Flang curling in profiled steel decks

Thin-Walled Structures Vol. 25, No. 1, pp. I 29, 1996

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263-8231/96 $15.00 ELSEVIER

0263-8231(95)00051

-8

Flange Curling in Profiled Steel Decks

E. S. B e r n a r d & R. Q. Bridge* Department of Civil Engineering, University of Western Sydney, Nepean, NSW 2747, Australia

G. J. Hancock School of Civil and Mining Engineering, University of Sydney, NSW 2006, Australia (Received 4 January 1995; accepted 6 July 1995)

ABSTRACT The occurrence and magnitude of flange curling deflections in profiled coldformed steel decking panels is investigated experimentally. The test panels examined were a roll-formed standing seam panel, and a brake-pressed trapezoidal panel. The test specimens were loaded in pure flexure and the load-response characteristics, and differential deflections of flanges relative to webs, were recorded for each specimen up to ultimate collapse. Estimates of flange curling deflections based on specifications contained within the Eurocode 3 (1993 draft) design standard are found to be comparable to experimentally determined deflections. A number of alternative curling models are developed and compared to the experimental curling behaviour. The effect of unrestrained longitudinal edges on curling deflections is determined and incorporated into the models. NOTATION The cross-sectional geometries o f the specimens under investigation are shown in Figs 1 and 2. Other principal notation are indicated below. A bs

Gross section area Width o f flange or half-flange

*To whom correspondence should be addressed.

2

E.S. Bernard, R. Q. Bridge, G. J. Hancock

E

Elastic modulus Yield stress in design formula H,S Height of intermediate stiffener Plate bending moment Mu Ultimate moment M y Moment at first yield r Radius of corner of test section t Thickness of base material u Curling deflection Umax Curling deflection at flange centre z Distance from neutral axis Z Section modulus V Poisson's ratio t7 a Mean stress in flange Buckling stress O'er O"u Ultimate tensile stress ~Ty Yield stress

Fy

1 INTRODUCTION Curling occurs in the flanges of thin-walled sections undergoing flexure. It involves a deflection of the portion of each flange farthest from supporting webs toward the neutral axis. Both tensile and compressive flanges are subject to it. The degree of deflection appears to be primarily a function of the width of the flange, but the thickness and distance of the flange from the neutral axis also contribute. Winter ~ conducted early work on the phenomenon, but little has been carried out since. Yu 2 briefly reviewed the subject and presented Winter's equation for the approximate calculation of flange deflections resulting from curling. A similar equation has been used in Eurocode 3. 3 A series of laboratory tests have been performed at the University of Sydney, on standing seam profiled cold-formed steel decking panels and trapezoidal panels. The specimens were all loaded in pure flexure. The differential deflections of flanges relative to webs were recorded, as well as other structural behaviour characteristics such as tangent stiffness, buckling moments and ultimate collapse moments. Whilst the stability and stiffness aspects of each specimen's behaviour are described in detail by Bernard, Bridge and Hancock, 4' s this paper will focus on the curling behaviour of these panels. A number of relatively simple theoretical models of curling behaviour are developed and compared with the measured results.

Flange curling in profiled steel decks

3

PrincipalDimensions Condeck

~~- ~

117

(Allradiimeasured approximately 3ramas to strip mid-plane)

___'

~---~7

- - ....

"

I

t

t~, w3

w6

1

6.~,~

PrincipalDimensions CondeckHP (Allradiimeasuredas approximately3mmto strip mid-plane)

3.75

.

dot

300

Fig. 1. Geometry and general dimensions of Condeck specimens.

W3

.45°{/N'~~ ' - ~ ~

\

i I_

W4

i _l -,

( W5

r

f

Fig. 2. Geometry and general dimensions of trapezoidal specimens with flat-hat stiffeners. 2 G E O M E T R Y OF TEST SPECIMENS The rolled cold-formed standing seam panels 4 consisted of two similarly designed products manufactured by Stramit Industries of Australia. They are known commercially as Condeck and Condeck HP. While available in a variety of strip thicknesses, only' one thickness in each product was investigated in the present tests. As depicted in Fig. 1, the panels consisted of two upstands, with either a simple or complex folded lip, arranged nonsymmetrically about a wide central flange. The flange, also known as the

4

E . S . Bernard, R. Q. Bridge, G. J. Hancock

'pan,' possessed two 'flat-hat' intermediate stiffeners, and a central depression dcf (as shown in Fig. 1). Panels were fastened together with selftapping screws via the interlocking upstand flanges. Each specimen was 3000 mm long and was arranged with the upstands pointing either up or down corresponding to positive and negative bending, respectively. In commercial practice, a wide floor deck can be produced by locking many panels together side-by-side and fastening the upstand flanges with screws. In this series o f tests, the results o f specimens consisting o f 1, 2 and 3 panels locked together in such a manner were examined. The objective was to determine whether the performance per panel varied as the number of panels increased and the possible influence of free-edge effects on panel behaviour. Labelling of the standing seam panels consisted of a prefix C (for Condeck) followed by a numerical denoting the number of panels in the specimen; an N or a P for negative or positive flexure; and an A or B describing the product used (A for Condeck, B for Condeck HP). Thus C 2 N B refers to a Condeck H P specimen with two panels bent in negative flexure. All Condeck specimens were nominally 1.0 mm thick and Condeck H P specimens were nominally 0,75 mm thick. The nominal and measured dimensions o f each type o f panel are listed in Table 1. Note that the nominal mid-plane corner radius has been taken to be 3 mm. T w o trapezoidal decking panels with flat-hat intermediate stiffeners 5 were also tested for the purpose o f comparison with the standing seam TABLE 1

Dimensions of Condeck Panels (mm) Dimension

H wl w2 w3 w4 dcf DI D2 w5 w6 w7 w8 * Not specified. t Not applicable.

