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Modeling and optimization of the sequential brakeforming processes
Journal of Materials Processing Technology 102 (2000) 153±163
Modeling and optimization of the sequential brakeforming processes Michael Yu Wang*, Sayidmasuthu Shabeer Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA Received 4 January 1999
Abstract The sequential bending process is often used to produce parts with simple curvatures (such as stringers and wing spars in airplane structures). The process involves a series of three-point bends, often overlapping, to produce the desired curvature pro®le along the arclength of the part. The objective of this work is to develop a bending model and an optimization method for obtaining the ``best'' sequence of multiple bends. The goal is to improve dimensional accuracy and to substantially reduce the rework iterations. In the analytical beam model developed, the part state is represented by its curvature and stress distribution. The model accounts for curved initial geometry, residual stresses and strain-hardening of the material. Results of multiple bends for parts with symmetric crosssections are presented. The synthesis of optimal bends was formulated as an optimal control problem that minimizes the curvature errors at speci®ed points along the part. Results are presented for the forming of parts with constant curvature as an example. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Sequential brakeforming processes; Bending model; Material properties; Strain-hardening
1. Introduction A sequential bending process is a common method to produce parts with simple curvatures in sheets, plates, or beams. It is used widely in the aerospace industry for a variety of parts such as stringers and spars of wing structures. Stringers and spars are long components with relatively simple cross-sectional shapes and with smoothly varying cross-sectional areas along the part length. Sequential bending is an important forming operation to achieve the speci®ed ®nal contour. This process is essentially a series of three-point bends, often overlapping. The process equipment is a simple press brake that is usually operated manually. The research work of this paper was initiated to address two issues raised in the manufacturing of aircraft stringers and spars with a sequential brakeforming process. The ®rst issue is the capability of process modeling and simulation. Of particular interest is a better understanding of the essen*
Corresponding author. Present address: Associate Professor (visiting), Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, 405 Mong Mau Building, Shatin N.T., Hong Kong. Tel.: 852-2609-8487; fax: 852-2603-6002. E-mail address: [email protected] (M.Y. Wang)
tial mechanics of the sequential brakeforming process that would lead to simulation of the capability of the process. A process model by itself would enable the process designer to explore the geometric outputs of curvature and residual stresses and their sensitivities to feasible process input parameters and process conditions. The second issue concerns the planning of an optimal bending sequence. In current practice, a manual ``control'' method is often adopted on the shop-¯oor. A bending sequence is typically devised by a manufacturing operator based on intuition. After an initial bending process, a new bending sequence may be needed for rework to meet the speci®ed curvature or shape tolerance. The process ef®ciency (i.e., the number of iterations) and the dimensional accuracy of the ®nal shape depend critically on the skill and the experience of the operator. The goal of optimal process planning is to devise the ``best'' bending sequence at any stage for reducing the number of iterations and improving dimensional accuracy. The objective of this work is to develop a bending model and an optimization algorithm that work in tandem to synthesize the ``best'' sequence of bends that would substantially reduce the rework iterations. In the analytical beam model developed, the state of the part is represented by its curvature and stress distribution. The model accounts for non-¯at initial geometry, residual stresses, and strain-
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 0 1 - X
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hardening of the overlapping bend material. Results for parts with symmetrical cross-sections are presented. The synthesis of bends was formulated as an optimization problem that minimizes the curvature errors at speci®ed critical points along the part. 2. Background The mechanics of simple bending have been thoroughly explored, both analytically (e.g. [1,2]) and using numerical methods [3]. An example of the pure modeling approach was presented by Hansen and Jannerup [4]. They developed an elastic±plastic process model to fabricate steel beams with rectangular cross-sections on a three-roll bending machine. Given the desired curvature, the model could be used to obtain the bending moment and springback curvature, which allow the determination of the center roll position. In a process with feedback corrections, the process relies on the part representation, process model, shape measurement, and the control algorithm. It may only require an approximate process model to generate corrective commands for the next iteration [5]. A recent example of the control approach is reported in [6,7], where process models for the open- and closed-loop control of the multi-axis bending process are developed. An earlier use of dynamic models in the closed-loop control of a three-roll bending machine was described in [8]. The process model was obtained through identi®cation experiments. The most recent work on the modeling of the sequential brakeforming process in particular is presented by Hardt et al. [8]. The model has been developed speci®cally for the forming of cylindrical shapes from plates. The process is modeled as a series of overlapping two-dimensional threepoint bends, where the overlap includes the plastic zone from previous bends. Further, the deformed zone is modeled as a curved initial geometry for the next bend and as a locally strain-hardened material with residual stresses. In the more general model developed in this paper for forming arbitrary shapes in two dimensions, some of the fundamental equations of mechanics that are derived in [8] are used. The manual sequential bending process typically used in the current aerospace manufacturing can be described as an off-line manual feedback control scheme as depicted in Fig. 1. The operator obtains information of the shape errors in the part, either by a visual inspection or through measurements on the checking ®xture. The operator then uses this information to intuitively generate the bend inputs for executing the actual bending process. In essence, the opera-
Fig. 1. Existing process scheme.
tor acts as the controller, and any algorithm being used to generate the bend commands has not been captured in any formal or explicit form. An optimization scheme is proposed in this paper to improve the methods used in the current practice of brakeforming. The optimization method takes the actual part shape from the measurement and the given desired shape as inputs. It uses the bending model to simulate the result of a sequence of bends. Starting with an initial bend sequence, it optimizes the bending process to minimize the shape errors with respect to the bending parameters including locations, spans, and punch forces. In this way, it arrives at an ``optimal'' sequence of bends to narrow the gap between the current and desired shapes. As a result, the number of rework iterations is greatly reduced. Optimization has been used in the past to address certain inverse problems in metal forming processes, such as the optimum die design to achieve a ®nal product with desired properties [9]. However, no work on optimization issues relating to the sequential bending process is evident in the literature. There exist a number of unique issues for the optimization scheme, including the determination of the number of bends with constraints on bend inputs, the effect of the bend interactions when overlapped, and the trade-off between dimensional accuracy and process cost. These issues offer a considerable scope for study. 3. Model development The sequential bending process is typically performed on a press brake, using tooling with a large nose radius, as shown schematically in Fig. 2. An analytical model is considered more appropriate for the present application. There are three compelling reasons in favor of developing an analytical bending model as opposed to a ®nite element model: (1) the essential mechanics of the sequential bending process is simple, (2) the cross-sectional shapes of the parts are typically simple (e.g., L-, J-, and Z-shaped), and (3) most importantly, a simpli®ed model for use with an optimization algorithm is needed, as if a computationally intensive model is used, the effectiveness of the method is severely compromised. 3.1. Part representation Consider a long beam component with a symmetric crosssection with curvature in a plane. The shape or contour of
Fig. 2. Schematic diagram of sequential brakeforming process.
