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A design method for prediction of dimensions of rectangularhollow sections formed in stretch bendingFrode Paulsen a,*,Torgeir Welo baHydro Automotive Structures Raufoss AS,Process Development,P .O.Box 15,N-2831Raufoss,NorwaybHydro Automotive Structures Holland Inc.,365West 24th Street,Holland,MI 49423,USAReceived 30August 2000;accepted 19March 2002AbstractLocal deformation of individual cross-sectional members is of great interest in bending of aluminium alloy extrusions for tight tolerance production such as automotive components.The primary concern is the impact of such distortions on manufacturability as well as the dimensional tolerances of the component.This paper presents analytical models for the determination of local post-buckling and suck-in deformations in stretch bending.The models are based on the deformation theory of plasticity combined with an energy method using appropriate shape functions.The analytical predictions are being veri®ed with experimental results.Based on the present ®ndings,a simpli®ed design method for evaluation of bendability of sections in industrial forming operations is being proposed.The results show that the slenderness (b /t )and the width of the ¯ange are the main parameters related to the bending radius at the onset of plastic buckling and the magnitude of local deformations,respectively.Material parameters have proven to be relatively more important to the former than to the latter.Although there is some discrepancy at tight nominal bend radii,the overall correlation between the experimental and theoretical results is surprisingly good.It is therefore concluded that the present method provides to be an ef®cient means to the evaluation of bendability of rectangular hollow sections.#2002Published by Elsevier Science B.V .Keywords:Bending;Stretch bending;Elastic springback;Buckling;Post-buckling;Suck-in;Sagging;Formability;Necking1.IntroductionThe behaviour of thin-walled sections in bending has frequently been examined for structural purposes,focusing on load carrying capacity.In today's forming of thin-walled components,however,new challenges have arisen including prediction of dimensional tolerances.One example from the automotive industry is the aluminium space frame concept,which consists of bent aluminium extrusions connected into a structural framework to reduce vehicle's weight.Here distortions of cross-sectional members in the form of pre-and post-buckling deformations during manufacturing have a major impact on the tolerance of the ®nal build.Local distortions also greatly in¯uence the robustness of the manufacturing process and hence the overall costs.Over the past several years,inelastic bending behaviour of tubes and columns have been investigated from a structural point of view.In forming of thin-walled extrusions,however,cross-sectional distortions are also important to aestheticsof exposed parts as well as functionality.In this connection,forming of thin-walled hollow sections also has certain similarities with sheet metal.The formability of extruded pro®les is closely related to material properties,geometry of cross-section and bending method.The material's hardening and plastic anisotropy determine the likelihood of ductile failure for a given section and bending method.The most common failure modes are wall-thinning,necking and ductile fracture,as illustrated in Fig.1.Strain hardening,strain rate sensitivity and plastic anisotropy are the more important formability parameters.Geometrical defects are essential to aesthetics,service capabilities and tolerances in regions where parts are joined.Cross-sectional distortions can be reduced by means of ¯exible mandrels or some other internal support.Pre-stretch-ing may be applied to reduce local buckling,but this may transfer the tolerance problem to the tensile ¯ange in the form of necking or a single sagging wave developing along the bent portion.Web de¯ections are usually a result of defor-mation taking place at the ¯anges.The sagging depth of the external ¯ange and the post-buckling deformation of the inner ¯ange are therefore usually of greaterimportance.Journal of Materials Processing Technology 128(2002)48±66*Corresponding author.E-mail address:frode.paulsen@ (F.Paulsen).0924-0136/02/$±see front matter #2002Published by Elsevier Science B.V .PII:S 0924-0136(02)00178-4V olume conservation of plastic deformation