# Precision Forming Technology for Crimping of Large Straight Welded Pipes

Authors: Li-feng Fan; Pei Liang; Ge Wang; Ying Gao; Benguo Zhang
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

Crimping plays a pivotal role in the production of large-diameter submerged-arc welded pipes. In the crimping forming process, predicting the springback while considering the real-time variation of material parameters is a major challenge faced by many researchers, which has a direct impact on the quality of the welded pipe. To address this problem, the precision forming technology of crimping was developed. The engineering theory of plastic bending was used to investigate the crimping forming process. Two methods, namely, the slope reverse method and the optimization method, were adopted to identify the material parameters. The results showed that the inverted material parameters can be evaluated in real time based on the analytical model of crimping. The maximum relative error of the identification value is less than 4%. Therefore, the springback and displacement during crimping can be predicted dynamically to control the crimping forming quality. Thus, this project provides an important opportunity to achieve precise forming through crimping.


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## Body

## 1. Introduction

The forming technology of crimping has been extensively used in the production of large-diameter straight seam submerged-arc welded (LSAW) pipes. To improve the geometry and dimensional accuracy of LSAW pipes, both sides of the sheet edge were bent into a certain nominal curvature by crimping, which can effectively prevent peach breaks and cracking during expansion.A considerable amount of literature has been published on crimping forming processes. These studies address the stress and strain state of the plate in crimping [1], the effects of technological parameters on the crimping forming quality [2], and the effects of crimping on the pipe quality [3–5] by the finite element analysis (FEA) method.The most significant disadvantages of FEA in the design of crimping parameters are its time required and the difficulty in identifying material parameters. However, the analytical method can improve the computational efficiency by idealizing the forming conditions. Scholars have analyzed the crimping forming process quality using analytical methods for the design parameters [6, 7], bending moment/forming force [8], stress/strain, and springback [9].It has previously been observed that the control of springback, which happens after the removal of the forming loads, remains a key issue in the crimping process. A number of intersectional studies have been published on springback calculations [10–12], springback reduction [13, 14], and the springback mechanism [15]. However, few studies have investigated the influence of the fluctuation of mechanical properties on the control accuracy of springback in applications. Hence, efforts have been made to reduce the influence of the fluctuation of mechanical material properties on springback by the robustness design method [16, 17], but this work still has not achieved the purpose of precise forming of crimping.Therefore, the purpose of this investigation is to propose a precision forming method for crimping. First, the quantitative relationship between the bending displacement and bending force was calculated by the analytical method of crimping. Second, the identification of material parameters was performed by the gray correlation method. Finally, crimping displacement predictions were achieved by the springback prediction model. The ideal crimping displacement can undergo dynamic adjustment by using the precision forming technology for crimping. The proposed technology has demonstrated that it is not only the forming accuracy that can be effectively improved in precision crimping but a lot of design times can be saved.


## 2. Analytical Model of Crimping

In this study, the analytical method as an efficient tool was used to calculate the quantitative relationship between the bending displacement and bending force. In the crimping forming process, the punch and the edges of the sheet are fixed. With the die loading progressively increasing, the edges of the sheet are gradually bent along the punch surface. Then, the process of unloading the sheet is begun by moving the die down when the sheet reaches the target bending angle. Crimping is completed after springback of the sheet (Figure1).Figure 1
Crimping sketch.The parameter equation of the involute can be used to describe the shape of the punch and die in Cartesian coordinates:(1)xφ=Rpcosφ+φRpsinφ,yφ=Rpsinφ+φRpcosφ,where Rp is the base radius and φ is the base angle.To facilitate subsequent calculation, a newXOY coordinate system, with point O as the original point that is the involute end point corresponding to the terminal base angle βp, is established. The parameter equation of the crimping punch in the XOY coordinate system is modified as follows:(2)Xφ=xφ−xβpcosβp+yφ−yβpsinβp,Yφ=−xφ−xβpsinβp+yφ−yβpcosβp.Therefore, the curvature radius of punchρ according to the involute property can be calculated as follows:(3)ρ=Rp⋅βp−θ,where θ is the crimping bending angle.In the analytical model of crimping, the description of the material behavior is very important. To simplify the calculation, it is assumed that a state of plane strain exists:(4)ε=yρ.The elastic behavior according to Hooke’s law can be written as follows for a state of plane strain:(5)σ=Eε.As the stress-strain relationship for plastic deformation, a linear hardening law is adopted here:(6)σ=σs+Epε.In this study, three deformed stages can be determined in the whole crimping forming process: (1) elastic deformation, (2) unwrapped stage in the elastoplastic regime, and (3) wrapped stage in the elastoplastic regime.