M.Y. Wang, S. Shabeer / Journal of Materials Processing Technology 102 (2000) 153±163
such a part can be represented by its centerline. The centerline passes through the cross-sectional centroid of the part. The cross-sectional orientation is assumed to be normal to the centerline. Hence, the part geometry is completely speci®ed by the absolute value of the curvature of its centerline as a function of its arc-length, K(s). Mathematically, curvature is de®ned as the reciprocal of the radius of curvature [10], given as K s
d2 y=dx2 1 d2 y=dx2 2 3=2
Fig. 3. Bending process.
(1)
With the support points of boundary conditions de®ned for the part, the intrinsic representation, K(s), can be translated into the shape of the part in the Cartesian coordinates, x(s) and y(s), by solving the following simultaneous equations from differential geometry [10]: d2 x dy K s 0; 2 ds ds
d2 y dx ÿ K s 0 2 ds ds
155
(2)
Since the geometry of a part with any reasonable curvature pro®le, K(s), is to be represented, this function is stored not as any analytical function but as a data array. Each entry in this array has two elements, the ®rst being the arc-length of a point on the centerline from one end of the part, and the second being the curvature of the centerline at this point of the part. This allows a part to be described coarsely with only a few entries in the array corresponding to a few points along the centerline, or ®nely with a large number of entries, as required. 3.2. Mechanics In this paper the mechanics for two-dimensional (or inplane) sequential bending is developed for simplicity. The basic mechanics can be extended to asymmetric sequential bending. For example, the basic mechanics and the analytical approach for simple asymmetric bending of L-shaped beams are developed by Xu et al. [11]. In such a case, the part representation also has to be modi®ed to include curvature as a vector. The fundamental relations of
Fig. 4. Bending inputs.
mechanics used for developing the sequential bending model are derived from [8,12]. Fig. 3 shows the scheme of the process model. The bend inputs for any single bend in the sequence of N bends are illustrated in Fig. 4. The inputs, outputs, and parameters for the process are summarized in Table 1. The model accounts for non-¯at initial geometry, residual stresses and strain-hardening of the overlapping material. The bending analysis proceeds at two levels: local analysis and global analysis. The local analysis determines the effect of a bending moment, M, on any cross-section of the part. The effect is essentially a change in curvature of the centerline, and in the residual stress distribution across the cross-sectional area. Note that since friction at the supports is almost negligible because of rollers, there is no net axial force on the cross-sections. The bending moment is the predominant cause of deformation. The material and geometric properties of the part are described by the initial stress distribution across the cross-section at any point along the part (si) and the initial curvature at that point (Ki). For a given bending moment applied in the loaded equilibrium state on that cross-section (Mload), the
Table 1 Process inputs, parameters, and outputs Inputs
Parameters
Outputs
For each of the N bends (i) Location of the bend (left support point) (ii) Span (iii) Punch force
Material properties (i) Young's modulus (ii) Yield strength (iii) Plastic modulus (iv) Initial residual stress in part Part geometry (i) Cross-sectional shape parameters (ii) Initial curvature/contour of part Machine conditions (i) Punch radius (ii) Friction at support points
Unloaded contour or curvature Final residual stresses
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Fig. 5. Local analysis.
local analysis determines the ®nal stress distribution (sf) and the ®nal unloaded curvature after springback (Kf) of that cross-section (Fig. 5). Two important assumptions are made in the local analysis [12], plane section (or Euler±Bernoulli) assumption and uniaxial stress assumption. Three fundamental relations are used to carry out the local analysis [2]: the constitutive stress±strain relation of the material together with its nature of hardening as it unloads is described as a function of s(e). The strain±curvature relation is de®ned by the true strain e ln 1 Ky, where K is the curvature and y the distance from the neutral axis. For bends that result in low curvatures (e.g., less than about 0.05 in.ÿ1 such as in spar-forming) and where the parts have a thickness less than 1 in. (25.4 mm), the quantity Ky is small. That is, Ky 1 and, therefore, the true strain can be approximated by e Ky. Finally, the bending moment at any cross-section along the part, M(s), is related to the curvature of the centerline of that cross-section by the following integral relation across the cross-section: Z M s
H=2 ÿH=2
Z s eyw y dy
H=2
ÿH=2
s e Kyw y dy
(3)
where H is the thickness of part cross-section, and w the width of the cross-section. For a given loading moment, M, the change in curvature is determined by solving the above equation. Since the relationship is not in general an analytical expression, it has to be solved by a numerical iteration scheme. When the part is unloaded, the net moment on the section vanishes and the curvature of the beam reduces: this phenomenon is known as springback. A springback analysis is equivalent to applying a moment equal and opposite in direction to the originally applied moment. Therefore, the above three relationships are solved again to transform the loaded state to one with a net bending moment of zero on the cross-sections. This gives the unloaded curvature and the residual stresses across the cross-sections. The global analysis constructs the bending moment pro®le along the part arc-length for a particular bend. It uses the local analysis to determine the effect of this moment pro®le on the resulting unloaded curvature and residual stress. The local analysis is repeated for the entire process for each bend of the bending sequence. Note that the ®nal curvature and residual stresses of the previous bends become the initial ones of the next bend. For a three-point bending, the bending moment M is simply described as in Fig. 6.
Fig. 6. Bending moment pro®le.
4. Simulation The model described above is implemented for a uniform rectangular cross-section part. The speci®c stress±strain and strain-hardening relationships used are based on the assumption that the shapes of the true stress vs. true strain diagrams for tension and compression are identical [2]. The generalized stress±strain and the kinematic strain-hardening relations are shown in Figs. 7 and 8. The usual assumption is that the unloading and reversed loading curves are linear. Fig. 9 is a ¯ow chart of the model computations. This closely follows the procedure outlined in Section 3. The program code is developed and implemented entirely in TM MATLAB . The secant method was used as the iteration scheme to solve Eq. (3) [13]. In the implementation, the part is discretized into a ®nite number of rectangular beam elements. The curvature of the centerline at the face of each element, and the stress distribution across the element crosssection are the quantities that represent the state of the element. The state representation of all the elements that
Fig. 7. Constitutive relationship.
Fig. 8. Strain-hardening relationship.
