FINITE ELEMENT SIMULATION OF ROLL FORMING OF HIGH STRENGTH STEEL

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1 FINITE ELEMENT SIMULATION OF ROLL FORMING OF HIGH STRENGTH STEEL Author: Joakim Lempiäinen Roll forming of high strength sheet steel, using explicit FE simulations has been analyzed in terms of the state of strain and stress, during forming and at relaxed state. A model with rotating tools and friction has been developed from a previous model with absence of friction and rotation tools. Various simulation parameters affect on the roll forming process and achieved results has been analysed. Specific behaviour such as springback and flange strain has also been studied. Abstract A roll forming model with friction and rotating tools has been developed and implemented in FE software ABAQUS to analyze the presence of friction in the process. The simulations have been performed using explicit FE calculations. Roll forming simulations using friction and rotating tools are more demanding in terms of computational cost compared to the model used without friction. Simulation time for a stable simulation using the friction roll forming model is ~33h whilst using the model without friction the simulation time is ~11h. Concluding that simulation with friction consumes three times the amount of time. Figure 1: Three set of tools in the roll forming process Mass scaling is indispensability in terms of computational cost has been varied between ~500 times the mass, up to ~6000 times the mass. Mass scaling also affects stability and accuracy of the results, and is recommended to be set as low as possible. Results has shown that little deviance exist between the roll forming model with normal friction µ=0.12, versus no friction. As little as ~0.15 difference in springback and negligible differences in residual true strain through the thickness of the sheet is present. Higher friction µ=0.65 shows increased flange strain and 1.2 decrease in springback. Stability is recommended not to exceed ~12% in stability quotient. This condition puts restrictions on the time increment and contact stiffness used. Contact stiffness affects simulation stability significantly with use of the friction roll forming model (see table below). High contact stiffness is recommended for a stable simulation which also minimizes overclosure between the sheet and the tools. Contact stiffness (N/m 3 ) (N/m 3 ) (N/m 3 ) Stability quotient 38% 32% 10% Table 1: Contact stiffness versus stability quotient for friction roll forming model. Corrosion and Metals Research Institute Drottning Kristinas väg 48, SE Stockholm, Sweden Tel. +46 (0) Fax +46 (0) info@kimab.com

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4 Table of Contents 1. INTRODUCTION THEORY ULTRA HIGH STRENGTH & HIGH STRENGTH STEEL FEM Solver methods Accuracy of explicit direct integration Time increment Mass scaling Elements Shear locking Hourglass affect ROLL FORMING General Forming setup Flange strains Springback FE SIMULATIONS THE MODELS Geometry The sheet geometry The tool geometry Assembly Contact interaction Boundary conditions Coordinate system Tool rotation Translation roll forming model Friction roll forming model Constraints Elements Mesh Friction Materials Contact stiffness Steps & damping RESULTS SIMULATION PARAMETERS Simulation cluster Mass scaling Contact stiffness Contact stiffness influence on simulation Mass scaling parameter influence on accuracy Shear locking Hourglass affect and hourglass control Mesh density evaluation FORMING PARAMETERS Gap size comparison Residual strain in centre of bend Residual stress in centre of bend Flange strain Comparison of residual true strain in centre of bend for different models and approximated Thickness of the sheet after roll forming Reaction force Sheet shape and contact pattern

5 4.2.9 Final sheet geometry Springback Accuracy in springback calculations DISCUSSION CONCLUSIONS AND ACHIEVEMENTS FUTURE WORK ACKNOWLEDGEMENT REFERENCES...93 APPENDIX A MATERIAL DATA...94 APPENDIX B ADDITIONAL RESULTS...95 B-1 SIMULATION TIMES...95 B-2 CONTACT STIFFNESS INFLUENCE ON SIMULATION...96 B-3 MASS SCALING INFLUENCE ON SIMULATION...99 B-4 SHEET SHAPE B-5 FINAL SHEET GEOMETRY B-6 VELOCITY DIFFERENCES

6 1. Introduction The roll forming process is by far the most cost efficient operation compared to stretch bending and press bending due to the high forming speed. Another advantage is the ability to vary the length of the sheet and therefore achieving different products without any effort needed to adjust machinery as well as the possibility to include secondary operations in the roll forming process. One must add that the initial machine cost is higher than traditional stretch and press bending at least for the time being. Advanced forming of high strength steel and ultra high strength steel has become possible with the roll forming process. The use of these roll formed products in the industry makes it possible to minimize weight due to the use of less material and reduce production time without deteriorating the structural strength. Not forgotten that the roll forming process offers production of profiles using high strength steel not possible with other means of forming as well as achieving close tolerances. Roll forming is yet a cutting edge area where little literature is available for industries as well as for researchers. The industries are used to applying recommendations and hand book solutions in their production process. There are yet little such aids available for roll forming. This thesis will analyze the roll forming process in terms of the specific strains and tensions appearing in the material through the process as well as comparing it to traditional press bending using finite element simulations. Finite element simulations enable fast and economic evaluations of roll forming processes, which can reduce the amount of experiments. Furthermore behaviours such as springback will be examined and finally an optimized platform for finite element calculations in roll forming analysis will be developed. Figure 1-1: A section of three forming steps during a roll forming process The starting position for a finite element platform was a model based on translational motion of the tools in the absence of friction and rotating tools. As a step closer to the actual real life process, the introduction of friction and rotating tools into the model has been performed and compared to results from the translation roll forming model using solid elements

7 2. Theory 2.1 Ultra high strength & high strength steel Ultra high strength steel and high strength steel can be defined several ways, one way is by a metallurgical designation, a second way is by material strength and the third way is by the mechanical properties of the material. With metallurgical designations one means the chemical compositions and phases present in the material. Mechanical properties are such as elongation to failure and work hardening definition constants while the material strength relates to yield and tensile strength. Since this thesis delimits itself to analysing mechanical properties and strength only, metallurgical designations and definitions will not be investigated nor commented. Due to this, the steel grades will be addressed as UHSS (ultra high strength steel) and HSS (high strength steel). High strength steel is defined as having yield strength between MPa and tensile strength between MPa, while ultra high strength steels have yield strength greater than 550 MPa and tensile strength greater than 700 MPa. These limits suggests discontinuity between steel grades, yet observing figure 1 below one can see that steel grades propagates in a continuous way and such firm definitions are thereby questionable.[3] Figure 2-1: Mapping of UHSS and HSS - 9 -

8 2.2 FEM Solver methods Finite element analysis software mainly offers two analysis modules, the implicit and the explicit solver. Provided that one is acquainted with the theory behind FE analysis the following chapter will provide a brief comparison of the two methods and also accentuate the choice of solver for the problem at hand in this thesis. Given a known distribution of load through time in a known space, one needs to calculate the response to the acting degrees of freedom. This means that velocities, accelerations and displacements need to be resolved. There are two major approaches to this, modal method and direct integration method. The modal solving method solves the problem by introducing an alternative set of degrees of freedom. Solving those as a function of time and then transforming them back to the original set of degrees of freedom. Direct integration on the other hand solves the problem by using the same set of degrees of freedom and solving them using a step-by-step integration with time increments t. This given the advantage of not being forced to change form of the dynamic equations. [1] Both solvers available in ABAQUS FE software uses direct integration, i.e. equilibrium is defined through applied internal and external forces and nodal accelerations (see below). Mu = P I (Equation 1) M = mass matrix u = nodal accelerations P = applied forces I = internal forces The solvers use the same technique and solve for nodal accelerations by calculating the internal forces for the element. The methods go apart by the way of withholding the accelerations. The implicit method solves a number of linear equations through Newton s iterative solution method and seeks to satisfy equilibrium at the end of each time increment i.e. equilibrium at t n = (t n-1 + t) while at the same time it calculates the displacements. The explicit method uses the kinematic conditions at the previous time increment to calculate the kinematic conditions at the next increment using a lumped mass matrix which for obvious reasons is more time efficient. Added to that not solving any equations simultaneously reduces computational cost. [8]

9 The accelerations are then integrated through time and the changes of velocities are calculated assuming constant accelerations. The change in velocity is then added to the velocity from the previous time increment to determine the resulting velocity at the actual increment. 1 ( M ) ( P I ( t ) u (Equation 2) = ) ( t) u ( t ( t + t ) t = u t + ( t+ ) ( t ) t ( t) ) u ( t ) (Equation 3) The velocities in their turn are then integrated through time to determine the displacement in end of the actual increment i.e. t 1 = t 0 + t. u ( t+ t) = u + t u ( t) ( t+ t ) t ( t+ ) 2 (Equation 4) The term explicit refers to that the state at the end of increment t n exclusively depends on the state at the beginning of that same increment. As noted above the explicit method uses constant accelerations through calculations which presupposes that the increments t are computed to minimize errors, i.e. small enough to justify the use of constant acceleration over that period of time. [1] Accuracy of explicit direct integration To evaluate the accuracy of explicit direct integration and error growth between steps in the calculation one should examine the central difference equations: 1 { } { D} { D} ( ) D = n n n (Equation 5) t 1 { D } { D} { D} { D} ( ) n = 2 n+ 1 2 n + n 1 (Equation 6) t Taylor expansion of { D } n and { } n D yields: 2 3 t t n n (Equation 7) 2! 3! { D} = { D} + t{ D} + { D } + { D }... n+ 1 n n t t n n n (Equation 8) 2! 3! { D} = { D} t{ D} + { D } { D }... n 1 n

10 From the Taylor expansions one can describe equation 5 and equation 6. Since the terms containing terms t larger than the power of two will vanish and therefore the error will be proportional to t 2. In other words { D } has a second order accuracy. Decreasing the step time by half hence reduces the error by three quarters. [1] Time increment Explicit methods are conditionally stable while the implicit method is unconditionally stable. This refers to calculations going unstable by assuming constant accelerations provoking a large error forcing the FE solver to increase the number of iterations to find an acceptable solution within the ratio of error. Since explicit method uses information at time t to calculate the output at t+ t, it is crucial that the amount of time t is carefully chosen in order to achieve an accurate result. Introducing the term stability limit and transforming it to the critical time increment gives us a value for t which is the maximum amount of time accepted in order to obtain a good result. The critical value for t is expressed in terms of the highest frequency in the system. Since it is a very large issue to determine this by looking at the whole system since the stiffness 2 matrix for the system has to be set up and the eigienvalues ω for equation 11 has to be solved. Therefore one limits to evaluate the natural frequency for one single element. The natural frequency of one single element is always lower than for the whole system making the calculations conservative. For a system with no damping the critical time increment t, 2 t (Equation 9) ω max 2π ω max = 2πf = (Equation 10) T min Corresponds to the highest natural frequency from the equation below which describes a free undamped vibration, 2 [ ] [ M ]{ ) D} = { 0} ( K ω (Equation 11) Whereas for a system with damping the critical time increment t, 2 2 t ( 1 ξ ξ ) (Equation 12) ω max ξ = damping ratio As easily realized equation 9 is more conservative than equation 10 when damping ratio differ from zero. [8]

11 ωmax is now to be stipulated and by solving equation 11 for a simple system, for example a one element bar of mass m (see below). u 1 u 2 x L Figure 2-2: Bar element with two degrees of freedom 2 An undamped free vibration is described by ([ ] [ M ]{ ) D} = { 0} K ω. For the system above provided the form functions N 1, N 2 and bar density ρ, yields the following: L x x N (Equation 13) L L [ ] = d dx 1 L 1 L [ B] = [ N ] = (Equation 14) L T [ K ] = [ B] E[ B] Adx = EA L (Equation 15) L 2 1 = ρ (Equation 16) T ρal [ M ] [ N] [ N ] Adx = ρal [ M ] lumped = 2 1 (Equation 17) 2 [ ] [ M ]{ ) D} = { 0} ( K ω EA 1 L ρal 2 1 u1 0 ω = (Equation 18) u

12 Two eigenvalues are then calculated through two modes: Mode 1: ω 1 = 0, u u =, rigid body translation. (Equation 19) 1 Mode 2: EA ω 2 = ρal, u u = 1 (Equation 20) Using the lumped mass matrix in eigenvalue calculations one get: 2 [ K ] [ M ] ){ D} = { 0} ( Mode 1: ω lumped EA 1 L ρal 1 0 u1 0 ω = (Equation 21) u 2 0 ω 1(lumped) = 0, u u =, rigid body translation. (Equation 22) 1 Mode 2: EA ω 2(lumped) = 2 2 ρal, u u = 1 (Equation 23) Since FE explicit method uses a lumped mass matrix ω max becomes: EA ω max = ω 2(lumped) = 2 2 ρal the material. 2 L E ρ where E c = is the speed of the sound wave in ρ This condition is called the CFL condition postulated by Courant, Friedrichs and Lewy. Note especially ω 2( lumped ) < ω 2 resulting in a smaller ω max which according to equation 7 increases the critical time increment t which accelerates the solution time. Note also that L is the critical element length, in FE modelling this is the limiting length throughout the total mesh of the model i.e. the element with the shortest side decides the critical time increment. This emphasizes the precision needed when establishing the mesh. Introduction of plasticity and yielding into the model does not invoke on the critical time increment. Since ω max remains constant in elastic regions and will not increase during plasticity and thus one does not have to take it in consider when calculating t. [1]

13 2.2.2 Mass scaling As a result of the fact that computational cost highly depends on the critical time increment it is crucial to partly use the optimal time increment as well as having a good mesh throughout the model. Apart from saving calculation time and accuracy through the above actions one can mass scale the model in order to increase the critical time increment. Mass scaling involves manipulating the density of the model in order to maximize the 2 2 E expression t where ω max =. ω L ρ max One immediately realizes that ω max decreases with an increased density which in turn increases the minimum critical time increment. Therefore to increase critical time increment and thereby decreasing calculation time one increases the density by a factor whose amplitude is chosen according to minimize the following affects: Solution stability Abnormal dynamic affects Deformation speed vs. sound wave speed Solution stability can be measured by examining the artificial strain energy against the total internal energy. If the kinetic energy is greater than 10% of the total energy it is an indication that the solution is not fully stable and calculation errors will increase. Dynamic affects are harder to examine analytically, one has to observe FE calculation results and by using the viewer tool and by analyzing values for displacements to ascertain if there is a great deal of dynamic affects as buckling or abnormal behaviour in the solution not likely to be accurate. This measure requires a great deal of experience in order to evaluate complicated dynamic course of events. E Since wave speed is calculated by c = the wave speed decreases significantly if the ρ density is increased. Therefore it can occur that the deformation speed where t invokes a great deal grows to the massive extent that it exceeds the wave speed by several factors causing the solution going unstable since the deformation occurs before the deformation wave reaches the element which is impossible and solution will collapse.[8] Elements Provided basic knowledge in FE-calculus this chapter will provide additional theory concerning the elements used in this thesis. There are basically five aspects to an element which distinguish them from each other whereas a few of them will be discussed further:

14 Element family Degrees of freedom Number of nodes Element behaviour Integration method One always has to analyze the interaction and application in order to choose an appropriate element. Due to the fact that roll forming of HSS and UHSS sheets are to be examined there are mainly two element families of interest, disregarding special use elements as springs and dashpots. These two element families are continuum elements and shell elements. During this thesis, solid elements have been used. Degrees of freedom are calculated at the nodes of the element and there after extrapolated to the location where results are required. If elements posses nodes only at the corners, FEsolvers uses a linear interpolation in each direction and are therefore called linear elements, or first order elements. Elements with mid-side nodes on the other hand, use second order interpolation and are therefore called quadratic or second order elements. The element behaviour is dominated by two specific descriptions, the Lagrangian and the Eulerian theory. [8] Lagrangian theory describes displacement as movement of material assembled to an element moving through space in time whilst Eulerian theory describes movement as loose material moving through space in time consisting of connected elements assembled to a specific point in space. Figure 2-3: Lagrangian and Eulerian theory of deformation The strength of the Lagrangian theory is its ability to easy tracking the movement of free surfaces as well as following material interfaces of models with different material definitions. The weakness of the Lagrangian theory is its inability to compute large distortions which leads to computational errors due to excessive distortions in the model. The Eulerian theory offers better description of distortions, yet lacking a good resolution of material flow and interfaces between material definitions. [10] There is also the possibility to use both descriptions combined ABAQUS FE solver used in this thesis uses Lagrangian theory or a combination of both using a routine called adaptive

