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Buckling of built up compression members in the plane of the connectors
M U R R A Y T E M P L E N D G H A D A L M A H D Y
Depa rfmenf of C ivil and Environmental Engineering Universify of Windsor Windsor
ON N B 3P4
Canada
Received July 24, 1992
Revised manuscript accepted March 2, 1993
An examination of the requirements for the design of built-up compression members in the North American and
European standards and specifications reveals a great variation in the allowable maximum slenderness ratio for an
individual main m ember , and also in the determinat ion of an equivalent s lenderness rat io. Th e requirements of the
Canad ian s tandard with regard t o the determinat ion of the maxim um al lowable slenderness rat io of a main member
between points of connection can be a bit confusing.
This research involved a s tudy of mo del bui l t-up members th at buckled a bou t an axis perpendicular to the plane
of the connectors. Twenty-four tests were conducted on model built-up members. The theoretical analysis consisted
of a f inite element analysis of the model built-up struts. In addition, an equivalent slenderness ratio was calculated
by several methods. These equivalent slenderness ratios were then used in conjunction with the requirements of the
Canadian standard to calculate a compressive resistance, which was compared with the experimental failure load.
From this research on built-up members that buckle about an axis perpendicular to the plane of the connectors it
was found that at least two connectors should be used, that the slenderness ratio of the main member between points
of con nection has a significant effect o n th e compressive resistance, a nd th at Timoshenko's equivalent slenderness ratio
when used in conjunction with the Ca nad ian stan dard gives results that a re in the best agreemen t with the experimental
results.
Key words:
battens, built-up members, compressive loads, connectors, equivalent slenderness ratio.
Un examen des exigences relatives a la conception des elements composes cornprimes contenues dans les normes
et les spkcifications europeennes et nord-amk ricaines permet d e constater un g rand e cart en ce qui concerne la determ i-
nation d e l 'klancement maximal adm issible pou r un ClCment principal et la determina tion d e I'elancement equivalent.
Les exigences de la no rme can adienn e en ce qui concerne la determ ination de 1'Clancement maximal admissible d'un
element principal entre les points de raccordement peuvent Cgalement pr ter a confusion.
Cette recherche incluait 1'Ctude d'elkments com poses qui subissent un flambe ment d ans un axe perpendiculaire au
plan des dispositifs d'assemblage. Vingt-quatre essais de mod tles d' tlCment com pose on t kt6 realises. La p artie theori-
que com portait une analyse par la mktho de des elements finis de mo dtles d7CtrCsilloncompose. De plus , l ' elancement
equivalent a kt6 calcu li selon plusieurs mkthodes. Ces elancements on t ensuite kt6 combines aux exigences de la norme
canadienne pour calculer une rksistance
a
la compression , qui a fait I 'objet par la suite d'un e com paraison av ec la
charge experimentale ultime.
Cette ttu de des ClCments composes q ui subissent un flam beme nt dan s un axe perpendiculaire a u plan de s dispositifs
d'assemblage a
permis de co nstater la nkcessite de recourir
a
deux dispositifs d'assemblage ainsi que I ' importance de
I 'effet de l 'elancement d e I'element principal ent re les points de raccorde ment sur la resistance
a
la compression. Les
auteurs ont en outre observe que l ' elancement equivalent de Timoshen ko co mbine aux exigences de la norm e cana-
dienne donnait des resultats qui correspondaient davantage aux rksultats experimentaux.
Mots clPs
:
latte, elements composes, charges en compression, dispositifs d'assemblage, elancement equivalent.
[Traduit par la r idact ion]
Can J . Civ Eng 20,
895-909
1993)
Introduction
Built-up compression members are used in structural
engineering for bridge and building colum ns, an d as bracing
and truss members. These built-up compression members
are composed of two o r more structural sections connected
by transverse members which can be batten plates, lacing
bars, o r perforated plates. The func tion of these transverse
members is to make the built-up member act as an integral
unit, to hold the main members apart so that a larger
moment of inertia is achieved, and to fo rm the shear con-
nection between the main mem bers. In this paper, built-up
members composed of two main members connected by
plates welded to the main members are studied. The trans-
verse plates welded to the m ain mem bers are often referred
to as connectors. Typical built-up members are illustrated
in Fig. 1.
