Roll Forming Handbook / Edited by George T. Halmos

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bendingsystems. Co ilspassing throughthe

preformer are repeatedly bent and bent back

longitudinally by the crowned or bent rolls. The

center and intermediate portions of the coilsare

subjectedtolongitudinal bending underthe

longitudinal tension because of utilization of

crown rolls or transversally bent rolls arranged in

azigzag. Thus,the centerand intermediate

portions of metalsheetsare longitudinally

elongated, whilethe edge portionkeeps its

original length. After preforming,metal strips

are formed by proﬁle rolls into products, where

the edge portions are longitudinally elongated

much morethan other portions.

The longitudinal elongation at the preforming

stage is adjusted so that it compensates the

nonuniform lo ngitudinal elongation, which

occurs at the forming stage. As aresult, when

the longitudinal elongation due to the forming is

superposed upon the longitudinal elongation

caused by the preforming,the elongation of sheet

metal becomes almost uniform in the transversal

direction. Thus the cause of the edge wave

is eliminated andthe occurrenceofitis

prevented.

3. Thethird meth od maybecalledthe

“smoothing ﬂow line method.”Inthis case, the

ﬂowline of the edge portion is made as smooth

and shortaspossible by using guide rollsand

shoes (Figure11.16). During roll forming,the

edge portion is repeatedly bent and bent back in

the longitudinal direction under the longitudinal

(+)

( − )

Metal Sheet

Preforming

Forming

Product

Crown Roll Roll Bending Partial Rolling

ECE

FIGURE 11.15 “Preforming method” for preventing edge wave.

Step Forming

(Main Roll)

Gage Forming

(Side Roll)

FIGURE 11.16 “Smoothingﬂow linemethod” for

preventing edge wave.

Roll Forming Handbook11-12

tensile stress. Especially when it passes through roll gaps, this repetition of forward and backward

bending becomes extensive and it contributes to the longitudinal elongation of the edge portion. It looks

likesthe longitudinal elongation of metal strip subjected to atension roller leveler.

This longitudinal bending can be effectively reducedbyadjusting the ﬂowline to reduce the edge

elongation, and thus preventing edge wave.

11.3.3 Center (Pocket) Wave

Apocket wave (usually called “center wave”or“oil canning”) is caused by slight elastic buckling,which

occurs mainly at webs, ﬂanges, and other nonbent portions of the cross-section. It is widely observedon

products that haveawide cross-section with comparatively low thickness, such as rooﬁng, siding, deck,

panels, automotive components, garage, and other doors, as well as other wideproducts. Their webs and

ﬂanges usually havelow stiffness against longitudinal buckling.

The mechanism of occurrenceofpocket wave can be explained as follows.

1. When metal coils are roll formed, the edge and intermediate portions are pulled toward the center of

the cross-section in the transversal direction. When they are pulled transversally,transversal tensile

stresses are induced in the metal strip.The transversal tensile stress induced in the center portion is larger

than those induced at the intermediate and edge portions. This is due to the fact that the transversal

tensile stress induced at one portion corresponds to the transversal force necessar yfor pulling the strip at

another portions. The portions of the strip,which become the bend lines (corners) of the products’ cross-

sections, are bent transversally under the inﬂuenceofsuch tensile stress. As aresult, the bend lines,

especially when they are located at the central

zone of the strip,are signiﬁcantly elongated in

the transversal direction and, at the same time,

following to transversal elongation, they shrink

in the longitudinal direction (Figure 11.17).

When the bend lines (corner portions) shrink

in the longitudinal direction, the longitudinal

compressivestresses are induced in webs, ﬂanges,

and other ﬂat (nonbent) portions. When the

compressivestress is excessive relativetoeach

portion’s stiffness, elastic buckling takesplace

there. This buckling appears as pocket(center)

wave on the ﬂat surfaceofthe product.

