CC BY
21A.V.Filimonov
Ltd Co «New Industrial technologies», city of N. Novgorod S.R.Dzhanaev, E.D.Kondratiev, V.I.Filimonov
JSC «Ulyanovsk Mechanical Plant», town of Ulyanovsk
FORCE DISTRIBUTION AT FLANGE FORMING OF ROLL-FORMED SECTIONS WITH MULTIPLE RIGIDITY ELEMENTS
Abstract: A new approach to calculate the linear force and expanding force between the rolls is proposed. The approach cumulate theoretical, finite-element and experimental methods. Some comparisons of the obtained results with those of previous authors are made. The proposed approach is obviously easier and exact.
Key words: forming roll, roll-formed section, rigidity element, linear force, load point.
These latest years the intensified roll-forming processes are of growing importance for constructional engineering and machine building [1-4]. Reduction of number of passes and diminution of forming roll diameters, use of closed-end openings and severe roll-forming modes - these are only few factors specific for the intensive deformation method (IDM) [5] which increase the level of the process force parameters, especially, if the folded flanges bear some rigidity elements. One of the most important factors is the market trend of commissioning new 3 mm coil stock with zinc coating [6]. If during roll-forming the contact stresses between the band coating and the forming tool are huge, the coat material is seized by the tool, reducing thus its service life due to premature wear. The task of force parameters calculation, as well as force distribution determination, are among the most important problems in roll-forming relating to further calculation of forming tool parameters, tool wear assessment and specialized roll-forming machine parameters computation.
It is most rational to consider this subject as a case study of roll-forming of a U-section with peripheral rigidity elements turned inwards. The last stipulation is of primary importance since the external rigidity elements are usually formed with their bending over round areas of the forming tool. This requires the consideration of the full stress diagram and renders the task rather complicated. The case study of U-section makes it possible to study the folding of a flat flange which is equivalent to any flange with rigidity elements turned inwards (fig. 1). The above equivalence is established through the use of local rigidity method exposed in paper [7].
Fig. 1. Reduction of a flange with rigidity elements to a smooth flange with equivalent thickness
The force distribution on the blank surface during roll-forming, broadly speaking, is far to be a trivial task, even for an angle or a U-section [5, 8]. The point is that previous hypotheses of uniform or linear distribution of bending forces, which were earlier used to design the analytical models, are not in full agreement with experimental data neither in intensive deformation method, nor in traditional roll-forming.
More precise data relating to force distribution may be obtained via the use of the finite element methods [9, 10] realized, for instance, in LS-Dyna program (being an Ansys program module) [11]. Thus, we simulated the U-section roll-forming with folding angles specific for respective combination of the shape cross-section parameters being as follow: flange width being from 10 to 60 mm with a step of 10 mm; blank thickness being from 1 to 4 mm with a step of 1 mm. The simulation parameters are as follows: finite element type was Shell 163 with single point reduced integration diagram spreading over the surface with Hourglass 4 checking; material model for the section was a bi-linear isotropic model (*MAT_PLASTIC_KINEMATIC), for rolls - tool steel - U8. The rolls were considered as perfectly rigid bodies with finite-element meshing on their surface only. The contact between the rolls and the blank (forming surface-to-surface contact) was taken in the form FSTS, while the one for the blank was ASTS (automatic surface-to-surface contact).
The simulation has shown that the distribution of normal forces across the folded flange has a form reminding a hyperbole or an exponential curve with its maximum value located near the bending zone. At the same time, up to 90 % of the shear area falls to a 1/3 of the flange width if counted from the bending zone. At this level, the force values for different blank thicknesses were 0.2 to 0.5 of the maximum force which were observed near the bending zone. Fig. 2 obtained in simulation process illustrates the character of the linear force distribution on the outer surface of the blank turned toward the lower roll.
Fig. 2. Linear force distribution along the folded flange at a total bending angle of ak
The finite element method is to be considered as a tool ad hoc, intended to solve a very specific problem. It differs from analytical methods by its limitations referring to the generalization of the results being applied to a large class of contact problems, including those existing in the roll-forming.
To make the transfer to analytical representation of linear force distribution along the flange width, it is convenient to tie-up the value of acting forces to the working surface of the forming rolls or to the points of the blank surface in the axial plane of forming rolls (fig. 3). It may be done through natural setting of the blank medium surface with the help of two parameters: r - current coordinate along the medium line of the folded flange, counted from the bending zone; a - resultant flange folding angle equal to the total bending angle of the current pass k (a = ak). In this case the folding angle in the current pass will be denoted as follows = ak — ak-1.
