NUMERICAL ANALYSES OF COLD-FORMED THIN-WALLED SECTIONS WITH CONSIDERATION OF IMPERFECTIONS DUE TO THE PRODUCTION PROCESS

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1 Advanced Steel Construction Vol. 5, No., pp (009) 151 NUMERICAL ANALYSES OF COLD-FORMED THIN-WALLED SECTIONS WITH CONSIDERATION OF IMPERFECTIONS DUE TO THE PRODUCTION PROCESS Albrecht Gehring 1 and Helmut Saal,* 1 Research assistant, Versuchsanstalt ür Stahl, Holz und Steine, Universität Karlsruhe (TH), German Proessor, Versuchsanstalt ür Stahl, Holz und Steine, Universität Karlsruhe (TH), German *(Corresponding author: helmut.saal@va.uka.de) ABSTRACT: The load bearing capacit o cold-ormed thin-walled sections strongl depends on deviations rom the nominal dimensions and the material properties. The ormer reduce the load bearing capacit. The latter enhance the load bearing capacit, because o work hardening during the manuacturing process. It is diicult to realisticall account or both eects in a inite-element analsis o the load bearing capacit o thin-walled sections. Toda, cost intensive testing is necessar, i a maximum utilization o the load bearing capacit is desired. The properties o a product can be determined during the product development process with a new simulation strateg, which covers the production process as well as the state o serviceabilit o a product. The roll orming process is simulated irst ollowed b a non-linear ultimate limit state analsis. The combination o both analsis steps gives the possibilit to determine the load bearing capacit realisticall as deviations rom the nominal value o dimensions and material properties areincluded in the analsis. The new analsis strateg is demonstrated or a U-section with respect to dierent aspects concerning work hardening and the load bearing capacit o a C-section. It is shown, that the new strateg leads to a realistic estimation o the load bearing capacit o thin gauged sections. Kewords: Numerical analses, cold-ormed sections, imperections, work hardening, ultimate limit state 1. INTRODUCTION Cold-ormed thin-walled sections are produced b roll orming. Roll orming is a technolog where a lat strip is ormed into a cross-section continuousl. The process involves a progressive bending o the metal strip as it passes through a series o orming tools, see Figure 1. The design o the rolling schedule is done b unolding the proile. From this, the lower diagram is obtained, see Figure 1. Dierent calibration methods are used to obtain the lower diagram. Arc bending or the constant radius method are in common use, but other methods are also applied, Halmos [1], Bogojawlenski et al []. The sheet is deormed b an intended transversal bending load and unavoidable reversed bending and shear loads in longitudinal and transverse direction during roll orming. The latter arise rom the curvature o a iber during the orming process and the signiicantl inluence the orming process and the inal shape o the cross-section, Halmos [1], Bogojawlenski et al []. Figure 1. Roll Former (Let); Flower Diagram o a U-Proile (Right)

