Residual stresses in cold-rolled profiles

Residual s t r e s s e s in cold-rolled profiles Jacques Rondal, Doctor in Applied Sciences, Civil Engineer, Associate Professor, University of Liege, Belgium.

Abstract Structural steel members obtained by cold.forming have become more and more popular in steel constructiort They allow for lightening the weight and thus provide appreciable savings over conventional hot.rolled sections. Cold. forming leads to changes in the main structural characteristics of the virgin materiaL Among them the increase of the yield strength due to strain.hardening and of the ultimate strength due to strain.ageing and the birth of residual stresses which play an important role in the behaviour of the profiles under compression loads. The aim of the paper is to propose a theoretical method for predicting residual stresses due to cold.rolling. The numerical results obtained lead to a better knowledge of the characteristics of the cold.rolling (elastic spring.back, effect of the ratio of the thickness versus the radius of the comer, distribution and size of the residual stresses with respect to the dimensions of the profiles).

I n t r o d u c t i o n

In steel construction, three main families of structural members are used. Hot-rolled shapes belong to the first family; they are of c o m m o n use and were born at the same time as the steel construction itself. The second family contains built-up members, ie. members c o m p o s e d by plates which are connected by bolts, rivets or welding. The third family is perhaps less familiar, though of growing importance: the cold-formed sections, obtained from steel sheets by roll forming or by press braking (11. Compared with hot-rolled shapes, cold-formed steel structural members provide interesting advantages in building construction(2~: - more especially for relatively low Ioadings and/or short spans, an important weight saving can be obtained; - it is easy to produce sections, the shapes of which are unusual but are appropriate to the function of the profile; - nestable sections can be obtained easily, allowing for compact packaging and economical transportation; the light weight simplifies handling and erection; - many secondary operations can take place which are associated directly to the forming (piercing, notching, marking, painting...). in contrast to hot-rolled steel m e m b e r s which are subject to residual cooling stresses, cold-formed profiles are affected by deformational residual stresses in both longitudinal and transverse directions (31. These residual stresses play an important role when the structure is subject to fatigue or stability problems and new EEC recommendations for the design of steel structures take -

150

account of the effects of residual stresses on the ultimate load of structure(4). ]'he distribution and the amplitude of the residual stresses much depend on the fabrication process of the section. Cold-formed sections can be produced: either by a discontinuous process, for small series of sections, with a leaf press brake (Figure I a) or a coin press brake (Figure l b); or by a continuous forming, for more important series, by passing through successive pairs of rolls (Figure lc). Major difficulties are met with when measuring residual stresses in cold-formed sections. On the one hand, measurements are allowed in a restricted number of points only at least for profiles of small dimensions. On the other hand, accurate measurements cannot be performed in the rounded parts of the sections, although these areas are subject to large residual stresses when cold-formed and are therefore worthwhile investigating. In the fiat areas, measurements can be made more easily; but the accuracy is rather small because of the reduced amplitude of the residual stresses. Any attempt to theoretical determining of residual stresses would therefore be welcome. The theoretical methods aimed to this purpose will not depend, contrary to experimental methods, on the size of the specimens and will allow for Darametric studies, with the result of a better understanding of the influence of the variables respectively and of the history of the onset of residual stresses. Theoretical methods may nevertheless require some calibration. Therefore, the validity of any theoretical method must be checked by comparing the theoretical -

CONSTRUCTION & BUILDING MATERIALS Vol, 1 No. 3 SEPTEMBER 1987

I II

\

iI

s~" 11\

/ " x~xx~

i!

~

~ •

a. leaf

press braking

"~x~

,

b.coin

S ~

~ ~s~

~

press braking

i

A

C. cold roll

forming

machine

Cold-forming machines; (a) leaf press braking; (b) coin press baking; (c) cold.roll forming machine results to the experimental; conclusions can then be drawn either about the accuracy of the method or about the range of validity of the latter. Predicting methods of residual stresses have been suggested by several authors for fabrication procedures other than cold-rolling: Plumier for welded ! sections ml, ingvarsson for sections produced by press-brakingl6) and Kato for welded hollow sections ~3~. In the frame of the present study, it will be referred to the

simple model of a rounded corner obtained by elastoplastic bending and a subsequent equilibrium of the stresses to account for the actual shape of the section. Several processes - uncoiling, levelling and side trimming - are not included here, although they result also with residual stresses. Roll-forming of a comer The method used to predict residual stresses due to cold-

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

151

0

fu

............................. •

mq)

•

m©

• \.~./ E

~e

~p

~u Fig 3

Fig 2

Forming of a corner by pure bending

Stress.strain diagram of the material

~

,~

/

~

/

~

/ f

, ~ . ~

/

.... -

/

!

