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Structural analysis and optimisation of press brakes
International Journal of Machine Tools & Manufacture 45 (2005) 1451–1460 www.elsevier.com/locate/ijmactool
Structural analysis and optimisation of press brakes Pedro G. Coelhoa,*, Luı´s O. Fariab, Joa˜o B. Cardosoa a
Department of Mechanical and Industrial Engineering, New University of Lisbon, 2829-516 Caparica, Portugal b IDMEC, Instituto Superior Te´cnico, 1049-001 Lisboa, Portugal Received 10 November 2004; accepted 20 January 2005 Available online 23 March 2005
Abstract A model of the bending process in Press Brakes is established using Timoshenko beam theory. Expressions for the workpiece bending error are derived that explicitly consider the influence of shape, dimensions and initial deformation of the machine structural components on its bending accuracy. The minimization of the bending error is formulated in terms of optimisation problems that are solved numerically using a genetic algorithm. The methodology presented in this paper permits the analysis of existing Press Brake design solutions, the optimisation of their performance and the development of new solutions. q 2005 Elsevier Ltd. All rights reserved. Keywords: Bending process; Press Brake; Design optimisation; Timoshenko beam
1. Introduction Flat metal plates are bent along a straight line to an angle in Press Brakes. A typical Press Brake is a C-frame design with a moving ram, which holds a punch, and a die located on a bed frame. Upon inserting the workpiece between bed and ram, a pair of hydraulic actuators forces the punch inside the die, bending the flat plate to the desired angle. The bending angle is very sensitive to the penetration, i.e. the relative displacement of punch and die. For example, a variation of 0.05 mm in the penetration will cause a variation of 18 in the bending angle for a 1 mm thick plate bent in a 10 mm die. The angular precision of the workpiece depends on the uniformity of the bending angle along the bending line. This uniformity is achieved with constant penetration of punch and die obtained through parallel deflections of ram and bed. The ram and the bed are long, narrow beams but their finite stiffness causes non-constant penetration and
non-uniform bending angle along the bending line—the ‘boat belly’ effect (Fig. 1a). The desirable parallel deflection of both beams is shown in Fig. 1b. The objective of this paper is to understand the source of deflection parallelism errors and minimize them through a structural optimisation methodology. Different Press Brake structural solutions are analysed and their performance and limitations explained. Recent work on this subject has been presented essentially in machine manufacturers’ magazines and patents [1–10] and in the research papers [11–13]. The paper is organized as follows: in Section 2 the analytical model for bending of a workpiece in a Press Brake is described; in the following Sections three structural optimisation problems are formulated: shape optimisation (Section 3), dimensional optimisation (Section 4) and initial deflection optimisation (Section 5). Section 6 comments on the results obtained in this work.
2. Bending model 2.1. Beam model
* Corresponding author. Tel.: C351 212948567; fax: C351 212948531. E-mail address: [email protected] (P.G. Coelho).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.01.030
The bending model is shown in Fig. 2a and is assumed to be symmetric. The ram and bed are modelled as simply supported beams and will be denoted, respectively, by Upper and Lower beam.
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Fig. 1. Influence of ram and bed deflection on the angular precision of the workpiece. (a) The ‘boat belly’ effect. (b) Uniform bending angle.
The bending length is defined by variable a and the maximum bending length is assumed to be the distance between machine columns and is denoted by L. In the last stages of the bending process the workpiece material plastifies completely along the bending line. Assuming small hardening, the reaction of the workpiece is almost independent of the deformation and may be modelled as a uniform load q. Variable t defines the location of a possible cross-section discontinuity in the Upper beam, used for mounting the hydraulic actuators. Variable d measures half the distance between the Lower beam supports. The case dZL/2 represents the conventional Press Brake with the bed and ram supports located in the machine columns. The case 0%d!L/2 models a design solution known as sandwich, in which the Lower beam is supported by two locking rods on two side plates fixed to the machine columns (see Fig. 2b). When dZ0 the rods are superposed. 2.2. Timoshenko theory of beams The Upper and Lower beams in a Press Brake have a length to height ratio lower than four. For such beams it is necessary to include the effect of shear deformations in the technical theory and the result is known as the Timoshenko theory of beams [14–16]. In this formulation the vertical displacement w of the beam is determined by the fourth-order equation: EI
d4 w EI d2 q Z Kq C 4 kGA dx2 dx
(1)
In Eq. (1), E is the Young’s modulus, G the Shear modulus, A and I the area of the cross-section and its moment of inertia. The dimensionless factor k is introduced to account for the non-uniform shear distribution in the crosssection while retaining the one-dimensional beam approach.
Fig. 2. Beam model. (a) Upper and Lower beam model. (b) Composite lower table.