typically increases the area of the compressive ¯ange and reduces that of the tensile ¯ange,see Fig.1.In extreme cases,i.e.tight radius bending,this has also to be considered in the design of the section.Elastic springback determines the overall dimensional tolerances of formed components.Since Young's modulus of aluminium is about three times lower than that of steel,dealing with springback represents a great industrial chal-lenge.The amount of springback,however,is important to tool design only.It is therefore essential to establish a pro-cess route including extrusion,heat treatment,cut-to-length,storage and forming that provides repeatable mechanical properties and forming conditions.Over the years,numerous authors have examined the bending behaviour of thin-walled sections from different perspectives.Examples of these authors are Ades [1],Lay [2],Reddy [3],Gellin [4],Shaw and Kyriakides [5],Kyr-iakides and Shaw [6],Ju and Kyriakides [7],Reid et al.[8],Yu and Johnsen [9],Welo [10],Paulsen and Welo [11]and Corona and Vaze [12].However,very few attempts have been made in trying to develop relatively simple analytical models for prediction of local deformations in industrial forming operations.From this point of view,the present work aims to identify the most essential material and geometry parameters,and incorporate these in user-friendly design models.An experimental program has been con-ducted to verify the applicability of the models.2.Theoretical models for cross-sectional deformations in pure bending2.1.Constitutive descriptionAccording to deformation theory of plasticity,the relation between stress and plastic strain in a state of plane stress is 1s ab E s C abgd e gd ;C abgd 13 d ag d bd d ad d bg 23d ab d gd(1)where E s s e =e e is the secant modulus,and s e and e e are theeffective stress and strain,respectively.Consider a long plate subjected to gradually increasing compressive stresses applied at the two shorter edges.At the onset of buckling,a slight distortion of the plate gives rise to a variation of the strain components e ab .Assume further that these bending strain components vary linearly acrossthe1Greek indices range from 1to 2and the co-ordinate directions 1equals x and 2equals y .F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6649plate's thickness,and also that they are unaffected by the membrane strain at the median plane of the plate.According to the variation theory of Stowell [13],integration of Eq.(1)across the thickness gives the following relation between the individual bending moment and curvature components:d M ab ÀD L abgd dk gd ;L abgd C abgd À1ÀE t E ss ab s gds 2e(2)where D E s t 3=12is the ¯exural rigidity,E t the tangent modulus,t the member's thickness and k ab is the local curvature (dk ab @2w =@x 2ab ).By adopting a power-law to represent the uniaxial stress±strain curve s e K e n e ,the ratio E t =E s n .2.2.Shape function for an inelastically built-in flange The mechanics of a bent pro®le's ¯ange (see Fig.2)resembles that of a long plate discussed above as long as the global deformations are reasonably small.A main challenge,however,is to establish a shape function that satis®es appropriate boundary conditions along the edges such that restraining effects from adjacent members can be taken into consideration.A suitable shape function to describe the de¯ections of an inelastically built-in ¯ange is proposed as follows:w w 0sin m p x l 0 c sinp y b 12c À1 cos 2p ybÀ1(3)Fig.1.Illustration of areas of concern related tobendability.Fig.2.Co-ordinate system for local deformations in bending.50F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±66where b is the width of the ¯ange,and m is the number of half-waves developing over the initial length l 0.The constraint factor c may be written in the formc p t 3fp t 3f 2Àn =h t 3wb (4)Here t f and t w are the ¯ange and web thicknesses,respec-tively,h the depth of the cross-section and n is the strain-hardening coef®cient of the material.It is noticed that c 0in case of completely ®xed edges t 3w b = t 3f h 3I ,andc 1in case of simply supported edges t 3w b = t 3f h 30 .A more detailed discussion on the fundamentals associated with Eqs.