~ 1 - ' /

~

,~

f / /

"

x,

/

/

i..'1" I

~

/ ~

"~"

¢;z.-. -~ - /

t

....",,/

/

I

/

/

/

/

/

/ /

•

/

\.,o~. /. j

Fig 4

Notations

forming is based on the following assumptions: the strain-stress relationship is represented by a trilinear law with an isotropic strain-hardening (F~lure 2); as sugessted by Karren (71, the forming of a corner is assumed to be obtained from a pure bending (Figure

.~).

-

the cold-roll forming used for a steel sheet of long length, justifies the assumption of plane strain, ie:

(Figure 4) de z = 0

(1)

with, because of the continuity of the manufacturing procedure: *z = 0

(2)

In addition the radial direct stresses are negligible:

152

~,

= 0

(3)

As shown in F i g u r e 4, a polar co-ordinates system is used, so that p can vary from 0 to the thickness t with a view to define any fibre. The method describes the history of the cold-roll forming of the comer (incremental increase of the curvature from 0 to the final inside radius r) with account taken of the elastic spring-back at the end of the process. The general flow-chart of the method is plotted in F i g u r e 5. T h e main steps are as follows:

aj Data and initial values: The data are: the inside radius r of the comer to be reached after forming; the coil thickness t;

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

Oafa : r, f, f y , f u , ~ p , ~ u ,

Nf

f

INc

go= f/2,

c=0

= Inf.[5.10 6, f/(r.fy)], dc=l/(r.N¢)

t

Ic--c d l t

t

d¢¢(i)=

(9 - % ) d e (l+9c)(l+goC+godC)

1

t

I

,

~N . @

d=l/c f Inl(d+f)/d]

~°-

'~ ¥

Do i=l,Nf

~

o¢1i)+o¢1i+1) . f --[( 2 Nf

: A He(i) = =

H e = ~ & He(i)

i=1

d~

"

He t. ( f/2-go) .210000

i-1

.~ +

Nf

2 Nt

r"= d+ go 1+6

=1~-- ~-I r

-~= (i-1)fin t

~)÷ r" Do i:l,Nt+l:

o~(i)=

He t(r+g)

ore~i) = ~ (i) ÷ oe(1) e O ¢

Fig 5

[n[(r+~)/r] t/2-~ r

t tn[{ r+t)/r]

ge~i) : oli)

z

z

General flow-chart of the method

CONSTRUCTION& BUILDINGMATERIALSVol. 1 No. 3 SEPTEMBER1987

153

the parameters of the stress-strain diagram of the material (f~, f,, ep, e,) (Figure 5); the number of layers, Nt, across the plate thickness. Newton and millimetre are the units used in the program. Several applications show that Nt = 40 is a suitable value. The computation is started with a fiat plate, ie. a zero curvature (c = 0) and a neutral axis at mid-depth of the plate (po -- t/2). The incremental curvature dc is given by -

I

-

dc = 1/rNc

\.

\

/.

