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An upper bound analysis for reshaping thick tubes to polygonal cross-section tubes through multistage roll forming process
International Journal of Mechanical Sciences 100 (2015) 90–98
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
An upper bound analysis for reshaping thick tubes to polygonal cross-section tubes through multistage roll forming process H.R. Farahmand a,n, K. Abrinia b a b
Research Group of Metallic Material Processing Technology, ACECR, Tehran University, Tehran, Iran School of Mechanical Engineering, University of Tehran, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2014 Received in revised form 15 June 2015 Accepted 17 June 2015 Available online 26 June 2015
In this investigation, an upper bound solution is presented for the multistage forming process to produce polygonal cross-section thick tubes by means of a roll forming rig with flat rolls. The modeling and simulation is presented so that it can be generalized for arbitrary number of sides and stages. Comparison of theoretical and experimental results for tube cross-section dimensions (height, wall thickness and outer corner radius) show good agreement for a square section tube and demonstrated the capabilities of the new formulation presented. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Upper bound Roll forming Tube reshaping Polygonal cross-section tubes Multistage deformation
1. Introduction Tubes with different cross sections are used widely in industries as structural components and building materials to home appliances. Recent applications of polygonal tubes, such as square and rectangular seamless tubes, have resulted in increased in industry (e.g. natural gas, chemical liquid transportation, construction of boilers, heat exchangers, etc.). In addition, the polygonal tubes have advantages such as high torque resistant in twist and weight reduction of structure parts compared to circular tubes. For producing these tubes, various methods such as extrusion, drawing, hot forging, roll forging, and roll forming might be used. The shape rolling method by passing a circular tube through flat or curve rolls is also one of the common methods and is more efficient. As yet, different analytical and numerical solutions have been obtained for reshaping of pipes and tubes. Kiuchi et al. [1] studied the process of reshaping thin circular pipes to non- circular tube sections in cold roll forming. He [2] also developed his study to produce square and the rectangular tubes from pipes by experimental investigations. In a published report by Wen [3] some details about an advanced tooling design which utilized numerical analysis to predict the material deformation and improved the tooling performance. Finite element numerical simulation was used to simulate the results and study effects of processing parameters. A computer-aided simulation program was developed
n
Corresponding author. Fax: þ98 21 88007277. E-mail address: [email protected] (H.R. Farahmand).
http://dx.doi.org/10.1016/j.ijmecsci.2015.06.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
by Kiuchi and Feizhou [4], based on 3-D elasto-plastic finite element method. Using this program they were able to analyze all of deformation features and mechanical characteristics of the reshaping process. Kiuchi et al. [5], presented theoretical methods to analyze and optimize the reshaping process. A mixed method composed of the finite element and finite difference was applied and called finite differential method. They also reported their work on reshaping of no-circular pipes, [6]. In an analytical work by Bayoumi [7] a solution was obtained for the problem of cold drawing through flat idle rolls of regular polygonal metal tubular sections from round tube. Bayoumi also gave an analytical solution for the problem of cold flattening of a round tube into an oblong shape through rolling between two flat rolls [8]. Abrinia and Farahmand [9] presented a new solution based on upper bound method that is used to solve the reshaping of thick square tube from a round tube in one stage. The influence of various effective process parameters was investigated. They generalized this method for polygonal tube sections, [10]. For determining the forming tool load in plastic shaping of a round tube into a square tubular section through a head comprising four idle flat rolls, Bayoumi and Attia [11] presented an analytical solution and finite element simulation using finite element code ABAQUS/STANDARD and LS-DYNA. Tajyar and Abrinia [12] studied reshaping of a circular thick tube into a square cross section by cold roll forming between four flat rolls in different passes. The influence of the amount of roll gap reduction in each pass on the final rolled product was investigated by FEM method. In the previous work [9], authors presented a new analytical solution for reshaping a pipe to square section in one stage basis
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
_i W _s W
Nomenclature half-width of flat side at exit be 0 f streamline function at contact region f ; g; h Components of Bezier function h normal distance from tube axis to flat side he normal distance from tube axis to flat side at exit Jn upper bound on total rolling power L bite length m friction factor 0 n number of sectors n number of contact surface division p, q, u parameters of Bezier velocity field at free region R radius of outer corner of tube at entrance r position vector r0 outer corner radii at final cross section r 0 and r 3 position vectors at initial and final cross section Δr roll gap reduction R1 roll radius S circumference length S0, Se circumference length at entry and exit, respectively t0 initial wall thickness te wall thickness at exit U_ constant circumferential velocity of rigid roll vx ; vy ; vz components of velocity field in free parts v0 entry velocity of tube V x ; V y ; V z velocity field components in contact part vn the velocity of material at contact surface _f frictional power dissipated over frictional boundary W
on upper bound method. It was the first time that upper bound method was used for roll forming process. In this paper the solution has been generalized so that can be used for multistage shape rolling of thick tubes to polygonal section with flat rolls. Dimensions of the deformed tube such as height, wall thickness and outer corner radius at each stage of reshaping process are determined by this solution and therefore can be used in pipe industry to produce polygonal tubes. The modeling and simulation can be generalized for arbitrary number of polygonal sides and forming stages. One of the most significant advantages of this solution is its simplicity and capability to find the answers at a very short time in comparison with the FEM solution. The influence of other parameters such as rolls radius, initial tube dimensions, friction coefficient and roll gap reduction, which had been studied for one stage in the previous work could be used for this situation. The tube material is Pb and is assumed to be isotropic, incompressible and follows a rigid-plastic behavior. To illustrate the integrity of analytical solutions some sample tubes were formed during a two-stage process to compare results to each other.
power dissipation due to internal deformation power consumption of shear boundary
Greek letter
α θ ϕ ϕc ψ σ0 σm ε_ ij ε_ τ δ Γ ξ
bite angle rolling angle angle of deforming region at entry angle of contact part of tube's cross section profile angle of deforming region at exit flow stress mean effective stress strain rate components effective strain rate shear stress height reduction of tube cross section during rolling boundary velocity field parameter at contact region
Subscripts C F CF i, e s
due to contact region due to free region due to common surface between contact and transition regions entry and exit planes, respectively due to shear
Fig. 1. The polygonal tube sections.
2. Material flow and strains analysis 2.1. Geometry of deformation zone 0
For a polygonal shape tube with “n ” sides, the section can be 0 divided into“2n ” equal sectors. Each sector has height he and radius ro with a flat side be . The primary input tube with radius R 0 and thickness t 0 also can be divided to “2n ” equal sectors, Fig. 1. At the first stage, the tube section is deformed to a polygonal shape with a flat side 2be ð1Þ , corner radius roð1Þ, and thickness
Fig. 2. A sector of the round tube and polygonal tube.
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Fig. 3. The deformation geometry of square tube in the first stage.