2 15 Numerical Analses o Cold-ormed Thin-Walled Sections with Consideration o Imperections Due to the Production Process. FINITE ELEMENT MODEL.1 General Considerations The roll orming process can be regarded as a quasi-static dnamic process. Thus an explicit solver is applicable. The explicit solver is advantageous in an analsis, where contact is involved, Abaqus [3]. An implicit solver is used or subsequent springback analses. The sotware package Abaqus [4] is used or all analses jobs. The analses are executed parallel with the domain-level method. The analses jobs are run mainl on 4 cpus or 8 cpus on the HP XC 6000 cluster and on HP XC 4000 cluster o the Universität Karlsruhe [5].. Model Details The orming tools are represented as rigid bodies in all analses. The blank is meshed with the 4 node shell element S4R. In the lat areas an element size o 3 x 3 mm with 5 integration points through thickness are used and in the bending area an element size o 1 x 3 mm with 9 integration points through thickness are used. The ends o the sheet are constrained in rolling direction to simulate an endless sheet. Fixed boundar conditions are applied on the orming tools in all analsis jobs. The general contact algorithm is used in the analsis, as no restrictions are placed on the domain decomposition or domain-level parallelization. Friction is neglected with all analses. The load is applied as velocit in rolling direction. The velocit is increased in the irst 0.05 s rom zero to a constant orming speed using the smooth step deinition o ABAQUS [4] to minimize inertia eects in this quasi-static analsis. Through this, longitudinal oscillations o the sheet are minimized and thus the probabilit to induce numericall nois or inaccurate results is reduced. The veriication o the model is described in detail b Gehring and Saal [6]..3 Material Model A Young s Modulus o E = 10 GPa and a Poisson s ratio o ν = 0.3 deine the elastic response in all analses. The plastic part is described b the ollowing material models, Khan and Huang [7]: Standard isotropic hardening with von Mises tpe ield unction and associated plasticit Yield unction according to the Hill stress potential and associated plasticit In all models, the isotropic hardening behaviour o the material is modelled with the modiied Ludwik-Hollomon equation, which is usuall applied in orming analsis, Lange [8]. In this Equation k n e n ( ε p ) = Φ ε p (1) n is the ield strength, Φ is the strength ratio o tensile strength to ield strength, ε p are true plastic strains, e is the Euler number and n is a constant. Eq. 1 gives a good approximation o the low curve or low allo steel and aluminium allos, i the exponent n is related to the uniorm elongation ε u in terms o n = ln (1+ ε u ), Lange [8]. The Hill stress potential is expressed as ( σ σ ) + G( σ σ ) + H ( σ σ ) + Lσ + Mσ N ( σ) = F + σ ()

3 Albrecht Gehring and Helmut Saal 153 where F, G, H, L, M and N are constants. These constants are deined as a unction o the ratios R ij o the reerence stress σ to measured ield stress value σ ij, when σ ij is applied as the onl nonzero stress component, Khan and Huang [7]. Here, the 1-axis constitutes the rolling direction and the -axis is transverse to the direction o rolling. The 3-axis is normal to the sheet plane. The reerence ield stress σ is determined in 1-direction and thus R 11 becomes Furthermore, the constants R 33, R 13 and R 3 are set to 1.00, because eects o a through-thickness anisotrop can be neglected in roll orming, Engl and Stich [9]. 3. DESIGN CONSIDERATIONS 3.1 General The application o cold-ormed sections in buildings is well established. The resistance o thin-walled sections can be determined with calculations based on the eective width approach according to international design codes, e.g. EN [10]. The application o the design codes leads to conservative resistance values. The ull beneit o the cross-sectional resistance can be achieved onl b cost intensive experimental investigations, e.g. according to the guidelines given in EN [10]. 3. Increased Yield Strength The strain distribution ater roll orming o an U-section can be seen in Figure. The strains in the corners are clearl higher than in the lat areas o the cross-section. This means, that work hardening takes place mainl in the corner areas. Man studies have been made to utilize the enhanced material properties or design. The mechanical properties were determined b tensile tests with specimen rom dierent parts o a proile. Also ull-section tensile or compressive tests were perormed to obtain speciic average ield strength or design application. A comprehensive list o reerences is contained in Halmos [1], Bogojawlenski et al [] and Yu [11]. Figure. Strain Hardening Due to Roll Forming o a U-Section

