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A design method for prediction of dimensions of rectangular

hollow sections formed in stretch bending

Frode Paulsen a,*,Torgeir Welo b

a

Hydro Automotive Structures Raufoss AS,Process Development,P .O.Box 15,N-2831Raufoss,Norway

b

Hydro Automotive Structures Holland Inc.,365West 24th Street,Holland,MI 49423,USA

Received 30August 2000;accepted 19March 2002

Abstract

Local 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 ˉange 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;Necking

1.Introduction

The 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 inˉuence 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 aesthetics

of 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 ˉexible 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 ˉange in the form of necking or a single sagging wave developing along the bent portion.Web deˉections are usually a result of defor-mation taking place at the ˉanges.The sagging depth of the external ˉange and the post-buckling deformation of the inner ˉange are therefore usually of greater

importance.

Journal of Materials Processing Technology 128(2002)48±66

*

Corresponding author.

E-mail address:frode.paulsen@https://www.doczj.com/doc/8517724132.html, (F.Paulsen).

0924-0136/02/$±see front matter #2002Published by Elsevier Science B.V .PII:S 0924-0136(02)00178-4

V olume conservation of plastic deformation typically increases the area of the compressive ˉange and reduces that of the tensile ˉange,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 bending

2.1.Constitutive description

According 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 2

3d ab d gd

(1)

where E s s e =e e is the secant modulus,and s e and e e are the

effective 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 across

the

1

Greek 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±6649

plate'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 à1à

E t E s

s ab s gd

s 2e

(2)

where D E s t 3=12is the ˉexural 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 ˉange (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 deˉections of an inelastically built-in ˉange is proposed as follows:

w w 0sin m p x l 0 c sin

p y b 12c à1 cos 2p y

b

à1

(3)

Fig.1.Illustration of areas of concern related to

bendability.

Fig.2.Co-ordinate system for local deformations in bending.

50F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±66

where b is the width of the ˉange,and m is the number of half-waves developing over the initial length l 0.The constraint factor c may be written in the form

c p t 3f

p t 3

f 2àn =h t 3w

b (4)

Here t f and t w are the ˉange 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 0

in case of completely ?xed edges t 3w b = t 3

f h 3I ,and

c 1in case of simply supporte

d edges t 3w b = t 3

f 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 2

For a uniformly curved,thin-walled ˉange loaded with a constant circumferential tensile stress s yy (no summation on y )across its thickness,equilibrium gives a radial stress component s zz

s yy t f R ?

(5)

where R ? R h =2 w s is the instantaneous bending radius at a given position of the ˉange (R is the nominal bending radius and w s is the local inward deformation of the ˉange,as de?ned in Eq.(3)with m 1(see Fig.2)).Notice that w s neglected in comparison with R ?and 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 ˉange is d I

s yy t f

R

?w s R ?d y d y àM yy k yy R ?d y d y (6)

where M yy à4=3D k yy and k yy @2w s =@x 2ab .By adopting

a 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 deˉection w 0s ,a third-order equation is obtained.At large radii,higher-order terms

of w 0s can be neglected such that a reasonably close approx-imation to the solution of w 0s is

w 0s à32b 4h 2f G 1

232f ;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 ˉange 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 ˉange.The strain energy due to local bending can be approximated by the following relation:

I int Z l 0x 0Z b y 0

D (C 1111w 20s @2w x @x 2 2 2C 1122w 20s

@2w x @x 2@2w y

@y 2 4C 1212@2w s

@x @y

2)d y d x (8)

where the ˉexural rigidity is D E s t 3

f =12and the corre-spondin

g membrane strain energy is

I ext Z l 0x 0Z b

y 0s yy t f

R

w 0s w y d y d x (9)

where s yy is the stress in the longitudinal or circumferential direction of the ˉange.By minimising the total

potential

Fig.3.Illustration of suck-in (sagging)for an SC section (left)and a DC section (right).

2

This 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±6651

energy 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 l40G4

2

05

4

06

(10)

The constraint functions G4;...;G6are

G4 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 bending

angle.The corrected sagging depth is simply Xw0s.