### 2.1. Elastic Deformation Stage

When the radius of curvature in crimping sectionρ is larger than the elastic limit radius of curvature ρe(7)ρe=tE2σs.In this phase, the crimping deformation occupies the whole elastic deformation stage. Thus, the bending moment can be expressed as(8)M=EIρ,where I is the section moment of inertia, I=2∫0t/2y2dy.In addition, the bending moment can be calculated according to the external force(9)M=Fxy1−y+Fyx1−x,Fx=Fsinθ1,Fy=Fcosθ1,where Fx is the horizontal forming force and Fy is the vertical forming force.According to the geometry, the following relationships can be obtained:(10)dsdθ=ρ,dx=dscosθ,dy=dssinθ.We simultaneously apply equations (8)–(10) and differentiate both sides of the equation:(11)−EId2θds2=Fsinθ1sinθ+Fcosθ1cosθ.Note:(12)d2θds2=12ddθdθds2.Therefore, the equation can become(13)−EI2ddθds2=Fsinθ1sinθ+Fcosθ1cosθdθ.The following equation can be obtained by integrating both sides of differential equation:(14)−EI2dθds2=−Fsinθ1cosθ+Fcosθ1sinθ+C.According to boundary conditions, whenθ=θ1, ds/dθ and the coefficient C = 0. Therefore,(15)dsdθ=EI2Fsinθ1cosθ−cosθ1sinθ.The curve equation can be obtained by solving the differential equation, and the crimping angleθ1 can be determined by the following formula:(16)s=∫0θ1ψθdθ=L,where ψθ=EI/2Fsinθ1−θ.


### 2.2. Unwrapped Stage in Elastoplastic Deformation

As the bending displacement increases, the bending radius of the curvature is smaller than the maximum elastic radius of the curvature but still larger than the radius of curvature of the punch. The stage enters the unwrapped stage in elastoplastic deformation. Therefore, the entire deformation section can be divided into an elastic deformation section and an elastic-plastic deformation section. The elastic-plastic bending momentM can be expressed as(17)M=Aρ2+B1ρ+C,where A=2σs3−2Epσs3/3E2−σs3/E21−Ep/E, B=Ept3/12, and C=σst4/41−Ep/E.The differential equations can be obtained by simultaneous application of equations (2) and (3):(18)Adsdθ2+Bdθds+C=Fxy1−y+Fyx1−x.Differentiate as follows:(19)Addsdsdθ2+Bddsdθds=−Fxdyds−Fydxds.Note that(20)2d2sdθ2=ddsdsdθ2.The differential equation can be obtained as follows:(21)2Ad2sdθ2+Bd2θds2=−Fcosθ1−θ.The conditionsθ=θp and s=lp mark the boundary point between the elastic section and the elastoplastic section. In addition, at M=Mmax−e, the following formula can be satisfied:(22)Mmax−e=2FEIsinθ1−θp=16σst2.From this boundary condition, the relationship betweenF and the position of the boundary point is given but depends only on the value that cannot be determined by the elastoplastic deformation section, and the elastic deformation section is needed to supplement the solution:(23)s=∫θpθ1ψθdθ=L−lp=le.


### 2.3. Elastoplastic Deformation Wrapped Stage

As the bending deformation increases, the radius of curvature becomes equal to the radius of curvature of the punch at the corresponding position, and the deformation is in the elastoplastic deformation wrapped stage. The whole sheet can be divided into three sections: elastic, elastoplastic unwrapped, and elastoplastic wrapped section.The relationship between the bending moment and the curvature of the elastic-plastic cladding section can be calculated according to the radius of curvature of the convex model surface. The origin of the coordinate system is translated to the centroid of the starting position of the bending of the sheet, and the downward translation is available:(24)x∗φ=xφ−xβpcosβp+yφ−yβpsinβp,y∗φ−xφ−xβpsinβp+yφ−yβpcosβp+t2.According to the involute property, the curvature radius of the sheet intermediate layerρθ can be known:(25)ρθ=Rp⋅βp−θ+t2.Therefore, the bending moment of the covered section is(26)Mθ=2σs3−2Epσs33E2−σs3E21−EpEρθ2+Ept3121ρθ+σst441−EpE.According to the involute arc length formula, the length of the wrapping section can be obtained as(27)lw=∫θθwdx∗dθ2+dy∗dθ2dθ.Whenθw≤θ≤θp, the length of the unwrapping section can be obtained from(28)2Ad2sdθ2+Bd2θds2=−Fcosθ1−θ.The elastic deformation section is needed to supplement the solution:(29)s=∫θpθ1ψθdθ=L−lp−lw.