NOTE:
Written discussion of this paper is welcomed and will be
received by the Editor until April 30, 1994 (address inside
front cover).
Prinlcd In Canada
Imprime
u
Canada
Built-up comp ression mem bers can be considered as eithe
simple or b uilt-up stru ts, depending o n the plane of bending
If buckling occurs abo ut the axis parallel to the connectors
the
X
axis in Fig. la , the connectors simply move with the
main members. The connectors maintain the separation
between the main members, provide rotational restraint to
the individual main mem bers, but transfer little or no forces
Thus this type of stru t may be referred t o as a simple strut
O n the other ha nd, if buckling occurs about the axis per
pendicular to the connectors, the Yaxis in Fig. la , the con
nectors deform and the effect of the shearing forces tha
occur in the built-up mem ber canno t be neglected. Th e con
sideration of the effect of shear results in the use of an
equivalent slenderness ratio. A n equivalent slenderness ratio
is an imaginary slenderness ratio used t o calculate the buck-
ling load of a built-up member when buckling involves a
relative deformation of the connectors. This type of stru
is considered to be a built-up stru t and will be considered
in this paper.
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b )
FIG.
1
Typical built-up members.
In this report , the requirements , as contained in several
s tandard s and specif ications, for the m aximum slenderness
ratio of the individual main mem bers between points of con-
nection an d the equivalent slenderness ratio of th e built-up
member are considered. Theoretical and experimental results
are presented. T his paper concludes with recom mendations
for the design of built-up compression members.
In this research, tests were carried out on welded model
built-up specimens. M odel strut specimens were used rather
tha n full-sized stru ts in orde r to have better co ntrol over the
fabrication and testing of each specimen so that more
accurate results could be obtained. As the purpose of the
research did not include a study of torsional-flexural buckl-
ing on the behaviour of built-up columns, rectangular mem-
bers were used for the main m embers . Figure 2 shows the
cross sections of the specimens.
Furthe r research with regard to the conn ection of built-up
members is planned. A study of the existing test results of
full-scale members, augmented, if necessary, with further
tests, is planned to help clarify, or modify, the various
requirements in Clause 19 of the Ca nadian s tandard (CSA
1989) that deal with the connection requirements of built-up
members . A study is also planned to determ ine the connec-
tion requirements of built-up members that buckle about
an axis parallel to the connectors .
The theoretical analysis consisted of a finite element anal-
ysis of the model b uilt-up struts. An equivalent slenderness
ratio was also calculated by several meth ods. T hese equiva-
lent slenderness ratios were then used to de termin e the com-
pressive resistance of the model built-up struts.
Th us the purpos e of this research is to examine the various
clauses of the Ca nadian s tandard (CSA 1989) that deal with
the connection of double members to form built-up com-
pression members , and to recommend changes to these
requirements so that the com pressive resistance of built-up
mem bers that buckle abou t an axis perpendicular to the con-
nectors can be predicted with greater accuracy.
FIG.2. Cross sections of built-up test specimens: a) zer
separat ion;
b)4.02
mm separat ion;
c)
7.85 mm separat ion.
tandards and specifications
Several steel standards and specifications, including th
Canadian , German, and Br i t i sh s tandards , and th
Ame rican specif ications, w ere examined in order t o deter
mine the requirements for built-up compression members
It was fou nd that there is a great variation in th e specif ie
maximum slenderness ratio of an individual m ain membe
between points of conn ection , and in the specified equivalen
slenderness ratio. Several examples of eq uivalent slendernes
ratio equations are given in the following sections.
Cana dian Sta nda rd CAN/CSA-S16.1-M89, Limit s tate
design o steel structure s (CSA 1989)
In Clause 19.1 the S tandard specifies two requirement
fo r the maximum slenderness ratio of an individual mai
member. Clause 19.1.3(c) requires that
where (KL/r)i is the maxim um slenderness ratio of a mai
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T E M P L E A N D E L M A H D Y
member between points of interconnection ; (KL/r), is the
slenderness ratio o f the integral memb er with respect to th e
axis perpendicu lar to the plane of the connectors; K is an
effective length factor; L is the length o f the m ember; and
r is the radius of gyration.