2. Another mechanism contributes to the

occurrence of pocket wave. When the corner

portions are bent transversally by the convexand

concave cornersofthe proﬁle rolls, they elongate

in thelongitudinal direction as well as the

transversal direction. In such cases, each corner

portion deforms like astrip subjected to so-called

“bulge forming.”However,when the semiformed

strip comes out from the roll gap,the long-

itudinally elongatedcornerportions should

shrink in the longitudinal direction to alength

equal to the length of other portions, because all

portions of the formed straight cross-section

should havethe same length. When the elongated

bend line portions shrink, longitudinal compres-

sive stress is induced in other nonbent portions

PocketWave

:Transversal Membrane Strain

:Longitudinal Membrane Strain

Shrinkage Elongation

(+)

(A) (B) (C) (D) (E) (F)

(+)

( − )

ABCDEF

Transversal

Tensile

Force

FIGURE 11.17 Mechanismofoccurrence of center

waviness (pocket wave) (shrink model).

Behavior of Metal Strip During Roll Forming 11-13

(Figure11.18). When compressivestress exceeds the critical value at the ﬂat sections, longitudinal

buckling (center waviness) occurs.

The following countermeasures can prevent or minimize center waviness.

1. Excessive transversal elongation induces excessivelongitudinal shrinkage or elongation/shrinkage

at the bend lines. Therefore, the transversal tensile stress affecting the bend lines and the

transversal elongation induced there should be reduced. This reduction can be accomplished by

reducing the transversal shifting of each portion of the strip by making the roll gap as large as

possible. An unnecessarily small roll gap,through which the edge and intermediate portions of the

coil havetoshift in the transversal direction, mayincrease the localized transversal elongation of

the bend lines.

2. At the same time, the sequence for bending corner portions should be optimized. In general, the

corners of the center portion should be bent prior to those of the intermediate and edge portions.

Moreover, the amount of bending (deformation) necessaryfor making up the required

corners should be divided and allotted appropriately to sequential bending steps performed by

proﬁle rolls.

3. In some cases, use of guide rolls and shoes is effective for promoting transversal shifting of the edge

and intermediate portions of the coil.

4. Moreover, decreasing the radii of the convexcorners of the rollsshould occur gradually,not

suddenly.The bending radii should not be too small at anypass. Small radii tend to induce

localized hightransversal elongation at the bend lines. Small radii also obstruct smooth transversal

shifting of the edge and intermediate portions.

11.3.4 Corner Buckling (Herringbone Effect)

Localized longitudinal elongation tends to occur frequently at corner portions when products haverather

wide cross-section and the material is thin. When these elongated corner portions shrink in the

longitudinal direction at the exits of roll gaps, in some cases, pocketwaves take placeatwebs, ﬂanges, and

PASS II

PASS I

PASS IPASS II

(+)

( − )

(+)

( − )

FABC DEF

FIGURE 11.18 Mechanism of occurrenceofcenter waviness (pocket wave) (elongate/shrink model).

Roll Forming Handbook11-14

other ﬂat (nonbent) portions.This happens

when the stiffness of those portions is relatively

low.When their stiffness is relatively highand

that of the corner portions is comparativelylow,

corner buckling deformation takes placeatthe

corners and adjacent areas (Figure 11.19). This is

asortofplastic shear buckling and small waves

appear periodically along the corners. Generally,

each waveinclines to the longitudinal direction,

hencethe term “herringbone”waviness. This

incl inationiscausedbythe shearbuckling

deformation.

To prevent corner buckling (herringbone

effect), similar precautions must be taken as in

the case of pocketwaves. The most important

objective is prevention of excessive longitudinal

elongation at corner portions by supporting and

enhancing the transversal shifting of the edge and

intermediate portions of the coil.

11.3.5 Edge Cracking and Splitting

When the transversal tensile stress or elongation

exceeds the critical value at the corner portions of

the cross-section, cracking or splitting of metal

takes place there. This kind of cracking or

splitting is often observedincases wherethe

coil is wide, the material is thin, and the product

hasmanysharp corners; this situation is

aggravated if the metal has poor formability.

Again, the countermeasures necessar ytoprevent cracking and splitting are almost the same

countermeasures for preventing pocket waves and corner buckling.

11.3.6 Nonuniform Springback Deformation of Panels

When strips are roll formed, they are subjected to various redundantdeformations, as well as transversal

bending. This means that the strips are bent transversallyunder the inﬂuencesofthe additivetransversal

and longitudinal tensile (or compressive) strains and stresses. Those inﬂuences are clearly observed when

springback deformations of formed sheet metals take place.