Fig. 4. The deformation geometry of square tube in the second stage.
t e ð1Þ. The first sector height reduction, δð1Þ; is equal to difference between Rð ¼ hi ð1ÞÞ and he ð1Þ; Fig. 2. Cartesian coordinate system origin is located at the intersection of the tube axis with the entry section of the deformation zone. The bite angle is p ð1Þ. Geometry of deformation zone for a 0 square section (polygonal with n ¼ 4) is shown in Fig. 3. This tube will be deformed again in the second stage. Thus at the exit, the flat side of the tube cross section,be ð2Þ, and the radius of the round corners, roð2Þ, will be respectively larger and smaller. Deformation zone geometries for the two consecutive stations such as the first and second stage for a square section are shown in
Fig. 4. Subscripts ‘e’ denotes exit plane of deformation zone and number ‘2’ shows the forming stage. Fig. 5 shows the deformation geometry for the first and second stages. If the second stage is considered, the entry section with the thickness t e ð1Þ can be taken apart to two parts: at the first part the corner arc of the cross section, r 0 ð1Þ, is converted to a smaller arc, roð2Þ, and a flat side, be ð2Þ; with thinner thicknesses, t e ð2Þ. So this is similar to the deforming zone geometry at the first stage. At the second part, the flat side of tube section which is contacting with the roll is converted to a thinner one. The thickness of these two parts at the exit will be equal, Fig. 6.
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
This deforming procedure will be repeated at the further stages similarly. At each stage such as j, the tube section can be divided into two parts in entry: 0 a flat side, be ðj 1Þ, and an arc RðjÞ ¼ r o ðj 1Þ with angle π =n . Then due to the deforming process, the flat side dimension will be larger and the corner radius will be smaller at the exit. Finally, the desired dimensions of flat side and corner radius will be obtained when tube exits from the last stage. Deformation energy is not dependent of coordinate system . Therefore, a new coordinate system such as X10 ðj ÞO10 ðj 1Þ Y10 ðj 1Þ could be considered for the j th deformation stage. The origin O10 ðj 1Þ is coincident with the center of the corner r o ðjÞ, Fig. 7. In the previous article of the authors [9], an analytical solution was presented for reshaping of the tube at one station, based on the upper bound technique. Now by considering the new coordinate axes, that analysis can be used for the tube deforming zone at the second or further stages. It is assumed that during a deformation stage, the outer perimeter of tube varies linearly from entry toward to exit, it verifies that SðjÞ ¼ S0 ðjÞ þ ðS0 ðjÞ S0 ðjÞÞ U
θðjÞ αðjÞ
ð1Þ
The deformation zone geometry is complex here and if material particle is considered to be moving from the entry to exit section
93
in a stage, it would pass through different paths. The deformation zone is divided into two areas: the “contact region” (the region which is in contact with the rolls during deformation) I and IV, and the “free region” (the region which is not in contact with the rolls during deformation), II and III, Fig. 8. At the contact regions (I and IV), the material meets the rolls and deforms, and at the free region (II and III) it deforms but does not touch the rolls. It is assumed that in the contact zone the tube surface is completely in contact with the roll surface and there is no material flow in the X direction. Curve f shows the boundary between regions I and II. By dividing the bite angle into n equal portions, curve f is also divided into n pieces. Now, the contact surface can be approximated with some longitudinal strips which are cylindrical surfaces with radius R1; Fig. 9. Therefore the outlet section also can be divided into n part. Besides, the entry section also can be divided into n parts such that each one can be related to its corresponding part at the exit section. Now, an article such as N in region II moves from point N1ðjÞ at entrance section towards the roll and enters region I at point N2ðjÞ, Fig. 10. Afterwards it travels on a circular curve under the roll to point N3ðjÞ. It is assumed that the entry and exit sections are perpendicular to the axis of rolling for all regions. 2.2. The kinematically admissible velocity fields In upper bound technique, it is necessary to define a kinematically admissible velocity field for the deformation region. Since
Fig. 5. The deformation geometry of the first and the second stages.
Fig. 7. New coordinate axes.
Fig. 6. Deformation of tube section in the second stage: (a) entry, (b) exit.
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Fig. 8. Division of deformation zone in the first and jth stages.
Fig. 9. Approximation of contact surface with cylinderical strips.
2.2.1. Admissible velocity field for contact regions; I and IV In the contact area, an elliptic single-parametric family curves, which is suitable to show the streamlines, is used to obtain the admissible velocity field, Fig. 11. An elliptic single-parametric family curves which achieved for the geometry of this problem is defined as ðz z0 Þ2 ðy R1 þhe 1 ξ e Þ2 þ ¼ 1 ; 0rξr1 ð2Þ ðR1 þ 1 ξ ðt e eÞÞ2 R12 where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 U t0 t e R12 z0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e¼ R1 R12 z0 2 thus Fig. 10. A flow line for a particle moves through regions II and I.
the material flow during the process was of a complicated nature due to the many regions of contact and non-contact between the workpiece and the tools, therefore a single generalized formulation could not be derived for the velocity field and three different velocity fields were formulated for the different parts of the deforming region and then incorporated into a single generalized formulation for the upper bound on power. These velocity fields are as follow:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y he R1 þ R12 ðz z0 Þ2 ffi ξ ¼ f ðy; zÞ ¼ 1 þ R1 U qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R12 ðz z0 Þ2 þ e UR1 tf e 0
ð3Þ
Referring to Fig. 11, ξ ¼ 1 gives the equation of roll contact surface or upper boundary streamline, and ξ ¼ 0 shows the equation of the inner surface of tube or lower boundary streamline. It is proved that if it is assumed that the entry surface is a plane perpendicular to the symmetric axis then the exit surface will also be perpendicular to that axis. Therefore, by supposing that the material under the roll surface has no flow in the X
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
direction, the velocity field is obtained as, [9] vx ¼ 0; 0
vy ¼
∂f ðz z 0 Þ F 0 ξ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂z R12 ðz z0 Þ2 ðt e eÞðy he Þ t e U R1 2 ð R1 U v0 U t 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðt e eÞ R1 ðz z0 Þ þ e UR1
vz ¼
0 ∂f F ξ ¼ ∂y
ðz z 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt e eÞ R1 ðz z0 Þ2 þ eU R1
ð4Þ
0 where F 0 ξ is a function to specify components of f in Cartesian coordinates system accordance to initial conditions, [9]. This velocity field satisfies the incompressibility condition; ε_ ii ¼ 0. 2.2.2. Admissible velocity field for free regions: II and III To obtain the velocity field in the non-contact areas, the Bezier curves is used. A cubic Bezier curve, Fig. 12, is defined as follow, [9]: r ¼ ð1 pÞ3 r 0 þ 3pð1 pÞ2 r 1 þ 3p2 ð1 pÞð1 pÞ3 r 2 þ p3 r 3 ;
0 rp r1 ð5Þ
By variation of Bezier curve parameters, the shape of the curve changes, but it still passes p0 and p3. The equation can be written as ! ! ! r ¼ f ðu; q; pÞ i þ g ðu; q; pÞ j þ ðu; q; pÞ k ; 0 r p; q; u r 1
ð6Þ
95
where ‘f’, ‘g’ and ‘h’ define the position of the particle on the streamline. ‘u’ and ‘q’ are parameters which are used to describe entry and exit surfaces and ‘p’ is a parameter used for z coordinate of the particle. The velocity vector in the Cartesian coordinates is given by ! ! ! V ¼ V x i þV y j þ V z k ð7Þ Since the particle of the material is moving on a streamline then the components of the velocity vector in the Cartesian directions will be coincident with the tangents of the streamline at that point. Thus Vx ¼
fp V z; hp
Vy ¼
gp V z; hp
V z ¼ Mðu; q; pÞ
ð8Þ
Mðu; q; pÞ is obtained from the incompressibility conditions as M ðu; q; pÞ ¼
C U hp
hu f q U g p f p U g q þ hq f p U g u f u U g p þ hp f u U g q f q U g u
ð9Þ where C is a constant and dependent on boundary conditions. The boundary conditions for regions II and III are different so the constant C must be verified for each region separately. 2.2.2.1. Velocity field in region II. A flow line is shown in Fig. 13. A particle such as P moves on a streamline from inlet toward outlet. This path can be divided into two pieces: a curve P 0 P 3 in region II and a curve P 3 P 4 in region I. The flow line should be perpendicular to entry section at point P 0 , and be tangent to a flow line of contact
Fig. 11. Streamlines model for contact region.