### 2.4. Verification for Analytical Model

In this study, an X80 gradeΦ1219×22×12000mm large-diameter straight seam welded pipe was taken as an example, and a #3 crimping mold was adopted as the research object. The mold parameters are shown in Table 1.Table 1
Parameters of the crimping mold.
Parameters
Value
Base radius of punchRp (mm)
303.20
Terminal angle of the punchβp (°)
88
Base radius of the dieRd (mm)
303.20
Terminal angle of the dieβd (°)
88
Length of crimpingB (mm)
190
Horizontal distance between the punch and dieD (mm)
80To evaluate the precision of the bending analysis model, the crimping forming process was conducted by a hydropress in the plant. The forming force and displacement can be collected with a pressure sensor and a displacement sensor. Then, the comparison between the measured values and those calculated from the analytical model is shown in Figure2.Figure 2
Verification with bending force.As seen from Figure2, the calculation results are in good agreement with the measurement results and their maximum relative error is less than 9.06%.


## 2.1. Elastic Deformation Stage

When the radius of curvature in crimping sectionρ is larger than the elastic limit radius of curvature ρe(7)ρe=tE2σs.In this phase, the crimping deformation occupies the whole elastic deformation stage. Thus, the bending moment can be expressed as(8)M=EIρ,where I is the section moment of inertia, I=2∫0t/2y2dy.In addition, the bending moment can be calculated according to the external force(9)M=Fxy1−y+Fyx1−x,Fx=Fsinθ1,Fy=Fcosθ1,where Fx is the horizontal forming force and Fy is the vertical forming force.According to the geometry, the following relationships can be obtained:(10)dsdθ=ρ,dx=dscosθ,dy=dssinθ.We simultaneously apply equations (8)–(10) and differentiate both sides of the equation:(11)−EId2θds2=Fsinθ1sinθ+Fcosθ1cosθ.Note:(12)d2θds2=12ddθdθds2.Therefore, the equation can become(13)−EI2ddθds2=Fsinθ1sinθ+Fcosθ1cosθdθ.The following equation can be obtained by integrating both sides of differential equation:(14)−EI2dθds2=−Fsinθ1cosθ+Fcosθ1sinθ+C.According to boundary conditions, whenθ=θ1, ds/dθ and the coefficient C = 0. Therefore,(15)dsdθ=EI2Fsinθ1cosθ−cosθ1sinθ.The curve equation can be obtained by solving the differential equation, and the crimping angleθ1 can be determined by the following formula:(16)s=∫0θ1ψθdθ=L,where ψθ=EI/2Fsinθ1−θ.


## 2.2. Unwrapped Stage in Elastoplastic Deformation

As the bending displacement increases, the bending radius of the curvature is smaller than the maximum elastic radius of the curvature but still larger than the radius of curvature of the punch. The stage enters the unwrapped stage in elastoplastic deformation. Therefore, the entire deformation section can be divided into an elastic deformation section and an elastic-plastic deformation section. The elastic-plastic bending momentM can be expressed as(17)M=Aρ2+B1ρ+C,where A=2σs3−2Epσs3/3E2−σs3/E21−Ep/E, B=Ept3/12, and C=σst4/41−Ep/E.The differential equations can be obtained by simultaneous application of equations (2) and (3):(18)Adsdθ2+Bdθds+C=Fxy1−y+Fyx1−x.Differentiate as follows:(19)Addsdsdθ2+Bddsdθds=−Fxdyds−Fydxds.Note that(20)2d2sdθ2=ddsdsdθ2.The differential equation can be obtained as follows:(21)2Ad2sdθ2+Bd2θds2=−Fcosθ1−θ.The conditionsθ=θp and s=lp mark the boundary point between the elastic section and the elastoplastic section. In addition, at M=Mmax−e, the following formula can be satisfied:(22)Mmax−e=2FEIsinθ1−θp=16σst2.From this boundary condition, the relationship betweenF and the position of the boundary point is given but depends only on the value that cannot be determined by the elastoplastic deformation section, and the elastic deformation section is needed to supplement the solution:(23)s=∫θpθ1ψθdθ=L−lp=le.