M.Y. Wang, S. Shabeer / Journal of Materials Processing Technology 102 (2000) 153±163
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ered during springback. This implies that although two bends might have been punched so that they physically overlap, their unloaded curvature or residual stress might not interact. Also notice that bend no. 5 is executed in a segment of the part that is completely within the span of a previous bend (no. 3). This bend only produces an elastic deformation during loading and leaves no permanent plastic change in the part. However, a similar bend (no. 4) does produce a permanent change in the part because it is partly executed in a region that was unaffected by any previous bend (no. 3). The curvature pro®le produced by the multiple bends is approximately piece-wise linear. This is because a linear stress±strain relationship was assumed in the plastic zone. If a non-linear plastic constitutive relation were assumed, this pro®le would be more non-linear. It is interesting to note the reversal of stresses near the top and bottom surfaces of the part that occurs due to springback. Under the loaded condition (e.g., loaded downwards as here), the bottom surface is in tension, whilst the top surface is in compression. When released, the effect of the negative springback moment results in the observed stress reversal at the surfaces. Also note that the residual stresses do interact when the bends have been overlapped. In this example, a peak residual stress of 20±25% of the yield stress is left in the part. 5. Optimization of sequential bending Fig. 9. Flow chart of model computations.
make up the part taken together represents the state of the entire part. The implementation of the model by simulating a sequence of bends on a part with a rectangular cross-section is illustrated. The inputs to the simulation program are: (1) the material properties, (2) the beam geometry, (3) the initial part shape (as curvature) as well as initial residual stresses, and (4) the bend inputs. In the following example, a sequence of ®ve bends on a part made of aluminum (2024-T351) is simulated. Note that bend nos. 1 and 2 overlap considerably, as do bend nos. 3, 4 and 5. For simplicity, an initially straight part (i.e., zero curvature throughout its length) that has negligible initial residual stress is considered. The results of the simulation are shown in Figs. 10 and 11, where Fig. 10 shows the ®nal unloaded curvature and shape of the part after the sequential bending, and Fig. 11 shows the ®nal residual stresses induced in the part. Notice that the curvature within the bend span but close to the support points is zero. This is because the bending moments near support points are low and the material deforms only elastically. This elastic deformation is recov-
The bending model developed in Section 3 essentially solves the ``forward problem''. That is, given the initial state of the part and the bend inputs, it predicts the ®nal state. The ``inverse problem'' of synthesizing the bend inputs, given the initial state and the desired ®nal state, is not straightforward. This is due to the factors of: (1) the non-linear mechanics and the history dependence of the process, and (2) the discrete nature of the bends with limits on bend inputs. The problem of optimization of a sequential bending process is to determine suitable bend inputs (i.e., the location, span and punch force of each of N bends), given the initial state of the part, the desired curvature pro®le, and the number of bends (N) to be executed. An appropriate objective for optimization would be minimizing the error between the actual part shape formed and the desired shape. 6. Formulation In this section, the problem of optimization of sequential bending is formulated. The objective is to minimize the error between the actual part shape produced by the process and the desired part shape. The design variables in the optimization problem are the bend inputs. It is proposed that the sum of squares of the difference between the actual curvature and
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Fig. 10. Final (unloaded) curvature and shape pro®le.
the desired curvature along the length of the part be used as the objective function. The reason to consider the curvature error and not the actual shape is that the bending model mechanics deal directly with curvature. To convert to the physical space of x±y coordinates necessitates a computationally expensive solution of differential equations (Eq. (2)). Transforming back and forth between curvature and coordinate space for bending simulation and objective function evaluation, respectively, ultimately makes the optimization program too time-consuming. If the curvatures are
matched exactly, it is ensured that the shapes will also match exactly in physical space (x±y coordinates). There are two main types of constraints in the optimization problem. The ®rst type is the upper and lower limits on the process inputs. The press can exert only a particular maximum amount of force. The guides on the bending machine allow only particular maximum and minimum spans to be used for the bends. These constitute simple bounds on the design variables. The second constraint is the material limit. The resulting strains induced in the part must
Fig. 11. Final residual stress distribution.
M.Y. Wang, S. Shabeer / Journal of Materials Processing Technology 102 (2000) 153±163
not exceed the fracture limit of the material. In aerospace manufacturing, spar forming involves only relatively small changes in shape and hence only low strain changes. In these cases, the strains would not be even near to the fracture limit. The optimization problem for sequential bending can be compactly formulated as an optimal control problem [14]. Notation to achieve this compact casting is de®ned as follows. Let the subscript n represent the stage after the completion of the nth bend in a sequence of N bends, with n taking on values 1, 2, . . ., N, where N is the total number of bends. Vector X1;n and matrix X2;n represent the state of the part after the nth bend. Here, X1;n represents the curvature at p speci®ed points along the centerline of the part. The size of this vector is therefore p1. Matrix X2;n represents the stress distribution across the cross-sections passing through the same set of p points. Suppose the stress distribution across each cross-section is represented by the longitudinal (axial) stress value at q different distances from the centerline. The size of X2;n is therefore pq. Further, let un be the bend input vector that transforms the state of the part from that before the nth bend to that after the bend. This vector is of size 31 and has the components of location, span, and punch force of the nth bend. Hence, the optimization of N sequential bends is mathematically described as follows: Xn fX1;n ; X2;n g
State variables : Process dynamics : Objective : Constraints :
Xn f Xnÿ1 ; un where f is the bending model
Minimize X1;N ÿ X1;spec 2 umin un umax
where X0 is the initial state, X1, spec the speci®ed ®nal curvature pro®le, and N the total number of bends. 6.1. Solution A globally optimal solution for the above problem can be found, in theory, by dynamic programming [15]. However, for reasonably long parts, the number of points along the part that adequately represent its state is not small. Hence, the dimension of the vector representing the part curvature pro®le and stress distribution is relatively large. Therefore, ®nding a global optimum by dynamic programming is computationally formidable. One way to handle this problem is to consider a ``moving horizon''. That is, instead of optimizing over all of the N bends at once, Nt (
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Considering the several dif®culties of ®nding a guaranteed global solution, a simpler approach that ensures a local solution is used in this paper. This can be effected by a conventional optimization procedure that depends on an appropriate gradient search method. An initial set of bend inputs must be provided, which could be the operator's guess. The optimization algorithm then searches locally using the bending model as a predictive tool for the objective function evaluation, until it reaches a minimum. To ensure global minimization, the optimization is repeated at different initial points. It was observed in the example presented later that for most starting points, the point of convergence with the minimum objective function value was identical. MATLABTM optimization toolbox is used to implement and numerically solve the problem. Speci®cally, a sequential quadratic programming (SQP) approach is used [16]. 6.2. Illustration A common use of the sequential brakeforming process is to make parts with a constant curvature (i.e., an arc of a circle). The optimal forming of a constant curvature part is illustrated as an example. Suppose there is an initially straight stock made of aluminum (2024-T351). A constant curvature of 0.05 in.ÿ1 is desired in a 5 in. segment of the part ranging from arc-length of 10±15 in. The strategy adopted to execute only the minimum number of bends, N, required to achieve a particular tolerance is simple. A low number of bends is ®rst used. The optimization procedure would yield an optimal bending sequence for this number of bends. If the resulting ®nal shape does not meet the tolerance requirements, the number of bends is increased one at a time and the optimization is repeated until the ®nal shape falls within the tolerance limits. This strategy is illustrated in Fig. 12. The tolerance requirement in this example is taken as 0.003 in. Upper limits were placed on span and punch force as 12 in. and 300 lb, respectively. Twenty equally spaced points in the region of interest were chosen as speci®ed points for the objective function evaluation. Thus, the sum of
Fig. 12. Finding the minimum number N.
