Int J Adv Manuf Technol (2009) 45:448–458 DOI 10.1007/s00170-009-1982-2

ORIGINAL ARTICLE

Study on multiple-step incremental air-bending forming of sheet metal with springback model and FEM simulation Zemin Fu & Jianhua Mo & Wenxian Zhang

Received: 14 September 2008 / Accepted: 12 February 2009 / Published online: 3 March 2009 # Springer-Verlag London Limited 2009

Abstract A mathematical model of springback radius was developed with dimensional analysis and orthogonal test. With this model, the punch radius could be solved for forming high-precision semiellipse-shaped workpieces. With the punch radius and other geometrical parameters of a tool, a 2D ABAQUS finite-element model (FEM) was established. Then, the forming process of sheet metal multiple-step incremental air bending was simulated with the FEM. The result showed that average errors of the simulated workpiece were +0.68/−0.65 mm, and provided the process data consisting of sheet feed rate, punch displacement and springback angle in each step. A semiellipse-shaped workpiece, whose average errors are +0.68/−0.69 mm, was made with the simulation data. These results indicate that the punch design method is feasible with the mathematical model, and the means of FEM simulation is effective. It can be taken as a new approach for sheet metal multiple-step incremental air-bending forming and tool design. Keywords Multiple-step air bending . Sheet metal incremental forming . Springback . Mathematical model . Dimensional analysis . FEM simulation

Z. Fu (*) : J. Mo State Key Laboratory of Material Processing and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan 430074 Hubei, People’s Republic of China e-mail: [email protected] W. Zhang Department of Mechanical Engineering, Xiamen University of Technology, Xiamen 361024 Fujian, People’s Republic of China

1 Introduction Air bending (Fig. 1b) is performed with a punch and a die having a pair of shoulders. The die gap is set according to the calculated requirements, and the sheet metal is placed on the shoulders. The punch at the mid-span of the die is given a displacement, and the die is deep enough to avoid the sheet from striking its bottom. This single-step airbending processing is suitable for workpieces of simple profile. However, parts with complex surface could not be bent by the single-step air bending. To solve this problem, a multiple-step incremental air-bending forming could be utilized [1–3]. This forming process is a flexible sheetmetal-forming technology that uses principles of stepped manufacturing. It transforms the complicated geometry information into a series of parameters of single-step, and then the plastic deformation is carried out step by step through the computer numerical controlled movements of the punch and sheet feed to get the desired part. Therefore, the process is cost-effective to form large complex parts in small to medium batches such as semiellipse-shaped workpieces which could be used as crane boom, telescopic arm of the concrete pump truck, boom of bridge, petroleum piping, etc. In the air-bending process, after release of the load by withdrawal of the punch, the metal tries to return to its original shape because of the elastic stresses. This phenomenon is called “springback”. Because of the existence of springback, the precision of products and subsequent assembly operations were severely affected. How to effectively control springback has been the key to precision forming and design of tools. The traditional methods at home and abroad are as follows: make several tools in advance, then modify the springback by experiment iteration, and obtain the process data. This practice would

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nal test, and check its applicability from the view point of making the production process more efficient. Though there are literatures on the FEM simulation of sheet metal single-step air-bending forming [6–9], little is reported on that of the multi-step one. Therefore, in order to reduce the number of forming experiments, the authors think that it is of great worth to study multi-step incremental air-bending forming with FEM simulation method to obtain the optimized process parameters of workpiece-forming. In the present paper, springback and FEM simulation of multiple-step incremental air-bending forming were taken as the subject investigated. Based on orthogonal test, a mathematical model of springback radius was developed by dimensional analysis. Then, a punch was designed according to the model, geometric planning of multi-step incremental-forming process was performed, and incremental-forming processes were simulated; finally, a semiellipse-shaped workpiece with 11,638 mm×609 mm× 395 mm was manufactured with the simulation results. Therefore, the above-work in the paper could provide theoretical and practical guide for sheet metal multiple-step incremental air-bending forming and tools design.