4 154 Numerical Analses o Cold-ormed Thin-Walled Sections with Consideration o Imperections Due to the Production Process Semi-empirical equations are introduced in international design standards or taking an enhanced ield strength into account. In EN [10], the increased ield strength a is expressed as a = min 0,5 (3) k n t ( + ) + ( ) u ; u Ag where and u are the design values o the ield strength and the tensile strength respectivel, k is a constant depending on the applied orming technolog, e.g. k = 7 or roll orming, n is the number o 90 bends and A g is the gross cross-sectional area. The increased ield strength a is approximated in North America, Yu [11], with a V c ( av ) = a + 1 (4) where c is the corner ield strength and a V is the ratio o corner areas to the gross cross-sectional area. The corner ield strength is calculated to c B c = (5) ( r / t) m where B c = 3,69 m = 0,19 u u 0,819 0,068 u 1,79 and r/t is the ratio o the bending radius to the thickness o the sheet. The application o Eq. 3 and Eq. 4 is restricted to a ull eective cross-sections, certain values o the ratios r/t and u / respectivel and also to certain load cases. 3.3 Numerical Analsis In EN , Annex C [1] a guidance is given or the use o inite-element analsis or ultimate limit state veriication, which includes appropriate assumptions or material models and imperections. In the ollowing, an analsis according to the guidance in EN , Annex C [1] is called conventional analsis strateg. A non-linear inite-element analsis is necessar or the determination o the ultimate limit state o thin-walled sections. Where the non-linearit takes into account geometric deviations as well as material aspects. The imperections can be introduced b appling equivalent geometric imperections deined in design codes, which are scaled eigenmodes obtained rom buckling analsis. Usuall the material properties are assumed to be distributed uniorml, which leads to a sae estimation o the ailure loads. General considerations on the characterisation o imperections in numerical analses are given b Schaer and Peköz [13]. Gardner and Nethercot [14] and Lecce and Rasmussen [15] introduce an enhanced ield strength due to cold-working b partitioning a cross-section. However, the problem is the lack o knowledge about the initial state o cold-ormed sections, Schaer and Peköz [13].

5 Albrecht Gehring and Helmut Saal New Analsis Approach The properties o a roll ormed section are determined with a new analsis strateg, which covers the production process as well as the state o serviceabilit o the section, Gehring et al [16]. The roll orming process is simulated irst. This is ollowed b a non-linear ultimate limit state analsis. The combination o both analsis steps gives the possibilit to determine the load bearing capacit realisticall as deviations rom the nominal value o dimensions and material properties are included in the analsis. The analsis is divided into the ollowing steps: Analsis o the manuacturing process Introduction o imperections and change o material properties obtained b the analsis o the manuacturing process as initial state to the new model Non-linear ultimate limit state analsis Veriication o results The new analsis strateg is demonstrated b two examples. Here, the ield strength is =30MPa, the tensile strength is u =390 MPa and the uniorm elongation is ε u = 0.1. These values compl with steel grade S30 according to EN 1036 [17]. 4. APPLICATION OF THE NEW APPROACH 4.1 Example 1: Degree o Work Hardening, Gehring and Saal [18] Roll orming o an U 35 / 100 / 35 x 1 section is simulated. The applied inner bending radius is 4.00 mm. The proile is obtained on base o three dierent lower diagrams 3 steps 30 /60 /90, 4 steps 0 /45 /70 /90 and 5 steps 15 /35 /55 /75 /90. The roll stands o the steps 15, 0, 30, 35 and 45 are built with two tools. An additional side tool is applied in the roll stands o steps 55, 60, 70, 75 and 90 respectivel, see Figure 3. The inter-pass distance is set to 500 mm. Figure 3. Roll Former with 5 Roll Stands Steps From Right to Let: 15 /35 /55 /75 /90 Experiments are perormed to measure the eects o variables on a response. The eect o a variable means the change in the response as a variable is moved rom a lower level to a higher level. Factorial designs are useul or this purpose, especiall two-level actorial designs, Box et al [19]. Here, the objective is to identi the eect o mechanical material properties on the distribution o the ield stress o a roll ormed section. Deviations rom the assumed linear relation in two-level designs can be identiied with an additional centre-point run. Then, the number m o experimental runs is m = k +1 (6)