(3)and (4)is presented in Paulsen and Welo [14].2.3.Suck-in type (sagging)deformations for a single chambered (SC)section 2For a uniformly curved,thin-walled ¯ange loaded with a constant circumferential tensile stress s yy (no summation on y )across its thickness,equilibrium gives a radial stress component s zzs yy t f R Ã(5)where R Ã R h =2 w s is the instantaneous bending radius at a given position of the ¯ange (R is the nominal bending radius and w s is the local inward deformation of the ¯ange,as de®ned in Eq.(3)with m 1(see Fig.2)).Notice that w s neglected in comparison with R Ãand is therefore only included in calculating the circumferential bending stress.Consider now the stress component s zz as an external load under the assumption local bending is the only internal action.The total potential energy of an element of the ¯ange is d Is yy t fRÃw s R Ãd y d y ÀM yy k yy R Ãd y d y (6)where M yy À4=3D k yy and k yy @2w s =@x 2ab .By adoptinga power-law to represent the uniaxial stress±strain curve,i.e.E t =E s n ,the global bending stress may be expanded in terms of a truncated Taylor series if w s is much smaller than the depth of the cross-section.By minimising the potential energy with respect to the maximum de¯ection w 0s ,a third-order equation is obtained.At large radii,higher-order termsof w 0s can be neglected such that a reasonably close approx-imation to the solution of w 0s isw 0s À32b 4h 2f G 1232f ;G 1 360p À1440 c À360p ;G 2 15p À16 c 2 16À24p c 12p ;G 3 315p À960 c 2 960À270p c 135p(7)Fig.3shows an example of a cross-section undergoing typical sagging deformations.Please notice that the equation describing the deformation depth of a double chambered (DC)section may be derived in a similar way.2.4.Correction of suck-in for small bending angles The sagging depth found from Eq.(7)is valid for large bending angles only.For relatively small bending angles,the suck-in depth is affected by the length of the bend.In pure bending or rotary draw bending for example (see [15]or [16]for a description of bending methods),the length of the bend is proportional to the bending angle.For an increasing bending angle,the sagging depth therefor gradually app-roaches the stationary value in Eq.(7).The effect of the length of the bend can be determined by use of the principle of virtual work.Let the shape function for a deformed ¯ange in bending in Eq.(3)be taken on the form w s w 0s w x w y ,where w x sin m p x =l 0 ,w y c sin p y =b 1=2 c À1 cos 2p y =b À1 ,w 0s is the stationary suck-in depth found in Eq.(7)and l 0is the length of the ¯ange.The strain energy due to local bending can be approximated by the following relation:I int Z l 0x 0Z b y 0D (C 1111w 20s @2w x @x 2 2 2C 1122w 20s@2w x @x 2@2w y@y 2 4C 1212@2w s@x @y2)d y d x (8)where the ¯exural rigidity is D E s t 3f =12and the corre-sponding membrane strain energy isI ext Z l 0x 0Z by 0s yy t fRw 0s w y d y d x (9)where s yy is the stress in the longitudinal or circumferential direction of the ¯ange.By minimising the totalpotentialFig.3.Illustration of suck-in (sagging)for an SC section (left)and a DC section (right).2This method applies to both the tensile and the compressive flange for pure bending.F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6651energy I int I ext with respect to the maximum sagging depth w0s and normalizing by the stationary sagging depth found in Eq.(7),the following correction factor is predicted:X 2p l40G4205406(10)The constraint functions G4;...;G6areG4 15pÀ16 c2 16À24p c 12p;G5 6p2À16p c2 48 16pÀ6p2 c 3p2;G6 30p2À32p c2 32pÀ48p2 c 24p2(11) and the relation between the bent part of the section and the bending angle is l0 y R h=2 ,where y is the bendingangle.The corrected sagging depth is simply Xw0s.2.5.Cambering that gives snap-through atthe tensile flangeA cambered¯ange is sometimes used to ensure a¯at ¯ange after bending,see Fig.4.The idea is to extrude a section with a convex¯ange,and let the sagging deforma-tion force the¯ange into a¯at con®ually,the section is supported by the bending tool at each side of the section to reduce distortions.If the cambering height is too high,however,snap-through may take place,producing signi®cant cross-sectional deformation.The method out-lined to determine snap-through is based on the method outlined by Timoshenko and Gere[17]for swallow,elastic arcs.A portion of a cambered¯ange is shown in Fig.5.A ``distributed load''in the form of an internal stress load ¯ange during bending as a result of the inward component of the bending stress(indicated by a distributed load in Fig.5). The initial cambering is w c and the sagging de¯ection w s is de®ned in Fig.5.In order to determine the critical cambering that gives snap-through at a speci®c bending radius,the following assumptions have been made:(1)the section is side-supported.