J Construct Steel Research 9 (1988) 195-216

Stability of Tapered I-Beams

M. A. Bradford

Department of Cwli Engmeenng, The Unlvers,ty of New South Wales, Kensmgton, NSW 2033, Austraha

(Recetved 29 Aprd 1987, revised version rece,ved 31 October 1987, accepted 2 November 1987)

SYNOPSIS

The Brmsh and Austrahan hmtt states design rules for the lateral buckhng hmlt state of tapered I-beams are revtewed A fimte element method ts descrtbed, and thts ts used to develop soluttons for the elasttc crtttcal loads of beams whtch cover an extenstve range of geometrtes and loading condtttons A destgn method ts proposed whtch makes use of the accurate elasttc crtttcal soluttons The code methods and the accurate proposal are compared by an example

NOTATION

The geometrical parameters are shown in Fig 1 Other pnncipal notation is as below E, G Young's modulus of elasticity and shear modulus respectively Fr Yield stress Iy, I~ Minor axis second moment of area and warping constant

respectively J Torsion constant K Beam parameter L Beam length ME Modified elastic critical moment Mob Elastic critical moment of tapered beam Moo Critical moment in uniform bending Mp Full plastic moment n Taper coefficient in BS 5950

195

J Construct Steel Research 0143-974X/88/$03 50 1988 Elsevier Apphed Soence Publishers Ltd, England Pnnted m Great Bntam

196 M A Bradford

~m

O~s

~st

7M, 3'. E

'r/LT

/~LT

Moment modification factor Slenderness reduction factor Taper coefficient in AS 1250 Moment gradient parameter Dimensionless critical moments and loads respectively Load height parameter Perry coeffioent in BS 5950 Equivalent slenderness in BS 5950

1 INTRODUCTION

Tapered I-beams fabricated by welding, such as that shown in Fig 1, have become a viable alternative to uniform beams because of the reduced costs of fabricating plated steel members The advantage of using a tapered beam instead of a uniform beam is that the member may be used In situations where the major axas bending moment vanes along the length of the beam, so that economy can be gained by reducing the member section In the regions of low bending moment Non-uniform I-beams may be tapered m their depth, or in their flange width, but rarely in their flange or web thicknesses

If a tapered beam does not have sufficient lateral stiffness or lateral support to allow its cross-sectional strength to be reached (which for

A

t d

. . . . . . . ~_ TT

B ElevcLtlon A - A .6.] A

l_ L _1 F" - I

PLa.n

Fig. 1. Dmaenslons of tapered beam

Stabthty of tapered I-beams 197

compact sections is the full plastic moment), then the strength of the beam IS governed by its resistance to flexural-torslonal buckling. However, significant economies in steel can still be achieved if the elastic critical load can be determined for the tapered beam This paper is concerned with design against such instability of tapered I-beams

A detailed review of research on the lateral stablhty of tapered I-beams prior to the early 1970s has been given by Kltlpornchal and Trahair Contributions in the field since then have included those by Lee, Morrell and Ketter, 2 Nethercot, 3 Prawel, Morrell and Lee, 4 Home, Shakir-Khahl and Akhtar, 5 Salter, Anderson and May, 6 Brown 7 and Shioml and Kurata 8 However, while several other papers on the lateral stability of tapered beams may be cited as well, there have been few general approaches to the problem contributed 9

In this paper, the proposals of the British and Australian limit states design (LSD) codes for the lnstablhty ILmit state of tapered I-beams are briefly reviewed A general finite element method, 1 suited to micro- computer apphcatlons, is then summansed Parametric solutions for the elastic lateral buckling of tapered beams, determined from the finite element program, are then given, and these are presented as an accurate alternatwe to the proposals In the LSD codes Finally, a design proposal is presented, and this is illustrated by an example

2 LSD CODE RULES

In the British Standard BS 5950,11 tapered doubly symmetnc 1-beams are designed by a modification of the rules for pnsmatlc members The elastic critical moment ME IS used as a basis, and is calculated from