15 meshing which combines both theories and evaluates the mesh after every time increment and adapts the mesh depending on the behaviour of the model. [7] Figure 2-4: ALE description of deformation As exemplified in figure 3 the mesh can move with the continuum, be held fixed or move an arbitrary motion to enhance and improve distortion control and material boundaries with improved resolution. [10] Each element consists of a number of integration points where the material response is calculated. Each element has a different configuration of integration points. Full and reduced integration can be used, where reduced integration commonly uses one integration point per element. If results are wanted between integration points, results are interpolated between integration points. ABAQUS uses Gaussian quadrature and Simpsons rule for integration which both is methods for resolving an integral by summation. [8] Shear locking Fully integrated linear elements suffer from shear locking behaviour. This behaviour increases the element stiffness, which prevents the element from deform in a correct way. Figure 2-5: Linear(left) and quadratic(right) full integrated solid elements subjected to bending Observing the figure 4 above which illustrates a linear and a quadratic fully integrated solid element subjected to a bending moment. As the linear element sides lacks the ability to curve due to the absence of mid-side nodes it will shape as a stretching motion, resulting in straight element sides but change in length of upper and lower fibre. Observing the dotted lines that intersect at each integration point in the linear element, one can see that the lines have changed in length as well as the angle between them has changed from

16 the initial 90. This indicates that shear stress is present commonly called parasitic shear, which conflict with the theory for bending where all cross sections during bending experience zero shear stress.[8][10] δ 1 δ 2 δ 1 > δ 2 Figure 2-6: Comparison in deflection between linear(right) and quadratic(left) elements subjected to bending Hourglass affect Consider a first order reduced integration solid element subjected to a bending moment. Due to the fact that it only possesses one integration point situated in the middle of the element, it can cause an unwanted affect called hourglassing due to the reduced amount of integration points. Since only one integration point is present, during bending illustrated in figure 6 it will experience no deformation of the dotted lines nor will the angles between them change from the initial 90. This will cause all the strains and stresses in the element to be assigned the value of zero. The element will offer no resistance in terms of stiffness during this type of deformation due to the zero deformation energy. This causes a large problem especially if this phenomenon propagates through the structure causing large errors in the results. Figure 2-7: Solid element subjected to bending This phenomenon will if it is propagated in the structure apart from producing incorrect results also distort the mesh net excessively. If first order reduced integration elements have to be used, this problem can be minimized by creating a very fine mesh and especially through the thickness of the sheet where it is preferred that a minimum of four elements are used. ABAQUS explicit analysis used in this thesis work uses hourglass control during simulations with linear solid reduced integration elements, nevertheless the problem can propagate before it is relieved and in order to find out weather hourglassing has been a problem during analysis one can observe the artificial strain energy in terms of total strain energy. The artificial strain energy is the energy ABAQUS builds up to prevent uncontrolled deformation and so it gives the critical elements a virtual stiffness. If the artificial strain energy is greater than approximately 5% of the total strain energy, hourglassing could be a problem and should be further analysed by changing mesh and elements to see weather results correlate.[8][10]

17 2.3 Roll forming General The roll forming process consists of several forming steps where the number of steps depends on the shape of the final geometry. Each forming step exhibits force to the object shaping it to requested geometry. The forming step consists of two rolls with shapes according to production geometry Forming setup The forming setup used in this analysis is eight forming steps turning a flat metal sheet into a V-shape. Each step forming the sheet 10 degrees i.e. the final shape is a V-shape with 80 degrees bend angle. The forming setup varies depending of the final geometry of the sheet where considerations to flange strains and springback behaviour has to be taken into account (for further reading on springback and flange strains see the following chapters) Figure 2-8: The 8 pairs of roll forming tools used in the model (forming starts with the toolset to the right moving to the left in the figure) A simple and distinct way to describe the complete forming process scheme is to visualize each forming step using a multiple plot with each step numbered in chronological order, this plot is commonly called the flower. [6] Figure 2-9: Forming process scheme expressed as a single image called the flower(note that this is the flower for a U-profile and not V-profile)

18 2.3.3 Flange strains During forming between one step to another the metal sheet will experience stretching of the flange since the flange has to travel a further distance than the centre of the sheet (see figure 9). During roll forming, especially during forming of deeper profiles, for instance the V-profile, it is exceedingly important to have full control over flange strains occurring in order not to develop any undesired wrinkles. Wrinkles appear in the flange if the strains exceed the elastic limit and turns into plastic strains. Wrinkles in the finished product are undesired. L L+ Figure 2-10: Flange length relative centre line distance during a forming step Due to flange strains there is a limitation in how close the forming steps can be placed in the roll forming process. In simple terms the shortest distance between two tool sets would be the distance just before the flange strains turn plastic, which can be calculated with use of the following data: The forming angle between the two forming steps The width of the sheet i.e. the flange from the centre line All this together with material properties gives the minimum distance between two forming steps. As mentioned above this was expressed in simple terms, in order to achieve and accurate result of the minimum distance one has to find out the deformation zone during forming as well as the exact shape of the deformation curve. This since the simple model assumes that deformation occurs in the whole zone between two tool sets. This is not the case and therefore use of the simple model would produce a generous value. Hence if: L + L + = 1 and = (presuming form angle is constant i.e. h = const) L L L L h

19 2.3.4 Springback When a sheet is subjected to a bending moment and formed to a certain curvature, and there after released, the sheet will springback due to elastic material properties. The amount of springback will mainly be affected by the amount of flow stress applied to the material, the elastic modulus, the bend ratio and the actual bending angle. The bend ratio is the bending angle relative to the thickness of the sheet. [5][2] α = 80 β < α α = Angle before springback β = Angle after springback Springback angle = α - β Figure 2-11: Springback of roll formed profile (note that springback is excessive for visual affect) Springback will also be greatly increased by the strength of the material, in other words by increased yield strength. The springback will also increase if thickness of the sheet is decreased. There is at this point of time no accurate way in describing springback during roll forming and the industry is at this point using the method to over-bend the work piece so it springback to desired shape or applying tension simultaneously to bending during final forming to relieve all stored residual stresses and therefore minimizing springback. The springback during the simulations are calculated by extracting two nodes along the flank of the sheet and by geometric correlations obtaining the angle after springback. This procedure is performed after the relaxation step which endures for one second after the main forming step to allow the sheet to springback fully (see figure 2-12). Two node coordinates extracted to obtain springback angle Figure 2-12: Node coordinate extraction for springback calculations

20 3. FE Simulations 3.1 The models The two models used during simulation and analysis will be presented in detail during this chapter. Both models are built using the same basic geometry of the tools as well as identical sheets. This is a requirement in order to compare the models quantitatively. The models developed are set with delimitations according to following: No heat is produced due to friction or deformation. Constant elastic modulus through entire simulation i.e. the elastic modulus does not increase due to plastic deformation. The tools are modelled as analytical rigid bodies and therefore tool deformation during forming is not accounted for. The sheet and tool surfaces topography is considered 100% homogeneous i.e. no matrix imperfections are present. Strain rate dependence is not accounted for. Short model description: 1. Translational roll forming model This model consists of the tool set accounted for in chapter and a sheet according to chapter During forming, the sheet is held fixed in the z-direction (the direction in which the tools are moving) by a constraint applied on a node in front and centre of the sheet, while the tools are given a certain velocity v in the negative z- direction. This procedure forms the sheet as the tools passes by with a velocity v. No friction is present and neither are the tools rotating. The sheet is simply formed by the contact pressure applied when it comes in contact with the tools. 2. Friction roll forming model As for the translational model, the friction roll forming model consists of the same set of tools and a similar sheet. The principal difference is the course of the forming. Where in this model the tools are fixed and set with an angular speed ω and friction is applied. Hence will the sheet be propelled by the tools and formed as it passes through each forming step according to figure 1. Deformation will occur due to contact with the rigid tools which enforces defined contact pressure onto the sheet. The results from these models has then as a step in understanding roll forming also been compared to traditional V-bending, using a V-bend model also modelled in ABAQUS. 3. Comparative V-bend model Two rigid tools, a punch and a die in V-shape using similar sheet geometries as the above models. This model has been used with and without friction in order to compare with both roll forming models. The sheet is formed by the stamping motion of the punch which enforces the sheet to form according to the die. The tools used are the last in the forming process according to the roll forming line. i.e. the 80 forming station is

21 copied and transformed into extruded tools instead of cylindrical due to pressing action and not rolling action (see figure 12). Transformed Figure 3-1: Transformation of 80 forming step to V-bend model Geometry The sheet geometry Only half the sheet is modelled due to symmetry along centre line. Line of symmetry 5 cm Several partitions for multiple mesh capabilities 70 cm 68 cm 2.5 cm 3.5 cm 2.5 cm 5.0 cm Figure 3-2: Sheet geometry

22 The tool geometry The tools are designed to roll form sheets with a width of maximum 10 cm and adjustable tool gap. The tools are as mentioned above designated for shaping a V-profile sheet. 80 forming step 70 forming step 60 forming step 50 forming step 40 forming step 30 forming step 20 forming step 10 forming step 0 forming step Final geometry of the sheet after forming i.e. a 80 V-profile Figure 3-3: Tool geometry and final geometry of the sheet (perspective from in front)

23 3.1.2 Assembly The forming setup is assembled according to the tool distribution presented in figure 14 where the sheet is gradually formed to an 80 V-shape. The tool sets are assembled at a distance of 30 cm between each centre point and since the sheet is 70 cm long it will have contact with a maximum of 3 tool sets and a minimum of 2 tool sets. This is highly important since if the sheet does not have contact with at least 2 tool sets, the sheet can commence a rotating motion around the tools and the risk for buckling affects are increased. This is due to the fact that when the sheet is exposed to contact by the tools it gains a certain velocity both due to the angular momentum as well as the velocity resulting from the penetrations of the master nodes onto the slave surface. Each tool set has a certain clearance between upper and lower tool. This clearance plays a significant role in the success of the forming process. This is usually pin pointed by the use of trail and error by machine operators, since it depends on many variables such as the free motion margin and the wear of the ball bearings. The clearance used in the FE model has been put to 10% between the tools in the 10 to 80 forming steps. This clearance has been determined through extensive simulation tryouts. The tools used to initiate the forming by giving the sheet its initial speed is built up using 3 tool sets which are flat i.e. no forming is performed during those steps (see geometry for 0 forming step in figure 14). The first of and second of these three initial tool sets uses the same clearance as the stations (i.e. 10%) and the third has zero clearance i.e. the tools are placed in each side of the sheet surface (i.e. 0%) Contact interaction The contact interaction used in the roll forming models is a penalty-overclosure finite sliding contact algorithm. The contact algorithm searches for penetration of master surface onto the slave surface (see figure 15a note that master surface is analytical and not discrete as in the figure although concept of contact is similar). The search algorithm is highly optimized in order to save time and not searching in areas that are not in contact. The algorithm is constructed in a way that it allows arbitrary sliding and separation of the surfaces. When in contact it searches nearby surface and possible contact points if sliding occurs which will be the case during a metal roll forming process (see figure 15b). If a small sliding formulation is used for contact it will not allow for arbitrary motions, the only contact allowed will be in the area in contact at first. This formulation saves time knowing that sliding will not occur otherwise the analysis will go unstable and interrupt. Figure 3-4a(left),b(right): Two surfaces A+B illustrating finite sliding concept with search algorithm steps

24 When the algorithm identifies a penetration of the slave surface, contact pressure which is a function of penetration depth is applied to the slave surface and an opposite and equal force acts on the master penetrated surface due to force equilibrium (see figure 3-5). There are several advantages using an analytical surface instead of a discrete surface when modelling the tools. The analytical surface contact is more efficient in terms of computational cost. The applied pressure acts along a surface instead of individual nodes which makes the contact more even, as well as contributing to a more stable process. Figure 3-5: Contact interaction between slave (sheet) and master (tool) Boundary conditions Coordinate system In order to interpret achieved results output, it is of high importance to have full control over applied coordinate systems. The FE software applies a fixed global coordinate system by default, and the user can thereafter apply several user defined local coordinate systems. Throughout this thesis the x-axis will be equal to 1-axis, y-axis equal to 2-axis and finally z-axis equal to 3-axis. It is very useful to define a coordinate system that runs along with the work piece which in this case is the sheet. This facilitates the interpretation of the results, as well as translations and deformations. Therefore a coordinate system was defined with its centre in the front left side on half the thickness (see figure 17 below). Figure 3-6: The three last forming steps with global coordinate system marked

25 The global coordinate system is applied as seen in figure 18, with the global z-axis is in the roll direction. As observed on the figures one can see that the local coordinate system is defined parallel to the global coordinate system first. During deformation of the sheet the local coordinate system will follow the material direction while the global will stay fixed. Therefore interpreting results when a local system is defined is very appropriate since the global x-axis might not coincide with local material deformation. Figure 3-7: Local coordinate system applied on sheet Tool rotation In the translational roll forming model the tools are as stated in the model definition not rotating and therefore this chapter is not applicable for that model. The second model addressed as friction roll forming model has both rotation and friction present in the model. The tools are set to rotate around its axis of revolution (i.e. the symmetry line in 3D perspective) and the sheet is forced through. Due to the discrepancy of geometry between each set of tools, angular velocity has to be calculated according to achieving equal velocity in the centre of the forming gap. Deviations in geometry between upper and lower tool are also accounted for. ω The forming of a V-profile according to figure 14 page 24, result in large deviation between obtained velocity in the gap centre versus flange velocity during forming as well as deviation gradients through the thickness of the sheet. This is especially critical during the last forming steps, as the flange moves further away from the upper tools centre of rotation as it approaches the centre of rotation for the lower tool. Figure 3-8: Rotation of tool set ω The velocity at the gap centre are set to equal on each side of the sheet, in this way velocity differences are eliminated in the sheet thickness in the centre of the sheet. Provided that upper and lower tool angular velocities are calculated according to above (resulting in equal velocities at each side of the thickness of the sheet), the centre velocity will stay the same during the whole forming process due to that the tool gaps are levelled with the forming direction (z-axis) and will therefore coincide

26 The table below presents calculated values for angular velocities and resulting velocities in the gap centre and the flange of the last tool set since that is where the most critical deviations are found. Angular velocity (ω rad/s) Resulting velocity (v m/s) Gap centre Resulting velocity (v m/s) Flange 80 forming step Upper tool Lower tool V upper and lower tool gap centre 0.0 V upper and lower tool flange 80 forming step Table 3-1: Tool velocity and velocity differences between upper and lower tool The values from the table above is plotted in the figure below, noticeably is the large difference between upper and lower tool velocity in the flange of the 80 forming step. This will result in the sheet experiencing a strong difference in velocity between the upper and lower side of the sheet. This is further discussed and analyzed in chapter 4. 0,4 Velocity gradients for gap centre and flange for upper and lower tool Velocity (m/s) 0,35 0,3 0,25 0,2 0,15 0,1 0,05 Lower tool Upper tool Point 1: Gap centre Point 2: Flange Figure 3-9: Velocity differences for gap centre and flange for upper and lower tool The angular velocity will be increased 0.5% for every tool set in order to prevent buckling during the forming process Translation roll forming model Roll forming simulation with Translation roll forming model uses translation of the tool set as forming procedure instead of friction and rotation of the tools (see figure 20). The sheet is therefore fixed and the tools are translated over the sheet, forming it due to tool contact. The tools are set to translate in the negative z-direction in a velocity of m/s and the velocity is kept constant. As discussed in the previous section