2.5.Cambering that gives snap-through at

the tensile flange

A camberedˉange is sometimes used to ensure aˉat ˉange after bending,see Fig.4.The idea is to extrude a section with a convexˉange,and let the sagging deforma-tion force theˉange into aˉat con?https://www.doczj.com/doc/8517724132.html,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 camberedˉange is shown in Fig.5.A ``distributed load''in the form of an internal stress load ˉange 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 deˉection 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 theˉange 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)Theˉange isˉat 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 initiallyˉatˉange.

The suck-in deformation will force the camberedˉange towards the neutral layer.Theˉangeˉattens and the width of the section will increase.The side supports,however,pre-vents theˉange from moving freely at the ends,producing a compressive stress shown in Fig.5.

Let the shape of the cambering and the sagging deˉection be de?ned by the shape function in Eq.(3),

w c w0c c sin

p y

b

1

2

cà1 cos

2p y

b

à1

(12) w s w0s c sin

p y

b

1

2

cà1 cos

2p y

b

à1

(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 ˉange from expanding is

e22

1

Z b

y 0

@w c

2

d yà

Z b

y 0

@ w càw s

2

d y

"#

à

1

e11

(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±66

where e11is the strain in the longitudinal direction of the ˉange due to global bending and stretching.Integration of Eq.(14)gives

e22 àp

12

w0c w0s

b2

G7

w0s

b

2

G8

à

1

2

e11(15)

where the constraint functions G7and G8are

G7 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 ˉange 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 aˉange with an initial imperfection in the form of w càw s .The principle of virtual work is applied to determine a critical deˉection,the external work is

I ext Z l

x 0

Z b

y 0

s22t f

@w b

@y

@

@y

w càw s

d y d x(17)

where the shape of the transverse buckling deˉection of the ˉange,w b,is assumed to be similar to the sagging deˉection, hence

w b w0b c sin

p y

b

1

2

cà1 cos

2p y

b

à1

(18) Here,w0b is the maximum buckling depth.At the onset of

buckling,curvatures in the longitudinal direction of the ˉange are neglected.The internal work is

I int Z l

x 0

Z b

y 0

D C2222 nà1

@2w b

@y2

2

d y d x(19)

where theˉexural 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,obtaining

w0b w0càw0s

1àe 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 p2

12

t f

b

2

C2222 nà1

G9

G10

(21)

where the constraint functions G9and G10are de?ned as

G9 15pà16 c2 16à24p c 12p;

G10 6pà16 c2 16à6p c 3p(22)

The transverse strain at the onset of snap-through is

determined from Eq.(14)with w càw s replaced by w b,

obtaining

e sn22

1

2b

Z b

y 0

@w c

@y

2

d yà

Z b

y 0

@w b

@y

2

d y

"#

à

1

2

e11

(23)

The strain at snap-through,e sn22,is compared with the uni-

form strain in Eq.(15),to determine the point of instability,

hence

e22 12e cr22(24)

By solving for the cambering height w0c,the minimum

cambering height that produce snap-through for a given

bending radius is

o0c

12

p w0s G7

p

12

w0s

b

2

G8à

1

4

h

R

à

p2

24

t f

b

2

C2222 nà1

G9

G10

b2

(25)

where w0s is the sagging depth determined from Eq.(7).

Notice that w0c has a minimum at a speci?c radius,meaning

that a check of two neighbouring radii is necessary in order

to determine if w0c is increasing or decreasing with decreas-

ing radius.If it is increasing,the minimum value occurs at a

larger radius.This value of w0c should be used as the

maximum cambering without snap-through.

2.6.Onset of local buckling at the compressive flange

Summarising the assumptions to be made in the analysis

to follow,there is no external load acting in the width

direction of theˉange such that uniaxial stress conditions

prevail in the median plane of theˉange plate prior to

buckling.The deˉection is assumed to follow Eq.(3)for

any arbitrary(integer)value of m,and the restraining

coef?cient c is given in Eq.(4).The local bending moments

are calculated in the state of plane stress.Referring to the co-

ordinate system in Fig.2,the potential energy of an element

may be calculated from the following:

d I s yy t f

@w b

@x

2

àd M ab dk ab

!

R d y d y(26)

where the circumferential bending stress in theˉange

s yy K h=2R n and w b is the buckling depth measured

relatively to the corner of the section,see Fig.2.