Mp "B "2 E ME = X2TF Y (1)

in which Mp is the full plastic moment of the section at the point where the factored applied moment IS greatest, and where hLT IS the 'equivalent slenderness' In BS 5950, the equivalent slenderness is calculated by modifying the beam slendemess ~, = Or, where r is the radius of gyration at the point of maximum applied moment and / is the effective length, by

~kLT = nvh (2)

In eqn (2), n is a coefficient related to the degree of tapenng, given by

n = 15-05Rf>10 (3)

198 M A Bradford

where Rf is the ratio of the flange area at the point of minimum moment to that at the point of maximum moment, and is always equal to unity when the flange does not taper The coefficient v in eqn (2) is a slenderness factor, which is related to h and to the torsional Index x by

v = [1 + (h/x)2/20] -u4 (4)

where

x = 0 566dx/(A/J) ~ D/T (5)

in which D is the overall beam depth, and where A and J are the area and torsion constant of the member at the point of maxnnum moment respectively

Finally, the elastic buckling moment ME IS related to the design strength Mb of the tapered beam by use of the Perry equation 12

ME Mp Mb = 6a + X/(6za - Mz Me) (6)

where

Mp + (r/LT + 1)ME 6B = 2 (7)

m which the Perry coefficient T~L T for fabncated sections is given by

tiLT = 0 005 6X/(~r2E/Fv) (8)

However, calculation of the resistance Mb of tapered beams in the BS 59501s not as difficult as eqns (1) to (8) would suggest, since most of these relationships are tabulated In fact, for a given slenderness ratio h, only the quantities h/x (eqn (5)) and n (eqn (3)) need to be calculated when the tables in BS 5950 are used

The draft Australian,Standard AS 125013 provides a somewhat more accurate rule than that of BS 5950 to account for the effects of section tapering in determining the design strength Mb Th~s IS again based on the elastic critical moment ME, which IS expressed as

ME = astMo (9)

where

Mo = x/[(rfl Ely/12)(GJ + 7r2 EIJ12)] (10)

Stablhty of tapered 1-beams 199

is the elastic critical load of a prismatic beam of effective length l, and where

:,0_0611_ (06+04m0 /m ] (11) In this equation, Am and Ac are the flange areas and Dr. and Dc are the section depths at the minimum section and at the cntlcal section where the ratio of the bending moment to the full plastic moment is greatest Equation (11) was designed to be an approximation to the limited results of Kltlpornchal and Trahalr, 1 and its basis is Illustrated in Fig 2,4 The minor axss second moment of area Iy, torsion constant J and warping constant I~ in eqn (10) should also be determined from the section properties at the critical section The effect of as, in reducing the elastic critical moment In eqn (9) Is slmllar to the reduction l /n 2 in the elastic critical moment afforded by the BS 5950

The design strength Mb is then obtained from ME as

Mb = CtmoqMp (12)

where the slenderness reduction factor t~. is given by

oq = 0 6{[(Mp/ME) 2 + 3] '/2 - Mp/ME} (13)

10

08

~06 b. C O ,3 U ~ 04 (X:

02

0 0

E,,,

D Depth T~perl~ W)d.th To.pered

o Th,ckness Tapered

I I I I I 02 04 06 05 I0

(0 6 "0 t., D,~ I D,: ) A, , IA

Fig. 2. Basis for AS 1250 rule

200 M A Bradford

and is analogous to eqns (6) to (8) above of the BS 5950 The moment distribution factor O~ m m eqn (12) is tabulated m the AS 1250, and may conservattvely be taken as unity, that latter approxmaat~on bemg the same as m the BS 5950

The AS 1250 also permits 'design by buckhng analysls',~2 m which eqns (12) and (13) are used, with the elastic cnUcal moment ME replaced by

