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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842

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etermination of the forming tool load in plastic shaping ofround tube into a square tubular section

aila S. Bayoumi ∗, Ahmed S. Attiaepartment of Mechanical Design and Production, Cairo University, Giza, Egypt

r t i c l e i n f o

rticle history:

eceived 10 November 2007

eceived in revised form

March 2008

ccepted 18 April 2008

eywords:

a b s t r a c t

The forming tool load in plastic shaping of a round tube into a square tubular section has

been determined analytically and also numerically by applying finite element simulation.

The theoretical results were compared to experimental measurements obtained in testing

tube specimens using a specially designed test-rig simulating actual production conditions.

The comparison has shown that the results obtained analytically were in fair agreement

with those obtained experimentally, while the finite element simulation results were in

agreement only at small deformation. The shaping load–displacement relationship obtained

analytically was applied to predict the roll load and drawing force in drawing a round tube

ube shaping

inite element

quare tubes

ubular sections

into a square tubular section through a head comprising four idle flat rolls. The results

obtained from this application proved to be in fair agreement with those obtained from a

previous more elaborate solution, which recommends the use of the analytical relationship,

obtained to direct industrial application.

(2003), investigated the interaction of material properties and

ube drawing

. Introduction

etal tubes of square section are widely used in severalngineering applications as lightweight structural elements,articularly in the automotive, aeronautical and aerospace

ndustries. The square tubular shape, compared to the roundubular shape, is known to offer several constructional advan-ages in the assembly and joining of structural elementsarticularly in the fields of lightweight frames constructionnd coachwork building.

The production of a square tubular section starts usuallyith a straight round tube. The shaping into a square section isplastic metal shaping process that has been investigated by

everal workers. Bayoumi (2001) presented an analytical solu-ion for cold drawing of a round tube through flat idle rolls to

roduce a regular polygonal metal tubular section. A veloc-

ty field has been obtained from kinematics considerationso yield the strain-rate components. The stress components

∗ Corresponding author.E-mail address: lsbayoumi@hotmail.com (L.S. Bayoumi).

924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.jmatprotec.2008.04.047

© 2008 Elsevier B.V. All rights reserved.

were obtained by combining the material constitutive lawwith Levy–Mises flow rule and integrating the equations ofequilibrium. The effects of section shape, friction at the rollinterface, roll radius and wall thickness on the roll load anddrawing force were investigated. Hwang and Altan (2002) hasanalyzed shaping a circular tube into a rectangular sectionby comparing two different ways; hydraulic expansion onlyand compression combined with hydroforming using a finiteelement code “DEFORM”. Manabe and Amino (2002) carriedout finite element simulation using nonlinear explicit finiteelement code LS-DYNA 3D, to investigate hydroforming of acircular tube into a square section in order to identify the fac-tors influencing the deformation behavior. Experimental workwas also performed to verify the simulated results. Kridli et al.

die geometry on the hydroforming of circular tube in a squaredie using two-dimensional plane-strain finite element codeABAQUS/STANDARD. Hwang and Chen (2003) explored the

1836 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e

Nomenclature

a section corner radius2b length of flat sideB tube lengthd tube mid-wall diameterd0 tube outer diameter2h square tube inner height2h0 square tube outer height (distance between

flats = d0 − 2u)2h1 section height at entry side2h2 section height at exit sideK flow stress coefficientL deformation zone length (bite zone)M1 unbending moment that straightens the edges

of the corner arcM2 bending moment that increases the curvature

of the arcn flow stress exponentN tool load acting on tube surfacePd drawing forcePr roll loadR roll radiust tube wall thicknessu tool displacementx arbitrary distance in the bite zone, measured

from the entry sidey distance from mid-wall

Greek symbolsı change in section height along the bite zone

angle in section height along the bite zoneness remains constant and the increase in the length of the flatpart is equal to the decrease in the length of the curved part.Such deformation pattern has been confirmed while runningthe experiments.