Fig. 13. A streamline relates region II to region I.
Fig. 12. Bezier curve as a streamline.
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Fig. 15. The Pb tubes were used for experiments.
Fig. 14. Bezier curve as streamline for free region III.
area I at point P 3 . Two curves joined at a common end point have at least C 0 continuity at their junctions. C 1 continuity requires a common tangent line at their junction. These conditions secure the smoothness of curves at P 3 . The velocity of the particle P at point P 3 on the curve P 0 P 3 must be the same as on the curve P 3 P 4 to prevent the velocity discontinuity. At the outlet section p ¼ 1; thus
M ðu; q; 1Þ ¼ vðIIÞ z z ¼ zi C ¼ C ðu; qÞ ¼ vðIÞ z z ¼ zi
hu f q Ug p f p U g q þ hq f p U g u f u U g p þ hp f u Ug q f q Ug u hp C ¼ 0:5 v0 U t 0 φ2 φ1 ð2R t 0 Þ 2.2.2.2. Velocity field in region III. In Fig. 14 a flow line is shown for region III. This is perpendicular to the inlet section. If the boundary condition at entry is considered, then p ¼ 0 and
M ðu; q; 0Þ ¼ vðIIIÞ z
Fig. 16. The experimental test rig with four flat rolls.
Since there are four deformation regions X
X
X
X
X _ i¼ _i þ _i þ _i _i W þ W W W W I
II
III
IV
ð14Þ
3.2. The power due to shear _ S is calculated from The shear power W Z _ S¼ Δv U τ UdA W
ð15Þ
z¼0
hq ¼ hu ¼ 0; hp ¼ 1 M ðu; q; 0Þ ¼ V z ¼ v0
In the regions in which the material experiences velocity discontinuities, such as the entry and exit sections of deformation, the power consumption due to the shear boundaries will exist.
therefore
C ¼ C ðu; qÞ ¼ f u Ug q f q U g u
p¼0
3. Upper-bound solution In accordance with the upper bound theorem, the following equation should be minimized for the actual velocity distribution in each stage X X X _ Sþ _ ; _J ¼ _ iþ W W ð12Þ W f _ i is where _J is an upper bound on the total power consumption, W _ s is the power consumption the internal power of deformation, W _ f is the frictional power due to the shear boundaries and W dissipated over the frictional boundaries. 3.1. Internal power of deformation The internal power of deformation is obtained from the following equation: Z U 0:5 2 2 0:5 ε_ 2x þ ε_ 2x þ ε_ 2x Þ þ ε_ 2xy þ ε_ 2yz þ ε_ 2zx ð13Þ W i ¼ pffiffiffiσ m 3 V
3.2.1. Contact regions; I and IV The velocity discontinuity does not exist on the entry exit boundary of region I, from the basic nature of its derived velocity field. Therefore the power of shear deformation consumed on this boundaries, vanishes. Similarly, there is no velocity discontinuity on the exit boundary of region IV. On the entry boundary of region I, or boundary between contact region I and
free region II, also _ S ¼ 0; and W _S there ¼
is no velocity discontinuity. Thus W I IV _S W . i IV
3.2.2. Free regions: II and III According Fig. 13 the shear deformation power on the entry and the exit boundaries of region II vanishes since there is no velocity
_ s ¼ 0; discontinuity on them. Thus W II Fig. 14 shows the region III. The velocity discontinuities exist on the entrance and exit boundaries. On the entrance, pi ¼ 0 and
_S _ Si ¼ W ð16Þ W III
III p ¼ 0
Similarly on the exit, pi ¼ z0 ¼ l and
_ _S W Se ¼ W III
III p ¼ z0
ð17Þ
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
Here, the interface friction is assumed to be given by
Table 1 Dimension of test specimens. Specimens
Outer diameter (mm)
Inner diameter (mm)
Thickness (mm)
Tube Tube Tube Tube Tube
85.6 85.4 85 83 83
68 68 65.5 65 63.6
8.8 8.7 9.7 9.0 9.7
1 2 3 4 5
Table 2 Dimensions of reshaped tubes after the first stage. Specimens dð1Þ Height decrease (mm)
Type t e ð1Þ Output thickness (mm)
r o ð1Þ Outer corner radius (mm)
2he ð1Þ Twice sector heighta (mm)
Tube 1
3
Tube 2
3
Tube 3
2.5
Tube 4
3
Tube 5
2.5
Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal.