2 Springback radius model

Fig. 1 Geometrical model of a die, a punch, and a workpiece (a) location (b) before unload (c) after unload

consume high mold cost, much time as well as labor cost. In addition, much work has also been investigated on models of air-bending springback. Two of the earliest works in the field were performed by Schroeder [4] and Gardiner [5]. Schroeder proposed a method to quantitatively predict the residual stress distribution and the magnitude of springback. He assumed the workpiece to be a narrow beam, that assumption can produce inaccurate results because transverse stresses present during forming are ignored. Gardiner developed a mathematical analysis for springback prediction of a workpiece whose thickness and length is assumed to remain unchanged, but there exists deficiency of the assumption, and the effect of practical application on the springback model has not been discussed. Therefore, this study attempts to establish a springback model with dimensional analysis and orthogo-

Air-bending springback is unavoidable in the sheet metal forming, and the springback leads to the change of bending radius. The bending radius of a sheet metal after unloading is called springback radius (Fig. 1). In the forming processes, there are parameters such as the material performance, geometric parameters of sheet metal, tool topological structure, tool gap, punch displacement, friction, etc. Although they all affect springback radius to some extent, the influence degree is various. On the base of experimental verification and orthogonal test analysis [10– 14], the four most significant influential factors were considered here: punch radius (r), sheet thickness (t), yield strength (σ) and Young’s modulus (E; Fig. 1). However, other parameters with comparatively minor influence on springback radius, like normal anisotropy, sheet length, friction, punch displacement, tool gap, etc., were ignored. Therefore, the model of springback radius (R) relating to the four factors is presented in Eq. (1): R ¼fðr; t; s; E Þ

ð1Þ

2.1 Dimensional analysis method As a method to establish mathematical model in the physical field, dimensional analysis could be used to

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confirm the relationship of the physical variables with the principle of dimensional homogeneity in physical laws and the Buckingham pi law [15]. Therefore, the functional relationship of the five variables in formula (1) could be solved with dimensional analysis method. The dimensions of every physical variable in the formula (1) can be expressed with three basic dimensions of length L, mass M, and time T. They are: ½R
¼ L;

½r
¼ L;

½t
¼ L;

½s
¼ L1 MT 2 ;

½E
¼ L1 MT 2

ð2Þ thus, the dimension matrix 2

1 A35 ¼ 4 0 0

1 1 0 0 0 0

1 1 2

Eq. (7) can be transferred to p 1 ¼ y ðp 2 ; p 3 Þ (where Ψ is some function).
r s  R So ¼y ; ð8Þ t t E i:e:

R¼ty


r s  ; t E

ð9Þ

The function Ψ could be derived with the orthogonal test. 2.2 The orthogonal test 2.2.1 Test condition

3

1 1 5 2

ð3Þ

The rank(A)=2, so the homogeneous equation has three fundamental solutions. Let be y1 =1, y2 =0, y4 =0, the first fundamental solution is y ¼ ð1; 0; 1; 0; 0ÞT ; Let be y1 =0, y2 =1, y4 =0, the second fundamental solution is y ¼ ð0; 1; 1; 0; 0ÞT ; Let be y1 =0, y2 =0, y4 =1, the third fundamental solution is y ¼ ð0; 0; 0; 1; 1ÞT .

The utilized sheet metal properties are shown in the Table 1. URSVIKEN OPTIMA 2200-t press brake was used for airbending tests of the sheet metal, the tool gap (c) between the punch and die is 20 mm (Fig. 1), the punch displacement is 45 mm in each test. The formed workpieces were measured with the Atos-II three-dimensional laser scanning system which possesses a ±0.005-mm precision. 2.2.2 Test methods and results

p 1 ¼ R1 r0 t 1 s 0 E 0 ¼

R t

ð4Þ

p 2 ¼ R0 r1 t 1 s 0 E 0 ¼

r t

ð5Þ

At first, the L25(53) factors level orthogonal tests were designed with the test parameters of t, r, and σ/E. Then, 25set tests were done under the different material properties and process parameters. The orthogonal test parameters were presented in Table 2. Lastly, the 3D data point cloud of the formed workpiece was obtained with the Atos-II three-dimensional laser measuring system. The point cloud data should be transferred into the CAD/CAM software for modeling. Springback radius of the formed workpiece can be obtained from the CAD model. The 25 sets of springback radius data are shown in Table 2.

p 3 ¼ R0 r0 t 0 s 1 E 1 ¼

s E

ð6Þ

2.3 Mathematical modeling of springback radius with dimensional analysis and orthogonal test

According to the Buckingham pi laws, three dimensionless parameters can be obtained with three fundamental solutions.

And there exists some function F that makes F ðp1 ; p 2 ; p3 Þ ¼ 0

ð7Þ

Table 1 Material properties

Mathematical expression of springback radius can be solved by combining Ψ function in section 2.1 with 25 sets of orthogonal test data (Table 2).

