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Sensitivity Analysis Of Technological And Material Parameters In Roll Forming Albrecht Gehring* and Helmut Saal* *Versuchsanstalt fr Stahl, Holz und Steine, Universitt Karlsruhe (TH), Kaiserstrae 12, 76128 Karlsruhe, Germany Abstract. Roll forming is applied for several decades to manufacture thin gauged profiles. However, the knowledge about this technology is still based on empirical approaches. Due to the complexity of the forming process, the main effects on profile properties are difficult to identify. This is especially true for the interaction of technological parameters and material parameters. General considerations for building a finite-element model of the roll forming process are given in this paper. A sensitivity analysis is performed on base of a statistical design approach in order to identify the effects and interactions of different parameters on profile properties. The parameters included in the analysis are the roll diameter, the rolling speed, the sheet thickness, friction between the tools and the sheet and the strain hardening behavior of the sheet material. The analysis includes an isotropic hardening model and a nonlinear kinematic hardening model. All jobs are executed parallel to reduce the overall time as the sensitivity analysis requires much CPU-time. The results of the sensitivity analysis demonstrate the opportunities to improve the properties of roll formed profiles by adjusting technological and material parameters to their optimum interacting performance. Keywords: Roll forming, sensitivity analysis, technological parameters, material models INTRODUCTION Roll forming is a technology where a flat strip is formed into a cross-section continuously. The process involves a progressive bending of metal strip as it passes through a series of forming tools, see Fig. 1. The design of the rolling schedule is done by unfolding the profile. From this, the flower diagram is obtained, see Fig. 1. Different calibration methods are used to obtain the flower diagram. Arc bending or the constant radius method are in common use, but other methods are also applied 1, 2. The sheet is deformed by an intended transversal bending load and unavoidable reversed bending and shear loads in longitudinal and transverse direction during roll forming. The latter arise from the curvature of a fiber during the forming process and they significantly influence the forming process and the final shape of the cross-section 1, 2. The unavoidable deformations were subject of many research works. The fundamental relations were investigated experimentally 1-4, where the technological aspects rolling schedule, inter-pass distance and tool diameter are identified as important parameters. The history of the longitudinal total strain is characteristic for roll forming, where the membrane strain at the edge of the flange exhibits a peak before each tool. The influence of the tribology are almost unknown. Some measured friction coefficients are reported in 2. Thus, the coefficient in the Coulomb friction model varies from = 0.05 to 0.30 depending on the lubrication and the roll load. FIGURE 1. Roll former (top); flower diagram of a U - profile (bottom), both from 5 781 ATTACHMENT I CREDIT LINE (BELOW) TO BE INSERTED ON THE FIRST PAGE OF EACH PAPER EXCEPT THE PAPERS ON PP. 301 306, 313 318, 331 336, 787 792, and 793 798. CP908, NUMIFORM 07, Materials Processing and Design: Modeling, Simulation and Applications edited by J. M. A. Csar de S and A. D. Santos 2007 American Institute of Physics 978-0-7354-0415-1/07/$23.00 The application of the finite-element method analyzing the roll forming process is increasing in recent years. This is due to the developments of the hard ware as well as improvements in finite-element formulations, especially in modelling continuously changing contact situations. Mainly simple profiles are analyzed with the aim to establish correlations of forming results and tool design 5-10. Both, an implicit solver and an explicit solver are used in these analyses. It is shown, that an explicit solver is applicable for the simulation of roll forming when geometric defects like the occurrence of edged waves and buckles are under consideration 6-8. Springback was examined in 11, 12 where both an implicit and an explicit solver were used. However, the interaction of technological and material parameters on the forming results are not investigated yet. The effect and the interaction of different parameters on roll formed sections is investigated in this paper. FINITE ELEMENT MODEL General Considerations The roll forming process can be regarded as quasi- static dynamic process. Thus an explicit solver is applicable. The explicit solver is advantageous in an analysis, where contact is involved 13. An implicit solver is used for subsequent springback analyses. The software package ABAQUS 14 is used for all analyses jobs. The analyses are executed parallel with the domain-level method. All analyses jobs are run on 8 cpus on the HP XC 6000 cluster 15. The parallel performance has a speed-up factor of approximately 5.5. Model Details Roll forming of an U 35 / 100/ 35 section in different gauges t is simulated. The applied inner bending radius is 4mm. The profile is obtained with steps 15/35/55/75/90. The roll stands of the steps 15 and 35 are built with two tools. An additional side tool is applied in the roll stands of steps 55, 75 and 90, see Fig. 2. The inter-pass distance is 500 mm respectively. The forming tools are represented as rigid bodies in the analysis. The blank is meshed with 4 node shell element S4R 13. In the flat areas an element size of 3 x 3mm with 5 integration points through thickness are used and in the bending area an element size of 1 x 3mm with 9 integration points through thickness are used. FIGURE 2. Roll former with 5 roll