## 2.3. Elastoplastic Deformation Wrapped Stage

As the bending deformation increases, the radius of curvature becomes equal to the radius of curvature of the punch at the corresponding position, and the deformation is in the elastoplastic deformation wrapped stage. The whole sheet can be divided into three sections: elastic, elastoplastic unwrapped, and elastoplastic wrapped section.The relationship between the bending moment and the curvature of the elastic-plastic cladding section can be calculated according to the radius of curvature of the convex model surface. The origin of the coordinate system is translated to the centroid of the starting position of the bending of the sheet, and the downward translation is available:(24)x∗φ=xφ−xβpcosβp+yφ−yβpsinβp,y∗φ−xφ−xβpsinβp+yφ−yβpcosβp+t2.According to the involute property, the curvature radius of the sheet intermediate layerρθ can be known:(25)ρθ=Rp⋅βp−θ+t2.Therefore, the bending moment of the covered section is(26)Mθ=2σs3−2Epσs33E2−σs3E21−EpEρθ2+Ept3121ρθ+σst441−EpE.According to the involute arc length formula, the length of the wrapping section can be obtained as(27)lw=∫θθwdx∗dθ2+dy∗dθ2dθ.Whenθw≤θ≤θp, the length of the unwrapping section can be obtained from(28)2Ad2sdθ2+Bd2θds2=−Fcosθ1−θ.The elastic deformation section is needed to supplement the solution:(29)s=∫θpθ1ψθdθ=L−lp−lw.


## 2.4. Verification for Analytical Model

In this study, an X80 gradeΦ1219×22×12000mm large-diameter straight seam welded pipe was taken as an example, and a #3 crimping mold was adopted as the research object. The mold parameters are shown in Table 1.Table 1
Parameters of the crimping mold.
Parameters
Value
Base radius of punchRp (mm)
303.20
Terminal angle of the punchβp (°)
88
Base radius of the dieRd (mm)
303.20
Terminal angle of the dieβd (°)
88
Length of crimpingB (mm)
190
Horizontal distance between the punch and dieD (mm)
80To evaluate the precision of the bending analysis model, the crimping forming process was conducted by a hydropress in the plant. The forming force and displacement can be collected with a pressure sensor and a displacement sensor. Then, the comparison between the measured values and those calculated from the analytical model is shown in Figure2.Figure 2
Verification with bending force.As seen from Figure2, the calculation results are in good agreement with the measurement results and their maximum relative error is less than 9.06%.


## 3. Identification of the Material Parameters

To adjust the forming parameters online, it is necessary to identify the mechanical properties of materials in real time considering the influence of fluctuations of the mechanical material properties on the springback. Therefore, first, the effects of the mechanical properties on the bending load curve are analyzed based on the crimping analytical model, and then the initial segment of the bending load curve is used to identify the elastic modulus. Finally, the objective function is constructed by the gray correlation method, and the yield strength and the plastic modulus are identified by the inverse optimization method. In this section, the #3 crimping mold remains the research object.


### 3.1. Effects of the Material Parameters on the Forming Force

To improve the identification accuracy and efficiency, the effects of the elastic modulus, yield strength, and plastic modulus on the bending displacement and bending load curve are analyzed, as shown in Figures3–6.Figure 3
Effects of the elastic modulus on the bending load.Figure 4
Partial enlargement of Figure3.Figure 5
Effects of the yield strength on the bending load.Figure 6
Partial enlargement of Figure5.It can be seen from Figures3 and 4 that the elastic modulus has a significant influence on the initial segment of the curve and has little effect on the subsequent plastic segment, and it is found that the initial segment of the curve is approximately linear.It can be seen from Figures5–8 that the change of the yield strength has no effect on the initial section of the curve, but the influence on the subsequent plastic section of the curve is more obvious, and the bending load increases as the yield strength increases with the same bending stroke.Figure 7
Effects of the plastic modulus on the bending load.Figure 8
Partial enlargement of Figure7.