6 156 Numerical Analses o Cold-ormed Thin-Walled Sections with Consideration o Imperections Due to the Production Process where k is the number o variables. Here the commercial statistic sotware Statistica [0] is used or the design o the runs and the analsis o the results. The material properties strength ratio Φ = u /, uniorm elongation ε u and the constant R are taken as variables. The constant R 1 is linked with R and thus not used as independent variable. On the upper level R 1 = 1.03 and on the lower level R 1 = In addition the technological variable number o roll stands n R is used. Thus, m = 17 experimental runs are necessar in this stud. The experimental design can be seen in Table 1. Plastic low occurs in Hill s stress potential, i the equivalent ield stress according to Eq. reaches the deined reerence ield stress value. The size o the ield surace changes with the equivalent plastic strain ε eppq isotropicall. From this ollows, that the ield stress,em can be calculated with Eq. 7 and the average ield stress a,em can be calculated with Eq. 8. e n ( ε ) = Φ ε n, em eppq eppq (7) n a, em 1 = l l 0, em ( ε )dl eppq (8) The eect o a variable on the increase o the ield strength is evaluated b statistical methods. The ratio ψ = a,em / is introduced or the evaluation. The eect C o a variable is calculated to C = m m ( ψ i {} ± ) i= 1 (9) where the subscript i reers to the run number and the sign in {} brackets to the low and high level respectivel. The interaction eects are calculated similarl. Since there were no replicated runs, a standard error s d ² has to be estimated using higher-order interactions with negligible inluence to s = k 1 d k C (10) The signiicance o an eect can be identiied b means o a conidence estimation based on the t distribution. Deviations rom the assumed linear relations o the two-level design can be identiied rom the centre-point run with a curvature test. Detailed inormation on the statistical evaluation methods is given in Box et al [19]. The average ield stress a,em and the maximum value o the ield stress,em # evaluated with Eq. 7 and Eq. 8 are given in Table 1. In addition, the ield strength a obtained rom Eq. 3 and 4 are given in Table 1. Eq. 3 leads to a sae estimation o the increased ield strength whereas Eq. 4 overestimates the increased ield strength or certain values o Φ slightl. The corner ield strength c obtained rom Eq. 5 overestimates the material response. This is evident b a comparison o c and the individual true tensile strength u,true = Φ (1 + ε u ), where c is greater than u,true or Φ = 1. and 1.3. The distribution o the ield stress obtained rom some runs with 3 roll stands are shown in Figure 4. The inluence o a change o Φ and ε u on the ield stress distribution is obvious rom the web o the U-section. The steeper gradient o the low curve with Φ = 1.3 and ε u = 0.10 causes more strain hardening than other combinations.

7 Albrecht Gehring and Helmut Saal 157 Table 1. Experimental Design and Results Run Variable 1) #,em a,em Results ψ ) a 3) a 4) c u,true no. Φ ε u R n R [MPa] [MPa] [-] [MPa] [MPa] [MPa] [MPa] {-} 0.10 {-} 1.00 {-} 3 {-} {+} 0.10 {-} 1.00 {-} 3 {-} {-} 0.15 {+} 1.00 {-} 3 {-} {+} 0.15 {+} 1.00 {-} 3 {-} {-} 0.10 {-} 1.05 {+} 3 {-} {+} 0.10 {-} 1.05 {+} 3 {-} {-} 0.15 {+} 1.05 {+} 3 {-} {+} 0.15 {+} 1.05 {+} 3 {-} {-} 0.10 {-} 1.00 {-} 5 {+} {+} 0.10 {-} 1.00 {-} 5 {+} {-} 0.15 {+} 1.00 {-} 5 {+} {+} 0.15 {+} 1.00 {-} 5 {+} {-} 0.10 {-} 1.05 {+} 5 {+} {+} 0.10 {-} 1.05 {+} 5 {+} {-} 0.15 {+} 1.05 {+} 5 {+} {+} 0.15 {+} 1.05 {+} 5 {+} ) The + and sign in {} brackets reer to the high and the low level respectivel. ) According to Eq. 3 3) According to Eq. 4 4) According to Eq. 5 Figure 4. Distribution o Yield Strength Runs 1, 4, 6 and 7