stands steps from right to left: 15/35/55/75/90. The ends of the sheet are constrained in rolling direction to simulate an endless sheet. Fixed boundary conditions are applied on the forming tools in all analysis jobs where friction is neglected. The rotation about the axis of each tool is free, when friction is considered in the analysis. In this case, additional rotary inertia elements are assigned to all tools. The general contact algorithm is used in the analysis, as no restrictions are placed on the domain decomposition for domain-level parallelization 13. Contact in normal direction is modelled with the hard contact pressure definition and the Coulomb friction model describes the tangential interaction of two surfaces. The load is applied as velocity in rolling direction. The velocity is increased in the first 0.05 s from zero to a constant forming speed using the smooth step definition of ABAQUS to minimize inertia effects in this quasi-static analysis. Through this, longitudinal oscillations of the sheet are minimized and thus the probability to induce numerically noisy or inaccurate results is reduced 13. For comparison and verification of the model, roll forming is also analyzed using the implicit solver provided by ABAQUS/Standard. Material Model AYoungs Modulus of E= 210GPa and a Poissons ratio of = 0.3 define the elastic response in all analyses. The plastic part is described by a combined isotropic / nonlinear kinematic hardening model 16 with a von Mises type yield function according to Equation (1). In Equation (1) is the stress tensor, X is the back-stress tensor and kfis the yield stress, which represents the isotropic hardening behaviour of the material. The yield stress kf is modelled with the modified Ludwik-Hollomon equation, which is usually applied in forming analysis 17, cp. Equation (2). In Equation (2) fy is the yield strength, is the strength ratio of tensile strength to yield strength, p are true plastic strains, e is the Euler number and n is a constant. Equation (2) leads to a good estimation of the flow curve for low alloy steel and aluminium alloys, if the exponent n is related to 782 the uniform elongation u in terms of n = ln (1+u) 17. 0= f kXfF)(1) n p n ypf n e fk =)(2) () )( )( p e C X p m m1=(3) The translation of the elastic domain in the stress space is given for the uniaxial case by Equation (3), where C and are material parameters. In all jobs the yield strength fy= 320MPa and the uniform elongation is u = 0.15. These values comply with steel grade S320 according to EN 10326 18. The strength ratio is set to different values. The parameters C and are taken from 12 depending on . Model Verification The strength ratio = 1.2 and isotropic hardening is applied for model verification. Also, friction is neglected with these analyses. The histories of kinetic energy and internal energy are considered to check if an analysis is numerically stable and if the assumption of a quasi-static analyses can be verified. The ratio of kinetic energy to internal energy should be less than 5% in a quasi-static analysis 13. Also the histories of kinetic energy and internal energy must be reliable technologically. The history of the longitudinal membrane strain at the edge of the flange and the degree of work hardening are examined as technological criteria. For comparison, the peak membrane strain epeak 3 is calculated according to Bhattacharyya to 11 4 3 1 +=)cos( a t peak (4) where a is the flange length and is the bend angle. The degree of work hardening is obtained from a cut through the profile. Plastic flow occurs according to the von Mises criterion, if the equivalent yield stress attains the defined reference yield stress value. The size of the yield surface changes with the equivalent plastic strain eppq. From this follows, that the yield stress fy,fem at any point of the profile can be expressed as () n eppq n yeppqfemy n e ff = , (5) ()dlf l f l eppqfemyfemya = 0 1 , (6) where the hardening behaviour according to Equation (2) is applied. Then the average yield stress fya,fem is obtained from Equation (6), where l is the developed length of the cross-section. The ratio of kinetic energy to internal energy is less than 5% which satisfies the criterion for a quasi-static analysis. It is worth to mention, that this criterion can be satisfied with a very coarse mesh. Hence other considerations concerning the reliability of the results must be taken into account. The length of the blank in the model has a significant influence on the history of internal energy, see Figure 3a. If the length is less than twice the inter-pass distance, discontinuities are arising from the change of the restraints. This behaviour does not reflect the real process. Therefore, the length of the blank is set to 2.25 times the inter-pass distance in the analysis jobs. The history of the longitudinal membrane strain at the edge of the flange obtained from the analysis is shown on Figure 3b. The characteristic peaks before each tool agree with experimental results 3, 4. However, the strain calculated with Equation 4 does not match the numerical results. This is due to the simplicity of Equation 4. The distribution of the yield strength after roll forming is shown on Figure 3c, which qualitatively complies with experimental results documented in 2. As expected, work hardening takes place in the bending area. The evolution of the springback angle is shown in Figure 3d. The results obtained from the implicit solution show a continuous development. The explicit solution isdiscontinuous, which is caused by oscillations. From this follows, that the explicit solutions does not reproduce the springback angle as clear as the implicit solution. This result corresponds to experiences with other forming simulations, e.g. deep drawing. However, there are only small differences between the absolute values of the springback angle . The results show, that the applied finite-element model is appropriate for the simulation of roll forming. The history of membrane strain at the edge of the flange and the yield stress distribution agree quite well with results documented in literature. However, the history of the springback angle suggests, that the difficulties known from other forming applications 13 shall be considered in further investigations. 