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Design methods for profiled steel decks with intermediate stiffeners
PII: S0143-974X(96)00010-7
J. Construct. Steel Res. Vol. 38, No. 1, pp. 61-88, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0143-974X/96 $15.00 + 0.00
ELSEVIER
Design Methods for Profiled Steel Decks with Intermediate Stiffeners E. Ste,fan Bernard", Russell Q. B r i d g e a & Gregory J. Hancock b aFaculty of Engineering, University of Western Sydney, Nepean, PO Box 52, Kingswood, NSW, Australia, 2747 bCentre for Advanced Structural Engineering, School of Civil and Mining Engineering, University of Sydney, NSW, Australia, 2006 (Received 31 October 1994; revised version received 25 September 1995; accepted 9 February 1996)
ABSTRACT Thin-~alled cold-formed profiled steel decking panels with intermediate stiffeners may exhibit distortional buckling involving transverse flexure of the stiffener. Local buckling in the flat plate elements between the flanges and webs may also occur. Varying degrees of post-buckled strength reserve may be encountered in sections undergoing local and~or distortional buckling. The extent of post-buckled strength reserve, as well as the ultimate moment capacity of the panel, are of interest to designers in their efforts to produce more efficient sections. Ultimate moment results for 27 profiled decking specimens tested in bending are compared with estimates derived using a number of design standards. The standards used were the AISI LRFD Coldformed Steel Structures Specification 1991, Eurocode 3/Part 1.3, and Australian Standard AS1538-1988. Several additional methods for predicting the ultima~!e moment capacity have been proposed recently and these are also compared with the test results. Copyright © 1996 Elsevier Science Ltd.
1 INTRODUCTION Thin-walled cold-formed steel sections have gained increasing acceptance as replaceme,nts for hot-rolled sections in structural applications in the last two decades. Floor decking panels manufactured from thin-gauge, high strength coated steel have featured as part of this growth in popularity. However, coldformed sections are subject to more complex forms of buckling and non-linear behaviour than hot-rolled sections, hence their use has necessitated greater developmental research. Design specifications (AISI, Eurocode, Australian 61
62
E. S. Bernard, R. Q. Bridge, G. ]. Hancock
Standards) governing cold-formed sections are still undergoing rapid development in an attempt to keep up with advances in manufacturing and structural design. A programme of tests of cold-formed floor decking panels has been undertaken in the School of Civil and Mining Engineering at the University of Sydney. Twenty-seven specimens were tested in flexure in a purpose-built test rig to determine the structural behaviour of the decking panels when subject to pure flexural loads. Tangent stiffness, stress distributions, buckling moments and ultimate moments were recorded for comparison with numerical predictions and design estimates. The tests have been described in detail in two reports by Bernard et al. ~,2 and published in Bernard et a/. 3"4 Comparisons with existing design methods are described. Furthermore, improved design methods are proposed and compared with the test results. As a result of differences in notation used in the various standards, notation is the same as the standards referred to and is described for each equation in the accompanying text rather than in a notation list. 2 GEOMETRY OF SECTIONS The test specimens examined in the present investigation consisted of brakepressed panels of a trapezoidal design with a single intermediate stiffener in each compression flange. The intermediate stiffener was in the shape of a 'V' or a 'flat-hat'. The two types of brake-pressed section are illustrated in Figs 1 and 2. The geometry of both the 'V' and 'flat-hat' stiffened deck panels was selected to exhibit both local and distortional buckling under pure bending. The buckling behaviour changed from distortional for small intermediate stiffeners to local for large stiffeners. The buckling modes for the flat-hat stiffened sections are shown in Fig. 3; similar buckling modes, excluding the anti-symmetric distortional mode, occurred in the V-stiffened sections and are described by Bernard et al. 3 Several specimens exhibited both modes of buckling, one occurring at a moment greater than the other. These were expected to show an 'interaction' of buckling waveforms. As depicted in Fig. 1, the intermediate stiffeners in the V-stiffened sections were in the shape of a 'V' with an enclosed angle of 90 ° at the base. As the height S (or depth) of the stiffener increased from 2 to 10 mm, the proportions of the 'V' remained the same. The measured dimension S has been listed in Table 1 for each specimen. The intermediate stiffeners were in the middle of the compression flanges. The full sections consisted of a trapezoidal shape repeated as four longitudinal ribs. Total width of the folded sections was approximately 785 mm while the length was 2000 mm. The system of identification for the 'V'-stiffened specimens has been described by Bernard et al. 1"3
Profiled steel decks I_
89mm
__1
5!I
Specimens with V-Stiffeners (IST)
I--
196
I _ 45- 135ram
55
63
T I
LI
Stiffeners (ST)
0.5
151.5-241.5
J
Fig. 1. Geometry and general dimensions of specimens with V-stiffeners (IST) and specimens with no stiffeners (ST).
The section geometry and intermediate stiffeners in the flat-hat stiffened sections a:re depicted in Fig. 2. The intermediate stiffeners were in the shape of a broad rectangle with webs inclined at 45 ° to the base, hence the name 'fiat hat' stiffeners. As the height S of the stiffener increased from 1 to 6 mm, the proportions of the 'fiat hat' changed. The widest portion b~ was maintained at 40 m m and the web inclination at 45 °, hence the stiffener base b2 decreased in width in inverse proportion to the height. The stiffener dimension S has been listed in Table 2 for each specimen. The intermediate stiffeners were in the middle of each of the compression flanges of a trapezoidal section with three longitudinal ribs. Total width of the folded sections was approximately
64
E. S. Bernard, R. Q. Bridge, G. J. Hancock
I"
551
135
~-0~'5 ~ ' - ~ ~ i I
241.5
~i
Specimenswith i Hat-hatStiffeners I (TS3) I
GeneralGeometry Fig. 2. Geometry and general dimensions of specimens with flat-hat stiffeners (TS3).
(a) Localbucklingsymmetric
/
(b) Distortionalbucklingsymmetric
\ .....
(c) Localbucklingantisymmetric
(d) Distortionalbucklingantisymmetric
BucklingModes Fig. 3. Buckling modes for specimens with flat-hat stiffeners.
Profiled steel decks
65
TABLE 1 Comparison of V-stiffened Specimen Test Results with Mu Calculated by AISI, 1991 Specimen
Stiffener height (mm)
Mode ~
Mu (test) (kNm)
Mub (AISI) (kNm)
Mr,..... Mumtst
ST21A ST21B ST22A ST22B ST23A ST23B IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B IST48A IST410A IST410B IST412B
------2.07 3.05 1.94 2.98 5.59 5.12 5.21 5.55 7.89 9.93 7.83 9.74
L L L L L L D+L D+L D+L D+L D+L D+L D+L D+L L L L L
3.39 2.97 3.56 3.16 2.65 2.38 3.85 4.00 3.68 3.88 4.59 4.56 4.57 4.54 5.42 5.75 5.14 5.59
2.790 2-799 2-924 2-939 2.162 2.163 3-716 3.895 3.843 3-877 4-735 4.410 4.467 4-647 5.814 6-101 5.672 6.059
1.215 1.061 1-218 1.075 1.226 1-100 1-036 1.027 0.958 1-001 0-969 1.034 1-023 0-977 0.932 0-942 0.906 0-923
mean o"
1-035 0.101
1.0 kNm = (I-73756 kip ft. L = Local only (observed in tests); D+L = Distortional then Local. b Calculated by AISI, 1991. a
720 m m while the length was 2000 m m . T h e s y s t e m o f identification for the flat-hat stiffened s p e c i m e n s has been described in B e r n a r d et al. 2"4 T h e stlSLp used in the m a n u f a c t u r e o f the specimens was G r a d e G 5 5 0 coldr e d u c e d coated steel c o n f o r m i n g to Australian Standard A S 1 3 9 7 . 5 The strip was n o m i a a l l y 0.60 m m thick, was c o a t e d with Zincalume, and had a n o m i n a l yield stre,;s o f 550 MPa. The m e a s u r e d base metal thickness was 0-585 m m . Further details regarding aspects o f manufacture, material properties, residual stresses and g e o m e t r i c imperfections are contained in the references described above, as well as the report b y Bernard and C o l e m a n 6 w h i c h gives a detailed study o f g e o m e t r i c imperfections in several o f the V-stiffened specimens.
E. S. Bernard, R. Q. Bridge, G. J. Hancock
66
TABLE 2
Comparison of Flat-hat Stiffened Test Results with Mu Calculated by AISI, 1991 Specimen
TS3A1 TS3A2 TS3A3 TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
Stiffener height (ram)
Mode a
M, (test) (kNm)
M, b (AISI) (kNm)
Mu.tes, MuzlS~
1.25 1.70 2.58 4.433 5.533 5.467 3.808 5.042 6.033
D+D2 D+D2 D+L L+D L+D L+D L+D L+D L
2.96 2-93 2-96 3.52 4.10 3.87 3.63 4-01 4.04
2.919 3.048 3.262 3.746 4.538 4-502 3-583 4.019 4.414
1.014 0.961 0.907 0.939 0-903 0-860 1-013 0.998 0-915
mean cr
0.946 0.055
1.0 kNm = 0.73756 kip ft. a L=Local only; L+D=Local then Distortional; D+D2= 1st & 2nd mode Distortional; D+L = Distortional then Local. o Calculated by AISI, 1991.
3 COMPARISON OF TEST RESULTS WITH EXISTING STANDARDS Estimates of the ultimate moment capacity of the test specimens were determined using several existing or draft design standards. The standards selected were the AISI LRFD Cold-formed Steel Structures Specification7 representing North American practice, Eurocode 3/Part 1.38 representing European practice and Australian Standard AS1538-1988. 9 The estimates have been compared with test results to determine the accuracy of the various methods. In each of the sets of comparisons described below, the dimensions used in the design calculations have been derived from the mean measured geometry of each specimen. The measured base metal thickness has been used as the design thickness in all calculations contained in this report. For the purpose of design comparisons, the measured yield (or 0.2% proof) stress, Fy, equal to 653 MPa, for the strip material has been used in all calculations. Although the stress-strain curve for the strip material was rounded, the 0-2% proof stress was very close to the ultimate stress. The specified elastic modulus, E, has been used instead of the measured value. This has been chosen because it is unusual for a designer to know the actual elastic modulus of the strip used and most standards explicitly specify a value to be used. Estimates of the elastic buckling moments for the mean measured geo-
Profiled steel decks
67
metries have been based on the finite strip elastic buckling analysis program BFINST (Hancock ~°) which forms part of a more recent computer program THIN-WALL (Papangelis and Hancockl~).
3.0.1 AISI Specification 1991 The American Iron and Steel Institute LRFD Cold-formed Steel Structures Specification, 1991 is based on the effective width approach to local buckling of cold-fo]aned sections pioneered by Winter. 12 This is applied to slender regions of a section that experience compressive stress. By Clause B2.1 (see Fig. 4), the effective width b is found as b = w
)t-<0-673
(la)
b = tgw
A>0-673
(lb)
where w i,; the fiat width of the compression element, and
(2) (3)
p = (1.0 - 0.22/A)/A A = (1.052/~/-k) (w/t) (~lf-/-E)
with t the strip thickness, k the plate buckling coefficient, a n d f t h e proof stress Fy when the nominal section strength is calculated and no lateral buckling is encountered. In the AISI Specification, E is taken to be 203.4 x 10 3 MPa. b0
"
/
w
I-
ffffliiiiiiiillllll
_1
_,
I
I
w/ lllllll
iiiiiiilIIIIl-
Stress f
Fig. 4. Effective width approach of AISI Cold-formed Steel Structures Specification CF-1 1991, Clause B2.1 for flange with intermediate stiffeners.