εe effective strain�e effective stress

plastic deformation of a circular tube hydraulically expandedinto a rectangular cross-section using analytical expressionsand the finite element code “DEFORM”. Leu and Wu (2005)examined squaring a circular tube between two V-shape dieby elasto-plastic finite element analysis, under plane-straincondition. The effects of various parameters of the process,on the occurrence of collapse and on the extent of asymme-try for the squaring process were discussed. Hwang and Chen(2005) predicted the forming pressure and thickness distri-bution of a circular tube that is hydraulically expanded in asquare die using a mathematical model and finite elementsimulation. The results of the mathematical model and finiteelement simulation were compared to experimental results.

Concerning industrial processing, the squaring of a roundtube by hydroforming is usually limited to the production ofspecial short tubular shapes of certain materials, whereas thebulk of industrial production of square tubular sections is per-formed in cold forming continuous lines in which the roundtube is drawn – or pushed – through a head comprising four

adjustable flat idle rolls, symmetrically arranged with theiraxes lying in one plane normal to the tube axis. In order todesign such a tool, it is necessary to assess the magnitude ofthe load to which the forming roll is subjected. In this respect,

c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842

the above literature review shows that, although several theo-retical studies have been carried out on the process of squaringround tubes through hydroforming, very few particular stud-ies are available for the process of drawing a round tube intoa square tubular section. Inspection of these particular stud-ies, have shown, however, that they are too elaborate to allowdirect application to determine the tool load in the industrialprocess. It is the objective of the present work to develop directrelationships to predict the roll load and drawing force, interms of the tube initial and final dimensions and the plas-tic flow characteristics of the tube material, to be applied tothe industrial process of shaping a round tube into a squaretubular section.

2. Deformation geometry

The tube is initially circular of an outer diameter d0 and a uni-form wall thickness t and length B. The ratio of t to d0 is smalland the tube is considered to remain stable during deforma-tion. As the circular tube is compressed laterally between foursymmetrical flat tool surfaces through equal displacementsu (Fig. 1) it deforms plastically into a tube of square cross-section with round corners. The deformation process consistsof straightening a part of the corner curve in contact with thetool by unbending to become flat, and bending of the remain-ing part of the corner curve to become more curved. There is nodisplacement along the surfaces of contact between the tubeexternal surface and the tools, which allows the deformationprocess to be considered frictionless. Such geometrical trans-figuration takes place smoothly with no constraint along thelength of the tube circumference. The strain normal to the wallthickness due to bending is negligible so that the tube thick-

Fig. 1 – Tube deformation on applying the lateralcompressive loads.

t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842 1837

idi

d

2

2

a

a

3

TraaMctar

b

Ttg

Ft

j o u r n a l o f m a t e r i a l s p r o c e s s i n g

Applying the constancy of volume principle, the follow-ng geometrical relations are obtained among the tube meaniameter d, flat side length 2b and tube corner radius a, approx-

mated to a quarter of a circle.

0 − 2u = 2h0 (1)

b = �

(d − 2h0 + t

4 − �

)(2a)

b = 2�

4 − �u (2b)

= −(�d − 4(2h0 − t))2(4 − �)

(3a)

= d

2− 4

4 − �u (3b)

. Analytical determination of tool load

he tool load is determined analytically from a direct equilib-ium approach. Fig. 2 shows the system of forces and momentscting on the corner curve of the deformed tube, that ispproximated to the form of a quarter circle of radius a, where

1 is an unbending moment that straightens the edges of theorner arc, M2 is a bending moment that increases the curva-ure of the arc and acts at the middle of the arc at point O,nd the horizontal and vertical force components N/2 are theeactions of the tool loads on the flat sides.

The moment equilibrium equation about point O is giveny

N√2

a

(1 − 1√

2

)= M1 + M2 (4)

he moments M1 and M2 are obtained through integratinghe bending stress over the cross-section, which is a rectan-ular cross-section of area B × t, and expressing the material

ig. 2 – System of forces and moments on a quadrant of theube.