783 FIGURE3. Histories of internal energy (a.), history of total longitudinal strain at edge of flange (b.), distribution of yield stress (c.), evolution of springback angle (d.), normal probability plot of fya,fem (e.), normal probability plot of peak (f.) SENSITIVITY ANALYSIS Experimental Design Experiments are performed to measure the effects of variables (factors) on a response. The effect of a factor means the change in the response as a factor is changed from a lower level to a higher level. Factorial designs are useful for this purpose, especially two- level factorial designs 19. The consideration of several factors leads to a large number of experimental runs. An estimation redundancy and negligible higher order interactions occur in full factorial designs. Through this, loss of information is insignificant with a fractional factorial design at two levels 19. Deviations from the assumed linear relation in two- level designs can be identified with an centre-point run. The number m of experimental runs is 12+= pk m(7) where k is the number of variables and p the degree of fractionation. Here the commercial statistic software 0.00 0.50 1.00 0.000.501.001.502.00 Normalized time - Normalized internal energy - Length of blank in model = n-times the inter-pass distance n = 1.5 n = 3.0 n = 2.25 a. 0.000 0.001 0.002 0.003 0.004 0.000.250.500.751.00 Normalized time - longitudinal membrane strain - b. Equation 4 numerical result 0.95 1.05 1.15 1.25 1.35 1.45 085 Developed length mm Yield stress fy,fem MPa c. strength ratio = 1.1 strength ratio = 1.2 strength ratio = 1.3 0 15 30 45 60 0500100015002000 Distance of roll forming mm Angle d. 1535 55 Implicit solution -9.0-6.0-3.00.03.06.09.012.015.0 Effect E on fya,fem - Cummulative frequency % 0,02 1 2,28 5 10 15,87 50 84,13 90 95 97,72 99 99,98 effect effect t interaction and t curvature e. -0.0005-0.0003 -0.0001 0.00010.00030.00050.00070.0009 Effect E on peak - Cummulative frequency % 0,02 1 2,28 5 10 15,87 50 84,13 90 95 97,72 99 99,98 curvature f. interaction and t effect effect D 784 Statistica 20 is used for the design of the runs and the analysis of the results. Here, the objective is to identify the effects and interactions of technological parameters and mechanical material properties on the properties of the roll formed section. The factors included in the investigation are the roll diameter, the rolling speed, the sheet gauge, friction between the tools and the sheet and the strain hardening behavior of the sheet material. In addition the application of an isotropic hardening model and a nonlinear kinematic hardening model is introduced as factor. In a full factorial design m = 26 = 64 analyses jobs would be necessary. Setting p = 2 with the fractional factorial design the number of necessary experimental runs can be reduced to m = 17 including a center point run. The experimental design including the values of the factors used in this study can be seen in Table 1. The parameters C and of the kinematic hardening are coupled to the strength ratio . On the lower level C = 110 GPa and = 600 and on the higher level C = 77.5 GPa and = 500. Evaluation Procedure The results of the analyses are evaluated depending on the peak of longitudinal strain at the edge of the flange and the degree of work hardening according to Equation (6). The effect of a variable on the defined criterions is evaluated by statistical methods. For this, the ratio i= Yi/ Y0 is introduced. In case of the longitudinal strain Yi is the maximum value occurring during the forming process and Y0 is the average from all runs. For the evaluation of the degree of work hardening Yi is set to fya,fem and Y0 is set to the yield strength fy of the virgin material. The effect E of a variable is calculated to () = = m i i m E 1 2 (8) where the subscript i refers to the run number and the sign in brackets to the low and high level respectively. The interaction effects are calculated similarly. Since there were no replicated runs, a standard error sd has to be estimated using higher- order interactions with negligible influence to = k d E k s 22 1 (9) The significance of an effect can be identified by means of a confidence estimation based on the t distribution. Deviations from the assumed linear relations of the two-level design can be identified from the centre-point run with a curvature test 19, 20. Results The individual results of the sensitivity analysis are given in Table 1. The normal probability plots of the calculated effects are shown in Figures 3e and 3f. The plots suggest that the values where text is added to the symbols require further interpretation. All other values are supposed to be negligible and thus are used for the estimation of the standard error sd, respectively. The confidence estimation reveals, that the degree of work hardening depends on the strength ratio and the TABLE 1. Experimetal design and results Run.Level of FactorResults no. - mat - t mm - v mm/s D mm peak - fya,fem MPa 11.1kin.0.750.0500900,0025322 21.3kin.0.750.01000900,0017329 31.1iso.0.750.010001800,0026322 41.3iso.0.750.05001800,0020327 51.1kin.1.250.010001800,0021324 61.3kin.1.250.05001800,0024344 71.1iso.1.250.0500900,0021324 81.3iso.1.250.01000900,0022338 91.1kin.0.750.25001800,0025322 101.3kin.0.750.210001800,0022329 111.1iso.0.750.21000900,0022322 121.3iso.0.750.2500900,0016327 131.1kin.1.250.21000900,0021324 141.3kin.1.250.2500900,0023347 151.1iso.1.250.2