(4) o

where N c is determined so that the associated increase of circumferential stress at each step does not exceed 2% of the yield stress; such a tdck is required by the necessity to detect the end of the elastic range in the fibres with a sufficient accuracy.

b) Increase of the circumferential strain and location of the netrual axis:

Fig 6

As demonstrated in (8), the increase of circumferential strain in the fibre i (Figure 6), that is associated to the curvature increment dc, is:

P,- Po d~,(O = 1 + p,c

dc 1 + poc + podc

c) Stresses and strains: In each fibre the behaviour is successively elastic, plastic and strain hardening. The flow-chart dealing with the computation of stresses and strains is drawn in Figure 7; the generalised Hooke's law and the Prandtl-Reuss equations are used(gL

(5)

where, Po, that determines the location of the neutral axis, shifts towards the inside fibre as a result of the curvature and is given by:

Po =

t

]/c + In i/c

d) Elastic spring-back: The 'active' part of the cold-roll forming is followed by an unloading (because of the bending moment vanishes), that yields a stress redistribution in the cross section and thus a change in curvature. This unloading is elastic and is produced by a bending moment equal but of opposite

1 t

c

(6)

y~

tasfi[

behav

Piasfi:

Effect of an increase of the curvature

N

behaviour ~

d~z(i) =69231 dE®(i}

de(i)= d~c(i}.o~[i)/lo~li}-~-oz{i))

d%{i)= 3.333 doz(i)

,"olil = dEPr(1) = ( d~p(i)/o,( i))[- ~ (%( i} +o, (~)) ]

dE~(;) : -0.429dE®(i)

P

Isotropic

ha I

LJ =(fo-fy)l{Eo-Ep) do~(i) = (LJ ,/~ O~3)dE¢(i) do [i)= 3.3~3 d~ ~) m

z'

~li) : ~(i)+ d%(i) ~I~l: ~{il, ~C(il

dj;) = d~ ~ • d~(~)

~E~r(~)=-0 L29 dz~(~)

~P~(i) : ~P~ (i) + d~P~(h

e~¢(i}=e~(~) .dee {~)

~(~) = E~([)~

~(i) =~ {i)~ ~P~(;)

~ {~}=~ {~}+ d~", ~)

Er(i)

E {~)=£e (~)+ Ep{~) r r r

E¢ (~} mE e¢ []) + Ep@ {~)

~ ( ~ ) = ~ (~)

~r (~) =Er~ (i} + erp (i)

: E~[E)

%(~) = %Ii)+ d%(i) ~zIi) = ~z(il + d~z(i)

% (i~ =% (i} +d% (~) £ep(i) : ~E~(;)+Er2{i ) -£~ (i} Zr (i)

az(i ) =a=t;)+da= (i)

~(i) : ~o~(i/+~zZIi)-%(i)'o:(;) o, { ~ ) = ~ ( ~ ) ~ ( ~ } - ~ = ~(~)~z (~)

~p(i) = &~d(i}+~(~)~ ~ ~ -~(;)'~(~[ ~p

Fig 7 154

(i):~(i)-~ r

~

(E)~r (;)

Flow-chart of the subroutine 'CALDECO' CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

sign to that reached at the end of the active part of the forming. The unloading bending moment is obtained by adding the elementary bending moments (Figure 8):

2

~

(i-l) ~

'

+

0

(7)

0

and

P

tv/e =

z

,~ Me (/)

(8)

i=] The associated relative angular variation is: A

"T & =

,~de

=

de

Me

t(~-

(9)

r+po l+5

e

(0

=Me

(P~- Po) t( ~ - Po) (r + p,.)

(11 )

or, account taken of (6): t

p~+rIn ~

e

(0 = Mo

r+t f

t(r+ ,,) ( ~ + r-

tt+ r) In

(12)

F

e) Residual stresses in the corner. The circumferential residual stresses in the comer are obtained by superimposing the stresses associated to the loading and the unloading respectively, ie.