### 3.2. Identification of the Elastic Modulus

In the elastic deformation stage, it was determined from the theoretical analytical model that only the elastic modulus has a direct influence on the bending load. The analysis of the influence rate of the elastic modulus on the forming load at different displacements is shown in Figure9. This demonstrated that in the elastic deformation stage (h < 5 mm), the degree of influence increases with increasing displacement and that linearity is satisfied.Figure 9
Effect of the rate elastic modulus on the bending load.However, in the plastic deformation stage, the plastic material parameters play a major role, and the influence of the elastic modulus on the forming load is weakened. To identify the elastic modulus in the elastic deformation stage, the effect of the elastic modulus on the slope of the loading curve at the elastic deformation stage is analyzed in Figure10, as fit by the following linear formula:(30)E=−0.0028Sy+402.07,E=−0.0053Sx+402.08,where E is the elastic modulus and Sx, Sy is the slope of the horizontal and vertical forming forces, respectively.Figure 10
Effects of the elastic modulus on the slope of the loading curve during the elastic deformation stage.Therefore, the elastic modulus can be identified by equation (30).


### 3.3. Plastic Parameter Inversion

It is found that both the yield strength and the plastic modulus have a significant influence on the plastic segment of the bending load curve, and its trends are to increase slowly and then rapidly in Figures11 and 12.Figure 11
Effect of the rate plastic modulus on the bending load.Figure 12
Effect of the rate yield strength on the bending load.Therefore, to identify the plastic parameters, the yield strength and plastic modulus are substituted into the crimping analytical model for trial calculation by the optimization algorithm, in which the calculated curve is refined to approximate the measured curve until it reaches the allowable error.The gray relational theory was applied to construct the objective function of inverse identificationfobject, which can comprehensively evaluate the approximation characteristics of the calculated curve and the measured curve as follows.


#### 3.3.1. Reference Sequence and Relative Sequence

The bending loadsFe=F1F2…Fn corresponding to the measured bending displacements h=h1h2…hn were selected as the reference sequence, and then the bending loads Fik=Fi1Fi2…Fin corresponding to the bending displacements h=h1h2…hn were calculated by the analytical model by a genetic algorithm for comparison.


#### 3.3.2. Dimensionless Data

To facilitate analysis, it is necessary to conduct dimensionless data processing on the original data before the comparison. When the target has visual characteristics, the data processing is as follows:(31)Fidk=Fik−mini∈NFikmaxi∈NFik−Fek−mini∈NFik−Fek,where Fidk is the Fik data processing results.


#### 3.3.3. Gray Correlation Coefficient

The gray relational coefficients were used to measure the difference between the theoretical and experimental results. The gray relational coefficientξik can be written as follows:(32)ξik=mini∈Nmini∈NFik+ρmaxi∈Nmaxi∈NFek−FikFek−Fikρmaxi∈Nmaxi∈NFek−Fik,where ρ = 0.5436.


#### 3.3.4. Gray Relational Grade

The gray relational grade corresponding to each performance characteristic is computed, and the overall evaluation of the multiresponse characteristics is based on the gray relational grade, which is given by [18]:(33)rFe,Fi=1n∑k=1nλkξik.A higher gray relational grade represents that the experimental result is closer to the ideally normalized value.


#### 3.3.5. Gray Difference Degree

In the optimization design, the minimum value of the objective function is usually obtained, so the concept of gray difference degree is introduced [19]:(34)di=1−rFe,Fi.Therefore, the above gray difference is used as the objective function of identification optimization:(35)fobject=mindiFe,Fi.Finally, the optimization model is constructed as follows:(36)design variableσs,Ep,constraint conditionσsl≤σs≤σsu,Epl≤Ep≤Epu,fobject=mindiFe,Fi.The relatively stable NLPQL (nonlinear programming by quadratic Lagrangian) algorithm is selected as the inversion optimization algorithm to solve the optimization model, which expands the objective function by the second-order Taylor series, linearizes the constraint condition, and obtains the next design point by solving the quadratic programming optimization problem. Finally, the plastic parameters, including the yield strength and plastic modulus, were identified by optimization.


### 3.4. Verification for Parameter Identification

To verify the reliability of the parameter identification technology, the material mechanical properties of X80 were calculated by the parameter identification technique described in 3.3 and compared with the results of the uniaxial tensile test in Table2.Table 2
Comparison with identification and test.
Parameters
Identification
Test
Relative error (%)
Elastic modulusE (GPa)
203
210
3.3
Yield strengthσs (MPa)
575
596
3.5
Plastic modulusEp (MPa)
5702
5500
3.67As seen from Table2, the maximum relative error of the identification value is less than 4%. This satisfies the requirements.


