The ability of ABAQUS Standard to obtain a non-steady implicit solution to the problem of cold roll forming a channel section is investigated. A solution can be found with careful selection of parameters, but solutions are unacceptably slow for commercial use. The implicit solutions show buckling on the first pass that does not develop into an edge wave, in contrast to a published explicit solution. Faster solutions to steady rolling can be obtained using ALE models that permit convection of stress in the direction of rolling

Daniel, W. J. T.

Meehan, P. A.

University of Queensland eSpace

5th Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
Implicit finite element study of non-steady effects in cold roll forming 
W. J.T. Daniel1 and P.A. Meehan 
 
1School of Engineering, University of Queensland, St Lucia Campus, Australia, 4072.  
 
Abstract: The ability of ABAQUS Standard to obtain a non-steady implicit solution to the problem of 
cold roll forming a channel section is investigated.  A solution can be found with careful 
selection of parameters, but solutions are unacceptably slow for commercial use.  The implicit 
solutions show buckling on the first pass that does not develop into an edge wave, in contrast 
to a published explicit solution.  Faster solutions to steady rolling can be obtained using ALE 
models that permit convection of stress in the direction of rolling.   
Keywords: roll forming, non-steady, finite element method 
1 Introduction 
Roll forming is a process whereby an initially flat strip of metal is formed into a tube or structural 
section by passing through a series of rolls.  Profiles of rolls have been estimated by using testing and 
simplified analyses.  For instance, analysis of the process by assuming the deformed shape of the 
sheet and predicting the strains corresponding to this shape, is available commercially [1,2] (COPRA 
and PROFIL).  If too much deformation is attempted in any one set of rolls, problems can result such 
as edge waves associated with local plastic buckling.  It is of interest to predict these situations.  Finite 
strip methods have been used [2] with limited accuracy.  Finite element analysis of roll forming 
processes has been performed by a number of researchers, and analysis is now available 
commercially.  An analysis using a single layer of assumed-strain solid elements is available in 
COPRA FEA [1].  COPRA automatically prepares data for the MSC/MARC solver.  The approach of 
COPRA FEA minimizes computation by modeling a limited length of the strip rolled, which is held in 
place at two roll-stands. PROFIL offers an interface to ABAQUS explicit [2].  Both explicit and implicit 
finite element approaches to modeling roll forming have been taken by researchers [3-7].  Analyses 
have been mainly Lagrangian.  Lagrangian analysis is a slow process, not only due to tracking the 
history of deformation, but also due to ever-changing contact between the strip being formed and the 
rolls. Boman and Ponthot [8,9] have published an arbitrary Euler-Lagrange (ALE) approach for 
predicting steady roll forming, which allows material to flow through the mesh in the rolling direction.  
This avoids revising the contact conditions as much as in a purely Lagrangian analysis, as the mesh 
does not move in the direction of rolling.  However, the approach in [8,9] does not simulate the correct 
history of loading, replacing it with a series of clamping deformations, occurring as rolls are 
successively brought into contact.  A difficulty of ALE analysis is that finite element stress fields in 
elements do not provide a good estimate of the local gradient of stress, which is needed to calculate 
how stress diffuses through the mesh.  Boman and Ponthot avoid this by means of a finite difference 
estimate comparing stresses at neighbouring Gauss points of adjacent elements.  Recently, a 
Lagrangian explicit approach was taken by Tehrani et al [10] to modeling rolling the channel section 
investigated experimentally by Bhattacharya [11].  They found edge waves if the angle on the first set 
of rolls exceeded 40 degrees.   
 
The present paper investigates this same problem using implicit finite element analysis with ABAQUS.  
It is found that this angle does indeed lead to local plastic buckling of the flange being formed, as the 
strip rolled passes through the second stand of rolls for the first time.  This does not necessarily 
produce an edge wave however, as a simulation assuming symmetry predicts that the buckled section 
passes through the rolls and the process stabilizes, which confirms observations in practice by 
Bhattacharya [11].  The on-going oscillations in strain reported by Tehrani et al [10] may be more to do 
with spurious inertia effects; a result of mass-scaling in the explicit analysis.  The implicit analysis 
described in this paper does however predict some oscillation in contact forces on the rolls.  This is 
associated with a cyclic variation in the extent of plastic deformation around each roll stand.  An 
analysis without assuming symmetry shows more sign of oscillation in residual strains along the 
flanges of the formed section.  Misalignment can also be modelled with such an analysis. 
 