ME = Mob~am (14)

where Mob ts the elastic crlUcal moment determined using the results of an elastic buckhng analysis that takes account of the support, restramt and loadmg condttlons, and of the tapenng of the member The results of such an analysm are gwen in Section 4 of this paper, and are dmcussed subsequently The moment mo&ficatlon factor am may agam be taken as unity in design by buckhng analysts, or as the values tabulated m AS 1250

The use of eqns (12) to (14) for tapered members tmphes that the mter- actton between elastic buckhng and yielding that determines the strength of non-prismatic members ~s the same as the interaction that governs the strength of prismatic members In the absence of sufficient relevant tests, the vahdlty of these equations for tapered beams was mvesugated herem by comparing the predictions of design by buckhng analysis to AS 1250 with the results of a materially and geometrically nonhnear analysis of web tapered beams with a moment at one end undertaken by Shloml and Kurata 8 These results are compared in Fig 3 for web taper constants aw in the range 0 4 to 0 7 The predlcttons of eqns (12) to (14) in Fig 3 are for O1~ m = 1 0 and

10

t -

?

t5 0

Eq 13 (et,,= 1 O) .Eq 13 (ot,~=1 75)

. . . . . . . . -~ - - - - \ \

~ \ ~Etc~st,c

1 I i I 05 10 15 20

Mot:hf~t~l, S ter~ l .e rness ~/Ma(t)/Me~,tt)

Fig 3. Strength predictions for tapered beams

Stabdtty o~ tapered 1-beams 201

am = 1 75 (which IS the approximation for the loading considered,12), with Mob(t) and Mp0) being the values of Mob and Mp at the largest section. It can be seen that the use of eqn (13) with am = 1 0 IS a conservative lower bound prediction of Shlomi and Kurata's results, while the use of this equation with am = 1 75 IS a conservative lower bound prediction of the mean results Because the strengths to AS 1250 are reduced In design by a capacity reduction factor of 0 9, which accounts in part for the scatter of the accurate results,iS it appears that the method of design by buckling analysis in the AS 1250 is suitable for tapered beams, provided that the elastic critical moment can be determined accurately

3 F INITE ELEMENT BUCKLING ANALYSIS

The use of finite elements to solve lateral buckling problems dates back to the work of Barsoum and Gallagher 16 and Powell and Kllnger 17 In 1970 The one-dimensional line elements developed by these researchers assumed the coincidence of the axis of twist with the shear centre axis which is parallel to the centroidal axis Uniform elements similar to these have been used by Nethercot 3 to study the flexural-torslonal buckling of tapered beams

However, application of uniform elements to approximate a tapered beam causes difficulties because of the artificial discontinuities introduced at the centroidal and shear centre axes at nodes In addition, the rate of convergence is very slow 1 because of the comparatively crude model provided for representing a tapered element In order to overcome these difficulties, a one-dimensional finite element has been developed 1 to provide an accurate and rapidly converging method of representing a tapered member, and which does not introduce any artificial discontinuities This element has been validated by comparisons with more complex, but less general, numerical treatments

The finite element method described in Reference 10 is superior to the use of uniform elements, In that it correctly caters for the effects of non- uniformity This is achieved by abandoning the usual shear centre and centroldal axis systems In the development of the line element The element uses a convenient and arbitrary Cartesian axis system passing through the mid-height of the web.as the reference axis for lateral displacements and twists The stiffness and stability matnces are easily calculated by making this assumption of an arbitrary axis of twist

The convergence of the finite element analysis which uses tapered elements has been demonstrated in Reference 10, where it was shown that very few elements are required to obtain an accurate solution Because of this, the global banded stiffness and stablhty matnces are of modest size,

202 M A Bradford

and a microcomputer may be used to achieve a rapid solution of the problem The elgenvalue routines given by Hancock TM are suitable for extracting the buckhng load and mode from these global matrices, and were employed m the study

4 PARAMETRIC STUDY

4.1 General

The finite element method ~ discussed in the previous section has been used to calculate the elastic critical loads or moments of tapered doubly symmetric I-beams for use in design Solutions are given for a beam with flange or web taper with concentrated end moments, and for a beam with flange or web taper acted upon by a uniformly distributed load

The differential equations for buckling derived by Kitlpornchat and Trahalr 1 Indicate that the beam parameter K is an Independent variable, where

77" K = --~%

Stabthty of tapered l-beams 203

8L .... , . .1 x~, ~, , / J

OI I I 0 ! 2 3

K

(a)

14

12

10

11.