## 3.1. Effects of the Material Parameters on the Forming Force

To improve the identification accuracy and efficiency, the effects of the elastic modulus, yield strength, and plastic modulus on the bending displacement and bending load curve are analyzed, as shown in Figures3–6.Figure 3
Effects of the elastic modulus on the bending load.Figure 4
Partial enlargement of Figure3.Figure 5
Effects of the yield strength on the bending load.Figure 6
Partial enlargement of Figure5.It can be seen from Figures3 and 4 that the elastic modulus has a significant influence on the initial segment of the curve and has little effect on the subsequent plastic segment, and it is found that the initial segment of the curve is approximately linear.It can be seen from Figures5–8 that the change of the yield strength has no effect on the initial section of the curve, but the influence on the subsequent plastic section of the curve is more obvious, and the bending load increases as the yield strength increases with the same bending stroke.Figure 7
Effects of the plastic modulus on the bending load.Figure 8
Partial enlargement of Figure7.


## 3.2. Identification of the Elastic Modulus

In the elastic deformation stage, it was determined from the theoretical analytical model that only the elastic modulus has a direct influence on the bending load. The analysis of the influence rate of the elastic modulus on the forming load at different displacements is shown in Figure9. This demonstrated that in the elastic deformation stage (h < 5 mm), the degree of influence increases with increasing displacement and that linearity is satisfied.Figure 9
Effect of the rate elastic modulus on the bending load.However, in the plastic deformation stage, the plastic material parameters play a major role, and the influence of the elastic modulus on the forming load is weakened. To identify the elastic modulus in the elastic deformation stage, the effect of the elastic modulus on the slope of the loading curve at the elastic deformation stage is analyzed in Figure10, as fit by the following linear formula:(30)E=−0.0028Sy+402.07,E=−0.0053Sx+402.08,where E is the elastic modulus and Sx, Sy is the slope of the horizontal and vertical forming forces, respectively.Figure 10
Effects of the elastic modulus on the slope of the loading curve during the elastic deformation stage.Therefore, the elastic modulus can be identified by equation (30).


## 3.3. Plastic Parameter Inversion

It is found that both the yield strength and the plastic modulus have a significant influence on the plastic segment of the bending load curve, and its trends are to increase slowly and then rapidly in Figures11 and 12.Figure 11
Effect of the rate plastic modulus on the bending load.Figure 12
Effect of the rate yield strength on the bending load.Therefore, to identify the plastic parameters, the yield strength and plastic modulus are substituted into the crimping analytical model for trial calculation by the optimization algorithm, in which the calculated curve is refined to approximate the measured curve until it reaches the allowable error.The gray relational theory was applied to construct the objective function of inverse identificationfobject, which can comprehensively evaluate the approximation characteristics of the calculated curve and the measured curve as follows.


### 3.3.1. Reference Sequence and Relative Sequence

The bending loadsFe=F1F2…Fn corresponding to the measured bending displacements h=h1h2…hn were selected as the reference sequence, and then the bending loads Fik=Fi1Fi2…Fin corresponding to the bending displacements h=h1h2…hn were calculated by the analytical model by a genetic algorithm for comparison.


### 3.3.2. Dimensionless Data

To facilitate analysis, it is necessary to conduct dimensionless data processing on the original data before the comparison. When the target has visual characteristics, the data processing is as follows:(31)Fidk=Fik−mini∈NFikmaxi∈NFik−Fek−mini∈NFik−Fek,where Fidk is the Fik data processing results.


### 3.3.3. Gray Correlation Coefficient

The gray relational coefficients were used to measure the difference between the theoretical and experimental results. The gray relational coefficientξik can be written as follows:(32)ξik=mini∈Nmini∈NFik+ρmaxi∈Nmaxi∈NFek−FikFek−Fikρmaxi∈Nmaxi∈NFek−Fik,where ρ = 0.5436.


### 3.3.4. Gray Relational Grade

The gray relational grade corresponding to each performance characteristic is computed, and the overall evaluation of the multiresponse characteristics is based on the gray relational grade, which is given by [18]:(33)rFe,Fi=1n∑k=1nλkξik.A higher gray relational grade represents that the experimental result is closer to the ideally normalized value.


