x
∈
-
d
2
,
d
1
N
1
l
1
1
-
x
l
1
1
cos
⁡
α
1
,
x
∈
d
1
,
l
1
N
1
l
1
-
d
1
cos
⁡
α
1
·
l
2
+
x
l
2
-
d
2
+
N
2
x
+
d
2
2
,
x
∈
-
l
2
,
-
d
2From (33), (34), (35), and (36), we obtain(37)d1=l1-kIpcos⁡α1N1rp+t/2n(38)d2=l2+2kIpN2rf+t/2nThe following discussion is based on the distribution of bending moment.Case 1.
The bending moment is a fixed value in-d2,d1.
(  1) The Bending Process. According to the geometric relationship in Figure 2, the deflection of particle x in the bending process is(39)yx=ymax-rp+t2-rp+t22-x2
The angle of rotation at particlex is(40)αx=-sin-1⁡xrp+t/2
(  2) The Springback Process. According to the unloading law, the radius of curvature at the particle x after springback is(41)1ρxu=1rp+t/2-MmaxEI
The rotation angle of particlex after unloading and springback is (42)axu=sin-1⁡x1rp+t/2-MmaxEICase 2.
Bending moment is in a linear distribution aread1,l1.
(  1) The Bending Process. According to Figure 3, the particle in d1,xt1 is in an elastic-plastic state, and the particle in xt1,l1 is in a fully elastic state. Then the radius of curvature at point x is(43)1ρx=l1-xMmax⁡1kIpl1-d11/nx∈d1,l1
Whenx=xt2, ρxt2=ρlim, the boundary point xt1 of elastic-plastic deformation can be obtained from (43). (44)xt1=l1-2σsEtnkIpl1-d1Mmax⁡1
Then the angle of rotation at particlex is(45)αx=sin-1⁡φ1xx∈d1,l1(46)φ1x=nn+1Mmax⁡1kIpl1-d11/nl1-x1+n/n-l1-d11+n/n-d1rp+t/2
According to the boundary conditionsyx=l=0, the deflection at the particle x is(47)yx=∫d1xφ1x1-φ12xdx-∫d1l1φ1x1-φ12xdx
The upper roller stroke is(48)h=ymax+rd+t21-cos⁡α1+rd+t2-rd+t22-d12
(  2) The Springback Process. According to the unloading law, the radius of curvature at particle x after springback is(49)1ρxu=Mmax⁡1l1-xkIpl1-d11/n-Mmax⁡1EIl1-xl1-d1x∈d1,xt1
The rotation angle at particlex∈d1,xt1 after unloading and springback is(50)αxu=sin-1⁡ϕ1x(51)ϕ1x=nn+1Mmax⁡1kIpl1-d11/nl1-xn+1/n-l1-d1n+1/n-Mmax⁡12EIl1-d1l1-x2-l1-d12-d11rp+t/2-Mmax⁡1EICase 3.
Bending moment is in a linear distribution area-l2,-d2.
(  1) The Bending Process. According to Figure 3, the particle in -l2,xt3∪xt2,-d2 is in an elastic-plastic state, and the particle in xt2,xt3 is in a fully elastic state.
The radius of curvature at particlex is(52)1ρx=l0+xMmax⁡1kIpl0-d21/nx∈-xt2,-d2where l0 is the distance from the point in x∈-l2,-d2 whose bending moment is zero to the origin point. And l0=(Mmax⁡1l1-Mmax⁡2l2)/(Mmax⁡1-Mmax⁡2).
Whenx=xt2, ρxt2=ρlim, the boundary point xt2 of elastic-plastic deformation is(53)xt2=-l0+2σsEtnkIpl0-d2Mmax
Whenx=-d2, the rotation angle of this particle can be obtained from (47) (54)αd=sin-1⁡d2rp+t/2
(  2) The Springback Process. According to the unloading law, the radius of curvature at particle x after springback is(55)1ρxu=Mmaxl0+xkIpl0-d21/n-MmaxEIl0+xl0-d2x∈-xt2,-d2
Then, the rotation angle at particlex∈xt2,-d2 after unloading and springback is(56)αxu=sin-1⁡ϕ2x(57)ϕ2x=-nn+1MmaxkIpl0-d21/nl0-xn+1/n-l0-d2n+1/n+Mmax2EIl0-d2l0-x2-l0-d22+d21rp+t/2-MmaxEI


## 5. Experimental Verification and Numerical Simulation

### 5.1. Material Parameters and Experimental Equipment

Tensile test was carried out with Q235b steel plate. The yield strengthσs, strength coefficient k, hardening index n, and elastic modulus E of the material were obtained after analysis and treatment, as shown in Table 1.Table 1
Material properties of sheet metal.
Name The yield strength Coefficient of strength Modulus of elasticity Hardening index σ s / M P a k / M P a E / G P a n Q235b steel plate 236.12 540.11 206.15 0.19The five-point bending test device is mainly composed of the following parts: electronic universal testing machine, bending mold, resistance strain gauge, data acquisition card, pressure sensor, and computer, shown in Figure4. The model of the electronic universal testing machine is AG-100, the capacity is 100 kN, and the resolution is 0.01 mm. The strain gauge is located on both sides of the bottom mold (Figure 4). The pressure sensor is located in the electronic universal testing machine. The electronic universal testing machine can collect the displacement and force of the upper die. The resistance strain gauge can collect the strain data of the bottom mold, by calibrating the bottom mold in advance, the relationship between deformation and force is obtained, and then the force of the bottom roll in bending process is obtained.Figure 4
The test device of five-point bending.