In this paper the definition of k given by Cowper [16] is chosen, giving kZ0.85 for a rectangular cross-section and kZ0.33 for a ‘T’ shaped one, with Poisson’s ratio nZ0.3. The curvature and slope equations for Timoshenko’s theory are, respectively: d2 w M q Z C EI kGA dx2 dw Z b C j; dx
where
(2)
b ZK
V kAG
(3)
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In Eq. (2) M is the bending moment and q the applied load and in Eq. (3) j is the rotation of the cross-section due to bending, b the shear deformation and V the shear force. Although Eq. (1) reduces to the technical theory equation for uniform load q, the boundary and interface conditions are different. Relevant to this study are the following two interface conditions: C bjK xZL=2Kd K bjxZL=2Kd Z
C bjK xZt K bjxZt Z
VjxZt ku G
qðL K 2aÞ 2kl Al G 1 1 K Au2 Au1
(4)
(5)
The indexes l and u refer to Lower and Upper beam, respectively. In Eq. (5) A takes the constant values Au1 for 0!x!t and Au2 for t!x!L/2. Eqs. (4) and (5) express the shear strain discontinuity related to the discontinuity of V at the supports of the Lower beam and to the variation of the Upper beam cross-sectional area. According to Eq. (3) these discontinuities in the shear strain will originate slope discontinuities. Since the Upper and Lower beams are statically determinate their deflections may be determined by integrating twice the curvature (Eq. (2)) with the adequate boundary and interface conditions: ðð M q wZ C dxdx C C1x C C2 (6) EI kGA
3. Shape optimisation In order to obtain parallelism between the deflected Upper and Lower beams their curvature must be equal. For each bending length this condition may be expressed as,
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using Eq. (2): Mu Kq M q C Z lC EIu ku GAu EIl kl GAl
(7)
Solutions for this equation are possible only for Press Brakes with sandwich Lower beams, 0%d!L/2, because when dZL/2, MuZKMl. For an illustrative example a rectangular cross-section with the same width for both beams is assumed with kuZ klZ0.85, a constant cross-section for the Upper beam with height of 1400 mm and bending length of 3200 mm. The Lower beam has variable height h(x). The shape optimisation problem of Eq. (7) consists in finding the Lower beam shape h(x) that makes the curvatures of the Upper and Lower beams match for this single bending length. Substituting the above values in Eq. (7) yields an implicit function of h in variable x. This function is shown in Fig. 3 for dZ0 and dZ400 mm. Although the curvatures of the Upper and Lower beams are the same in the above solutions, parallelism will only occur for the case in which the effect of shear deformations is disregarded. As expressed in Eq. (4), the shear force discontinuities that occur at the supports will generate slope discontinuities in the Lower beam deflection. In order to guarantee slope continuity the Lower beam may be supported by a uniformly distributed reaction, like the one shown in Fig. 4a. In this case shear force and shear angle are continuous and q2 is determined to satisfy static equilibrium. As shown in Fig. 4b the load discontinuity applied to the Lower beam implies, from Eq. (7), an abrupt variation of h for curvature continuity, but the deflections of both beams become parallel. What kind of support for the Lower beam will produce a distributed reaction? A technical solution existent in the market and patented [3] (see Fig. 5) resembles the computed
Fig. 3. Variation of height and lower beam shape (units: mm).
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4. Dimensional optimisation 4.1. Penetration and bending error The bending error of the workpiece is defined as the amplitude of its angle variation along the bending length. It is proportional [17] to the amplitude of the penetration p(x) of the punch into the die, or oscillation u(p) defined by: L uðpÞ Z max pðxÞ K min pðxÞ; D0 Z x 2R = a% x% x2D0 x2D0 2 (8) The penetration p(x) is defined as the difference between the vertical displacements of the Upper and Lower beams, plus a constant translation d to ensure that their deflection is the same at a point, here taken as point a: p ðxÞ Z wu ðxÞ K wl ðxÞ K d;
d Z wu jxZa K wl jxZa
(9)
Analytical expressions for the penetration were established using Timoshenko’s beam theory. The expressions for the particular case tZdZ0 and constant cross-section dimensions are the following: sffiffiffiffiffiffi Iu Au ku a x Il rI Z ; rA Z ; rk Z ; j Z ; z Z ; rl Z L L Il Al kl A1 (10)
Fig. 4. Optimal Shape for Lower beam (units: mm). (a) Beam model. (b) Height variation and shape.
pb Z optimal shape for the Lower beam. This solution is used for low force requirements. Another solution of this type but applied to the Upper beam can be seen in patent [7]. The formulation in this Section achieves a null bending error for one bending length and is very sensitive to its variation. To take into account all bending lengths an optimisation problem is formulated in the next Section that keeps the shape of the beams fixed but not its dimensions.
qL4 f r ; z; j 24EIl b I
fb rI ; z; j Z
ps Z
(11)
1 2 C 1 ðz K jÞ4 C ð1 K 2jÞðj3 K z3 Þ rI rI 3
1 3 1 ð1 K 2jÞ K 4ð1 C rI Þ K j C ðz K jÞ rI 2 2 (12)
qL2 f ðr ; r ; z; jÞ 24Gkl Al s k A
1 1 fs ðrk ;rA ;z;jÞZK Crk ðzKjÞ2 C ð1K2jÞðzKjÞ rA rA
(13)
(14)
p Z pb C ps
24ð1 C nÞ rl 2 qL4 Z f r ; z; j C f ðr ; r ; z; jÞ ku 24EIl b I L s k A (15)
Fig. 5. Patented solution for Lower beam.
The Eq. (15) for the penetration is given by the product of a factor proportional to the Lower beam maximum displacement and a non-dimensional factor. This includes the bending contribution fb and the shear contribution, proportional to fs. The influence of the shear term grows proportionally to (rl/L)2, as it is well known.
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The cross-sectional shape of the beams and its influence in shear stiffness is present through rk and ku. 4.2. Formulation of the optimisation problem The dependence of the penetration on selected design variables is explicitly considered by writing pZp(s, a, x). The components of sZ(s1, s2, ., sn) are the n variables related to cross-section dimensions of the beams and the distance between locking rods, which have lower and upper bounds, sK and sC i i , iZ1,., n, respectively. Variable a defines a bending length, x is the position variable and both retain their previous meaning. The formulation of the dimensional optimisation problem is the following: for a given bending force q and maximum bending length L, find the design s0 of a Press Brake for which the maximum oscillation of the penetration function is minimum. This can be expressed as: find the (optimal) design s02D1, the bending length a02D2 and the penetration pso ;ao such that,
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was used instead of Timoshenko’s theory, ‘shear only’ as if the bending stiffness of the beams was very high and all the contribution to the bending error came from shear and a third one with both contributions taken into account. A rectangular cross-section for the Upper beam was considered and both a rectangular and ‘T’ shape cross-sections for the Lower beam. The last column gives uðps0 ;a0 Þ for Upper beam dimensions of 3200!90!1600 mm3 and bending force per unit length q of 180 N/mm. Fig. 6 shows the penetration curves for each case in Table 1.
uðps0 ;a0 Þ Z min max uðps;a Þ; s2D1 a2D2
(16)
uðps;a Þ Z max pðs; a; xÞ K min pðs; a; xÞ x2D0
x2D0
C D1 Z s 2Rn =sK i % si % si ; i Z 1; .; n ; L D2 Z a 2R = 0% a% 2
(17)
From the different expressions for the penetration (see, for example, Eq. (15)) it can be concluded that the optimal design s0 is not dependent on q and that the bending error is proportional to it. To evaluate the solutions of the minimax in Eq. (16) a genetic algorithm [12,13,18] is used since it does not depend on the analytical properties of the objective function. 4.3. Unconstrained dimensional optimisation In this Section the optimisation in Eq. (16) is solved with three design variables: d, and ratios rI and rA, defined in Eq. (10). The upper bounds on these ratios are high enough to reach an unconstrained optimal solution. Table 1 presents the optimum values for three types of analysis: ‘bending only’, as if the technical theory of beams
Table 1 Optimal values for bending and shear contributions separately and together d (mm)
Bending only Shear only BendingCshear
0 471 163
rI
2 – 2.87
rA klZ0.33
klZ0.85
– 0.74 0.14
– 1.88 0.40
uðpso ;ao Þ (mm) 0.0080 0.0085 0.0129
Fig. 6. Penetration curves for each case presented in table 1 (units: mm). (a) Bending only. (b) Shear only. (c) BendingCShear.