Material

E(Mpa)

σ(Mpa)

Poisson’s ratio

Density(kg/m3)

Q235A WELDOX700-1 WELDOX700-2 WELDOX900-1 WELDOX900-2

206378.3 217069.7 178846.6 205856.3 200405.4

239.783 776.287 699.842 950.812 956.009

0.3 0.266 0.243 0.284 0.276

7850 7830 7830 7830 7830

σ/E 0.00116 0.00358 0.00391 0.00466 0.00477

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Table 2 Orthogonal test design and air-bending test results Test number

σ/E

R (mm)

25 50 70 142 193 25 50 70

0.00116 0.00358 0.00391 0.00466 0.00477 0.00358 0.00391 0.00466

25.6 57.9 88.4 285.3 640.3 26.3 55.6 83.9

142 193 25 50 70 142 193 25 50 70 142 193 25 50 70 142 193

0.00477 0.00116 0.00391 0.00466 0.00477 0.00116 0.00358 0.00466 0.00477 0.00116 0.00358 0.00391 0.00477 0.00116 0.00358 0.00391 0.00466

216.11 217.6 26.3 55.7 81.9 152.9 275.6 26.4 55.1 72.3 175.9 270.4 26.2 51.2 76.6 174.8 277.2

t (mm)

r (mm)

1 2 3 4 5 6 7 8

4 4 4 4 4 6 6 6

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9

  Because y rt ; Es function is dimensionless, it is assumed that y

h

1 ri ti

; ðs=E Þi

i ¼ a25  

ti24 t23 þ a24  i23 þ       þ a2 24 ri ri

ti þ a1  ðs=E Þi ri

ð10Þ Replace formula (8) by (10), then

ðs=E Þi

R¼

t  5  4  3  2 12:36ð rt Þ7 þ 8:03ð rt Þ6  13:49 rt þ5:1 rt 0:82 rt þ0:04 rt þ rt  3:03 Es

ð13Þ When t<
t r

t  3:03 Es

ð14Þ

The mathematical model formula (14) can be used to design punch tool, which is described in Section 3. 2.4 Model results The relationship between various process parameters and springback radius (Fig. 2) can be obtained with the model formula (14). The punch radius was found to be the significant influencing factor on springback radius. For air bending of sheet metal, the springback error can be reduced by selecting those materials with smaller yield strength, bigger Young’s modulus and thicker sheet metal, which can be taken as a reference for design. From the model results, the relationship between R/r (ratio of springback radius to punch radius) and R/t (ratio of springback radius to sheet thickness) is nearly linear.

3 Punch design with the springback radius mathematical model

ði ¼ 1 . . . . . . 25Þ:

ti t 24 t 23 ti ¼ a25  i24 þ a24  i23 þ       þa2  þ a1 Ri r ri ri i


t 7
t 6
t 5
t 4 1  r s  ¼ 12:36 þ8:03 13:49 þ5:1 r r r r y t;E
t 3
t 2 t s  0:82 þ0:04 þ  3:03 r r r E ð12Þ

ð11Þ

ði ¼ 1 . . . . . . 25Þ:

Twenty-five equations can be established with 25 sets of orthogonal test data in Table2. The over 8-order polynomials in the equations could be neglected for their values approximating to zero. Consequently, the formulae (10) and (11) can be simplified into Eqs. (12), (13), respectively.

The section of the third arm of one crane telescopic boom was designed (Fig. 3), the workpiece length is 11,638 mm, which is made from WELDOX 900-1 (Table 1). Since the length of the workpiece neutral plane is constant before and after springback, the incrementalforming process can be planned according to geometric shape and size of the workpiece neutral plane (Fig. 4), which is detailed in Section 4. The workpiece is to be bent through 11 steps. The circle center of the springback arc in each step ensures to be on the small semiellipse, and the center of springback arc segment in each step ensures to be on the big semiellipse. The developed length for two adjacent center points of springback arc segments is

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Fig. 2 Relationships between various process parameters and springback radius

regarded as sheet feed rate. For example, the developed length of pr (Fig. 4) is the sheet feed rate of the second step bending. The biggest offset between the two concentric semiellipse is 194 mm, which can make springback arc segment in each step to be the most ideal curve approximating semiellipse. Therefore, the springback radius R=194−t/2=194−4= 190 mm in each step, and the punch radius r=142 mm could be solved with the mathematical model formula (14).