37.8 38.38 37.7 38.27 37.7 38.72 36.0 37.26 36.1 37.81
82.6 82.60 81.9 82.40 83.1 82.50 80.2 80.00 80.2 80.50
a
8.6 8.68 8.0 8.58 9.6 9.59 8.8 8.87 9.5 9.58
For square tube, the section height is twice the sector height.
therefore
_ Si _ Se _S ¼ W þ W W III
III
I
II
III
mσ τ ¼ pffiffiffim
ð22Þ
3
where m is constant during the deformation process and takes the values between 0 and 1. Thereby with minimizing the total deformation power in each stage, _J, the independent problem unknowns consist of r o ðjÞ, t e ðjÞ, and zn ðjÞ obtained and the dependent unknowns can be calculated. The dependent unknowns are: be ðjÞ, he ðjÞ. These values are used as input data for the next stage so that: r i ðjþ 1Þ ¼ r o ðjÞ, t i ðj þ 1Þ ¼ t e ðjÞ, bi ðj þ 1Þ ¼ be ðjÞ and hi ðj þ 1Þ ¼ he ðjÞ. The Complex method (M. J. Box-1965) which is applied for constraint optimization in nonlinear problems is used to minimize the total deformation power. The constrained Simplex (Complex) method searches for the maximum value of a function f(x1;. xn) subject to p constraints of the form gk rxk rhk, k¼ 1,. p, where xn þ i,. xp are functions of x1,. , xn, and the lower and upper constraints gk and hk are either constants or functions of x1,. , xn. (To find a minimum, f is maximized.) It has been developed from the Simplex method of Spendley et al. It is assumed that an initial point xi1,…, xn1, which satisfies all the p constraints is available. In this method, k Zn þ1 points are used, of which one is the given point. The further (k 1) points required to set up the initial configuration are obtained one at a time by the use of pseudorandom numbers and ranges for each of the independent variables, viz. xi ¼gi þai(hi gi) where ai is a pseudo-random deviate rectangularly distributed over the interval (0, 1).
ð18Þ
III
Table 3 Dimensions of reshaped tubes after the second stage.
and the total shear power will be equal to
X _S þ W _S þ W _S _S _ s¼ W þ W W
_S _S ¼ W þ W
97
III
IV
ð19Þ
IV
3.2.3. The power due to friction In the free region of deformation there is no contact between the tube and rolls, and then the frictional power for that region vanishes. But in the contact region, the frictional power dissipated over the roll surface is given by Z _f¼ W τ U Δv UdA ð20Þ
Specimens dð2Þ Height decrease (mm)
Type t e ð2Þ Output thickness (mm)
r o ð1Þ Outer corner radius (mm)
2he ð2Þ Twice sector height (mm)
Tube 1
1.5
Tube 2
3.5
Tube 3
1.5
Tube 4
1.5
Tube 5
1
Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal.
36.5 35.89 32.9 34.09 37.3 36.31 35.0 34.84 37.8 36.11
81.4 81.1 78.6 78.9 81.3 81.0 79.1 78.5 79.5 79.56
8.2 8.62 8.7 8.43 9.5 9.52 8.4 8.81 9.5 9.54
contact S
But
thus
2
_ ¼ be 6 W 4 f
Z
zn 0
Z þ
that
z0
zn
R1 ds ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R12 ðz z0 Þ2
Outer corner radius after roll shaping 44
τ U_ v_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidz R1
2
R1 ðz z0 Þ
2
3
τ v_ U_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidz7 5 R1
R12 ðz z0 Þ2
ð21Þ
where τ is the frictional shear stress along the contact surface of the rolls, U_ is constant circumferential velocity of the rigid roll, v_ is the velocity of material at contact surface, and zn is the abscissa of the neutral point on the contact interface.
Outer corner radius, mm
dA ¼ dsU dx
Outer corner radius after the 1st stage (exp.)
42 40
Outer corner radius after the 1st stage (cal.)
38
Oyter corner radius after 2nd stage (exp.)
36 34 32
0
1
2
3
Tube No.
4
5
Outer corner radius after the 2nd stage (cal.)
Fig. 17. Comparison between outer corner radius of tubes by experiment and calculation after stages 1 and 2.
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5. Conclusion
Tube thickness after roll shaping 9.75
Tube thickness after the first stage (cal.)
Tube thickness, mm
9.5 9.25 9
Tube thickness after the first stage (exp.)
8.75 8.5
Tube thickness after the 2nd stage (cal.)
8.25 8 7.75
0
1
2
3
4
5
Tube thickness after the 2nd stage (exp.)
Tube No. Fig. 18. Comparison between tube thicknesses after roll forming in stages 1 and 2.
In this study, an analytic solution based on upper bound theory is presented for reshaping of a thick tube to a square tube in multistage roll forming by using a roller with four flat rolls. This method can be generalized for reshaping process of other regular sections. The material is considered as rigid perfectly plastic. To evaluate the analytical model results, some tube samples were reshaped at two sequential stages by a rolling machine. The results obtained for the thickness and the radius corner in both theoretical and practical procedures are very close together. Using this method and considering the next sequential steps, the multistage deformation process for producing polygonal tube can be simulated and the effects of various factors on its quality be predicted.
Acknowledgment The authors would like to acknowledge with gratitude for the support and laboratory services given to the research by the Metallic Material Processing Technology Research Group of ACECR, Tehran university branch.
References
Fig. 19. Comparison between tubes section heights after stages 1 and 2.
4. Experiments and results To evaluate the validity and efficiency of the mentioned solution, the essential experimental and theoretical results were compared. Carrying out the experiments, some thick Pb tubes with different dimension were passed through two stations under different conditions and were reshaped to square section (n¼4), Fig. 15. Experimental investigation of the process has been carried out using a test rig with 4 flat rolls which had been designed and built for previous work, Fig. 16. The rolls were made of CK45 and with the outer diameter of 107.5 mm. Two coupled gearboxes and motors with an output power of 1.1 KW were used. The diameter and speed of rolls were constant for all experiments. The output speed of 18 rpm was used. No lubricant was used. The friction factor m for the dry condition between Pb tube and rolls is 0.35. Table 1 shows dimensions of initial tubes. Dimensions of reshaped tubes in the first and the second stations are presented in Tables 2 and 3, respectively. Comparison of theoretical and experimental results showed good agreement. The results are shown as diagrams in Figs. 17–19.