8 158 Numerical Analses o Cold-ormed Thin-Walled Sections with Consideration o Imperections Due to the Production Process Table. Statistical Evaluation Eect 1) C 10² [-] Φ (1) 6.96 ε u () -.58 R (3) 0.17 n R (4) ) Interactions are denoted with digits, e.g. the nd-order interaction o Φ and ε u is denoted with 1. The calculated eects and interactions C are given in Table. The comparison o the values suggests that the values in bold igures require urther interpretation. All other values are supposed to be negligible and thus are used or the estimation o the standard error s d ², which becomes to s d ² = From this ollows the absolute value c t,p o a conidence interval. The values o c t,p on conidence levels o 95 %, 99 % and 99.9% are c t,95% = 0.356, c t,99% = and c t,99.9% = 0.73 respectivel. Herewith the assumption is conirmed that the bold values are signiicant. The curvature test reveals, that the value o ψ does not depend linear on the variables. Thus, it is necessar to appl a higher-order statistical model or urther investigations. It is obvious that the value o the strength ratio Φ has a major eect on the increase o the ield strength. Raising Φ rom 1.1 to 1.3 leads to an increase o the average ield strength o 6.9 % in this example. The eect o ε u is opposite to that o Φ, i.e. a smaller value o the uniorm elongation ε u results in a higher average ield strength. The increase o the ield strength decreases with increase o the number n R o roll stands. This is due to smaller incremental bending in longitudinal direction. However, the signiicance o the interactions 1 and 14 shows, that the eect o the strength ratio Φ must also be considered jointl with the uniorm elongation and the number o roll stands. The interaction o Φ and ε u needs no urther explanation as it was alread discussed above. It is expected, that the interaction o Φ and n R becomes less important with an increasing number o roll stands. But this has to be conirmed. It is worth to point out, that the inluence o the anisotrop can be neglected. 4. Example : Load Bearing Capacit o a C-section, Gehring et al [16] The roll orming process o a C-section 10 / 35 / 100 / 35 / 10 x 1 is simulated in the irst analsis step. The applied inner bending radius is 4 mm. The C-section is obtained in 9 steps. First the edge

9 Albrecht Gehring and Helmut Saal 159 stiener o the C-section is ormed with 4 roll stands in steps 0 /40 /60 /90. Then the web is olded with 5 roll stands in steps 15 /35 /55 /75 /90. All roll stands or the stiener and the roll stands o the steps 15 and 35 are built with two tools. An additional side tool is applied in the roll stands o steps 55, 75 and 90 respectivel. The inter-pass distance is set to 500 mm. The second step o the analsis is to transer the results o the preceding analsis as initial state into the new model. In this step the out-o-balance orces o the explicit analsis are removed and static equilibrium is achieved. This accounts or springback eects. Thus dimensional deviations and cold-working eects due to roll orming are considered in the initial state. The ailure load is calculated in the last step o the analsis. For this, the proile is divided into 6 specimen with a length o 50 mm each. This value is less than 0 times the radius o gration and thus agrees with the length o stub column specimen recommended in EN [10]. The virtual stub column tests are perormed with the 4 specimen rom the middle part o the proile. New boundar conditions are applied to the model. At one end o the sections all translation degrees o reedom are ixed and at the other end the out-o plane degrees o reedom are ixed. The sections are subjected to constant translation in longitudinal direction. For comparison, the ultimate limit state o the C-section is determined with the conventional analsis strateg. The eigenmode which corresponds to the lowest eigenvalue is used as imperection. The amplitude o the eigenmode is scaled to a value o b/00 which complies with the value recommended in EN [1], Annex C. The material properties are assumed to be constant across the cross-section. The boundar conditions are deined as in the last step o the advanced analsis. Also, the eective cross-sectional area A e o the C-section is calculated in accordance with provisions given in EN [10]. The characteristic resistance N c,ec3 o the cross-section or uniorm compression according to EN [10] is N c, EC3 = A e (11) The ailure load N u obtained rom the inite-element analsis is calculated to k N u = n1, (1) i= 1 i where n 1,i is the reaction orce o the i-th node in loading direction and k is the number o nodes across the section. The variation o material properties and geometric imperections due to the orming process is taken into account in the determination o the load bearing capacit o the 4 individual specimen. The deormed shape obtained rom the new analsis exhibits tpical geometric deects o roll ormed sections, like the lare at the ends o the proile, see Figure 5. The geometric imperections obtained rom this analsis are shown in Figure 5, where the values are taken rom a path along the arrows in Figure 5 respectivel.