On the other hand, Clause 19.1.16 contains slenderness
rat io requirements for bat tened columns, which can be
summarized as follows:
If (KL/r), 0.8(KL/r),,
[2a] 0 an d e ) i < ~ . 7 e )
I X
[Zbl
e)
40 and < 0 . 6 e )
I Y
where (KL/r), is the slenderness ratio of the integ ral mem-
ber with respect to the axis parallel to th e plane of th e
connectors. Fo r the model struts tested, (KL/r), was
greater than (KL/r), so the requirements of [2b] are
applicable. Thu s [ l] and [2b] seem to contradict each othe r.
That is, the maximum slenderness ratio of an individual
main member must not be greater than 40 or 60% of
(KL/r),, and at the same time (KL/r) i must be equal to or
less than (KL/r),.
The Canadian standard, in Clause 19.1.4, requires that
the equivalent slenderness ratio, (KL/r),,, for comp ression
members composed of two or more shapes in contact or
separated from one another by welded connectors shall be
where r is the radius of gy ration of th e integral member with
respect to the axis about which buckling occurs, which in
this research is the axis perpendicular to the plane of the
connectors, that is, the Y axis; a is the centre-to-centre
distance between connectors; and is the minimum radius
of gyrat ion for one of the main members.
Am erican Specification, Specificat ion fo r stru ctu ral steel
buildings, allow able stress design an d plastic design
(AISC 1989)
In the allowable stress specification Chapter E specifies
that
and th at at least two intermediate connecto rs shall be used
along the length of the built-up mem ber. Th ere is no require-
ment for an equivalent slenderness ratio. T hus it seems the
factor 0.75 was added to cover the case of buckling about
the Y axis, as well as buckling about the X axis.
American Specvication, Load a nd resistance facto r design
speci$cation fo r structu ral steel buildings (AISC 1986)
Chapte r E has the same requirement as [I] , that is , the
slenderness ratio of the individual mem ber between points
of connection cannot exceed the slenderness ratio of the
built-up m ember. This specification also states the following
requirements for welded connectors when buckling involves
relat ive deformation that produce shear force in the
where (KL/r), is the mod ified slenderness ratio of the
built-up membe r and (KL/r), is the column slenderness
rat io of the bui l t -up member act ing as a uni t .
German Standard IN
4114-1952
German buckling
specification DI N 1952)
The Germ an stan dard gives two criteria for the m aximum
slenderness ratio of an individual member between points
of connection, which are
provided th at a t least two connectors a re used, one at each
of the third points of the bui l t -up column.
Clause 8.212 of the German standard requires that the
equivalent slenderness ratio for built-up columns tha t buckle
at right angles to th e axis perpendicular to the connectors
be taken as
where m is the number of main members.
Bri tish Stan dard BS
5950
Stru ctura l use of steelw ork in
building (BSI 1985)
The requirements of the British standard for the m aximum
slenderness ratio of an individual main member, given in
Clause 4.7.9(c), are
Clause 4.7.9(c) also specifies the equivalent slenderness
rat io of bui l t -up columns, a bo ut the axis perpendicular to
the plane of the connectors, as
where is the clear distance between adjacent connectors.
This equat ion is the sam e as that specified by the Germ an
standard w hen there are two main mem bers, except for the
second term where the length is measured from centre to
centre of adjacent connectors in the German stan dard and
from the ends of adjacent connectors in the British standard.
'1 t should be noted tha t an u pdated Germ an s tandard has just
been released but to dat e this standard has no t been translated into
English.
onnectors:
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J . C I V . ENG.
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n
dl
A = Amer i can
B
=
Br i t i s h
C = Canadian
G
=
German
8
a/ri
FIG. 3. Comparison of equivalent slenderness ratios according
to various standards.