  
An alternative approach to finding a faster solution to the steady roll forming problem may be to use a 
relaxation method to correct the deformed shape, starting from an assumed profile of the strip rolled.  
This is easier said than done, however.  A difficulty with conventional elasto-plastic finite element 
analysis, is that any plastic deformation that is implied in an initial profile is accepted by such an 
algorithm - the solution depends on the initial state, even with use of a tangent elasto-plastic modulus.  
On the other hand, the formulation of a rigid-plastic finite element analysis does explicitly minimize 
plastic work, and a rigid-plastic analysis can be used as a relaxation method.  Strictly, nodal velocities 
are found that minimize the rate of plastic work. These can be used to iteratively revise the shape of 
the deformed sheet being rolled. This approach has been applied to other metal forming problems 
since 1982 [12-17], and has been applied to roll forming by Hong et al [18].  Hong et al approach the 
problem in two stages, solving a generalized plane strain problem to determine the deformed cross-
section at each roll stand, and then using these solutions as boundary conditions on a 3D rigid-plastic 
model of the sheet between roll stands.  Some allowance for elasticity is possible in a rigid-plastic 
analysis.  Strains found from updated nodal coordinates can be separated into elastic and plastic 
strains, using the radial-return algorithm. Where stresses are below yield, the rigid-plastic response 
can be replaced with an elastic response [19].  If this approach is combined with an ALE treatment 
where the material flows through the mesh, then a more efficient numerical method is possible.  A 
relaxation approach has a fundamental problem, that a thin sheet is much more flexible out-of-plane.  
Hence the deformation assumed initially tends to be correcte d by changing out-of-plane deflections, 
but in-plane strains  like the edge strain that could cause plastic buckling, are more reluctant to 
change from their initial values.  If initial strains are too large, they may be relieved by spurious 
buckling, sending the solution on an incorrect path.  Hence it is best if the initial condition of a 
relaxation solution underestimates the membrane strains, if not starting from zero strains. 
 
2 ABAQUS models of roll forming 
Many studies have used the experimental data for roll-forming a channel section of reference 11, as a 
simple case to validate the methods of analysis.  The geometry of the rolls for bending the sheet 
through a 40 degree angle is shown in Figure 1.  To investigate the transient behaviour as a sheet of 
mild steel is fed through first a 40 degree roll and then one or more 60 degree rolls, ABAQUS models 
have been set up that actually feed the strip through the rolls, with the rolls driving the strip.  However, 
the rolls cannot drive the strip through until they have first made contact.  To avoid this difficulty, two 
approaches have been taken.  Firstly, for each stand, the lower roll of Figure 1 has been moved up to 
clamp onto the strip, in an initial step before rolling commences.  Secondly, the strip has been pulled 
into the first set of already rotating rolls by imposing a velocity on its leading edge for the time needed 
for it to make adequate contact.  The first of these approaches is taken in the following model that 
exploits symmetry and the second in the following full model. 
 
 
 
Figure 1  Geometry of the rolls (400 case) 
 
Rolls were modeled as rigid bodies, and the strip being rolled was modeled with shell elements.  
Contact modeling required some care.  ABAQUS provides a number of options for contact in implicit 
analysis.  The augmented Lagrangian method of enforcing a normality constraint is available, but was 
  
found to have convergence problems applied to the roll forming problem.  The alternative is a penalty 
method, which can have a linear or an exponential variation of normal contact force.  The exponential 
approach, which provides soft contact initially, was found to work much better.  Any of the approaches 
are likely to have difficulty if only one node is in contact, or if the normal to the contact surface rotates 
more than by a very small angle in one increment, or if the inter-penetration increases too much in one 
increment.  The algorithm gives up on its contact search rather easily in fact.  The contact modeling 
constrained what time interval could be used for simulation.  The automatic adjustment of timestep 
size in the program was found to react too late to convergence difficulties  to solve them, and needed 
assistance from the user, to stop it increasing the timestep too much.  When simulating rolling, a 
timestep limit was set that implied that each element was only allowed to move about 1/5  of its length 
through the rolls in each update.  On the other hand, simulating clamping the rolls onto the strip 
permits use of much larger timesteps, as contact conditions do not continually change.  The automatic 
timestepping of ABAQUS works well on the clamping problem. 
 
To improve the robustness of the contact calculations, by further softening the contact situation, one of 
each pair of rolls was mounted on a spring.  This also reflects reality, as the actual roll and its 
mounting is not perfectly rigid.  Coulomb friction with a coefficient of friction of 0.2 was used in the 
simulations. 
 
3 Simulations conducted with ABAQUS 
 
Tehrani et al correctly point out that the passage through the first roll stand stretches the flanges of a 
channel being formed more than the web, causing the strip to bend down until it encounters the 
second roll stand, when it is bent up again.  The encounter with the second roll stand can lead to 
plastic buckling of the flange of the channel.  A simulation with a half model and three roll stands of 40, 
60 and 60 degree angles confirms that plastic buckling does indeed occur.  The buckled section 
continues to move through the rolls, as shown in the screen snapshot of Figure 2.  The plots of 
membrane edge strain of Figure 3, obtained from a simulation with two 40 degree roll stands, confirm 
this.  Figure 3 shows the edge strain settling down to a steady value, with no evidence of an edge 
wave.  This is in conflict with Tehrani et al, who show large oscillations in strain developing after the 
first roll.  A possible explanation of the conflict is that the mass scaling used by Tehrani, while not 
causing the total kinetic energy to be a very large fraction of the energy, could have been enough to 
set off spurious dynamics.  Explicit analysis is attractive  as the contact problem becomes much more 
robust.  The present authors attempts at explicit analysis were found to be unsatisfactory due to 
similar mass-scaling problems, as it does not take much false mass added to a thin sheet to excite 
out-of-plane modes of vibration. 
 
