allowance and developed length for a strain hardening sheet metal formed by ..... shape of the sheet, see Fig. 4(c), will be discussed. Next, this general case will ...
Bend Allowance and Developed Length Calculation for Pressbrake Bending
F. Pourboghrat Alcoa Laboratories, Alcoa Center, PA 15069-0001
Developed length refers to the length of the unstretched fiber measured over both bent and straight sections of a bent sheet. Bend allowance is a term coined by Sachs as a measure of the length of the unstretched fiber in the bent section. Sachs' empirical equation for calculating bend allowance is not physically based and is independent of material and forming conditions. A physics-based model for calculating bend allowance and developed length for a strain hardening sheet metal formed by pressbrake bending is presented. Effects of material properties and tooling geometry on the calculation of these parameters are considered. It is shown that unlike Sachs' assumption, it is the deformed shape and not the neutral axis shift or thinning that is important for calculating the developed length in pressbrake bending. It is also shown, by comparing calculated and measured data, that better accuracy can be obtained when the proposed method is used instead of Sachs' empirical equation.
K. A. Stelson Professor. University of Minnesota, Minneapolis, MN 55455
1
Introduction and Previous Work
The need for accurate bend allowance calculations often dictates the manufacturing method for sheet metal parts and may determine whether those parts can be fabricated with sufficient accuracy in a flexibly automated sheet metal fabrication cell. For example, to manufacture the two piece assembly shown in Fig. 1, one would have two choices. One choice is to first bend the sheets and then drill the holes, and the other choice is to punch the holes first and then bend the sheets. The first method requires the use of special fixtures to drill the holes in the bent sheets, which results in an increase in the manufacturing cost and requires manual operation. The advantage of this method is that the drilled holes will be at the desired locations, and there will be no mismatched holes in the assembly. The advantage of the second method is that it is very easy and economical to punch the holes in the flat sheets first and then bend the sheets. This method is also amenable to flexible automation. The problem with this method is that because of the uncertainty in the bend allowance the holes may not be at the desired locations once the sheet is bent, making the parts difficult to assemble, of lower quality or perhaps unacceptable or requiring rework. If one can obtain an accurate model for pressbrake bending which predicts the exact shape of the bent sheet, then one can accurately calculate the developed length of the sheet. This model can then be used in the second method described above to manufacture precision sheet metal parts economically for aerospace and electronic applications, with the possibility of eventual automation. Currently, in pressbrake bending, the developed length of the sheet is calculated based on the following empirical equation proposed by Sachs (1976) to calculate bend allowance. B = 6-{rp
+
k-t)
(1)
Equation (1) would be applicable if the shape of the deformed sheet is undeformed (straight part) in the free section and bent (circular part) in the punch-sheet contact region (see Fig. 2). This is called the straight-circular sheet shape assumption. The Mactor, a constant related to the neutral axis shift, shown in Eq. (1) can vary from 5 to 5 depending upon the bend Contributed by the Manufacturing Engineering Division for publication in the Journal of Manufacturing Science and Engineering. Manuscript received Dec. 1993; revised Feb. 1996. Associate Technical Editor: D. Durham.