E. S. Bernard, R. Q. Bridge, G. J. Hancock
68
The AISI Specification has additional provisions for uniformly compressed elements with intermediate stiffeners (Clause B4.1). The code has no requirement for an intermediate stiffener to be adequate. However, a continuum of stiffener stiffnesses is treated identically by reducing the cross-section of the stiffener to an effective area As, and reducing the width of the flange to an effective width b. A limiting slenderness S is calculated as S = 1.28 ~/E--~
(4)
where f is the maximum stress (normally Fy) encountered using the effective design width. With the width of the complete compression flange, bo, equal to either 90 or 135 m m for all the test specimens, bo/t >- 3S, hence Case III of Clause B4.1 is used. Thus b = w
A -< 0.673
(5a)
b = pw
)t > 0.673
(5b)
where w is the fiat width of the buckled element, and p = (1-0 - 0.22/)t)/)t
(6)
)t = (1.052/qk) (w/t) (qf-7-E)
(7)
k = 33~[Is/Ia + 1 -- 4 As = a's (Is/Ia) <-- a's Ia = ( 1 2 8 ( b o l t )
-
285)t 4
(8a) (8b) (9)
A~ = full area of the intermediate stiffener As = reduced area of stiffener to be used in computing effective section properties = second moment of area of the intermediate stiffener about its own axis. The webs of most specimens were found to be partially effective according to Clause B2.3. The width of the compressed region of each web is reduced in a similar manner to the flanges. The partial effective widths bl and b2 (see Fig. 5) are determined by Clause B2.3(a) to be b, = bd(3 -
4~)
where ~b =f2/fl with f~ the compression stress at the tip of the web, and
(10)
Profiled steel decks
69 m
(compression)
Fig. 5. Effective web width by AISI Cold-formed Steel Structures Specification CF-1, Clause B2.3 for compression region of web.
b2 = be~2
~b -< - 0 . 2 3 6
(lla)
b2 = be - bl
4, > - 0 - 2 3 6
(llb)
where be is the effective width of the web such that be = yc
)t --< 0.673
(12a)
be = p y c
)t > 0-673
(12b)
where f = f ~ and Yc is the compressed portion of the web (see Fig. 5), and
70
E. S. Bernard, R. Q. Bridge, G. J. Hancock
A = (1-052P[k) (yJt) (~/fl----E)
(13) (14)
k=4+2(1
(15)
p = (1.0 - 0.22/A)/A
- ~b)3 + 2 ( 1 - ~b).
The effective web widths b~ and b2 are substituted for the actual width into the cross-section together with the effective width of the flange and effective area of intermediate stiffener. The section properties are determined, whereupon it is commonly found that the neutral axis changes position. Several iterations of the procedure are normally required to converge to a stable estimate of b e. The effective section modulus Ze is then calculated and the ultimate moment capacity determined as Mu = FyZe.
(16)
The code-based estimates of the ultimate moment capacity Mu are listed in Table 1 for V-stiffened and unstiffened specimens, and Table 2 for flat-hat stiffened specimens. Comparison with test results indicates that the method is slightly unconservative for the flat-hat stiffened sections and the V-stiffened sections with large intermediate stiffeners. The method proved conservative for the sections without stiffeners and the V-stiffened sections with small stiffeners. Those specimens with large intermediate stiffeners that displayed only local buckling prior to ultimate failure proved to have the most unconservative result. In Table 1 and all subsequent tables, the designated modes qf buckling (L for local, D for distortional) were based on empirical observations. 3.0.2 Eurocode 3~Part 1.3 Eurocode 3/Part 1.3 for cold-formed steel structures also uses the effective width concept. This is used for flat plate elements without stiffeners in Clause 3.2. However, in this procedure the plate buckling coefficient k~ is determined by more elaborate means than in the AISI Specification. The equations for determining k~ in Table 3.1 of Eurocode 3/Part 1.3 are a substitute for a rational elastic buckling analysis, Distortional (or stiffener) buckling within compressed flanges with one or two intermediate stiffeners is treated in Clauses 3.3.3 and 3.3.4.2. The intermediate stiffener is treated as a beam on an elastic foundation with restraint against out-of-plane displacement being provided by the flange and web (as well as the stiffness of the stiffener). The critical axial load capacity Ncr,s of the stiffener is calculated as N c r , s -~-
Kw4.2E ~/ld3/[8b 3 (1 + 3bs/(2bp)]
and the critical stress O'cr,s found as
(17)
Profiled steel decks
71
(18)
Orcr,s =/~Tcr, s/A s
where, as shown in Fig. 6, As Is bp bs E Kw
= = = = = =
effective area of stiffener second moment of area of stiffener width of fiat plate elements length of perimeter of stiffener Elastic Modulus (210,000 MPa) restraint coefficient.
The area stiffener As includes the stiffener itself and the effective parts of the flat plate elements on either side of the stiffener as illustrated in Fig. 6. The restraint coefficient Kw is dependent on the restraining action of the web on the flange. It is a function of the interaction of buckling half-wavelengths in both elements. It can conservatively be taken as equal to unity, or else calculated as Kw = K,~.o
lb/Sw
I_
I-
beff,1.1
bo
I
b~ff,1.2
--> 2
(19a)
-~
-I-
beff,2.2
-I
b~ff,2.1
Fig. 6. Effective widths of compressed flange with intermediate stiffeners according to Eurocode 3/Part 1.3. The area of the stiffener, As, includes the adjacent effective portions of the flange, beff,1.2 and beff.2.2.
E. S. Bernard, R. Q. Bridge, G. J. Hancock
72
Kw = Kwo - (Kwo - 1)[21b/Sw -- (lb/Sw) 2]
lb/Sw < 2
(19b)
where Kwo = q(Sw + 2be)/(Sw + bd2)
(20)
Ib = 3-65 4~/Isb3(1 + 3bJ(2bp))/t 3
(21)
Sw = width o f w e b shown in Fig. 7 be = bs + beff, l.2 + beff,2.2 (see Fig. 6). Once O'cr,s has been found, it is used in Clause 5.1.2 to determine the buckling coefficient X
where ~b = 0-511 + a(X~ - 0-2) + X 21
(23)
X s = ~/O'cr, s
(24)
o~ = 0.13. The term oz is an out-of-plane imperfection factor and fy is the yield (or proof) stress. The effective stress within the stiffener is then found as
ec
/il/.~ ' ~
Fig. 7, Effective width of compressed region of web according to Clause 3.3.4.4 of Eurocode 3/Part 1.3.
Profiled steel decks O'com ~- ) ( L
73 (25)
and it is used to determine a new estimate of the effective widths beff, l.2, beff,2.2 of the plate elements forming the effective stiffener as shown in Fig. 6. Note that bCff,L1 and beff,2.1 (the edges of the flange adjacent to the webs) are maintained at their initial values determined taking O'com=fy. Each of bell,L1, beff,/.~ and beff,2. 2 a r e calculated as boff/2, where beef = p bp
(26)
in which bp is the full width of the flange flat element and p = 1.0
hp - 0.673
(27a)
p = (1.0 - 0.22/,~p)/)tp
Xp > 0"673
(27b)
where hp
=
O'com =
1.052(bp/t)
~O'com[(Eko.)
(28)
X fy
k~ = plate buckling coefficient found from Table 3.1 of EC3.
(29)
This is an iterative procedure that requires at least two iterations to obtain a convergent estimate of O'comand thus beef. Estimates of the effective width of the webs are determined by Clause 3.3.4.4 for sheeting in bending. The effective widths of the webs are calculated after the flange calculations but are otherwise independent. A pair of equations is provided for the estimation of the effective parts of the web Seff,, = 0.76t ~/~fy
(30)
Serf,2 = l'5Seff,1
(31)
where Seff I and Self, 2 a r e as indicated in Fig. 7. These effective widths are used in the seclion geometry together with the effective flange widths and stiffener area to anSve at an effective section modulus Ze. In the determination of Ze, the reduction in O'comfrom fy is taken into account by a proportional reduction in the thickness of the effective intermediate stiffener. The ultimate moment capacity is then found as Mu = ZefyComparisons of EC3-based estimates of the ultimate moment capacity and
74
E. S. Bernard, R. Q. Bridge, G. ]. Hancock
test results are listed in T a b l e s 3 and 4. This c o m p a r i s o n indicates that E u r o c o d e 3/Part 1.3 p r o d u c e s a c o n s e r v a t i v e result with a g o o d level o f c o n s i s t e n c y a m o n g all the s p e c i m e n s . T h e c o n s e r v a t i v e e s t i m a t i o n m o s t likely results f r o m the failure to include the p o s t - b u c k l i n g reserve in the strength o f the stiffener as the b u c k l i n g coefficient X is b a s e d on a c o l u m n design curve. 3.0.3 Australian S t a n d a r d A S 1 5 3 8 T h e Australian Standard A S 1 5 3 8 - 1 9 8 8 9 is similar to the A I S I C o l d - f o r m e d Steel Structures Specification o f 198013 w h e n designing i n t e r m e d i a t e stiffened elements. It uses the effective width c o n c e p t b a s e d on the W i n t e r f o r m u l a to d e t e r m i n e the e f f e c t i v e n e s s o f the c o m p r e s s i o n flange. V a r i o u s reductions in the e f f e c t i v e width o f the flat portions o f the flange are then m a d e . Since A S 1 5 3 8 is an a l l o w a b l e stress code, the strength equation 2.4.1.3(1) incorpor-
TABLE 3
Comparison of V-stiffened Test Results with Mo Calculated by Eurocode 3/Part 1.3 Specimen
ST21A ST21B ST22A ST22B ST23A ST23B IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B IST48A IST410A IST410B IST412B
Mode a
L L L L L L D+L D+L D+L D+L D+L D+L D+L D+L L L L L
M, (test) (kNm)
M, h (EC3) (kNm)
M,,Ec3
3.39 2.97 3.56 3.16 2,65 2,38 3.85 4.00 3-68 3,88 4-59 4.56 4.57 4.54 5.42 5.75 5-14 5-59
2-780 2.791 2.907 2.922 2.151 2-152 3.056 3.258 3-043 3.226 3-917 3-917 3.819 4.007 4.322 4.739 4.254 4-747
1.219 1.064 1.225 1-081 1.232 1,106 1.260 1.228 1.209 1-203 1.173 1-228 1.197 1-133 1.254 1.213 1.208 1.178
mean o-
1-189 0.057
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. o Calculated by EC3/Part 1.3.
Mu .....
Profiled steel decks
75
TABLE 4 Comparison of Flat-hat Stiffened Test Results with Mu Calculated by Eurocode 3/Part 1.3
Specimen
Mode"
M. (test) (kNm)
M~b (EC3) (kNm)
Mu,test M..Ec3
TS3A1 TS3A2 TS3A3 TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
D+D2 D+D2 D+L L+D L+D L+D L+D L+D L
2.96 2.93 2.96 3.52 4.10 3.87 3.63 4-01 4.04
2-468 2.592 2-736 3.010 3.182 3.164 2.952 3.234 3-253
1.199 1.130 1.082 1.169 1-288 1.223 1-230 1.240 1.242
mean cr
1-200 0.064
1.0 kNm = I).73756 kip ft. " L = L o c a l only; L+D=Local then Distortional; D + D 2 = l s t & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by EC3/Part 1.3.
ates a maximum design stress of 0.6Fy. Hence, for unfactored strength calculations the effective width bo is found from eqn 2.4.1.3(3) to be be t -
4::8 ~/h" [ ~--
[1
93-5 ~]~'] (--~-) ~ ]
(32)
where t is the strip thickness, b the actual flat element width, K the plate buckling factor (taken to be 4.0), andfis the yield stress Fy. By Clause 2.4.1.4, the effectiveness of the compressed flange is further reduced if b/t > 60. The width of the fiat element is reduced by (b - 600/10
(33)
and then the area of the intermediate stiffener is reduced to Aef as follows:
(i) For b/t between 60 and 90: (34)
Aef = £gAst
where
Ast is
the actual area of the intermediate stiffener, and
76
E. S. Bernard, R. Q. Bridge, G. J. Hancock
ot=(3-~)-~0(1-~)(~
).