### 5.2. Establishment of Finite Element Model

Abaqus is one of the most advanced large-scale general-purpose nonlinear finite element analysis software. It has powerful calculation functions and extensive simulation performance. In this paper, Abaqus software is used to simulate loading and unloading process of five-point bending. The five-point bending process belongs to plane strain deformation, so this paper uses a two-dimensional model to simulate its bending process. In addition, because the boundary conditions and geometric models are simple, static implicit algorithms can be used to simulate the loading and unloading process. The plate parameters are the same as the experimental plate parameters. Define the die as a discrete rigid body. Set 3 elements on the thickness of the sheet. The element type is CPE4 (quadrilateral plane strain element), and the number of nodes is 1200. Use the general contact to simulate contact conditions between the sheet and the mold. The upper roll die can only move in the y direction, and the lower roll die and the middle roll die are fixed. The y direction displacement of the sheet section element when the upper die is pressed 8mm is as shown in Figure5. The unit is millimeter.Figure 5
Finite element analysis displacement diagram.


### 5.3. Comparison of Theoretical Results, Numerical Simulation, and Experimental Results of Bending Force

Using the five-point bending theory and the same finite element geometry to calculate the bending force, compare and analyze the displacement of the upper roll die. The relevant calculation parameters are plate widthB=200mm, thickness t=2mm, the yield strength σs=236.12MPa, coefficient of strength k=540.11MPa, hardening index n=0.19, modulus of elasticity E=206.15GPa, the radius of upper roll die rp=12.5mm, the radius of bottom roll die rd=15mm, the radius of the lower roll die rf=20mm, the span between the lower roll die, and the bottom roll die L3=95mm. According to the experimental results in reference [24], the friction coefficient has little effect on the rebound angle. By simulating the whole process of FFX forming with explicit method, it is found that when the friction coefficient is 0.15, the strip can enter the roll gap smoothly; thus, the friction coefficient in the theoretical calculation is 0.15. Figure 6 shows the comparison between theoretical calculation value of bending force and experimental results.Figure 6
Comparison of bending force between the experimental results, numerical simulation, and theoretical analysis.From Figure6 we can see that the bending force increases as the bending stroke increases. There is a certain error between the theoretical calculation result, numerical simulation, and the experimental result, and the error increases as the bending stroke increases. The maximum relative error is about 9.79%. At the same time, it can be found that the results of theoretical analysis are generally smaller than the experimental results and numerical simulation. The reasons for the errors are mainly concentrated in two aspects. On the one hand, the derivation of theoretical calculation formula is carried out under some assumptions, such as material hardening model, tension-free assumption, the assumption that the neutral layer, and the geometric center line always coincide; on the other hand, there is also a certain deviation in testing. On the whole, the theoretical value and experimental value are basically consistent, and the trend of change is consistent.


### 5.4. Comparison of Theoretical Results, Numerical Simulation, and Experimental Results of Springback Angle

In order to verify the correctness of the analytical results of springback theory, theoretical values of springback angle under different bending stroke were calculated according to the analytical formula derived above and compared with numerical simulation and experimental results as shown in Figure7.Figure 7
The comparison of springback angle between the theoretical results, numerical simulation, and the experimental results.From Figure7 we can see that the springback angle increases with the increase of bending stroke, when the bending stroke is 8 mm, the corresponding rebound angle is about 3.7°. At the same time, experimental and numerical simulation results show that the theoretical calculation of the rebound angle agrees well with the experimental results, the overall trend is the same, and the largest angular deviation is 0.43°. It shows that the five-point bending rebound model established in this paper is reasonable and reliable.


## 5.1. Material Parameters and Experimental Equipment

Tensile test was carried out with Q235b steel plate. The yield strengthσs, strength coefficient k, hardening index n, and elastic modulus E of the material were obtained after analysis and treatment, as shown in Table 1.Table 1
Material properties of sheet metal.
Name The yield strength Coefficient of strength Modulus of elasticity Hardening index σ s / M P a k / M P a E / G P a n Q235b steel plate 236.12 540.11 206.15 0.19The five-point bending test device is mainly composed of the following parts: electronic universal testing machine, bending mold, resistance strain gauge, data acquisition card, pressure sensor, and computer, shown in Figure4. The model of the electronic universal testing machine is AG-100, the capacity is 100 kN, and the resolution is 0.01 mm. The strain gauge is located on both sides of the bottom mold (Figure 4). The pressure sensor is located in the electronic universal testing machine. The electronic universal testing machine can collect the displacement and force of the upper die. The resistance strain gauge can collect the strain data of the bottom mold, by calibrating the bottom mold in advance, the relationship between deformation and force is obtained, and then the force of the bottom roll in bending process is obtained.Figure 4
The test device of five-point bending.