radius to thickness ratio (rp/t) • k = 5 is suggested for cases where the ratio of the punch radius to sheet thickness (rp/t) is less than 2 and k = j is suggested for cases where this ratio is equal to or larger than 2. No further guidelines for choosing a value for k is provided as material properties, frictional conditions and tooling geometry are changed. As a result, once suitable values of k are found by trial-and-error, they are protected as proprietary information. According to Sachs' (straight-circular) model, developed length in pressbrake bending can be calculated as the sum of the length of the straight (undeformed), Lf„, and curved (bent), B, sections where the length of curved sections are given by Eq. (1) and L/u can be calculated as follows: Lfu =
(/-,, + rtl + 0*sin 1 cos 9
(2)
To correctly calculate developed length, one needs to calculate the length of the unstretched fiber along the entire deformed sheet. According to Hill (1950), the unstretched fiber refers to a fiber between the neutral axis and the centerline and which experiences a zero total strain. The neutral axis on the other hand refers to the location in the cross section where tangential stresses vanish. Hill (1950) derives, an exact equation for the amount of the neutral axis shift from the centerline when a rigid perfectly plastic sheet is bent and stretched around a circular punch. Hill's equation shows that neutral axis shift is influenced more by stretching than by bending and that thinning occurs due to stretching alone. Using Hill's approach (1950), an analysis of Sachs' Eq. (1) was done (Pourboghrat and Stelson, 1988a), where it was concluded that Eq. (1) is not physically sound, since proposed values of k differed greatly from those predicted by Hill. Therefore, it was concluded that the proposed values of k compensate for other effects that are not included in Eq. (1) such as: free section rotation and deformation, friction and strain hardening. In a recent paper by Prasad and Somasundaram (1993), the authors attribute the major source of error in bend allowance calculation, using Eq. (1), to the neglect of thinning. In their work, they describe an approximate bending analysis model, similar to that of Dadras and Majlessi (1982) but without the Bauschinger's effect, that accounts for thinning of strain hardening sheets in pure bending. Thinning and neutral axis shift are shown to increase with increased bending curvature. A smaller
Journal of Manufacturing Science and Engineering
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Fig. 1 A sheet metal part with two pieces to be fastened at location shown
bend allowance and a better match with some published data, in comparison with Sachs' model and other models without thinning, are also reported. Although this model improves upon Sachs' assumption in the contact region, it still neglects to take into account the deformation of the entire sheet and its unloaded shape due to springback when calculating bend allowance and developed length. It will be shown in this paper that when thinning and straight-circular sheet shape are used to calculate bend allowance, results do not compare as well with experimental results as it will when the deformation of the entire sheet is considered. In pressbrake bending, where the sheet is hardly stretched, the amount of the neutral axis shift is relatively small (Pourboghrat, 1992). This suggests that it is the shape of the sheet and not the neutral axis shift that accounts for most of the variations in bend allowance and developed length predictions. Figure 3 shows, schematically, the difference in location for a part feature (e.g. a hole), measured a distance L0 from the centerline (C.L.), along two different curves. Curve ( • ) represents a sheet with no strain hardening and a straight free section (SC), and curve (O) represents a sheet with strain hardening and a deformed (SH) free section. The observed difference in feature location is accompanied by different punch travels, YSH versus YSc, required to bend the two sheets with different strain hardening to the same flank angle, 6„. To accurately predict the shape of the sheet, a pressbrake bending model which accounts for strain hardening, friction, tooling geometry and possible formation of a gap between the sheet and the punch is needed. This model must be able to predict the unloaded shape, due to springback, accurately. All pressbrake bending models proposed until now (West, 1980;
Fig. 2 A straight circular sheet shape and definition of /(-factor used in Eq.(1)
Stelson and Gossard, 1982; Gossard and Allison, 1980; Stelson, 1986; Pourboghrat and Stelson, 1988b and 1988c), except for finite element models (Oh and Kobayashi, 1980; Makinouchi, 1984), will not meet all of the above requirements. All analytical models except for Pourboghrat and Stelson (1988b and 1988c) do not account for non uniform deformation that occurs in the punch-sheet contact region, which in turn makes them unable to predict the possible formation of a gap region under the punch. Although the bending model proposed by Pourboghrat and Stelson (1988b and 1988c) can account for non uniform deformation in the punch-sheet contact region and predict the onset of gap formation, its use is limited to rigid-perfectly plastic cases. Therefore, such a model can not be applied to real materials considered in this paper which exhibit strain hardening behavior. This short coming was recently remedied with a new pressbrake bending model for strain hardening sheet metals. The accuracy of this model's predictions are verified and documented (Pourboghrat, 1992). These verifications include comparisons of predicted free section, contact, gap and springback angles and developed length of the sheet with experimentally measured data. In the next section, a brief description of this new pressbrake bending model will be presented.