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the supports, independently of their location and the penetration curve has the form shown in Fig. 6b. Therefore, the optimal solution for ‘shear only’ selects locking rods far apart, with dzL/3.5, taking into account all possible bending lengths. Changes in the beams’ cross-sectional shapes are compensated by changes in its dimensions to maintain the support optimum location. – When both bending and shear effects are present and since the penetration is negative for bending and essentially positive for shear, the optimum solution increases each absolute value in order to minimise their sum (see Fig. 6c). 4.4. Examples of dimensional optimisation Dimensions currently used in industry for a Press Brake with 175 tons of bending force and LZ3200 mm are now considered and two possible geometries for each Upper and Lower beam, as shown in Fig. 7. UB1 and UB2 are designs for Upper beams, without and with inertia variation to allow the mounting of the actuators. LB1 and LB2 are designs for Lower beams, with and without ditch, considering a height above ground of 834 mm. The optimisation problem in Eq. (16) was solved for all four combinations of Upper and Lower beam geometries. The design variables si, iZ1,.,7 and their bounds are identified in Fig. 7 and s8 is the distance d as defined in Fig. 2a. The optimum values are presented in Table 2, in bold when the upper bounds of the design variables were reached. The penetration curves at the optimum for two cases are shown in Fig. 8 for qZ180 N/mm. The UB1/LB1 combination is the solution that minimizes the bending error. The Upper beam reaches both specified upper bounds on cross-section dimensions and the Lower beam attains the maximum specified width of 70 mm. This relatively low value for the upper bound on the width is a design requirement for the sandwich construction, since the Lower beam has to leave room for two side plates as shown in Fig. 2b. For the same Upper beam, the geometry of the Lower beam at the unconstrained optimum in the previous Section required a cross-section geometry of 600!600 mm2— clearly an infeasible design.
Fig. 7. Upper (UB) and Lower (LB) geometries and design variables with bounds (units: mm). (a) UB1. (b) UB2. (c) LB1. (d) LB2.
The following comments on the above results are in order: – The influence of bending and shear on bending error are of the same order. Any model of the bending process needs to include both effects. – The optimal solution for ‘bending only’ selects vertically aligned locking rods and halves the inertia of the Lower beam to compensate for the stiffness derived from the midspan location of its support. The penetration curve ps0 ;a0 , shown in Fig. 6a, is negative indicating that the Lower beam deforms more than the Upper beam. – For ‘shear only’ the central support of the Lower beam does not help since the curvature will always have the sign of the distributed load q (see Eq. (2)). Shear deformation will cause Upper and Lower beams to have always opposite curvatures and slope discontinuities at Table 2 Optimal solutions for each combination of geometries (units: mm) Geometries combination
s1
s2
s3
s4
s5
s6
s7
s8
uðpso ;ao Þ
UB1/LB1
1600 1600 1600 1379 – –
90 90 90 90 – –
1375 1228 1560 – 950 –
70 70 70 – 70 –
– – – 70 – 47
– – – 130 – 130
– – – 600 – 600
290 400 0 436 566 576
0.0139 0.0168 0.0183 0.0256 0.0282 0.0308
UB1/LB2 UB2/LB1 UB2/LB2
P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (2005) 1451–1460
Fig. 8. Penetration curves with Shear and Bending contributions (units: mm). (a) UB1/LB1. (b) UB1/LB2.
The bending error increases 8% for this constrained solution in relation to the error for the unconstrained optimum. A stress analysis reveals that the distance dZs8 needs to be increased to avoid too high local contact stresses between the locking rods and the Lower beam. Considering an allowable stress of 320 MPa for the beam material and a maximum bending force of 175 ton, a Finite Element analysis determined a lower bound of 400 mm for the design variable s8. The bending error with this constraint (see Table 2, second line) is 30% larger than the unconstrained optimum. The solution with superposed rods (dZ0) is also feasible but the error increases (see Table 2, third line). The UB1/LB2 is the best solution without ditch but the bending error almost doubles in comparison with the previous case. The Lower beam stiffness has to be supplied by a ‘T’ shape cross-section and in this case the unconstrained solution requires a flange width of 4300 mm, instead of the specified 600 mm upper bound. At the optimum, all Lower beam cross-section dimensions reach their upper bounds while the Upper beam does not attain the allowable height. As seen in Fig. 8b most of the bending error comes from shear and since the ‘T’ crosssection is weak in shear (low k) it uses all allowable area while the larger k of the Upper beam is compensated by a smaller cross-sectional area. The influence of a design constraint for the Upper beam is assessed in the optimization for the UB2/LB1 and UB2/LB2
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combinations. Both solutions generate larger errors but the increase is moderate (13 and 23%, respectively) when compared with the UB2/unconstrained Lower beam design, for which a uðpso ;a0 ÞZ0.025 mm was calculated. The unconstrained dimensional optimisation solutions presented in the previous Section provide the lowest bending errors, but correspond to cross-section dimensions that violate the limits accepted in industry. The above examples show that the currently used dimensional constraints do not penalise significantly the bending error. The sandwich design solution is also seen to be preferable to the conventional one (dZL/2). In Section 5 it is shown that the error for the latter is an order of magnitude larger than for the sandwich design obtained in every constrained or unconstrained optimal solution in this Section. The penetration curves in Fig. 8 for two constrained optimum designs reveal that the main source of bending error in sandwich Press Brakes is shear deformation. This unexpected result explains why locking rods horizontally aligned are preferred over the vertically aligned optimal solutions of the technical theory of beams: shear deformations cause opposite curvatures for the Upper and Lower beams instead of the desirable matching curvatures generated from bending. 4.5. Comparison with numerical solutions from the theory of elasticity The UB2/LB2 combination was used to compare analytical results from Eq. (6) with numerical results from a Finite Element analysis. The Upper and Lower beams were modelled with PLANE 82 2-D 8-Node Structural Solid ANSYS Finite Elements and elements CONTACT 52 3-D Point-to-Point were used to model contact between locking rods and Lower beam. The deflections for each beam for maximum bending length are presented in Fig. 9. For all bending lengths the differences do not exceed 4% for the Upper beam and 10% for the Lower beam. The main difference in the results occurs at the slope discontinuities, which appear as rapid variations in the Elasticity results. It can also be observed in Fig. 9 that the curvatures of the Upper and Lower beams are in opposition due to the predominance of shear over bending.