27 (rotation of tools), velocity differences appears during forming which will not be the case when the tools are translated hence all the points on the tools are travelling at equal speed. v = m/s The tool set translates the distance of the total length of the forming steps in addition to the sheet length. The total translated distance is 3.25 meter with margin. In order for the sheets symmetry line to stay parallel to the translating tools a constraint is applied on the front side which discussed in the next chapter. Figure 3-10: Translation roll forming model Friction roll forming model The friction roll forming model is an attempt to simulate the roll forming process as close to the real process as possible. Since discrepancies has been performed using the translation roll forming model where no friction is present as well as the sheet being fixed in space while the tools are translated across the sheet. During the friction roll forming simulation the sheet is formed with the force of the rotation motion of the tools, with the help of a coulomb friction. The sheet interaction with the tools is using the same penalty contact algorithm in both models. Another difference between the models is the initiating tools which are indispensable in the friction roll forming model. This is due to the fact that mass scaling is being used as well as the rotation of the tools exerts a sudden force on the sheet which cause buckles and waves on the sheet. This affect has been eliminated with the use of these extra initiating tools which are not present in the translation model (see figure 20). Adapting smooth 1 step option to the rotation has also been used to manage the negative affect caused by the exerted force in the commence of the process but did not succeed. Therefore the extra tool sets are at this point a must, although this is a more 2 extra tool sets expensive measure in terms of computational cost than it would have been using a smooth step. Figure 3-11: Friction roll forming assembly 1 Smooth step option increases the velocity on the tools gradually instead of going from magnitude zero to full in one step

28 Constraints Due to the fact that during forming operation using both the translation model and the friction model, it is critical that the sheet enters the forming step in the centre of the gap. If not the sheet risks to crash into the tool in arbitrary position ending up with the sheet greatly deformed and the simulation is bound to crash due to excessive distortion. In the figure below an example of excessive distortion during simulation is illustrated. Figure 3-12: Excessive distortion of sheet Figure 3-13: Constraint on symmetry line As a measure to control this, a constraint is applied to a section of the front of the symmetry line of the sheet so that the sheet will always enter the forming step in the gap centre. This measure will consequently reduce the amount of reality in the forming process but in a very fine way. The sheet is constrained to move along the y-axis and therefore forcing it to move parallel to gap centre. It is of great importance to place the constraint in a way that it does not interfere with the forming process and prevents any deformation. The constraint is simply applied so that the forming proceeds in a sequential way. It should act as an invisible hand, acting without the forming simulation noticing Elements Hexahedral solid brick elements with 8 nodes with full and reduced integration (C3D8 and C3D8R) have been used during the simulations. These elements have the three degrees of freedom per node, u x, u y and u z. The used elements are linear elements (see chapter 2.2.3) and equipped with hourglass control. The nodes on an element face in 2D is always numbered anti clockwise due to the fact that the determinant to the Jacobian would become negative if chosen otherwise, and so will the stiffness matrix which indicate a negative element area causing the FE simulation to fail. The node numbering in the 3D follows the same principle but is more complicated due to more layers in the element. Figure 3-14: Numbering of the nodes and faces of C3D8/R

29 3.1.6 Mesh Acquired results are highly dependent of the resolution of the model, speaking in simple terms - the fewer elements used the more generalized results you get. In order to achieve good results and yet not using an extensive number of elements at a high computational cost requires a thoughtful mesh. This is possible using a low resolution model to analyze where the areas with critical deformation and large strains and tensions are, as well as other areas on interest. After collecting this information a new mesh is developed with high resolution on specific focus areas and a lower resolution on the remaining. Using this method will cost a minimum amount of computational time as well as not deteriorating on the final results Friction The friction used during simulations is a coulomb friction model with no elastic slip allowance. The coulomb friction model is defined as figure 24, where the friction coefficient represents the gradient between shear stress and contact pressure acting on the surface. In other words the friction force acting on the surface due to contact pressure will make the surface stick as long as the shear stress does not exceed the critical limit. Figure 3-15: Coloumb friction model Figure 3-16: Elastic slip illustrated The elastic slip is illustrated in figure 3-16, since the elastic slip in the simulations equals zero, the stiffness coefficient κ = which will make the curve in figure 3-16 follow the shear stress axis to critical shear stress where slipping will initiate. Otherwise sticking will be acting

30 3.1.8 Materials The material used in FE simulations is: Docol 1200M SSAB cold-rolled sheet metal Docol 1200M is a work hardening ultra high strength steel produced by SSAB with steel grade name Docol 1200M. The material expresses isotropic hardening and has yield strength of approximately MPa (~1197 MPa is the experimental value used in simulations). The ultimate tensile strength is MPa (1585 MPa is the experimental value). The number stated in the steel grade is the lowest ultimate tensile strength for this material. The experimental data has been acquired during earlier projects and not within this thesis. See appendix A for complete material input. Material %C %SI %Mn %P %S %Nb %Al Docol 1200M 0,11 0,20 1,6 0,015 0,002 0,015 0,04 Table 3-2: Chemical composition of Docol 1200M used in simulations. For complete material input data used in the plastic strain definition in ABAQUS please see appendix A. The material work hardening is described by: [2] [2] [4] σ f pl ( ε ) n = Κ (Equation 21) K,n = constants pl ε = plastic strain σ = flow stress f Contact stiffness As described in the contact algorithm description, penetration of master surface into the slave surface is identified during every time increment. The penetrated distance is adjusted so the slave surface aligns with the master surface. In order to do so, a pressure has to be applied to the slave surface. This pressure divided by the penetration depth is defined as the contact stiffness. The overclosure is defined as the penetration depth of master surface into the slave surface (the sheet). A positive clearance is a negative overclosure and the other way around. The contact stiffness is critical during the roll forming process. The contact stiffness determines the force which will be applied to the metal sheet when it passes through the tools during forming. If the contact is to stiff, the forming will be very noisy in terms of reaction forces and node velocities and displacements as well as the contact acting to hard which can cause the sheet to deform in an unwanted way. If the contact is to soft, the force exerted will not be sufficient for the sheet to form after the tools i.e. the penetration depth into the sheet will be to large and the forming process will not be valid. In mathematical terms it would be preferred that the contact stiffness would be infinite, due to the fact that an ideal simulation would not have any overclosure present at all, yet this in not possible in terms of software simulation. The pressure is increased linear to the penetration depth and therefore it acts as a spring in the way that if you compress/depress a spring it generates a counter force linear to the pressed och pulled distance

31 Steps & damping In FE simulations, every process is divided into one or several steps. This allows multiple setups and configurations between different steps. The roll forming simulations has been divided into two main steps except for the initial step which is a default start up step. The two main steps are: 1. Main forming step ( seconds) 2. Relaxation step (1 second) The main forming step performs the roll forming process just as the name implies. The relaxation step lets the sheet relax and gain time to springback. This is necessary to be able to calculate springback since if this value would be measured just after the sheet was leaving the forming station, the sheet would not have time to relax from elastic state and springback value would be lower. The sheet would also vibrate due to the forming and absence of damping. A default small amount of damping is built in the system and therefore the oscillations are successively decreased with time and coordinates will converge towards accurate values. The sheet is not constrained in any way during the relaxation step. The velocity it has gained during forming will still be exhibited during this step and continue in weightless condition as if in space as no gravity is present in the model. If the sheet would not be damped it would oscillate in infinity and never come at rest where as the results such as residual stress, tensions and springback values would contain errors. The amount of damping which is used is a linear bulk viscosity factor 0.06 and the quadratic bulk viscosity 1.2. The linear bulk viscosity is a pressure linear to the volumetric strain rate: p = b ρ C L ε (Equation 22) BV d e The quadratic bulk viscosity is a pressure quadratic to the volumetric strain rate: 2 p BV = ρ b Cd Le ε (Equation 23) This pressure is not included in node stress results output hence it only acts as a numerical affect only and is not considered part of the material's constitutive response. Bulk viscosity will prevent the structure from vibration at its highest natural frequency which causes noise in the solution results as well as exceedingly high amplitudes at times. Bulk viscosity also prevents the element from collapsing, since deformation speed cannot exceed the dilatational wave speed, and the dilatational wave speed is fixed by the mass and elasticity modulus and the critical stable time increment is the transit time through an element, then if the velocity of the nodes on one side would correlate to the dilatational wave speed it would cross the element in a single transit time which would cause the element to collapse in a single increment. This is prevented by the bulk viscosity as well as warnings produced by a guard which checks for deformation speed larger than 0.3 of the dilatational speed. [8]

32 4. Results 4.1 Simulation parameters Simulation cluster Simulations are performed using a cluster consisting of two connected computers with two (2) CPU s each. The CPU s are of type 2.4 GHz Dual Core AMD Optron which therefore acts as eight (8) processors. The cluster is running a 64-bit Linux Red Hat system. Most simulations has been performed using four (4) processors on the above specified computer cluster Mass scaling When using explicit analysis on this type of simulation, mass scaling is necessary to reduce simulation time within reasonable limits. During simulations the mass has been scaled with time increments from 1e-7 to 2e-5 which is equivalent to multiplying the mass by a factor 500 for t = 1e-7 and a factor 6000 for t = 2e-5. A typical simulation endures for approximately hours for reduced integrated elements and about the double for fully integrated elements. More simulation times can be found in annex B-1. Model Time increment (s) Simulation time (h) Stability quotient (%) Friction model 5e-6 s 47h 7% Friction model 7e-6 s 33h 12% Friction model 1.0e-5 s 26h 25% Friction model 2e-5 s 11h-12h 38% Translation model 2e-5 s 10h-12h <1% Table 4-1: Stability quotient versus simulation time and used time increment (i.e. mass scaling) for reduced integrated solid elements (33880 elements) The negative affect consist of the risk of achieving dynamic phenomenons in terms of buckling and large inertias causing the process to evolve in an improbable way. As an alternate way to decrease calculation time is to reduce total time of process from T 1 to T 2 where T 2 < T 1, this causes the same affect as using mass scaling. For obvious reasons it is recommended to have one factor fixed while manipulating the second factor. Therefore only mass scaling has been experimented with while the total process time has been fixed and therefore also the velocity of which the sheet is formed through the tools (i.e. 0,16667 m/s). A norm for maximum mass scaling does not exist, it is therefore up to the FE-analyst to decide what is a reasonable mass scaling to use, when balancing calculation time against accuracy and possible dynamic affects on the results. Recommendations are that the kinetic energy should be less or equal to ~10% of the total internal energy [8], when this quotient is growing so is the error on the results

33 Although the results are still reasonably accurate they are not accurate enough for precise springback analysis which is of great importance in roll forming. The ratio kinetic energy over total internal energy is referred to as the stability quotient and since the condition is <10% the analysis is more stable the lower the quotient is Simulation time versus time increment Friction roll forming model (reduced and full integration) Data values (Full integration) Linear f it to data values (Full integration) Data values (Reduced integration) Linear f it to data values (Reduced integration) Simulation time (h) Time increments in one second (dt -1 ) Figure 4-1: Simulation time versus time increment used in simulation for friction roll forming model with reduced and full integration solid elements (C3D8R & C3D8) Contact stiffness During analysis different contact stiffness has been used to analyse and to determine the stiffness which is most adequate for the roll forming process in terms of minimizing noise and bad geometry. The contact stiffness will also affect the stability quotient, an increase of the contact stiffness will decrease the stability quotient, although not affecting the translation model in the same extent as the friction model due to the model being more stable. In the tables below one can observe the influence of contact stiffness on the stability quotient for both roll forming models (friction and translation), all values are achieved using the same mass scaling and time increment since these parameters also have a significant affect on the stability. Contact stiffness (N/m 3 ) Stability quotient (%) 38% 32% 10% Table 4-2: Contact stiffness and stability quotient for different simulations for friction roll forming model using t=1e-5 s The translation model is much more stable than the friction model and has a stability <1% independent of contact stiffness (in the simulated range). Contact stiffness (N/m 3 ) Stability quotient (%) <1% <1% <1% <1% Table 4-3: Contact stiffness and stability quotient for different simulations for translation roll forming model using t=1e-5 s

34 Contact stiffness influence on simulation The contact stiffness has influence on the simulation in terms of stability, contact interaction, geometry and springback. It is therefore of interest to determine a suitable contact stiffness in order to calibrate the model in achieving reasonable and correct result. Contact stiffness (N/m 3 ) Springback Translation model ( ) Springback Friction model ( ) Springback Translation model (%) Springback Friction model (%) Table 4-4: Springback for friction and translation roll forming model after the relaxation step for different contact stiffness Springback is influenced by the contact stiffness as noted in the above table. The springback differs by ~75% in the translation roll forming model between the lower and the higher contact stiffness. Table 4-5 and table 4-6 contain values for residual true strain and accumulated plastic strain through the thickness of the sheet, in centre of bend after the relaxation step (i.e. at ~21 seconds). The nodes are placed according to figure. Symmetry plane (centre of bend) Node 6 Node 5 Node 4 Node 3 Node 2 Node 1 Figure 4-2: Node position through thickness of the sheet in centre of bend 2 Time increment used ( t) is smaller than the other used in this comparison. Springback decreases with stability and therefore with lower time increment. A lower value is therefore achieved. See table 5-9 or 5-18 for further information

35 The deviation between the soft ( N/m 3 ) and the hard ( N/m 3 ) contact simulation results for the residual strain is 29% and for the accumulated plastic strain 19 %. The hard contact results in a smaller residual strain though the thickness as well as accumulated plastic strain. Residual strain (ε 11 -through thickness) Soft contact simulation (ε 11 ) Hard contact simulation (ε 11 ) Inner node nr Node nr Node nr Node nr Node nr Upper node Mean deviation 29 % Table 4-5: Residual true strain through the thickness of the sheet in centre of bend for soft and hard contact simulations after relaxation step at ~21 seconds. Accumulated plastic strain (PEEQ -through thickness) Soft contact simulation (ε PEEQ ) Hard contact simulation (ε PEEQ ) Inner node nr Node nr Node nr Node nr Node nr Upper node Mean deviation 19 % Table 4-6: Accumulated plastic strain through the thickness of the sheet in centre of bend for soft and hard contact simulations after relaxation step at ~21 seconds. Residual stress differ more than residual strain and accumulated plastic strain. The mean deviation is 50% which is a large value in terms of absolute value but small in reference to the yield stress which is ~1200 MPa. Residual stress strongly influences springback. Residual stress (σ 11 -through thickness) Soft contact simulation (MPa) Hard contact simulation (MPa) Inner node nr Node nr Node nr Node nr Node nr Upper node Mean deviation 50 % (for surface nodes only) Table 4-7: Residual stress through the thickness of the sheet in centre of bend for soft and hard contact simulations after relaxation step at ~21 seconds

36 The contact pressure on the sheet is extracted for 30 nodes along the symmetry line in the centre of the bend for upper and inner nodes. The contact pressure for the translation roll forming model for upper nodes for all forming steps (10-80 ) are plotted in the figure below. Both simulations using high and low contact stiffness show contact during the two last forming steps. The contact pressure for the higher contact stiffness is much higher than the low contact stiffness. The higher contact stiffness has a magnitude of ~ MPa mean while the lower stiffness has values between MPa Contact pressure on upper nodes (30 nodes) in centre of bend for different contact stiffness for translation roll forming model High contact stiffness 1166*10 11 N/m 3 Low contact stiffness 16*10 11 N/m 3 Contact pressure (MPa) Forming step (degrees) Figure 4-3: Contact pressure on upper surface nodes (30 nodes) for translation roll forming model at each forming step for different contact stiffness The contact pressure for the friction roll forming model using a high and low friction and the translation model for upper surface nodes during the forming process, using a low contact stiffness yields that there is contact for the high friction model (see figure 4-4) during the first and last forming step (10 and 80 ) and for the normal friction only at the last forming step. The translation model with low contact stiffness has contact pressure on upper surface nodes during forming step 70 and Contact pressure on upper nodes (30 nodes) in centre of bend for friction and translation roll forming model at each forming step (contact stiffness 16*10 11 N/m 3 ) Normal friction model High friction model Translation model Figure 4-4: Contact pressure on upper surface nodes (30 nodes) for friction (high & low) and translation roll forming model at each forming step with low contact stiffness