Taking into account the results of the mathematical simulation, it is supposed to describe the roll-forming process with the help of a hypothetical exponential force distribution function reflecting the active forces in the current pass:
{
P( r) = P0 exp
r - x
V
n
J
(1)
where: P0 - is the maximum value of the acting force; r - is the current coordinate counted along the generatrix of the forming roll tapering part; n is a dimension parameter determining the exponent form; x is the dimension of the «dead zone» (lack of contact in the angle zone), i.e. distance to the junction point of the generating lines of cylindrical and conical parts of the lower forming roll extending to the contact point between the blank and the tool.
Fig. 3. Local coordinate system and cross-sectional parameters of the shape
In formula (1) the pass indexing is conventionally omitted, although in reality the acting forces are related to the passes which may be easily identified according to total folding angles ak, being part of the respective dependencies. The «dead zone» dimension is easily obtained through geometric consideration of the blank location in the angle zone of roll opening.
x = s(^ + 1)tg
2
(2)
J
where: s - is the blank thickness; ^ - is the relative bending radius of the blank equal to r^/s ; rb - is the inner bending radius of the blank.
The «dead zone» dimension in equation (2) increases with the growth of the total bending angle. The angle zone of the roll opening represents so called «dead zone», i.e. the zone where the blank is not limited by the tool. The bent part of the blank may move from the rounded area of the upper roll in radial direction under the action of the folded flange edge compression, if the intensive deformation method is used.
At the level of 1/3 of the flange width, the acting force value in equation (1) may be determined by the relation:
P(b /3) = kPo, (3)
where: b - is the folded flange width; k - is the attenuation ratio of the maximum force.
The equation (3) allows to easily find the parameter n, depending on the «dead zone» given by the equation (2).
3x - b
n =
ln k
(4)
The maximum value of the acting force may be obtained through comparing the moments of inner and outer forces:
J rdP( r ) = -T Ln
(5)
where: T - is a linear moment of inner forces bringing the angle zone adjacent to the folded flange into plastic state on the flange deformation length L ; n - is an extension ratio of the angle plastic zone.
The ratio n in equation (5) is used to take into account the relationship between the deformation length and the extension of the plastic angle zone along the roll-forming axis.
The integration by parts in the left part of equation (5) and easy algebraic transformations yield the following relation to determine the value of the acting force:
Po =-(TL)
(b + n )exp
b - x
n
( x + n)
-1
(6)
The moment of inner forces and the deformation length appearing in relation (6) are given as analytical dependencies in the paper [8]. Particularly, when the strain hardening is absent, the moment of the inner forces, as it is known, is given by the following formula:
T = —Ts2, 4
where —T - is the yield stress of the blank material.
The relations (1), (4) and (6) give the complete linear force distribution across the folded flange. Fig. 4 and 5 illustrate the linear force distribution according to formula (1).
200
150
E 100
£
pu 50
b = 100 mm;
4 a - CTs L о ; о -= 260 M и , Pa
3 2 1
__ /
20
40 60 r, mm -
80
100
Fig. 4. Distribution of linear force along the flange width: 1, 2, 3, 4 - s = 1, 2, 3 u 4 mm respectively
Fig. 5. Distribution of linear force along the flange width: 1, 2, 3, 4 - a = 10, 20, 40 u 90° respectively
It may be seen in fig. 4 that the linear force increases as the blank thickness grows. The maximum linear force also augment if the folding angle tends to rise (see fig. 5). Practically, it means that a big amount of load is localized on the area adjacent to the angle zone. The loading point (cen-troid) on the flange surface rR may be calculated as follows:
Г
| rP( r )dr
R
x
J P( r )dr
(7)
Fig. 6 shows the calculation results obtained with the aid of MathCAD2001Pro pack according to relation (7). The centroid curves move toward the angle zone as the folding angle grows. It is also seen that the loading point is located near the angle zone. These results are very different from those of paper [12] alleging that the loading point should be at a distance of (2b)/3, if it is measured from the angle zone.
Fig. 6. Load point position of the force acting on the flange: 1, 2, 3, 4 - b = 40, 60, 80 h 100 mm respectively
The erroneous character of this conclusion may be seen if we take into account the following reasoning. At the flange folding the resistance moment of the plastic joint does not depend on the
loading point, thus a resulting force F should be proportional to the value (1/r ). Further calculations show that rR « b/ln(b/x). The evaluation of the above expression for the flanges 20 to 100
mm large gives the values being 1/4 to 1/3 of the flange width when measured from the angle zone.
It is interesting to compare these results with experimental and calculated data obtained by other authors. A direct measurement of the linear force on the tapering part of the forming roll with the help of pin sensors is sophisticated and expensive, at any rate, we are not aware of some other direct methods applied to measure the linear force. Nevertheless an indirect verification of the model is still possible if we use the integral indicator which is the blank expanding force acting on the lower and upper rolls during the roll-forming. This parameter is often taken as main value for strength calculation of the tooling and roll-forming machines. It may be used to calculate the required power of a roll-forming machine as well.