5. Optimisation of the initial deformation In this Section the previous results are improved by adding an initial deformation to one of the beams to allow for a better parallelism between their deflections. Such a function may be a Spline and is introduced as a design variable in the formulation of the optimisation design problem. An initial deformation may be introduced in the Upper beam by shimming as shown in Fig. 10.
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The optimisation problem is reformulated into a sequential optimisation problem. Problem (16) is solved first giving an optimal design s0. Then the following problem is solved for a particular load value q: find the initial deformation v02D3, the bending length ao2D2 and the penetration pv0 ;s0 ;a0 such that, uðpv0 ;s0 ;a0 Þ Z min max uðpv;s0 ;a Þ v2D3 a2D2
Fig. 9. Comparison between analytical (Timoshenko theory) and Finite Element numerical results for UB2/LB2 using maximum bending length. (a) Upper beam. (b) Lower beam.
Let S(v,x) be a cubic spline [19] with nodes (x1, xi,., xm), m being the number of equally spaced points along half of the Upper beam. The components of the vector vZ (v1,vi,.,vm) are the initial deformation values at the points xi and their bounds are given by: C D3 Z v 2Rn =vK (18) i % vi % vi ; i Z 1; .; m S(v,x) is added to penetration given by Eq. (9) in the following way p ðxÞ Z ½wu ðxÞ K wl ðxÞ K d C Sðv; xÞ K SjxZa
(19)
(20)
The previous geometry combinations with optimal dimensions UB1/LB1 and UB2/LB2 were selected to illustrate this sequential optimisation. Assuming mZ9 and C KvK i Zvi Z0.4 mm in Eq. (18) one obtains, respectively, 0.0129 and 0.0253 mm for uðpv0 ;s0 ;a0 Þ—a gain in bending accuracy of 7 and 18% (see Table 2). Problem (16) was also solved for UB1/LB1 and UB2/LB2 but setting s8ZdZL/2 and augmenting the bounds on width for the Lower beam because there are no side plates to accommodate. At the optimum (see Table 3) all the bounds on dimensions were attained and the uðps0 ;a0 Þ values calculated are an order of magnitude higher for this conventional solution than for the corresponding sandwich designs. The introduction of an optimal initial deformation for this case achieves a significant decrease in the bending error, as shown in Table 4. In the case UB1/LB1 the bending error decreases from 0.1092 to 0.0025 mm and from 0.2658 to 0.0064 mm in the case UB2/LB2. The improvement in bending accuracy is radical for these conventional designs because both beams deform with the same shape for all bending lengths. In the case of the sandwich solution for the Lower beam, different bending lengths originate very different deformations and there is no unique initial deformation that reduces appreciably the error for all lengths. The correction introduced by an initial deformation is associated with a load value q. If the load changes and the initial deformation remains constant, the error will not change proportionally to its value at q. In order to introduce the optimal initial deflection for the actual load q, it seems advantageous to use an automatic
Fig. 10. Modelling shimming with a Spline function.
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Table 3 Optimal dimensions for UB1/LB1 and UB2/LB2 with conventional location of the supports (units: mm) Case
s1
s2
s3
s4
s5
s6
s7
s8
uðpso ;ao Þ
UB1/LB1 UB2/LB2
1600 –
90 –
1600 –
130 –
– 130
– 130
– 600
1600 1600
0.1092 0.2658
Table 4 Optimal initial deformation for UB1/LB1 and UB2/LB2 with conventional location of the supports (units: mm) Case
v1
v2
v3
v4
v5
v6
v7
v8
v9
uðpvo ;so ;ao Þ
UB1/LB1 UB2/LB2
0 0
K0.023 K0.065
K0.045 K0.125
K0.064 K0.168
K0.079 K0.202
K0.092 K0.228
K0.100 K0.244
K0.105 K0.256
K0.107 K0.259
0.0025 0.0064
system like the crowning devices that some manufactures have been proposing [1–10], instead of the time-consuming trial–and–error shimming process.
6. Conclusions Based on the model of the bending process in Press Brakes defined in Section 2 it has been found that it is not possible to design a machine that achieves uniform bending angles for every bending length. This happens because there are no optimal shapes or dimensions for the bed and ram that lead to parallel deflections for all bending lengths. Shape optimisation makes possible parallelism, but only for one bending length and is very sensitive to its variations; furthermore the optimal shape is not simple to manufacture. Dimensional optimisation leads to a composite Lower beam supported at the middle, known as sandwich design. Some manufacturers have been praising this solution without noticing the unexpected and negative influence of shear deformations. These deformations cause opposite curvatures of bed and ram, independently of the location of their supports. However, the errors due to bending and shear have opposite signs and may be made to almost cancel at the optimum, making this solution an attractive compromise between bending precision and design simplicity. The introduction of an optimised initial deflection for each bending length and load value is essential in a conventional Press Brake, where the bed and ram supports are located in the machine columns. This is an interesting solution if it can be computed and introduced in an automatic way each time the bending conditions are changed. The methodology presented in this paper proved well suited to analyse the structural behaviour and bending precision of existent Press Brakes and should be useful to optimise their performance and assist in the design of new solutions.
Acknowledgements The support of Adira—A. Dias Ramos Company in Porto, Portugal and the useful discussions with Eng. Jose´ Bessa Pacheco and Eng. Miguel Costa are gratefully acknowledged. The support of FCT through Project POCTI/36055/ECM/99 is gratefully acknowledged.