37 The contact pressure on the inner surface nodes for the translation model with low contact stiffness experience pressure during the first five forming steps (10 to 50 ) with a mean pressure of ~650 MPa which can also be seen in the iso-plot patterns in chapter During the high contact stiffness simulation, nodes only have contact during the first forming step with a magnitude of ~1100 MPa. Contact pressure on inner nodes (30 nodes) in centre of bend for different contact stiffness for translation roll forming model (at each forming step) Low contact stiffness (16*10 11 N/m 3 ) High contact stiffness (1166*10 11 N/m 3 ) Contact pressure (MPa) Forming step (degrees) Figure 4-5: Contact pressure on inner surface nodes (30 nodes) for translation roll forming model at each forming step for different contact stiffness Contact pressure for the friction roll forming model using a high and low friction and the translation model for inner surface nodes during the forming process, using a low contact stiffness, yields that there is contact for the normal friction model (see figure 4-6) during all forming steps except for the first and the last step (10 and 80 ). These pressures are between ~ MPa. For the high friction model the nodes experience pressure during all forming steps except the last one (80 ) with a magnitude ~ MPa. The translation model with low contact stiffness has contact pressure on inner surface nodes throughout forming step 10 until 50 of a magnitude of ~ MPa. Contact pressure (M Pa) Contact pressure on inner nodes (30 nodes) in centre of bend for friction and translation model with low contact stiffness (at each forming step) Normal friction model High friction model Translation model Forming step (degrees) Figure 4-6: Contact pressure on inner surface nodes (30 nodes) for friction (high & low) and translation roll forming model at each forming step with low contact stiffness

38 Maximum in plane stress for the friction roll forming model and the translation roll forming model with different contact stiffness at each forming step is presented in the table below. All simulations obtain approximately the same values ~1450 MPa which is ~20 % above yield limit. σ Max (MPa) σ Max (MPa) σ Max (MPa) σ Max (MPa) σ Max (MPa) σ Max (MPa) Friction model Translation model Model µ=0.12 C-Stiffness µ=0.65 C-Stiffness C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) Form Step (N/m 3 ) (N/m 3 ) Mean value Table 4-8: Maximum in plane stress for upper surface node along symmetry line in centre of bend for friction and translation roll forming model at each forming step. Values for the mean principal stress (hydrostatic pressure) for upper surface node along symmetry line in centre of bend is presented in the table below. No difference between the models or the use of different contact stiffness is present. Model σ Mean σ Mean (MPa) (MPa) Friction model µ=0.12 µ=0.65 C-Stiffness C-Stiffness (N/m 3 ) (N/m 3 ) σ Mean (MPa) C-Stiffness (N/m 3 ) σ Mean σ Mean (MPa) (MPa) Translation model C-Stiffness C-Stiffness (N/m 3 ) (N/m 3 ) σ Mean (MPa) C-Stiffness (N/m 3 ) Form Step Mean value Table 4-9: Mean principal stress for upper surface node along symmetry line in centre of bend for friction and translation roll forming model at each forming step

39 Plastic strain at each forming step for the friction (high and low friction coefficient) and translation roll forming model, is presented in the table below. The plastic strain is not influenced by the friction coefficient, but decrease with increased contact stiffness. The plastic strain value for the translation model at the last forming step is decreased by ~14 % with use of the highest contact stiffness versus the lowest. Model ε pl ε pl ε pl ε pl ε pl ε pl Friction model Translation model µ=0.65 C-Stiffness C-Stiffness C-Stiffness C-Stiffness (N/m 3 ) (N/m 3 ) (N/m 3 ) (N/m 3 ) µ=0.12 C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) Form Step Table 4-10: Maximum in plane stress for upper surface node along symmetry line in centre of bend for friction and translation roll forming model at each forming step. Figure 4-7 below illustrates the sheet shape for the translation roll forming model using two different contact stiffness s. The figure is produced by extracting the 7 first nodecoordinates starting from the centre of the sheet at the symmetry line. As noted the higher contact stiffness has a lower upper plateau than the lower stiffness. Along the right flange of the figure the lower contact stiffness aligns closer to the lower tool. -0,046-0,047 Tool pliancy for translation roll forming model with different contact stiffness (high versus low) during forming step 80 degrees Contact stiffness 16.0*10 11 N/m 3 Contact stiffness *10 11 N/m 3-0,048 X coordinate (m) -0,049-0,050-0,051-0,052-0, ,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 0,0040 Y coordinate (m) Figure 4-7: Sheet shape for translation roll forming model for different contact stiffness (high versus low) during forming step 80 The contact stiffness influence velocity for upper flange node for the friction roll forming model with different contact stiffness is presented in the figure below. The velocity is much larger for the higher contact stiffness (see figure 4-8)

40 0.06 Velocity in y-direction for flange node for different contact stiffness 0.04 Contact stiffness 33.3e11(N/m 3 ) Contact stiffness 66.6e11(N/m 3 ) Velocity in y-direction (m/s) Time (s) Figure 4-8: Velocity in y-direction for flange node for different contact stiffness ( versus (N/m 3 )) Contact stiffness influence on the forming process in terms of displacements and velocity in the cross section plane (i.e. x-y plane) is further presented in annex B Mass scaling parameter influence on accuracy Using different time increments during simulation, produces deviations in obtained results. As a measure of estimating this affect, a number of selected values have been compared with the use of different time increments and therefore a change in the stability quotient. The stability quotient is calculated as the ratio between the kinetic energy over the total internal energy. The stability quotient is recommended to be below 10% for the simulation to be guaranteed more accurate results and for the simulation to be stable. For information about how mass scaling parameters are calculated see introductory chapters for FE calculations. Time increment t= t= t= t= Mass scaling Calculation time 12h 26h 33h 47h Stability quotient 38% 25% 12% 7% Table 4-11: Calculation time versus mass scaling and stability quotient Table 4-12 below yields springback values for different stability quotient. Increased stability decreases springback. Stability quotient Springback degrees Springback percent 7% % 38% % Relative deviation -springback = 9% Table 4-12: Springback values for different stability quotients after the relaxation step at ~21 seconds

41 Springback is highly dependent on the residual stress through the thickness of the sheet, below figure shows the residual stress through the thickness of the sheet for different stability using friction roll forming model after the relaxation step. The residual stress values for the more instable simulation (38%) show smaller values at the sheet surface and approximately the same through the thickness of the sheet. 1,5 Residual stress through thickness of sheet in centre of bend 1,5 Thickness coordinate (mm) 1,0 0,5 1,0 0,5 Stability quotient 7% Stability quotient 38% Residual stress (MPa) Figure 4-9: Residual stress through thickness of sheet in centre of bend Residual strain and accumulated plastic strain show little deviation between high and low stability with a deviation within 2% (figures for these values are available in appendix B- 4). Node position Deviation (%) Residual strain Deviation (%) Residual stress Deviation (%) Acc plastic strain Node 1 (inner node) Node Node Node Node Node 6 (upper node) Mean deviation (%) Table 4-13: Relative deviation in residual stress, strain and accumulated plastic strain at nodes through the thickness of the sheet in centre of bend for different stability quotas (7% and 38%) after the relaxation step for friction roll forming model Principal stress in x,y and z direction for a surface node on the upper side in the centre of the bend as well as true strain in length direction in the flange of the sheet is presented in table These values are calculated by extracting the values at every forming step (i.e ) and the calculating the mean value for each simulation with different stability quotient. The values are then relatively compared to the different values at each stability limit (i.e. different stability quotient)

42 Deviations are between ~13% - 19% for principal stress during forming between simulations with stability 12% and 7%, which increase to between ~33% - 52% when comparing simulations with 7% and 38% stability showing large deviations. Worth mentioning is that the largest deviation in residual stress value is no more than 17% of yield stress value. The deviations are large in terms of relative values but small compared to yield stress. The true strain values in x (i.e. cross wise direction) deviates from ~1% to 5% between simulation stability 7% and 38%. Flange strain in the length direction deviates between ~11% to ~24% between the different stabilities, none of them exceeding the plastic limit. Results Mean relative deviation between 7% 12% stability (%) Mean relative deviation between 7% 38% stability (%) σ 11 - centre upper node σ 22 - centre upper node σ 33 - centre upper node ε 11-centre upper node ε 33-flange upper node Table 4-14: Mean relative deviation between results using friction model with different stability quotient during the forming process (~0-19 seconds) using friction roll forming model Shear locking Shear locking as recalled from chapter is a term that refers to linear, full integrated elements. Since the element sides are not able to curve the element acts in a stiffer way during bending. Due to the fact that the element can not curve, the sides of the element will not be right-angled to the element bottom side, and shear stress will be present which opposes against bending theory. This is shown in the figure below where the shear stress for the fully integrated element is shown versus a reduced integrated element. As one can see the results are striking Shear stress for reduced and full integration method Friction model - reduced integration Translation model - reduced integration Friction model - full integration Shear stress (MPa) Time (s) Figure 4-10: Shear stress (σ 12 in centre of sheet for upper node) for reduced and full integration method during the forming process (forming step)

43 To find out if this induced stiffness has any influence on the forming process, springback, residual stress and strain will be examined. Parameter / Result Full integration Reduced integration Time increment (s) Simulation time (h,m) 64h 0m 33h 0m σ 11 (inner fibre) (MPa) σ 11 (upper fibre) (MPa) ε 11 (inner fibre) (%) ε 11 (upper fibre) (%) PEEQ (inner fibre) (%) PEEQ (upper fibre) (%) Springback ( ) & (%) 4.64 / 5.8 % 4.25 / 5.3 % Stability quotient (%) 22% 12% Table 4-15: Residual data after the relaxation step for upper and inner fibre in the centre of bend, stability quotient and simulation time for reduced and full integration simulation with the use of friction roll forming model Excessive stiffness in the linear fully integrated elements results in higher residual stress through the thickness and also shows larger plastic strain in the inner fibre of the bend which would indicate that the neutral line would have moved towards the upper side of the bend which is incorrect. Due to the extensive bending during this roll forming process the linear fully integrated elements are to poor in their description. Springback values are increase, even though not by much ~9% due to increased stiffness leaving higher residual stress especially in the inner fibre were also the residual true strain as well as the accumulated plastic strain is very high compared to the reduced integrated element Hourglass affect and hourglass control Recalling hourglassing from chapter 2.2.4, which is a problem for reduced integration elements with only one integration point. When this element is subjected to pure bending it will not offer any deformation resistance, as a way to counteract this, artificial strain resistance is accumulated by the Abaqus. This way one can observe the magnitude of hourglassing by looking at the energy levels for the strain and artificial strain. If the artificial strain energy is more than ~5-10% of the strain energy, hourglassing could be a problem and the model should be further evaluated with different meshes in order to validate the accuracy of the results. During this chapter different hourglass modes (normal and enhanced hourglass control) have been evaluated with the use of 8 node solid elements with reduced integration (C3D8R). These are then compared to results where instead of using enhanced hourglass control, the time increment has been reduced in order to see if hourglassing reduces in proportion with the stability quotient decreasing. The different hourglass control modes do not influence the stability quotient unless the time increment is changed

44 Energy (J) Energy levels for artificial strain and strain for different hourglass modes Artificial strain energy Normal hourglass control (time inkr = 2*10-5 ) Artificial strain energy Enhanced hourglass control (time inkr = 2*10-5 ) Artificial strain energy Enhanced hourglass control (time inkr = 7*10-6 ) Artificial strain energy Normal hourglass control (time inkr = 7*10-6 ) Artificial strain energy Normal hourglass control (time inkr = 5*10-6 ) Strain energy Time (s) Figure 4-11: Energy levels for artificial strain and strain for different hourglass modes As one can see in the above figure, the artificial strain energy is very high in comparison with the strain energy using normal hourglass control and time increment t = but is reduced efficiently with the use of enhanced hourglass control using the same time increment. Using normal hourglass control and instead reducing the time increment produces almost the same energy levels. Worth noticing is that by reducing the time increment from s to s for normal hourglass control reduces the artificial strain energy prominently but with the use of enhanced hourglass control and the same reduction in time increment does not influence the artificial strain energy in a striking way. One can therefore draw the conclusion that if using enhanced hourglass control, it is not efficient reducing the time increment which increases the computational cost (see table 4-16). Simulation nr Hourglass mode Time increment Simulation time 1 Normal hourglass control h 2 Normal hourglass control h 3 Normal hourglass control h 4 Enhanced hourglass control h 5 Enhanced hourglass control h Table 4-16: Computational time for different hourglass modes with different time increments The striking fact is that enhanced hourglass control is efficient in terms of reducing artificial strain energy levels compared to achieving the same affect by reducing the time increment. The use of enhanced hourglass control reduce the artificial energy without adding any time to the simulation which is clearly not the case when achieving the same low energy levels with use of time increment reduction. Even though the facts point in a positive direction using enhanced hourglass control, there exists problems. The use of enhanced hourglass control during intense bending does not produce accurate and good results. This affect can be seen analysing the plotted curves in the above figure were one can see that the energy levels are kept low during the first 8-10 seconds and is way below the curves for the normal hourglass control with reduced time increment. After these seconds the curve drastically rises and the enhanced hourglass control looses control and rises above the levels for the curves with normal hourglass control with reduced time increment. Worth noticing is also that the affect from decreasing the time increment is reduced as it stabilizes

45 To find out more about how this affects the forming process, springback, residual stresses and strains for the above simulations have been analysed. The data is collected into the table below. Simulation Nr PEEQ (inner/upper fibre) ε 11 (inner/upper fibre) σ 11 (MPa) (inner/upper fibre) Springback ( ) & (%) / / / / 5.9 % / / / / 5.3 % / / / / 5.4 % / / / / 7.7 % / / / / 7.2 % Table 4-17: Accumulated plastic strain, residual true strain and stress through the thickness of the sheet in centre of bend as well as springback for the different hourglass simulations after the relaxation step at ~21 seconds Results achieved using enhanced hourglass control are inaccurate. The residual stress for the simulations with enhanced hourglass control produce values high over the yield limit which is clearly not possible. Since the use of smaller time increment in simulation number 2 and 3 produced in terms of artificial strain energy rates approximately the same levels but are on the other hand correlating to the value achieved with original settings and with residual stress that are in the correct magnitude indicate that the simulations are trustworthy. The residual strain for the models with normal hourglass control (simulation nr 1-3) indicates that the neutral line in the sheet is almost not moved which should be the case for thin sheet bending. On the other hand observing the values for the enhanced hourglass control indicates that the neutral line has moved significantly towards the inside of the bend. The springback values are approximately 40% higher for the enhanced hourglass simulations and differ from the normal hourglass simulations which correlate well. Concluding above discussion and results yields that enhanced hourglass control is not recommended when using reduced integrated brick elements (C3D8R) subjected to large bending irrespective of reduction of the time increment Mesh density evaluation During this chapter a mesh density evaluation using hexahedral continuum elements with reduced integration (Abaqus element label: C3D8R) and normal hourglass control have been used. As a measure of the performance using different meshes, a number of results will be compared and analysed between the different configurations stated in below table. These results compared are: Springback Accumulated plastic strain (for upper and inner fibre only) Residual stress and strain in centre of bend (for upper and inner fibre only) Simulation time Stability