[1] Kiuchi M, Shintani K, Tozawa M. Investigation into forming process of square pipe—experimental study on roll forming process of non-circular pipe—II. J Jpn Soc Technol Plast 1980;21:73–80. [2] Kiuchi M. Overall study on roll forming process of square and rectangular pipes. In: Proceedings of the second international conference on rotary metal working process Straford-Upon_Avon; 1982 pp. 213–226. [3] Wen B. Using advanced tooling designs to reshape round tube into square and rectangular tubes. Ohio, U.S.A: Roll-Kraft, Inc.; 1998. [4] Kiuchi M, Feizhou W. Reshaping of round pipes into square and rectangular pipes. J Tube Pipe Technol 1999:72–5. [5] Kiuchi M, Moslemi Naeni H, Shintani K. Numerical analysis of reshaping processes of pipes with non-circular cross section. Adv Technol 1999; III:2399–404. [6] Moslemi Naeni H, Kiuchi M, Kitawaki T, Kuromatsu R. Design method of rolls for reshaping processes of pipes with pentagonal cross-sections. J Mater Process Technol 2005;169:5–8. [7] Bayoumi Laila S. Cold drawing of regular polygonal tubular sections from round tubes. Int J Mech Sci 2001;43:2541–53. [8] Bayoumi Laila S. Analysis of flow and stresses in flattening a circular tube by rolling. J Mater Process Technol 2002;128:130–5. [9] Abrini K, Farahmand HR. An upper bound analysis for the reshaping of thick tubes with experimental verification. Int J Mech Sci 2008;50:342–58. [10] Abrinia K, Farahmand HR. Manufacturing of polygonal tubes from round pipes by shaped rolling. In: Proceedings of the 40th CIRP International Seminar on Manufacturing Systems. Liverpool, England, June; 2007. [11] Bayoumi Laila S, Attia AS. Determination of the forming tool load in plastic shaping of a round tube into a square tubular section. J Mater Process Technol 2009;209(4):1835–42. [12] Tajyar A, Abrinia K. FEM simulation of reshaping of thick tubes in different passes. Int J Recent Trend Eng 2009;1(5):82–5.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
An upper bound analysis for reshaping thick tubes to polygonal cross-section tubes through multistage roll forming process H.R. Farahmand a,n, K. Abrinia b a b
Research Group of Metallic Material Processing Technology, ACECR, Tehran University, Tehran, Iran School of Mechanical Engineering, University of Tehran, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2014 Received in revised form 15 June 2015 Accepted 17 June 2015 Available online 26 June 2015
In this investigation, an upper bound solution is presented for the multistage forming process to produce polygonal cross-section thick tubes by means of a roll forming rig with flat rolls. The modeling and simulation is presented so that it can be generalized for arbitrary number of sides and stages. Comparison of theoretical and experimental results for tube cross-section dimensions (height, wall thickness and outer corner radius) show good agreement for a square section tube and demonstrated the capabilities of the new formulation presented. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Upper bound Roll forming Tube reshaping Polygonal cross-section tubes Multistage deformation
1. Introduction Tubes with different cross sections are used widely in industries as structural components and building materials to home appliances. Recent applications of polygonal tubes, such as square and rectangular seamless tubes, have resulted in increased in industry (e.g. natural gas, chemical liquid transportation, construction of boilers, heat exchangers, etc.). In addition, the polygonal tubes have advantages such as high torque resistant in twist and weight reduction of structure parts compared to circular tubes. For producing these tubes, various methods such as extrusion, drawing, hot forging, roll forging, and roll forming might be used. The shape rolling method by passing a circular tube through flat or curve rolls is also one of the common methods and is more efficient. As yet, different analytical and numerical solutions have been obtained for reshaping of pipes and tubes. Kiuchi et al. [1] studied the process of reshaping thin circular pipes to non- circular tube sections in cold roll forming. He [2] also developed his study to produce square and the rectangular tubes from pipes by experimental investigations. In a published report by Wen [3] some details about an advanced tooling design which utilized numerical analysis to predict the material deformation and improved the tooling performance. Finite element numerical simulation was used to simulate the results and study effects of processing parameters. A computer-aided simulation program was developed
n
Corresponding author. Fax: þ98 21 88007277. E-mail address: [email protected] (H.R. Farahmand).
http://dx.doi.org/10.1016/j.ijmecsci.2015.06.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
by Kiuchi and Feizhou [4], based on 3-D elasto-plastic finite element method. Using this program they were able to analyze all of deformation features and mechanical characteristics of the reshaping process. Kiuchi et al. [5], presented theoretical methods to analyze and optimize the reshaping process. A mixed method composed of the finite element and finite difference was applied and called finite differential method. They also reported their work on reshaping of no-circular pipes, [6]. In an analytical work by Bayoumi [7] a solution was obtained for the problem of cold drawing through flat idle rolls of regular polygonal metal tubular sections from round tube. Bayoumi also gave an analytical solution for the problem of cold flattening of a round tube into an oblong shape through rolling between two flat rolls [8]. Abrinia and Farahmand [9] presented a new solution based on upper bound method that is used to solve the reshaping of thick square tube from a round tube in one stage. The influence of various effective process parameters was investigated. They generalized this method for polygonal tube sections, [10]. For determining the forming tool load in plastic shaping of a round tube into a square tubular section through a head comprising four idle flat rolls, Bayoumi and Attia [11] presented an analytical solution and finite element simulation using finite element code ABAQUS/STANDARD and LS-DYNA. Tajyar and Abrinia [12] studied reshaping of a circular thick tube into a square cross section by cold roll forming between four flat rolls in different passes. The influence of the amount of roll gap reduction in each pass on the final rolled product was investigated by FEM method. In the previous work [9], authors presented a new analytical solution for reshaping a pipe to square section in one stage basis
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
_i W _s W
Nomenclature half-width of flat side at exit be 0 f streamline function at contact region f ; g; h Components of Bezier function h normal distance from tube axis to flat side he normal distance from tube axis to flat side at exit Jn upper bound on total rolling power L bite length m friction factor 0 n number of sectors n number of contact surface division p, q, u parameters of Bezier velocity field at free region R radius of outer corner of tube at entrance r position vector r0 outer corner radii at final cross section r 0 and r 3 position vectors at initial and final cross section Δr roll gap reduction R1 roll radius S circumference length S0, Se circumference length at entry and exit, respectively t0 initial wall thickness te wall thickness at exit U_ constant circumferential velocity of rigid roll vx ; vy ; vz components of velocity field in free parts v0 entry velocity of tube V x ; V y ; V z velocity field components in contact part vn the velocity of material at contact surface _f frictional power dissipated over frictional boundary W
on upper bound method. It was the first time that upper bound method was used for roll forming process. In this paper the solution has been generalized so that can be used for multistage shape rolling of thick tubes to polygonal section with flat rolls. Dimensions of the deformed tube such as height, wall thickness and outer corner radius at each stage of reshaping process are determined by this solution and therefore can be used in pipe industry to produce polygonal tubes. The modeling and simulation can be generalized for arbitrary number of polygonal sides and forming stages. One of the most significant advantages of this solution is its simplicity and capability to find the answers at a very short time in comparison with the FEM solution. The influence of other parameters such as rolls radius, initial tube dimensions, friction coefficient and roll gap reduction, which had been studied for one stage in the previous work could be used for this situation. The tube material is Pb and is assumed to be isotropic, incompressible and follows a rigid-plastic behavior. To illustrate the integrity of analytical solutions some sample tubes were formed during a two-stage process to compare results to each other.
power dissipation due to internal deformation power consumption of shear boundary
Greek letter
α θ ϕ ϕc ψ σ0 σm ε_ ij ε_ τ δ Γ ξ
bite angle rolling angle angle of deforming region at entry angle of contact part of tube's cross section profile angle of deforming region at exit flow stress mean effective stress strain rate components effective strain rate shear stress height reduction of tube cross section during rolling boundary velocity field parameter at contact region
Subscripts C F CF i, e s
due to contact region due to free region due to common surface between contact and transition regions entry and exit planes, respectively due to shear
Fig. 1. The polygonal tube sections.