 
 
 
Figure 2    Deformed shape of a strip rolled  for a half model of 3 stands (400, 400 and 600). 
  
 
 
Figure 3.  Distribution of strain along the edge of the strip rolled at four times, showing progress of 
buckled section. 
 
An implicit simulation avoiding use of symmetry used the mesh shown in a deformed screen snapshot 
in Figure 4.  The strain contours on the mesh reveal oscillation of edge strain.  These could develop 
into an edge wave, if it became more severe.  This model can represent the effect of misalignment.  
For instance, a sheet fed into the rolls displaced to one side does emerge twisted.  The models also 
reveal the contact pressure distributions on the rolls, and hence give a guide to the prediction of roll 
wear. 
 
 
 
Figure 4.  Full model of a channel section being rolled showing oscillation of edge strain. 
 
4 Simulation by relaxation 
 
Subsequent to the ABAQUS study, the problem of roll-forming a channel has also been investigated 
by finite element relaxation methods.  An ALE approach is used: that is, the membrane stresses and 
  
bending moments are corrected each iteration for convection of stress through the mesh.  While 
ABAQUS can do some implicit ALE simulations, the program only applies inflow/outflow boundary 
conditions for explicit analysis with solid elements. Hence the authors own software was used instead. 
To minimize computation a membrane model enhanced with bending terms based on the work of Yoo 
[16] was used.  This model ignores membrane-bending coupling, and uses a preset elasto-plastic 
tangent modulus to describe bending.  It has no rotation degrees of freedom, using a central 
difference of lateral displacements to estimate curvature. This model reduces computational time by 
an order of magnitude, but the accuracy suffers, as shown by the comparison of edge strain with 
ABAQUS Standard for 20 and 40 degree rolls, given in Figure 5.  The mesh is a half model with 8 
elements transversely by 108 elements longitudinally.  The model starts wit h a guess of the deformed 
shape, of the sheet rolled, obtained by fitting a cubic transition in the direction of rolling, between two 
roll profiles, and a quadratic transition to the first roll-stand.  It is first run with elastic membrane 
behaviour, correcting the error in equilibrium for 20 iterations.  This obtains a much more realistic 
deformed shape of the sheet rolled, tending to lower the strains assumed initially.  The material is then 
switched to rigid-plastic membrane behaviour at any Gauss point that exceeds yield, for further 
iterations.  The resulting strains after 40 iterations are qualitatively similar to those of ABAQUS but 
stray too far negatively.  This could be a problem with contact modeling, but also suggests that real 
shell elements, or assumed-strain solid elements, while less efficient, may be needed to ge t more 
acceptable answers.  The ABAQUS solution may also be in error due to its soft modeling of contact.  
The relaxation solution, if fully converged, should lead to zero inter-penetration, as it uses constraint 
equations for normal contact.  The relaxation solution ignores friction.  A problem of the relaxation 
solution that has been observed is local buckling of the mesh at the rolls, in an unphysical manner. 
This can occur if the step size is too large. 
 
Figure 5.  Comparison of transient implicit ABAQUS solution with a rigid-plastic ALE relaxation 
solution. 
3 Conclusions and recommendations 
There are a number of computational alternatives to modelling cold roll forming of a thin sheet.  Both 
implicit and explicit approaches are available commercially, but explicit analysis can easily suffer from 
spurious inertia effects, and implicit analysis with ever-changing contact boundary conditions is very 
slow, and not robust.  Implicit solutions obtained with ABAQUS revealed that non-steady effects in roll 
forming are mainly associated with feeding a sheet through the rolls the first time.  Buckling that 
occurs on the first pass through the rolls may not continue once steady conditions of rolling are 
achieved.  If only the steady solution is of interest, an ALE approach is attractive, but trials with a 
bending-enhanced membrane model show that converging successfully to the steady state is tricky 
  
and requires a very good initial guess of the deformed shape.  While the stressing of a thin sheet is 
mainly due to membrane stress, a proper description of the nonlinear stress variation through the 
sheet thickness that accounts for both membrane and bending strains is needed for good accuracy. 
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Implicit finite element study of non-steady effects in cold roll forming

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Implicit finite element study of non-steady effects in cold roll forming

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