(35)
(ii) For b/t greater than 90:
aef ( )Zst In these equations, the centroid of the stiffener is retained at the centroid of the full area of the stiffener. The Australian Standard uses the concept of a 'minimum effective stiffener' in Clauses 2.4.2.1 and 2.4.2.2. The minimum second moment of a r e a Imin of an edge stiffener about its own centroidal axis is found as
Imin = 1"83t4
~/[(~)2
27,600] Fyy / "
(37)
If the second moment of area of the intermediate stiffener is greater than 2Imin, the stiffener may be included in the section geometry subject to the reductions of Clause 2.4.1.4. If the second moment of area is less than 21min, the stiffener is disregarded and the effective width of the compressed flange calculated as for a flat element of the same width [Clause 2.4.2.2(d)]. Several test specimens proved to have inadequate intermediate stiffeners by this Clause. They are marked with an asterisk in Tables 5 and 6. The effective widths of flanges and area of the intermediate stiffener are then used to determine an effective section modulus Ze. The effectiveness of the webs is determined in Clause 3.4.2. A reduced stress, Fbw, with the factor of safety removed, is calculated by the equation Fbw = F y [ l ' 2 1 - 0"00013 ( ~ ) "~Fy] --< Fy
(38)
where dl is the width of the web, and used in the expression Mu = FbwZe.
(39)
The ultimate moments Mu calculated by AS1538-1988 for all the V and flat-hat stiffened sections are listed in Tables 5 and 6. Comparison with test results indicate conservative values of 1.095 and 1.045 for the mean ratio
Profiled steel decks
77
TABLE 5 Comparison of V-stiffened Test Results with Mu Calculated by AS1538
Specimen
ST21A ST21B ST22A ST22B ST23A ST23B IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B IST48A IST410A IST410B IST412B
Mode ~
L L L L L L D+L D+L D+L D+L D+L D+L D+L D+L L L L L
M~ (test) (kNm) 3.39 2.97 3.56 3.16 2.65 2-38 3.85 4-00 3.68 3.88 4.59 4.56 4.57 4.54 5.42 5.75 5.14 5.59
M~b (AS1538) (kNm) 3-276 3.289 3.400 3.418 2-502 2.504 3.369c 3-352c 3.352 c 3-367c 3.466c 3.391 c 3.408 c 3-427~ 5.549 5.855 5.438 5.825 mean o-
Mu,test Mu~s15~8
1-035 0.903 1.047 0.924 1.059 0.950 1-143 1.193 1.098 1.152 1-324 1.345 1.341 1-325 0.977 0.982 0.945 0.959 1.095 0-154
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. b Calculated by AS1538-1988. c Stiffener inadequate by AS1538. a
Mu,test]MuAs1538. T h e
coefficient o f variation in this ratio was f o u n d to be about 0.16, hence several u n c o n s e r v a t i v e ratios were observed. T h e s e tended to o c c u r for those specimens with large intermediate stiffeners. All the specimens with inadequate intermediate stiffeners p r o d u c e d conservative results. In general, the m e t h o d o f Australian Standard A S 1 5 3 8 was easy to p e r f o r m and p r o d u c e d satisfactory results.
3.1 The application of proposed design methods to trapezoidal sections Several n e w m e t h o d s for the determination o f the ultimate m o m e n t capacity Mu for c o l d - f o r m e d decking panels h a v e been p r o p o s e d ( H a n c o c k et al., 14 Bernard et a/.3"4). C o m p a r i s o n s with the full set o f test results are described below.
E. S. Bernard, R. Q. Bridge, G. 1. Hancock
78
TABLE 6 Comparison of Flat-hat Stiffened Test Results with Mu Calculated by A51538
Specimen
TS3A 1 TS3A2 TS3A3 TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
Mode a
D+D2 D+D2 D+L L+D L+D L+D L+D L+D L
M. (test) (kNm) 2.96 2.93 2.96 3-52 4.10 3.87 3.63 4.01 4-04
M. b (AS1538) (kNm) 2.611 c 2.642 ¢ 2.630 c 4.287 4.210 4.220 2.628 c 4.140 4.140 mean t~
Mu,test Mu~sls3s
1-134 1.109 1.125 0.821 0.974 0-917 1.38 t 0.969 0.976 1.045 0.163
1.0 kNm = 0-73756 kip ft. a L = L o c a l only; L + D = L o c a l then Distortional; D + D 2 = 1st & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by AS 1538-1988. "Stiffener inadequate by AS1538.
3.1.1 Distortional buckling The Winter Formula is commonly used in design procedures to determine an estimate of the ultimate load carrying capacity of plates in compression (e.g. AISI-1991; 7 AS1538-19889). In its usual form it is expressed
~-=
(40)
1 - 0.22
where be is the effective part of the plate width b, o'je is the elastic buckling stress and Fy is the yield stress. Kwon and Hancock 15 proposed a modification of this formula to permit application to the symmetric distortional mode of buckling for columns undergoing essentially uniform compression. This was done in two stages. At first, the elastic local buckling stress (O'le) was replaced by the elastic distortional buckling stress (O'de) to arrive at
be
--= 1 b
)t --< 0.673
(41a)
Profiled steel decks
~-= ~ / ~ y
1 - 0-22
~ --> 0.673
79
(41b)
where
In this form, it was assumed that all plate elements forming the crosssection we,re reduced to effective widths in the same proportions. This was equivalent to
a -¥,~yy
1 -0-22
=~yy
(42)
where A e is the effective part of the gross section area A and Ou is the stress at ultimate load based on the gross area A. In the method described above, O'le and O'de could be derived using either an elastic finite strip 11 or finite element analysis, or could be based on plate buckling coefficients, where available. Kwon and Hancock found that this approach yielded unconservative estimates under certain circumstances and thus produced their 'Design Proposal 2' in which the exponent of the (O'de/Fy) t e r m in eqn (41b) was changed from 0.5 to 0-6 and the 0.22 coefficient increased to 0.25, thus
be
b = 1
b e (O'd~/0"6
o-6)
)t --< 0.561
(43a)
where
As for eqn (41), this reduction of the effective widths was performed for
E. S. Bernard, R. Q. Bridge, G. J. Hancock
80
all the elements of the section and thus the reduction was equivalent to reducing the gross area to an effective area. A similar approach was applied to members in flexure by Bernard e t al. L,3 By direct analogy with eqn (42), the effective section modulus (Ze) was calculated as
=
1 - 0.22
(44)
fy
or alternately from eqn (43),
~e (~#)06
1 - 0.25
(~)o6)
~
(45)
=~
leading to the ultimate moment capacity (46)
M u = F y Z e = o-uZ
where Z is the section modulus of the gross section. This latter method [using eqn (46)] has presently been called the Modified Winter Formula Method. Note that the numerically determined distortional buckling stress Ode is used in these calculations irrespective of the actual mode of buckling seen in the tests. Kwon and Hancock 15 also investigated a Maximum Strength approach for the design of sections with intermediate stiffeners. Their 'Design Proposal 1' used a curve that displayed a good empirical fit to a wide range of test results. The curve is composed of two parts,
36t ~n ~y(1 4~e)
(47a)
F_ o'de > ~ . 2
(47b)
Comparisons of the test results with ultimate moments derived using eqns (45) and (46) are listed in Tables 7 and 8. These tables reveal a satisfactory conservative result for the specimens with small V-shaped intermediate stiffeners which displayed predominantly distortional buckling. For these speci-
Profiled steel decks
81
TABLE 7
Comparison of V-stiffened Test Results with M. Calculated using Modified Winter Formula
Specimen
Mode a
M, (test) (kNm)
M, b Mu..... ( M W F ) M,.MWF (kNm)
Mde
Mle
(kNm)
Mle
(kNm)
Mde
IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B
D+L D+L D+L D+L D+L D+L D+L D+L
3-85 4.00 3.68 3.88 4.59 4-56 4.57 4-54
3.127 3.609 2.986 3.459 4.767 4.175 4.223 4-582 mean cr
1.231 1-108 1.232 1.122 0-963 1.092 1.082 0.991 1-103 0-097
-2.940 --3.597 3-410 3-440 3.550
1.930 2.588 1-780 2.413 3.999 3.771 3-385 4.067
-1.14 --0.90 0.90 1.02 0.87
IST48A IST410A IST410B IST412B
L L L L
5.42 5.75 5.14 5.59
5.881 6.753 5.753 6.671 mean ~r
0.922 0.851 0.972 0.838 0.896 0.063
3.980 4.364 3.878 4.416
6.533 8.952 6.305 8.861
0.61 0.49 0.62 0-50
total mean cr
1-034 0.132
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. b Calculated by Modified Winter Formula. mens, the ratio o f the calculated local buckling m o m e n t to calculated distortional buckling m o m e n t M~/Moe was f o u n d to e x c e e d 0.85. H o w e v e r , an u n c o n s e r v a t i v e result was f o u n d for those specimens with larger stiffeners that displayed local-then-distortional or purely local buckling. T h e s e specimens displayed a M~ffMde ratio less than 0-69. This is to be e x p e c t e d since the m e t h o d was o n l y intended to be applied to specimens that e x p e r i e n c e predominantly distortional buckling. Tables 7 and 8 have therefore been g r o u p e d into two sets o f specimens: those that displayed distortional buckling b e f o r e local buckling or soon after local buckling (upper group) and those that did not (lower group). Note that the numerically d e t e r m i n e d distortional buckling stress O'de has b e e n used for all specimens, including those that only displayed local buckling prior to ultimate failure. T h e distortional buckling stress O'de has b e e n used together with the full section m o d u l u s Z to obtain the distortional buckling m o m e n t Mde. F o r those specimens that did not display a local
E. S. Bernard, R. Q. Bridge, G. J. Hancock
82
TABLE
8
Comparison of Flat-hat Stiffened Test Results with Mu Calculated using Modified Winter Formula
Specimen
Mode a
Mu,test
Mu (test) (kNm)
M, b (MWF) (kNm)
Mu,MWF
TS3A1 TS3A2 TS3A3
D+D2 D+D2 D+L
2.96 2.93 2.96
2.784 2-922 3.821 mean ~r
1.063 1-003 0-775 0.947 0.152
TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
L+D L+D L+D L+D L+D L
3.52 4.10 3.87 3.63 4.01 4.04
4.398 4-976 4.836 4.296 4.767 5.104 mean o,
0-800 0.824 0.800 0.845 0.841 0.792 0.817 0.023
total mean ~r
0.860 0.102
Mle (kNm)
Mde (kNm)
Mt~ Mae
2-296 2.612
1-498 1.598 2-763
-1.44 0.95
2.414 2.367 2-398 2.423 2.374 2.365
3.572 4.709 4:350 3.502 4.344 5.073
0.68 0.50 0.55 0-69 0.55 0.47
--
1.0 kNm = 0.73756 kip ft. a L=Local only; L+D=Local then Distortional; D+D2= 1st & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by Modified Winter Formula. mode of buckling, no value of Mle has been included in Tables 7 and 8. The numerical estimates of Mle and M ~ are based on the mean measured geometry including the central depression o f the stiffener relative to the plane of the flange. The ultimate moment capacity of specimen TS3A3 was somewhat lower than anticipated for those specimens that first displayed distortional buckling. This may have been due to interaction occurring between the local and distortional modes o f buckling, which occurred simultaneously for this specimen. This specimen may have been grouped with specimens TS3A4 through to TS3B6, but has instead been included with TS3A1 and TS3A2 because the distortional mode o f buckling appeared to predominate.