### 3.3.5. Gray Difference Degree

In the optimization design, the minimum value of the objective function is usually obtained, so the concept of gray difference degree is introduced [19]:(34)di=1−rFe,Fi.Therefore, the above gray difference is used as the objective function of identification optimization:(35)fobject=mindiFe,Fi.Finally, the optimization model is constructed as follows:(36)design variableσs,Ep,constraint conditionσsl≤σs≤σsu,Epl≤Ep≤Epu,fobject=mindiFe,Fi.The relatively stable NLPQL (nonlinear programming by quadratic Lagrangian) algorithm is selected as the inversion optimization algorithm to solve the optimization model, which expands the objective function by the second-order Taylor series, linearizes the constraint condition, and obtains the next design point by solving the quadratic programming optimization problem. Finally, the plastic parameters, including the yield strength and plastic modulus, were identified by optimization.


## 3.3.1. Reference Sequence and Relative Sequence

The bending loadsFe=F1F2…Fn corresponding to the measured bending displacements h=h1h2…hn were selected as the reference sequence, and then the bending loads Fik=Fi1Fi2…Fin corresponding to the bending displacements h=h1h2…hn were calculated by the analytical model by a genetic algorithm for comparison.


## 3.3.2. Dimensionless Data

To facilitate analysis, it is necessary to conduct dimensionless data processing on the original data before the comparison. When the target has visual characteristics, the data processing is as follows:(31)Fidk=Fik−mini∈NFikmaxi∈NFik−Fek−mini∈NFik−Fek,where Fidk is the Fik data processing results.


## 3.3.3. Gray Correlation Coefficient

The gray relational coefficients were used to measure the difference between the theoretical and experimental results. The gray relational coefficientξik can be written as follows:(32)ξik=mini∈Nmini∈NFik+ρmaxi∈Nmaxi∈NFek−FikFek−Fikρmaxi∈Nmaxi∈NFek−Fik,where ρ = 0.5436.


## 3.3.4. Gray Relational Grade

The gray relational grade corresponding to each performance characteristic is computed, and the overall evaluation of the multiresponse characteristics is based on the gray relational grade, which is given by [18]:(33)rFe,Fi=1n∑k=1nλkξik.A higher gray relational grade represents that the experimental result is closer to the ideally normalized value.


## 3.3.5. Gray Difference Degree

In the optimization design, the minimum value of the objective function is usually obtained, so the concept of gray difference degree is introduced [19]:(34)di=1−rFe,Fi.Therefore, the above gray difference is used as the objective function of identification optimization:(35)fobject=mindiFe,Fi.Finally, the optimization model is constructed as follows:(36)design variableσs,Ep,constraint conditionσsl≤σs≤σsu,Epl≤Ep≤Epu,fobject=mindiFe,Fi.The relatively stable NLPQL (nonlinear programming by quadratic Lagrangian) algorithm is selected as the inversion optimization algorithm to solve the optimization model, which expands the objective function by the second-order Taylor series, linearizes the constraint condition, and obtains the next design point by solving the quadratic programming optimization problem. Finally, the plastic parameters, including the yield strength and plastic modulus, were identified by optimization.


## 3.4. Verification for Parameter Identification

To verify the reliability of the parameter identification technology, the material mechanical properties of X80 were calculated by the parameter identification technique described in 3.3 and compared with the results of the uniaxial tensile test in Table2.Table 2
Comparison with identification and test.
Parameters
Identification
Test
Relative error (%)
Elastic modulusE (GPa)
203
210
3.3
Yield strengthσs (MPa)
575
596
3.5
Plastic modulusEp (MPa)
5702
5500
3.67As seen from Table2, the maximum relative error of the identification value is less than 4%. This satisfies the requirements.


## 4. Crimping Displacement Prediction

### 4.1. Springback Prediction Model

To predict the crimping displacement, a calculation model of the springback must be constructed by the analysis method. Thus, according to unloading laws, the curvature after springback can be obtained by(37)1ρθ′=1ρθ−1ρθe.The springback curvature can be expressed as follows:(38)1ρθe=MθEI.For the elastoplastic deformation wrapped stage, the bending moment can be calculated by equations (25) and (26). For the unwrapped stage in elastoplastic deformation, the bending moment can be calculated by equation (16). In addition, the differential equations can be established:(39)1ρθ′=dsdθ.WhenM=Mmax−e=1/6σst2, θ=θp, and s=lp, the angle after springback θ=θ∗ can be obtained by solving the above differential equations. Then, we evaluate whether the crimping angle after springback satisfies the target value, and if this condition is satisfied, the predicted displacement is obtained. If not, the displacement is increased and recalculated until the criterion is satisfied.To evaluate the precision of the springback analysis model, the crimping forming process is conducted by a hydropress in the plant. The forming force and displacement can be collected with a pressure sensor and a displacement sensor. Then, the comparison between measurement and calculation from the analytical model is shown in Figure13.Figure 13
Verification with the bending angle.As seen from Figure13, the calculation results are in good agreement with the measurement results and their maximum relative error is 4.76%.