res (i) =¢q~ (i) + ~r ~e (i) ~q~

- in accordance with the theory of curved beams, the neutral axis, shifts towards the inside fibre but remains, nevertheless, rather close to the mid-depth because of plastic yielding in the cross-section; the ~ and ~rz stresses before elastic unloading are nearly identical, but of opposite sign, in the extreme fibres; the ¢, and ~z stresses before elastic unloading on the one hand, and the circumferential residual stresses o~ on the other hand are nearly proportional to the yield stress of the material. Therefore, the diagrams hereafter were only drawn for fy = 355 N / m m 2. Circumferential stresses reached before and after elastic unloading are given in Figure 9 for four steel grades and two values of the radius of curvature. Figure I0 shows the evolution of the circumferential residual stresses in the four extreme points with respect to the relative radius of curvatures, when the o-- ¢ diagram is assumed to be elastic-perfectly plastic. It is observed that the location where the maximum residual tensile stress is reached depends on the relative radius of curvature. The results obtained can be of great interest not only for the researcher, who has thus a better knowledge on the onset of residual stresses due to cold-roll forming, but also for the manufacturer, who is now able to specify a minimum ductility of the material with respect to forming requirements (maximum strain that will be reached), to know the amplitude of the bending moment to be developed and of the elastic spring-back with the result of better dimensional accuracy. For an elastic-perfect plastic material, the maximum strain in the corner is reached in the inside fibre and does not depend on the material yield stress fy (Figure 1 laj. On base of numerous simulations, the maximum strain can be expressed as a simple function of the relative radius r/t: -

Po

When (r' - r) is in excess with respect to the fabrication tolerances, an additional forming is applied (Figure 5). The stresses associated with the unloading bending moment are:

~

Calculation of the bending moment

po) E

The radius of curvature after unloading is then:

r' --

Fig 8

(13)

-

Some conclusions can be drawn from the numerical results obtained accordingly:

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

e~ (~)

=

40 (r/t )-o.~

(]4)

155

O(plfy

9It 1.00 . . . . . . .

_----

I

0.50

O(Nlmm

2SOU

~-,o

I 0.0(

0 I

I

!

I

I

.

10

20

0.50

5!~'°1 , ~

30

0.00 ,~

!

~

. . . .

1.00 .....

500

0.50

I

•

0

0.00 I

~

0.50

~.(%)

,

,

10

20

• ~ , ~

30

0.00

I

0+1

0 +I - I

-I

'x

/

',

I

I

0¢ / fy

~/t 1.00 . . . . . . .

O" Nlmm2 )

2 S 0 ~

I

!

0 .1

r=3Okmm

o(plfy

!

I

• 1 -1

r=Bmm 9It

:

-~

0+1

0 +I -I

-1

.

~

oe/fy

91t 1.00. . . . . . . .

_

1~

;.~

r=Bmm

_~ ; .4

r = 30./+ m m

o'¢ 1 fy

o'@ / fy 91t

Pit _--

°"ot

0.0

0 l

-1

I

i

0 ,1

I

-1

I

10

I

,

20

30

I_

0.00 . . . . .

I

-I

0 ÷1

I

I

9It

__--ii iii __'x_

0.0~

I

-4

I

~ +I-I

0

10

20

30

0.00

. . . . . . . ~

~

~÷I

-

Celt: before right: after

156

I

!

I

0 +I -I

!

0

I~

.I

r = 30,/+ rn rn

r = 8 mm

Fig 9

!

°¢1 fy

~¢ / fy )It 1.00

0.5(

I

0 +I

30./+mm

r =

r=Smm

I

0 ÷1 -I

spring-back spring-back

teft :before right: after

spring-bat k spring-back

Circumferential stresses before and after spring, back for four grades of steel (t = 6 rnm)

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

For a strain-hardening material (Figure lib), the maximum strain is increasing with the strain value ~ at the end of the yield plateau. Then the maximum strain can be obtained by linear interpolation between both following values: - steel without yield plateau:

e,,~(%) = 33

(r/t)

OSres / fy

i•,•'•'•,c

1.(

i:iiiiiiii-i

° ~ fiber ° ~ o A externat

(!5)

°.~

02 -

elastic-perfect plastic material:

(16) For an elastic-perfect plastic material, the relative maximum bending moment of forming decreases when r/t increases but seems to be nearly independent on the yield stress. Usually, one has: e ~ ( % ) = 4 0 ( r / t ) -o.~

I

~

5 ° ~ ' ~ o ~ ,

t0

,D

(17)

M~.~/M~ = 1.16 to 1.20

r/f

~__

where the plastic moment of the section is: .0

t2

M~ = f. -~-

(18)

This increase of the bending moment that is required for the forming, in respect to the plastic moment, results from the von Mises plasticity criterion in the plane strain state when computing the stresses in the corner dudng the forming. With a strain hardening material the maximum bending moment will depend on the strain hardening modulus Ep and on the value of the strain epat