3.1.2 Interaction o f local and distortional buckling A method by P e kr z, 16 called the 'Unified Approach to the Design of Coldformed Steel Members', has previously been used to estimate ultimate collapse
Profiled steel decks
83
loads if an interaction of local and flexural-torsional buckling has occurred. Serrette and P e k t z ~7 also applied the method to the interaction of local and distortional buckling in roof sheeting panels undergoing flexure. This method has been applied to the V and fiat-hat stiffened specimens as follows. Estimates of the elastic local and distortional buckling moments (Mcr~l), Mcr~d)) are determined using an elastic finite strip buckling analysis program (Papangelis and Hancock11). In this formulation, Mere1)and Mcr(d ) are equivalent to Mle and Mde, respectively. A stress ratio m-
mcr(d) Mcr(l)
(48)
is calculated. The elastic distortional buckling stress O'de is found as
O'de--
/~cr(d)
(49)
Z
where Z i,; the section modulus based on the gross section. A nominal stress Fn is then determined depending on the value of m. If m > 12.5, then Fn = Fy; however for this test series m < 2.5 in all cases. Hence Fn is calculated as
Fy
Fn =tra,,
Fn=Fy
1
4~de
O'ae <-- - 2
(50a)
O'de>2 "
(50b)
Once an estimate of F. is determined, it is used in the Winter formula,
(51)
to determine the effective width of the compression flange be accounting for the interaction of both local and distortional buckling. In this case, K = 4 and t is taken as the base metal thickness. The effectiveness of the webs is determined by Clause B2.3 of the AISI Specification as usual. The ultimate moment is then given by
E. S. Bernard, R. Q. Bridge, G. J. Hancock
84
(52)
Mu = F.Ze
where Ze is the effective section modulus incorporating be from eqn (51). The resulting estimates of ultimate moment Mu(unmed~have been listed in Table 9 for the specimens with 'fiat-hat' intermediate stiffeners. The mean ratio of Mu(testy/Mu(unified) w a s found to be 1-543, hence the method is very conservative. The method was not checked for the V-stiffened specimens on account of its poor suitability for the other specimens. It was suggested by Bernard e t a l . 2 that the method by Serrette and PekOz is too conservative because the curve used for Fn [eqn (50)] does not make an allowance for post-buckling. It was suggested that Fn should instead be calculated according to an equation based on test results of specimens that experienced distortional buckling alone, or distortional buckling prior to the onset of local buckling. This has been done using the effective section modulus formula [eqn (44)]. Thus, Fn = Fy
o'a~ -->
022#yt
Fy
(53a)
0.453
(53b)
O'de < 0"453 "
TABLE 9 Comparison of Flat-hat Stiffened Test Results with Mu Calculated by Unified Approach (Pek6z) (with Partially Effective Webs by AISI)
Specimen"
TS3A4 TS3A5 TS3A6 TS 3B4 TS3B5 TS3B6
Mode b
M, (test) (kNm)
M," (Unified) (kNm)
M ...... M,,v,,i¢~,,d
L+D L+D L+D L+D L+D L
3.52 4.10 3.87 3.63 4-01 4.04
2-189 2.929 2.980 1.810 2.447 3.095
1-608 1-400 1.299 2.006 1-639 1.305
mean o-
1.543 0.270
1-0 kNm = 0.73756 kip ft. a No local buckling moment found in numerical study for specimens TS3A1-3. b L = Local only; L+D = Local then Distortional. e Calculated by Unified Approach.
85
Profiled steel decks
Estimate, s o f ultimate m o m e n t s b y this M o d i f i e d E f f e c t i v e Section ( M E S ) M e t h o d are listed in T a b l e s 10 and 11. T h e e f f e c t i v e n e s s o f the w e b s w a s d e t e r m i n e d a c c o r d i n g to the A I S I Specification 7 as P e k 6 z h a d done. T h e m e a n ratios o f Mu~teso/Mu~MES) w e r e f o u n d to b e 1.328 and 1.160 for the t w o sets o f s p e c i m e n s with V stiffeners and flat-hat stiffeners, respectively. T h e ratios calculated b y this M E S m e t h o d a p p e a r to b e o v e r l y c o n s e r v a t i v e for the sections with s m a l l e r stiffeners w h e r e m e a n ratios o f 1.395 and 1.184 w e r e f o u n d for the t w o sets o f s p e c i m e n s . H o w e v e r , the ratios w e r e 1.194 and 1.148 for those s p e c i m e n s with larger stiffeners that d i s p l a y e d only local buckling, or local b u c k l i n g well b e f o r e distortional buckling. It can be c o n c l u d e d that the M o d i f i e d W i n t e r F o r m u l a m e t h o d should b e u s e d for s p e c i m e n s that display distortional buckling, and the M o d i f i e d E f f e c t i v e Section m e t h o d should b e used for s p e c i m e n s that display local buckling, or local b u c k l i n g well b e f o r e TABLE 10
Comparison of V-stiffened Test Results with Mu Calculated by Modified Effective Section Method (with Partially Effective Webs by AISI) Specimen
Mode~
M, (test) (kNm)
M, b (MES) (kNm)
IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B
D+L D+L D+L D+L D+L D+L D+L D+L
3.85 4.00 3-68 3-88 4.59 4.56 4.57 4.54
2.506 2.838 2.445 2-752 3-568 3.376 3.281 2.568 mean tr
IST48A IST410A IST410B IST412B
L L L L
5-42 5.75 5.14 5.59
M ...... M~.Mes
Mle (kNm)
Mde (kNm)
1.536 1-409 1-505 1-410 1.286 1-351 1.393 1.272 1.395 0-094
-2.940 --3.597 3.410 3-440 3.550
1.930 2-588 1.780 2.413 3.999 3.771 3.385 4.067
-1.14 --0.90 0.90 1-02 0.87
4.317 4.945 4.200 4.925 mean tr
1.255 1.163 1.224 1.135 1.194 0-055
3.980 4-364 3-878 4.416
6.533 8-952 6.305 8-861
0.61 0.49 0.62 0.50
total mean o"
1.328 0.127
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. b Calculated by MES Method.
Mle Mae
86
E. S. Bernard, R. Q. Bridge, G. J. Hancock TABLE
lI
Comparison of Flat-hat Stiffened Test Results with Mu Calculated by Modified Effective Section Method (with Partially Effective Webs by AISI) Specimen
TS3A1 TS3A2 TS3A3
TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
Mu (test) (kNm)
M, ~ (MES) (kNm)
Mu,test
Mle
Md~
Mle
Mu.Mes
(kNm)
(kNm)
Male
D+D2 D+D2 D+L
2,96 2.93 2.96
2.229 2-424 2.916 mean o-
1.328 1-209 1.015 1.184 0-158
2.296 2.612
1.498 1.598 2.763
-1.44 0-95
L+D L+D L+D L+D L+D L
3-52 4.10 3.87 3.63 4.01 4.04
3.256 3.486 3.470 3-116 3-341 3.510 mean o-
1.081 1.176 1.115 1-165 1.200 1.151 1.148 0.043
2.414 2-367 2.398 2.423 2.374 2.365
3.572 4.709 4-350 3.502 4.344 5.073
0-68 0-50 0.55 0-69 0.55 0.47
total mean o-
1-160 0.088
Mode a
--
1.0 kNm = 0-73756 kip ft. a L = Local only; L+D = Local then Distortional; D+D2 = 1st & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by MES Method. distortional buckling. The effective division b e t w e e n the two m e t h o d s and m o d e s o f b e h a v i o u r appears to lie near the ratio o f MlJMde equal to 0.85, although this is a provisional value as there is an insufficient a m o u n t o f data for ratios b e t w e e n 0.69 and 0-87. A m e a n ratio o f Mu(tesJMu(design) that is closer to unity would o c c u r for most specimens, based on this procedure. It must be b o r n e in mind that the tested specimens were c o m p o s e d o f 650 M P a steel and therefore the design m e t h o d is at present applicable only to high strength steel.
4 CONCLUSIONS T h e ultimate m o m e n t results f r o m a series o f experimental investigations o f thin-walled profiled steel decking panels have been c o m p a r e d with estimates d e t e r m i n e d using existing design standards and p r o p o s e d design methods. All the m e t h o d s relied on the effective width concept for c o m p r e s s i o n elements
Profiled steel decks
87
that experience buckling prior to ultimate failure. The specimens tested in the experimental investigation were all of a trapezoidal design with a single 'V' or 'fiat-hat' intermediate stiffener in each compression flange. The comparisons revealed different degrees of success in the ability of the existing methods to predict the ultimate moment capacity. The method of Eurocode 3/Part 1.3 proved to give the most consistent results, but which were all conservative by 20%. The predictions of the AISI Cold-formed Steel Structures; Specification and the Australian Standard AS1538 were closer to the test results, but with less consistency than Eurocode 3/Part 1.3. The recently proposed Modified Winter Formula method gave results that were good for sections that experienced predominantly distortional buckling prior to ultimate failure. However, the method was unconservative for sections that displayed local buckling, or local buckling well before distortional buckling. A proposed method by Serrette and Pekrz, called the Unified Approach, which accounts for the interaction of local and distortional buckling modes, proved highly conservative. A recently proposed modification to this Unified Approach, called the Modified Effective Section method, has been used for specimen,; that displayed local buckling, or local buckling well before distortional buckling. This method produced somewhat conservative results that were similar to those obtained using Eurocode 3/Part 1.3, but involved only a fraction as many design calculations. A proposal has been made to combine the Modified Winter Formula method and the Modified Effective Section method into one procedure, with a ratio of local to distortional buckling stresses, ~ ' l l e ] M d e , equal to 0.85, defining the boundary between the two methods and modes of behaviour. ACKNOWLEDGEMENTS This paper 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 E[aterials and Structures. Technical and material support for the experimental programme was provided by Lysaghts Building Industries Pty Ltd. Financial support for laboratory and computer equipment was provided by BHP Research-Melbourne Laboratories Pty Ltd. REFERENCES 1. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Intermediate Stiffeners in Coldformed Profiled Steel Decks Part I--'V' Shaped Stiffeners. Research Report
88
2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
E. S. Bernard, R. Q. Bridge, G. J. Hancock R653, School of Civil and Mining Engineering, University of Sydney, NSW, Australia. Bernard, E. S., Bridge, R. Q. & Hancock, G..I., Intermediate Stiffeners in Coldformed Profiled Steel Decks Part 2--'Flat-hat' Shaped Stiffeners. Research Report R658, School of Civil and Mining Engineering, University of Sydney, NSW, Australia. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Tests on profiled steel decks with V-stiffeners. J. Struct. Engng, ASCE, 119 (1993) 2277-2293. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Tests on profiled steel decks with fiat hat stiffeners. J. Struct. Engng, ASCE, 121 (1995) 1175-1182. Standards Association of Australia, AS1397, Steel Sheet and Strip--Hot-Dipped Zinc Coated or Aluminium/Zinc Coated. Standards Association of Australia, NSW, Australia, 1984. Bernard, E. S. & Coleman, R., Precise Geometric Measurement of Profiled Steel Decks. Research Report R669, School of Civil and Mining Engineering, University of Sydney, NSW, Australia. American Iron and Steel Institute, LRFD Specification for Cold-formed Steel Structural Members. AISI, Washington. Eurocode 3, Design of Steel Structures, Part 1.3. Commission of the European Communities, Brussels, 1992 draft. Standards Association of Australia, AS1538, Cold-formed Steel Structures. Standards Association of Australia, NSW, Australia, 1988. Hancock, G. J., Local, distortional and lateral buckling of I-beams. J. Struct. Div., ASCE, 104(ST11) (1978) 1787-1798. Papangelis, J. P. & Hancock, G. J., Computer analysis of thin-walled structural members. Computers Structures, 56 (1995) 157-176. Winter, G., Stress Distribution in and Equivalent Width of Flanges of Wide, ThinWalled Steel Beams. NACA Technical Note 784, 1940. American Iron and Steel Institute, Specification for the Design of Cold-formed Steel Structural Members. AISI, Washington, 1980. Hancock, G. J., Kwon, Y. B. & Bernard, E. S., Strength design curves for thinwalled sections undergoing distortional buckling. J. Construct. Steel Res., 31 (1994) 169-186. Kwon, Y. B. & Hancock, G. J., Tests of cold-formed channels undergoing local and distortional buckling. J. Struct. Engng, ASCE, 117 July (1992) 1786-1803. Pektiz, T., Development of a unified approach to the design of cold-formed steel members. Eighth Int. Specialty Conf. on Cold-formed Steel Structures. St. Louis, MO, 1986, pp. 77-84. Serrette, R. & Pekrz, T., Behavior of thin-walled sections with laterally unsupported compression flanges. Canadian Soc. of Civil Engineers Conf., May 1991.