The paper by Trishevsky I.S. and co-authors [12] considers the 4 mm U-section roll-forming to obtain the rated relationships to calculate the blank expanding force of the rolls and experimental data for three U-sections having flange widths 60, 80 h 100 mm with their bottom width equal to 80 mm. The rated dependencies are given in the form of semi-empirical model for blank expanding using the same symbols as above:
PT = ^ ln
2
C - x ( b
2 x v 2x
\ cosa
+
+ 3,52 • 10-4Ев1,4 s2'6
2
b
4 x 2 (C - 4 x )С
cos« + —
2,6
C
2,6
(8)
where E - is Young modulus.
In paper [13] by Bogoiavlenski K.N. and co-authors a purely empirical model of blank elastic-plastic swaging was derived from experimental data processed statistically. The latter included several hundreds of angle and web sections having wall thickness 1 to 6 mm.
Рб =
0,059a°'42 B °'30
^5^0,43
(9)
where: B - is the blank width (B « C + 2b); 5 - is a relative gap in the roll opening.
The expanding force in formula (9) is expressed in «ton-force» which is easily convertible into Newtons. In this model we find no sign of mechanical property descriptors, although in the paper itself it was indicated that various sections with different mechanical properties had been used. Moreover, this model takes no account of the folding angle, i.e. forming mode. In our model the expanding force is calculated as follows:
РФ = 2(l + cos ak )J P0 exp
' r- x^
n
dr
(10)
The MathCAD2001Pro pack being applied to relationships (8) - (10) allowed us to illustrate them through fig. 7. In the same figure one can see experimental values of the expanding force for web sections 80*b*4 mm made of low-carbon steel.
t
21.0 19.5 18.0 16.5 15.0
13.5
12.0 10.5 9.0 7.5 6.0
NK b 60 mm -----O b SO mm -----Д b 100 mm—-a
\ * * * У у # * РБ Рт Рф
□ -£ \*\ Л ' / v *
/p * в
3 * s *
Л ; ! f • ' 1 '
1 \ 1 ' 1 f 3
Ay
2 /
i/
10
20
30
40
50
60
70
80
90
a, deg.
Fig. 7. Comparative curves of the expanding force when forming the web section 80*b*4 mm according to the models: by Trishevsky (Pf), Bogoiavlensky (Pb ) and proposed model (Pp): 1, 2, 3 - b = 60, 80 and 100 mm respectively. Tree markers are referring to experimental data of paper [12] for the same shape in the second stand (0 =10°; a=18°)
The Trishevsky model has some deviations from experimental data within 7-30%, while model (10) remains within the limits of 2-11%. The expanding force in models (8) and (10) depends on the folding angle similarly: the bigger is the folding angle, the less is the expanding force. In Bogoiavlensky model the elastic swaging was taken as big as 12% (compensation of elastic strain in the roll-forming stand). In calculations according to model (9), we have taken the gap between the forming rolls equal to the blank thickness (i.e. no swaging). If we use the intensive deformation method, the contact between the lower and upper rolls is effected on their mounting surfaces at the level of the roll collars and circular grooves.
The deviations of model (9) from experimental data lie within the limit of 12%, nevertheless the growth of the expanding force with the total folding angle rise has no agreement with the general diagram of force distribution (see fig. 7): as the total folding angle rises the vertical component of the expanding force diminishes, whereas the curves of model (9) tend to become greater.
The curves of fig. 8 obtained according to model (10) and showing the dependence between the expanding force and the blank thickness illustrate the influence of the forming mode on the force parameters of the roll-forming process.
2.0
I"
¿T 0.5 0
1.0 1.5 2.0 2.5 3.0
s, mm-►
Fig. 8. Dependence of expanding force of the flange folding mode: 1, 2, 3, 4 - 6 = 5, 10, 15 and 20° respectively
The paper [14] presents a model to calculate the expanding force between the rolls on the basis of infinitesimal increments for the first forming stand (when the folding angle is equal to 15 degrees). The revealed experimental expanding forces are referring to a web section 60*60*3 mm. The calculated value was 7.5 kN, while the experimental one was 10.17 kN. Our model gives the value of 11.21 kN. The calculation errors were 26% and 10% respectively. It should be bear in mind that the realization of the first model requires much labor to carry out the blank segmentation and further partial force summation. This example clearly confirms sufficient correctness of relations (1), (6) and (10). Finally, we refer the reader to some extended studies in this field [15-17].
Thus, the proposed model embracing the force distribution across the flange is in good agreement with experimental data and may be used to calculate contact stresses, forming roll wear, forming roll and roll-forming machine strength on the stage of their design.
References
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