$/o

"~"

/

e

~

Fig 10 Maximum values of the circumferential residua slresses uersus the relative radius of the corner the end of the yield plateau. When the material is elastic-perfect plastic, the relative change of the radius of curvature during the spring-back

Cnax(%) * fy =235N/ram 2

E~ax(%) .~p :~e

:355 N/ram~

• fy

e

• ~p=¢%

I

¢0

x f). =800 N/m~

¢0

30

x%

30

~~ \ \

•

max(%)=¢0(r/t)-0"85 20

20

=

10%

~max (%)=¢0( r/ t) -0~S

; ,o, o

10

0

:

I

I

:

;

1

2

3

¢

5

r/t 0

;

!

:

;

I

1

2

3

¢

5

a.

=-

r/f

b.

Fig 11 Maximum strain.. (a) without strain-hardening; (b) with strain-hardening

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

157

In order to simplify the computation, the 'equilibrium coefficients' are computed at the level of the mid-depth of the plate and the so obtained values are used when considering any fibre. The resulting force in the comer is:

increases with the material yield stress and with the relative radius of curvature (Figure 12a). This change is obtained approximately by following expresssion

,~r/r (~) = 0.0015 f, (1.3 + r / t ) '.°e

(19)

with f~ in N/mm ~. For a strain hardening material, the spring-back depends on ~ strain. The value a r/r can be obtained by linear interpolation between both following limit cases: - steel without yield plateau:

F(i) = ~ (i ).

rr

-E . t (r +

~_)

The residual stresses having been equilibrated can be computed from the knowledge of the force F, by means of the normal force and the bending moment. By using the symbols in Figure 13 and by restricting to five values on each fiat side, one obtains:

Ar/r (~) = 0.0023 f, (~.3 + r/t ) o.~ strain-hardening material:

~,r/r (56) = 0.0015 f~ (1.3 + r/t ),.oe

2

1 YG creS (i) = ~z(i) - F(i) [ -~ + ~ ]

Residual stresse~ in the whole profile

In the previous section no account is taken of elastic spring-back of the longitudinal stresses that occur in the corner during forming. These stresses are not in equilibrium with the centroid of the whole section, which can be composed of several comers. Therefore the stresses must be equilibrated. As the latter depend on the shape of the section, it is not possible to give general equilibrium equations, which would be valid for any shape. Such equations were established for three common shapes (L, U, C) and are given in reference(~; only the angle will be considered hereafter. Ar/r(%)

(22)

res

I

(23)

Y~" Yl

(24)

~fl (0=~'(0 /-~+ 722 / I

~(O=F(O

/-7+

Yo.

(25)

Y2

77~,2 ]

At/r(%) + fy = 235N/ram2

/ x

•

/

:

x f~ : 800N/ramz

,/

x

/

o r )1.o6 A t / r ( V o ) = 0.0015fy (1.3 +

T

~ n0

÷

)=0.0o2af,

1o.~

J ÷

• ¢= / . %

/

P

v.

0

A r/r(%)= 0.0015fy(1.3+ ~)'

,~

!

I

;

."

1

2

3

~+

5

a.

,~ ~=10 % P

r/t ~ 0

I

I

I

&

~"

1

2

3

~,

5

r/f

~--

b.

Fig 12 Variation of the radius of the comer due to spring-back: (a) without strain-hardening; (b) with strain.hardening 158

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

t

a res(O 1 YO" YZ f~ = F(O [ - - ~ + l~,-------~I

(26)

ares(o 1 Yo. Y~ f, = F(i) [ - - ~ + -!~,,~ l

(27)

r+~ ~u,,- - - , ~c _ ~ v'~ _ ~

i.~. =

c~t + ct3 + ct (-~- 2 dz + r + 12

~

(do - r - t)

~r 1 4 )2 + t(r + t)z ( --~-+-~- - -~ )

2.Vfff (r + t)3 p ] 2 3,, t(r +--~)

+ - ~ t (r +-~ ) [ ' ~ (d~ - r - t) + - -

a r~r~es(O= F(O [--~1 + ...uo. E~~5 ]

(28)

(39)

(40)

with:

b.t.r

(29)

7 b-t.r 92 = Y . - I0 X~-

(30)

Y' =Y"

~3=~"

9

10 ~

5

U

,

b-t.r

lO ~

(31)

t- t

li

% mean

fiber

Fig I3 Notations for the calculation of the residual stresses of an angle Y4 = Y .