46 The results from this mesh configuration test are accomplished using continuum hexahedral 3D elements and are therefore only 100% valid for that specific element. The results can yet in some extent give information for configurations using different kind of elements, although with caution. Mesh configuration Elements in thickness Elements in width (centre section) Elements in length Time increment (s) Simulation time (h) ~31h ~30h ~33h ~36h ~40h ~45h Table 4-18: Mesh configuration table, time increment and simulation time used in mesh evaluation Different time increment is used for the simulations in this mesh evaluation, this is due to the fact that there will be affect on the stability quotient due to the change of elements through the thickness which is increased from 5 to 7 in some of the simulations. This results in reduced critical element length which decreases the stable time increment. In total this concludes that the mass scaling parameter has to increase in order for the deformation wave not to overrun any elements, and therefore a loss in stability. The time increments have therefore been reduced in order for all 6 mesh configurations reaches stability quotient ~10%. The amount of element specified in the sheet in table 4-18 above results in the following ratios between width, height and depth. Bending is acting in the x-y plane which is width, height. Mesh configuration Ratio width (x) Ratio height (y) Ratio depth (z) Final ratio (w,h,d) (1,2,15) (1,4,38) (1,4,26) (1,7,53) (1,3,26) (1,5,53) Table 4-19: Mesh size ratio used in mesh evaluation (x-y is the bending plane) Values for springback, accumulated plastic strain, residual true stress and residual true strain through the thickness of the sheet in centre of bend after the relaxation step is presented in table 4-20 and Mesh configuration 1 show a very low springback as well as low residual true strain values. Mesh configuration 2 and 3 shows similar values for all results indicating the elements in length direction does not influence in the same extent. Mesh configuration 4 shows identical springback value as configuration 3 but shows increased residual true strain values by almost 0.1. Mesh configuration 5 and 6 deviates in springback values between each other but shows the same tendencies in terms of residual true strain being larger for the inner fibre than the outer fibre indicating that the neutral line would have

47 travelled outwards in the sheet. This in not possible during this forming process. Furthermore are the residual true stress very high for configuration 5 and 6 which is at levels about 45-55% of yield strength and very high in terms of residual stress indicating unreliable results. Mesh configuration Springback ( ) & (%) PEEQ Inner fibre (%) PEEQ Upper fibre (%) / 2.6 % / 6.2 % / 5.8 % / 5.8 % / 5.0 % / 6.5 % Table 4-20: Stability quotient, springback values and accumulated plastic strain for the different mesh configurations Mesh configuration ε 11 Inner fibre (%) ε 11 Upper fibre (%) σ 11 Inner fibre (MPa) σ 11 Upper fibre (MPa) Table 4-21: residual strain (ε 11 ) and residual stress across the sheet (σ 11 ) in centre of bend for upper and inner fibre for the different mesh configurations after the relaxation step ~21 seconds The final mesh resolution chosen for the simulations is stated in the table below. Model / Element Thickness resolution Length resolution Width resolution Solid (C3D8/R) Table 4-22: Final mesh resolution

48 4.2 Forming parameters Gap size comparison In order to evaluate the gap size influence on the roll forming process this chapter will present results from simulations using two different gap sizes, 1.54 mm and 1.65 mm which with the sheet thickness of 1.5 mm results in a clearance of 2.7% and 10%. Residual stresses and strains as well as springback and sheet shape will be compared. Gap size Figure 4-12: Visualisation of gap size for a tool set in the roll forming process When the gap is set to 1.65 mm respectively 1.54 mm, that distance is set in the centre of the symmetry line which means that the clearance between upper and lower tool will increase at the flanges of the sheet and is also influenced by the angle of the forming step. This affect arises since the gap Gap 1 size is set by translating the tools in y direction and since adjustment is done according to Gap 2 the distance in the centre of the tool, the flanges will have a Gap 3 different gap size due to geometrical properties of the tools. The gap distance between Gap 4 upper and lower tools for four different points are listed in the table below(see left figure 4-13). Figure 4-13: Gap point indices for distance comparison Tool gap point 1.65 mm gap size 50 forming step 1.65 mm gap size 70 forming step Distance from centre line of rotation mm 1.65 mm 79.5 mm mm 1.60 mm 78.0 mm mm 1.80 mm 54.9 mm mm 2.01 mm 31.7 mm Table 4-23: Gap size for different gap points with distance between measure point to centre line of rotation of the lower tool

49 As one can see the gap increases from gap point 1 to gap point 4 by a total of 19 %. This extra space influence the forming process in terms of sheet shape during the forming steps which deteriorates 0.3 mm at the flange which corresponds to an angle variation of ~ x 10-3 Residual strain through thickness of sheet x=0 mm inside on bend x=1.5 mm outside of bend 1.5 Thickness coordinates (mm) , mm gap 1.65 mm gap True strain (%) Figure 4-14: Residual strain through thickness of sheet in centre of bend for different gap size after the relaxation step at ~21 seconds 1.5 Residual stress through thickness of sheet x=0 mm inside of bend x=1.5 mm outside of bend 1,5 Thickness coordinates (mm) 1.0 0,5 1,0 0, mm gap 1.65 mm gap Residual stress (MPa) Figure 4-15: Residual stress through thickness of sheet in centre of bend for different gap size after the relaxation step at ~21 seconds 1,5 Accumulated plastic strain through thickness of sheet Thickness coordinates (mm) 1,0 0, mm gap 1.65 mm gap Accumulated plastic strain (%) Figure 4-16: Accumulated plastic strain through thickness of sheet in centre of bend for different gap size after the relaxation step at ~21 seconds

50 The three figures above shows residual stress, residual true strain and accumulated plastic strain (through the thickness of the sheet in centre of bend) after the relaxation step at ~21 seconds. These results correlate well with exception for the residual stress which has higher values for the larger gap size simulation. Flange strain values in length direction (z direction) shows no pronounced difference between the two models (see figure 4-17 below). 0,004 True strain in length direction in flange for upper fibre 0,003 0,002 True strain 0, ,001-0, mm gap 1.65 mm gap -0, Time (s) Figure 4-17: True strain in length direction (σ 33 ) in flange for upper fibre for different gap size during the forming process Springback values are affected by residual stress through the thickness of the sheet and also on the gap size, since a larger gap means more springback space during each forming step. As table 4-24 below states, the difference in springback angle is 0.23 between the two gap size models. Gap size Springback angle Springback percent 1.54 mm (2.7% clearance) % 1.65 mm (10% clearance) % Table 4-24: Springback in degrees and percent for different gap size after the relaxation step at ~21 seconds Reaction forces on the tools for the forming process are plotted in figure 4-17, as the figure state, the reaction forces are a higher for the model with smaller gap. A smaller forming space between the tools is going to cause a harder contact which will produce slightly higher reaction forces. Table 4-25 present the mean values for the reaction force during forming step 60 degrees and 8 degrees, and as the table shows the reaction forces are increased by approximately 25% for the 80 degree forming step and 5% for the 60 degrees forming step. The final cross section sheet geometry and sheet shape during the 80 forming step is showed in figure 4-18 below and as discussed earlier due to increased gap size the larger gap model has more space to springback during the forming step which is also the proven case in figure

51 7000 Reaction force on tools for forming steps 50 degrees and 70 degrees mm gap 70 degrees 1.65 mm gap 70 degrees 1.54 mm gap 50 degrees 1.65 mm gap 50 degress Reaction force (N) Time (s) Figure 4-18: Reaction force on tools during 50 and 70 forming steps for different gap size Tool gap Element Mean reaction force 80 forming step Mean reaction force 60 forming step 1.54 mm Solid 2919 N 1757 N 1.65 mm Solid 2334 N 1693 N Table 4-25: Mean reaction force during two forming steps for two different gap sizes Figure 4-19 below presents contact pressure on upper side nodes at each forming step for the two different gap simulations. The smaller gap shows larger contact pressure (increase of ~400 MPa) during the steps in contact. During the other forming steps no contact on the upper centre nodes along the symmetry line is present Contact pressure on upper nodes (30 nodes) in centre of bend for different gap size with translation roll forming model (contact stiffness 16*10 11 N/m 3 ) Tool gap 1.54 mm Tool gap 1.65 mm Contact pressure (MPa) Forming step (degrees) Figure 4-19: Contact pressure on upper nodes in centre of bend during each forming step for different gap size using the same contact stiffness Contact pressure for inner side nodes in centre of bend during each forming step is presented in figure Both gap models have contact during the first 5 forming steps. The contact pressure values are at the same level during the first two forming steps and slightly higher during the other three forming steps for the smaller gap

52 Contact pressure on inner nodes (30 nodes) in centre of bend for different gap size with translation roll forming model (contact stiffness 16*10 11 N/m 3 ) Tool gap 1.54 mm Tool gap 1.65 mm 700 Contact pressure (MPa) Forming step (degrees) Figure 4-20: Contact pressure for inner nodes in centre of bend at each forming step for simulations with different gap size (1.54 mm versus 1.65 mm) Final cross section in figure 4-21 shows similar final shape and geometry for both gaps. Final cross section sheet geometry for different gap size -0,04-0,04-0,05 Tool 1.65 mm gap size 1.54 mm gap size -0,05-0,06-0,06 y coordinate (m) -0,07-0,08 y coordinate (m) -0,07-0,08-0,09-0,09-0,10 0 0,002 0,004 0,006 0,008 0,010 0,012 0,014 x coordinate (m) -0, x coordinate (m) Figure 4-21: Final cross section sheet geometry for different gap size (1.65 mm versus 1.54 mm) Sheet shape during the 80 degree forming step for the different gap size differ at the flange of the sheet. This is due to the fact that there is more space for the larger gap allowing larger springback during the forming steps (see table 4-23 for distance at flange between upper and lower tool for different gap sizes) Tool docility during 80 degree forming step for different gaps (1.65 mm versus 1.54 mm) with contact stiffness 16.0e mm gap 1.54 mm gap Tool y coordinate (m) y coordinate (m) x coordinate (m) x coordinate (m) Figure 4-22: Sheet shape during 80 forming for different gaps (1.65 mm versus 1.54 mm) with contact stiffness (N/m 3 )

53 4.2.2 Residual strain in centre of bend Residual true strain at nodes through thickness in centre of bend for translation model with different contact stiffness is presented in table Residual true strain decreases with increased contact stiffness. Node position C-Stiffness (N/m 3 ) Translation roll forming model (ε true ) C-Stiffness C-Stiffness C-Stiffness (N/m 3 ) (N/m 3 ) (N/m 3 ) C-Stiffness (N/m 3 ) Point Point Point Point Point Point Table 4-26: Residual true strain through the thickness of the sheet in centre of bend for translation roll forming model for different contact stiffness after the forming process at last frame in step 2 (relaxation step) Residual true strain at nodes through thickness in centre of bend for friction model with different contact stiffness is presented in table Residual true strain stays intact independent of contact stiffness with exception for inner node with the highest contact stiffness where the value decreases by ~21%. Node position C-Stiffness (N/m 3 ) Friction roll forming model (ε true ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) Point Point Point Point Point Point Table 4-27: Residual true strain through the thickness of the sheet in centre of bend for friction roll forming model for different contact stiffness after the forming process at last frame in step 2 (relaxation step)

54 Residual true strain at nodes through thickness in centre of bend for friction model with different friction coefficient is presented in table Almost no difference exists between the two simulations with different friction coefficient. Friction roll forming model (ε true ) Node position Friction coeff µ = 0.12 Friction coeff µ = 0.65 Point Point Point Point Point Point Table 4-28: Residual true strain through the thickness of the sheet in centre of bend for friction roll forming model for different friction coefficient after the forming process at last frame in step 2 (relaxation step) Residual true strain at nodes through thickness in centre of bend for friction model with different stability quotient is presented in table The results are equal independent of stability quotient for residual true strain. Node position Stability quotient 38 % C-Stiffness (N/m 3 ) Friction roll forming model (ε true ) Stability quotient 12 % C-Stiffness (N/m 3 ) Stability quotient 7 % C-Stiffness (N/m 3 ) Point Point Point Point Point Point Table 4-29: Residual true strain through the thickness of the sheet in centre of bend for friction roll forming model for different stability quotient after the forming process at last frame in step 2 (relaxation step)

55 4.2.3 Residual stress in centre of bend Figure 4-30 shows residual stress through the thickness of the sheet in centre of bend after the relaxation step. Stresses reach maximum values of 0-35% of yield strength. One can see that compressive residual stress in point 2 increases with contact stiffness from -100 to -300 MPa, as well as the tensional stress in point 4 increases by increased contact stiffness. Node position C-Stiffness (N/m 3 ) Translation roll forming model- Residual stress (MPa) C-Stiffness C-Stiffness C-Stiffness (N/m 3 ) (N/m 3 ) (N/m 3 ) C-Stiffness (N/m 3 ) Point Point Point Point Point Point Table 4-30: Residual stress through the thickness of the sheet in centre of bend for translation roll forming model for different contact stiffness after the forming process at last frame in step 2 (relaxation step) Residual stress field though the thickness of the sheet after the relaxation step is shown in figure 4-23 below. The stress at the sheet surface is approximately the same between the different contact stiffness s. The stress peaks inside the thickness of the sheet are on the other hand are firmly increased for increased contact stiffness with 300 MPa between the lowest the highest contact stiffness which is 25% of yield strength and large difference in terms of residual stress. 1,5 Residual stress through the thickness of the sheet in centre of bend for translation roll forming model with different contact stiffness Thickness coordinate (mm) 1,0 0,5 Contact stiffness 8.0*10 11 N/m 3 Contact stiffness 16.0*10 11 N/m 3 Contact stiffness 21.3*10 11 N/m 3 Contact stiffness 66.6*10 11 N/m 3 Contact stiffness *10 11 N/m Residual stress (MPa) Figure 4-23: Residual stress through the thickness of the sheet in centre of bend for translation roll forming model for different contact stiffness after the forming process at last frame in step 2 (relaxation step)

56 Residual true strain through the thickness of the sheet in centre of bend for friction roll forming model is shown in the table 4-31 below. Increased stress can be observed except for point 3 and 4 between the lowest and the highest contact stiffness, although not as clearly as the translation roll forming model in table Friction roll forming model- Residual stress (MPa) Node position C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) C-Stiffness (N/m 3 ) Point Point Point Point Point Point Table 4-31: Residual stress through the thickness for friction roll forming model for different contact stiffness after the forming process at last frame in step 2 (relaxation step) Residual stress through the thickness of the sheet in centre of bend for the friction model with different friction coefficient does not change more than marginally (see table 4-32 below). Friction roll forming model Node position Friction coefficient µ=0.12 Friction coefficient µ=0.65 Residual stress (MPa) Residual stress (MPa) Point Point Point Point Point Point Table 4-32: Residual stress through the thickness of the sheet in centre of bend for friction roll forming model for different friction coefficient after the forming process at last frame in step 2 (relaxation step) Residual stress through the thickness of the sheet in centre of bend for the friction model with different stability quotient from 7% (more stable) to 38% (less stable) shows small deviations and differences are not more than marginal (see table 4-33 below). Friction roll forming model σ 11 (MPa) Node position Stability quotient 38 % C-Stiffness (N/m 3 ) Stability quotient 12 % C-Stiffness (N/m 3 ) Stability quotient 7 % C-Stiffness (N/m 3 ) Residual stress (MPa) Residual stress (MPa) Residual stress (MPa) Point Point Point Point Point Point Table 4-33: Residual stress through the thickness of the sheet in centre of bend for friction roll forming model for different stability quotient after the forming process at last frame in step 2 (relaxation step)

57 Residual stress through the thickness of the sheet in centre of bend for the friction model, translation model and V-bend model are presented in table 4-34 below. V-bend model shows a different stress field pattern than the friction and translation roll forming model, which could be as a result of a reliving affect of the V-bend as it hits the die. The difference between the friction roll forming model with the translation model are marginal and they show the same stress field pattern. Comparison between friction and translation roll forming model with V-bend model with contact stiffness (N/m 3 ) Node position V-bend model Stability quotient <1% Translation model Stability quotient <1% Friction model Stability quotient 7% Residual stress (MPa) Residual stress (MPa) Residual stress (MPa) Point Point Point Point Point Point Table 4-34: Comparison in residual stress through the thickness of the sheet in centre of bend for friction and translation roll forming model with V-bend model after the forming process at last frame in step 2 (relaxation step) Flange strain 3 True flange strain in length direction (ε 33 ) for inner element at integration point during each forming step for different models is presented in the table below. No plastic strain is present during the forming process independent of model used. True strain at forming step (ε true ) Model Friction model µ = 0.12 Stability 7% Low stiffness Friction model µ = 0.12 Stability 38% Low stiffness Friction model µ = 0.12 Stability 38% High stiffness Translation model Stability <1% Low stiffness Translation model Stability <1% High stiffness Table 4-35: Flange strain for upper node in flange of sheet at each forming step for friction and translation roll forming model with different stability quotient 3 Plastic limit is ε ~