J. Construct. Steel Res. Vol. 38, No. 1, pp. 61-88, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0143-974X/96 $15.00 + 0.00
ELSEVIER
Design Methods for Profiled Steel Decks with Intermediate Stiffeners E. Ste,fan Bernard", Russell Q. B r i d g e a & Gregory J. Hancock b aFaculty of Engineering, University of Western Sydney, Nepean, PO Box 52, Kingswood, NSW, Australia, 2747 bCentre for Advanced Structural Engineering, School of Civil and Mining Engineering, University of Sydney, NSW, Australia, 2006 (Received 31 October 1994; revised version received 25 September 1995; accepted 9 February 1996)
ABSTRACT Thin-~alled cold-formed profiled steel decking panels with intermediate stiffeners may exhibit distortional buckling involving transverse flexure of the stiffener. Local buckling in the flat plate elements between the flanges and webs may also occur. Varying degrees of post-buckled strength reserve may be encountered in sections undergoing local and~or distortional buckling. The extent of post-buckled strength reserve, as well as the ultimate moment capacity of the panel, are of interest to designers in their efforts to produce more efficient sections. Ultimate moment results for 27 profiled decking specimens tested in bending are compared with estimates derived using a number of design standards. The standards used were the AISI LRFD Coldformed Steel Structures Specification 1991, Eurocode 3/Part 1.3, and Australian Standard AS1538-1988. Several additional methods for predicting the ultima~!e moment capacity have been proposed recently and these are also compared with the test results. Copyright © 1996 Elsevier Science Ltd.
1 INTRODUCTION Thin-walled cold-formed steel sections have gained increasing acceptance as replaceme,nts for hot-rolled sections in structural applications in the last two decades. Floor decking panels manufactured from thin-gauge, high strength coated steel have featured as part of this growth in popularity. However, coldformed sections are subject to more complex forms of buckling and non-linear behaviour than hot-rolled sections, hence their use has necessitated greater developmental research. Design specifications (AISI, Eurocode, Australian 61
62
E. S. Bernard, R. Q. Bridge, G. ]. Hancock
Standards) governing cold-formed sections are still undergoing rapid development in an attempt to keep up with advances in manufacturing and structural design. A programme of tests of cold-formed floor decking panels has been undertaken in the School of Civil and Mining Engineering at the University of Sydney. Twenty-seven specimens were tested in flexure in a purpose-built test rig to determine the structural behaviour of the decking panels when subject to pure flexural loads. Tangent stiffness, stress distributions, buckling moments and ultimate moments were recorded for comparison with numerical predictions and design estimates. The tests have been described in detail in two reports by Bernard et al. ~,2 and published in Bernard et a/. 3"4 Comparisons with existing design methods are described. Furthermore, improved design methods are proposed and compared with the test results. As a result of differences in notation used in the various standards, notation is the same as the standards referred to and is described for each equation in the accompanying text rather than in a notation list. 2 GEOMETRY OF SECTIONS The test specimens examined in the present investigation consisted of brakepressed panels of a trapezoidal design with a single intermediate stiffener in each compression flange. The intermediate stiffener was in the shape of a 'V' or a 'flat-hat'. The two types of brake-pressed section are illustrated in Figs 1 and 2. The geometry of both the 'V' and 'flat-hat' stiffened deck panels was selected to exhibit both local and distortional buckling under pure bending. The buckling behaviour changed from distortional for small intermediate stiffeners to local for large stiffeners. The buckling modes for the flat-hat stiffened sections are shown in Fig. 3; similar buckling modes, excluding the anti-symmetric distortional mode, occurred in the V-stiffened sections and are described by Bernard et al. 3 Several specimens exhibited both modes of buckling, one occurring at a moment greater than the other. These were expected to show an 'interaction' of buckling waveforms. As depicted in Fig. 1, the intermediate stiffeners in the V-stiffened sections were in the shape of a 'V' with an enclosed angle of 90 ° at the base. As the height S (or depth) of the stiffener increased from 2 to 10 mm, the proportions of the 'V' remained the same. The measured dimension S has been listed in Table 1 for each specimen. The intermediate stiffeners were in the middle of the compression flanges. The full sections consisted of a trapezoidal shape repeated as four longitudinal ribs. Total width of the folded sections was approximately 785 mm while the length was 2000 mm. The system of identification for the 'V'-stiffened specimens has been described by Bernard et al. 1"3
Profiled steel decks I_
89mm
__1
5!I
Specimens with V-Stiffeners (IST)
I--
196
I _ 45- 135ram
55
63
T I
LI
Stiffeners (ST)
0.5
151.5-241.5
J
Fig. 1. Geometry and general dimensions of specimens with V-stiffeners (IST) and specimens with no stiffeners (ST).
The section geometry and intermediate stiffeners in the flat-hat stiffened sections a:re depicted in Fig. 2. The intermediate stiffeners were in the shape of a broad rectangle with webs inclined at 45 ° to the base, hence the name 'fiat hat' stiffeners. As the height S of the stiffener increased from 1 to 6 mm, the proportions of the 'fiat hat' changed. The widest portion b~ was maintained at 40 m m and the web inclination at 45 °, hence the stiffener base b2 decreased in width in inverse proportion to the height. The stiffener dimension S has been listed in Table 2 for each specimen. The intermediate stiffeners were in the middle of each of the compression flanges of a trapezoidal section with three longitudinal ribs. Total width of the folded sections was approximately
64
E. S. Bernard, R. Q. Bridge, G. J. Hancock
I"
551
135
~-0~'5 ~ ' - ~ ~ i I
241.5
~i
Specimenswith i Hat-hatStiffeners I (TS3) I
GeneralGeometry Fig. 2. Geometry and general dimensions of specimens with flat-hat stiffeners (TS3).
(a) Localbucklingsymmetric
/
(b) Distortionalbucklingsymmetric
\ .....
(c) Localbucklingantisymmetric
(d) Distortionalbucklingantisymmetric
BucklingModes Fig. 3. Buckling modes for specimens with flat-hat stiffeners.
Profiled steel decks
65
TABLE 1 Comparison of V-stiffened Specimen Test Results with Mu Calculated by AISI, 1991 Specimen
Stiffener height (mm)
Mode ~
Mu (test) (kNm)
Mub (AISI) (kNm)
Mr,..... Mumtst
ST21A ST21B ST22A ST22B ST23A ST23B IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B IST48A IST410A IST410B IST412B
------2.07 3.05 1.94 2.98 5.59 5.12 5.21 5.55 7.89 9.93 7.83 9.74
L L L L L L D+L D+L D+L D+L D+L D+L D+L D+L L L L L
3.39 2.97 3.56 3.16 2.65 2.38 3.85 4.00 3.68 3.88 4.59 4.56 4.57 4.54 5.42 5.75 5.14 5.59
2.790 2-799 2-924 2-939 2.162 2.163 3-716 3.895 3.843 3-877 4-735 4.410 4.467 4-647 5.814 6-101 5.672 6.059
1.215 1.061 1-218 1.075 1.226 1-100 1-036 1.027 0.958 1-001 0-969 1.034 1-023 0-977 0.932 0-942 0.906 0-923
mean o"
1-035 0.101
1.0 kNm = (I-73756 kip ft. L = Local only (observed in tests); D+L = Distortional then Local. b Calculated by AISI, 1991. a
720 m m while the length was 2000 m m . T h e s y s t e m o f identification for the flat-hat stiffened s p e c i m e n s has been described in B e r n a r d et al. 2"4 T h e stlSLp used in the m a n u f a c t u r e o f the specimens was G r a d e G 5 5 0 coldr e d u c e d coated steel c o n f o r m i n g to Australian Standard A S 1 3 9 7 . 5 The strip was n o m i a a l l y 0.60 m m thick, was c o a t e d with Zincalume, and had a n o m i n a l yield stre,;s o f 550 MPa. The m e a s u r e d base metal thickness was 0-585 m m . Further details regarding aspects o f manufacture, material properties, residual stresses and g e o m e t r i c imperfections are contained in the references described above, as well as the report b y Bernard and C o l e m a n 6 w h i c h gives a detailed study o f g e o m e t r i c imperfections in several o f the V-stiffened specimens.
E. S. Bernard, R. Q. Bridge, G. J. Hancock
66
TABLE 2
Comparison of Flat-hat Stiffened Test Results with Mu Calculated by AISI, 1991 Specimen
TS3A1 TS3A2 TS3A3 TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
Stiffener height (ram)
Mode a
M, (test) (kNm)
M, b (AISI) (kNm)
Mu.tes, MuzlS~
1.25 1.70 2.58 4.433 5.533 5.467 3.808 5.042 6.033
D+D2 D+D2 D+L L+D L+D L+D L+D L+D L
2.96 2-93 2-96 3.52 4.10 3.87 3.63 4-01 4.04
2.919 3.048 3.262 3.746 4.538 4-502 3-583 4.019 4.414
1.014 0.961 0.907 0.939 0-903 0-860 1-013 0.998 0-915
mean cr
0.946 0.055
1.0 kNm = 0.73756 kip ft. a L=Local only; L+D=Local then Distortional; D+D2= 1st & 2nd mode Distortional; D+L = Distortional then Local. o Calculated by AISI, 1991.
3 COMPARISON OF TEST RESULTS WITH EXISTING STANDARDS Estimates of the ultimate moment capacity of the test specimens were determined using several existing or draft design standards. The standards selected were the AISI LRFD Cold-formed Steel Structures Specification7 representing North American practice, Eurocode 3/Part 1.38 representing European practice and Australian Standard AS1538-1988. 9 The estimates have been compared with test results to determine the accuracy of the various methods. In each of the sets of comparisons described below, the dimensions used in the design calculations have been derived from the mean measured geometry of each specimen. The measured base metal thickness has been used as the design thickness in all calculations contained in this report. For the purpose of design comparisons, the measured yield (or 0.2% proof) stress, Fy, equal to 653 MPa, for the strip material has been used in all calculations. Although the stress-strain curve for the strip material was rounded, the 0-2% proof stress was very close to the ultimate stress. The specified elastic modulus, E, has been used instead of the measured value. This has been chosen because it is unusual for a designer to know the actual elastic modulus of the strip used and most standards explicitly specify a value to be used. Estimates of the elastic buckling moments for the mean measured geo-
Profiled steel decks
67
metries have been based on the finite strip elastic buckling analysis program BFINST (Hancock ~°) which forms part of a more recent computer program THIN-WALL (Papangelis and Hancockl~).
3.0.1 AISI Specification 1991 The American Iron and Steel Institute LRFD Cold-formed Steel Structures Specification, 1991 is based on the effective width approach to local buckling of cold-fo]aned sections pioneered by Winter. 12 This is applied to slender regions of a section that experience compressive stress. By Clause B2.1 (see Fig. 4), the effective width b is found as b = w
)t-<0-673
(la)
b = tgw
A>0-673
(lb)
where w i,; the fiat width of the compression element, and
(2) (3)
p = (1.0 - 0.22/A)/A A = (1.052/~/-k) (w/t) (~lf-/-E)
with t the strip thickness, k the plate buckling coefficient, a n d f t h e proof stress Fy when the nominal section strength is calculated and no lateral buckling is encountered. In the AISI Specification, E is taken to be 203.4 x 10 3 MPa. b0
"
/
w
I-
ffffliiiiiiiillllll
_1
_,
I
I
w/ lllllll
iiiiiiilIIIIl-
Stress f
Fig. 4. Effective width approach of AISI Cold-formed Steel Structures Specification CF-1 1991, Clause B2.1 for flange with intermediate stiffeners.