6

I !

4

0 L 0 1 2

(b)

Fig. 4. Beam with end moments (a)/3 = - 1, (b)B = -0 5, (c)/3 = O, (d)/3 = 0 5, (e)/3 = 1

204 M A Brad]oral

20

16

12

&

i !

0 L I 0 1 2

K

(c)

2&

24

20

16

12

gf.= !

p:+05

I

0 25 ~.

OL 0

I I

K

(d) Fig 4.--contd

Stabthty of tapered/-beams 205

28 ! 1

"1M

24

2(3

12

OI 0

~. M ~M ~

~=1

Otf= 1

~w= 1

J f / f f

/ / J

I

I 1 K

(e) Fig. 4.--contd

~'~'~"~ 0 25 t I 2 3

those due to increasing web taper are much less Also of interest is the observation that for stocky beams the elastic cnt~cal moment is higher for the/3 = 0 5 loading case than for the/3 = 1 0 loading case (the latter bemg the safest loading condmon for umform beams~2), and that this trend increases as the taper constants af and aw decrease

The use of Figs 4a to 4e represents an accurate alternative to the codified design approaches for tapered beams with end moments, since many more parameters are treated than m eqns (3) and (11)

4.3 Beam with uniformly distributed load

The lateral stabdlty of the tapered beam loaded by a umformly distributed load w shown m Fig 5 has been studied For this, the distributed load is assumed to act at a distance ~ below the web mid-height For an molated simply supported determinate beam, the end moment parameter fl ~s zero, and values of the dimensionless elastic cnucal load

~/w = wL3/X/(Ely GJ(l)) (17)

206 M A Bradford

[..~A

el .d,= B '0wL2112 ~ "~wL=II2 el d,elfO

1 Ete~bon

l_ L 5 r - I

Pto.n

Fig. 5. Tapered beam w=th UDL

et,B

T A-A

are shown m the form of design curves m F~gs 6a to 6c as functions of K and the dimensionless load height parameter

/(H,) E - - -~ ~ GJ (18)

120

100

80

6O

/.(3

20

I I

'~' ~ '~ 0 25 ~

" - _ - ~ - _ ~ ~2g- -~-~

=05

0 1 I 0 1 2

K

Ca)

Fig. 6. SS beamwl thUDL (a) E = 0 5, (b) = 0, (c) e = -0 5

Stabthn oJ tapered i-beam~ 207

On the other hand, for 'continuous' beams with the end moment parameter t3 being taken as umty, the corresponding plots of the dimensionless elastic crmcal load are shown m Figs 7a to 7c In both Fags 6 and 7, the sohd hnes are for c~f = 1 with o~w varying, whde the dashed hnes are for ~ = 1 with c~f varying

It can be seen from the figures that placing the load above the web mid-height (~ < 0) results m a significant destabdlslng effect and reduces the buckhng load, whilst placing the load below the web mid-height tends to stabdlse the beam against lateral buckhng The reduction m 3~w below that

IO0 '- otf=l i

F 20 _._~2 . . . . . . . . . . . . . . ---- E=OU zo

0 0 l 2 3

(b)

5O

60

~,,

40

2C

0 0

l I

cC,t =1

~,,,= 1

\(3 ~ '

0 25 -

L I I 2 3

K

(c) Fig. 6 --contd

208 M A BradJord

I.,

K~=I

4O0

300 - ~-~

/ / /

/ 200 - / /

/ / 025_ . . . .

/

I00

~ ~ =0 5

0 I I 0 1 2 3

(a)

300

25O

200

t00

50

_ nr f=|

( - - -') J .S .5 :~ ..