58 4.2.5 Comparison of residual true strain in centre of bend for different models and approximated theoretical calculated value Plastic strain at upper and inner node for different models as well as an approximated theoretical value is presented below. The roll forming models correlate well with each other and are in the same range as the theoretical calculated approximation. V-bend value is larger, this is due to the V-bend not having contact with any upper tool. V-bend model furthermore shows that the neutral line has moved further towards the inner side of the bend than the roll forming models. Contact stiffness (N/m 3 ) Plastic strain in centre of bend for upper node (ε true ) Plastic strain in centre of bend for inner node (ε true ) Friction model Stability quotient 7% Friction model Stability quotient 38% Translation model Stability quotient <1% V-bend model Stability quotient <1% Theoretical approximation Table 4-36: Comparison of plastic strain in centre of bend for upper and inner surface node for different models and theoretical calculated value after the forming process at last frame in step 2 (relaxation step) with low contact stiffness ( N/m 3 ) Plastic strain at upper and inner node for different models as above with higher contact stiffness is presented in the table below. All models (except for the theoretical value which does not include contact stiffness) show a decreasing plastic strain with increased contact stiffness. This reduction is in the range 5% between contact stiffness to N/m 3. Contact stiffness (N/m 3 ) Plastic strain in centre of bend for upper node (ε true ) Plastic strain in centre of bend for inner node (ε true ) Friction model Stability quotient 38% Translation model Stability quotient <1% V-bend model Stability quotient <1% Theoretical approximation Table 4-37: Comparison of plastic strain in centre of bend for upper and inner surface node for different models and theoretical calculated value after the forming process at last frame in step 2 (relaxation step) with high contact stiffness ( N/m 3 ) 4 Springback angle is not accounted for in this value (i.e. 80 is used as final angle)

59 4.2.6 Thickness of the sheet after roll forming The thickness of the sheet in the centre of bend which is the critical position for the different roll forming models do not change more than 0.7%. The V-bend model reduced in thickness by almost 6%. Model (Solid elements C3D8R) Contact stiffness (N/m 3 ) Thickness (mm) Change in thickness (%) Friction model (µ=0.12) Friction model (µ=0.12) Friction model (µ=0.65) Translation model V-bend model Translation model Table 4-38: Final thickness of the sheet in centre of bend for different models after the forming process at last frame in step 2 (relaxation step) Reaction force The reaction force increases with contact stiffness, in the below figure the reaction force on the 30 forming tool set and 80 forming tool set is plotted for different contact stiffness for the friction roll forming model. Each reaction force lives for approximately 4 seconds since the sheet is 70 cm long and travels at a speed of ~0.17 m/s. Reaction force (N) Friction roll forming model Reaction force on tools during forming step 30 degrees and 80 degrees for different contact stiffness Contact stiffness 8e11 (N/m 3 ) - Forming step 80 degrees Contact stiffness 16e11 (N/m 3 ) - Forming step 80 degrees Contact stiffness 21.3e11 (N/m 3 ) - Forming step 80 degrees Contact stiffness 8e11 (N/m 3 ) - Forming step 30 degrees Contact stiffness 16e11 (N/m 3 ) - Forming step 30 degrees Contact stiffness 21.3e11 (N/m 3 ) - Forming step 30 degrees Contact stiffness 66.6e11 (N/m 3 ) - Forming step 80 degrees Contact stiffness 66.6e11 (N/m 3 ) - Forming step 30 degrees Time (s) Figure 4-24: Reaction force on tools during friction roll forming (30 and 80 forming step) with solid elements using different contact stiffness 5 Stability quotient 7% 6 Stability quotient 38%

60 The figure below presents the reaction force during forming step 80, the reaction force shows increasing pattern for increased contact stiffness. Reaction force (N) Friction roll forming model Reacton force on tools during forming step 80 degrees for different contact stiffness Contact stiff ness 8e11 (N/m 3 ) Contact stiff ness 16e11 (N/m 3 ) Contact stiff ness 21.3e11 (N/m 3 ) Contact stiff ness 66.6e11 (N/m 3 ) Time (s) Figure 4-25: Reaction force on tools during friction roll forming (80 forming step) with solid elements using different contact stiff Stability quotient influences the appearance of the reaction force although not differing substantially in magnitude. The figure below presents the reaction force on the tool set during the 80 forming step for the friction roll forming model with different Friction roll forming model Reaction force on tools during forming step 80 degrees for equal contact stiffness and different stability quotient Stability quotient 7% Stability quotient 12% Stability quotient 38% Reaction force (N) Time (s) Figure 4-26: Reaction force on tools during friction roll forming (80 forming step) with solid elements with different stability quotient

61 Summarising the results in tables yields the following, as one can see the mean reaction force increases with contact stiffness and decreases with stability. Friction roll forming model Contact stiffness Element Mean reaction force Solid 4619 N Solid 6350 N Solid 7079 N Solid 9367 N Table 4-39: Mean reaction force for friction roll forming model with different contact stiffness using solid elements Friction roll forming model Stability quotient Element Mean reaction force 38 % Solid 6350 N 12 % Solid 6125 N 7 % Solid 5710 N Table 4-40: Mean reaction force for friction roll forming model with different stability quotient and equal contact stiffness Comparing reaction force between the friction and translation roll forming model using the same contact stiffness yields an increase of ~70% for the friction model. Contact stiffness Element Mean reaction force Model Solid 6350 N Friction model Solid 3719 N Translation model Table 4-41: Mean reaction force for friction and translation roll forming model with equal contact stiffness

62 4.2.8 Sheet shape and contact pattern Node nr 1 Upper side Perspective P1 Inner side Perspective P2 Node nr 23 Figure 4-27: Description of node numbering during chapter and and definition of upper and inner side of sheet The sheet shape is analysed by extracting coordinates for 23 nodes alongside the inner side of the sheet (see above figure) these coordinates are furthermore plotted together with coordinates for the tool surface for the lower tool. It is then possible to visually see where the sheet is in contact with the tool. To enhance the apprehension of how and where the sheet is in contact with the tools (upper and lower), iso-plots showing contact surfaces seen from two perspectives. One perspective referenced as P1 shows the contact surface between the upper tool and the sheet for the different forming steps, the perspective referenced P2 shows the contact surface between the lower tool and the sheet for the different forming steps. The iso-plots do not contain any information regarding the pressure in actual numbers, hence only information regarding the magnitude relative other surface elements can be interpreted. The applied scale legend is approximate. The contact pattern is similar (but of different magnitude) between all simulations regardless of contact stiffness, tool gap size and stability except for the last forming step (80 ) Se figure Tool docility during 80 degree forming step for friction model and translation model with contact stiffness 16.0e11 Friction model Stability quotient 7% Tool Friction model Stability quotient 38% Tool Translation model Stability quotient <1% Tool y coordinate (m) y coordinate (m) y coordinate (m) x coordinate (m) x coordinate (m) x coordinate (m) Figure 4-28: Sheet shape during 80 forming step for friction model and translation model with contact stiffness (N/m 3 )

63 2.5 cm 1.0 cm 2.5 cm 1.0 cm Figure 4-29: Iso-plot showing contact area with perspective P1 left and P2 right during forming step 10 Above figure shows the contact area during the first forming step (10 ). The upper tool exhibits pressure alongside almost the whole flange while the lower tool just exhibits pressure in the centre of bend. Note that no contact exist on the outside of the bend in centre of the sheet (perspective P1). 2.5 cm 2.5 cm 1.0 cm 1.0 cm Figure 4-30: Iso-plot showing contact area with perspective P1 left and P2 right during forming step 20 During forming step 20 the contact zone for the lower tool stays intact while the contact zone for the upper tools changes geometry and form two contact islands. 2.5 cm 2.5 cm 1.0 cm 1.0 cm Figure 4-31: Iso-plot showing contact area with perspective P1 left and P2 right during forming step

64 During forming step 30 the contact zone for the upper tool have the same pattern except for a narrow partition, and the contact zone for the lower tool decreases but still in centre. 2.5 cm 2.5 cm 1.0 cm 1.0 cm Figure 4-32: Iso-plot showing contact area with perspective P1 left and P2 right during forming step 40 During forming step 40 the contact zone for the upper side shows the same pattern. The lower tool contact has moved away from the centre of the bend indicating less contact along symmetry line of the sheet. 2.5 cm 1.0 cm 2.5 cm 1.0 cm Figure 4-33: Iso-plot showing contact area with perspective P1 left and P2 right during forming step 50 During forming step 50 the contact zone for the lower continues to move towards the flange of the sheet, leaving the centre of the sheet with no contact. The upper tool changes in form as the previous line contact along the flange has ended up with two separate islands where the right island has moved all the way to the end of the flange and further back indicating contact earlier with the tool set

65 2.5 cm 1.0 cm 2.5 cm 1.0 cm Figure 4-34: Iso-plot showing contact area with perspective P1 left and P2 right during forming step 60 During forming step 60 the contact zone for both upper and lower tools shows the same pattern as previous forming step (i.e. 50 ). Note the clear difference in contact pattern for the lower tool between this forming step and first four forming steps. 2.5 cm 2.5 cm 1.0 cm 1.0 cm Figure 4-35: Iso-plot showing contact area with perspective P1 left and P2 right during forming step 70 During forming step 70 the contact zone for both upper and lower tools shows the same pattern and behaviour as previous forming step. 2.5 cm 1.0 cm Contact at symmetry line 2.5 cm 1.0 cm Figure 4-36: Iso-plot showing contact area with perspective P1 left and P2 right forming step

66 During forming step 80 the contact zone for the lower tools show the same pattern as the previous forming step with the exception of the contact zone getting wider and narrower than previous steps. The contact zone for the upper tool now has also contact along the symmetry line in centre of the sheet (to the left of circular field near the centre with perspective P1) Final sheet geometry Node nr 1 Upper side Inner side Node nr 23 Figure 4-37: Description of node numbering during chapter and and definition of upper and inner side of sheet In order to evaluate the final cross section sheet geometry for the sheet it is necessary to stipulate certain designations. The geometry of the sheet is extracted to results by collecting the coordinates for 23 nodes along the inner side of the sheet. These values are then interpolated by linear interpolation method. The same procedure is done for the tool geometry (80 tool, since the concern is evaluating final geometry which is after leaving the last tool set i.e. last forming step 80 ). The upper side of the sheet is the side of the V-profile is experiencing contraction whilst the outside is the side experiencing extraction. With final geometry one means the cross section geometry after leaving the last forming step and after the relaxation step. The features of interest is of course springback which has been discussed earlier as well as wrinkles along the flange is due to plastic flange strain and has been also been discussed in earlier chapters. During this chapter buckles and irregularity present in the cross section geometry will be analysed

67 Final cross section sheet geometry for friction model and translation model for contact stiffness 16e11(N/m 3 ) Tool (80 degrees) Tool (80 degrees) Tool (80 degrees) Friction model Friction model Translation model Stability quotient 38% Stability quotient 7% Stability quotient <1% y coordinate (m) y coordinate (m) y coordinate (m) x coordinate (m) x coordinate (m) x coordinate (m) Figure 4-38: Final cross section sheet geometry for friction model and translation model for contact stiffness (N/m 3 ) As a tool to evaluate the above figures more quantitative, a reference evaluation line has been inserted to the plots in order to magnify any buckles or irregularity along the cross section. There are no irregularities or buckles present when analysing figure 4-38 showing the final cross section geometry for a stable friction roll forming simulation. All simulations show the same result. The conclusion is therefore that the final geometry is satisfactory and that both the stable and unstable frictional roll forming model produces satisfactory cross section geometries (see appendix B-4 for all evaluation figures). Final cross section sheet geometry with reference evalution line -0,040-0,045-0,050 Friction model Stability quotient 7% Contact stiffness 16e11(N/m^3) Reference evaluation line -0,055 y coordinate (m) -0,060-0,065-0,070-0,075-0,080-0,085-0, ,002 0,01 0,006 0,008 0,01 0,012 0,014 x coordinate (m) Figure 4-39: Final cross section sheet geometry after relaxation step at ~21 seconds with reference evaluation line plotted along the sheet flange for friction roll forming model with stability quotient 7% and contact stiffness N/m

68 Springback Springback behaviour is among the most important behaviour to examine during roll forming process. Springback occurs as explained in chapter due to unloading of the elastic state causing the sheet to relax and springback a certain amount. During simulations various parameters affect on springback has been examined. Residual strains and stresses are used in this chapter as it offers some explanation to the different behaviour due to changes in studied parameters. Parameters examined in this thesis that affect springback are: Contact stiffness Gap size Friction coefficient Model dependent (translation roll forming model / friction roll forming model) Mass scaling (i.e. time increment used) In the table below springback for the friction roll forming model and the translation roll forming model is presented in degrees and as percent (deviation from 80 which is the last forming step) with the use of different contact stiffness. Springback decreases by increased contact stiffness. The difference between the two models (friction roll forming model and translation roll forming model) are ~ 1-3% which is marginal and the springback between the different models can with tolerance be concluded as equal. Contact stiffness (N/m 3 ) Springback Translation model ( ) Springback Friction model ( ) Springback Translation model (%) Springback Friction model (%) Table 4-42: Springback angle for different contact stiffness for friction and translation roll forming model after the relaxation step at ~21 seconds Contact stiffness (N/m 3 ) Relative deviation in springback between friction and translation roll forming model % % % % Table 4-43: Deviation in springback angle for different contact stiffness between friction and translation roll forming model after the relaxation step at ~21 seconds 7 Time increment used ( t) is smaller than the other used in this comparison. Springback decreases with stability and therefore with lower time increment. A lower value is therefore achieved. See table 4-1 and 4-45 for t influence on stability and springback values

69 In lack of experimental values in terms of springback, the final evaluation and final calibration of the model is not possible. As a tool for final adjustment of the model when experimental values are achieved it can be used to pin point a springback value and the contact stiffness can be chosen with use of the figures below. Below figure shows a fitted model to the springback values versus the contact stiffness. The data is fitted using a power model y = Ax B +C and the goodness of the fit (r 2 = ). The data is plotted using logarithmic scale for the contact stiffness axle. This procedure is done for both models i.e. the translation and the friction roll forming model (see figure 4-40 and figure 4-41). 7 6 Sprinback angle versus contact stiffness fitted to power model y=a*x b +c for friction roll forming model Springback angle at a certain contact stiffness Power model fitted to data: y = a*x^b+c R-square = Sprinback angle (degrees) Contact stiffness (N/m 3 ) Figure 4-40: Springback angle after the relaxation step at ~21 seconds versus contact stiffness fitted to power model for friction roll forming model 7 6 Springback angle versus contact stiffness fitted to power model y=a*x b +c for translation roll forming model Sprinback angle at a certain contact stiffness Power model fitted to data: y = a*x^b+c R-square = Springback angle (degrees) Contact stiffness (N/m 3 ) Figure 4-41: Springback angle after the relaxation step at ~21 seconds versus contact stiffness fitted to power model for translation roll forming model

70 The springback values for different tool gap size are presented in the table below. The deviation in springback between gap size 1.65 mm and 1.54 mm is ~5%, this is due to the extra gap space available in the larger gap model. Tool gap size Springback ( ) Springback (%) 1.65 mm mm Table 4-44: Springback angle after the relaxation step at ~21 seconds for different tool gap size The springback values for different stability quotients are presented in the table below. The deviation in springback between stability quotient 38% and 12% is ~6% and between stability quotient 12% and 7% is ~1.2%. Model Stability (%) t (s) Springback ( ) Springback (%) Friction model 38% Friction model 12% Friction model 7% Table 4-45: Springback angle after the relaxation step at ~21 seconds for different mass scaling (i.e. t) for friction roll forming model Using different friction coefficients has also an affect on springback which will be further discussed in the friction chapter As the values present in the below table the deviation in springback between an extremely high friction coefficient µ=0.65 and a normal friction coefficient µ=0.12 is ~17% lower value for the high friction model. Values for the low friction model could not be achieved due to the fact that the sheet did not pass all the tool sets due to the lack of sticking contact between the sheet and the tool which has its explanation in the low friction coefficient applied. There is a critical friction needed in order for the sheet to gain enough force to pass each tool set. This force is exerted by the tools behind as well as the tool it is entering. The force needed is the force necessary to bend the sheet 10 for each forming step and some force to pull it forward. Friction coefficient t (s) Springback ( ) Springback (%) µ = µ = µ = Table 4-46: Springback angle after the relaxation step at ~21 seconds for different friction coefficients for friction roll forming model