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[16] G.R. Cowper, The shear coefficient in Timoshenko’s beam theory, J. Appl. Mech. 33 (2) (1966) 335–340. [17] J.B. Pacheco, Utilizac¸a˜o de Quinadoras e Guilhotinas, APTCP, Porto, 1992. [18] D.E. Goldberg, Genetic Algorithms in Search, Optimisation, and Machine Learning, Addison-Wesley, Reading, MA, 1989. [19] H. Pina, Me´todos Nume´ricos, McGraw-Hill, Portugal, 1995 (pp. 77–88).
Structural analysis and optimisation of press brakes Pedro G. Coelhoa,*, Luı´s O. Fariab, Joa˜o B. Cardosoa a
Department of Mechanical and Industrial Engineering, New University of Lisbon, 2829-516 Caparica, Portugal b IDMEC, Instituto Superior Te´cnico, 1049-001 Lisboa, Portugal Received 10 November 2004; accepted 20 January 2005 Available online 23 March 2005
Abstract A model of the bending process in Press Brakes is established using Timoshenko beam theory. Expressions for the workpiece bending error are derived that explicitly consider the influence of shape, dimensions and initial deformation of the machine structural components on its bending accuracy. The minimization of the bending error is formulated in terms of optimisation problems that are solved numerically using a genetic algorithm. The methodology presented in this paper permits the analysis of existing Press Brake design solutions, the optimisation of their performance and the development of new solutions. q 2005 Elsevier Ltd. All rights reserved. Keywords: Bending process; Press Brake; Design optimisation; Timoshenko beam
1. Introduction Flat metal plates are bent along a straight line to an angle in Press Brakes. A typical Press Brake is a C-frame design with a moving ram, which holds a punch, and a die located on a bed frame. Upon inserting the workpiece between bed and ram, a pair of hydraulic actuators forces the punch inside the die, bending the flat plate to the desired angle. The bending angle is very sensitive to the penetration, i.e. the relative displacement of punch and die. For example, a variation of 0.05 mm in the penetration will cause a variation of 18 in the bending angle for a 1 mm thick plate bent in a 10 mm die. The angular precision of the workpiece depends on the uniformity of the bending angle along the bending line. This uniformity is achieved with constant penetration of punch and die obtained through parallel deflections of ram and bed. The ram and the bed are long, narrow beams but their finite stiffness causes non-constant penetration and
non-uniform bending angle along the bending line—the ‘boat belly’ effect (Fig. 1a). The desirable parallel deflection of both beams is shown in Fig. 1b. The objective of this paper is to understand the source of deflection parallelism errors and minimize them through a structural optimisation methodology. Different Press Brake structural solutions are analysed and their performance and limitations explained. Recent work on this subject has been presented essentially in machine manufacturers’ magazines and patents [1–10] and in the research papers [11–13]. The paper is organized as follows: in Section 2 the analytical model for bending of a workpiece in a Press Brake is described; in the following Sections three structural optimisation problems are formulated: shape optimisation (Section 3), dimensional optimisation (Section 4) and initial deflection optimisation (Section 5). Section 6 comments on the results obtained in this work.
2. Bending model 2.1. Beam model
* Corresponding author. Tel.: C351 212948567; fax: C351 212948531. E-mail address: [email protected] (P.G. Coelho).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.01.030
The bending model is shown in Fig. 2a and is assumed to be symmetric. The ram and bed are modelled as simply supported beams and will be denoted, respectively, by Upper and Lower beam.
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Fig. 1. Influence of ram and bed deflection on the angular precision of the workpiece. (a) The ‘boat belly’ effect. (b) Uniform bending angle.
The bending length is defined by variable a and the maximum bending length is assumed to be the distance between machine columns and is denoted by L. In the last stages of the bending process the workpiece material plastifies completely along the bending line. Assuming small hardening, the reaction of the workpiece is almost independent of the deformation and may be modelled as a uniform load q. Variable t defines the location of a possible cross-section discontinuity in the Upper beam, used for mounting the hydraulic actuators. Variable d measures half the distance between the Lower beam supports. The case dZL/2 represents the conventional Press Brake with the bed and ram supports located in the machine columns. The case 0%d!L/2 models a design solution known as sandwich, in which the Lower beam is supported by two locking rods on two side plates fixed to the machine columns (see Fig. 2b). When dZ0 the rods are superposed. 2.2. Timoshenko theory of beams The Upper and Lower beams in a Press Brake have a length to height ratio lower than four. For such beams it is necessary to include the effect of shear deformations in the technical theory and the result is known as the Timoshenko theory of beams [14–16]. In this formulation the vertical displacement w of the beam is determined by the fourth-order equation: EI
d4 w EI d2 q Z Kq C 4 kGA dx2 dx
(1)
In Eq. (1), E is the Young’s modulus, G the Shear modulus, A and I the area of the cross-section and its moment of inertia. The dimensionless factor k is introduced to account for the non-uniform shear distribution in the crosssection while retaining the one-dimensional beam approach.
Fig. 2. Beam model. (a) Upper and Lower beam model. (b) Composite lower table.