10 160 Numerical Analses o Cold-ormed Thin-Walled Sections with Consideration o Imperections Due to the Production Process Figure 5. Predicted Imperections rom Simulation o Roll Forming Process The numbers in Figure 5 are reerring to the specimen number or the virtual stub column tests. The maximum value o the geometric imperection o the lange is approximatel 0.4 mm and o the web approximatel 0.5 mm. These values agree quantitativel with measured values o similar sections, Lecce and Rasmussen [15] [16]. Also residual stresses and work hardening eects due to roll orming are considered in the initial state. The distribution o the ield stress ater roll orming is shown in Figure 5. In addition, the average ield strength a is determined according to Eq. 7 and Eq. 8. The deormed C-sections ater ailure are shown in Figure 6. The ailure modes deviate rom each other, because the applied imperections are dierent. In the conventional analsis, the wave length o the edge stiener is longer than in the advanced analsis. It is obvious rom Figure 6, that the new analsis includes dierent imperection modes and thus reveals dierent ailure loads or each virtual test. The ailure loads determined with inite-element analses are given in Table 3. The subscripts con and new reer to the conventional and new analsis respectivel. The characteristic resistance according to EN [10] is N c,ec3.

11 Albrecht Gehring and Helmut Saal 161 a. b. c. d. e. Figure 6. Failed Specimen rom Conventional Analsis (a.) and New Analsis (b. to e.) Table 3. Ultimate Loads N c,ec3 [kn] 37,0 N u,con [kn] 39,8 N u,new,1 [kn] 41,4 N u,new, [kn] 4,4 N u,new,3 [kn] 43,7 N u,new,4 [kn] 4,4 The higher ailure loads are due to the increased ield strength and the smaller geometric imperections with the new analsis strateg. The average ield strength a is 18 % higher than the ield strength o the virgin material. The highest ailure loads are obtained rom the new analsis strateg. These are approximatel 4 % to 10 % higher than the ailure load obtained rom the conventional analsis strateg and approximatel 11 % to 18 % higher than characteristic resistance N c,ec3 according to EN [10]. This example shows, that the new analsis strateg gives the opportunit to simulate the structural perormance o cold-ormed sections realisticall. This is due to the act, that deviations rom the nominal dimensions and material properties are taken into account. The results are reasonable and

12 16 Numerical Analses o Cold-ormed Thin-Walled Sections with Consideration o Imperections Due to the Production Process within the expected range. With this model, the eect o slight modiications o, e.g. material speciication, tool design, on the ailure load, can be determined easil. However, in an case the results have to be veriied with some tests beore introducing them as characteristic resistance values in design. 5. SUMMARY AND CONCLUSIONS A new simulation strateg or cold-ormed thin-walled components, which covers the production process as well as the state o serviceabilit o a roll ormed section was presented. The roll orming process is simulated irst. This is ollowed b a non-linear ultimate limit state analsis. The combination o both analsis steps gives the possibilit to determine the load bearing capacit realisticall as deviations rom the nominal value o dimensions and material properties are included in the analsis. The new analsis strateg is demonstrated or a U-section with respect to dierent aspects concerning work hardening and the load bearing capacit o a C-section. From the examples, the ollowing conclusions can be drawn: The ormulas or the average ield strength a given in Eurocode 3 lead to a sae estimation o the increase o the ield stress due to roll orming. The strength ratio Φ = u / is identiied as the material propert with the strongest inluence. However, the eect o Φ interacts with the value o the uniorm elongation ε u. The anisotrop o the sheet material can be neglected. An increasing number o roll stands reduces the increase o the average ield stress. The enhanced average ield strength can be determined as a unction o the material properties and the parameters o the roll ormer. This gives the opportunit to utilize the enhanced ield stress or speciic sections in design. Deviation rom the nominal dimensions and material properties are estimated realisticall with the new analsis strateg and the obtained ailure loads are reasonable compared to the values according to design standards. The characteristic resistance obtained rom this process oriented analsis is higher than the characteristic values or general use in design standards. The new analsis strateg gives the chance to optimize the dimensions, the material speciication and the manuacturing process o a section without a cost intensive experimental trial and error procedure. It has to be kept in mind that the new analsis strateg will onl lead to sae results i its application is based on a proound knowledge o the manuacturing process and the structural behaviour o the component. This demands substantial experience with experimental and numerical analsis o the object. REFERENCES [1] Halmos, G.T. (Edit.), Roll Forming Handbook, CRC Press Talor & Francis Group, Boca Raton, 006 [] Bogojawlenskij, K.N., Neubauer, A. and Ris, V.W., Technologie der Fertigung von Leichtbauproilen, VEB Deutscher Verlag ür Grundstoindustrie, Leipzig, [3] ABAQUS Documentation Version 6.6.1, Copright 006, ABAQUS, Inc. [4] ABAQUS (Explicit & Standard); Version 6.6.1, Copright 006, ABAQUS, Inc. [5]