By integrating over theˉange area,and minimising the

potential energy with respect to the maximum buckling

F.Paulsen,T.Welo/Journal of Materials Processing Technology128(2002)48±6653

deˉection w0b,after some manipulation the global bend radius at the onset of buckling is found to be

R c

18hb4m2l20C1

2

f

4

032

2

01

;

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 deformations

Upon 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 theˉange is being used in the constitutive equations,meaning that local bending moments do not affect the instantaneous stiffness of theˉange.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 buckledˉange during folding must be included,

s yy K h

2R

à

1

2

@w s

d x

2!n

(28)

Expanding this equation is terms of a truncated Taylor series, and integrating over the area of theˉange 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 ˉange plate,whereas the two other solutions represent the folding of theˉange,

where the constraint constants C4;...;C6are quoted as follows:

C4 33915pà106496 c4 204800à65100p c3 47250pà147456 c2 49152à14700p c 3675p; C5 1225pà3840 c4 7936à2520p c3

1960pà6144 c2 2048à700p c 175p;

C6 7665pà23808 c4 53504à17010p c3

14385pà44544 c2 14848à5880p c 1470p

(30)3.Extension to stretch bending and global behaviour 3.1.Extension to stretch bending

All 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 requires

àV sin b N cos b à

Z y

j b

qR sin j d j

àm

Z y

j b

qR cos j d jàN c cos y V c sin y 0

(31) Equilibrium in the vertical direction gives

V cos b N sin b à

Z y

j b

qR cos j d j

àm

Z y

j b

qR 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:3

q 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: N

N c

sin y 1 m sin y àcos y àl sin y

sin y m cos y à1 l cos y

cos y

(34)

w0b ?

1120

v u

u u

t

3The contact force distribution has a very small impact on the

equilibrium equations.Results from numerical simulations indicate a

constant distribution of contact forces along the profile.

54F.Paulsen,T.Welo/Journal of Materials Processing Technology128(2002)48±66

where l

V 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 c

sin y 1 m sin y àcos y

sin y m cos y à1

cos 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 the

above ?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 a

1

1 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 necking

In 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 is

reduced.

Fig.6.Stretch bending with two rotating

dies.

Fig.7.Forces acting on one-half of the profile during stretch bending with two rotating

dies.

Fig.8.Relation between the axial force at the symmetry plane and at the

clamp (N /N c )for various friction coefficients (m )and bending angles (y ).

F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6655

A 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 l

l 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 ˉange 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:

n

sin y 1 m sin y àcos y

sin y m cos y à1

cos y

1=n l l 0

à1

(39)

3.3.Springback along the length of the profile

A 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),hence

D l K

E l l 0à1 h 2R n

l (40)

where E is the Young's modulus.

3.4.Elastic springback for heavy tension

In 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 t

h W (41)

The elastic reversion of the bending moment upon unloading is M e E

h 2R e

W (42)

where W is the sectional modulus,E t the tangent modulus

taken 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,hence

by

Fig.9.Illustration of global necking in stretch bending with two

dies.

Fig.10.Illustration of springback along the length of the

profile.Fig.11.Elastic springback.

56F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±66

subtracting Eq.(42)from Eq.(41)and solving for the released radius R rel (1=R rel 1=R à1=R e ),one obtains R rel

R 1à E t =E

(43)

4.Experimental setup for pure bending

Experiments were performed in the bending setup shown in Fig.12[18].The setup is designed to provide a constant moment along the beam with minimum effects of axial forces and end clamps.An angle sensor is used to register the end rotation (global curvature)during bending.A three-point curvature sensor is placed in the pro?le's centre to determine the local curvature in the corner of the section.A scanner is being developed to measure the deformation of the ˉanges at different stages of bending,moving the pro?le against a small wheel connected to a plate that follows the ˉange's contour.A position sensor (LVDT)is mounted on

the plate to register the vertical movement while an incre-mental pulse transmitter is connected to one of the rollers to record the longitudinal position.Strain gauges are used to detect the onset of buckling as deduced from any change of strain relative to that in the pro?le's corner.