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Table 2 Optimal sequential bending results N
Bending sequence
Start point (in.)
Span (in.)
Punch force (lb)a
Objective function
1
1
6.57
12.00
258
1.1810ÿ2
2
1 2
5.76 8.46
11.06 10.79
259 265
3.3010ÿ3
3
1 2 3
5.54 7.48 8.98
10.67 10.29 10.30
258 261 265
1.4910ÿ3
a
1 lb4.45 N.
squares of the curvature error at these 20 points was minimized. The results of the optimization program for up to three bends are presented in Table 2. Figs. 13±15 show the curvature and part shape obtained by the optimal single, two and three bend(s), respectively. In the same ®gures, the desired curvature and part shapes are shown as dotted lines. Fig. 16 shows the actual geometric deviation (in y coordinate) between desired and actual shape obtained by the optimal sequence of bends, respectively. Fig. 17 shows the iteration for the single bend case. The decrease in objective function value is relatively rapid during the ®rst few iterations, but subsequently the rate decreases.
The objective function evaluation for three bends takes approximately 20 s of CPU time in MATLAB running on a Sun Ultra workstation, and several such objective evaluations are needed for an iteration. Hence, there is a trade-off between approaching an optimum and the time needed for reaching such a point. It was observed for the example that about 8±10 iterations produce results quite close to the optimum within a reasonable time. The number of iterations for convergence depends on how close the initial bend inputs are to the optimum: if the initial guess is close to an optimum, early convergence results. Other interesting observations include the following: 1. It is clear from Fig. 16 that to achieve a tolerance of 0.003 in., three bends are adequate. If the tolerance limit were to be set higher at 0.01 in., then two bends would be suf®cient. With N1 (a single bend), as expected, the upper bound on the span is reached. 2. An interesting result was obtained when the in¯uence of order of execution of the bends, with N3, was investigated. The order of the three bends was permuted and the optimal bends were determined. However, the ®nal curvature was not affected signi®cantly. In other words, although the overlapping portion of the bends do interact, their order does not appear to be signi®cant.
Fig. 13. Optimal single bend: (Ð) actual shape; (. . .) desired shape.
M.Y. Wang, S. Shabeer / Journal of Materials Processing Technology 102 (2000) 153±163
Fig. 14. Optimal two bends: (Ð) actual shape; (. . .) desired shape.
Fig. 15. Optimal three bends: (Ð) actual shape; (. . .) desired shape.
161
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Fig. 16. Geometric deviations (in the y coordinate).
Fig. 17. Iteration history for a single bend.
3. The results show how geometric differences are magni®ed in the intrinsic curvature representation. Although the deviations between the obtained shape and the desired shape are small even with one bend, the curvature pro®les indicate considerable difference. 7. Conclusions A bending model and an optimization algorithm are proposed in this paper to improve dimensional accuracy and reduce rework iterations in sequential brakeforming.
Simulation of two-dimensional bending process has been achieved. The resulting shape and residual stresses of a symmetric part with arbitrary initial shape and residual stresses after a sequence of bends can be determined. This is a useful tool that allows a designer to explore different sequential bending plans. A primary objective of the bending process is to achieve a desired curvature pro®le along the part. This led to the formulation of an optimization problem that minimized the curvature error at any number of speci®ed points along the part. Constraints are placed on the bend inputs, and the problem is solved using optimization algorithm that employs local gradient-based methods. As an illustration, optimal bend sequences are synthesized for forming constant curvature parts using multiple bends. The improvement in dimensional accuracy with increasing number of bends is quantitatively demonstrated. Symmetric sequential bending has been implemented in this paper. Many actual spars may have an asymmetric crosssectional shape. Therefore, the sequential bending simulation needs to be extended for asymmetric bending. Although the primary objective of the paper is to achieve a desired curvature pro®le, an equally important concern is to minimize variation in the curvature obtained by sequential bending. Variation of the initial residual stresses in a part is known to in¯uence the curvature output variation. It seems feasible to use multi-objective optimization methods for simultaneously minimizing curvature error and curvature variation. These aspects are worthy of investigation in the future.
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References [1] C.A. Queener, R.J. DeAngelis, Elastic springback and residual stresses in sheet metal forming by bending, Trans. ASME (1968) 757±768. [2] O.M. Sidebottom, C.F. Gebhardt, Elastic springback in plates and beams formed by bending, Exp. Mech. 19 (1979) 371±377. [3] S.I. Oh, S. Kobayashi, FEM analysis of brake bending, Final Report to Batelle Columbus Labs, Contract No. F33615-77-C-5059, UCAL, Berkeley, CA, 1979. [4] N.E. Hansen, O. Jannerup, Modeling of elastic±plastic bending of beams using a roller bending machine, ASME J. Eng. Ind. 101 (3) (1979) 304±310. [5] D.E. Hardt, B. Chen, Control of a sequential brakeforming process, ASME J. Eng. Ind. 107 (1985) 141±145. [6] K.A. Stelson, J.X. Luo, J.L. Daniel, Control of the fabrication of long slender workpieces. I. Open-loop control of multi-axis bending process, ASME J. Dyn. Syst. Measure. Control. 118 (1996) 540±548. [7] K.A. Stelson, J.X. Luo, J.L. Daniel, Control of the fabrication of long slender workpieces. II. Closed-loop control of multi-axis bending process, ASME J. Dyn. Syst. Measure. Control 118 (1996) 549±558.