Condeck

Condeck HP

Nominal

Measured

Nominal

Measured

--* --* 38.0 300.0 -- * 5.0 54.0 55.0 23.0 21.0 t t

1.87 27.12 37.99 300.3 79.48 4.72 54-50 55.75 22.92 21.28 t t

8.5 23.0 37-6 298.0 -- * 5.2 54.0 55.0 30-0 27-5 , ,

6-35 21-80 40.16 297.9 74.61 2.55 53.17 54-34 30.77 26.17 19-94 19.17

Flange curling in profiled steel decks

5

panels. These specimens consisted of three trapezoidal ribs with flat-hat intermediate stiffeners in the centre of each compressed flange. These specimens are referred to as F C 1 and FC2 and consisted o f 2000 mm long specimens brake-pressed using 0.60 mm Zincalume coated strip. The average measured dimensions o f each trapezoidal panel are listed in Table 2. Corner radii are a b o u t 0.5 mm. The mechanical properties of the strip used in each set of specimens are described in Table 3. The designation Z275 refers to the strip coating of 275 g/m 2 of zinc, while Z200 denotes 200 g / m 2. The strip in the trapezoidal specimens was aluminium/zinc (Zincalume) coated steel with 150 g/m 2 of Zincalume coating.

3 FLEXURAL TESTING Testing was performed in a purpose built test rig that had previously been used to investigate trapezoidal decking panels. 46 A schematic diagram of the apparatus is included in Fig. 3. This shows the location TABLE 2

Dimensions of Trapezoidal Panels (mm) Dimension

FC1

FC2

D S wl w2 w3 w4 w5

54.87 6.18 40.17 27.42 135.17 84.50 244.7

55.57 5.22 41.08 29.42 136.00 83.38 242.5

TABLE 3

Strip Properties for Test Specimens

Grade * 0.2% Proof stress (MPa) Ultimate tensile stress (MPa) Elastic modulus (GPa) Base metal thickness (mm) Coating thickness (mm) *As specified by AS1397--1984.

Condeck

Condeck HP

Trapezoidal decks

G55ff-Z275 578 580 202 0.994 0.037

G550-Z200 651 654 203 0.752 0.031

G550-AZ 150 653 656 230 0-585 0.045

E. S. Bernard, R. Q. Bridge, G. J. Hancock

North PointLoad Counterweight

~ ~

PivotPoint (~

South PointLoad DetailB Specimen

i] ) x . ~ f _ )1000 DetailA !_ _ North Support StiffMoment TransferFrame

~

ScrewJack

t~

~

|,

[I lO00..~ LI - t LoadCell I South I Support

2000

TestSpecimen

I I

StiffMoment TransferFrame

Support LoadCells

I I L

"Outer"LoadCells I

¢ + *e North Pivot Axis

,7"II

Axis

j

G ~/~

Pivot

"Inner"LoadCells

Screwjack shaft

a~../Loadcell Pivot

~_.l~~ernstone ~

-

]

~

block Specimen

}~/Loadcell Frictionless bearing

r~

~ ~"-- HSFG bolts

Fig. 3. Diagram of testing apparatus.

of the test specimen in the centre of the rig between two cantilevered moment transfer frames. Each of the tests involved the application of point loads at four locations, two adjacent to each end of the specimen. Moment was thereby imparted into the specimen through the rigid connections with moment transfer frames. Loading was applied using manually operated screwjacks. As the displacement of the screwjacks was increased, the loads and associated moment also increased, resulting in flexure of the specimen. Displacements were applied equally at both ends of the specimen so as to induce a uniform moment. As each increment was applied, measurements of load, strains and displacements

Flange curling in profiled steel decks

7

were made. If a discrepancy in load between the two ends of the specimen (known as the North and South ends) occurred, a correction was made in the next increment so as to maintain uniformity longitudinally. Equality of load between each of the jacks in the transverse direction was not as important because the great stiffness of the m o m e n t transfer frames ensured that torsion did not occur in the specimen. Load could be applied in approximately equal increments up to ultimate failure in such a manner as to maintain uniform moment while retaining the benefits of displacement controlled loading. The rigid connection between m o m e n t transfer frames and the specimen was achieved through the use of cast gypsum blocks post-tensioned with High Strength Friction Grip bolts. The resulting clamping stress ensured an even transfer of m o m e n t up to ultimate collapse.