In the test program,SC and DC aluminium alloy AA6060-T4sections were used.These two section types provide different constraint conditions along the edge of the ˉange plate.The pro?les were 600mm long,of which a 120mm portion at each side was clamped,leaving a length of 360mm be subjected to a constant bending moment.5.Results and discussion

The buckling radius (R c )normalised with the radius R e ss predicted by assuming the ˉange as a simply supported elastic plate is shown for different constraint conditions in Fig.13.Since the normalised buckling radius is relatively independent of the ˉange slenderness,the

conventional

Fig.12.Bending setup (left);SC and DC cross-sections

(right).

Fig.13.Normalised buckling radius vs.the flange slenderness factor (b /t f )2.

F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6657

factor(b/t f)2is also the major factor for buckling in the inelastic range.As expected,the present model predicts abuckling radius between the one for a simply supported and clamped edges using the so-called effective modulus theory presented by Bleich[19].Predicting a constraint factor of c 0:63for the sections used by Corona and Vaze [12],the present analytical model is in excellent agreement with their more involved numerical model as well as their experiments shown as?lled circles.The analytical results are also in reasonably good agreement with the experimental results obtained herein(c 0:44).It is concluded that the factor(b/t f)2is as important for inelastic as for elastic buckling.

Fig.14shows the maximum suck-in of the externalˉange for SC and DC sections.Notice that neither of the curves uses material parameters as variables since these are of less importance to local deformations of tensile members.It appears that the curve for an SC section increases more rapidly with the parameter b4= t2f R2 due to a relatively smaller restraint coef?cient c provided by the two inelasti-cally built-in edges,compared to?xed and inelastically built-in edges.For each section type,the predicted curves for three differentˉange thicknesses(2.0,2.5and3.0mm) merge perfectly together,indicating that the parameter b4= t2f R2 is the main parameter with respect to distortions of the tensileˉange.It is further observed that the predic-tions agreed quite well with the experimental results,espe-cially if the deformation w0=h is less than0.1where the relationship is almost linear.The sagging depth of the DC section tends to deviate somewhat for large values of w=h, most likely as a result of strong non-linear effects provided by localised post-buckling deformations of the internal ˉange.

The graph on the right-hand side in Fig.14shows the maximum wave depth w b=b along the compressiveˉange (full lines)versus the parameter

h= nR

p

for different width-to-thickness ratios.Noticing that the predicted curves for the internalˉange follow those of the externalˉange in the beginning,continuing relatively linear after the onset of local buckling,it is suggested that the parameter

h= nR

p

has been identi?ed as the main inˉuential parameter with respect to post-buckling wave https://www.doczj.com/doc/8517724132.html,paring the pre-dicted curves with the experimental results for an SC section with b/t ratio of31(open square markers),the agreement is remarkably good.The experimental post-buckling plots for a DC section(b=t 18:8),however,are found to resemble the predicted uniform sagging mode of the externalˉange rather than the wavy appearance predicted for the internal ˉange.The reason for the discrepancy between theoretical and experimental results is not exactly known.

The theoretical predictions have been converted into a more practical method for design of rectangular hollow sections.Although rectangular hollow sections have been used to exemplify the method,the model is general in the sense that it can easily be modi?ed to other types of cross-sections by changing the shape function and boundary conditions in Eqs.(3)and(4).See Paulsen and Welo[20] for diagrams for pure bending only.

5.1.Onset of global necking in the mid-section of the beam Global necking is typically seen as a contraction of the centre region of the pro?le,see Fig.9.In case of large friction,the stretch increases from the clamps towards the centre of the beam.Reduced friction produce a more uni-form distribution of the elongation of the pro?le,reducing the risk of obtaining a global https://www.doczj.com/doc/8517724132.html,e:(1)Enter the diagram in Fig.15with the bending angle(y).(2)Go vertically to a representative friction coef?cient m(Coulomb friction).(3)A horizontal line can now be drawn to

the

Fig.14.Suck-in of tensile of tensile flange(left);buckling wave depth of compressive flange for six width-to-thickness ratios and suck-in of tensile flange for three width-to-thickness ratios(DC)(right).

58F.Paulsen,T.Welo/Journal of Materials Processing Technology128(2002)48±66

representative hardening coef?cient.(4)The critical elonga-tion producing necking in the area centre of the section are found at the left part of the diagram.If the elongation at the mid-depth of the section is larger than the critical value found in the diagram,undesirable contraction of the pro?le is likely to take place.It is also possible to reverse the steps (1)±(4)in order to determine the bending angle (y )where necking takes place.