Fig. 3 – Tube material flow curve (K = 202 MPa and n = 0.172).

constitutive law by a simple power law as follows:

M1 = 2B

∫ t/2

0

�e1 y d y �e1 = Kεne1

(5)

M2 = 2B

∫ t/2

0

�e2 y d y �e2 = Kεne2

(6)

The axial strain due to N/√

2 is negligible and only the bend-ing strains are considered. The effective strains εe1 and εe2 , areexpressed by the bending strains; in terms of the initial curva-ture 1/Ri, final curvature 1/Rf and the distance from the neutralaxis y as

ε = y

(1Rf

− 1Ri

)(7)

Noting that the flat part was originally a part of the cornercurve of radius a then: for εe1 : 1/Ri = 1/a and 1/Rf = 0 whichgives

εe1 = y

(1a

)(8)

For εe2 : 1/Ri = d/2 and 1/Rf = 1/a which gives

εe1 = y

(1a

− 1d/2

)(9)

Substituting into Eqs. (5) and (6), and then in Eq. (4) gives

N = 4KB(t/2)n+2

a(n + 2)(√

2 − 1)

[(1a

)n

+(

1a

− 2d

)n]

(10)

Replacing a in terms of u from Eq. (3b) gives

∴ N = 4KB(t/2)n+2

((d/2) − (4/(4 − �))u)n+1(√

2 − 1)(n + 2)

×(

1 +(

1 − 2((d/2) − (4/(4 − �))u)d

)n)(11)

The values of the strength coefficient K and the strain

hardening exponent n in Eqs. (10) and (11) are obtained exper-imentally from tension tests using ASTM-E8 standard tensiontest specimens cut longitudinally from the tubes. The resultsobtained are shown in Fig. 3.

1838 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e

Fig. 4 – Analytical tool load in relation to tool displacement.

the experimental work. The deformed shape of the tube at theend of the shaping process is shown in Fig. 7.

The contact force between the tool and the tube obtainedfrom plane stress finite element simulation is shown in Fig. 8.

Fig. 5 – Moving tools applying four lateral compressiveforces on the circular tube.

Substituting the values of K and n from Fig. 3 into Eq. (11)gives a plot in Fig. 4 of the analytical tool load in relation tothe tool displacement.

4. Finite element simulation

The general purpose finite element code ABAQUS/STANDARDv.6.5-1 is used to analyze the plastic flow pattern of a circu-lar tube shaped into a square cross-section, by applying fourlateral compressive forces using four, steel, moving tools asshown in Fig. 5.

The dimensions and material properties of the finite ele-

ment simulation tube are given in Table 1.

Upon exceeding the material yield strength, the materialflow relationship, which has been obtained experimentally, isexpressed by the simple power law �e = 202ε0.172

e .

Table 1 – Tube dimensions and material properties

Outer diameter (mm) 25Thickness (mm) 1Length (mm) 30Modulus of elasticity (E, MPa) 70,000Poisson’s ratio (�) 0.33Yield strength (MPa) 85Strength coefficient (K, MPa) 202Strain hardening exponent (n) 0.172

c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842

The CPS4R elements were used in the FE simulation; theseelements are 4-node bilinear plane stress quadrilateral ele-ments with reduced integration, and hourglass control. Dueto symmetry conditions, only a quarter of the tools and tubeare modeled.

The tube is meshed using continuum elements and thetools are modeled as analytic rigid surfaces. The analytic rigidsurface option in ABAQUS is computationally less expensivethan the use of rigid elements to model the tools, and hasa more efficient contact algorithm. To capture the effect ofbending the tube is meshed with six layers of elements in thethickness direction. A total of 1200 elements are used to meshthe quarter tube model. A master/slave contact approach isused in the analysis where the tools (rigid surfaces) are con-sidered the master surfaces and the outer surface of the tubecontacting the tools is considered to be the slave surface. Theinteraction between the tube outer surface and the tools isformulated using the finite sliding approach, which allows forthe possibility of separation between the two surfaces.

During the finite element simulation the following assump-tions were made:

(1) The initial tube cross-section is perfectly circular.(2) The initial tube wall thickness is uniform at all locations.(3) The deformation behavior of the material is isotropic.(4) The tube is elastic-strain hardening plastic.(5) The tools are rigid.

The initial mesh of the tube with the six layers of elementsthrough the thickness is shown in Fig. 6. Symmetry bound-ary conditions are applied along the X and Y axes, also theupper and side tools are allowed to move in the vertical andhorizontal directions only.