4 D E R I V A T I O N OF T H E O R E T I C A L C U R L I N G ESTIMATES The curling phenomena was first investigated by Winter j for hot-rolled sections available in the 1930s. His derivations have been used in all subsequent design standards, including Eurocode 3. 3 In this standard, the curling deflection, u, of the flange toward the neutral axis is found to be u = 2

(¢Ta~2b 4s

(l)

where bs is the width of half the flange as indicated in Fig. 4(a) and (b); t is the flange thickness; z the distance of the flange from the neutral axis; aa the mean stress in the flange and E is Young's modulus. Figure 4(c) shows the same dimensions for the trapezoidal FC specimens. Winter based his original derivations on/-sections and box sections. He assumed that the flanges of the/-section deflected symmetrically and thus that each could be considered a cantilever fixed at the base. In Fig. 5(a), a segment ds of a n / - b e a m in pure flexure is depicted according to Winter's representation. The tensile flange of this segment is shown in Fig. 5(b). A tensile force H per unit width of flange gives rise to a force R acting normal to the flange toward the neutral axis as a result of the curvature of the beam. Because H is approximately evenly distributed across the width of the flange, R is as well. The normal force R acts to bend the flange in cantilever action toward the neutral axis [Fig. 5(c)]. Winter assumed that the deflection was small and hence did not account for the varying distribution of the force R near the ends of the flanges. The magnitude of the force R per unit length of flange was found as

E. S. Bernard, R. Q. Bridge, G. J. Hancock

i

.

bs

bs (a) Positive bending

L

..--.-----

i

uI! bs

_ I_

[

bs

-I-

(b) Negative bending

I_

bs

bs

_l_

I-

u

_i

-I

--~

~

L_

I

I . . . . . . . . . . .

i

r

I

(c)

Fig. 4. Flange curling parameters for Condeck and trapezoidal specimens.

R = H d ¢ _ H _ o a/' ds

r

(2)

r

where r is the radius of curvature o f the bent beam, a a is the stress in the flange, and t is the flange thickness. The differential equation for the bending of a wide rectangular plate is d2u _

1 _

dy 2

r

Mp

(3)

D

where u is the deflection of the flange, Mp is the m o m e n t per unit width of plate, and D is the flexural rigidity of a plate where D = Et3/(12(1

- v2)).

(4)

Flange curling in profiled steel decks

(a) Beam Elemental Length

(b) Flange Element Length

t ~

~Um2

(c) Beam Cross-Section Fig. 5. Winter's original model for the determination of flange curling deflections in I-beams.

By assuming that the flange acts as a cantilever, the maximum deflection o f the flange tip is found as

Rb~

Umax- 8D

~5)

10

E. S. Bernard, R. Q. Bridge, G. J. Hancock

where b~ is the 'length' of the cantilever (i.e. width of flange). The radius of curvature r and the bending m o m e n t M for the beam segment can be expressed as

E1

r=--

and

M

M--

2a a I h

(6)

and hence r =

Eh

(7)

2O'a

where I is the second m o m e n t of area of the beam and h is the height of t h e / - b e a m . Substituting r from eqn (7) into eqn (2) gives t R = 2aZa E h "

(8)

Then from eqns (4), (5) and (8) Um~x=2a 2

b:t

8Deh

-3 (E:b:

( 1 - v 2)

(9)

F o r the more general case where the distance of the flange from the neutral axis is taken as z(z = h/2 for the/-section), and taking v = 0-3 for steel, then the m a x i m u m deflection of the flange for a given applied stress aa is

( O'a'~2bs4

Um~, = 1-37 \ ~ ]

t-57.

(10)

Winter also examined the case of a rectangular box section of height h and flange thickness t where he assumed that the webs made a negligible contribution to the flexural rigidity of the flange. This is equivalent to modelling the flange as a simply supported beam. The m a x i m u m deflection at midspan for a simply supported beam with a uniformly distributed load is

5 RL 4 Umax--384

D

(11)

or, since L = 2bs

5 Rb 4 um~ - 24

(12)

D

and then, substituting eqns (4) and (8)

(O'a'~2 b4

(1 - v 2)

(13)

Flange curling in profiled steel decks

11

which, for the more general case where the distance of the flange from the neutral axis is taken as z ( z = h / 2 for the box section), and taking v = 0.3 for steel, becomes um~,= 2.28 ( E ) 2 t2 b4sz •

(14)

In order to describe all flange curling deflections with a single equation, the constants in eqns (10) and (14) could be 'averaged' to 2.0. This is used in the expression described by eqn (1) and in Eurocode 3. For the present tests, O'a is based on engineering bending theory, thus O"a -~-

Mz

I

( 1 5)

where M is the imposed moment and I the second moment of area. Using eqn (l), the magnitude of curling deflection experienced by each specimen can be estimated, according to Eurocode 3, to be Um~x = 2 z M E l t ]

(16)

for both negative and positive bending, and both supported internal and unsupported edge flanges.

5 TEST RESULTS FOR S T A N D I N G SEAM PANELS The differential deflection of flange centre to web was recorded for each flange of each specimen in the present test programme. The details of the specific specimens identified (e.g. C2NA) can be found in Refs 4 and 5. Only the wide stiffened flange of each Condeck and Condeck HP panel was investigated, not the lip of each upstand since the top flange width was not regarded as sufficient to result in curling deflections. The curling deflection results are shown in Figs 6-9. In these figures, the arrows shown in the inset diagrams indicate the location of transducers used for displacement measurements. The fine dotted lines indicate the deflections predicted by eqns (19)-(36), with lines identified according to the schedule in Tables 4 and 5. Using eqn (16), estimates of the flange curling deflections according to Eurocode 3 can be made. For the case of Condeck E = 202.0 GPa, t-0-994mm, I is 159642mm 4 per panel, b s = 150.15mm and z = 12-0 mm. Thus um~, = 11-8725 M 2

where u is in millimetres and M in kNm.