E. S. Bernard, R. Q. Bridge, G. J. Hancock
68
The AISI Specification has additional provisions for uniformly compressed elements with intermediate stiffeners (Clause B4.1). The code has no requirement for an intermediate stiffener to be adequate. However, a continuum of stiffener stiffnesses is treated identically by reducing the cross-section of the stiffener to an effective area As, and reducing the width of the flange to an effective width b. A limiting slenderness S is calculated as S = 1.28 ~/E--~
(4)
where f is the maximum stress (normally Fy) encountered using the effective design width. With the width of the complete compression flange, bo, equal to either 90 or 135 m m for all the test specimens, bo/t >- 3S, hence Case III of Clause B4.1 is used. Thus b = w
A -< 0.673
(5a)
b = pw
)t > 0.673
(5b)
where w is the fiat width of the buckled element, and p = (1-0 - 0.22/)t)/)t
(6)
)t = (1.052/qk) (w/t) (qf-7-E)
(7)
k = 33~[Is/Ia + 1 -- 4 As = a's (Is/Ia) <-- a's Ia = ( 1 2 8 ( b o l t )
-
285)t 4
(8a) (8b) (9)
A~ = full area of the intermediate stiffener As = reduced area of stiffener to be used in computing effective section properties = second moment of area of the intermediate stiffener about its own axis. The webs of most specimens were found to be partially effective according to Clause B2.3. The width of the compressed region of each web is reduced in a similar manner to the flanges. The partial effective widths bl and b2 (see Fig. 5) are determined by Clause B2.3(a) to be b, = bd(3 -
4~)
where ~b =f2/fl with f~ the compression stress at the tip of the web, and
(10)
Profiled steel decks
69 m
(compression)
Fig. 5. Effective web width by AISI Cold-formed Steel Structures Specification CF-1, Clause B2.3 for compression region of web.
b2 = be~2
~b -< - 0 . 2 3 6
(lla)
b2 = be - bl
4, > - 0 - 2 3 6
(llb)
where be is the effective width of the web such that be = yc
)t --< 0.673
(12a)
be = p y c
)t > 0-673
(12b)
where f = f ~ and Yc is the compressed portion of the web (see Fig. 5), and
70
E. S. Bernard, R. Q. Bridge, G. J. Hancock
A = (1-052P[k) (yJt) (~/fl----E)
(13) (14)
k=4+2(1
(15)
p = (1.0 - 0.22/A)/A
- ~b)3 + 2 ( 1 - ~b).
The effective web widths b~ and b2 are substituted for the actual width into the cross-section together with the effective width of the flange and effective area of intermediate stiffener. The section properties are determined, whereupon it is commonly found that the neutral axis changes position. Several iterations of the procedure are normally required to converge to a stable estimate of b e. The effective section modulus Ze is then calculated and the ultimate moment capacity determined as Mu = FyZe.
(16)
The code-based estimates of the ultimate moment capacity Mu are listed in Table 1 for V-stiffened and unstiffened specimens, and Table 2 for flat-hat stiffened specimens. Comparison with test results indicates that the method is slightly unconservative for the flat-hat stiffened sections and the V-stiffened sections with large intermediate stiffeners. The method proved conservative for the sections without stiffeners and the V-stiffened sections with small stiffeners. Those specimens with large intermediate stiffeners that displayed only local buckling prior to ultimate failure proved to have the most unconservative result. In Table 1 and all subsequent tables, the designated modes qf buckling (L for local, D for distortional) were based on empirical observations. 3.0.2 Eurocode 3~Part 1.3 Eurocode 3/Part 1.3 for cold-formed steel structures also uses the effective width concept. This is used for flat plate elements without stiffeners in Clause 3.2. However, in this procedure the plate buckling coefficient k~ is determined by more elaborate means than in the AISI Specification. The equations for determining k~ in Table 3.1 of Eurocode 3/Part 1.3 are a substitute for a rational elastic buckling analysis, Distortional (or stiffener) buckling within compressed flanges with one or two intermediate stiffeners is treated in Clauses 3.3.3 and 3.3.4.2. The intermediate stiffener is treated as a beam on an elastic foundation with restraint against out-of-plane displacement being provided by the flange and web (as well as the stiffness of the stiffener). The critical axial load capacity Ncr,s of the stiffener is calculated as N c r , s -~-
Kw4.2E ~/ld3/[8b 3 (1 + 3bs/(2bp)]
and the critical stress O'cr,s found as
(17)
Profiled steel decks
71
(18)
Orcr,s =/~Tcr, s/A s
where, as shown in Fig. 6, As Is bp bs E Kw
= = = = = =
effective area of stiffener second moment of area of stiffener width of fiat plate elements length of perimeter of stiffener Elastic Modulus (210,000 MPa) restraint coefficient.
The area stiffener As includes the stiffener itself and the effective parts of the flat plate elements on either side of the stiffener as illustrated in Fig. 6. The restraint coefficient Kw is dependent on the restraining action of the web on the flange. It is a function of the interaction of buckling half-wavelengths in both elements. It can conservatively be taken as equal to unity, or else calculated as Kw = K,~.o
lb/Sw
I_
I-
beff,1.1
bo
I
b~ff,1.2
--> 2
(19a)
-~
-I-
beff,2.2
-I
b~ff,2.1
Fig. 6. Effective widths of compressed flange with intermediate stiffeners according to Eurocode 3/Part 1.3. The area of the stiffener, As, includes the adjacent effective portions of the flange, beff,1.2 and beff.2.2.
E. S. Bernard, R. Q. Bridge, G. J. Hancock
72
Kw = Kwo - (Kwo - 1)[21b/Sw -- (lb/Sw) 2]
lb/Sw < 2
(19b)
where Kwo = q(Sw + 2be)/(Sw + bd2)
(20)
Ib = 3-65 4~/Isb3(1 + 3bJ(2bp))/t 3
(21)
Sw = width o f w e b shown in Fig. 7 be = bs + beff, l.2 + beff,2.2 (see Fig. 6). Once O'cr,s has been found, it is used in Clause 5.1.2 to determine the buckling coefficient X
where ~b = 0-511 + a(X~ - 0-2) + X 21
(23)
X s = ~/O'cr, s
(24)
o~ = 0.13. The term oz is an out-of-plane imperfection factor and fy is the yield (or proof) stress. The effective stress within the stiffener is then found as
ec
/il/.~ ' ~
Fig. 7, Effective width of compressed region of web according to Clause 3.3.4.4 of Eurocode 3/Part 1.3.
Profiled steel decks O'com ~- ) ( L
73 (25)
and it is used to determine a new estimate of the effective widths beff, l.2, beff,2.2 of the plate elements forming the effective stiffener as shown in Fig. 6. Note that bCff,L1 and beff,2.1 (the edges of the flange adjacent to the webs) are maintained at their initial values determined taking O'com=fy. Each of bell,L1, beff,/.~ and beff,2. 2 a r e calculated as boff/2, where beef = p bp
(26)
in which bp is the full width of the flange flat element and p = 1.0
hp - 0.673
(27a)
p = (1.0 - 0.22/,~p)/)tp
Xp > 0"673
(27b)
where hp
=
O'com =
1.052(bp/t)
~O'com[(Eko.)
(28)
X fy
k~ = plate buckling coefficient found from Table 3.1 of EC3.
(29)
This is an iterative procedure that requires at least two iterations to obtain a convergent estimate of O'comand thus beef. Estimates of the effective width of the webs are determined by Clause 3.3.4.4 for sheeting in bending. The effective widths of the webs are calculated after the flange calculations but are otherwise independent. A pair of equations is provided for the estimation of the effective parts of the web Seff,, = 0.76t ~/~fy
(30)
Serf,2 = l'5Seff,1
(31)
where Seff I and Self, 2 a r e as indicated in Fig. 7. These effective widths are used in the seclion geometry together with the effective flange widths and stiffener area to anSve at an effective section modulus Ze. In the determination of Ze, the reduction in O'comfrom fy is taken into account by a proportional reduction in the thickness of the effective intermediate stiffener. The ultimate moment capacity is then found as Mu = ZefyComparisons of EC3-based estimates of the ultimate moment capacity and
74
E. S. Bernard, R. Q. Bridge, G. ]. Hancock
test results are listed in T a b l e s 3 and 4. This c o m p a r i s o n indicates that E u r o c o d e 3/Part 1.3 p r o d u c e s a c o n s e r v a t i v e result with a g o o d level o f c o n s i s t e n c y a m o n g all the s p e c i m e n s . T h e c o n s e r v a t i v e e s t i m a t i o n m o s t likely results f r o m the failure to include the p o s t - b u c k l i n g reserve in the strength o f the stiffener as the b u c k l i n g coefficient X is b a s e d on a c o l u m n design curve. 3.0.3 Australian S t a n d a r d A S 1 5 3 8 T h e Australian Standard A S 1 5 3 8 - 1 9 8 8 9 is similar to the A I S I C o l d - f o r m e d Steel Structures Specification o f 198013 w h e n designing i n t e r m e d i a t e stiffened elements. It uses the effective width c o n c e p t b a s e d on the W i n t e r f o r m u l a to d e t e r m i n e the e f f e c t i v e n e s s o f the c o m p r e s s i o n flange. V a r i o u s reductions in the e f f e c t i v e width o f the flat portions o f the flange are then m a d e . Since A S 1 5 3 8 is an a l l o w a b l e stress code, the strength equation 2.4.1.3(1) incorpor-
TABLE 3
Comparison of V-stiffened Test Results with Mo Calculated by Eurocode 3/Part 1.3 Specimen
ST21A ST21B ST22A ST22B ST23A ST23B IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B IST48A IST410A IST410B IST412B
Mode a
L L L L L L D+L D+L D+L D+L D+L D+L D+L D+L L L L L
M, (test) (kNm)
M, h (EC3) (kNm)
M,,Ec3
3.39 2.97 3.56 3.16 2,65 2,38 3.85 4.00 3-68 3,88 4-59 4.56 4.57 4.54 5.42 5.75 5-14 5-59
2-780 2.791 2.907 2.922 2.151 2-152 3.056 3.258 3-043 3.226 3-917 3-917 3.819 4.007 4.322 4.739 4.254 4-747
1.219 1.064 1.225 1-081 1.232 1,106 1.260 1.228 1.209 1-203 1.173 1-228 1.197 1-133 1.254 1.213 1.208 1.178
mean o-
1-189 0.057
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. o Calculated by EC3/Part 1.3.
Mu .....
Profiled steel decks
75
TABLE 4 Comparison of Flat-hat Stiffened Test Results with Mu Calculated by Eurocode 3/Part 1.3
Specimen
Mode"
M. (test) (kNm)
M~b (EC3) (kNm)
Mu,test M..Ec3
TS3A1 TS3A2 TS3A3 TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
D+D2 D+D2 D+L L+D L+D L+D L+D L+D L
2.96 2.93 2.96 3.52 4.10 3.87 3.63 4-01 4.04
2-468 2.592 2-736 3.010 3.182 3.164 2.952 3.234 3-253
1.199 1.130 1.082 1.169 1-288 1.223 1-230 1.240 1.242
mean cr
1-200 0.064
1.0 kNm = I).73756 kip ft. " L = L o c a l only; L+D=Local then Distortional; D + D 2 = l s t & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by EC3/Part 1.3.
ates a maximum design stress of 0.6Fy. Hence, for unfactored strength calculations the effective width bo is found from eqn 2.4.1.3(3) to be be t -
4::8 ~/h" [ ~--
[1
93-5 ~]~'] (--~-) ~ ]
(32)
where t is the strip thickness, b the actual flat element width, K the plate buckling factor (taken to be 4.0), andfis the yield stress Fy. By Clause 2.4.1.4, the effectiveness of the compressed flange is further reduced if b/t > 60. The width of the fiat element is reduced by (b - 600/10
(33)
and then the area of the intermediate stiffener is reduced to Aef as follows:
(i) For b/t between 60 and 90: (34)
Aef = £gAst
where
Ast is
the actual area of the intermediate stiffener, and
76
E. S. Bernard, R. Q. Bridge, G. J. Hancock
ot=(3-~)-~0(1-~)(~
).