Nomenclature r„= punch radius
rd = die radius t = thickness w = half of die width do,e„ = loaded and unloaded flank angles 4>o„ = loaded and unloaded free section angles *l*o i/»« = loaded and unloaded contact angles Jo yu = loaded and unloaded gap angles K = strength coefficient of material n = strain hardening exponent E = Young's modulus V = Poisson's ratio Ku = unloaded centerline curvature k, = loaded centerline curvature &n = curvature of neutral fiber B = bend allowance as defined in Eq. (1)
228 / Vol. 119, MAY 1997
k = a constant related to neutral axis shift in Eq. (1) = flank angle (radian) defined in Eq. e (1) Lhu = length of straight section from diesheet contact point up to the center of the hole (unloaded) Lu = length of straight section from diesheet contact point up to the center of the hole (loaded) SDI = arc length from bend line up to the die-sheet contact point (loaded) SHI = arc length from bend line up to the center of the hole (loaded) = arc length from bend line up to the •JDU die-sheet contact point (unloaded) SHU "-= arc length from bend line up to the center of the hole (unloaded) ^Dnl -= arc length from bend line up to the die-sheet contact point (neutral fiber)
Lfu = length of straight section within the die cavity (unloaded) B/u = length of bent section of the free section within the die cavity (unloaded) Bc„ = length of bent section of contact region within the die cavity (unloaded) Bgu = length of bent section of the gap region within the die cavity (unloaded) *- envl = length of straight line enveloping the sheet up to the center of the hole '-'die = length of straight line enveloping the sheet up to the die-sheet contact point L„ = distance from bend line up to the center of the hole, before bending J
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Fig. 3 Difference in location for a part feature (e.g., a hole) measured a distance LD from the centerline (C.L.), along two different curves
2 A Pressbrake Bending Model For Strain Hardening Sheet Metals An analytical model to describe the plastic deformation of a strain hardening sheet metal, undergoing large rotation and deflection in pressbrake bending, has recently been developed. The details of this model can be found in Pourboghrat (1992). This model takes into account the effects of the neutral axis shift due to tension, friction and tooling geometry on the deformation of the sheet. In this model, the most general shape of the sheet is assumed to comprise of the three deformation regions shown in Fig. 4(c): namely 1) the free (II), 2) the contact (III) and 3) the gap (IV) regions. Depending upon the bending geometry rr/t, and flank angle 0, the shape of the sheet may consist of the contact and the free section regions (see Fig. 4(b)) or the free section region (see Fig. 4(a)) only. The undeformed region (I) is assumed to exist for all modes of deformation and will remain outside of the die cavity. For the most general shape of the loaded sheet, see Fig. 4(c), there will be two points of contact between the sheet and the die and two areas of contact between the sheet and the punch. In Fig. 4 ( c ) , that portion of the sheet which is under the punch (region IV) and is sandwiched between the two areas of the punch-sheet contact (region III) is called the gap region. For the other two cases shown in Figs. 4 ( a ) and 4(b), besides the two points of contact between the sheet and die, there is only one point of contact between the sheet and the punch in the nowrap case, while in the wrap-around case there is an area of contact between the sheet and the punch. For the most general shape of the loaded sheet, the bending solution is obtained by simultaneously solving three plane strain boundary value problems for the three deformation regions shown in Fig. 4 ( c ) . Due to the simplicity of the deformation domain, i.e. the centerline of the sheet, the finite difference method is used to solve the two point boundary value problems.
3 Calculating Developed Length For Strain Hardening Sheet Metals The developed length results reported in this paper are all generated from the pressbrake bending model described in the previous section and given in Pourboghrat (1992). It is found that predicted developed length results are a function of the flank angle 0, tool and workpiece geometry, relt, material properties and friction. For rPlt = 1, a gap region will form at very large flank angles, e.g. 0 > 45 deg, while for r,,lt > 1, a gap region may form at very small flank angles, e.g. 0 < 10 deg, (Pourboghrat and Stelson, 1988a). In bending with "typical" setups, that is r,,/t = 1.0, rdlt = 1.0 and wit = 5.0, the use of Journal of Manufacturing Science and Engineering
Eq. (1) proposed by Sachs to calculate developed length will cause errors due to large free section rotation which renders the straight-circular sheet shape assumption inappropriate. In bending with "non-typical" setups, i.e. rdlt > 1.0 and wit > 5.0, the use of Eq. (1) to calculate developed length will cause noticeable errors due to the formation of a large gap region. In this paper, bend allowance calculation will be based on the predicted loaded and unloaded shapes of the sheet. To assess predicted developed length results, a parameter L,m, measuring the distance between the die-sheet contact point and the center of a hole punched into the flat sheet (see Fig. 5(b)), will be used. Parameter L,m is chosen since its accurate prediction depends upon the accurate prediction of the loaded and unloaded shapes of the sheet. In appendix A, the experimental set up used for measuring L,,„ is discussed. In the following section the numerical method for calculating L,,„ for the most general shape of the sheet, see Fig. 4 ( c ) , will be discussed. Next, this general case will be simplified for the two special cases of the wrap-around and the no-wrap respectively.