Y5 = Y .

3 b-t-r I0 M~-

(32)

1 b-t-r 10 V ~

(33)

where:

33.

c=b-t-r

A=2ct+

So --

(34)

~-t(r+

2~/~ 3~r

The numerical results demonstrate that the residual stresses are nearly identical, though of opposite sign, on the fibres located symmetrically to the mid-depth. Residual stresses on the outer and inner fibres of an angle are represented in Figures 14 and 1.5 for three values of the wall thinness b / t Figure 14 deals with the following set of data: fy = 3 5 5 N / m m 2, t = 6 rmT¢ r/t = 1; Figure 15 is concerned with the same data except r/t

)

(35)

(r + t)3- P t ( r +~)

(36)

In both figures, residual stresses in the comer are shown to increase when the thinness b i t grows up and when the radius r decreases. The reverse is observed for that about the residual stresses in the fiat parts of the walls. When r/t = 3 and b i t -----5, that being a limiting case in practice, the sign of the residual stresses remains the same across the wall width, while in the other cases a sign reversal is observed in the zone located in the vicinity of the free edges. The results can be presented somewhat differently, as shown in Figures 16 and 17. It can be easily observed that: in the corner, the absolute value of the residual stresses decreases when b / t decreases and r/t increases; in the fiat parts, the absolute value of the residual stresses first increases and then decreases, when b i t decreases and r/t increases; residual stresses are obviously larger for the material with strain hardening. Last, the influence of the shape of the section on the distribution and the amplitude of residual stresses is emphasised in Figure 18. More especially, one can -

c2_Zt 2 + c t r + ~ c F + - ~ t (~rr

+-~) --~ [(r + 0 J -rJ]

2 ct +-~ t (r +

)

(37)

-

-

y o = So + V ~ (d¢- r- t)

(38)

CONSTRUCTION & BUILDING MATERIALS VoI. 1 No. 3 SEPTEMBER 1987

159

-

~

Hlmm 2

f

~

I! " ' ; - '

' "=' i

~"~'~~

~

~1_ I I ! I I IH21I I I I I I_~

r / f =1 b/f=5

-

~

.

.

0

~

~

C~

o

~ ~

~

~

o

~ 0

0

t

•~

H/mm?

~

I I I _[~ /~ i II _-~ i

-

II;It | I l L

-

[ ~ - i b

:

-

~

I

r/'r = 1 b/f

= 10

;i,llIll~

~

~

a

~

O

t~ ~ mm 2

r/f=1

r

b/f=

20

I I I

~:~ z

~ ~

O ~

external

E

O

f,iber

interna~

fiber

Fig 14 Longitudinal residual stresses in a cold-rolled angle (r/t = 1).

160

CONSTRUCTION & BUILDING MATERIALS V01. 1 No. 3 SEPTEMBER 1987

Nlmm 2

i

..,,j~~

" r/f: b/L:

i

.x'

g.

~

~

3 5

~

N/mm2

~"

~

I~

o

-- ~---~~ i :! ! !I I"l ! ! I]l'

~ ] ] ]

: ,. :~ ~_~2~ ~ ~ ~ ~_~ "

~

I I I 1-

r/f=3 b/f = 10

~

_

.

~

~

~

e

~

Nim~ 2

2

~

~

I

_

-

•

,,"

_

_

I

I

I

I

1~

~

r/t: 3 bit : 20

~

~~ ~

~

~ ~

~

~~

~ ~

~

exferna[

~

f;ber

;nferna[

N

~

~~ ~ ~

f;ber

Fig 15 Longitudinal residual stresses in a cold-rolled angle (r/t = 3).