2. Material flow and strains analysis 2.1. Geometry of deformation zone 0
For a polygonal shape tube with “n ” sides, the section can be 0 divided into“2n ” equal sectors. Each sector has height he and radius ro with a flat side be . The primary input tube with radius R 0 and thickness t 0 also can be divided to “2n ” equal sectors, Fig. 1. At the first stage, the tube section is deformed to a polygonal shape with a flat side 2be ð1Þ , corner radius roð1Þ, and thickness
Fig. 2. A sector of the round tube and polygonal tube.
91
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Fig. 3. The deformation geometry of square tube in the first stage.
Fig. 4. The deformation geometry of square tube in the second stage.
t e ð1Þ. The first sector height reduction, δð1Þ; is equal to difference between Rð ¼ hi ð1ÞÞ and he ð1Þ; Fig. 2. Cartesian coordinate system origin is located at the intersection of the tube axis with the entry section of the deformation zone. The bite angle is p ð1Þ. Geometry of deformation zone for a 0 square section (polygonal with n ¼ 4) is shown in Fig. 3. This tube will be deformed again in the second stage. Thus at the exit, the flat side of the tube cross section,be ð2Þ, and the radius of the round corners, roð2Þ, will be respectively larger and smaller. Deformation zone geometries for the two consecutive stations such as the first and second stage for a square section are shown in
Fig. 4. Subscripts ‘e’ denotes exit plane of deformation zone and number ‘2’ shows the forming stage. Fig. 5 shows the deformation geometry for the first and second stages. If the second stage is considered, the entry section with the thickness t e ð1Þ can be taken apart to two parts: at the first part the corner arc of the cross section, r 0 ð1Þ, is converted to a smaller arc, roð2Þ, and a flat side, be ð2Þ; with thinner thicknesses, t e ð2Þ. So this is similar to the deforming zone geometry at the first stage. At the second part, the flat side of tube section which is contacting with the roll is converted to a thinner one. The thickness of these two parts at the exit will be equal, Fig. 6.
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
This deforming procedure will be repeated at the further stages similarly. At each stage such as j, the tube section can be divided into two parts in entry: 0 a flat side, be ðj 1Þ, and an arc RðjÞ ¼ r o ðj 1Þ with angle π =n . Then due to the deforming process, the flat side dimension will be larger and the corner radius will be smaller at the exit. Finally, the desired dimensions of flat side and corner radius will be obtained when tube exits from the last stage. Deformation energy is not dependent of coordinate system . Therefore, a new coordinate system such as X10 ðj ÞO10 ðj 1Þ Y10 ðj 1Þ could be considered for the j th deformation stage. The origin O10 ðj 1Þ is coincident with the center of the corner r o ðjÞ, Fig. 7. In the previous article of the authors [9], an analytical solution was presented for reshaping of the tube at one station, based on the upper bound technique. Now by considering the new coordinate axes, that analysis can be used for the tube deforming zone at the second or further stages. It is assumed that during a deformation stage, the outer perimeter of tube varies linearly from entry toward to exit, it verifies that SðjÞ ¼ S0 ðjÞ þ ðS0 ðjÞ S0 ðjÞÞ U
θðjÞ αðjÞ
ð1Þ
The deformation zone geometry is complex here and if material particle is considered to be moving from the entry to exit section
93
in a stage, it would pass through different paths. The deformation zone is divided into two areas: the “contact region” (the region which is in contact with the rolls during deformation) I and IV, and the “free region” (the region which is not in contact with the rolls during deformation), II and III, Fig. 8. At the contact regions (I and IV), the material meets the rolls and deforms, and at the free region (II and III) it deforms but does not touch the rolls. It is assumed that in the contact zone the tube surface is completely in contact with the roll surface and there is no material flow in the X direction. Curve f shows the boundary between regions I and II. By dividing the bite angle into n equal portions, curve f is also divided into n pieces. Now, the contact surface can be approximated with some longitudinal strips which are cylindrical surfaces with radius R1; Fig. 9. Therefore the outlet section also can be divided into n part. Besides, the entry section also can be divided into n parts such that each one can be related to its corresponding part at the exit section. Now, an article such as N in region II moves from point N1ðjÞ at entrance section towards the roll and enters region I at point N2ðjÞ, Fig. 10. Afterwards it travels on a circular curve under the roll to point N3ðjÞ. It is assumed that the entry and exit sections are perpendicular to the axis of rolling for all regions. 2.2. The kinematically admissible velocity fields In upper bound technique, it is necessary to define a kinematically admissible velocity field for the deformation region. Since
Fig. 5. The deformation geometry of the first and the second stages.
Fig. 7. New coordinate axes.
Fig. 6. Deformation of tube section in the second stage: (a) entry, (b) exit.
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Fig. 8. Division of deformation zone in the first and jth stages.
Fig. 9. Approximation of contact surface with cylinderical strips.
2.2.1. Admissible velocity field for contact regions; I and IV In the contact area, an elliptic single-parametric family curves, which is suitable to show the streamlines, is used to obtain the admissible velocity field, Fig. 11. An elliptic single-parametric family curves which achieved for the geometry of this problem is defined as ðz z0 Þ2 ðy R1 þhe 1 ξ e Þ2 þ ¼ 1 ; 0rξr1 ð2Þ ðR1 þ 1 ξ ðt e eÞÞ2 R12 where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 U t0 t e R12 z0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e¼ R1 R12 z0 2 thus Fig. 10. A flow line for a particle moves through regions II and I.