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[8] D.E. Hardt, E. Constantine, A. Wright, A model of the sequential bending process for manufacturing simulation, ASME J. Eng. Ind. 114 (1992) 181±187. [9] N. Zabaras, S. Badrinarayanan, Inverse problems and techniques in metal forming processes, Inverse Problems in Engineering: Theory and Practice, ASME, New York, 1993, pp. 65±76. [10] D.J. Struik, Lectures on Classic Differential Geometry, 2nd Edition, Addison-Wesley, Reading, MA, 1961. [11] Y. Xu, L.C. Zhang, T.X. Yu, The elastic±plastic pure bending and springback of L-shaped beams, Int. J. Mech. Sci. 29 (1987) 425± 433. [12] T.X. Yu, L.C. Zhang, Plastic bending: theory and applications, Series on Engineering Mechanics, Vol. 2, World Scienti®c, Singapore, 1996. [13] J.M. Maron, R.J. Lopez, Numerical Analysis: A Practical Approach, Wadsworth, Belmont, CA, 1991. [14] R.F. Stengel, Stochastic Optimal Control: Theory and Application, Wiley, New York, 1986. [15] R.E. Bellman, S.E. Dreyfus, Applied Dynamic Programming, Princeton University Press, Princeton, NJ, 1962. [16] M.A. Branch, A. Grace, MATLAB Optimization Toolbox, User's Guide, Version 1.5, The MathWorks Inc., 1996.
Modeling and optimization of the sequential brakeforming processes Michael Yu Wang*, Sayidmasuthu Shabeer Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA Received 4 January 1999
Abstract The sequential bending process is often used to produce parts with simple curvatures (such as stringers and wing spars in airplane structures). The process involves a series of three-point bends, often overlapping, to produce the desired curvature pro®le along the arclength of the part. The objective of this work is to develop a bending model and an optimization method for obtaining the ``best'' sequence of multiple bends. The goal is to improve dimensional accuracy and to substantially reduce the rework iterations. In the analytical beam model developed, the part state is represented by its curvature and stress distribution. The model accounts for curved initial geometry, residual stresses and strain-hardening of the material. Results of multiple bends for parts with symmetric crosssections are presented. The synthesis of optimal bends was formulated as an optimal control problem that minimizes the curvature errors at speci®ed points along the part. Results are presented for the forming of parts with constant curvature as an example. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Sequential brakeforming processes; Bending model; Material properties; Strain-hardening
1. Introduction A sequential bending process is a common method to produce parts with simple curvatures in sheets, plates, or beams. It is used widely in the aerospace industry for a variety of parts such as stringers and spars of wing structures. Stringers and spars are long components with relatively simple cross-sectional shapes and with smoothly varying cross-sectional areas along the part length. Sequential bending is an important forming operation to achieve the speci®ed ®nal contour. This process is essentially a series of three-point bends, often overlapping. The process equipment is a simple press brake that is usually operated manually. The research work of this paper was initiated to address two issues raised in the manufacturing of aircraft stringers and spars with a sequential brakeforming process. The ®rst issue is the capability of process modeling and simulation. Of particular interest is a better understanding of the essen*
Corresponding author. Present address: Associate Professor (visiting), Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, 405 Mong Mau Building, Shatin N.T., Hong Kong. Tel.: 852-2609-8487; fax: 852-2603-6002. E-mail address: [email protected] (M.Y. Wang)
tial mechanics of the sequential brakeforming process that would lead to simulation of the capability of the process. A process model by itself would enable the process designer to explore the geometric outputs of curvature and residual stresses and their sensitivities to feasible process input parameters and process conditions. The second issue concerns the planning of an optimal bending sequence. In current practice, a manual ``control'' method is often adopted on the shop-¯oor. A bending sequence is typically devised by a manufacturing operator based on intuition. After an initial bending process, a new bending sequence may be needed for rework to meet the speci®ed curvature or shape tolerance. The process ef®ciency (i.e., the number of iterations) and the dimensional accuracy of the ®nal shape depend critically on the skill and the experience of the operator. The goal of optimal process planning is to devise the ``best'' bending sequence at any stage for reducing the number of iterations and improving dimensional accuracy. The objective of this work is to develop a bending model and an optimization algorithm that work in tandem to synthesize the ``best'' sequence of bends that would substantially reduce the rework iterations. In the analytical beam model developed, the state of the part is represented by its curvature and stress distribution. The model accounts for non-¯at initial geometry, residual stresses, and strain-
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 0 1 - X
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hardening of the overlapping bend material. Results for parts with symmetrical cross-sections are presented. The synthesis of bends was formulated as an optimization problem that minimizes the curvature errors at speci®ed critical points along the part. 2. Background The mechanics of simple bending have been thoroughly explored, both analytically (e.g. [1,2]) and using numerical methods [3]. An example of the pure modeling approach was presented by Hansen and Jannerup [4]. They developed an elastic±plastic process model to fabricate steel beams with rectangular cross-sections on a three-roll bending machine. Given the desired curvature, the model could be used to obtain the bending moment and springback curvature, which allow the determination of the center roll position. In a process with feedback corrections, the process relies on the part representation, process model, shape measurement, and the control algorithm. It may only require an approximate process model to generate corrective commands for the next iteration [5]. A recent example of the control approach is reported in [6,7], where process models for the open- and closed-loop control of the multi-axis bending process are developed. An earlier use of dynamic models in the closed-loop control of a three-roll bending machine was described in [8]. The process model was obtained through identi®cation experiments. The most recent work on the modeling of the sequential brakeforming process in particular is presented by Hardt et al. [8]. The model has been developed speci®cally for the forming of cylindrical shapes from plates. The process is modeled as a series of overlapping two-dimensional threepoint bends, where the overlap includes the plastic zone from previous bends. Further, the deformed zone is modeled as a curved initial geometry for the next bend and as a locally strain-hardened material with residual stresses. In the more general model developed in this paper for forming arbitrary shapes in two dimensions, some of the fundamental equations of mechanics that are derived in [8] are used. The manual sequential bending process typically used in the current aerospace manufacturing can be described as an off-line manual feedback control scheme as depicted in Fig. 1. The operator obtains information of the shape errors in the part, either by a visual inspection or through measurements on the checking ®xture. The operator then uses this information to intuitively generate the bend inputs for executing the actual bending process. In essence, the opera-
Fig. 1. Existing process scheme.