-

~'=0

i 1 ! 2 3

K

(b)

F]g. 7. BI beamw]th UDL (a) = 0 5, (b) E = O, (c) e = -0 5

140

Stablht, o] tapered I-beams

I I

209

I .

120

I00

80

40

~,f =1

Or,,= 1

f f

f

/ /

20

OL 0

=-0 5

1 Z 3 K

(c) Fig. 7 --contd

for the corresponding uniform beam, expressed as a ratio, is shown in Fig 8 for the beam with fl = 0 loaded at the centrold The figure demonstrates that increasing the degree of flange taper reduces the ratio of the resistance of the tapered beam to that of the corresponding umform beam The reduction in the lateral buckling resistance for web tapered beams is less dramatic, however, with web tapenng having little effect for the more slender beams In all cases, the reductions in the lateral buckhng resistance below that of the corresponding umform beam increases as the beam parameter K increases and the beam becomes more stocky

As for the previous sub-sections, the elastic solutions in Figs 6 and 7 are based on a rational analysis, and are therefore of higher accuracy than the code approximations which are extrapolations from results of limited scope

5 DESIGN PROPOSAL

The resistance of tapered beams may be calculated by the previously discussed method of design by buckling analysis from the accurate elastic

210 M A Bradlold

I0

08

06

. j J "0 cO4

n-

02-

0 0

I I I l

~. ~ I ~"

~ 0 5 ~. . . . .~s ~,

\ \ \ \ \

Ft~n9e taper a~

Web t(1per R .

W

I~=0

=0

I I 1 [ 02 04 06 08

To l~r constant ~. IK.~

Fig 8 Reduction m elastic buckhng load due to tapering

buckhng design curves presented in the previous section The proposal advocated here ~s essentmlly that of the AS 1250 LSD code, and ~s also apphcable, with minor modification, to the design formulation of the BS 5950

F~rstly, the design curves presented hereto are used to calculate the elastic lateral buckhng moment Mob at the crmcal section, that Is, the section where the ratio of the moment resulting from the factored load effects to the plastic moment ~s greatest The crmcal moment Mob includes the effects of off-shear centre loading and non-uniform moment d~stnbut~on Secondly, the elastic critical moment Mobo IS obtained from the figures for shear centre loading and incorporating non-umform moments, and the moment d~stnbut~on

StabdlW o] tapered 1-beams 2 l 1

tactor a m then calculated from

Om = Mobo/Moo (19)

where the elastic critical moment Moo for uniform bending and centroldal loading is obtained from the design curves in Fig 4a

Finally, the design resistance Mb at the critical section Is obtained from eqns (12) and (13), with am determined from eqn (19) above and with ME determined from eqn (14) The phdosophy of using this approach for relating the elastlcal critical moment to inelastic buckling and strength is discussed more fully m Reference 12 The method may also be apphed tentatively to design in accordance with the BS 5950, with ME determined as above and with eqn (6) modified to

am ME Mp Mb = ch~ + \ (cb~ - MEMp) (20)

so as to conform with the strength curve In the BS 5950, where (hB and "OLv are given m eqns (7) and (8) The use ofeqn (20) is somewhat more rational than eqns (12) and (13) for fabricated tapered members, because it allows for an empirical adjustment of the Perry coefficient "0ev to make the theoretical predictions fit test results more closely

6 DESIGN EXAMPLE

Prob lem

Calculate the bending resistance Mb of the tapered beam shown m Fig 9 by

(1) the proposed design method, (n) the method of the AS 1250 LSD code,

(Ill) the method of BS 5950

Solut ton (1) Assume the factored moment M* = 10 000 kNm

At the larger end M*/Mp = 10 000/3457 = 2 89 At the smaller end M*/Mp = 0 5 10 000/753 = 5 18

The smaller end is therefore critical

100007r / / /20010364301013 ) K - ~ \- 7-~ -~ x T0x ~ 37--~-0->~ ~-~6

=210

From Fig 4b, yM = 69

212 M A Brad]ord

~25

V/ / / / / /A

V / / / / / /A ~, 350 _1_1

1 ?&6 '10 '~ ram"

3 730 ,,106 mm 4

mme 6 430 I01~

2 455 "10 ~ mm 2

&5 3 mm

3457 kNm

|y

J

A

ry

Mp

E =200,103 MPo. G =76 92,103MPo.