4 Calculating Llm for the Most General Shape of the Sheet (With Gap) To numerically calculate L,m (after springback), Sw (sum of SDi and Lh, in Fig. 5(a)) is first calculated from the loaded shape of the sheet. Next, SH„ (sum of SD„ and Llm in Fig. 5(b)) is calculated from the unloaded shape of the sheet. Then, by assuming that the length of the sheet remains unchanged after the elastic unloading, that is Sflll = SHl, Lh„ is calculated by subtracting SDu from Sm:
In the following sections, the method for calculating SH, and S,,„ in Eq. (3), using the new strain hardening pressbrake bending model (Pourboghrat, 1992), will be described. 4.1 Calculating SHl. 51/,, (sum of S^andL/,, in Fig. 5 ( a ) ) , represents the distance, along the centerline fiber, between the bend line (C.L.) and the center of a hole in loaded sheet. This distance can not be calculated from the knowledge of the deformed centerline fiber alone, since location of the center of the hole along this fiber is unknown after the bending. The only known information available is L„, which is the distance from the bend line up to the center of the hole before bending, see Fig. A. 1 (a). It is also known that there exists a fiber, the unstretched fiber, that lies somewhere between the centerline fiber and the neutral axis that undergoes zero total strain. The length of the deformed sheet measured over this fiber always will be equal to L„. Deriving the exact location of the unstretched fiber, along the thickness, is not trivial and will be omitted here, but, in light of the small neutral axis shift in pressbrake bending (Pourboghrat, 1992), it is assumed that the unstretched fiber coincides with the neutral axis. S„„i, shown in Fig. 6, represents the distance, along the neutral axis, between the bend line (C.L.) and the die-sheet contact point, in the loaded state. This distance can be calculated using the curvature of the neutral axis, knl, as follows (see appendix B):
rs"»< S,M =
r"' do dS=
Jo
\
—
(4)
Jo k„i
By subtracting the value of SD„, from L„, the length of the straight section extending from the die-sheet contact point up to the center of the hole, L,lh can be found: LM = L„ — SD„i
(5)
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(a)
(b)
(c) Fig. 4 Three modes of deformation, (a) three point bending, (b) two points and one area of contact bending and (c) two points and two areas of contact bending
The length of the straight section, Lhh will be common for all fibers in the cross section since the sheet in this region is undeformed. Finally, total length of the sheet, both loaded and unloaded, along the centerline fiber, extending from the bend line up to the center of the hole, can be found by adding Lu to SDISHI — SHU — SDI +
L/,i
— La
+
(SDI
SD„I)
(6)
where the term in parentheses is the difference in length measured along the two fibers. To calculate SD: in Eq. ( 6 ) , centerline curvature of the sheet, &,, is integrated, as shown in Eq. (B2), from the bend line up to the die-sheet contact point. For the most general shape of the sheet, this calculation will involve integration over the gap, contact and free section regions. In the following section, the method for calculating SDu in Eq. (3), will be described for the most general shape of the sheet. 4.2 Calculating SDu- For the most general mode of pressbrake bending, the loaded shape of the sheet is comprised of the free ( / ) , contact (c) and gap (g) regions. The arc length of the unloaded sheet, extending from the bend line (C.L.) up to the die-sheet contact point, is comprised of one straight and three curved sections, as shown in Fig. 1(a). The unloaded curved sections (Bfu, Bcu, Bg„) are those portions that were plastically deformed in the loaded sheet, while the unloaded straight section (Lfu) is that portion which was elastically de230 / Vol. 119, MAY 1997
formed. SDu, is the sum, along the centerline fiber, of all these arc lengths. SDU = Bg„ + Bm + Bfu + Lfl, (7) After the elastic unloading, it is assumed that each section of the sheet retains its loaded length, though its centerline curvature becomes smaller. It is also assumed that after the elastic unloading, the gap (Bg„) and contact regions (Bcu) remain inside, but only part of the free section remains inside and the rest slides outside of the die cavity. That portion of the free section that remains inside is assumed to split into a curved (fi/„) and a straight section (Lfu). The amount of the free section that slides outside depends upon the geometry of the die and the shape of the unloaded sheet. Using Fig. 1(a), the following relationship between the geometry of the die and the shape of the unloaded sheet can be expressed: t
+ 2
w = Xm
t
+ 2
•sin (y„ + t//„)
•sin yu + x max , /u -cos (y„ + if/u)
)W,/«• sin (y„ + i/>„) + L/„-cos 6„
+ (rrf + -)-sin0„
(8)
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In Eq. (10), \l(rp + t/2) represents the centerline curvature of the loaded sheet in the contact region. To calculate the unloaded contact angle, \pu, it is assumed that no change will occur in the length of the sheet in the contact region after the elastic unloading:
Center of Hole C.L.