Comparison of equivalent slenderness ratio equations
Figure 3 compares the various equivalent slenderness
ratios specified in the previous section in standards and
specifications as a part of this research. The g raph shows
two sets of curves, one for each of the integral slenderness
ratios used in this study, which are 120 and 80. Each set
of curves can be identified by looking at the graph when
the slenderness ratio of the main member, a/c, is equal to
zero. Th e curves show how the equivalent slenderness ratio
varies as a/rl is increased.
The Canadian Standard S16.1, in Clause 10.2.1, limits
the slenderness ratio of a com pression member to 200. Som e
of th e individual slenderness ratios in the tests exceeded this
value. Thus, in the graph, some of the curves have been
extended beyond a n a/r l of 200, but the curves are broken
to indicate that these slenderness ratios are not allowed by
the Canadian standard.
It should be noted that each standard contains additional
limits on the slenderness ratio of the individual main m ember
between points of connection. The Canadia n stand ard, for
example, as shown in [2], limits a/ c to either 40 or 50,
depending on the ratio of the slenderness ratios abo ut the
two axes shown in Fig. la . These limits are not shown in
Fig.
3
Theoretical analysis
The finite element method was used to calculate the ulti-
mate compressive load-carrying capacity of the model struts,
and also their compressive behaviour as given by the non-
linear load-deflection curves. Th e equivalent slenderness
ratio of e ach model strut was determined using Timoshenko s
method (Timoshenko and Gere 1961), Bleich s method
(Bleich 1952), and th e requirements of the C anadian stan-
dard (CSA 1989) and the AISC load and resistance factor
design (LRFD) specification (AISC 1986). The theoretical
Member
4 (37,38,39)
number
(0 ,0 ,33)
FIG.4 Finite element model.
compressive resistance was then calculated according to th
requirements of the Canadian standard using each of th
equivalent slenderness ratios, except when the equivalen
slenderness ratio was determined using the AISC LRF
specification, in which case the com pressive resistance wa
then calculated using the requirements of the sam
specification.
inite element method
The finite element method was used to predict the theo
retical load-carrying capacity of the model struts. This wa
don e using a comp uter progra m, first as an eigenvalue pro
gram t o predict the critical load, and second as an iterativ
incremental program to predict the nonlinear load
deflection behaviour of the model struts.
A commercial finite element package,
ABAQUS
(Hibbi
Karlsson and Sorenson, Inc. 1989), was used. Geometr
imperfections, the initial out-of-straightness of the mode
stru t, were included in the input for th e analysis as the coo
dinates of the nodes used t o define the initial geometric shap
of the strut. T he initial shape of the unloaded main m embe
was defined such that each main m ember was parabolic
shape with the maximum out-of-straightness at mid-heigh
A linear elastic, perfec tly plastic type of analysis wa s use
to model the material properties. Deformed geometry wa
used, as this is a large deflection problem with a nonline
load-deflection response.
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T E M P L E N D E L M H D Y
FIG.
5.
Model for
framework.
+ d ,
replacing redundant system with determinate
A two-dimensional finite element model was used, since
buckling was confined to the plane of the connectors. The
main members, connectors, and end plates were all modelled
using two-dimensional Euler-Bernoulli beam elements with
rectangular cross sections.
Th e axes of the finite element model were taken such tha t
the cross section of the main members were in the X-Y
plane. Buckling was modelled to take place about the Yaxis,
that is, in the X-Z plane. Figure 4 shows the elements,
nodes, and degrees of freedom for th e finite element model
for a strut with three connectors.
Th e connectors were modelled as one beam element con-
nected at its ends to the corresponding node o n each of the
main members. Each connector was assigned the properties
of two parallel connectors together and, as the distance
between the centroids of the two m ain m embers was very
small, the connectors were modelled as rigid beams.
The boundary conditions were taken as pin-ended. The
top bo undar y was free to displace in the Z direction in order
to allow for the shortening of the built-up strut under the
application of load, while the bottom was prevented fr om
FIG.
6.
Model fo r Timoshenko s equivalent length form ula.
T o determine the load-deflection curve, the load was
applied, in increments, to the finite element model on the
middle node of the to p end plate in the negative Z direction.