the material flow during the process was of a complicated nature due to the many regions of contact and non-contact between the workpiece and the tools, therefore a single generalized formulation could not be derived for the velocity field and three different velocity fields were formulated for the different parts of the deforming region and then incorporated into a single generalized formulation for the upper bound on power. These velocity fields are as follow:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y he R1 þ R12 ðz z0 Þ2 ffi ξ ¼ f ðy; zÞ ¼ 1 þ R1 U qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R12 ðz z0 Þ2 þ e UR1 tf e 0
ð3Þ
Referring to Fig. 11, ξ ¼ 1 gives the equation of roll contact surface or upper boundary streamline, and ξ ¼ 0 shows the equation of the inner surface of tube or lower boundary streamline. It is proved that if it is assumed that the entry surface is a plane perpendicular to the symmetric axis then the exit surface will also be perpendicular to that axis. Therefore, by supposing that the material under the roll surface has no flow in the X
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
direction, the velocity field is obtained as, [9] vx ¼ 0; 0
vy ¼
∂f ðz z 0 Þ F 0 ξ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂z R12 ðz z0 Þ2 ðt e eÞðy he Þ t e U R1 2 ð R1 U v0 U t 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðt e eÞ R1 ðz z0 Þ þ e UR1
vz ¼
0 ∂f F ξ ¼ ∂y
ðz z 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt e eÞ R1 ðz z0 Þ2 þ eU R1
ð4Þ
0 where F 0 ξ is a function to specify components of f in Cartesian coordinates system accordance to initial conditions, [9]. This velocity field satisfies the incompressibility condition; ε_ ii ¼ 0. 2.2.2. Admissible velocity field for free regions: II and III To obtain the velocity field in the non-contact areas, the Bezier curves is used. A cubic Bezier curve, Fig. 12, is defined as follow, [9]: r ¼ ð1 pÞ3 r 0 þ 3pð1 pÞ2 r 1 þ 3p2 ð1 pÞð1 pÞ3 r 2 þ p3 r 3 ;
0 rp r1 ð5Þ
By variation of Bezier curve parameters, the shape of the curve changes, but it still passes p0 and p3. The equation can be written as ! ! ! r ¼ f ðu; q; pÞ i þ g ðu; q; pÞ j þ ðu; q; pÞ k ; 0 r p; q; u r 1
ð6Þ
95
where ‘f’, ‘g’ and ‘h’ define the position of the particle on the streamline. ‘u’ and ‘q’ are parameters which are used to describe entry and exit surfaces and ‘p’ is a parameter used for z coordinate of the particle. The velocity vector in the Cartesian coordinates is given by ! ! ! V ¼ V x i þV y j þ V z k ð7Þ Since the particle of the material is moving on a streamline then the components of the velocity vector in the Cartesian directions will be coincident with the tangents of the streamline at that point. Thus Vx ¼
fp V z; hp
Vy ¼
gp V z; hp
V z ¼ Mðu; q; pÞ
ð8Þ
Mðu; q; pÞ is obtained from the incompressibility conditions as M ðu; q; pÞ ¼
C U hp
hu f q U g p f p U g q þ hq f p U g u f u U g p þ hp f u U g q f q U g u
ð9Þ where C is a constant and dependent on boundary conditions. The boundary conditions for regions II and III are different so the constant C must be verified for each region separately. 2.2.2.1. Velocity field in region II. A flow line is shown in Fig. 13. A particle such as P moves on a streamline from inlet toward outlet. This path can be divided into two pieces: a curve P 0 P 3 in region II and a curve P 3 P 4 in region I. The flow line should be perpendicular to entry section at point P 0 , and be tangent to a flow line of contact
Fig. 11. Streamlines model for contact region.
Fig. 13. A streamline relates region II to region I.
Fig. 12. Bezier curve as a streamline.
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Fig. 15. The Pb tubes were used for experiments.
Fig. 14. Bezier curve as streamline for free region III.
area I at point P 3 . Two curves joined at a common end point have at least C 0 continuity at their junctions. C 1 continuity requires a common tangent line at their junction. These conditions secure the smoothness of curves at P 3 . The velocity of the particle P at point P 3 on the curve P 0 P 3 must be the same as on the curve P 3 P 4 to prevent the velocity discontinuity. At the outlet section p ¼ 1; thus
M ðu; q; 1Þ ¼ vðIIÞ z z ¼ zi C ¼ C ðu; qÞ ¼ vðIÞ z z ¼ zi
hu f q Ug p f p U g q þ hq f p U g u f u U g p þ hp f u Ug q f q Ug u hp C ¼ 0:5 v0 U t 0 φ2 φ1 ð2R t 0 Þ 2.2.2.2. Velocity field in region III. In Fig. 14 a flow line is shown for region III. This is perpendicular to the inlet section. If the boundary condition at entry is considered, then p ¼ 0 and
M ðu; q; 0Þ ¼ vðIIIÞ z
Fig. 16. The experimental test rig with four flat rolls.
Since there are four deformation regions X
X
X
X
X _ i¼ _i þ _i þ _i _i W þ W W W W I
II
III
IV
ð14Þ
3.2. The power due to shear _ S is calculated from The shear power W Z _ S¼ Δv U τ UdA W
ð15Þ
z¼0
hq ¼ hu ¼ 0; hp ¼ 1 M ðu; q; 0Þ ¼ V z ¼ v0
In the regions in which the material experiences velocity discontinuities, such as the entry and exit sections of deformation, the power consumption due to the shear boundaries will exist.
therefore
C ¼ C ðu; qÞ ¼ f u Ug q f q U g u
p¼0
3. Upper-bound solution In accordance with the upper bound theorem, the following equation should be minimized for the actual velocity distribution in each stage X X X _ Sþ _ ; _J ¼ _ iþ W W ð12Þ W f _ i is where _J is an upper bound on the total power consumption, W _ s is the power consumption the internal power of deformation, W _ f is the frictional power due to the shear boundaries and W dissipated over the frictional boundaries. 3.1. Internal power of deformation The internal power of deformation is obtained from the following equation: Z U 0:5 2 2 0:5 ε_ 2x þ ε_ 2x þ ε_ 2x Þ þ ε_ 2xy þ ε_ 2yz þ ε_ 2zx ð13Þ W i ¼ pffiffiffiσ m 3 V
3.2.1. Contact regions; I and IV The velocity discontinuity does not exist on the entry exit boundary of region I, from the basic nature of its derived velocity field. Therefore the power of shear deformation consumed on this boundaries, vanishes. Similarly, there is no velocity discontinuity on the exit boundary of region IV. On the entry boundary of region I, or boundary between contact region I and
free region II, also _ S ¼ 0; and W _S there ¼
is no velocity discontinuity. Thus W I IV _S W . i IV
3.2.2. Free regions: II and III According Fig. 13 the shear deformation power on the entry and the exit boundaries of region II vanishes since there is no velocity
_ s ¼ 0; discontinuity on them. Thus W II Fig. 14 shows the region III. The velocity discontinuities exist on the entrance and exit boundaries. On the entrance, pi ¼ 0 and
_S _ Si ¼ W ð16Þ W III
III p ¼ 0
Similarly on the exit, pi ¼ z0 ¼ l and
_ _S W Se ¼ W III
III p ¼ z0
ð17Þ
H.R. Farahmand, K. Abrinia / International Journal of Mechanical Sciences 100 (2015) 90–98
Here, the interface friction is assumed to be given by
Table 1 Dimension of test specimens. Specimens
Outer diameter (mm)
Inner diameter (mm)
Thickness (mm)
Tube Tube Tube Tube Tube
85.6 85.4 85 83 83
68 68 65.5 65 63.6
8.8 8.7 9.7 9.0 9.7
1 2 3 4 5
Table 2 Dimensions of reshaped tubes after the first stage. Specimens dð1Þ Height decrease (mm)
Type t e ð1Þ Output thickness (mm)
r o ð1Þ Outer corner radius (mm)
2he ð1Þ Twice sector heighta (mm)
Tube 1
3
Tube 2
3
Tube 3
2.5
Tube 4
3
Tube 5
2.5
Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal.