Fy = 275 MPa

25

V,~/ / / / /A T

1/ / / /1 I- 3so J

- I

~ .=0 25

I 786 106 men 4

3652 106 mm 4

4 019 " 10 lz mme

1 915 " I0 ~" mm 2

96 6 rnm

753 kNm

Fig 9 Design example

At the larger end

69 Mob =

10 000 \ (200 x 103 1 786 X 10 s X 76 92 x 103 3 730 X 10 ) Nmm

= 2209 kNm

Hence Mob at the smal ler end = 0 5 x 2209 = 1104 kNm From Fig 4a, YM = 5 0, thus

OL m = 6 9/5 0 = 1 38

Us ing the AS 1250 st rength curve,

ME = 1104/1 38 = 800 kNm

o~, = 0 6 [[(753/800) 2 + 3] t/2 - 753/800} = 0 618

Stabdttv o] tapered l-beam~

Hence

Mb = 1 380618753 = 642kNm

Thus at the larger end,

Mb = 642/0 5 = 1284 kNm

Us ing the BS 5950 st rength curve,

T]LT = 0 005 6\ (71-2 X 2(X) x 103/275) = 0 474

753 + (1 + 0 474) x 800 d~B = = 966 kNm

2

Hence

mb =

213

1 38 x 800 x 753 = 539 kNm

966 + ,~ (9662 - 800 x 753)

Thus at the larger end,

Mb = 539/0 5 = 1079 kNm

This result IS 6% lower than the Austra l ian predict ion (1284 kNm) based on the accurate curves The reduct ion ~s due pr imari ly to the d i f ferent fo rms o f eqns (6) and (13)

(n) S ince the mlmmum sect ion is the critical sect ion, Dm= De, Am = Ac, so thatast = 1 0

Hence

Mo = \ [(T/"2 X200X 103 X 1 786X 108/100002)(76 92X 103 X3 652X 106)

+ 7r: X 200 X 103 X 4 019 X 1012/10 0002)] Nmm

= 1129 kNrn

so that

ME = 1 0x1129 = 1129kNm

and

as = 0 6{[(753/1129) 2 + 3] 1/2 - 753/1129} = 0 713

For /3 = - 0 5, AS 1250 predicts am = 1 30, so that

Mb = 1300713753 = 698kNm

214 M A Brad]ord

Thus at the larger end,

Mb = 698/0 5 = 1396kNm

This result is 9% unconservat lve when compared with the more accurate Aust rahan solutaon (1284 kNm) based on the design curves

(Ill) Using the approx imate British method,

~t = 10000/85 3 = 117

n = 1 0 since there is no flange tapering

r = (1200 + 25)/25 = 49

= [1 + (117/49)2/20] -~/4 = 0 940

Hence

)tcx = 1 00940x117 = 110

3457 7r 2 200 x 10 ~ ME = 1102 275 Nmm = 2051 kNm

~cT = 0 474 asbetore

3457 + (1 + 0 474) x 2051 d)B =

2 = 3240 kNm

2051 3457 Mb 1394 kNm

3240 + \ '(3240 z - 2051 3457)

This result is 29% unconservat lve when compared w~th the more accurate Brit ish solut ion (1079 kNm) based on the design curves

7 CONCLUSIONS

The new Aust rahan and Brmsh hmit states steel codes provide tor the buckhng resistance of tapered I -beams fabricated by welding These provis ions are based on limited analyses of only a few geometr ical and loading condmons , and are therefore approx imate

A finite e lement method of analysis statable for studying the lateral buckhng of tapered I-section beam-co lumns is described briefly This