tor acts as the controller, and any algorithm being used to generate the bend commands has not been captured in any formal or explicit form. An optimization scheme is proposed in this paper to improve the methods used in the current practice of brakeforming. The optimization method takes the actual part shape from the measurement and the given desired shape as inputs. It uses the bending model to simulate the result of a sequence of bends. Starting with an initial bend sequence, it optimizes the bending process to minimize the shape errors with respect to the bending parameters including locations, spans, and punch forces. In this way, it arrives at an ``optimal'' sequence of bends to narrow the gap between the current and desired shapes. As a result, the number of rework iterations is greatly reduced. Optimization has been used in the past to address certain inverse problems in metal forming processes, such as the optimum die design to achieve a ®nal product with desired properties [9]. However, no work on optimization issues relating to the sequential bending process is evident in the literature. There exist a number of unique issues for the optimization scheme, including the determination of the number of bends with constraints on bend inputs, the effect of the bend interactions when overlapped, and the trade-off between dimensional accuracy and process cost. These issues offer a considerable scope for study. 3. Model development The sequential bending process is typically performed on a press brake, using tooling with a large nose radius, as shown schematically in Fig. 2. An analytical model is considered more appropriate for the present application. There are three compelling reasons in favor of developing an analytical bending model as opposed to a ®nite element model: (1) the essential mechanics of the sequential bending process is simple, (2) the cross-sectional shapes of the parts are typically simple (e.g., L-, J-, and Z-shaped), and (3) most importantly, a simpli®ed model for use with an optimization algorithm is needed, as if a computationally intensive model is used, the effectiveness of the method is severely compromised. 3.1. Part representation Consider a long beam component with a symmetric crosssection with curvature in a plane. The shape or contour of
Fig. 2. Schematic diagram of sequential brakeforming process.
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such a part can be represented by its centerline. The centerline passes through the cross-sectional centroid of the part. The cross-sectional orientation is assumed to be normal to the centerline. Hence, the part geometry is completely speci®ed by the absolute value of the curvature of its centerline as a function of its arc-length, K(s). Mathematically, curvature is de®ned as the reciprocal of the radius of curvature [10], given as K s
d2 y=dx2 1 d2 y=dx2 2 3=2
Fig. 3. Bending process.
(1)
With the support points of boundary conditions de®ned for the part, the intrinsic representation, K(s), can be translated into the shape of the part in the Cartesian coordinates, x(s) and y(s), by solving the following simultaneous equations from differential geometry [10]: d2 x dy K s 0; 2 ds ds
d2 y dx ÿ K s 0 2 ds ds
155
(2)
Since the geometry of a part with any reasonable curvature pro®le, K(s), is to be represented, this function is stored not as any analytical function but as a data array. Each entry in this array has two elements, the ®rst being the arc-length of a point on the centerline from one end of the part, and the second being the curvature of the centerline at this point of the part. This allows a part to be described coarsely with only a few entries in the array corresponding to a few points along the centerline, or ®nely with a large number of entries, as required. 3.2. Mechanics In this paper the mechanics for two-dimensional (or inplane) sequential bending is developed for simplicity. The basic mechanics can be extended to asymmetric sequential bending. For example, the basic mechanics and the analytical approach for simple asymmetric bending of L-shaped beams are developed by Xu et al. [11]. In such a case, the part representation also has to be modi®ed to include curvature as a vector. The fundamental relations of
Fig. 4. Bending inputs.
mechanics used for developing the sequential bending model are derived from [8,12]. Fig. 3 shows the scheme of the process model. The bend inputs for any single bend in the sequence of N bends are illustrated in Fig. 4. The inputs, outputs, and parameters for the process are summarized in Table 1. The model accounts for non-¯at initial geometry, residual stresses and strain-hardening of the overlapping material. The bending analysis proceeds at two levels: local analysis and global analysis. The local analysis determines the effect of a bending moment, M, on any cross-section of the part. The effect is essentially a change in curvature of the centerline, and in the residual stress distribution across the cross-sectional area. Note that since friction at the supports is almost negligible because of rollers, there is no net axial force on the cross-sections. The bending moment is the predominant cause of deformation. The material and geometric properties of the part are described by the initial stress distribution across the cross-section at any point along the part (si) and the initial curvature at that point (Ki). For a given bending moment applied in the loaded equilibrium state on that cross-section (Mload), the
Table 1 Process inputs, parameters, and outputs Inputs
Parameters
Outputs
For each of the N bends (i) Location of the bend (left support point) (ii) Span (iii) Punch force
Material properties (i) Young's modulus (ii) Yield strength (iii) Plastic modulus (iv) Initial residual stress in part Part geometry (i) Cross-sectional shape parameters (ii) Initial curvature/contour of part Machine conditions (i) Punch radius (ii) Friction at support points
Unloaded contour or curvature Final residual stresses
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Fig. 5. Local analysis.
local analysis determines the ®nal stress distribution (sf) and the ®nal unloaded curvature after springback (Kf) of that cross-section (Fig. 5). Two important assumptions are made in the local analysis [12], plane section (or Euler±Bernoulli) assumption and uniaxial stress assumption. Three fundamental relations are used to carry out the local analysis [2]: the constitutive stress±strain relation of the material together with its nature of hardening as it unloads is described as a function of s(e). The strain±curvature relation is de®ned by the true strain e ln 1 Ky, where K is the curvature and y the distance from the neutral axis. For bends that result in low curvatures (e.g., less than about 0.05 in.ÿ1 such as in spar-forming) and where the parts have a thickness less than 1 in. (25.4 mm), the quantity Ky is small. That is, Ky 1 and, therefore, the true strain can be approximated by e Ky. Finally, the bending moment at any cross-section along the part, M(s), is related to the curvature of the centerline of that cross-section by the following integral relation across the cross-section: Z M s
H=2 ÿH=2
Z s eyw y dy
H=2
ÿH=2
s e Kyw y dy
(3)
where H is the thickness of part cross-section, and w the width of the cross-section. For a given loading moment, M, the change in curvature is determined by solving the above equation. Since the relationship is not in general an analytical expression, it has to be solved by a numerical iteration scheme. When the part is unloaded, the net moment on the section vanishes and the curvature of the beam reduces: this phenomenon is known as springback. A springback analysis is equivalent to applying a moment equal and opposite in direction to the originally applied moment. Therefore, the above three relationships are solved again to transform the loaded state to one with a net bending moment of zero on the cross-sections. This gives the unloaded curvature and the residual stresses across the cross-sections. The global analysis constructs the bending moment pro®le along the part arc-length for a particular bend. It uses the local analysis to determine the effect of this moment pro®le on the resulting unloaded curvature and residual stress. The local analysis is repeated for the entire process for each bend of the bending sequence. Note that the ®nal curvature and residual stresses of the previous bends become the initial ones of the next bend. For a three-point bending, the bending moment M is simply described as in Fig. 6.