5.2.Springback along the length of the profile

A section bent under the action of simultaneous external tension contracts at the onset of unloading,as illustrated in Fig.10.The contraction of the length will increase for higher hardening and larger yield strength,and is also a function of the stretch introduced during https://www.doczj.com/doc/8517724132.html,e :(1)Enter the diagram in Fig.16with the relative elongation (l /l 0

)

Fig.15.Onset of global

necking.

Fig.16.Contraction of length at unloading.

F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6659

measured at the area centre of the section.(2)Go vertical to a suitable hardening coef?cient.(3)Find the yield stress (s 0.2)divided by Young's modulus in the right-hand side of the diagram.(4)The relative contraction of the section at unloading is found by drawing a vertical line down from the curves representing the yield stress divided by Young's modulus.

5.3.Elastic springback for heavily stretched profiles When a large tension is applied,springback curvature is effectively reduced.The effect of the geometry is also cancelled out,because the stress distribution over the height of the section becomes fairly linear.This can be utilized to draw the curves in Fig.17,which should be considered as an indication of springback in cases where the neutral layer is located outside the https://www.doczj.com/doc/8517724132.html,e :(1)Determine the relative elongation (l /l 0)measured at the area centre of the section and enter the diagram in Fig.17.(2)Go vertically to the appropriate hardening coef?cient.(3)Find the yield stress (s 0.2)divided by Young's modulus in the right-hand side of the diagram.(4)Finally,the ratio between the released radius (R el )and the die radius (R d )can be found.5.4.Maximum sagging depth for SC sections

Suck-in (sagging)of ˉanges is strongly related to the width of the section.A small bending radius and a large stretch also provide large sagging.Suck-in is almost inde-pendent of yield stress and https://www.doczj.com/doc/8517724132.html,e:(1)Determine the inˉuence factor of external tension (a )in the lower diagram in Fig.18by using the elongation at the area centre

of the section (l /l 0).The constant a 1for pure bending and a 1=2if the neutral layer located at the interior ˉange.(2)Go from web thickness-to-ˉange thickness ratio to the curves representing the width (b ),the height (h )and the

elongation factor (a ).(3)After the suck-in factor (b 4= t 2f R 2

)is calculated,the suck-in depth w 0s is easily determined by the multiplying w 0s =h a by h /a .

5.5.Maximum sagging depth for DC sections

The suck-in depth is effectively reduced by an internal wall.The diagrams for DC sections similar to the curves drawn for an SC section.The only difference is the con-straint conditions at the centre of the ˉange,which gives a stiffer ˉange as compared to an https://www.doczj.com/doc/8517724132.html,e:(1)Determine the inˉuence factor of external tension (a )in the lower diagram in Fig.19by using of the elongation of the inner ˉange (l /l 0).The constant a 1for pure bending and a 1=2if the neutral layer is located in the interior ˉange.(2)Go from the web thickness-to-ˉange thickness ratio to the curves repre-senting the width of one chamber (b ),the height (h )and the

elongation factor (a ).(3)After the suck-in factor b 4= t 2f R 2

is calculated,the suck-in depth w 0s is easily determined by multiplying w 0s =h a by h /a .

5.6.Correction factor for sagging for small bending angles

For small bending angles (b = y R !0:2),the suck-in depth is reduced as compared to value found in the previous diagrams.In the diagram shown in Fig.20,a correction factor that can be multiplied by the sagging depth found

in

Fig.17.Elastic springback at large tension.

60F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±66

Figs.18and https://www.doczj.com/doc/8517724132.html,e:(1)Determine the inˉuence factor of external tension (a )in Figs.18or 19.(2)Go from the web thickness-to-ˉange thickness ratio to the curves representing the width (b ),the height (h )and the elongation factor (a ).(3)Calculate the ratio of the width (b ),the bending angle (y )and the die radius (R )(b /(y R ))and ?nd the correction factor (X ).(4)The corrected sagging depth is Xw 0s ,where w 0s is the sagging depth found in Figs.18or 19.