### 4.2. Application

In this study, aΦ1219×22×12000mm large-diameter straight seam welded pipe was taken as an example. First, the elastic modulus E, yield strength σs, and plastic modulus Ep of the mechanical properties of the material were determined by tensile testing. Then, a #3 mold was used for crimping forming.To verify the feasibility of precision forming technology for crimping, the crimping forming process was performed using different displacements: 23, 28, 33, 38, and 42 mm. The forming displacement can be obtained from the displacement sensor, and the crimping angle after springback was measured by three-coordinate measuring instruments. Finally, a comparison between the calculations and measurements is shown in Figure14.Figure 14
Verification with bending angle and bending displacement.It can be seen from Figure14 that the proposed precise forming technology can dynamically adjust the bending displacement in real time and effectively control the quality of crimping forming.


## 4.1. Springback Prediction Model

To predict the crimping displacement, a calculation model of the springback must be constructed by the analysis method. Thus, according to unloading laws, the curvature after springback can be obtained by(37)1ρθ′=1ρθ−1ρθe.The springback curvature can be expressed as follows:(38)1ρθe=MθEI.For the elastoplastic deformation wrapped stage, the bending moment can be calculated by equations (25) and (26). For the unwrapped stage in elastoplastic deformation, the bending moment can be calculated by equation (16). In addition, the differential equations can be established:(39)1ρθ′=dsdθ.WhenM=Mmax−e=1/6σst2, θ=θp, and s=lp, the angle after springback θ=θ∗ can be obtained by solving the above differential equations. Then, we evaluate whether the crimping angle after springback satisfies the target value, and if this condition is satisfied, the predicted displacement is obtained. If not, the displacement is increased and recalculated until the criterion is satisfied.To evaluate the precision of the springback analysis model, the crimping forming process is conducted by a hydropress in the plant. The forming force and displacement can be collected with a pressure sensor and a displacement sensor. Then, the comparison between measurement and calculation from the analytical model is shown in Figure13.Figure 13
Verification with the bending angle.As seen from Figure13, the calculation results are in good agreement with the measurement results and their maximum relative error is 4.76%.


## 4.2. Application

In this study, aΦ1219×22×12000mm large-diameter straight seam welded pipe was taken as an example. First, the elastic modulus E, yield strength σs, and plastic modulus Ep of the mechanical properties of the material were determined by tensile testing. Then, a #3 mold was used for crimping forming.To verify the feasibility of precision forming technology for crimping, the crimping forming process was performed using different displacements: 23, 28, 33, 38, and 42 mm. The forming displacement can be obtained from the displacement sensor, and the crimping angle after springback was measured by three-coordinate measuring instruments. Finally, a comparison between the calculations and measurements is shown in Figure14.Figure 14
Verification with bending angle and bending displacement.It can be seen from Figure14 that the proposed precise forming technology can dynamically adjust the bending displacement in real time and effectively control the quality of crimping forming.


## 5. Conclusions

To calculate the springback considering the real-time variation of material parameters, the identification of material parameters was performed by an analytical model of crimping and the inverse optimization method. Then, the crimping displacement was predicted based on the springback model. The conclusions were as follows:(1)
Three phases compose the whole crimping forming process: the elastic deformation stage, the unwrapped stage in elastoplastic deformation, and the wrapped stage in elastoplastic deformation. The quantitative relationship between the bending displacement and bending force was calculated by an analytical model of crimping.(2)
The elastic modulus has a significant effect on the bending force curve, and the initial segment of the bending force curve is approximately linear. The elastic modulus can be identified by the linear formula.(3)
The yield strength and the plastic modulus have a significant effect on the plastic segment of the bending load curve, and the trends are to increase slowly and then rapidly. The plastic parameters, including the yield strength and plastic modulus, can be identified by the gray correlation method.(4)
Precision forming technology for crimping was achieved by identification of the material parameters and by crimping displacement prediction technology.



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*Source: 1376164-2019-10-31.xml*