In this paper the definition of k given by Cowper [16] is chosen, giving kZ0.85 for a rectangular cross-section and kZ0.33 for a ‘T’ shaped one, with Poisson’s ratio nZ0.3. The curvature and slope equations for Timoshenko’s theory are, respectively: d2 w M q Z C EI kGA dx2 dw Z b C j; dx
where
(2)
b ZK
V kAG
(3)
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In Eq. (2) M is the bending moment and q the applied load and in Eq. (3) j is the rotation of the cross-section due to bending, b the shear deformation and V the shear force. Although Eq. (1) reduces to the technical theory equation for uniform load q, the boundary and interface conditions are different. Relevant to this study are the following two interface conditions: C bjK xZL=2Kd K bjxZL=2Kd Z
C bjK xZt K bjxZt Z
VjxZt ku G
qðL K 2aÞ 2kl Al G 1 1 K Au2 Au1
(4)
(5)
The indexes l and u refer to Lower and Upper beam, respectively. In Eq. (5) A takes the constant values Au1 for 0!x!t and Au2 for t!x!L/2. Eqs. (4) and (5) express the shear strain discontinuity related to the discontinuity of V at the supports of the Lower beam and to the variation of the Upper beam cross-sectional area. According to Eq. (3) these discontinuities in the shear strain will originate slope discontinuities. Since the Upper and Lower beams are statically determinate their deflections may be determined by integrating twice the curvature (Eq. (2)) with the adequate boundary and interface conditions: ðð M q wZ C dxdx C C1x C C2 (6) EI kGA
3. Shape optimisation In order to obtain parallelism between the deflected Upper and Lower beams their curvature must be equal. For each bending length this condition may be expressed as,
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using Eq. (2): Mu Kq M q C Z lC EIu ku GAu EIl kl GAl
(7)
Solutions for this equation are possible only for Press Brakes with sandwich Lower beams, 0%d!L/2, because when dZL/2, MuZKMl. For an illustrative example a rectangular cross-section with the same width for both beams is assumed with kuZ klZ0.85, a constant cross-section for the Upper beam with height of 1400 mm and bending length of 3200 mm. The Lower beam has variable height h(x). The shape optimisation problem of Eq. (7) consists in finding the Lower beam shape h(x) that makes the curvatures of the Upper and Lower beams match for this single bending length. Substituting the above values in Eq. (7) yields an implicit function of h in variable x. This function is shown in Fig. 3 for dZ0 and dZ400 mm. Although the curvatures of the Upper and Lower beams are the same in the above solutions, parallelism will only occur for the case in which the effect of shear deformations is disregarded. As expressed in Eq. (4), the shear force discontinuities that occur at the supports will generate slope discontinuities in the Lower beam deflection. In order to guarantee slope continuity the Lower beam may be supported by a uniformly distributed reaction, like the one shown in Fig. 4a. In this case shear force and shear angle are continuous and q2 is determined to satisfy static equilibrium. As shown in Fig. 4b the load discontinuity applied to the Lower beam implies, from Eq. (7), an abrupt variation of h for curvature continuity, but the deflections of both beams become parallel. What kind of support for the Lower beam will produce a distributed reaction? A technical solution existent in the market and patented [3] (see Fig. 5) resembles the computed
Fig. 3. Variation of height and lower beam shape (units: mm).
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4. Dimensional optimisation 4.1. Penetration and bending error The bending error of the workpiece is defined as the amplitude of its angle variation along the bending length. It is proportional [17] to the amplitude of the penetration p(x) of the punch into the die, or oscillation u(p) defined by: L uðpÞ Z max pðxÞ K min pðxÞ; D0 Z x 2R = a% x% x2D0 x2D0 2 (8) The penetration p(x) is defined as the difference between the vertical displacements of the Upper and Lower beams, plus a constant translation d to ensure that their deflection is the same at a point, here taken as point a: p ðxÞ Z wu ðxÞ K wl ðxÞ K d;
d Z wu jxZa K wl jxZa
(9)
Analytical expressions for the penetration were established using Timoshenko’s beam theory. The expressions for the particular case tZdZ0 and constant cross-section dimensions are the following: sffiffiffiffiffiffi Iu Au ku a x Il rI Z ; rA Z ; rk Z ; j Z ; z Z ; rl Z L L Il Al kl A1 (10)
Fig. 4. Optimal Shape for Lower beam (units: mm). (a) Beam model. (b) Height variation and shape.
pb Z optimal shape for the Lower beam. This solution is used for low force requirements. Another solution of this type but applied to the Upper beam can be seen in patent [7]. The formulation in this Section achieves a null bending error for one bending length and is very sensitive to its variation. To take into account all bending lengths an optimisation problem is formulated in the next Section that keeps the shape of the beams fixed but not its dimensions.
qL4 f r ; z; j 24EIl b I
fb rI ; z; j Z
ps Z
(11)
1 2 C 1 ðz K jÞ4 C ð1 K 2jÞðj3 K z3 Þ rI rI 3
1 3 1 ð1 K 2jÞ K 4ð1 C rI Þ K j C ðz K jÞ rI 2 2 (12)
qL2 f ðr ; r ; z; jÞ 24Gkl Al s k A
1 1 fs ðrk ;rA ;z;jÞZK Crk ðzKjÞ2 C ð1K2jÞðzKjÞ rA rA
(13)
(14)
p Z pb C ps
24ð1 C nÞ rl 2 qL4 Z f r ; z; j C f ðr ; r ; z; jÞ ku 24EIl b I L s k A (15)
Fig. 5. Patented solution for Lower beam.
The Eq. (15) for the penetration is given by the product of a factor proportional to the Lower beam maximum displacement and a non-dimensional factor. This includes the bending contribution fb and the shear contribution, proportional to fs. The influence of the shear term grows proportionally to (rl/L)2, as it is well known.
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The cross-sectional shape of the beams and its influence in shear stiffness is present through rk and ku. 4.2. Formulation of the optimisation problem The dependence of the penetration on selected design variables is explicitly considered by writing pZp(s, a, x). The components of sZ(s1, s2, ., sn) are the n variables related to cross-section dimensions of the beams and the distance between locking rods, which have lower and upper bounds, sK and sC i i , iZ1,., n, respectively. Variable a defines a bending length, x is the position variable and both retain their previous meaning. The formulation of the dimensional optimisation problem is the following: for a given bending force q and maximum bending length L, find the design s0 of a Press Brake for which the maximum oscillation of the penetration function is minimum. This can be expressed as: find the (optimal) design s02D1, the bending length a02D2 and the penetration pso ;ao such that,
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was used instead of Timoshenko’s theory, ‘shear only’ as if the bending stiffness of the beams was very high and all the contribution to the bending error came from shear and a third one with both contributions taken into account. A rectangular cross-section for the Upper beam was considered and both a rectangular and ‘T’ shape cross-sections for the Lower beam. The last column gives uðps0 ;a0 Þ for Upper beam dimensions of 3200!90!1600 mm3 and bending force per unit length q of 180 N/mm. Fig. 6 shows the penetration curves for each case in Table 1.
uðps0 ;a0 Þ Z min max uðps;a Þ; s2D1 a2D2
(16)
uðps;a Þ Z max pðs; a; xÞ K min pðs; a; xÞ x2D0
x2D0
C D1 Z s 2Rn =sK i % si % si ; i Z 1; .; n ; L D2 Z a 2R = 0% a% 2
(17)
From the different expressions for the penetration (see, for example, Eq. (15)) it can be concluded that the optimal design s0 is not dependent on q and that the bending error is proportional to it. To evaluate the solutions of the minimax in Eq. (16) a genetic algorithm [12,13,18] is used since it does not depend on the analytical properties of the objective function. 4.3. Unconstrained dimensional optimisation In this Section the optimisation in Eq. (16) is solved with three design variables: d, and ratios rI and rA, defined in Eq. (10). The upper bounds on these ratios are high enough to reach an unconstrained optimal solution. Table 1 presents the optimum values for three types of analysis: ‘bending only’, as if the technical theory of beams
Table 1 Optimal values for bending and shear contributions separately and together d (mm)
Bending only Shear only BendingCshear
0 471 163
rI
2 – 2.87
rA klZ0.33
klZ0.85
– 0.74 0.14
– 1.88 0.40
uðpso ;ao Þ (mm) 0.0080 0.0085 0.0129
Fig. 6. Penetration curves for each case presented in table 1 (units: mm). (a) Bending only. (b) Shear only. (c) BendingCShear.