5.7.Cambering that gives snap-through

Cambering has commonly been used to control the ?nal shape of the exterior ˉange.A cambered ˉange can be used successfully to obtain a ˉat ˉange after bending and stretch-ing.If the cambering height is too large,however,snap-through may occur.During bending,sagging forces the cambered ˉange towards the interior ˉange.This

gives

Fig.18.Maximum sagging depth for SC sections.

F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6661

compressive forces in the ˉange that may lead to buckling where the ˉange suddenly snaps through.In order to obtain a ˉat ˉange after bending,the cambering should reˉect the shape of a deformed ˉange without cambering.This shape can be determined from Eqs.(3)and (4).The diagram to be presented in this section assumes that the pro?le is side-supported and that the shape of the cambering resembles the deformed shape of an initially ˉat ˉhttps://www.doczj.com/doc/8517724132.html,e:(1)Enter the diagram in Fig.21with the hardening (n ).(2)Determine the width-to-height ratio and the elongation factor (a )from Fig.18(ba /h ).(3)Find the width-to-thickness ratio of the ˉange (b /t f ),so that the critical strain at snap-through can be predicted.(4)Go to the diagram in Fig.22,calculate the ratio of the maximum suck-in depth (w 0s )at the desired radius from Fig.18and ?nd the ratio of the suck-in depth to the bending radius (R ).(5)Use the critical strain from the previous ?gure and determine the maximum cambering that can be applied without snap-through (w 0c

).

Fig.19.Maximum sagging depth for DC sections.

62F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±66

Fig.20.Correction factor for sagging at small bending angles.

Fig.21.Cambering height that gives snap-through(part1).

Fig.22.Cambering height that gives snap-through(part2).

Fig.23.Critical bending radius at the onset of buckling for SC sections.

5.8.Critical radius at the onset of local flange buckling for SC sections

The diagram shown in Fig.23gives an estimate of the critical radius at the onset of ˉange buckling.Important parameters are the width-to-thickness ratio of the ˉange and the hardening of the material.Increasing tension may reduce or even eliminate the possibility of local buckling of the internal ˉange.The constraint at the corner of the section also has some https://www.doczj.com/doc/8517724132.html,e :(1)Calculate the ratio of the web-to-ˉange thickness (t w /t f ).This can be set to 3in case of side support.(2)Determine the elongation factor (a )from Fig.18and ?nd the correct curve representing b ,h and a .(3)Go to the correct hardening coef?cient and determine the width-to-thickness ratio of the ˉange (b /t f ),go vertically to resolve the critical buckling radius (R c ).5.9.Post-buckling depth for SC sections

In this ?nal diagram in Fig.24,the post-buckling depth of an SC section can be estimated.For DC sections,the

buckling depth follows more or less the sagging depth [16]and can be found from https://www.doczj.com/doc/8517724132.html,e :(1)Calculate the web-to-ˉange thickness (t w /t f )ratio.This can be set to 3in case of side support.(2)Determine the inˉuence factor of additional elongation (a )from Fig.18and determine the correct curve representing b ,h and a .(3)Find the width-to-thickness ratio of the ˉange (b /t f ).(4)Go horizontally to the desired value of the buckling parameter

h 2a à1 = nR p .(5)Finally,an estimate for the relative buckling depth w b =b can be obtained.6.Conclusions

The present investigation aims to predict local deforma-tions and buckling of rectangular hollow sections in order to better understand factors affecting dimensional tolerances in industrial bending https://www.doczj.com/doc/8517724132.html,ing the deformation theory of plasticity and the energy method,analytical models that provide simple closed-form solutions for various shapes and materials are being developed.The results revealed a

very

Fig.24.Post-buckling depth for SC sections.

F .Paulsen,T.Welo /Journal of Materials Processing Technology 128(2002)48±6665

strong inˉuence ofˉange width on local deformations.The instantaneous bending stiffness(constraint condition)of the web also seriously affects the local deˉection of connected ˉange members.Unlike local buckling,the suck-in of the externalˉange is almost independent of strain hardening and the initial yield stress of the material.The models developed have proven to provide results that are in good agreement with experimental?ndings.Their validity is found to cover most practical cases;though shortcomings are revealed at large local deformations owing to localisation of post-buckling deformations in the experiments.It is,therefore, concluded that the developed models provide to be an ef?cient,simple tool to the evaluation of bendability of thin-walled sections.

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