The transversal and longitudinal membrane strains and stresses usually affect the transversal

springback deformation of thestrip signiﬁcantly.The sh apeofthe productchanges in

correspondencetothe amount of membrane strains and stresses. In general, the membrane strains

and stresses, irrespectiveofwhether they are tensile or compressive, signiﬁcantly reduce the amount

of springback deformation, which takesplacefollowing transversal bending. For instance, if the

transversal or longitudinal membrane strain is sufﬁciently large, then the springback deformation

becomes zero.

When wide coils are formed into roof decks, sidings, and similar products, transversal tensile strain is

induced at everycorner of the cross-section. However,the transversal tensile strain induced at the corners

located at the center portion of the panels is usually larger than that induced at the corners located at

intermediate and edge portions. This means that the amount of springback deformation, which takes

placeatthe corners at center portion, is smaller than that at the corners at intermediate and edge

Corner Buckling

At roll gap

BC DEFGH

At exit

BC DEFGH

Longitudinal Membrane Strain

Elongate

Shrink

FIGURE 11.19 Mechanism of occurrenceofherring-

bone effect (corner buckling).

Behavior of Metal Strip During Roll Forming 11-15

portions. In other words, the amount of springback deformation at the edge portion is expected to be

larger than those at other portions. Thus, nonuniform springback deformation takesplaceinthe

transversal direction (Figure11.20).

Nonuniform springback deformation causes dimensional inaccuracy and, in some cases, cross

(transversal) bow (Figure11.21). Flare like this is inevitable; therefore, the rollproﬁles should be designed

by taking such nonuniform springback deformation into consideration.

Similar problems and defects in products are observedwhen precut sheets are roll formed. In these

cases, the front and tail ends of the strip are transversally bent without the inﬂuence of effects of

restrictionsimposed by neighboring rolls. They

are formed (i.e., transversally bent) under the

effects of longitudinal and transversal stress and

strains smaller than those for the longitudinal

midportion of the strip.Atthe front and tail

ends (entryand exit ends) of the strip,the

longitudinal stress is released, because those

ends are not clamped (held) simultaneously by

neighboring proﬁlerolls. On the other hand, the

midportion of the strip is formed under the

restriction imposed by the simultaneous clamp-

ing of plural pairs of proﬁle rollsarranged in

tandem.

Therefore, in general, the amount of spring-

back deformation at the entryand exit ends of

cutoffmetal sheet is larger than that of the

midportion. This difference causes the longitudi-

nal nonuniformityofheight and width of the

product’scross-section.

The longitudinal nonuniformity of the spring-

back deformation maybereduced or eliminated

by usingagapcontrol systematseveral

stands, installed at the ﬁnal stage of forming

After Spring Back

Before Spring Back

FIGURE 11.20 Schematic illustration of ﬂare (nonuniform springback deformation).

< q

FIGURE 11.21 An example of ﬂarecausing cross-bow.

Roll Forming Handbook11-16

(Figure11.22). At these stands, the roll gaps are adjusted to be small when the entryand exit ends of the

sheets are formed, and are slightly opened when the midportions are formed. The small roll gap enhances

the restriction imposed upon the top and bottom ends of metal sheet by the rolls and, consequently,the

amount of springback deformation is reduced. Thus, the longitudinal nonuniformity of springback

deformation is corrected and the geometrical inaccuracy of productisimproved.

The sequence to adjust the roll gap should be studied and decided upon by taking into account the

geometryofthe product’scross-section, the mechanical properties of the strip,the number of available

stands, and other forming conditions.

11.3.7 Flare After Cutting to Length

When roll formed products are cut offinto pieces after forming,the entryand exit cross-sections of each

pieceare frequently ﬂared due to springback deformation. In manycases, the distortion of both ends

exceeds tolerance. The mechanism of occurrence of this springback distortion of cutoffends can be

explained as follows.