Bcu — Ba
(11)
By assuming that the shape of the sheet in the contact region is circular and its length does not change after the elastic unloading, the following expression can be written:
+
2
'*' dO 12-(1
«0
-v2)-M{kdY
(12)
E-t3
(a) r„ +
The left hand term in Eq. (12) represents the length of the sheet in the contact region in the loaded state, Bch and the right hand term represents the same length in the unloaded state, Bcu. To calculate ip,, from the above expression, the right hand side is integrated until at 6 = iff,, its magnitude becomes equal to that given on the left hand side. Similarly, to calculate the unloaded gap angle, y„, the arc length of the sheet in the gap region will be assumed to not change after the elastic unloading, that is:
Center of Hole C.L.
(13)
B„u - B„
By substituting for Bg, and Bm„ from Eqs. (B2) and (B4), we will have:
F'd6= Jo
ks i
(b) Fig. 5 Definition of various arc length used for calculating developed length in (a) loaded sheet and (6) unloaded sheet
In the above expression subscript u signifies the unloaded state (e.g. i//„ represents the unloaded contact angle). XmaXi8„ represents the projection of the arc length Bs„ onto the X-axis and it is calculated by integrating Eq. ( B l ) over the unloaded gap angle, that is 0 < 8 < yu, as follows:
•f
Xm
Jo
cos 8 • d6
[*tf -
v2)-M{k„)^
12-(1 -
de hi
-
v2)-M(ksl)
12-(1 -
(14)
E-t3
Again, the right hand side is integrated until at 6 = y„, its magnitude becomes equal to that given by the left hand side expression integrated over 0 < 8 < yt. Once, i//„ and y„ are calculated from Eqs. (12) and ( 1 4 ) , the unloaded free section angle, „, is calculated by subtracting their sum from 6„, the unloaded flank angle. The unloaded free section angle, cf>u, is then used to calculate the arc length of the
Center of Hole.
(9)
E-t3
p" Jo
CL
Similarly, xmaXi/„ and ymm,fu represent projections of Bfu onto the x- and v-axes respectively and are calculated as shown in Eq. (9), except limits of integration are taken over the unloaded free section angle, that is 0 < 6 < „, and cosine is replaced with the sine function when calculating vmax,/„. In Eq. (8), 1/ (rp + f/2)|„ represents the centerline curvature of the contact region after the elastic unloading and is calculated, using Eq. (B4), as follows: 1
1
-v2)-M(kc))
12-(1 3
/"
+
2
rP
+
-2
E-t
Journal of Manufacturing Science and Engineering
(10)
Centerline Fiber Fig. 6 Shape of the loaded sheet showing both neutral fiber and centerline fiber used for calculating developed length
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curved section of the unloaded free section, Bfll, see Fig. 7 ( a ) , as follows: Centerline Fiber.
de 12-(1 - v2)-M(kf„, could reach 30. deg when typical pressbrake bending setups are used, i.e. rn/t = 1.0, rdlt = 1.0 and wit = 5.0. The difference between predicted and measured data, shown in Figs. 10 through 14, correspond to only half of the total error (since half of the sheet was considered) in predicted final location of a part feature. If an Journal of Manufacturing Science and Engineering
the error in predicted final location of the center of the hole, as measured from the left hand edge, will be 6 times the error obtained from Fig. 11, i.e. 6*(0.4775 mm) = 2.865 mm (0.1125"). Now, if a thicker aluminum sheet [t = 3.11 mm (0.1225")] is used instead, but other bending characteristics remained the same, this error using Sachs' Eq. (1) will increase to 6 *( 0.9347 mm) = 5.608 mm (0.22").