Fig. 6. Bending moment pro®le.
4. Simulation The model described above is implemented for a uniform rectangular cross-section part. The speci®c stress±strain and strain-hardening relationships used are based on the assumption that the shapes of the true stress vs. true strain diagrams for tension and compression are identical [2]. The generalized stress±strain and the kinematic strain-hardening relations are shown in Figs. 7 and 8. The usual assumption is that the unloading and reversed loading curves are linear. Fig. 9 is a ¯ow chart of the model computations. This closely follows the procedure outlined in Section 3. The program code is developed and implemented entirely in TM MATLAB . The secant method was used as the iteration scheme to solve Eq. (3) [13]. In the implementation, the part is discretized into a ®nite number of rectangular beam elements. The curvature of the centerline at the face of each element, and the stress distribution across the element crosssection are the quantities that represent the state of the element. The state representation of all the elements that
Fig. 7. Constitutive relationship.
Fig. 8. Strain-hardening relationship.
M.Y. Wang, S. Shabeer / Journal of Materials Processing Technology 102 (2000) 153±163
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ered during springback. This implies that although two bends might have been punched so that they physically overlap, their unloaded curvature or residual stress might not interact. Also notice that bend no. 5 is executed in a segment of the part that is completely within the span of a previous bend (no. 3). This bend only produces an elastic deformation during loading and leaves no permanent plastic change in the part. However, a similar bend (no. 4) does produce a permanent change in the part because it is partly executed in a region that was unaffected by any previous bend (no. 3). The curvature pro®le produced by the multiple bends is approximately piece-wise linear. This is because a linear stress±strain relationship was assumed in the plastic zone. If a non-linear plastic constitutive relation were assumed, this pro®le would be more non-linear. It is interesting to note the reversal of stresses near the top and bottom surfaces of the part that occurs due to springback. Under the loaded condition (e.g., loaded downwards as here), the bottom surface is in tension, whilst the top surface is in compression. When released, the effect of the negative springback moment results in the observed stress reversal at the surfaces. Also note that the residual stresses do interact when the bends have been overlapped. In this example, a peak residual stress of 20±25% of the yield stress is left in the part. 5. Optimization of sequential bending Fig. 9. Flow chart of model computations.
make up the part taken together represents the state of the entire part. The implementation of the model by simulating a sequence of bends on a part with a rectangular cross-section is illustrated. The inputs to the simulation program are: (1) the material properties, (2) the beam geometry, (3) the initial part shape (as curvature) as well as initial residual stresses, and (4) the bend inputs. In the following example, a sequence of ®ve bends on a part made of aluminum (2024-T351) is simulated. Note that bend nos. 1 and 2 overlap considerably, as do bend nos. 3, 4 and 5. For simplicity, an initially straight part (i.e., zero curvature throughout its length) that has negligible initial residual stress is considered. The results of the simulation are shown in Figs. 10 and 11, where Fig. 10 shows the ®nal unloaded curvature and shape of the part after the sequential bending, and Fig. 11 shows the ®nal residual stresses induced in the part. Notice that the curvature within the bend span but close to the support points is zero. This is because the bending moments near support points are low and the material deforms only elastically. This elastic deformation is recov-
The bending model developed in Section 3 essentially solves the ``forward problem''. That is, given the initial state of the part and the bend inputs, it predicts the ®nal state. The ``inverse problem'' of synthesizing the bend inputs, given the initial state and the desired ®nal state, is not straightforward. This is due to the factors of: (1) the non-linear mechanics and the history dependence of the process, and (2) the discrete nature of the bends with limits on bend inputs. The problem of optimization of a sequential bending process is to determine suitable bend inputs (i.e., the location, span and punch force of each of N bends), given the initial state of the part, the desired curvature pro®le, and the number of bends (N) to be executed. An appropriate objective for optimization would be minimizing the error between the actual part shape formed and the desired shape. 6. Formulation In this section, the problem of optimization of sequential bending is formulated. The objective is to minimize the error between the actual part shape produced by the process and the desired part shape. The design variables in the optimization problem are the bend inputs. It is proposed that the sum of squares of the difference between the actual curvature and
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Fig. 10. Final (unloaded) curvature and shape pro®le.
the desired curvature along the length of the part be used as the objective function. The reason to consider the curvature error and not the actual shape is that the bending model mechanics deal directly with curvature. To convert to the physical space of x±y coordinates necessitates a computationally expensive solution of differential equations (Eq. (2)). Transforming back and forth between curvature and coordinate space for bending simulation and objective function evaluation, respectively, ultimately makes the optimization program too time-consuming. If the curvatures are
matched exactly, it is ensured that the shapes will also match exactly in physical space (x±y coordinates). There are two main types of constraints in the optimization problem. The ®rst type is the upper and lower limits on the process inputs. The press can exert only a particular maximum amount of force. The guides on the bending machine allow only particular maximum and minimum spans to be used for the bends. These constitute simple bounds on the design variables. The second constraint is the material limit. The resulting strains induced in the part must
Fig. 11. Final residual stress distribution.