Fig. 3 Semiellipse-shaped workpiece

4 Geometric planning for multi-step incrementalforming process Eleven springback arcs and ten tangent lines in Fig. 4 were utilized to approximate the semiellipse, the concrete geometric planning shown in Fig. 5 are as follows: First step:

The larger semiellipse is offset by −194 mm, and a smaller one is generated. Then, the larger one is divided into ten equations. The neighboring two points among 11 equal

Fig. 4 Process planning of 11-step incremental air-bending forming of the third arm

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Fig. 5 Geometric planning of the third arm

diversion points of the outline are made up of lines. The adjacent lines consist of angles where angular bisectors intersect with the small semiellipse and intersecting points are obtained (Fig. 5a). Second step: In Fig. 5b, the circles of 194 mm radius are obtained with intersecting points as their circle centers. Then, the lines of ab and cd are obtained by the tangency of neighboring circles of A, B and C .With the tangent lines of ab and cd trimming the circle B, the arc bc is retained to approximate the semiellipse curve with the tangents ab and cd of the circle B. Other arcs are obtained with the same method. The arcs and the tangent lines consist of the large semiellipse D (Fig. 5c). Third step: In Fig. 5c, the arcs of bc, de, …… are divided into two equals, whose center points j, k, …… are assumed to be the centers of springback arc segment in each step. The intersecting points f, g, h, …… of the small semiellipse are hypothesized as the circle center of springback arc in each step. The developed length of jk, …... is supposed as the sheet feed rate in each step, angles between line E and the tangent lines of ab, cd, …… are assumed as springback angle each step (Fig. 5d).

5 FEM simulation To reduce the number of experiments for multiple-step incremental air-bending forming of sheet metal, process parameters have to be optimized with numerical simulation. 5.1 Algorithm in ABAQUS software The sheet metal multiple-step incremental air-bending forming is an extremely complicated physical process. It is a high non-linear problem containing geometric nonlinear, material non-linear, and contact non-linear. To solve this, the explicit algorithm ABAQUS/explicit module fitted for dynamic and non-linear analysis could be used to simulate the sheet metal forming process, and the implicit algorithm ABAQUS/standard module adapted to static and steady analysis to simulate the springback process. Due to the result of any time during the ABAQUS/ Standard module running could be treated as initial condition run in the ABAQUS/Explicit module for further calculating and analyzing, and vice versa. Therefore, ABAQUS software function is very suitable for multiplestep air-bending process. We integrated ABAQUS/Explicit and ABAQUS/Standard to carry out the mixed operations. This method can make the non-linear behavior of each step in multiple-step air-bending forming be solved with higher precision and faster convergence, compared with algorithms of other FEM software.

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Fig. 6 Stress–plastic strain curve of WELDOX900-1 sheet

5.2 Simulation setup 5.2.1 Material and process parameters The sheet metal with 11,638 mm ×1153 mm ×8 mm dimensions are made from WELDOX 900-1(Table 1) whose property curve of the stress–plastic strain is shown in Fig. 6. The punch-pressing speed of 8 mm/s is chosen according to the operating requirements of URSVIKEN 2200-ton press brake, and the mass scaling factor is set as 10. The punch radius of 142 mm (r=142 mm) was attained by the mathematical model, and then the punch width of 200 mm (b=200 mm) and the die width of 240 mm (w=240 mm; Fig. 1) are determined. The sheet feed rate and springback angle in each step could be observed in Table 3. 5.2.2 Finite-element model A two-dimensional finite-element model (Fig. 7) was established for the multiple-step incremental air-bending

Fig. 7 FEM model of incremental air-bending forming process (a) bending (b) before bending

forming process. The sheet metal was meshed with 7,667 nodes and 2,188 elements (ABAQUS type CPS8R, namely the eight-node biquadratic plane stress quadrilateral, reduced integration shell elements). Its material was assumed

Table 3 Process parameters in incremental air-bending forming of the third arm Processing sequence 1 2 3 4 5 6 7 8 9 10 11

Step

Geometric planned sheet feed rate (mm)

Optimized sheet feed rate (mm)

Punch displacement (mm)

Springback angle (°)

1 2 3 4 5 6 7 8 9 11 10

189 75 70 79 83 84 84 83 79 142 -72

189 64.5 62.5 84 84 86 85.5 83 78 151 -62

33.4 48.5 48.9 48 47.6 47 46.8 47.5 48.4 33.4 48.9

157.08 140.19 124.18 109.67 96.41 83.59 70.33 55.82 39.81 16.89 0

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Fig. 8 Nodes’ residual stress centers coinciding with centers of arc segments in geometric planning

to be planar anisotropic following Hill yield criterion. The punch and die were modeled as rigid surfaces. Coulomb friction principle was applied with a friction coefficient of 0.12 between the blank and the punch and die. The contact condition was implemented through a pure master–slave contact-searching algorithm and penalty contact force algorithm.