(35)
(ii) For b/t greater than 90:
aef ( )Zst In these equations, the centroid of the stiffener is retained at the centroid of the full area of the stiffener. The Australian Standard uses the concept of a 'minimum effective stiffener' in Clauses 2.4.2.1 and 2.4.2.2. The minimum second moment of a r e a Imin of an edge stiffener about its own centroidal axis is found as
Imin = 1"83t4
~/[(~)2
27,600] Fyy / "
(37)
If the second moment of area of the intermediate stiffener is greater than 2Imin, the stiffener may be included in the section geometry subject to the reductions of Clause 2.4.1.4. If the second moment of area is less than 21min, the stiffener is disregarded and the effective width of the compressed flange calculated as for a flat element of the same width [Clause 2.4.2.2(d)]. Several test specimens proved to have inadequate intermediate stiffeners by this Clause. They are marked with an asterisk in Tables 5 and 6. The effective widths of flanges and area of the intermediate stiffener are then used to determine an effective section modulus Ze. The effectiveness of the webs is determined in Clause 3.4.2. A reduced stress, Fbw, with the factor of safety removed, is calculated by the equation Fbw = F y [ l ' 2 1 - 0"00013 ( ~ ) "~Fy] --< Fy
(38)
where dl is the width of the web, and used in the expression Mu = FbwZe.
(39)
The ultimate moments Mu calculated by AS1538-1988 for all the V and flat-hat stiffened sections are listed in Tables 5 and 6. Comparison with test results indicate conservative values of 1.095 and 1.045 for the mean ratio
Profiled steel decks
77
TABLE 5 Comparison of V-stiffened Test Results with Mu Calculated by AS1538
Specimen
ST21A ST21B ST22A ST22B ST23A ST23B IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B IST48A IST410A IST410B IST412B
Mode ~
L L L L L L D+L D+L D+L D+L D+L D+L D+L D+L L L L L
M~ (test) (kNm) 3.39 2.97 3.56 3.16 2.65 2-38 3.85 4-00 3.68 3.88 4.59 4.56 4.57 4.54 5.42 5.75 5.14 5.59
M~b (AS1538) (kNm) 3-276 3.289 3.400 3.418 2-502 2.504 3.369c 3-352c 3.352 c 3-367c 3.466c 3.391 c 3.408 c 3-427~ 5.549 5.855 5.438 5.825 mean o-
Mu,test Mu~s15~8
1-035 0.903 1.047 0.924 1.059 0.950 1-143 1.193 1.098 1.152 1-324 1.345 1.341 1-325 0.977 0.982 0.945 0.959 1.095 0-154
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. b Calculated by AS1538-1988. c Stiffener inadequate by AS1538. a
Mu,test]MuAs1538. T h e
coefficient o f variation in this ratio was f o u n d to be about 0.16, hence several u n c o n s e r v a t i v e ratios were observed. T h e s e tended to o c c u r for those specimens with large intermediate stiffeners. All the specimens with inadequate intermediate stiffeners p r o d u c e d conservative results. In general, the m e t h o d o f Australian Standard A S 1 5 3 8 was easy to p e r f o r m and p r o d u c e d satisfactory results.
3.1 The application of proposed design methods to trapezoidal sections Several n e w m e t h o d s for the determination o f the ultimate m o m e n t capacity Mu for c o l d - f o r m e d decking panels h a v e been p r o p o s e d ( H a n c o c k et al., 14 Bernard et a/.3"4). C o m p a r i s o n s with the full set o f test results are described below.
E. S. Bernard, R. Q. Bridge, G. 1. Hancock
78
TABLE 6 Comparison of Flat-hat Stiffened Test Results with Mu Calculated by A51538
Specimen
TS3A 1 TS3A2 TS3A3 TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
Mode a
D+D2 D+D2 D+L L+D L+D L+D L+D L+D L
M. (test) (kNm) 2.96 2.93 2.96 3-52 4.10 3.87 3.63 4.01 4-04
M. b (AS1538) (kNm) 2.611 c 2.642 ¢ 2.630 c 4.287 4.210 4.220 2.628 c 4.140 4.140 mean t~
Mu,test Mu~sls3s
1-134 1.109 1.125 0.821 0.974 0-917 1.38 t 0.969 0.976 1.045 0.163
1.0 kNm = 0-73756 kip ft. a L = L o c a l only; L + D = L o c a l then Distortional; D + D 2 = 1st & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by AS 1538-1988. "Stiffener inadequate by AS1538.
3.1.1 Distortional buckling The Winter Formula is commonly used in design procedures to determine an estimate of the ultimate load carrying capacity of plates in compression (e.g. AISI-1991; 7 AS1538-19889). In its usual form it is expressed
~-=
(40)
1 - 0.22
where be is the effective part of the plate width b, o'je is the elastic buckling stress and Fy is the yield stress. Kwon and Hancock 15 proposed a modification of this formula to permit application to the symmetric distortional mode of buckling for columns undergoing essentially uniform compression. This was done in two stages. At first, the elastic local buckling stress (O'le) was replaced by the elastic distortional buckling stress (O'de) to arrive at
be
--= 1 b
)t --< 0.673
(41a)
Profiled steel decks
~-= ~ / ~ y
1 - 0-22
~ --> 0.673
79
(41b)
where
In this form, it was assumed that all plate elements forming the crosssection we,re reduced to effective widths in the same proportions. This was equivalent to
a -¥,~yy
1 -0-22
=~yy
(42)
where A e is the effective part of the gross section area A and Ou is the stress at ultimate load based on the gross area A. In the method described above, O'le and O'de could be derived using either an elastic finite strip 11 or finite element analysis, or could be based on plate buckling coefficients, where available. Kwon and Hancock found that this approach yielded unconservative estimates under certain circumstances and thus produced their 'Design Proposal 2' in which the exponent of the (O'de/Fy) t e r m in eqn (41b) was changed from 0.5 to 0-6 and the 0.22 coefficient increased to 0.25, thus
be
b = 1
b e (O'd~/0"6
o-6)
)t --< 0.561
(43a)
where
As for eqn (41), this reduction of the effective widths was performed for
E. S. Bernard, R. Q. Bridge, G. J. Hancock
80
all the elements of the section and thus the reduction was equivalent to reducing the gross area to an effective area. A similar approach was applied to members in flexure by Bernard e t al. L,3 By direct analogy with eqn (42), the effective section modulus (Ze) was calculated as
=
1 - 0.22
(44)
fy
or alternately from eqn (43),
~e (~#)06
1 - 0.25
(~)o6)
~
(45)
=~
leading to the ultimate moment capacity (46)
M u = F y Z e = o-uZ
where Z is the section modulus of the gross section. This latter method [using eqn (46)] has presently been called the Modified Winter Formula Method. Note that the numerically determined distortional buckling stress Ode is used in these calculations irrespective of the actual mode of buckling seen in the tests. Kwon and Hancock 15 also investigated a Maximum Strength approach for the design of sections with intermediate stiffeners. Their 'Design Proposal 1' used a curve that displayed a good empirical fit to a wide range of test results. The curve is composed of two parts,
36t ~n ~y(1 4~e)
(47a)
F_ o'de > ~ . 2
(47b)
Comparisons of the test results with ultimate moments derived using eqns (45) and (46) are listed in Tables 7 and 8. These tables reveal a satisfactory conservative result for the specimens with small V-shaped intermediate stiffeners which displayed predominantly distortional buckling. For these speci-
Profiled steel decks
81
TABLE 7
Comparison of V-stiffened Test Results with M. Calculated using Modified Winter Formula
Specimen
Mode a
M, (test) (kNm)
M, b Mu..... ( M W F ) M,.MWF (kNm)
Mde
Mle
(kNm)
Mle
(kNm)
Mde
IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B
D+L D+L D+L D+L D+L D+L D+L D+L
3-85 4.00 3.68 3.88 4.59 4-56 4.57 4-54
3.127 3.609 2.986 3.459 4.767 4.175 4.223 4-582 mean cr
1.231 1-108 1.232 1.122 0-963 1.092 1.082 0.991 1-103 0-097
-2.940 --3.597 3-410 3-440 3.550
1.930 2.588 1-780 2.413 3.999 3.771 3-385 4.067
-1.14 --0.90 0.90 1.02 0.87
IST48A IST410A IST410B IST412B
L L L L
5.42 5.75 5.14 5.59
5.881 6.753 5.753 6.671 mean ~r
0.922 0.851 0.972 0.838 0.896 0.063
3.980 4.364 3.878 4.416
6.533 8.952 6.305 8.861
0.61 0.49 0.62 0-50
total mean cr
1-034 0.132
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. b Calculated by Modified Winter Formula. mens, the ratio o f the calculated local buckling m o m e n t to calculated distortional buckling m o m e n t M~/Moe was f o u n d to e x c e e d 0.85. H o w e v e r , an u n c o n s e r v a t i v e result was f o u n d for those specimens with larger stiffeners that displayed local-then-distortional or purely local buckling. T h e s e specimens displayed a M~ffMde ratio less than 0-69. This is to be e x p e c t e d since the m e t h o d was o n l y intended to be applied to specimens that e x p e r i e n c e predominantly distortional buckling. Tables 7 and 8 have therefore been g r o u p e d into two sets o f specimens: those that displayed distortional buckling b e f o r e local buckling or soon after local buckling (upper group) and those that did not (lower group). Note that the numerically d e t e r m i n e d distortional buckling stress O'de has b e e n used for all specimens, including those that only displayed local buckling prior to ultimate failure. T h e distortional buckling stress O'de has b e e n used together with the full section m o d u l u s Z to obtain the distortional buckling m o m e n t Mde. F o r those specimens that did not display a local
E. S. Bernard, R. Q. Bridge, G. J. Hancock
82
TABLE
8
Comparison of Flat-hat Stiffened Test Results with Mu Calculated using Modified Winter Formula
Specimen
Mode a
Mu,test
Mu (test) (kNm)
M, b (MWF) (kNm)
Mu,MWF
TS3A1 TS3A2 TS3A3
D+D2 D+D2 D+L
2.96 2.93 2.96
2.784 2-922 3.821 mean ~r
1.063 1-003 0-775 0.947 0.152
TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
L+D L+D L+D L+D L+D L
3.52 4.10 3.87 3.63 4.01 4.04
4.398 4-976 4.836 4.296 4.767 5.104 mean o,
0-800 0.824 0.800 0.845 0.841 0.792 0.817 0.023
total mean ~r
0.860 0.102
Mle (kNm)
Mde (kNm)
Mt~ Mae
2-296 2.612
1-498 1.598 2-763
-1.44 0.95
2.414 2.367 2-398 2.423 2.374 2.365
3.572 4.709 4:350 3.502 4.344 5.073
0.68 0.50 0.55 0-69 0.55 0.47
--
1.0 kNm = 0.73756 kip ft. a L=Local only; L+D=Local then Distortional; D+D2= 1st & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by Modified Winter Formula. mode of buckling, no value of Mle has been included in Tables 7 and 8. The numerical estimates of Mle and M ~ are based on the mean measured geometry including the central depression o f the stiffener relative to the plane of the flange. The ultimate moment capacity of specimen TS3A3 was somewhat lower than anticipated for those specimens that first displayed distortional buckling. This may have been due to interaction occurring between the local and distortional modes o f buckling, which occurred simultaneously for this specimen. This specimen may have been grouped with specimens TS3A4 through to TS3B6, but has instead been included with TS3A1 and TS3A2 because the distortional mode o f buckling appeared to predominate.