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not exceed the fracture limit of the material. In aerospace manufacturing, spar forming involves only relatively small changes in shape and hence only low strain changes. In these cases, the strains would not be even near to the fracture limit. The optimization problem for sequential bending can be compactly formulated as an optimal control problem [14]. Notation to achieve this compact casting is de®ned as follows. Let the subscript n represent the stage after the completion of the nth bend in a sequence of N bends, with n taking on values 1, 2, . . ., N, where N is the total number of bends. Vector X1;n and matrix X2;n represent the state of the part after the nth bend. Here, X1;n represents the curvature at p speci®ed points along the centerline of the part. The size of this vector is therefore p1. Matrix X2;n represents the stress distribution across the cross-sections passing through the same set of p points. Suppose the stress distribution across each cross-section is represented by the longitudinal (axial) stress value at q different distances from the centerline. The size of X2;n is therefore pq. Further, let un be the bend input vector that transforms the state of the part from that before the nth bend to that after the bend. This vector is of size 31 and has the components of location, span, and punch force of the nth bend. Hence, the optimization of N sequential bends is mathematically described as follows: Xn fX1;n ; X2;n g
State variables : Process dynamics : Objective : Constraints :
Xn f Xnÿ1 ; un where f is the bending model
Minimize X1;N ÿ X1;spec 2 umin un umax
where X0 is the initial state, X1, spec the speci®ed ®nal curvature pro®le, and N the total number of bends. 6.1. Solution A globally optimal solution for the above problem can be found, in theory, by dynamic programming [15]. However, for reasonably long parts, the number of points along the part that adequately represent its state is not small. Hence, the dimension of the vector representing the part curvature pro®le and stress distribution is relatively large. Therefore, ®nding a global optimum by dynamic programming is computationally formidable. One way to handle this problem is to consider a ``moving horizon''. That is, instead of optimizing over all of the N bends at once, Nt (
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Considering the several dif®culties of ®nding a guaranteed global solution, a simpler approach that ensures a local solution is used in this paper. This can be effected by a conventional optimization procedure that depends on an appropriate gradient search method. An initial set of bend inputs must be provided, which could be the operator's guess. The optimization algorithm then searches locally using the bending model as a predictive tool for the objective function evaluation, until it reaches a minimum. To ensure global minimization, the optimization is repeated at different initial points. It was observed in the example presented later that for most starting points, the point of convergence with the minimum objective function value was identical. MATLABTM optimization toolbox is used to implement and numerically solve the problem. Speci®cally, a sequential quadratic programming (SQP) approach is used [16]. 6.2. Illustration A common use of the sequential brakeforming process is to make parts with a constant curvature (i.e., an arc of a circle). The optimal forming of a constant curvature part is illustrated as an example. Suppose there is an initially straight stock made of aluminum (2024-T351). A constant curvature of 0.05 in.ÿ1 is desired in a 5 in. segment of the part ranging from arc-length of 10±15 in. The strategy adopted to execute only the minimum number of bends, N, required to achieve a particular tolerance is simple. A low number of bends is ®rst used. The optimization procedure would yield an optimal bending sequence for this number of bends. If the resulting ®nal shape does not meet the tolerance requirements, the number of bends is increased one at a time and the optimization is repeated until the ®nal shape falls within the tolerance limits. This strategy is illustrated in Fig. 12. The tolerance requirement in this example is taken as 0.003 in. Upper limits were placed on span and punch force as 12 in. and 300 lb, respectively. Twenty equally spaced points in the region of interest were chosen as speci®ed points for the objective function evaluation. Thus, the sum of
Fig. 12. Finding the minimum number N.
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Table 2 Optimal sequential bending results N
Bending sequence
Start point (in.)
Span (in.)
Punch force (lb)a
Objective function
1
1
6.57
12.00
258
1.1810ÿ2
2
1 2
5.76 8.46
11.06 10.79
259 265
3.3010ÿ3
3
1 2 3
5.54 7.48 8.98
10.67 10.29 10.30
258 261 265
1.4910ÿ3
a
1 lb4.45 N.
squares of the curvature error at these 20 points was minimized. The results of the optimization program for up to three bends are presented in Table 2. Figs. 13±15 show the curvature and part shape obtained by the optimal single, two and three bend(s), respectively. In the same ®gures, the desired curvature and part shapes are shown as dotted lines. Fig. 16 shows the actual geometric deviation (in y coordinate) between desired and actual shape obtained by the optimal sequence of bends, respectively. Fig. 17 shows the iteration for the single bend case. The decrease in objective function value is relatively rapid during the ®rst few iterations, but subsequently the rate decreases.
The objective function evaluation for three bends takes approximately 20 s of CPU time in MATLAB running on a Sun Ultra workstation, and several such objective evaluations are needed for an iteration. Hence, there is a trade-off between approaching an optimum and the time needed for reaching such a point. It was observed for the example that about 8±10 iterations produce results quite close to the optimum within a reasonable time. The number of iterations for convergence depends on how close the initial bend inputs are to the optimum: if the initial guess is close to an optimum, early convergence results. Other interesting observations include the following: 1. It is clear from Fig. 16 that to achieve a tolerance of 0.003 in., three bends are adequate. If the tolerance limit were to be set higher at 0.01 in., then two bends would be suf®cient. With N1 (a single bend), as expected, the upper bound on the span is reached. 2. An interesting result was obtained when the in¯uence of order of execution of the bends, with N3, was investigated. The order of the three bends was permuted and the optimal bends were determined. However, the ®nal curvature was not affected signi®cantly. In other words, although the overlapping portion of the bends do interact, their order does not appear to be signi®cant.
Fig. 13. Optimal single bend: (Ð) actual shape; (. . .) desired shape.
M.Y. Wang, S. Shabeer / Journal of Materials Processing Technology 102 (2000) 153±163
Fig. 14. Optimal two bends: (Ð) actual shape; (. . .) desired shape.
Fig. 15. Optimal three bends: (Ð) actual shape; (. . .) desired shape.
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Fig. 16. Geometric deviations (in the y coordinate).
Fig. 17. Iteration history for a single bend.
3. The results show how geometric differences are magni®ed in the intrinsic curvature representation. Although the deviations between the obtained shape and the desired shape are small even with one bend, the curvature pro®les indicate considerable difference. 7. Conclusions A bending model and an optimization algorithm are proposed in this paper to improve dimensional accuracy and reduce rework iterations in sequential brakeforming.
Simulation of two-dimensional bending process has been achieved. The resulting shape and residual stresses of a symmetric part with arbitrary initial shape and residual stresses after a sequence of bends can be determined. This is a useful tool that allows a designer to explore different sequential bending plans. A primary objective of the bending process is to achieve a desired curvature pro®le along the part. This led to the formulation of an optimization problem that minimized the curvature error at any number of speci®ed points along the part. Constraints are placed on the bend inputs, and the problem is solved using optimization algorithm that employs local gradient-based methods. As an illustration, optimal bend sequences are synthesized for forming constant curvature parts using multiple bends. The improvement in dimensional accuracy with increasing number of bends is quantitatively demonstrated. Symmetric sequential bending has been implemented in this paper. Many actual spars may have an asymmetric crosssectional shape. Therefore, the sequential bending simulation needs to be extended for asymmetric bending. Although the primary objective of the paper is to achieve a desired curvature pro®le, an equally important concern is to minimize variation in the curvature obtained by sequential bending. Variation of the initial residual stresses in a part is known to in¯uence the curvature output variation. It seems feasible to use multi-objective optimization methods for simultaneously minimizing curvature error and curvature variation. These aspects are worthy of investigation in the future.
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