71 Accuracy in springback calculations Due to the fact that only a small amount of damping (see chapter 2.4.9) is introduced during simulation the sheet will oscillate after exiting the forming process. During the relaxation step which lasts for one second, the sheet is meant to stabilize in order to achieve accurate results from springback coordinates extracted according to chapter This chapter will evaluate achieved results versus error-calculations as a measure of how affective the relaxation step is. The flange node coordinates are extracted from element number 60 counted from in front of the sheet. That certain element leaves the forming process at approximately 17 seconds into the main forming step which lasts for 20.5 seconds. The end of the sheet leaves the forming process at approximately 19.5 seconds. Therefore the extracted node has been under relaxation during 4.5 seconds but under influence from the remaining sheet which is being formed until 19.5 seconds, which leaves us with a total relaxation time of 2 seconds (i.e. when residual time from step 1 is added to relaxation step 2). In the figure below one can see the x coordinate plotted over the period of time seconds for two different analyses with different stability quotient. As noted the coordinate are showing a planar shape but with traces of oscillations. It can be identified that the x coordinate for the less stable run (38%) oscillates more explicitly. -0,040-0,028 Coordinate in x direction for flange node at end of friction roll forming process for different stability quotient Stability quotient 7% Stability quotient 38% -0,030 x coordinate (m) -0,032-0,034-0,036-0,038-0, Time (s) Figure 4-42: Coordinate in x direction for flange node at end of friction roll forming process during relaxation for different stability quotient Examining the coordinates more closely and extracting the minimum and maximum values of the oscillations an error range can be calculated for each of the two stability quotients (7% and 38%). Looking at figure 4-43 and 4-44, the maximum and minimum coordinate has been marked and the same procedure has been repeated for both stability quotients as well as for the y coordinate. The information is presented in table 4-47 and table 4-48 below. The error limits are calculated by applying maximum and minimum coordinates to the flange node. Then calculating the change of angle for the sheet with these new coordinates representing maximum error gives the maximum springback error. These values are presented in table 4-49 and

72 Coordinate in x direction for flange node at end of friction roll forming process for stability quotient 7% -0, ,03510 Maximum x coordinate Stability quotient 7% x coordinate (m) -0, , , ,03530 Minimum x coordinate -0, , Time (s) Figure 4-43: Coordinate in x direction for flange node at end of friction roll forming process during relaxation step (~ seconds) for stability quotient 7% Coordinate i x direction for flange node at end of friction roll forming process for stability quotient 38% Stability quotient 38% x coordinate (m) Minimum x coordinate Maximum x coordinate Time (s) Figure 4-44: Coordinate in x direction for flange node at end of friction roll forming process during relaxation for stability quotient 38% Stability quotient Extracted max x coordinate Extracted min x coordinate 7% % Table 4-47: Extracted flange node x coordinates at max and min position during relaxation step These coordinate values are then calculated in terms of springback error for both simulations, i.e. 7% and 38% stability and the results can be found in table 4-49 and 4-50 below

73 Coordinate in y direction for flange node at end of friction roll forming process for different stability quotient Stability quotient 7% Stability quotient 38% y coordinate (m) Time (s) Figure 4-45: Coordinate in y direction for flange node at end of friction roll forming process for different stability quotient Looking at figure 4-45 above one can notice that the y-coordinate for the simulation with 7% stability quotient is increasing by time in an oscillating motion. One immediate thought would be that springback increases uncontrolled during relaxation step by observing the coordinates. This is not the case, the graph shows the same oscillating motion as for the x- coordinate with the difference that it is climbing. The reason for the climbing is the sheet translating upwards in positive y-direction, it has therefore nothing to do with springback. This is caused by the velocity and moment that is exerted in the rear of the sheet as it exits the last forming step. This affect would be partly eliminated by applying a boundary condition to the relaxation step to prevent the symmetry line of the sheet to move in y- direction but as the aim is to manipulate as little as possible with real life parameters it was chosen not to. This has no influence in the springback calculations or on any other results and is therefore not a problem Coordinate in y direction for flange node at end of friction roll forming process for stability quotient 7% Stability quotient 7% y coordinate (m) Maximum y coordinate for one cycle Minimum y coordinate for one cycle Time (s) Figure 4-46: Coordinate in y direction for flange node at end of friction roll forming process during relaxation step process (~ seconds) for stability quotient 7%

74 -0,0420-0,0425 Coordinate in direction for flange node at end of friction roll forming process for stability quotient 38% Stability quotient 38% -0,0430 y coordinate (m) -0,0435-0,0440-0,0445-0,0450-0,0455 Maximum y coordinate Minimum y coordinate -0,0460-0,0465-0, Time (s) Figure 4-47: Coordinate in y direction for flange node at end of friction roll forming process for stability quotient 38% Stability quotient Extracted max y coordinate Extracted min y coordinate 7% % Table 4-48: Extracted flange node y coordinates at max and min position Stability x coord y coord Max Springback error (m) (m) springback error mean value ε =ε Max -ε Min 2 ε Mean =ε Max -ε Min (ε ) (ε ) Stability 7% Stability 38% Table 4-49: Relative error in springback due to sheet flange vibration Stability Springback angle ( ) Max deviation of springback value (ε Max /Springback) Stability 38% % 5.2% Table 4-50: Relative error in springback as percentage of actual springback angle due to sheet flange vibration Looking at the values in table 4-49 one can see that the error in springback is in the range 0.24 for the analysis with 7% stability and 0.49 for the analysis with 38% stability. By running with 7% stability the error is thus reduced by ~50% which is a significant reduction. One must of course account for the expense in computational cost which is 35 hours on the reference computer used in this thesis. These values are then presented as percentage of actual springback and one can see that we are in the range ~5-11%. This is within reasonable limits for the more stable analysis (7% stability) but in the upper boundary for the less stable analysis (38% stability) Mean deviation of springback value (ε Mean /Springback) Stability 7% % 2.8%

75 Friction In terms of evaluating the roll forming process using friction and rotating tools, simulations has been performed using different friction coefficients in order to identify the extent of influence of this parameter. The different configurations used are: Low friction analysis (µ=0.01) Normal friction analysis (µ=0.12) High friction analysis (µ=0.65) Possible affects of the variation of friction coefficient could be that flange strain is increased as the sheets sticks (slipping occurs for a higher shear force) in a higher extent to the tools and therefore enforces a stronger pulling motion when the sheet is between two tools which could be a danger if the flange strains turn plastic. This is most critical for the flange since it travels the furthest distance, but of course plastic strain in z direction is unwanted anywhere on the sheet, even in the centre since plastic strain in z direction results in bad geometry and wrinkles along the sheet. Springback could also be affected due to changes in residual strains and stresses. The acceleration of the sheet in the beginning of the process should be increased as less slipping will be present and therefore the sheet will gain full velocity in a shorter period of time. This could affect the process in terms of presenting dynamic affects resulting in buckling and distorted elements during simulation. 0,18 Velocity in z direction for sheet with different friction coefficients 0,16 0,14 Velocity in z direction (m/s) 0,12 0,10 0,08 0,06 0,04 0,02 0 Normal friction (coeff = 0.12) High friction (coeff = 0.65) -0,02 0 0,5 1,0 1,5 2,0 2,5 3,0 Time (s) Figure 4-48: Velocity in z direction for sheet with different friction coefficients during the beginning of the forming process (first 3 seconds) The above figure shows a plot over the velocities for the sheet for the two different friction coefficients and as noted the acceleration for the high friction simulation is larger than the acceleration for the normal friction. In the table below the acceleration is calculated by extracting the time stamps and the velocity for both curves in the area from acceleration from zero to full velocity. Model t=t 2 -t 1 (s) v=v 2 -v 1 (m/s) Mean acceleration (m/s 2 ) Normal friction µ = High friction µ = Table 4-51: Table for time stamp and velocity extraction from curves and accelerations

76 The flange strain for the two simulations with different friction coefficients are plotted in below figure, as one can see the high friction simulation results in higher strain values when observing the peaks for each forming step (eight peaks excluding the first peak which is during the initiation of motion for the sheet at the three flat rolls). 0,008 0,007 0,006 True strain in flange of sheet for inner fibre for different friction coefficients Normal friction (coeff = 0.12) High friction (coeff = 0.65) Plastic limit 0,005 0,004 True strain 0,003 0,002 0, ,001-0,002-0, Time (s) Figure 4-49: Flange strain for simulations with different friction coefficients Extracting the strain at each forming step peak for the two simulations and calculating the deviation in strain and in percent increase from normal friction to high friction strain, the following data is obtained (see table 4-52). Form step High friction Max true strain Normal friction Max true strain Deviation ε = ε µ=0.65 -ε µ=0.12 Deviation (%) % % % % % % % % Mean deviation (µ = 0.65 vs. µ = 0.12) % Table 4-52: Peak strain for each forming step for simulations with different friction coefficients with deviation in value and percent presented The flange strain between the two friction models is changed by 17.38% which is a distinct increase due to a higher friction coefficient. Also noticeable is that the flange strain exceeds the plastic limit at one point (during forming step 30 ) which is critical in terms of producing wrinkles along the flange and obtaining a weak final geometry. Springback values for the different frictions are presented in table As discussed in the springback chapter, the springback occurs after unloading of the elastic state resulting in residual strain and stress through the thickness of the sheet forcing the sheet to springback. In order to evaluate the difference in springback the residual data has been collected through the thickness and presented in table

77 Friction coeff t (s) Springback ( ) Springback (%) µ = µ = µ = Table 4-53: Springback values for the different friction simulations after the relaxation step at ~21 seconds The table above presents the springback values for the different friction simulations and as presented in the table there are differences between obtained values for the high and the normal friction. This is probably due to the fact that the high friction simulation forces the sheet through the forming process in a very fixed and distinct way and allows little slip resulting in higher residual stress through the thickness as presented in table Thickness coordinate PEEQ µ=0.12 (%) PEEQ µ=0.65 (%) σ 11 µ=0.12 (MPa) σ 11 µ=0.65 (MPa) ε 11 µ=0.12 (%) ε 11 µ=0.65 (%) 0.00 mm mm mm mm mm mm Mean deviation 2.5 % 67.2 % 2.6 % Table 4-54: Residual data for the two different friction simulations (µ = 0.65 and µ = 0.12) after the relaxation step at ~21 seconds The accumulated plastic strain as well as the strain across the sheet is not changed more than marginally. The main deviation is found in the residual stress through the thickness of the sheet where a mean deviation of 67.2% is obtained. This is the main reason for the deviation in springback between the friction models. Results from the simulations with extremely low friction coefficient have not been presented due to the fact that the simulation did not complete successfully. The sheet did not pass forming step number three (30 forming step) as a result of the friction being to low and therefore the sheet does not gain enough contact force for it to stick to the tools and maintain velocity. Instead the sheet slips and when it finally stops at the third forming step it is because of the force gets critical since the tools can not deliver the force necessary for the sheet to bend further as a lack of grip. Summarising above results and discussions in a table yields: Friction coefficient Deviation in springback Increased strain ε 33 (flange) Increased acceleration (start of step) Successful simulation Simulation time µ = No - µ = (0 %) 0 % 0 % Yes 11h 10m µ = (-16.8 %) % 8 92 % Yes 12h 48m Table 4-55: Summary of the deviations between normal and high friction simulations 8 Exceeds plastic limit (see figure 4-49)

78 Velocity differences (applies only to the friction roll forming model) Recalling chapter concerning rotation of the tools in the friction roll forming model. As discussed, due to difference in geometry between upper and lower tools, velocities will diverge as the sheet is being formed. This is especially explicit during the last forming steps in the flange as the difference increase by forming angle (see figure 3-29, page 27 chapter ). As the velocity of the tool surfaces differs by ~0.3 m/s during the last forming step (80 forming step) it is of vital importance to investigate how this affects the sheet during forming and if this leads to radical divergence in velocities on the nodes in the flange in the sheet which would result in large shear stresses or translational shear of the sheet during forming because of the velocity difference between the centre and flange of the sheet which would cause the sheet to deviate from normal forming procedure. Large deviations in velocities between centre and flange could also have an affect on flange strain which theoretically could increase if the flank velocity would increase dramatically as the sheet has contact with two forming steps (as the sheet has during the whole forming process, except for when exiting the forming process at the 80 tool) and therefore a stretching motion could be created. The figure below shows plots over velocities for upper and lower node in the flange of the sheet followed by a plot over the divergence between the two velocities marked with the maximum diversity which in this case is m/s at time stamp 2 seconds. These plots are made from output from friction roll forming model with a time increment of resulting with a stability quotient of 7%. Integrating the area under this graph between the two time stamps at zero diversity gives the total extra travelled distance for the node with the higher velocity. This calculations gives the distance m. This would indicate that the sheet flange has sheared 0.9 mm through the thickness. This corresponds 60% of the thickness. In order to investigate this further, the translation for the same two nodes in z direction has been plotted, and as noted there is no indication of the nodes reaching ahead of each other (see figure 4-50 below). Therefore this issue is not regarded as a problem. In appendix E the same comparison for a similar analysis with a higher stability quotient (38%) can be found where the results indicate the same conclusion. Velocity in z direction for upper and inner node in flange of sheet Velocity in z direction (m/s) 0,200 0,195 0,190 0,185 0,180 0,175 0,170 Velocity for upper node Velocity for inner node 0,165 0, Time (s) Figure 4-50: Node velocity in flange for upper and inner node

79 Divergence in velocity in z direction between upper and inner node in flange of sheet 0,003 Max diversity Divergence in velocity between upper and inner node in flange of sheet Divergence in velocity (m/s) 0,002 0, ,001-0, Time (s) Figure 4-51: Divergence in velocity in flange of sheet for upper and inner node with maximum diversity marked with arrow 3,5 Translation in z direction for upper and inner node in flange of sheet 3,0 Translation in z direction (m) 2,5 2,0 1,5 1,0 0,5 Translation for upper node Translation for inner node Time (s) Figure 4-52: Node translation in z direction in flange of sheet for upper and inner node The second issue was to investigate the velocity difference influence between centre and flange velocities. In figure 4-53 below the plot is showing the velocity for a node on the upper surface in the centre and in the flange in the same cross section. Maximum deviation in velocities is as marked in figure 4-54 is m/s. This after integration over time results in a shear of 1.4% of the sheet width which can be concluded as marginal. Concerning the flange strain theory no indication of increased strain due to velocity differences can be found in flange strain that could appear due to a stretching motion as the sheet has contact with two tool sets (see table 4-35, chapter 4-24 for flange strain values during the forming steps)

80 Summing achieved results, no indication of critical shearing due to velocity differences can be found between upper and lower node in flange of sheet as well as between node in centre and in flange of sheet. Velocity in z direction for upper node in centre and flange of the sheet 0,205 0,200 Velocity for upper centre node Velocity for upper flange node Velocity in z direction (m/s) 0,195 0,190 0,185 0,180 0,175 0,170 0,165 0, Time (s) Figure 4-53: Node velocity in centre and flange of sheet (upper side nodes only) 0,004 0,003 Divergence in velocity in z direction between centre and flange upper nodes Divergence in velocity between centre and flange upper nodes Divergence in velocity (m/s) 0,002 0, ,001-0,002-0,003 Max diversity -0, Time (s) Figure 4-54: Divergence between upper nodes in flange and centre of sheet with maximum diversity marked with arrow