Springback distortion is mainly caused by the residual shear stress distribution in the thickness

direction at the edge and intermediate portions of product’scross-section. As shown in Figure11.23,

the strip especially its edge portion, usually enters into the rollgap along the ﬂowline, which is the

lead-in partofthe concave roll. The lead-in partofthe rollsguide the semiformed strip into the roll

Process

Controller

Servo

Actuator

Shape

Meter

Speed Meter

Forming Roll

Metal Sheet

FIGURE 11.22 Conceptual illustration of aroll gap control system for preventing ﬂare in sheet fed lines.

Tail End

Cut-Off

Lead End

Forming

Direction

FIGURE 11.23 Flare of cutoffends.

Behavior of Metal Strip During Roll Forming 11-17

gap,asshown in the ﬁgure,althoughthe section

is wider than that of the roll gap.Owing to the

ﬂowline following the shape of the concave roll,

shear deformation occur in the longitudinal

section of the edge por tion of the strip.This

shear deformation takes placesothat the outer

layer of the strip,which is close to the surface of

the concave roll, enters the rollgap faster than

the inner layer,which faces the surface of the

convexroll (Figure 11.24). However,atthe exit

end of the rolls, the inverse shear deformation

occurs. As aresult, all of the layers of metal

sheet havetocome out simultaneously and

straight in the longitudinal direction. This shear

deformation at the entrystage and inverse shear

deformation at the exit stage generate residual

shear stress in the metal sheet, especially at the

edge portion.

When the formed product is cut offinto

pieces, the released residual shear stress distorts

the entryand the exit ends of the formed

products. In general, aspringback deformation

(ﬂare)takesplace, and, as aresult, the shape of

the product changes from the present state to

the previous state. Therefore, both ends of the

product tend to deform from the present state

wherethe inverse shear deformation is ﬁnished

to the previous state where they weresubjected

to the shear deformation at the inletting stage.

That is, the state of deformation wherethe edge

portion, and sometimes the intermediate por-

tion, rounds to the concaveroll.

Thus, springback distortion (ﬂare) of the

cutoffends takes place. It should be noted that

the distortion of the entryend and that of the

exit end usually occur in opposite directions. For

instance, as shown in Figure 11.24,the entryend

tends to distort to close the cross-section and the

exit endtends to open it. This interesting

characteristic of the springback distortion of

the cutoffends is widely observed.

To prevent or minimize the ﬂare springback

distortion of the cutoffends, the methods of (a)

“overbending andbending back”and (b)

“inversebending by inner rolls”are used.

1. By the “overbending and bending back”

method, the excessivetransversal bending (over-

bending) is applied to the corners (bend lines) of

the cross-section just before the ﬁnal stage of

forming.Afterward, those corners are bent back

to the required shape or angle (Figure11.25).

Spring

Back

Spring

Back

Cut-Off

FIGURE 11.24 Schematic illustration of mechanism of

occurrenceofﬂare (distortion of cutoffends) caused by

reciprocal shear deformation in the longitudinal cross-

section.

Cut-Off

Bend Back

Required

Profile

Over Bend

FIGURE11.25 “Over bendingand bendingback”

method to prevent ﬂare.

Roll Forming Handbook11-18

The residual shear stress accumulated in the

corner portions at preceding forming stages

is canceled by thesufﬁciently largeinverse

shear deformation given at the bending back

stage. Becauseofthis“over bendingand

bending back,”the springback distortion of the

cutoffends of the product can be reduced or

prevented.

2. In the inverse bending method, as shown

in Figure11.26,the semiformed metal sheet is

bent transversally to the inverse (opposite)

direction by side rolls installed at the appro-

priate stage of forming process. The inverse

bending rolls can be used effectivelytogive

sufﬁciently large inverse shear deformation to

the metal sheet. The inverse shear deformation,

like the shear deformation given by the above

mentioned bending back method, cancels the

effects of the residual shear stress accumulated

in the preceding forming stages, and eliminates

thecauses of springback distortion of the

cutoffends.

11.4 Mathematical Simulation of the Deformation

of aMetal Strip

In the roll forming process, metal strips are continuously and progressively formed into products with

required transversal cross-sections and longitudinal shapes by aseries of proﬁle rolls arranged in tandem.

In everyprocess, each pair of proﬁlerolls plays aparticular role in forming the product’s cross-section

and shape.