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the supports, independently of their location and the penetration curve has the form shown in Fig. 6b. Therefore, the optimal solution for ‘shear only’ selects locking rods far apart, with dzL/3.5, taking into account all possible bending lengths. Changes in the beams’ cross-sectional shapes are compensated by changes in its dimensions to maintain the support optimum location. – When both bending and shear effects are present and since the penetration is negative for bending and essentially positive for shear, the optimum solution increases each absolute value in order to minimise their sum (see Fig. 6c). 4.4. Examples of dimensional optimisation Dimensions currently used in industry for a Press Brake with 175 tons of bending force and LZ3200 mm are now considered and two possible geometries for each Upper and Lower beam, as shown in Fig. 7. UB1 and UB2 are designs for Upper beams, without and with inertia variation to allow the mounting of the actuators. LB1 and LB2 are designs for Lower beams, with and without ditch, considering a height above ground of 834 mm. The optimisation problem in Eq. (16) was solved for all four combinations of Upper and Lower beam geometries. The design variables si, iZ1,.,7 and their bounds are identified in Fig. 7 and s8 is the distance d as defined in Fig. 2a. The optimum values are presented in Table 2, in bold when the upper bounds of the design variables were reached. The penetration curves at the optimum for two cases are shown in Fig. 8 for qZ180 N/mm. The UB1/LB1 combination is the solution that minimizes the bending error. The Upper beam reaches both specified upper bounds on cross-section dimensions and the Lower beam attains the maximum specified width of 70 mm. This relatively low value for the upper bound on the width is a design requirement for the sandwich construction, since the Lower beam has to leave room for two side plates as shown in Fig. 2b. For the same Upper beam, the geometry of the Lower beam at the unconstrained optimum in the previous Section required a cross-section geometry of 600!600 mm2— clearly an infeasible design.
Fig. 7. Upper (UB) and Lower (LB) geometries and design variables with bounds (units: mm). (a) UB1. (b) UB2. (c) LB1. (d) LB2.
The following comments on the above results are in order: – The influence of bending and shear on bending error are of the same order. Any model of the bending process needs to include both effects. – The optimal solution for ‘bending only’ selects vertically aligned locking rods and halves the inertia of the Lower beam to compensate for the stiffness derived from the midspan location of its support. The penetration curve ps0 ;a0 , shown in Fig. 6a, is negative indicating that the Lower beam deforms more than the Upper beam. – For ‘shear only’ the central support of the Lower beam does not help since the curvature will always have the sign of the distributed load q (see Eq. (2)). Shear deformation will cause Upper and Lower beams to have always opposite curvatures and slope discontinuities at Table 2 Optimal solutions for each combination of geometries (units: mm) Geometries combination
s1
s2
s3
s4
s5
s6
s7
s8
uðpso ;ao Þ
UB1/LB1
1600 1600 1600 1379 – –
90 90 90 90 – –
1375 1228 1560 – 950 –
70 70 70 – 70 –
– – – 70 – 47
– – – 130 – 130
– – – 600 – 600
290 400 0 436 566 576
0.0139 0.0168 0.0183 0.0256 0.0282 0.0308
UB1/LB2 UB2/LB1 UB2/LB2
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Fig. 8. Penetration curves with Shear and Bending contributions (units: mm). (a) UB1/LB1. (b) UB1/LB2.
The bending error increases 8% for this constrained solution in relation to the error for the unconstrained optimum. A stress analysis reveals that the distance dZs8 needs to be increased to avoid too high local contact stresses between the locking rods and the Lower beam. Considering an allowable stress of 320 MPa for the beam material and a maximum bending force of 175 ton, a Finite Element analysis determined a lower bound of 400 mm for the design variable s8. The bending error with this constraint (see Table 2, second line) is 30% larger than the unconstrained optimum. The solution with superposed rods (dZ0) is also feasible but the error increases (see Table 2, third line). The UB1/LB2 is the best solution without ditch but the bending error almost doubles in comparison with the previous case. The Lower beam stiffness has to be supplied by a ‘T’ shape cross-section and in this case the unconstrained solution requires a flange width of 4300 mm, instead of the specified 600 mm upper bound. At the optimum, all Lower beam cross-section dimensions reach their upper bounds while the Upper beam does not attain the allowable height. As seen in Fig. 8b most of the bending error comes from shear and since the ‘T’ crosssection is weak in shear (low k) it uses all allowable area while the larger k of the Upper beam is compensated by a smaller cross-sectional area. The influence of a design constraint for the Upper beam is assessed in the optimization for the UB2/LB1 and UB2/LB2
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combinations. Both solutions generate larger errors but the increase is moderate (13 and 23%, respectively) when compared with the UB2/unconstrained Lower beam design, for which a uðpso ;a0 ÞZ0.025 mm was calculated. The unconstrained dimensional optimisation solutions presented in the previous Section provide the lowest bending errors, but correspond to cross-section dimensions that violate the limits accepted in industry. The above examples show that the currently used dimensional constraints do not penalise significantly the bending error. The sandwich design solution is also seen to be preferable to the conventional one (dZL/2). In Section 5 it is shown that the error for the latter is an order of magnitude larger than for the sandwich design obtained in every constrained or unconstrained optimal solution in this Section. The penetration curves in Fig. 8 for two constrained optimum designs reveal that the main source of bending error in sandwich Press Brakes is shear deformation. This unexpected result explains why locking rods horizontally aligned are preferred over the vertically aligned optimal solutions of the technical theory of beams: shear deformations cause opposite curvatures for the Upper and Lower beams instead of the desirable matching curvatures generated from bending. 4.5. Comparison with numerical solutions from the theory of elasticity The UB2/LB2 combination was used to compare analytical results from Eq. (6) with numerical results from a Finite Element analysis. The Upper and Lower beams were modelled with PLANE 82 2-D 8-Node Structural Solid ANSYS Finite Elements and elements CONTACT 52 3-D Point-to-Point were used to model contact between locking rods and Lower beam. The deflections for each beam for maximum bending length are presented in Fig. 9. For all bending lengths the differences do not exceed 4% for the Upper beam and 10% for the Lower beam. The main difference in the results occurs at the slope discontinuities, which appear as rapid variations in the Elasticity results. It can also be observed in Fig. 9 that the curvatures of the Upper and Lower beams are in opposition due to the predominance of shear over bending.