37.8 38.38 37.7 38.27 37.7 38.72 36.0 37.26 36.1 37.81
82.6 82.60 81.9 82.40 83.1 82.50 80.2 80.00 80.2 80.50
a
8.6 8.68 8.0 8.58 9.6 9.59 8.8 8.87 9.5 9.58
For square tube, the section height is twice the sector height.
therefore
_ Si _ Se _S ¼ W þ W W III
III
I
II
III
mσ τ ¼ pffiffiffim
ð22Þ
3
where m is constant during the deformation process and takes the values between 0 and 1. Thereby with minimizing the total deformation power in each stage, _J, the independent problem unknowns consist of r o ðjÞ, t e ðjÞ, and zn ðjÞ obtained and the dependent unknowns can be calculated. The dependent unknowns are: be ðjÞ, he ðjÞ. These values are used as input data for the next stage so that: r i ðjþ 1Þ ¼ r o ðjÞ, t i ðj þ 1Þ ¼ t e ðjÞ, bi ðj þ 1Þ ¼ be ðjÞ and hi ðj þ 1Þ ¼ he ðjÞ. The Complex method (M. J. Box-1965) which is applied for constraint optimization in nonlinear problems is used to minimize the total deformation power. The constrained Simplex (Complex) method searches for the maximum value of a function f(x1;. xn) subject to p constraints of the form gk rxk rhk, k¼ 1,. p, where xn þ i,. xp are functions of x1,. , xn, and the lower and upper constraints gk and hk are either constants or functions of x1,. , xn. (To find a minimum, f is maximized.) It has been developed from the Simplex method of Spendley et al. It is assumed that an initial point xi1,…, xn1, which satisfies all the p constraints is available. In this method, k Zn þ1 points are used, of which one is the given point. The further (k 1) points required to set up the initial configuration are obtained one at a time by the use of pseudorandom numbers and ranges for each of the independent variables, viz. xi ¼gi þai(hi gi) where ai is a pseudo-random deviate rectangularly distributed over the interval (0, 1).
ð18Þ
III
Table 3 Dimensions of reshaped tubes after the second stage.
and the total shear power will be equal to
X _S þ W _S þ W _S _S _ s¼ W þ W W
_S _S ¼ W þ W
97
III
IV
ð19Þ
IV
3.2.3. The power due to friction In the free region of deformation there is no contact between the tube and rolls, and then the frictional power for that region vanishes. But in the contact region, the frictional power dissipated over the roll surface is given by Z _f¼ W τ U Δv UdA ð20Þ
Specimens dð2Þ Height decrease (mm)
Type t e ð2Þ Output thickness (mm)
r o ð1Þ Outer corner radius (mm)
2he ð2Þ Twice sector height (mm)
Tube 1
1.5
Tube 2
3.5
Tube 3
1.5
Tube 4
1.5
Tube 5
1
Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal.
36.5 35.89 32.9 34.09 37.3 36.31 35.0 34.84 37.8 36.11
81.4 81.1 78.6 78.9 81.3 81.0 79.1 78.5 79.5 79.56
8.2 8.62 8.7 8.43 9.5 9.52 8.4 8.81 9.5 9.54
contact S
But
thus
2
_ ¼ be 6 W 4 f
Z
zn 0
Z þ
that
z0
zn
R1 ds ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R12 ðz z0 Þ2
Outer corner radius after roll shaping 44
τ U_ v_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidz R1
2
R1 ðz z0 Þ
2
3
τ v_ U_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidz7 5 R1
R12 ðz z0 Þ2
ð21Þ
where τ is the frictional shear stress along the contact surface of the rolls, U_ is constant circumferential velocity of the rigid roll, v_ is the velocity of material at contact surface, and zn is the abscissa of the neutral point on the contact interface.
Outer corner radius, mm
dA ¼ dsU dx
Outer corner radius after the 1st stage (exp.)
42 40
Outer corner radius after the 1st stage (cal.)
38
Oyter corner radius after 2nd stage (exp.)
36 34 32
0
1
2
3
Tube No.
4
5
Outer corner radius after the 2nd stage (cal.)
Fig. 17. Comparison between outer corner radius of tubes by experiment and calculation after stages 1 and 2.
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5. Conclusion
Tube thickness after roll shaping 9.75
Tube thickness after the first stage (cal.)
Tube thickness, mm
9.5 9.25 9
Tube thickness after the first stage (exp.)
8.75 8.5
Tube thickness after the 2nd stage (cal.)
8.25 8 7.75
0
1
2
3
4
5
Tube thickness after the 2nd stage (exp.)
Tube No. Fig. 18. Comparison between tube thicknesses after roll forming in stages 1 and 2.
In this study, an analytic solution based on upper bound theory is presented for reshaping of a thick tube to a square tube in multistage roll forming by using a roller with four flat rolls. This method can be generalized for reshaping process of other regular sections. The material is considered as rigid perfectly plastic. To evaluate the analytical model results, some tube samples were reshaped at two sequential stages by a rolling machine. The results obtained for the thickness and the radius corner in both theoretical and practical procedures are very close together. Using this method and considering the next sequential steps, the multistage deformation process for producing polygonal tube can be simulated and the effects of various factors on its quality be predicted.
Acknowledgment The authors would like to acknowledge with gratitude for the support and laboratory services given to the research by the Metallic Material Processing Technology Research Group of ACECR, Tehran university branch.
References
Fig. 19. Comparison between tubes section heights after stages 1 and 2.
4. Experiments and results To evaluate the validity and efficiency of the mentioned solution, the essential experimental and theoretical results were compared. Carrying out the experiments, some thick Pb tubes with different dimension were passed through two stations under different conditions and were reshaped to square section (n¼4), Fig. 15. Experimental investigation of the process has been carried out using a test rig with 4 flat rolls which had been designed and built for previous work, Fig. 16. The rolls were made of CK45 and with the outer diameter of 107.5 mm. Two coupled gearboxes and motors with an output power of 1.1 KW were used. The diameter and speed of rolls were constant for all experiments. The output speed of 18 rpm was used. No lubricant was used. The friction factor m for the dry condition between Pb tube and rolls is 0.35. Table 1 shows dimensions of initial tubes. Dimensions of reshaped tubes in the first and the second stations are presented in Tables 2 and 3, respectively. Comparison of theoretical and experimental results showed good agreement. The results are shown as diagrams in Figs. 17–19.
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