CONSTRUCTION & BUILDING MATERIALS VoL 1 No. 3 SEPTEMBER 1987

161

~

~

a

~

c°rner face I ce 5

External fiber :

corner

,

E xt'erna[

fiber: -corner

+

corner

+

....

fate 1

• -.--

face 1

× .....

face 5

x .....

face5

~z r e s l f y blf:30 ÷

az res If),

0.3

0.3

"

0.2

0.2

"

0,1

0.1

'

b/t= 30

2oi\

\\

10

2~,~. --

20 ~x "" 30

*'"

r/~ ,

0.0

~ t

I

,,

bit:30

rlt

--

0,0

1;

S

I

I

I

1

2

3

I

,~

A-

5

.-_

~ bit = 30 ,~.

20 "X 'x..

~,

~. "~.

X

~

.

%.

•

o.1 I

~o ~k

~ .

-0,1

~

.. ~

"~

. •

~

,

~

•

~.~

•~..

~.

~. • ~

.~ -~.

~

"~

.

~ ,-~. i ~*

~,.

'

"~ .

~

~.

20"-

,

~

"~

~0".

~,

-~

'~

~.

~. ~.

~

.~ .

0,2

a cold.rolled angle (steel without strain, hardening, f, = 355

162

~ ~ "~

~. ~ ' ~ "

.~ ~

F~g 16 Longitudinal residual stresses/n

~'~.

~

,~"

-0.2 ff

"

N/rnm2)

"~ "~. ~ .

~

~"

Fig 17 Longitudinal residual stresses in a cold.rolled angle (steel with strain-hardening, f~ = 355 N / m m ~, f, = 515 N / m m ~, ~ = 4~, E, = 500 N/mm~).

CONSTRUCTION & BUILDING MATERIALS VOI. 1 NO. 3 SEPTEMBER 1987

~ Z

°

~

~-"

yietd

sfrength

: fy=355N/mm 2

-50

r a d i u s of

fhe

corner: r / f =

3

0 50 100 ~o Z

,=,

o

I

o I

~

Nlmm 2 '

100

-

Fig 18 Longitudinal residual stresses for three types of profiles (exemal fibre)

CONSTRUCTION & BUILDING MATERIALS Vol. 1 No. 3 SEPTEMBER 1987

163

observe the stress evolution in the corners when going from an angle to a U section and then to a C section.

Conclusions The method presented in this paper provides a better

knowledge, very empirical at present, of the effects of the forming method on the structural characteristics of the final product. This theoretical proposal gives results which, on the one hand, are of interest for the producers of cold-profiles, for what regards especially the amplitude of the springback and the required minimum ductility of the material. It is also of importance, when studying the behaviour of cold-profiles under loads. Nurnedcal results show that the residual stresses due to cold-forming are quite different from those observed and measured in hot-rolled common sections.

164

References I. Rondal J, Thin-walled structures. SecondRegional Colloquium on the Stability of Steel Structures, Final Report, Hungary, September 1986, pp. 269-285. 2. Wei-Wen Yu, Cold.formed steel design, John Wiley, 1985. 3. Kato B, Aoki H, 'residual stresses in cold-formed tubes', Journal of Strain Analysis. VoL 13, No 4, 1978, pp. 193-204. 4. Eurocode 3, Common unified code of practice for steel structures, Commission of the European Communities, Report EUR 8849 EN, 1984. 5. Rumier A, Baus I~ 'Etude param&rique de la r~sistance au flambement de colonnes en H d'impeffection structural ~lev~e', Construction M~tallique, No 1, 1981, pp. 3-20. 6. lngvarsson L, 'Cold-forming residual stresses and box columns built up by two cold-formed channel sections welded together',

Royal Institute of Technology, Dpt_ of Building Statics and Structural Engineering, Bulletin No 121, Stockholm 1977. 7.

Karren K W, 'Comer properties of cold-formed steel shapes',

Journal of the Structural Division, A.S.C.E., Vol. 93, No ST1,967, 8. 9.

pp. 401-432. Rondal J, Contraintes r~siduelles dans les 616ments en acier profiles ~ froid, University of Liege, Laboratory of Mechanics of Materials and Stability of Constructions, Report 161, 1987. Mendelson A, Plasticity - theory and application, MacMillan, 1968.

CONSTRUCTION & BUILDING MATERIALS VoI. 1 No. 3 SEPTEMBER 1987