3.1.2 Interaction o f local and distortional buckling A method by P e kr z, 16 called the 'Unified Approach to the Design of Coldformed Steel Members', has previously been used to estimate ultimate collapse
Profiled steel decks
83
loads if an interaction of local and flexural-torsional buckling has occurred. Serrette and P e k t z ~7 also applied the method to the interaction of local and distortional buckling in roof sheeting panels undergoing flexure. This method has been applied to the V and fiat-hat stiffened specimens as follows. Estimates of the elastic local and distortional buckling moments (Mcr~l), Mcr~d)) are determined using an elastic finite strip buckling analysis program (Papangelis and Hancock11). In this formulation, Mere1)and Mcr(d ) are equivalent to Mle and Mde, respectively. A stress ratio m-
mcr(d) Mcr(l)
(48)
is calculated. The elastic distortional buckling stress O'de is found as
O'de--
/~cr(d)
(49)
Z
where Z i,; the section modulus based on the gross section. A nominal stress Fn is then determined depending on the value of m. If m > 12.5, then Fn = Fy; however for this test series m < 2.5 in all cases. Hence Fn is calculated as
Fy
Fn =tra,,
Fn=Fy
1
4~de
O'ae <-- - 2
(50a)
O'de>2 "
(50b)
Once an estimate of F. is determined, it is used in the Winter formula,
(51)
to determine the effective width of the compression flange be accounting for the interaction of both local and distortional buckling. In this case, K = 4 and t is taken as the base metal thickness. The effectiveness of the webs is determined by Clause B2.3 of the AISI Specification as usual. The ultimate moment is then given by
E. S. Bernard, R. Q. Bridge, G. J. Hancock
84
(52)
Mu = F.Ze
where Ze is the effective section modulus incorporating be from eqn (51). The resulting estimates of ultimate moment Mu(unmed~have been listed in Table 9 for the specimens with 'fiat-hat' intermediate stiffeners. The mean ratio of Mu(testy/Mu(unified) w a s found to be 1-543, hence the method is very conservative. The method was not checked for the V-stiffened specimens on account of its poor suitability for the other specimens. It was suggested by Bernard e t a l . 2 that the method by Serrette and PekOz is too conservative because the curve used for Fn [eqn (50)] does not make an allowance for post-buckling. It was suggested that Fn should instead be calculated according to an equation based on test results of specimens that experienced distortional buckling alone, or distortional buckling prior to the onset of local buckling. This has been done using the effective section modulus formula [eqn (44)]. Thus, Fn = Fy
o'a~ -->
022#yt
Fy
(53a)
0.453
(53b)
O'de < 0"453 "
TABLE 9 Comparison of Flat-hat Stiffened Test Results with Mu Calculated by Unified Approach (Pek6z) (with Partially Effective Webs by AISI)
Specimen"
TS3A4 TS3A5 TS3A6 TS 3B4 TS3B5 TS3B6
Mode b
M, (test) (kNm)
M," (Unified) (kNm)
M ...... M,,v,,i¢~,,d
L+D L+D L+D L+D L+D L
3.52 4.10 3.87 3.63 4-01 4.04
2-189 2.929 2.980 1.810 2.447 3.095
1-608 1-400 1.299 2.006 1-639 1.305
mean o-
1.543 0.270
1-0 kNm = 0.73756 kip ft. a No local buckling moment found in numerical study for specimens TS3A1-3. b L = Local only; L+D = Local then Distortional. e Calculated by Unified Approach.
85
Profiled steel decks
Estimate, s o f ultimate m o m e n t s b y this M o d i f i e d E f f e c t i v e Section ( M E S ) M e t h o d are listed in T a b l e s 10 and 11. T h e e f f e c t i v e n e s s o f the w e b s w a s d e t e r m i n e d a c c o r d i n g to the A I S I Specification 7 as P e k 6 z h a d done. T h e m e a n ratios o f Mu~teso/Mu~MES) w e r e f o u n d to b e 1.328 and 1.160 for the t w o sets o f s p e c i m e n s with V stiffeners and flat-hat stiffeners, respectively. T h e ratios calculated b y this M E S m e t h o d a p p e a r to b e o v e r l y c o n s e r v a t i v e for the sections with s m a l l e r stiffeners w h e r e m e a n ratios o f 1.395 and 1.184 w e r e f o u n d for the t w o sets o f s p e c i m e n s . H o w e v e r , the ratios w e r e 1.194 and 1.148 for those s p e c i m e n s with larger stiffeners that d i s p l a y e d only local buckling, or local b u c k l i n g well b e f o r e distortional buckling. It can be c o n c l u d e d that the M o d i f i e d W i n t e r F o r m u l a m e t h o d should b e u s e d for s p e c i m e n s that display distortional buckling, and the M o d i f i e d E f f e c t i v e Section m e t h o d should b e used for s p e c i m e n s that display local buckling, or local b u c k l i n g well b e f o r e TABLE 10
Comparison of V-stiffened Test Results with Mu Calculated by Modified Effective Section Method (with Partially Effective Webs by AISI) Specimen
Mode~
M, (test) (kNm)
M, b (MES) (kNm)
IST43A IST44A IST44B IST45B IST46A IST47A IST47B IST48B
D+L D+L D+L D+L D+L D+L D+L D+L
3.85 4.00 3-68 3-88 4.59 4.56 4.57 4.54
2.506 2.838 2.445 2-752 3-568 3.376 3.281 2.568 mean tr
IST48A IST410A IST410B IST412B
L L L L
5-42 5.75 5.14 5.59
M ...... M~.Mes
Mle (kNm)
Mde (kNm)
1.536 1-409 1-505 1-410 1.286 1-351 1.393 1.272 1.395 0-094
-2.940 --3.597 3.410 3-440 3.550
1.930 2-588 1.780 2.413 3.999 3.771 3.385 4.067
-1.14 --0.90 0.90 1-02 0.87
4.317 4.945 4.200 4.925 mean tr
1.255 1.163 1.224 1.135 1.194 0-055
3.980 4-364 3-878 4.416
6.533 8-952 6.305 8-861
0.61 0.49 0.62 0.50
total mean o"
1.328 0.127
1.0 kNm = 0.73756 kip ft. L = Local only; D+L = Distortional then Local. b Calculated by MES Method.
Mle Mae
86
E. S. Bernard, R. Q. Bridge, G. J. Hancock TABLE
lI
Comparison of Flat-hat Stiffened Test Results with Mu Calculated by Modified Effective Section Method (with Partially Effective Webs by AISI) Specimen
TS3A1 TS3A2 TS3A3
TS3A4 TS3A5 TS3A6 TS3B4 TS3B5 TS3B6
Mu (test) (kNm)
M, ~ (MES) (kNm)
Mu,test
Mle
Md~
Mle
Mu.Mes
(kNm)
(kNm)
Male
D+D2 D+D2 D+L
2,96 2.93 2.96
2.229 2-424 2.916 mean o-
1.328 1-209 1.015 1.184 0-158
2.296 2.612
1.498 1.598 2.763
-1.44 0-95
L+D L+D L+D L+D L+D L
3-52 4.10 3.87 3.63 4.01 4.04
3.256 3.486 3.470 3-116 3-341 3.510 mean o-
1.081 1.176 1.115 1-165 1.200 1.151 1.148 0.043
2.414 2-367 2.398 2.423 2.374 2.365
3.572 4.709 4-350 3.502 4.344 5.073
0-68 0-50 0.55 0-69 0.55 0.47
total mean o-
1-160 0.088
Mode a
--
1.0 kNm = 0-73756 kip ft. a L = Local only; L+D = Local then Distortional; D+D2 = 1st & 2nd mode Distortional; D+L = Distortional then Local. b Calculated by MES Method. distortional buckling. The effective division b e t w e e n the two m e t h o d s and m o d e s o f b e h a v i o u r appears to lie near the ratio o f MlJMde equal to 0.85, although this is a provisional value as there is an insufficient a m o u n t o f data for ratios b e t w e e n 0.69 and 0-87. A m e a n ratio o f Mu(tesJMu(design) that is closer to unity would o c c u r for most specimens, based on this procedure. It must be b o r n e in mind that the tested specimens were c o m p o s e d o f 650 M P a steel and therefore the design m e t h o d is at present applicable only to high strength steel.
4 CONCLUSIONS T h e ultimate m o m e n t results f r o m a series o f experimental investigations o f thin-walled profiled steel decking panels have been c o m p a r e d with estimates d e t e r m i n e d using existing design standards and p r o p o s e d design methods. All the m e t h o d s relied on the effective width concept for c o m p r e s s i o n elements
Profiled steel decks
87
that experience buckling prior to ultimate failure. The specimens tested in the experimental investigation were all of a trapezoidal design with a single 'V' or 'fiat-hat' intermediate stiffener in each compression flange. The comparisons revealed different degrees of success in the ability of the existing methods to predict the ultimate moment capacity. The method of Eurocode 3/Part 1.3 proved to give the most consistent results, but which were all conservative by 20%. The predictions of the AISI Cold-formed Steel Structures; Specification and the Australian Standard AS1538 were closer to the test results, but with less consistency than Eurocode 3/Part 1.3. The recently proposed Modified Winter Formula method gave results that were good for sections that experienced predominantly distortional buckling prior to ultimate failure. However, the method was unconservative for sections that displayed local buckling, or local buckling well before distortional buckling. A proposed method by Serrette and Pekrz, called the Unified Approach, which accounts for the interaction of local and distortional buckling modes, proved highly conservative. A recently proposed modification to this Unified Approach, called the Modified Effective Section method, has been used for specimen,; that displayed local buckling, or local buckling well before distortional buckling. This method produced somewhat conservative results that were similar to those obtained using Eurocode 3/Part 1.3, but involved only a fraction as many design calculations. A proposal has been made to combine the Modified Winter Formula method and the Modified Effective Section method into one procedure, with a ratio of local to distortional buckling stresses, ~ ' l l e ] M d e , equal to 0.85, defining the boundary between the two methods and modes of behaviour. ACKNOWLEDGEMENTS This paper 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 E[aterials and Structures. Technical and material support for the experimental programme was provided by Lysaghts Building Industries Pty Ltd. Financial support for laboratory and computer equipment was provided by BHP Research-Melbourne Laboratories Pty Ltd. REFERENCES 1. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Intermediate Stiffeners in Coldformed Profiled Steel Decks Part I--'V' Shaped Stiffeners. Research Report
88
2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
E. S. Bernard, R. Q. Bridge, G. J. Hancock R653, School of Civil and Mining Engineering, University of Sydney, NSW, Australia. Bernard, E. S., Bridge, R. Q. & Hancock, G..I., Intermediate Stiffeners in Coldformed Profiled Steel Decks Part 2--'Flat-hat' Shaped Stiffeners. Research Report R658, School of Civil and Mining Engineering, University of Sydney, NSW, Australia. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Tests on profiled steel decks with V-stiffeners. J. Struct. Engng, ASCE, 119 (1993) 2277-2293. Bernard, E. S., Bridge, R. Q. & Hancock, G. J., Tests on profiled steel decks with fiat hat stiffeners. J. Struct. Engng, ASCE, 121 (1995) 1175-1182. Standards Association of Australia, AS1397, Steel Sheet and Strip--Hot-Dipped Zinc Coated or Aluminium/Zinc Coated. Standards Association of Australia, NSW, Australia, 1984. Bernard, E. S. & Coleman, R., Precise Geometric Measurement of Profiled Steel Decks. Research Report R669, School of Civil and Mining Engineering, University of Sydney, NSW, Australia. American Iron and Steel Institute, LRFD Specification for Cold-formed Steel Structural Members. AISI, Washington. Eurocode 3, Design of Steel Structures, Part 1.3. Commission of the European Communities, Brussels, 1992 draft. Standards Association of Australia, AS1538, Cold-formed Steel Structures. Standards Association of Australia, NSW, Australia, 1988. Hancock, G. J., Local, distortional and lateral buckling of I-beams. J. Struct. Div., ASCE, 104(ST11) (1978) 1787-1798. Papangelis, J. P. & Hancock, G. J., Computer analysis of thin-walled structural members. Computers Structures, 56 (1995) 157-176. Winter, G., Stress Distribution in and Equivalent Width of Flanges of Wide, ThinWalled Steel Beams. NACA Technical Note 784, 1940. American Iron and Steel Institute, Specification for the Design of Cold-formed Steel Structural Members. AISI, Washington, 1980. Hancock, G. J., Kwon, Y. B. & Bernard, E. S., Strength design curves for thinwalled sections undergoing distortional buckling. J. Construct. Steel Res., 31 (1994) 169-186. Kwon, Y. B. & Hancock, G. J., Tests of cold-formed channels undergoing local and distortional buckling. J. Struct. Engng, ASCE, 117 July (1992) 1786-1803. Pektiz, T., Development of a unified approach to the design of cold-formed steel members. Eighth Int. Specialty Conf. on Cold-formed Steel Structures. St. Louis, MO, 1986, pp. 77-84. Serrette, R. & Pekrz, T., Behavior of thin-walled sections with laterally unsupported compression flanges. Canadian Soc. of Civil Engineers Conf., May 1991.