The upper and side tools are set in contact with the tube atthe beginning of the shaping process, as shown in Fig. 6. Thetools are then moved 1.525 mm as the distance they moved in

Fig. 6 – Mesh configuration in the tube before the shapingprocess.

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842 1839

afte

5

Ttstjis

dwDfloc

m2a

Fig. 7 – Tube deformed shape

. Experimental work

he shaping process was carried out on a specially designedest-rig that simulates actual production conditions. Fig. 9hows the experimental setup of the shaping process. Theube is equally compressed laterally by four movable flataws that move inwards during the rotation of the load-ng arm block by means of four pins engaged in inclinedlots.

The test-rig has been calibrated to obtain the load appliedirectly on the tube during the shaping process. A load cellas specially designed to be used in the test-rig calibration.uring the shaping process the tube is deformed equally

rom four sides by applying stepped loading. The flat sideength and the corner radius were measured at each stepf loading, and then compared to those calculated theoreti-ally.

The tubes used in the shaping process are made of com-ercially pure aluminum (Al ≥ 99%), which was annealed at

99 ◦C for 1 h. The tube dimensions and the material flow char-cteristic are shown in Table 1.

Fig. 8 – Load on tools as obtained from FE simulation.

r moving the tools 1.525 mm.

6. Results and discussion

The corner radius is plotted against the flat side length, asobtained theoretically and experimentally in Fig. 10, whichshows a good agreement, as the maximum deviation is 5%.

Fig. 11 shows that there is no significant difference betweenthe deformed tube shapes under load obtained experimen-tally and through FE simulation. The two curves show slightinward curvature at the planes x = 0 and y = 0 which has beenobserved to occur even at small loads. This indicates that theunbending moment has exceeded the moment required forflattening and has thus caused reverse bending; resulting inshifting the contact between the tool and tube surface towardsthe junction of the flat part with the curved corner. The exper-imentally deformed tube shape, after it has been extractedfrom the loading test-rig (Fig. 11) reveals a pronounced elasticspring-back. From Eqs. (7) and (8), the total strains in shapingthe tube are so small that such a pronounced spring-back hasshown up.

The tool load versus tool displacement as obtained ana-lytically, experimentally and from ABAQUS finite elementsimulation is plotted in Fig. 12, which shows a fair agree-ment between the analytical and experimental results. Suchagreement is also indicated in Fig. 13, by expressing the tubedeformation in terms of the tool load versus the corner radiusa.

On the other hand, the results from FE simulation showto be much lower than the experimental results. The devi-ation in the results increases as the deformation increasesdespite the fact that, there is no significant difference between

the deformed tube shapes obtained experimentally and thatfrom FE simulation (Fig. 11). Inspection of the FE simulationresults has shown that tool contact with the tube outer sur-face (Fig. 14) was confined to only four elements, located at the

1840 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842

Fig. 9 – Test-rig setup.

Fig. 10 – Corner radius versus flat side length, theoreticallyand experimentally.

Fig. 11 – Deformed tube shapes obtained analyticallyversus that obtained from FE simulation.

Fig. 12 – Tool load versus tool displacement, theoreticallyand experimentally.

Fig. 13 – Tool load versus tube corner radius.

junction of the flat part with the corner curve. The total lengthof these contact elements is 0.3927 mm, which looks to beunrealistic and too small when compared to the considerablelength of contact observed in the experiments.

Further FE simulation has been carried out using LS-DYNAsoftware applying the same type of elements and mesh andadopting the same procedure as used previously in ABAQUSsoftware. The results obtained are plotted in Fig. 15 togetherwith those obtained analytically, experimentally and from FE

simulation using ABAQUS software. The tool load, obtainedfrom LS-DYNA is higher than that obtained from ABAQUSbut is still lower than the experimental load. Contact is con-fined, to the same four elements as for ABAQUS, which,

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842 1841

Fig. 14 – Distribution of contact pressure obtained from FEsimulation.

Fig. 15 – Tool load versus tool displacement frome

at

7r

Deibl

Table 2 – Roll load at different roll radii

Roll radii (R, mm) Roll load (Pr, N) Drawing force (Pd, N)

xperimental, analytical and FE simulations results.

gain, is too small when compared to experimental observa-ions.