(2)The pressure(s22)that is introduced as the¯ange is forced downwards is assumed to be larger than the longitudinal bending stress at the onset of snap-through. This means that only the transverse stress(s22)is involved in the constitutive model.(3)The¯ange is¯at at the onset of snap-through such that the height of section is taken at the supported sides of the section.(4)The shape of the cambering resembles the shape produced by sagging of an initially¯at¯ange.The suck-in deformation will force the cambered¯ange towards the neutral layer.The¯ange¯attens and the width of the section will increase.The side supports,however,pre-vents the¯ange from moving freely at the ends,producing a compressive stress shown in Fig.5.Let the shape of the cambering and the sagging de¯ection be de®ned by the shape function in Eq.(3),w c w0c c sinp yb12cÀ1 cos2p ybÀ1(12) w s w0s c sinp yb12cÀ1 cos2p ybÀ1(13) where w0c and w0s are the maximum initial cambering height and the maximum suck-in depth,respectively.The com-pressive strain developing as the side support prevents the ¯ange from expanding ise221Z by 0@w c2d yÀZ by 0@ w cÀw s2d y"#À1e11(14) Fig.4.Illustration of an SC section with an initial cambering height w0.Fig.5.Schematic illustration of the flange in tension used for the snap-through model.52F.Paulsen,T.Welo/Journal of Materials Processing Technology128(2002)48±66where e11is the strain in the longitudinal direction of the ¯ange due to global bending and stretching.Integration of Eq.(14)givese22 Àp12w0c w0sb2G7w0sb2G8À12e11(15)where the constraint functions G7and G8areG7 16 4p c2À 16 4p cÀ6p;G8 À 2p 8 c2 2p 8 c 3p(16) and c is de®ned in Eq.(4).For side-supported sections,the webs are prevented from bending outwards,stiffening the ¯ange and reducing the suck-in deformation.This effect may be included for by replacing the current t w/t f ratio in Eq.(4)by a larger ratio,for example,3or higher.In order to derive an expression for the critical strain at the onset of snap-through,it is convenient to study transverse buckling of a¯ange with an initial imperfection in the form of w cÀw s .The principle of virtual work is applied to determine a critical de¯ection,the external work isI ext Z lx 0Z by 0s22t f@w b@y@@yw cÀw sd y d x(17)where the shape of the transverse buckling de¯ection of the ¯ange,w b,is assumed to be similar to the sagging de¯ection, hencew b w0b c sinp yb12cÀ1 cos2p ybÀ1(18) Here,w0b is the maximum buckling depth.At the onset ofbuckling,curvatures in the longitudinal direction of the ¯ange are neglected.The internal work isI int Z lx 0Z by 0D C2222 nÀ1@2w b@y22d y d x(19)where the¯exural rigidity now is D E s t3f=12and E s is the secant modulus.The total potential energy is minimized with respect to the maximum buckling deformation,obtainingw0b w0cÀw0s1Àe cr22=e22(20)where e22is the transverse strain from Eq.(15)and e cr22is the critical strain at the onset of buckling with no imperfections, i.e. w0cÀw0s 0 .The latter strain component may be presented on the following form:e cr22 p212t fb2C2222 nÀ1G9G10(21)where the constraint functions G9and G10are de®ned asG9 15pÀ16 c2 16À24p c 12p;G10 6pÀ16 c2 16À6p c 3p(22)The transverse strain at the onset of snap-through isdetermined from Eq.(14)with w cÀw s replaced by w b,obtaininge sn2212bZ by 0@w c@y2d yÀZ by 0@w b@y2d y"#À12e11(23)The strain at snap-through,e sn22,is compared with the uni-form strain in Eq.(15),to determine the point of instability,hencee22 12e cr22(24)By solving for the cambering height w0c,the minimumcambering height that produce snap-through for a givenbending radius iso0c12p w0s G7ÂÀp12w0sb2G8À14hRÀp224t fb2C2222 nÀ1G9G10b2(25)where w0s is the sagging depth determined from Eq.(7).Notice that w0c has a minimum at a speci®c radius,meaningthat a check of two neighbouring radii is necessary in orderto determine if w0c is increasing or decreasing with decreas-ing radius.If it is increasing,the minimum value occurs at alarger radius.This value of w0c should be used as themaximum cambering without snap-through.2.6.Onset of local buckling at the compressive flangeSummarising the assumptions to be made in the analysisto follow,there is no external load acting in the widthdirection of the¯ange such that uniaxial stress conditionsprevail in the median plane of the¯ange plate prior tobuckling.The de¯ection is assumed to follow Eq.