6
Conclusions
A more exact method for calculating the developed length of the sheet in pressbrake bending was described. This method assumes the most general deformed shape for the bent sheet which includes the free section, contact and gap regions. Developed length is calculated as the sum of all curved (bent) and straight sections comprising the shape of the sheet after unloading (springback). The length of the curved (bent) sections are calculated by integration of centerline curvatures over the arc length of loaded and unloaded sheets. Centerline curvatures are obtained from a new pressbrake bending model which can predict deformation of the sheet as a function of tooling geometry, friction and material properties. An experimental method for measuring the location of a hole (part feature) punched into the flat sheet after bending was described in appendix A. A measurable and calculable parameter, Lhu, was introduced, representing the straight section extending from the die-sheet contact point up to the center of the hole, to verify different models' prediction of the developed length. Variations of Llmlt with respect to variations of rplt, rjt, wit and 90 were measured and compared against those predicted by Sachs' Eq. (1), Eq. (22) with thinning included and the method described in this paper, based on the new pressbrake bending model (Pourboghrat, 1992). In regards to bend allowance calculation, it was shown that unlike Sachs' claim, the magnitude of the neutral axis shift in the bent section is small and can be neglected for calculation of the developed length in pressbrake bending. The large neutral axis shift suggested by Sachs is unrealistic and that k in Eq. (1) is simply a compensation factor to mask the errors caused by neglecting strain hardening and rotation of the free section. It was shown that although including thinning improves bend allowance and developed length calculations, it does not completely remedy shortcomings of the Sachs' Eq. (1). Accurate prediction of the shape of the sheet is of great importance when calculating developed length. When the free section rotation is under estimated or neglected altogether as in the case of Sachs' model, the actual punch stroke will be over estimated and L,m under estimated. This is the case for Sachs' model as shown in Figs. 10 through 14. Finally, it was shown that the difference between predicted and measured data, shown in Figs. 10 through 14, represent only half of the expected total error in predicted final location of a part feature for a part with single bend. For parts with multiple bends, this error increases correspondingly. MAY 1997, Vol. 1 1 9 / 2 3 5
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Acknowledgment This research was funded in part by a Presidential Young Investigation Award from the National Science Foundation.
Bend Line
Punched Hole
C.L.
*nr
/
References Sachs, G., Principles and Methods of Sheet-Metal Fabrication, Reinhold, 1976, pp. 95-115. Hill, R„ The Mathematical Theory of Plasticity, Clarendon, Oxford, 1950, pp. 287-294. Pourboghrat, F., Stelson, K. A., "Bend Allowance Calculation in a Flexibly Automated Sheet Metal Fabrication System," Proceedings of the U.S.A.-JAPAN Symposium on Flexible Automation, July 18-20, 1988a, Minneapolis, Minnesota. West, J. S., "Adaptive Stroke Reversal Control in Brakeforming Process," S. M. Thesis, MIT, Cambridge, MA, July 1980. Stelson, K. A., and D. C. Gossard, "An Adaptive Pressbrake Control Using an Elastic-Plastic Material Model," ASME Journal of Engineering for Industry, Vol. 104, November 1982, pp. 389-393. Gossard, D. C , and Allison, B. T., "Adaptive Brakeforming," Eighth North American Manufacturing Research Conference Proceedings, May 1980, pp. 252256. Stelson, K. A., "An Adaptive Pressbrake Control for Strain-Hardening Materials," ASME Journal of Engineering for Industry, Vol. 108, May 1986, pp. 127132. Pourboghrat, F., and Stelson, K. A., "Pressbrake Bending in the Punch-Sheet Contact Region, Part 1: Modeling Nonuniformities," ASME Journal of Engineering for Industry, Vol. 110, May 1988b, pp. 124-130. Pourboghrat, F., and Stelson, K. A., "Pressbrake Bending in the Punch-Sheet Contact Region, Part 2: Gap Formation and the Direction of Sheet Motion," ASME Journal of Engineering for Industry, Vol. 110, May 1988c, pp. 131-136. Oh, S. I., and Kobayashi, S., "Finite Element Analysis of Plane-Strain Sheet Bending," Int. J. Mech. Sci„ Vol. 22, 1980, pp. 583-594. Makinouchi, A., "Elastic-Plastic Stress Analysis of U-Bend Process of Sheet Metal," Advanced Technology of Plasticity, 1984, Vol. 1, pp. 672-677. Pourboghrat, F., "A Model of Pressbrake Bending for Gap Prediction and Bend Allowance Calculation," Ph.D. Thesis, University of Minnesota, 1992. Prasad, Y. K. D. V., and Somasundaram, S., "A Mathematical Model for BendAllowance Calculation in Automated Sheet-Metal Bending," J. Materials Processing Technology, Vol. 39, 1993, pp. 337-356. Dadras, P., and Majlessi, S. A., "Plastic Bending of Work Hardening Materials," ASME Journal of Engineering for Industry, Vol. 104, August 1982, pp. 224-230. ABAQUS, User's Manual, Theory Manual, version 4.7, Hibbit, Karlsson, and Sorenson, Providence, Rhode Island, 1988. Story, J. M., and Engelmann, D. M., "Surface Topography Effects on Friction in the Forming of Aluminum Sheet," Proceedings of IDDRG'94-Recent Developments in Sheet Metal Forming Technology, M. J. M. Barata Marques, ed., Lisbon, Portugal, May 16-17, 1994, pp. 165-176.