(17)

12

E. S. Bernard, R. Q. Bridge, G. J. Hancock

24 22

I

20

0 .~,

I

I

I

I

-

Pan deflection

-

u

I ,' ~ ," I ",~,. t

*. ,. /

18 16 14

c

....

\.

I

~,

•

I

,,"

"

oATr."

:

,,'

Z

-

,'

, / V.~)K/~K?.." k.

12 10 8 6 4 2 0

0

0.2

0.4 0.6 0.8 1.0 1.2 1.4 1.6 Normalised Midspan Moment (kNm)

1.8

2.0

Fig. 6. Condeck, negative bending, differential deflections.

24

I

22 20 -

"

co

18 16 14

~D

12

_~, __ •

I

I

L

Pan deflection Edge deflection ~

~. •

~

.... •

.I .,: ~ll.~,c~,'/

L'

I

"~:: ',: ,~/~,~ :':// , ~ / / ~ /t :. /,~-..,

/

~

1

o -~ "%.q ~*

/

-'

E

-~

10 8 _ 6

,..'(.~Z>,"

...-

/7

\o,

4 2 L i I 0 .,_~~__.---~L 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Normalised Midspan Moment (kNm)

i

1.8

I 2.0

Fig. 7. Condeck, positive bending, differential deflections. F o r C o n d e c k H P , E = 202.8 G P a , t = 0.752 m m , I is 140652 m m 4 p e r p a n e l , bs = 148-95 m m a n d z = 12.58 m m . T h u s Um~× = 26.9159 M 2 w h e r e u is in m i l l i m e t r e s a n d M in k N m .

(18)

Flange curling in profiled steel decks 24

'

!

.

13

,'L~

1,'

22 E r.. o

0 e~

20 18 16 14

_-

12

_

,~/'~"//"I /~~~8'"' ,'" .,' ../

, ' ~me

/ ,,:/..-x--...

c,,,

-°..,..........

..."//y/;

10

;'YZ Y.-

8 6

4 2 I 1 1 1 I 0 ~ " 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Normalised Midspan Moment (kNm) Fig. 8. Condeck HP, negative bending, differential deflections.

24 I

r

t

.

22 20 18 g

16

~

]4

I-k

,.':~// /;/,f'

-

110

..I/Ill

I/

/, I

,,,Ii

/TK

r

,

.

,,

I1 ," - /.": t7 .¢~:. ,: "

i'." Is,'

r

I':..:

, ,: / /

[

.r

---.~. ,: / '~t * ~ c ~ , x :,'L~'-D ..

X

Pan deflection Edge deflection

i

8 u

. .£ . : ,i~/ i

.. . .

\ ., .~ ./ . . . .

~

.,

4 2 0

~.~.::-"

0

0.2

"

V

!

0.4 0.6 0.8 1.0 1.2 1.4 1.6 Normalised Midspan Moment (kNm)

1.8

2.0

Fig. 9. Condeck HP, positive bending, differential deflections.

These equations are compared to the recorded differential deflections between flange centres and webs for all the test specimens in Figs 6--9 and are shown as the heavy dashed line. For both Condeck and Condeck HP tested in negative bending (Figs 6 and 8), the predicted deflection was reasonably accurate up to approx. 50% of ultimate load.

14

E. S. Bernard, R. Q. Bridge, G. J. Hancock TABLE 4 Line Identifiers for Condeck and Condeck HP Specimens in Negative Bending

Line

Equation

Restraint conditions

A B C D E F

1 19 22 25 28 30

EC3 * sway sway no sway sway no sway

"

Panel(s) all 1 2, 3 (edge 2, 3 (edge 3 (central 3 (central

panel) panel) panel) panel)

Coefficient of A 2.00 2.28 1.37 1-11 0.91 0.64

* As in Eurocode 3. TABLE5 L i n e l d e n t i f i e r s ~ r C o n d e c k a n d C o n d e c k HPSpecimensinPositiveBending

Line

Equation

Restraint conditions

Panel

Coefficient of A

A B C D E

1 33 36 28 30

EC3* sway no sway sway no sway

all edge edge pan pan

2.00 3.20 1-98 0-91 0.64

*As in Eurocode 3.

Several observations can be made from Figs 6-9. Condeck HP of 0-75 mm thickness was found to experience more curling than 1.00 mm Condeck. The primary reason was the lesser thickness of the Condeck HP base metal. This occurred despite the fact that the intermediate stiffeners were much larger in the Condeck HP specimens. The presence of these longitudinal stiffeners has been found to have little effect on curling deflections. The number of panels comprising each specimen did not significantly influence the degree of curling experienced. This was a rational result of the fact that interconnection of panels was achieved through the upstands and so stresses in one pan could not affect the pans of other panels to an appreciable extent. The upstands themselves did not display significant curling. However, they ultimately failed in such a manner that the lip-like flanges experienced deflection toward the neutral axis. This may have been the result of curling deflections at an advanced stage of loading. For the internal pans in Condeck (those supported on both edges), compressive stresses resulting from negative bending produced significantly greater curling deflections than tension (which occurred in positive bending). This can be seen by comparing the solid lines shown in Fig. 6 with those in Fig. 7. However, the corresponding curling deflections