(3)forany arbitrary(integer)value of m,and the restrainingcoef®cient c is given in Eq.(4).The local bending momentsare calculated in the state of plane stress.Referring to the co-ordinate system in Fig.2,the potential energy of an elementmay be calculated from the following:d I s yy t f@w b@x2Àd M ab dk ab!R d y d y(26)where the circumferential bending stress in the¯anges yy K h=2R n and w b is the buckling depth measuredrelatively to the corner of the section,see Fig.2.By integrating over the¯ange area,and minimising thepotential energy with respect to the maximum bucklingF.Paulsen,T.Welo/Journal of Materials Processing Technology128(2002)48±6653de¯ection w0b,after some manipulation the global bend radius at the onset of buckling is found to beR c18hb4m2l20C12f4032201;C1 c2 21pÀ64 c 64À18p 9p;C2 c2 6pÀ16 c 16À6p 3p;C3 c2 15pÀ16 c 16À24p 12p(27) where the number of half-waves that maximises the radius is m 2l0 =b f C3= C1 3n 1 g1=4.2.7.Post-buckling deformationsUpon further bending after buckling,the deformation will continue to increase until the section collapses.In the further analysis,it is assumed that no localised deformation takes place,i.e.the buckling waves are regularly shaped along the length,and the number of half-waves m remains constant during further bending.For simplicity,the strain in the median plane of the¯ange is being used in the constitutive equations,meaning that local bending moments do not affect the instantaneous stiffness of the¯ange.Like in the other models presented above,the global bending stress(s yy)used in the constitutive equation is assumed to be uniaxial.In order to include large deformations,the increased length of a buckled¯ange during folding must be included,s yy K h2RÀ12@w sd x2!n(28)Expanding this equation is terms of a truncated Taylor series, and integrating over the area of the¯ange and®nally minimising the potential energy with respect to the buckling wave depth w0b,a third-order equation is obtained.One of the solutions represents further uniform contraction of the ¯ange plate,whereas the two other solutions represent the folding of the¯ange,where the constraint constants C4;...;C6are quoted as follows:C4 33915pÀ106496 c4 204800À65100p c3 47250pÀ147456 c2 49152À14700p c 3675p; C5 1225pÀ3840 c4 7936À2520p c31960pÀ6144 c2 2048À700p c 175p;C6 7665pÀ23808 c4 53504À17010p c314385pÀ44544 c2 14848À5880p c 1470p(30)3.Extension to stretch bending and global behaviour 3.1.Extension to stretch bendingAll the models for predicting cross-sectional deformations shown above,are based on the Navier±Bernoulli hypothesis, i.e.initially plane cross-sections sections remains plane after bending.This hypothesis can be used to include the effect of tension on cross-sectional deformations.For processes like stretch bending and rotary stretch bending,see Fig.6,it will be shown that the distribution of tension along the bend is relatively constant.For moderately large bending angles,it is possible to include the effect of axial force by applying a correction factor a to be multiplied by the height h of the section.Consider the bending method shown in Fig.6,where one of the pivoting dies is shown in Fig.7.Equilibrium in the horizontal direction requiresÀV sin b N cos b ÀZ yj bqR sin j d jÀmZ yj bqR cos j d jÀN c cos y V c sin y 0(31) Equilibrium in the vertical direction givesV cos b N sin b ÀZ yj bqR cos j d jÀmZ yj bqR sin j d jÀN c sin y ÀV c sin y 0(32)where N and V are the forces at b,N c and V c the clamping forces and m is the friction coef®cient(Coulomb friction) between the pro®le and the tool surface.The distributed load q is assumed to be constant along the bend:3q q0(33)The shear force V,is close to zero at the slip point between the pro®le and the tool.The latter assumptions in combina-tion with Eqs.(31)and(32)give the following relation between the tension at the symmetry line and at the clamp: NN csin y 1 m sin y Àcos y Àl sin ysin y m cos y À1 l cos ycos y(34)w0b Æ1120v uu ut3The contact force distribution has a very small impact on theequilibrium equations.Results from numerical simulations indicate aconstant distribution of contact forces along the profile.54F.Paulsen,T.Welo/Journal of Materials Processing Technology128(2002)48±66where lV c q 0R(35)The above relation between the shear force V c and the uniformly distributes contact force q 0is determined from moment equilibrium at the centre of the bending die and combined with compatibility equations.Two comparability equations can be established by means of an interaction curve between axial force and bending moment.For exam-ple,the bending radius is equal to the die radius at the clamp and at the symmetry plane,i.e.at b 0,obtaining l 0for m 0.Adopting this to the relation between the axial force in Eq.