13 Albrecht Gehring and Helmut Saal 163 [6] Gehring, A. and Saal, H., Sensitivit Analsis o Technological and Material Parameters in Roll Forming, Proceedings o the 9th International Conerence on Numerical Methods in Industrial Forming Processes, Porto, 007, pp [7] Khan, A.S. and Huang, S., Continuum Theor o Plasticit, John Wile & Sons, Inc., New York, [8] Lange, K. (Edit.), Umormtechnik Handbuch ür Industrie und Wissenschat, Band 1: Grundlagen, Berlin: Springer Verlag, [9] Engl, B. and Stich, G., Neue Stahlsorten ür die Kaltormung, Tagungsband 7, Studiengesellschat ür Stahlanwendung e.v., [10] EN :006, Eurocode 3 - Design o Steel Structures - Part 1-3: General Rules - Supplementar Rules or Cold-ormed Members and Sheeting. [11] Yu, W.-W., Cold-ormed Steel Design, 3rd Edition, John Wile & Sons, New York, 001. [1] EN :006, Eurocode 3 - Design o Steel Structures - Part 1-5: General Rules - Supplementar Rules or Cold-ormed Members and Sheeting, Annex C. [13] Schaer, B.W. and Peköz, T., Computational Modeling o Cold-ormed Steel: Characterizing Geometric Imperections and Residual Stresses, Journal o Constructional Steel Research, 1998, Vol. 47, pp [14] Garnder, L. and Nethercot, D.A., Numerical Modeling o Stainless Steel Structural Components A Consistent Approach, Journal o Structural Engineering, 004, Vol. 130, pp [15] Lecce, M. and Rasmussen, K., Distortional Buckling o Cold-ormed Stainless Steel Sections: Experimental Investigation, Journal o Structural Engineering, 006, Vol. 13, pp [16] Gehring, A., Kathage, K. and Saal, H., A New Strateg or Finite-element Analsis o the Load Bearing Capacit o Cold-ormed Sections, Proceedings o the 3rd International Conerence on Structural Engineering, Mechanics and Computation, Capetown, 007, pp [17] EN 1036:004, Continuousl Hot-dip Coated Strip and Sheet o Structural Steels - Technical Deliver Conditions. [18] Gehring, A. and Saal, H., Yield Strength Distribution in Thin-walled Sections Due to Roll Forming A Finite-Element Analsis, Proceedings o the 6th International Conerence on Steel and Aluminium Structures, Oxord, 007, pp [19] Box, G.E.P., Hunter, W.G. and Hunter, J.S., Statistics or Experimenters, nd Edition, New York: John Wile & Sons, 005. [0] Statistica 7.1, Copright , StatSot, Inc.