Flange curling in profiled steel decks

15

in negative and positive bending for Condeck HP were approximately the same (compare the solid lines in Figs 8 and 9). Distortional buckling 4"7 was observed in all the Condeck negative bending specimens, while it did not occur in the pans of the Condeck HP specimens due to the deeper intermediate stiffeners. This leads to the suggestion that a form of interaction of distortional buckling and curling occurred in the compressed pans of the Condeck specimens, resulting in substantially greater differential deflections than would have been caused by curling alone. Eurocode 3 states that deflection in each of the flanges should be identical. While all the Condeck negative bending specimens displayed distortional buckling, only one was found to have deflection readings affected by buckling because of the placement of the transducer on the buckle crest (C3NA in Fig. 6). Unsupported edge flanges were found to experience greater deflection than the supported internal flanges. This is contrary to Eurocode 3 which regards the two as identical provided bs is the same (see Fig. 4). For both Condeck and Condeck HP in positive bending, the edge flanges were exactly half the width of the internal flanges, and so bs was the same for the unsupported edge flanges and supported internal flanges. In Figs 7 and 9, the two types of flange (edge and pan) clearly did not deflect by the same amount. In fact, for Condeck HP, the positive bending specimens experienced so much curling in the edge flanges as to cause them to go into compression and experience buckling. This is shown for Edge 1 flanges in Fig. 9. The mean longitudinal strains measured in the pans of each positive bending Condeck HP specimen are shown in Fig. 10 (positive strain denotes tension). The arrows shown in the inset schematic diagram indicate the location of the strain gauges. The graph reveals the similarity of behaviour for specimens consisting of I, 2 and 3 panels joined together. As the moment increased, there was a prominent strain reversal in the centre of the tensile pans. The strain at the edges of the pan increased at an accelerated rate at approximately the same moment as reversal of strain occurred in the pan centres. The reversal of strain was a result of a redistribution of stress away from the pan centres accompanying the deflection of the pan toward the neutral axis as a result of flange curling.

6 REVISED D E R I V A T I O N OF T H E O R E T I C A L C U R L I N G ESTIMATES The simplification of the curling formula [eqn (1)] has been necessary to provide one equation to model all flange curling situations. Somewhat

16

E. S. Bernard, R. Q. Bridge, G. J. Hancock

,-, 1200

I

t

1000 . . . . 800

-

t

_

600

E

O

I

I

T

/

I

/

///"

r

....

. . . . . .

/ // /'./

.=. "~ 400 c-

I

Pan centre Pan edge

- -

~

200

o

c2eB

~D

-2oo

I

0

0.2

I

I

I

I

I

0.4 0.6 0.8 1.0 1.2 Normalised Midspan Moment (kNm)

I 1.4

1.6

Fig. 10. Longitudinal strains in pans of Condeck HP specimens in positive bending.

better estimates of the curling deflections encountered in the present investigation may be derived if the condition of simply supported edges is replaced by more realistic assumptions. The flange boundaries have instead been taken to be rigid joints, capable of rotation, linking the flanges to the webs of the upstands which can provide restraint. The Condeck and Condeck HP geometries will be considered together and will be examined first, followed by the trapezoidal decks. 6.1 Condeck and Condeck H P in negative bending - - pan in compression

For the case of one panel, as shown in Fig. 11 (a), it can be seen that for negative bending, assuming that the upstands do not provide any torsional restraint, a single pan acts like a simply supported beam. This is precisely the condition that Winter assumed for any doubly-supported flange, thus from eqn (14) Um~ = 2.28A

(19a)

where A = ( - - ~ ) 264t2z .

(19b)

For the case of two panels, two restraint conditions at the junctions of the panels must be considered. The first restraint condition is depicted in Fig. 11(b). This shows the condition in which the flanges of the panels are

Flangecurlingin profiledsteeldecks

17

L

(a) One Panel Webs free to expand Me Me

............... ;IIIIZ

........................

(b) Two Panels Central Webs free to expand

"z"::"'?"?:'?:'~":'2~""'2~"-":''"Z:'::':':F' "ixed "?"

".":'~? ~'~'~'i"2"?i"~"-""'::':":".... , ~ j

(c) Two Panels Central Webs not free to expand

(d) Three Panels M~ ~CentralWebsfreet°expan~te~

~e

(e) Three Panels Central Webs not free to expand

Fig. 11. Approximationsused in the modellingof Condeck and Condeck HP specimensin negative bonding.

assumed incapable of providing sufficient in-plane rigidity to prevent lateral displacements of the flanges (i.e. the 'sway' condition). The central web is therefore taken to be capable of spreading. It is also assumed that snug interlock of the upstand flanges occurs and that fixity is provided by the self-tapping screw in the flange. The junction of the two webs at the

18

E. S. Bernard, R. Q. Bridge, G. J. Hancock

upstand flange has therefore been taken to be fixed against displacement a n d rotation. The maximum deflection of the pan, due to the force R

acting normal to the flange, is then 5

Um~ - - 384

RL 4

MeL

D

16D

(20a)

5

RL 4

k

RL 4

384

D

128

D

(20b)

where Me is the bending moment in the plate at the edge of the pan Me=k--

RL 2

and

8

D-

E1 (1 -

v 2)

leading to 5

RL 4

u'n~x--384

D

[1-0.6k].