Stabdtty o] tapered l-beams 215

method has been validated elsewhere, where it was shown to be accurate and to converge rapidly The formulation Is particularly stated to micro- computer apphcat~ons

The finite element method has been used to derwe accurate elastic buckhng resistances for tapered doubly symmetric I-beams loaded by end moments or by a uniformly dmtnbuted load A method of design is proposed, based on inelastic buckhng, which transforms the accurate elastic solutions into member strengths An example is given, and this demon- strates the inaccuracies of the LSD code approximations, particularly that of the BS 5950 The example illustrates that little additional effort is required to use the accurate deslgn curves, than is needed in present design to the Australian or British codes

ACKNOWLEDGEMENT

The comments of Professor N S Trahair of the Umverslty of Sydney, Australia, following a review of the manuscript, are appreciated

REFERENCES

1 Kitlpornchal, S and Trahalr, N S , Elastic stabdlty of tapered I-beams, J Struct Dtv, ASCE, 98, No ST3 (1972) 713-28

2 Lee, G C , Morrell, M L and Ketter, R L , Allowable Stress/or Web Tapered Beams, Bulletin No 192, Welding Research Councd, 1974

3 Nethercot, D A , The effectwe length of cantdevers as governed by lateral buckhng, The Structural Engmeer, 57, No 5 (1973) 161-8

4 Prawel, S P , Morrell, M L and Lee, G C , Bending and buckhng strength of tapered structural members, Welding Journal, 53, No 2 (1974) 75s-84s

5 Horne, M R , Shaklr-Khahl, H and Akhtar, S , The stabdity of tapered and haunched beams, Proc Insn Ctv Engrs, 67, Part 2 (Sept 1979) 677-94

6 Salter, J B , Anderson, D and May, I M, Tests on tapered steel columns, The Structural Engineer, 58A, No 6 (1980) 18%93

7 Brown, T G , Lateral torsional buckhng of tapered I-beams, J Struct Dry, ASCE, 107, No ST4 (1981) 68%97

8 Shlomi, H and Kurata, M, Strength formula for tapered beam-columns, J Struct Engng, ASCE, 110, No 7 (1984) 1630--43

9 Structural Stability Research Councd, Gutde to Stabthty Design Crtterta for Metal Structures, New York, John Wdey and Sons, 1976

10 Cuk, P E and Bradford, M A , Lateral buckling of tapered monosymmetnc I-beams, J Struct Engng, ASCE (1988) (m press)

11 British Standards Institution, Structural Use of Steelwork tn Bulldtng, BS 5950 Part 1, London, BSI, 1985

12 Trahair, N S , The Behavlour and Design of Steel Structures, London, Chapman and Hall, 1977

216 M A Brad]oral

13 Standards Association of Austraha, Draft Ltmtt State Steel Structure~ Code, A S 1250, Sydney, SAA, 1987

14 Bradford, M A , Bridge, R Q , Hancock, G J , Rotter, J M andTrahalr, N S , Austrahan hmtt state destgn rules for the stablhty o steel structures, International Conference on Steel and Alummmm Structures, Cardiff, UK, 1987

15 Pham, L , Bridge, R Q and Bradford, M A , Cahbratlon of the proposed hmlt states destgn rules for steel beams and columns, Ctvtl Engineering Transacuons, lnstttutton o~ Engineers, Austraha, CE28, No 3 (1986) 268-74

16 Barsoum, R S and Gailagher, R H , Flmte element analysis of torsional and torstonal-flexural stabthty problems, lnternattonal Journal ~or Numerical Methods In Engineering, 2 (1970) 335-52

17 Powell, G and Khnger, R , Elastic lateral buckhng of steel beams, J Struct Dlv , ASCE, 96, No ST9 (1970) 1630--43

18 Hancock, G J , Structural buckhng and vibration analyses on micro- computers, Ctvll Engineering Transacttons, Institution of Engineers, Austraha, CE26, No 4 (1984) 327-32