. Adaptation of theoretical result to predictoll load and drawing force in tube drawing

uring tube shaping using the test-rig the tube length wasqual to the tool length, so that the length was only a scal-ng factor, but herein the pipe is being continuously deformed

etween the rolls over a distance L which is the bite zone

ength (Fig. 16).

Fig. 16 – Deformation zone of length L (bite zone).

50 983 37165 1122 37280 1246 372

Referring to Fig. 16 the following geometric relations arededuced:

L =√

2R(h1 − h2) − (h1 − h2)2 (12)

h0 = h2 + R −√

R2 − (L − x)2 (13)

h1 − h2 = R −√

R2 − L2 (14)

ı =√

R2 − (L − x)2 −√

R2 − L2 (15)

In order to obtain the roll load making use of the previ-ous theoretical work, namely the analytical approach, Eq. (11)is written per unit length then integrated over L, taking intoconsideration to replace u in Eqs. (2b) and (3b) by ı, the rollload Pr is expressed as

Pr =∫ L

0

N

Bd x (16)

The drawing force per each roll Pd can be obtained by takingthe summation of moments about the roll axis of rotation,taking into consideration that the roll is idle (i.e. no torqueexists); which gives the drawing force for the four rolls as

Pd = 4R

∫ L

0

N

B(L − x) d x (17)

To check the effectiveness of the predicted roll load and draw-ing force introduced by Eqs. (16) and (17); the results obtainedare compared to the results obtained from the previous workby Bayoumi (2001) in Tables 2 and 3, respectively.

On comparing the results in Tables 2 and 3; it is found thatthe predicted roll loads obtained from Eq. (16), and the rollloads from the previous work are in good agreement as themaximum deviation is less than 5%. On the other hand thepredicted drawing forces obtained from Eq. (17) are higherthan that obtained from previous work by about 15%. This

may be attributed to the difference in the roll load distributionalong the roll bite length (Fig. 17) due to assuming the stateof stress to be uniaxial in the analytical solution, while it isthree-dimensional in the previous work. It also indicates that

Table 3 – Previous work results

Roll radii (R, mm) Roll load (N) Drawing force (N)

50 1043 33065 1163 32180 1306 323

1842 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e

r

Manufacturing Technology 25, 691–699.

Fig. 17 – Roll load distribution along the bite zone,R = 50 mm.

the effect of the drawing force may be ignored in the predictionof the roll load.

8. Conclusions

An analytical relationship is obtained to predict the roll loadand drawing force, in terms of the tube initial and final dimen-sions and the material plastic flow characteristics, which canbe applied directly to the process of round tube drawing

into a square tubular section through four flat idle rolls. Theresults obtained from the relationship are in fair agreementwith those obtained experimentally, while the finite elementsimulation results were in agreement only at small defor-

c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1835–1842

mation. The results obtained from this application provedto be in fair agreement with those obtained from a previousmore elaborate solution, which recommends the use of theanalytical relationship, obtained to direct industrial applica-tion.

e f e r e n c e s

Bayoumi, L.S., 2001. Cold drawing of regular polygonal tubularsections from round tubes. International Journal ofMechanical Sciences 43, 2541–2553.

Hwang, Y.M., Altan, Y., 2002. Finite element analysis of tubehydroforming processes in a rectangular die. Finite Elementsin Analysis and Design 39, 1071–1082.

Hwang, Y.M., Chen, W.C., 2003. Analysis and finite elementsimulation of tube expansion in a rectangular cross-sectionaldie. Journal of Engineering Manufacture, Part B 217, 127–135.

Hwang, Y.M., Chen, W.C., 2005. Analysis of tube hydroforming ina square cross-sectional die. International Journal of Plasticity21, 1815–1833.

Kridli, G.T., Bao, L., Malliek, P.K., Tian, Y., 2003. Investigation ofthickness variation and corner filling in tube hydroforming.Journal of Materials Processing Technology 133, 287–296.

Leu, D.K., Wu, J.Y., 2005. Finite element simulation of thesquaring of circular tube. International Journal of Advanced

Manabe, K., Amino, M., 2002. Effects of process parameters andmaterial properties on deformation process in tubehydroforming. Journal of Materials Processing Technology123, 285–291.