(20c)

From eqns (11) and (14), then Um~, = 2.28A [1 -- 0.6k]

(20d)

where

A ~ ( E ) 2 b.;

(20e)

12Z

and from simple moment distribution between the pan and the upstand, (21)

k = L/(3Lr + L).

For the Condeck and Lr = 50 mm, thus Um~x= 1'37A.

Condeck

HP

panels,

L = 300mm

and (22)

Figure 1 l(c) depicts the condition in which it is assumed that the pans of the panels provide sufficient in-plane flexural rigidity to prevent lateral displacements of the flanges and thus prevent the central web from spreading apart. This is commonly known as the 'no sway' condition. The webs of each panel are taken to be rigidly restrained against lateral displacement, but not rotation, at the intersection with the upstand flange. As the flanges deflect toward the neutral axis in this model, the webs bend toward each other and possibly interfere. Although this will change the structural model in practice, the consequences have at present been ignored. The webs provide a measure of flexural rigidity that must be

Flange curling in profiled steel decks

19

included in the derivation. This has been done by performing a simple m o m e n t distribution at the edges o f the pan, thus, as above Umax = 2.28A [1 - 0.6k]

(23)

where k = L / ( L + Lr)

(24)

hence u~,,a~ = 1.11 A.

(25)

For the case o f a specimen incorporating three panels, the two edge panels can be treated in the same m a n n e r as the two panel case, but the central panel requires a different derivation. For the sway condition [see Fig. 1 l(d)], where the webs are assumed to be free to expand, the structural stiffness changes and the m a x i m u m deflection o f the central panel becomes

u~,~x

5

RL 4

MeL 2

384

D

8D

(26a)

As for eqns (20a)-(20d) this reduces to un,~x = 2.28A [1 - 0-8k]

(26b)

where k = L / ( L + 2Lr)

(27)

thus Um~x= 0.9 l A.

(28)

In reference to Fig. 1 l(e), for the no sway condition in which it is assumed that the webs are prevented from spreading, the m a x i m u m deflection of the central panel is Um~x = 2"28A [1 - 0.8k] where k = 3 L / ( 3 L + 2Lr).

(29)

Thus, for both Condeck and Condeck HP, Um~x = 0.64A.

(30)

Using eqns (15) and (19b), the expressions for Um~ have been developed in terms of the m o m e n t M applied to the member. The results have been plotted

E. S. Bernard, R. Q. Bridge, G. J. Hancock

20

in Fig. 8 for comparison with the experimental results ofCondeck HP loaded in negative flexure. Equation (1) from Eurocode 3 is plotted as line A. The derivations for alternate edge restraints, eqns (19), (22), (25), (28) and (30), are plotted as lines B to F, respectively (as listed in Table 4). Comparison with the test results indicates that the assumption that the pans provide little in-plane flexural rigidity, and thus the webs are free to expand (the sway condition), appears to be valid (line C for two panels and the edge panel of three panels, line E for the central panel of three panels in Fig. 8). The deflections of the pans are greater than predicted by the stiffer no sway condition, especially for the multiple panel specimens. The comparisons with the results of the Condeck specimens are shown in Fig. 6. These indicate that, unlike the results for Condeck HP (Fig. 8), the code approximation (line A) or eqn (19) (line B) appear to be better. The reason for the difference in behaviour between the Condeck and Condeck HP specimens is unknown. The comparisons have been made mainly at low moments since these are believed to be less affected by second-order considerations not included in the curling model. 6.2 Condeck and Condeck H P in positive bending - - pan in tension

For all the positive bending specimens, the edges consisted of pans that had been cut in half. These will be analysed similarly to the full pans. A single half-pan is shown in Fig. 12(a). This depicts the sway condition in which the pan is assumed to provide insufficient in-plane flexural restraint to prevent sway. The web is then free to spread apart. As before, the junction of the two webs at the upstand flange has been taken to be fixed against displacement and rotation. The maximum flange deflection, as a cantilever, is found as

RL4 Um~x- 8D-- ~-

Le

(31a)

RL 4 kRL 4 -- 8D + 2---D---

(31b)

where k --- Lr/Le and Me is the bending moment in the plate at the edge of the pan

Me - R L 2 2

and D -

E/ (1 - v 2)

leading to Umax-

R L 4 [1 + 4k]

8D

(31c)

Flange curling in profiled steel decks

21

\? (a) One Edge Web free to expand

Le

Fixed//~

(b) One Edge Web not free to expand

Fig. 12. Approximations used in the modelling of Condeck and Condeck HP specimens in positive bonding.

From eqns (4) and (10), Umax= 1-37A [1 +4kJ

(31d)

where

For both Condeck and Condeck HP, Lr = 50 mm and L~ = 150 mm, thus um~, = 3.20A.

(33)

E. S. Bernard, R. Q. Bridge, G. J. Hancock

22

Alternately, the no sway condition, in which the pan is assumed to provide sufficient in-plane flexural rigidity to prevent spreading of the webs, is depicted in Fig. 12(b). The web is also assumed to be restrained against displacement but not rotation, at the upstand flange corner just as for the case of full width pans in negative bending. Using m o m e n t distribution, the maximum flange deflection is calculated as Um~x--- 1.37A [1 + 1.33k]

(34)

where k = Lr/Le.

(35)

For both Condeck and Condeck HP, Lr = 50 m m and Le = 150 mm, thus Um~x= 1"98A.