## 5.2. Establishment of Finite Element Model

Abaqus is one of the most advanced large-scale general-purpose nonlinear finite element analysis software. It has powerful calculation functions and extensive simulation performance. In this paper, Abaqus software is used to simulate loading and unloading process of five-point bending. The five-point bending process belongs to plane strain deformation, so this paper uses a two-dimensional model to simulate its bending process. In addition, because the boundary conditions and geometric models are simple, static implicit algorithms can be used to simulate the loading and unloading process. The plate parameters are the same as the experimental plate parameters. Define the die as a discrete rigid body. Set 3 elements on the thickness of the sheet. The element type is CPE4 (quadrilateral plane strain element), and the number of nodes is 1200. Use the general contact to simulate contact conditions between the sheet and the mold. The upper roll die can only move in the y direction, and the lower roll die and the middle roll die are fixed. The y direction displacement of the sheet section element when the upper die is pressed 8mm is as shown in Figure5. The unit is millimeter.Figure 5
Finite element analysis displacement diagram.


## 5.3. Comparison of Theoretical Results, Numerical Simulation, and Experimental Results of Bending Force

Using the five-point bending theory and the same finite element geometry to calculate the bending force, compare and analyze the displacement of the upper roll die. The relevant calculation parameters are plate widthB=200mm, thickness t=2mm, the yield strength σs=236.12MPa, coefficient of strength k=540.11MPa, hardening index n=0.19, modulus of elasticity E=206.15GPa, the radius of upper roll die rp=12.5mm, the radius of bottom roll die rd=15mm, the radius of the lower roll die rf=20mm, the span between the lower roll die, and the bottom roll die L3=95mm. According to the experimental results in reference [24], the friction coefficient has little effect on the rebound angle. By simulating the whole process of FFX forming with explicit method, it is found that when the friction coefficient is 0.15, the strip can enter the roll gap smoothly; thus, the friction coefficient in the theoretical calculation is 0.15. Figure 6 shows the comparison between theoretical calculation value of bending force and experimental results.Figure 6
Comparison of bending force between the experimental results, numerical simulation, and theoretical analysis.From Figure6 we can see that the bending force increases as the bending stroke increases. There is a certain error between the theoretical calculation result, numerical simulation, and the experimental result, and the error increases as the bending stroke increases. The maximum relative error is about 9.79%. At the same time, it can be found that the results of theoretical analysis are generally smaller than the experimental results and numerical simulation. The reasons for the errors are mainly concentrated in two aspects. On the one hand, the derivation of theoretical calculation formula is carried out under some assumptions, such as material hardening model, tension-free assumption, the assumption that the neutral layer, and the geometric center line always coincide; on the other hand, there is also a certain deviation in testing. On the whole, the theoretical value and experimental value are basically consistent, and the trend of change is consistent.


## 5.4. Comparison of Theoretical Results, Numerical Simulation, and Experimental Results of Springback Angle

In order to verify the correctness of the analytical results of springback theory, theoretical values of springback angle under different bending stroke were calculated according to the analytical formula derived above and compared with numerical simulation and experimental results as shown in Figure7.Figure 7
The comparison of springback angle between the theoretical results, numerical simulation, and the experimental results.From Figure7 we can see that the springback angle increases with the increase of bending stroke, when the bending stroke is 8 mm, the corresponding rebound angle is about 3.7°. At the same time, experimental and numerical simulation results show that the theoretical calculation of the rebound angle agrees well with the experimental results, the overall trend is the same, and the largest angular deviation is 0.43°. It shows that the five-point bending rebound model established in this paper is reasonable and reliable.


## 6. Conclusion

(1) According to the five-point bending characteristics of “W” antibending pass in FFX preforming of ERW pipe, under the premise of various assumptions, the five-point free bending model was obtained by simplifying the contact state model of roll bending process. The bending process is divided into two stages: fully elastic bending stage and elastic-plastic bending stage. At the same time, the mechanical model was established piecewise, and the mathematical expressions of bending curvature and turning angle of a particle on the strip under any state were derived.(2) According to the theory of plane elastic-plastic bending, the formulas for calculating the curvature of particles on the steel plate after unloading were derived, and the formulas for calculating the bending angle and rebound angle after springback were obtained.(3) The law of springback angle of five-point bending changing with stroke is revealed. At the same time, the springback angle of the theoretical maximum angular deviation is 0.43°. The experimental results show that the five-point bending model is reasonable, which provide theoretical support for the realization of setting model of FFX forming for roll bending process.



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*Source: 1703739-2019-06-23.xml*