(b) Fig. A.1 A schematic of devices used to measure Lhu (a) bent sheet placed into the vice to measure L,imL (b) various arc length defined for an unloaded sheet
APPENDIX B Calculating Loaded and Unloaded Length of the Sheet In Cartesian coordinate system, curvature k is expressed as a function of the variation of the slope, 6, over the arc length, S, or its components X or Y, as follows:
APPENDIX A Experimental Method for Calculating Parameter Lhu A circular hole was chosen to be a part feature punched into the flat sheet, a distance L„ from the bend line (C.L.), before bending. Parameter L,m was chosen to represent the length of the straight section between the center of the hole and the diesheet contact point. This definition allowed for a simple and accurate method to measure Lh„ since it did not require measurements over curved surfaces. In this method, first the length of the straight line (OC in Fig. AA(b)) enveloping the sheet, Lmvl, was measured by placing the bent sheet into a V-shaped vice, as shown in Fig. A.l (a). Then, the length of the straight line, Lhu, extending from the die-sheet contact point up to the center of the hole (BC in Fig. A,1(b)), was calculated by subtracting Ldk from Lenvi. (w - o-sin 8„)
(Bl)
The total length of a loaded sheet, S,, is expressed as the sum of the lengths of the straight, Lh and curved (bent), Bh sections, fi, is calculated for the loaded sheet by integrating its centerline curvature, k,, using Eq. ( B l ) , over the curved sections as follows: =
dS = Jo
Jo
k,
(B2)
Similarly, the length of an unloaded sheet, S„, is expressed as the sum of the lengths of the straight, L„, and curved (bent), Bu, sections, and Bu is calculated by integrating the unloaded centerline curvature ku, using Eq. (B1), over the curved sections after springback as follows:
(Al) Bu=
This experimentally measured value of Lhl, was then compared with the numerically calculated values of Lhu, using Eqs. (1) and (22) and the strain hardening pressbrake bending model. 236 / Vol. 119, MAY 1997
d6 . sin 0 dY
, dO dO k =— = cos dS dX
\'KdS= Jo
f"^ Jo k„
(B3)
It is assumed that after the elastic unloading (springback), loaded and unloaded centerline curvatures of the sheet, k, and k„, are related by this expression: Transactions of the ASME
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k„ = k, -
M{k,) 8M dk
(B4)
In Eq. (B4), M(kt) is the bending moment about the centerline of the sheet per unit width, and is a function of the centerline curvature, k,, and tension in the sheet. The term in the denominator is the slope of the moment-curvature curve in the elastic range and is equal to Ef 3 /[12(1 - v2)] for a sheet with rectan-
Journal of Manufacturing Science and Engineering
gular cross section. E and t are the modulus of elasticity and the thickness of the sheet respectively. By substituting for k„ from Eq. (B4) into (B3), an expression for B„ as a function of the centerline curvature, k,, can be obtained.
Jo
dd 12-(1 v2)-M{k,) 3 E-t
(B5)
MAY 1997, Vol. 1 1 9 / 2 3 7
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