-
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
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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
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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)
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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)
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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
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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
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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,
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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 = --~%
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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