5. Optimisation of the initial deformation In this Section the previous results are improved by adding an initial deformation to one of the beams to allow for a better parallelism between their deflections. Such a function may be a Spline and is introduced as a design variable in the formulation of the optimisation design problem. An initial deformation may be introduced in the Upper beam by shimming as shown in Fig. 10.
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The optimisation problem is reformulated into a sequential optimisation problem. Problem (16) is solved first giving an optimal design s0. Then the following problem is solved for a particular load value q: find the initial deformation v02D3, the bending length ao2D2 and the penetration pv0 ;s0 ;a0 such that, uðpv0 ;s0 ;a0 Þ Z min max uðpv;s0 ;a Þ v2D3 a2D2
Fig. 9. Comparison between analytical (Timoshenko theory) and Finite Element numerical results for UB2/LB2 using maximum bending length. (a) Upper beam. (b) Lower beam.
Let S(v,x) be a cubic spline [19] with nodes (x1, xi,., xm), m being the number of equally spaced points along half of the Upper beam. The components of the vector vZ (v1,vi,.,vm) are the initial deformation values at the points xi and their bounds are given by: C D3 Z v 2Rn =vK (18) i % vi % vi ; i Z 1; .; m S(v,x) is added to penetration given by Eq. (9) in the following way p ðxÞ Z ½wu ðxÞ K wl ðxÞ K d C Sðv; xÞ K SjxZa
(19)
(20)
The previous geometry combinations with optimal dimensions UB1/LB1 and UB2/LB2 were selected to illustrate this sequential optimisation. Assuming mZ9 and C KvK i Zvi Z0.4 mm in Eq. (18) one obtains, respectively, 0.0129 and 0.0253 mm for uðpv0 ;s0 ;a0 Þ—a gain in bending accuracy of 7 and 18% (see Table 2). Problem (16) was also solved for UB1/LB1 and UB2/LB2 but setting s8ZdZL/2 and augmenting the bounds on width for the Lower beam because there are no side plates to accommodate. At the optimum (see Table 3) all the bounds on dimensions were attained and the uðps0 ;a0 Þ values calculated are an order of magnitude higher for this conventional solution than for the corresponding sandwich designs. The introduction of an optimal initial deformation for this case achieves a significant decrease in the bending error, as shown in Table 4. In the case UB1/LB1 the bending error decreases from 0.1092 to 0.0025 mm and from 0.2658 to 0.0064 mm in the case UB2/LB2. The improvement in bending accuracy is radical for these conventional designs because both beams deform with the same shape for all bending lengths. In the case of the sandwich solution for the Lower beam, different bending lengths originate very different deformations and there is no unique initial deformation that reduces appreciably the error for all lengths. The correction introduced by an initial deformation is associated with a load value q. If the load changes and the initial deformation remains constant, the error will not change proportionally to its value at q. In order to introduce the optimal initial deflection for the actual load q, it seems advantageous to use an automatic
Fig. 10. Modelling shimming with a Spline function.
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Table 3 Optimal dimensions for UB1/LB1 and UB2/LB2 with conventional location of the supports (units: mm) Case
s1
s2
s3
s4
s5
s6
s7
s8
uðpso ;ao Þ
UB1/LB1 UB2/LB2
1600 –
90 –
1600 –
130 –
– 130
– 130
– 600
1600 1600
0.1092 0.2658
Table 4 Optimal initial deformation for UB1/LB1 and UB2/LB2 with conventional location of the supports (units: mm) Case
v1
v2
v3
v4
v5
v6
v7
v8
v9
uðpvo ;so ;ao Þ
UB1/LB1 UB2/LB2
0 0
K0.023 K0.065
K0.045 K0.125
K0.064 K0.168
K0.079 K0.202
K0.092 K0.228
K0.100 K0.244
K0.105 K0.256
K0.107 K0.259
0.0025 0.0064
system like the crowning devices that some manufactures have been proposing [1–10], instead of the time-consuming trial–and–error shimming process.
6. Conclusions Based on the model of the bending process in Press Brakes defined in Section 2 it has been found that it is not possible to design a machine that achieves uniform bending angles for every bending length. This happens because there are no optimal shapes or dimensions for the bed and ram that lead to parallel deflections for all bending lengths. Shape optimisation makes possible parallelism, but only for one bending length and is very sensitive to its variations; furthermore the optimal shape is not simple to manufacture. Dimensional optimisation leads to a composite Lower beam supported at the middle, known as sandwich design. Some manufacturers have been praising this solution without noticing the unexpected and negative influence of shear deformations. These deformations cause opposite curvatures of bed and ram, independently of the location of their supports. However, the errors due to bending and shear have opposite signs and may be made to almost cancel at the optimum, making this solution an attractive compromise between bending precision and design simplicity. The introduction of an optimised initial deflection for each bending length and load value is essential in a conventional Press Brake, where the bed and ram supports are located in the machine columns. This is an interesting solution if it can be computed and introduced in an automatic way each time the bending conditions are changed. The methodology presented in this paper proved well suited to analyse the structural behaviour and bending precision of existent Press Brakes and should be useful to optimise their performance and assist in the design of new solutions.
Acknowledgements The support of Adira—A. Dias Ramos Company in Porto, Portugal and the useful discussions with Eng. Jose´ Bessa Pacheco and Eng. Miguel Costa are gratefully acknowledged. The support of FCT through Project POCTI/36055/ECM/99 is gratefully acknowledged.
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