(36)

For the case of two or more pans, the doubly-supported pans are modelled in the same manner as for negative bending involving the central pan of a three-panel specimen. The maximum deflection is therefore described by eqns (28) and (30) for the sway and no sway conditions respectively. Comparisons with the results for Condeck HP loaded in positive flexure are shown in Fig. 9. The results have been affected by the influence of buckling in the edge flanges. Curling deflections became so large that the edge flange in two of the specimens went into compression and buckled, leading to softening of the specimen and rotation about the longitudinal axis. This has tended to accentuate the difference in deflection between one edge and the other. Nevertheless, the assumption that the pans do not provide sufficient in-plane flexural rigidity to prevent the webs spreading apart appears to be valid (line B for edges and line D for pans in Fig. 9). This is the same result as for negative bending. The comparisons with the results of the Condeck specimens are shown in Fig. 7. This figure indicates that the approximations developed to account for web restraint in eqns (31)-(36) [and eqns (28) and (30)] gave rise to better agreement between test and numerical results than the simple code formula, particularly for the pan deflections. 6.3 Advanced behaviour

Beyond approx. 50% of ultimate moment, the experimental deflections slowed in their rate of increase while the code-based estimates continued their parabolic rise with moment. Winter, l in the derivation of eqn (1), assumed that the stress in the flange was uniform across the width. This was an acceptable approximation provided that the maximum curling

23

Flange curling in profiled steel decks

deflection experienced by the flange during loading was small. Although this proved admissible for the hot-rolled sections produced in 1940, this assumption does not hold true for the very thin-walled specimens in this study. A corollary of Winter's assumption that the maximum curling deflection remained small is that the section second m o m e n t of area 1 remained constant. This is clearly not the case in the present investigation. Equation (1) is thus based on a first-order model leading to unusable results at large strains. The tapering of the experimental result near ultimate collapse may have been due to the reduction of in-plane stress in the flanges as they approached the neutral axis. According to Winter's model, curvature of the flanges results in a component of the flange force acting normal to the flange plane and toward the neutral axis. This force is assumed to be responsible for deflection of the flange toward the neutral axis. Measured strains in the present tests indicate that stress within the flange returns to near zero levels once the flange approaches the neutral axis. The intuitive result of this would be a reduction of the tendency of the flange to deflect further. 6.4 Test results for trapezoidal panels

In the trapezoidal sections with flat-hat intermediate stiffeners (FC1 and FC2) the curling deflections were almost the same for each rib but differed between the compressive and tensile flanges (see Figs 13 and 14, and an

I0

I

9 E

I

Top flange test result ----- Bottom flange test result

I

-"

- -

8 _

Topna.ge

Top

v ~D

I

7

/

6 -

Bot!om

4 -..~,

= o~

!

,s.s-~///

.,'///

3 a 1

o

0

1.0

2.0 3.0 Midspan Moment (kNm)

Fig. 13. Specimen FCI differential deflections.

4.0

24

E. S. Bernard, R. Q. Bridge, G. J. Hancock

10

I

9-

o

O ¢~

I

t

Top flange test result - - - - - Bottom flange test result

8~9

I

•

7-

/

t_~

6 -

/,:topn.~

4 E

g2om

3 2

-

1

-

~

flange

t~ 0

..... i

0

1.0

2.0 3.0 Midspan Moment (kNm)

4.0

Fig. 14. Specimen FC2 differential deflections.

TABLE

6

Line Identifiers for Trapezoidal Specimens in Negative Bending Line

Equation

Restraint conditions

Flange

Coefficient of A

A B C D E F G

l (and 37/39) 48 43 51 1 (and 38/40) 45 53

EC3* sway sway no sway EC3* sway no sway

all comp flanges outer comp flanges inner comp flanges all comp flanges tension flanges tension flanges tension flanges

2-000 1.984 1.885 0.678 2.000 1.791 0.794

*As in Eurocode 3.

identification of graph lines in Table 6). This occurred because each of the compressive flanges had a width of 135 mm and each of the tensile flanges were 82.5 mm wide. The tensile and compressive ribs therefore had different average widths and thus displayed markedly different mean deflections. The estimated curling deflections according to eqn (1) are shown as the heavy dashed lines. These estimates are somewhat more accurate than for the Condeck panels. For specimens FC1 and FC2, the code-based predictions were fairly accurate for both the tensile and compressive flanges. For the compression flange of FC1, E = 230-0 GPa, t = 0.585 mm, I is 316404 mm 4 for the

Flange curling in profiled steel decks

25

whole specimen, b s - 67-59 m m and z = 22.2 mm. Thus, using eqns (1) and (16), the deflection o f the compression flange is Umax, c =

2-0A

(37a)

where t2g

as earlier, thus Umax,c -- 0.5111 M 2

(37b)

where Umax,c is in millimetres and M in kNm. For the tensile flange, bs = 42.25 m m and z = 31-30 mm, hence the curling deflection Umax.t is /gmax, t ~--- 2 - 0 A

(38a)

//max, t = 0-1 l 0 M 2 .

(38b)

or

These estimates are compared with test results in Fig. 13. For specimen FC2, the corresponding equations were found to be Umax,c = 0.5018 M 2

(39)

Umax,t -- 0" 1006 M

(40)

2.