5.3 Optimization of process parameters with simulation

Fig. 9 Residual stress field of the formed shape with sheet feed rate of 75 mm in second step

Fig. 10 Residual stress field of the formed shape with sheet feed rate of 64.5 mm in second step

In order to form workpiece with high-dimensional precision, the sheet feed rate in each step needs to be adjusted accurately so that centers of the arc-shaped residual stress fields by simulation coincide with centers of arc segments by geometric programming (Fig. 8). For

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Fig. 11 Errors in chromatography graph. a Matching result of simulation model and CAD model. b Matching result of measured point cloud model of the cut one with 500 mm length from the scanned part and CAD model

example, sheet feed rate for the second step obtained by the 2D CAD is 75 mm, the geometric center of arc segment is located at node 1065. While the simulation results show that center of the arc-shaped residual stress field is located at node 1068 (Fig. 9). The two centers are not coincident and the error is 7.47 mm. The reason causing this error is that one end of the sheet metal touched with the die shoulders is line-segment-shape, while the other one without touching the shoulders is arcsegment-shape (Fig. 7b), which leads to offset of the bending center when loaded (Fig. 7a). Therefore, the sheet feed rate needs adjusting. In simulation, we note that when the sheet feed rate is 64.5 mm, the center of the arc-shaped residual stress field is located at node 1065 (Fig. 10), the

two center points are identical. As a result, the highdimensional precision of the formed workpiece could be obtained. The optimal sheet feed rate of each step obtained by simulation is shown in Table 3. In Fig. 7b, when sheet metal is moved in the opposite feed direction, the sheet feed rate is smaller and the gap is bigger. In other words, the smaller the sheet feed rate is, the bigger the gap is and the more the overlap area between the punch and workpiece arc segment formed by the last step is; and much more overlap area can bring smoother surface of the workpiece and higher precision shape. Therefore, the number of air-bending steps is set as much as possible to achieve smaller sheet feed rate when we do the process planning. However, we should choose the optimum number

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6 Application examples A new punch was manufactured with the above study results. Then, a semiellipse-shaped workpiece (Fig. 12) was formed at the URSVIKEN 2200-ton press brake with the punch and the process data shown in Table 3 The surface profile of the workpiece was measured with Atos-II three-dimensional laser measuring system, and a 3D point cloud model could be obtained. The 3D point cloud model was matched with CAD model with Geomagic Qualify software, and the result showed that the average dimension errors of the part were +0.68/−0.69 (Fig. 11b), which can completely meet the requirements of workpiece form and position accuracy.

7 Conclusions

Fig. 12 The third arm formed at the URSVIKEN 2200-ton press brake

of air-bending steps, which takes the production efficiency of workpiece-forming and process simplification in consideration as well. To avoid interference, the air-bending processing sequence needs to be partially adjusted according to the simulation results. The optimum sequence is from step 1 to step 9, step 11, and step10, which could be observed in Fig. 4 and Table 3. 5.4 Simulation results The simulated workpiece model was matched with CAD model with Geomagic Qualify software, and the results show that the average dimension errors in the simulated workpiece were +0.68/−0.65 (Fig. 11a). Therefore, the process planning method of forming high-precision semiellipse-shaped workpiece with numerical simulation is feasible and effective. Lastly, the processing parameters (in Table 3) are obtained with the method.

The mathematical model of the springback radius in sheet metal air bending is constructed with dimensional analysis and orthogonal test in this paper. Then, the sheet metal multistep incremental air-bending forming process is planned and designed using FEM simulation method. The present work generalized the variation law of springback radius, designed a punch, built a FEM model, and simulated and manufactured a workpiece. From the foregoing discussion, the following conclusions can be made: 1. The established model of springback radius is capable of providing optimum parameters for punch design, speeding up the progress of tool design and improving production efficiency. 2. From the model results, lower springback value could be obtained with the material of smaller yield strength, bigger Young’s modulus, thicker sheet metal, and smaller punch radius. 3. The simulation results can provide optimum processing parameters for incremental air-bending forming, which was successfully applied in manufacturing of one semiellipse-shaped workpiece used in crane boom. 4. The manufacturing results show that the methods of punch design, geometric planning, and FEM simulation for sheet metal incremental air-bending forming are reasonable and effective. Acknowledgments This work was supported by the National Nature Science Fund of People’s Republic of China, under 50175034 and Innovation Fund by the SANY Heavy Industry Co. Ltd, Changsha, China.

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