Generally,the number of rolls and their contours are designed by taking account not only of the

geometrical intricacy of product’s cross-section, but also the dimensions and mechanical properties of

the metal strip to be formed. Moreover,the design of the rolls depends on the available forming passes,

adjustabilityofroll position, changeabilityofthe pass line, controllability of the roll gap,convenience of

the roll changing system, and other inﬂuencing factors.

Until now,veryfew investigations havebeen conducted on the methodologyfor atheoretical approach

to the design of rolls, pass schedules, equipment, and processes. This is par tly due to the fact that

deformation of the strip during roll forming is one of the most difﬁcult and complicated deformations

for theoretical analysis. Therefore, for along time, the approach to rollforming technologyhas been

based on past experience and trial-and-error methods. As aresult, the details and theoryofdeformations

of the strip remainunclariﬁed.

In ordertoremedy this situation, in recent years, the mathematical theoryofroll forming has been

actively investigated and has made remarkable progress. Computerized simulation techniques for

analyzing details of deformation features of the strip in the rollforming processes are now being

developed from various aspects. Furthermore,utilizing such simulation techniques, automated roll

design systems are also being developed. These techniques and systems are going to be used to make

diagnosis of rolls, equipment, processes and working conditions, determine the required modiﬁcations to

them, and discovermeasures to improveproduct quality. The outline of the most advanced simulation

technique and automated design system is explained in the following sections.

FIGURE 11.26 “Bending back by side (inner) roll”

method to prevent ﬂare.

Behavior of Metal Strip During Roll Forming 11-19

11.4.1 Mathematical Expression of

Deformed Metal Strip

Figure11.27 shows the coordinate systems used

for the mathematical formulation and analysis as

well as atypical deformed curvedsurface of the

strip positioned between tworollstands. Here,

the ð X ; Y ; Z Þ coordinates represent the orthog-

onal coordinate system, which is ﬁxed at an

appropriate position in the concerning space. On

the other hand, ð x ; y ; z Þ coordinates express the

curvilinearcoordinatesystemlocated on

theneutral curved surface of thedeformed

strip.The neutral curved surfacemeans the

curvedsurface,which is included in the middle

of the thickness of the deformed strip.Hereafter,

this neutral curved surfacewill be referred to as

the “deformed curvedsurface of the strip.”

The x -axis is deﬁned as the horizontal axis and coincideswith the forming (longitudinal) direction. In

Figure11.27, X ¼ X

and X ¼ X

represent the positions of #(i )-roll stand, and # ð i þ 1 Þ -roll stand,

respectively.

To express the 3-D deformed curved surface of the strip between tworoll stands, anormalized function

S ð X Þ ; named as “shape function”isintroduced. Forthe present, S ð X Þ is tentatively deﬁned by the next

equation.

S ð X Þ¼sin½ðp = 2 Þ · ð X

= L Þ



¼ X 2 X

; L ¼ X

2 X

ð 11: 1 Þ

Here, L is the (horizontal) distancebetween the tworollstands. The shape function S ð X Þ represents the

3-D pattern of spatial ﬂowlines along which each portion (or element) of the strip movesfrom#(i )-rolls

to # ð i þ 1 Þ -rolls. It changes from 0.0 to 1.0 as X changes from X

to X

Figure11.28 shows the calculated cur vesof S ð X Þ : The value of S ð X Þ is afunction of aparameter “ n .” As

shown in this ﬁgure, the pattern of those curvesisremarkably affected by the value of “ n .” When “ n ”

is small, the value of S ð X Þ increases from 0.0 to 1.0 gradually as X changes from X ¼ X

at the location of

#(i )-rolls to X ¼ X

at the # ð i þ 1 Þ -rolls. However,when “ n ”islarge, S ð X Þ increases very slowlyatthe

initial stage, but approaches 1.0 very rapidly at the ﬁnal stage, just beforethe # ð i þ 1 Þ -rolls. When S ð X Þ is

( X

, Y

)

X =(X

,#i -Roll

X = X

,#( i +1)-Roll

i + 1

( X

, Y

)

Neutral Curved

Surface of

Metal Sheet

Spatial Flow-Line

of an Element

of Metal Sheet

FIGURE 11.27 The coordinate systems for analysis and

schematic illustration of the deformed curved surface

(deformed neutral surface in thickness direction) of strips

between #(i )and (i þ 1)-roll stands.