These are compared with test results in Fig. 14. As with the Condeck and Condeck HP specimens, more accurate predictions of the flange curling deflections may be derived, as shown below, if the simple plate boundary conditions used by Winter 1 are replaced by more realistic assumptions.

6.5 Flanges free to sway For the central rib, in which it is assumed that the flanges are free to sway [Fig. 15(a)], the m a x i m u m deflection o f a compression flange bounded on both sides by other flanges is, as given by eqn (26b) Um~x= 2.28A [1 - 0-8k]

(41)

where, from simple m o m e n t distribution between the pan and webs k -

Lc

-

Lc +

and

Lt

2Ls

(42)

E. S. Bernard, R. Q. Bridge, G. J. Hancock

26

/

Me

Me

\

\

/

\"~; ............ J

\~- .....

7"

(a) Sway condition

I--

I_ I--

L,

Lc

_ I

_I -7

(b) No sway condition

Fig. 15. Approximations used in the modelling of inner portion of trapezoidal specimen in negative bending.

I22

thus Urea× = 1.885 A

(43a)

un,ax = 0"4819 M 2 .

(43b)

or

F o r the tension flange k -

Lt - Lc Lt + 2Ls

(44)

thus Uma~ = 1"791A

(45a)

Um~, = 0" 1524 M 2 .

(45b)

or

F o r the two ribs o n either side o f the central rib, there is little restrain a f f o r d e d b y the outside webs and a d j o i n i n g h a l f tensile flanges. T h e c o n d i t i o n s are t h e r e f o r e similar to those used in deriving eqn (23), hence

27

Flange curling in profiled steel decks

(46)

u ~ , = 2.28A [1 - 0.6k] where k -

t c -- t t

(47)

Lc + 3Ls

for the compression flange only, and thus Um~×= 1"984A

(48a)

Um~x = 0"5071 M 2.

(48b)

or

6.6 Flanges restrained against sway Figure 15(b) depicts the trapezoidal decks for the no sway condition in which it is assumed that the flanges are restrained against lateral deflections. Only the central portion o f the three rib decking panel is shown. The m a x i m u m deflection o f each flange (bounded on both sides by other flanges) for specimens FC1 and FC2 is again found as u~,~ax= 2.28A [1 - 0.8k].

(49)

For the compression flange (possessing the intermediate stiffeners) 6Lc 6Lc + 2Ls

(50)

Umax = 0.678 A

(51 a)

u,nax = 0" 1734 M e

(51 b)

k thus

or

while for the tension flange at the bottom 6Lt 6Lt + 2Ls

(52)

U~x = 0.794A

(53a)

Um~x = 0.04367 M 2 .

(53b)

k and

or

28

E. S. Bernard, R. Q. Bridge, G. J. Hancock

Equations (48), (43), (51), (45) and (53) have been plotted as lines B, C, D, F and G (see Table 6) and compared to the test results in Figs 13 and 14 for specimens FC1 and FC2. The predictions of Eurocode 3, using eqn (1) and eqns (37)-(40), are included as lines A and E in each figure. The comparisons indicate that the assumption that the flanges are free to 'sway' is valid (line C for inner compression flanges and line F for tension flanges).

7 CONCLUSIONS Flange curling deflections can be significant in cold-formed steel sections subject to bending, particularly profiled decking sections with wide flanges. It is possible to have deflections of such a magnitude that the unsupported centres of panels can deflect up to and beyond the neutral axis of the cross-section. Winter's formula for the determination of flange curling deflections was derived for the outstands of hot rolled/-sections and the supported flanges of box sections. The Eurocode 3 expressions for these two cases have been generalised into one expression. This has been found to predict deflections satisfactorily in cold-formed thin-walled decking sections. However, by modifying the expressions to incorporate the edge restraint conditions imposed on the flanges by the webs, curling deflections can be estimated more accurately than by the use of the present general expression.

ACKNOWLEDGEMENTS This report forms part of a programme of research into the stability of steel structures being carried out in the School of Civil and Mining Engineering at the University of Sydney. Tests were performed in the J. W. Roderick Laboratory for Materials and Structures. Technical and material support for the present experimental programme was provided by Stramit Industries Pty Ltd. Financial support for laboratory and computer equipment was provided by BHP Research - Melbourne Laboratories Pty Ltd.

REFERENCES . Winter, G., Stress distribution in and equivalent width of flanges of wide, thin-walled steel beams. NACA Technical Note 784, 1940.

Flange curling in profiled steel decks

29

2. Yu, W-W., Cold-Formed Steel Design, 2nd edn. John Wiley, New York, 1991. 3. Eurocode 3, Design of Steel Structures, Part 1.3. European Commission, Brussels, 1993. 4. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Intermediate stiffeners in cold-formed profiled steel decks. Part 3 - - Condeck rolled panels. Research Report R674, School of Civil and Mining Engineering, University of Sydney, Australia, March 1993. 5. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Intermediate stiffeners in cold-formed profiled steel decks. Part 2 - - flat-hat shaped stiffeners. Research Report R658, School of Civil and Mining Engineering, University of Sydney, Australia, August 1992. 6. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Tests on profiled steel decks with V-stiffeners. J. Struct. Engng, ASCE, 119(8) (1993) 2277-93. 7. Hancock, G. J., Local, distortional and lateral buckling of/-beams. J. Struct. Div. ASCE, 104, (ST11) (1978) 1787 98.