81 5. Discussion Due to the complexity of roll forming, an extensive amount of simulations have been performed during this thesis enabling the study of parameters affect on the forming process and achieved results. The analysis of the roll forming process can basically be divided into two areas, one is the simulation parameters affects and the other is forming parameters affect on the roll forming process. Simulation parameters 1. Contact stiffness Contact stiffness influences the roll forming simulations greatly in terms of springback values and residual true strain through the thickness of the sheet in centre of bend, after the relaxation step. Increasing contact stiffness decreases springback and residual true strain through the thickness of the sheet. As the residual stress state strongly influences springback, the stress after the relaxation step has been extracted through the thickness of the sheet in centre of bend. As one can see in the figure below the residual stress in the thickness is strongly increased by contact stiffness which could counteract springback leaving the sheet with as little as 1.27 springback. This increased residual stress indicates that the stress at these points must have been larger during the forming process leaving larger residual stress after unloading of the elastic state. Contact pressure for the high contact pressure shows that contact pressure is 7 times larger for the higher stiffness during forming step 70 and 80 degrees where contact exists on the upper centre nodes. Figure 5-2 shows the sheet shape for a high contact stiffness simulation versus a low contact simulation. One can see the difference geometry of the lines showing that bend geometry between the two simulations are different. 1,5 Residual stress through the thickness of the sheet in centre of bend for translation roll forming model with different contact stiffness Thickness coordinate (mm) 1,0 0,5 Contact stiffness 8.0*10 11 N/m 3 Contact stiffness 16.0*10 11 N/m 3 Contact stiffness 21.3*10 11 N/m 3 Contact stiffness 66.6*10 11 N/m 3 Contact stiffness *10 11 N/m Residual stress (MPa) Figure 5-1: Residual stress through the thickness of the sheet in centre of bend for translation roll forming model for different contact stiffness after the forming process at last frame in step 2 (relaxation step)

82 -0,046-0,047 Tool pliancy for translation roll forming model with different contact stiffness (high versus low) during forming step 80 degrees Contact stiffness 16.0*10 11 N/m 3 Contact stiffness *10 11 N/m 3-0,048 X coordinate (m) -0,049-0,050-0,051-0,052-0, ,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 0,0040 Y coordinate (m) Figure 5-2: Sheet shape for translation roll forming model for different contact stiffness (high versus low) during forming step 80 Contact stiffness is recommended to be as high as simulation allows hence in mathematical terms the contact stiffness would be infinite to prevent any overclosure. Due to that the tools are analytically rigid no deformation is possible and contact stiffness has to be adjusted for the forming process to proceed. High contact stiffness results in a harder contact which results in a higher contact pressure, distinct higher reaction forces and increased node velocities during the forming steps. 2. Mass scaling Mass scaling in terms of changing time increment influences stability of the roll forming simulations. Stability quotient is recommended to stay around 10% for more accurate results. Difference between results with stability 38% and 7% show a 9% deviation in springback and marginal difference in residual true strain and accumulated plastic strain through the thickness of the sheet after relaxation (approx 2%). Differences in stress at upper and inner node through the process deviates with 50% and flange strain values deviates with 25% between stabilities. Values for principal stress during the forming process deviate from 13% -52% depending on how stable the process is. Reaction forces decrease for increased stability by approximately 10% which can partly be explained by the forming process going smoother due to smaller time increments resulting in less overclosure between each time increment, decreasing reaction force. Concluding this 12% in stability quotient is recommended for accurate results. 3. Shear locking Shear locking occurring using fully integrated linear 8-node elements making the element over stiff for extensive bending, leaving large residual stress (50% of yield strength) as well as residual strains increasing and indicating that neutral line moving towards the outer surface of the bend. Therefore linear fully integrated solid elements (C3D8) are not recommended during roll forming

83 4. Hourglass modes Enhanced hourglass mode using hexahedral 8-node reduced integrated first order (C3D8R) elements has shown bad performance independent on simulation stability. Enhanced hourglass option does not add any time to simulation and lowers artificial strain affectively although the achieved results are not truthful. Normal hourglass control is recommended when using C3D8R element subjected to large bending operations. Reducing the amount of artificial strain energy is also possible using a lower time increment although with extensive computational cost. 5. Element evaluation Element evaluation shows that 5 elements through the thickness are better than 7 which yield results such as the neutral line moving towards the outer fibre of the bend which is not possible during this forming process. Increasing the amount of elements in width from 7 to 14 increases residual true strain by approximately 0.1 which is a large deviation and far from the theoretical value. Residual stress is also largely increased to a magnitude of 50% of yield strength, which is large in terms of residual stress indicating weak results. Decreasing down to 4 elements give large deviations in springback which reduces to 2.5 degrees. Decreasing the amount of elements in length direction does not influence the results more than marginal showing that the process is not so 3D dependent. Concluding the above discussion together with simulation times 5 elements in the thickness and 7 in the centre section was chosen as the final mesh configuration. 6. Simulation time Simulation time is highly dependant on the mesh configuration, the time increment (i.e. mass scaling) and the type of element used (i.e. reduced and full integration as well as number of nodes and degrees of freedom). During this thesis two roll forming models have been simulated (translation roll forming model and friction roll forming model). Since the translation model is more stable than the friction model a smaller time increment has to be used for the friction roll forming model in order to achieve the same stability quotient. Simulation time for a stable simulation using the friction roll forming model is ~33h whilst using the model without friction the simulation time is ~11h. Simulation using the friction roll forming model therefore consumes three times as much time as the translation roll forming model, in the case of a stable simulation. Forming parameters 1. Friction High friction versus normal friction (µ=0.65 versus µ=0.12) shows significantly increased acceleration (92% increase) in the beginning of the forming process. This must be monitored closely in not achieving any unwanted dynamic affect. The use of smooth step option is possible. Furthermore high friction causes a 19% increase in flange strain and flange strain passes plastic strain limit during one forming step. The difference in springback is 9% (~1 ) between the models. Using higher friction does

84 not impose any additional computational cost. Using lower friction (µ=0.01) causes the sheet to stop during the forming process due to lack of friction force preventing the sheet from passing all the tools. Results are summarised in the table below: Friction coefficient Deviation in springback Increased strain ε 33 Increased acceleration (start of step) Successful simulation Simulation time (flange) µ = No - µ = (0%) 0 % 0 % Yes 11h 10m µ = (-17%) % 9 92 % Yes 12h 48m Table 5-1: Results for different friction coefficient with the friction roll forming model 2. Gap size Different gap sizes have been simulated in order to analyze the influence of clearance between the sheet and the tool. The two clearances used are 10% (1.65 mm gap) and 2.7% (1.54 mm gap). Gap size influences in terms of springback, reaction force and contact pressure, and have marginal affects on residual true strain and stress as well as flange strain. Reaction forces increase by 25% during forming step 80 and contact pressure at upper surface nodes in centre of bend with ~400MPa or 40% during the same forming step. Springback deviation is 0.23 (~4%) between the two gap sizes which is mainly due to the fact that the larger gap size model allows for larger springback during each forming step (i.e. not forcing it to 80 bend due to tool gap), the difference is yet very small and therefore negligible. As mentioned above, residual true strain and stress as well as plastic strain values contain marginal differences. Gap size does however influence the forming process and should be chosen carefully. Gap size in real life process is among other things affected by bearing play and how worn they are except for the adjusted gap set. Therefore there are no handbook solutions for FE simulations, hence trial and error has to be used. 3. Velocity differences between tools The introduction of a friction model also introduced new questions such as the influence of velocity differences between the upper and lower tools which have proven not to have caused critical shearing of the sheet for both normal and high friction as well as no increased flange strain due to different velocity of the sheet between two tool sets. This proves that the coloumb friction model works well with simulations and slipping occurs naturally. Contact Contact pattern for the roll forming process has been presented using iso-plots as well as the sheet shape figures. Important noticing is that only during the last forming steps (70 and 80 ) show contact along the symmetry line in the centre of bend, which indicate that the sheet is air bended through the other steps. Contact pressure is increased by increased contact stiffness and decreased tool gap size. The contact distribution between the different 9 Exceeds plastic limit (see figure 4-45)

85 forming steps changes between the translation and friction model. The friction model using normal friction (µ~0.1) results in a larger contact pressure during forming step Contact pressure on upper nodes (30 nodes) in centre of bend for friction and translation roll forming model at each forming step (contact stiffness 16*10 11 N/m 3 ) Normal friction model High friction model Translation model Figure 5-3: Contact pressure on upper surface nodes (30 nodes) for friction (high & low) and translation roll forming model at each forming step with low contact stiffness The contact on the inner side of the bend is shown in the figure below and it can be seen that the friction model has contact during all the forming steps except for the last step (80 ) while the translation model looses contact at forming step 60 and forward. Also noticing that the contact pressure for the friction model is larger than for the translation model. The friction model therefore has a more propagated contact pattern than the translation model. Contact pressure (MPa) Contact pressure on inner nodes (30 nodes) in centre of bend for friction and translation model with low contact stiffness (at each forming step) Normal friction model High friction model Translation model Forming step (degrees) Figure 5-4: Contact pressure on inner surface nodes (30 nodes) for friction (high & low) and translation roll forming model at each forming step with low contact stiffness Final geometry Final cross section sheet geometry did not show any difference between any of the models. All models have good final geometries without buckling or wrinkle alongside the flange. Tolerances are dependent on the individual process and there is no overall standard to be applied. Buckling or wrinkles along the cross section is yet always unwished for and should be eliminated. Final thickness Final thickness of the sheet for the translation and the friction model is changed by ~ % and is a marginal. The sheet thickness can therefore be said to keeping its thickness throughout the process and no thinning is present

86 Flange strain No plastic strain is present in any of the models except for the high friction model (µ=0.65) where plastic strain is achieved during one forming step. Flange strain is dependent on tool set and material properties which influences the deformation zone of the sheet. Plastic flange strain is one important parameter which has to be controlled since plastic strain produces wrinkles along the flange which are unwanted. Friction roll forming model versus translation roll forming model Friction present in the roll forming model has shown small deviance in results and therefore justifying the translation model as a plausible simplification that has not diminished significantly on the results. The friction roll forming model is not more time consuming using the same time increment (which gives a lower stability for the friction model) than the translation roll forming model. If the same stability is desired the friction model will cost about 3 times as much in simulation time. The translation model is less sensitive to simulation parameters compared to the friction model, enabling use of a higher contact stiffness which should portray reality since in real life process no overclosure between the tool and sheet is present. Springback Springback is strongly influenced by contact stiffness. As a tool to adjust the model to the correct springback value when experimental results are achieved, a diagram with a power function fitted to springback data versus contact stiffness has been produced (see below). Springback is furthermore affected by residual stress through the thickness. Therefore a more narrow stress field can counteract springback which was discussed during the contact stiffness topic above, where a large difference in springback could be observed with a possible explanation in the residual stress field through the thickness. Accuracy in the springback values are good and show a mean deviation of not more than ~2% using a stable simulation (stability quotient ~10%) and ~5% using a unstable simulation (stability quotient >12%). Conclusion is that default damping is enough for springback extraction during the relaxation step which should last for~ 1 second. Springback values between the friction roll forming model and the translation model only differs by ~1-3% which once again show good agreement between the models. 7 6 Springback angle versus contact stiffness fitted to power model y=a*x b +c for translation roll forming model Sprinback angle at a certain contact stiffness Power model fitted to data: y = a*x^b+c R-square = Springback angle (degrees) Contact stiffness (N/m 3 ) Figure 5-5: Springback angle after the relaxation step at ~21 seconds versus contact stiffness fitted to power model for translation roll forming model

87 6. Conclusions and achievements Successful transformation and implementation of a translational roll forming model into a fully functional friction roll forming model with rotating tools. Friction present in the roll forming model (0.1<µ<0.65) has shown small deviance in results, therefore justifying the translation model as a plausible simplification that has not diminished the results significantly. Therefore allowing the translational roll forming model to be used as a simplified roll forming model. No plastic strain is present in the flanges using the current tool assembly with normal friction (µ 0.12) and translation roll forming model with steel grade Docol 1200M. Stability is recommended to be ~ 12% in terms of the stability quotient in order to achieving accurate results and a stable simulation. Simulation times using the computer specified vary between ~11h ~47h for friction roll forming model depending on stability, and ~11h for the translation roll forming model. Simulation time for a stable simulation using the friction roll forming model is ~33h whilst using the model without friction the simulation time is ~11h. Concluding that simulation with friction consumes three times the time. High friction (µ~0.65) causes increased acceleration in the beginning of the forming process invoking possible dynamic affects as well as increased flange strain resulting in plastic strain during the forming process. Mass scaling is necessary to achieve reasonable simulation times. Mass scaling affects simulation stability and achieved results in terms of stresses and strains during the forming process. Deviations in results are yet reasonable small relative yield strength and plastic strain limit. Springback values have a relative deviation of 0.39 (~9%) between a stable and unstable simulation (according to stability recommendations). Mass scaling coefficient is recommended to be set a low a possible for maximum accuracy within the limit of available time for the simulation. Increased contact stiffness increases simulation stability and is recommended to be set as large as possible without deteriorating on the simulation results. Contact stiffness also affects springback greatly and are preferably validated and calibrated against experimental results. Gap size influence (within 3%-10% clearance) affects the forming process marginally in terms of springback, residual stresses and strains. The difference in springback (0.23 ) is mainly as an affect of the extra space between the tools allowing the sheet to form to a smaller angle (~0.3 )

88 Normal hourglass control is preferred compared to enhanced hourglass control which with the use of reduced integrated 8-node solid elements is not recommended during extensive bending operations such as roll forming irrespective of the amount of mass scaling. Fully integrated solid 8-node elements are not recommended during roll forming simulations due to the element suffering from extensive shear locking making the element over stiff who produces inaccurate results. Mesh density plays a significant role during simulation. 5 elements are recommended through the thickness and the ratio (1,4) in the bending plane showed the best results. Ratio in depth direction did not influence more than marginally within the tested ratio (width,height,depth), (1,4,26) to (1,4,38)

89 7. Future work Future work to be done on this project would consist of the following tasks: Achieve experimental results in order to compare to simulations and to establish final contact penalty stiffness. The most important results are springback values and reaction force(s) on tool(s). Implementation of strain rate dependency. Implementation and validation of a fracture criteria. Simulation of the roll forming process using implicit FE code also making it possible using C3D20 elements which can describe bending behaviour very well although with extensive computational cost

90 8. Acknowledgement The author would like to thank the following people for their contribution and support. Maria Lundberg (Supervisor) Prof. Arne Melander (Supervisor and examiner) Everyone at the vehicle strength group at KIMAB

91 9. References [1] COOK ROBERT D. et al, Concepts and applications of finite element analysis 4 th ed. John Wiley & Sons, Inc, United States (2001). [2] PEARCE R., Sheet metal forming, Adam Hilger (1991). [3] INTERNATIONAL IRON AND STEEL INSTITUTE., Advanced high strength steel application guidelines, version III (2006). [4] PETTERSSON K., Materials mekaniska egenskaper. Royal institute of technology, Department of material science, Stockholm (2001). [5] SSAB,. Plåthandboken Att konstruera och tillverka i höghållfast plåt 2 nd Borlänge (1990). ed, SSAB Tunnplåt, [6] BLAZYNSKI T. Z., Plasticity and modern metal forming technology. Elsevier applied science, United States (1989). [7] BOMAN R. et al, Application of the Arbitrary Lagrangian Eulerian formulation to the numerical simulation of cold roll forming process. Journal of Materials Processing Technology 177 (2006). [8] HIBBIT, KARLSSON & SORENSEN, ABAQUS user s manual, version 6.4 (2003). [9] SSAB,. Formingshandboken, SSAB Tunnplåt, Borlänge (1997). [10] DONEA J. et al, Encyclopaedia of computational mechanics, Wiley (2006)

92 Appendix A Material data Figure A-1: Tensile test for Docol 1200M, thickness 1.5 mm O degrees to the direction of rolling (Used in FE simulations) True strain (ε true ) True stress (MPa)