(34)yields,N N csin y 1 m sin y Àcos ysin y m cos y À1cos y (36)The relation between the tension at the clamp (N c )and the tension at the symmetry plane (N )is illustrated in Fig.8for different friction coef®cients.An important aspect of theabove ®ndings is that for zero friction,the shear force is zero in the stretch bending method with two rotating dies.The process is therefore less affected of shear as compared to stretch bending methods where heavy shear forces are introduce by transverse die movement.Fig.8indicates that for moderately small bending angles,the variation of axial force is small along the section even for heavy friction (m 0:4).For hardening materials,the posi-tion of the neutral layer will therefore be almost constant along the bend.The effect of tension may be incorporated in the models by relating the location of the neutral layer to the original height of the section.This is done by multiplying the following factor (found from geometrical considerations)to the height of the section a11 2R =h l =l 0 À1(37)where the corrected height is h /a and l is the length (mea-sured at the mid-depth)of the section.Notice that in pure bending,l =l 0 1.By slightly adjusting the equilibrium equations,it is possible to perform a similar analysis of rotary draw bend-ing,giving almost similar curves as the one shown in Fig.8.3.2.Global neckingIn stretch bending,global necking is typically seen as a contraction of the centre region of the pro®le,see Fig.9.If friction is present,the stretch increases from the clamps towards the centre of the beam.Reduced friction gives a more uniform elongation of the section,and the risk of obtaining global necking isreduced.Fig.6.Stretch bending with two rotatingdies.Fig.7.Forces acting on one-half of the profile during stretch bending with two rotatingdies.Fig.8.Relation between the axial force at the symmetry plane and at theclamp (N /N c )for various friction coefficients (m )and bending angles (y ).F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6655A conservative estimate for the stretch in the centre region of the bend (e )can be found if the stretching force (N c )is taken as the average axial force,N c AK ll 0 n (38)where A is the cross-sectional area,K the constant in a power-law hardening model s K e n ,l the ®nal length at the mid-height of the section,l 0the initial pro®le length and h is the pro®le.The terms inside the brackets is the strain along the inner ¯ange of the section.If a simple diffuse necking criterion is employed (e n )together with Eqs.(36)and (38),global necking takes place if the following condition is satis®ed:nsin y 1 m sin y Àcos ysin y m cos y À1cos y1=n l l 0À1(39)3.3.Springback along the length of the profileA large axial force gives springback in the longitudinal direction of the pro®le upon unloading,see Fig.10.This can be estimated by using the global deformation at the mid-depth of the section (area centre),Eq.(37),henceD l KE l l 0À1 h 2R nl (40)where E is the Young's modulus.3.4.Elastic springback for heavy tensionIn cases where the neutral layer is located outside the section,the distribution of bending stress is almost linear over the cross-section before unloading,i.e.the stress gra-dient equals the slope of the stress±strain curve.This may be used to obtain an estimate for the released bending radius after unloading.For a linear stress distribution just prior to unloading,the bending moment is M p E th W (41)The elastic reversion of the bending moment upon unloading is M e Eh 2R eW (42)where W is the sectional modulus,E t the tangent modulustaken at the mid-depth of the section,1/R e the curvature change due to elastic springback and R is the radius at the area centre of the section ( R d h =2),see Fig.11.No bending moment is present after unloading,hencebyFig.9.Illustration of global necking in stretch bending with twodies.Fig.10.Illustration of springback along the length of theprofile.Fig.11.Elastic springback.56F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±66。