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Structural analysis and optimisation of press brakes
Pedro G. Coelhoa,*, Luı&s O. Fariab, Joa˜o B. Cardosoa
aDepartment of Mechanical and Industrial Engineering, New University of Lisbon,
Caparica, Portugal
bIDMEC, Instituto Superior Te&cnico,
Lisboa, Portugal
Received 10 November 2004; accepted 20 January 2005
Available online 23 March 2005
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: B Press B D 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
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
B 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
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.
International Journal of Machine Tools & Manufacture 45 (–1460
/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.
* Corresponding author. Tel.: C351 ; fax: C351 .
E-mail address:
(P.G. Coelho).
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:
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.
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:
ZbCj; where bZK
Fig. 1. Influence of ram and bed deflection on the angular precision of the
workpiece. (a) The ‘boat belly’ effect. (b) Uniform bending angle.
Fig. 2. Beam model. (a) Upper and Lower beam model. (b) Composite
lower table.
1452 P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–1460
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:
xZL=2KdKbjC
q&LK2aÞ
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:
dxdxCC1xCC2 (6)
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,
using Eq. (2):
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).
P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–
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.
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:
u&pÞZmax
p&xÞKmin
p&xÞ; D0 Z x2R = a%x%
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 &xÞZwu&xÞKwl&xÞKd; dZwujxZaKwljxZa (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:
ffiffiffiffiffiffi
&zKjÞ4C
&1K2jÞ&j3Kz3Þ
&1K2jÞK4&1CrI Þ
&zKjÞ
fs& rA; jÞ (13)
fs&rA;z;jÞZK
&zKjÞ2C
&1K2jÞ&zKjÞ (14)
24&1CnÞ
fs& rA; jÞ
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.
Fig. 4. Optimal Shape for Lower beam (units: mm). (a) Beam model. (b)
Height variation and shape.
Fig. 5. Patented solution for Lower beam.
1454 P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–1460
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, si
C, 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
ao such that,
u&ps0;a0 ÞZmin
u&aÞ;
u&aÞZmax
p&s; xÞKmin
p&s; xÞ
D1 Z s2Rn=sK
D2 Z a2R = 0%a%
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
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 u&ps0;a0 Þ for
Upper beam dimensions of 0 mm3 and
bending force per unit length q of 180 N/mm. Fig. 6
shows the penetration curves for each case in Table 1.
Optimal values for bending and shear contributions separately and together
d (mm) rI rA u&ao Þ
klZ0.33 klZ0.85
Bending only 0 2 – – 0.0080
Shear only 471 – 0.74 1.88 0.0085
BendingCshear 163 2.87 0.14 0.40 0.0129
Fig. 6. Penetration curves for each case presented in table 1 (units: mm). (a)
Bending only. (b) Shear only. (c) BendingCShear.
P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–
The following comments on the above results are in
– 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
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
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
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.
Optimal solutions for each combination of geometries (units: mm)
Geometries
combination
s1 s2 s3 s4 s5 s6 s7 s8 u&ao Þ
UB1/LB1 5 70 – – – 290 0.0139
8 70 – – – 400 0.0168
0 70 – – – 0 0.0183
UB1/LB2 1379 90 – – 70 130 600 436 0.0256
UB2/LB1 – – 950 70 – – – 566 0.0282
UB2/LB2 – – – – 47 130 600 576 0.0308
1456 P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–1460
The bending error increases 8% for this constrained
solution in relation to the error for the unconstrained
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
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 u&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
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.
Fig. 8. Penetration curves with Shear and Bending contributions (units:
mm). (a) UB1/LB1. (b) UB1/LB2.
P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–
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:
D3 Z v2Rn=vK
S(v,x) is added to penetration given by Eq. (9) in the
following way
p &xÞZ½wu&xÞKwl&xÞKdCS&v; xÞKSjxZa (19)
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,
u&pv0;s0;a0 ÞZmin
u&s0;aÞ (20)
The previous geometry combinations with optimal
dimensions UB1/LB1 and UB2/LB2 were selected to
illustrate this sequential optimisation. Assuming mZ9 and
CZ0.4 mm in Eq. (18) one obtains, respectively,
0.0129 and 0.0253 mm for u&pv0;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 u&ps0;a0 Þ
values calculated are an order of magnitude higher for this
conventional solution than for the corresponding sandwich
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. 9. Comparison between analytical (Timoshenko theory) and Finite
Element numerical results for UB2/LB2 using maximum bending length.
(a) Upper beam. (b) Lower beam.
Fig. 10. Modelling shimming with a Spline function.
1458 P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–1460
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
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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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 u&ao Þ
UB1/LB1 0 K0.023 K0.045 K0.064 K0.079 K0.092 K0.100 K0.105 K0.107 0.0025
UB2/LB2 0 K0.065 K0.125 K0.168 K0.202 K0.228 K0.244 K0.256 K0.259 0.0064
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 u&ao Þ
UB1/LB1 0 130 – – – 2
UB2/LB2 – – – – 130 130 600 8
P.G. Coelho et al. / International Journal of Machine Tools & Manufacture 45 (–
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