1.0

0.5

0.2 0.4 0.6 0.8 1.0

X = X

: S ( X )=0, S '(X )=0

X = X

: S (X)=1,S '(X )=0

L = X

- X

=Inter-Stand Distance

n:Parameter decided by the energy method

( X

/ L )(X

/ L )X/L

n >1

n =1

n =2

n =3

S(X) =sin( ()

S(X)

=sin( ()

)

FIGURE 11.28 Calculated curves of shape function “ S ð X Þ ”with different values of parameter in “ n : ”

Roll Forming Handbook11-20

used for actual analysis, the value of “ n ”isdetermined throughthe mathematical procedure based on the

so-called “energymethod,”wherethe value of “ n ”istheoretically searched so as to makethe total power

of deformation dissipated in the metal strip minimum.

Using the shape function S ð X Þ ; the deformed curvedsurface of the strip between #(i )-rolls at X ¼ X

and # ð i þ 1 Þ -rollsat X ¼ X

is expressed by the following equation:

X ¼ X ð x ; y Þ

Y ¼ Y

ð y Þþ½ Y

ð y Þ 2 Y

ð y Þ·Sð X Þ

Z ¼ Z

ð y Þþ½ Z

ð y Þ 2 Z

ð y Þ·Sð X Þ

X ¼ X

, X

;

ð 11: 2 Þ

Here,

; Z

: Y and Z coordinates of the proﬁle of # ð i Þ -rolls projectedtothe Y–Z plane, which is

perpendicular to x -axis at X ¼ X

, Z

: Y and Z coordinates of the proﬁle of # ð i þ 1 Þ -rolls projectedtothe Y–Z plane, which is

perpendicular to x -axis at X ¼ X

From Equation 11.2, the following relations are derived

When X ¼ X

; S ð X Þ¼0 : 0 ; then Y ¼ Y

ð y Þ and Z ¼ Z

ð y Þ :

When X ¼ X

; S ð X Þ¼1 : 0 ; then Y ¼ Y

ð y Þ and Z ¼ Z

ð y Þ :

These relations show that Equation 11.2 is formulated on the following assumptions concerning the

boundaryconditions.

1. The cross-sectional proﬁle of semiformed strip at each rollgap center (roll position, X ¼ X

and/or X ¼ X

)coincides with the roll proﬁle, respectively.

2. Everyportion of the strip passes by the roll gap center ( X ¼ X

and/or X ¼ X

)

along ahorizontal ﬂowline.

11.4.2 Incremental Treatment of Steady-State Deformation

of Moving Metal Strip

Hereafter,the deformed curvedsurface of the strip between the adjacent two rollsstands is denoted

as D.C.S. (deformed curved surface). The general mathematical expression of D.C.S. is described

by Equation 11.2. However,Equation 11.2 expresses only the approximated 3-D shape of the semiformed

strip without providing anydetails about the deformation features of it.

During rollforming,the metal strip is subjected to aso-called “steady-state deformation.”That is to

say, between everyforming stand, each portion of metal strip is considered to move from #(i )-rolls to #

ð i þ 1 Þ -rolls along D.C.S. On the D.C.S., everyportion of the strip is bent, bent back, elongated or

shrunk in various ways, under the complicated inﬂuence of working conditions. In order to calculate

the stresses and strain occurring in the moving strip,the deformation of those portions should be

analyzed incrementally by taking account of various mechanical constraints acting on them while they

move (or ﬂow) along D.C.S.

In ordertoperform an analysis like this, an incremental method is introduced, where the deformation

of astrip having an original length of D ‘

and width equal to the original strip width, is followed up and

analyzed (Figure11.29). The strip is considered to move from #(i )-rolls to # ð i þ 1 Þ -rollsand deform

along D.C.S. Here, the following assumptions and mechanical restrictions acting on the sheet strip are

introduced.

1. At everyforming stage from#( i )-rolls to # ð i þ 1 Þ -rolls, the front (entryside) and rear (exit side)

cross-sections of thedeformed stripare respectively included in theplaneswhich are

perpendicular to x -axis.

Behavior of Metal Strip During Roll Forming 11-21