Timoshenko s method (Timoshenko and Gere 1961)
The effect of shear forces on the deflection of a co lumn
is greater fo r a built-up column than for a solid column and
thus decreases the buckling load. The se shear forces bend
the main members and connectors. T o account for these
effects on the buckling load of a built-up column, the
concept of an equivalent slenderness ratio is used.
The analysis of a battened column is based o n the assump-
tion that there are hinges at the midpoints of the main
members between batten plates, and at the midpoints of the
connectors, as illustrated in Fig. 5. It is also assumed that
the deflected shap e is sinusoidal.
T o derive the expression f or the equivalent slenderness
ratio, Timoshenko determined the effect the shear force
would have on the lateral deflection of a built-up column.
Th e lateral deflection caused by the bending of th e connec-
tors, 6 and by the bending of the main members, A2 as
shown in Fig. 6 are computed. T he effect of shear defor-
mation in th e main m embers an d connectors is neglected.
The equivalent slenderness ratio, as determined by
Timoshenko, for a battened column, when the battens are
of practical proportions, is
displacing in the Z direction. Each end plate was modelled
as two beam elements with large moments of inertia con-
nected to each other at a common middle node.
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CAN J C I V . ENG
VOL. 20,
1993
I
d
F I G 7
Model for Bleich s equ ivalent length formula.
Equation [lo ] shows that the equivalent slenderness ratio
is composed of two parts. The first is the square of the
slenderness ratio of the integral column, and the second is
the squ are of the slenderness ratio of a single mai n member
between adjacent connectors, multiplied by a factor of
7r2/12, which is 0.82.
Bleich s method (Bleich 1952)
Bleich developed a formula to calculate the equivalent
slenderness ratio of a pin-ended batten ed colum n. Bleich s
derivation is based on an energy ap proach. Th e elastic strain
energy of the distorted column consists of three energy
terms, which are due to
( i)
the axial shortening in the two
main mem bers due to the axial force
F
(ii) the local bend-
ing of the two main members, and (iii) the local bending
of the connectors. The forces and mom ents on the battened
colum n, in one panel, are shown in Fig. 7. The total energy,
that is, the sum of energy from all the panels, is then used
to determine the buckling load; and, subsequently, the
equivalent slenderness ratio is derived as
where
I
is the mom ent of inertia of the integral column a bou t
the axis perpendicular to the plane of the connecto rs, tha t
is, the axis abo ut which buckling occurs; and
I
is the same
mom ent of inertia, neglecting the momen t of inertia of the
individual main members about their own centroidal axis.
Thus
I
2 ~ ~ ( d / 2 ) ~~ ~ d ~ / 2 ,here Ai is the cross-
sectional area of one main member and
d
is the distance
from centroid to centroid of the main members.
F IG
8. Member with rigid ends.
Bleich s equation differs from Timoshenko s only in th
second term where the ratio
I o / I
appears. This ratio o
mom ents o f inertia is less tha n 1; as a result, Bleich s equiva
lent slenderness ratio is less than that given by Tim oshenko
and hence a slightly higher buckling load results.
Effect of the length of th e en d pl ate s
Th e effect of th e length of th e end plates on the bucklin
load of t he mod el strut specimens was investigated using th
modified stability functions derived by Livesley an
Chandler (1956). These modified stability functions wer
derived on the assu mption that the end plates were perfectl
rigid, which is close to the real case but not precise, as th
end p lates possess a little flexu ral flexibility. Thus th e resul
obtained using these modified stability functions indicat
a slightly greater effect on the load-carrying capacity tha
is actually th e case. It sho uld be em phasized tha t this so
of analysis applies to elastic behaviour only.
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T E M P L E A N D E L M A H D Y
T BLE
Results of tension tests
Specimen No.
Property
1 2 3 4
Average
Modulus of elasticity (MPa) 209 000
199 000 21 1 000
199 000 204 000
Yield stress (MPa)
321
327
339 327
328
Th e pin-ended model stru t was modelled as one element
with rigid end plates of length g at each end as shown in
Fig. 8. Applying these modified stability functions to the
model bu ilt-up strut s that buckled elastically, it was deter-
mined that the maximum increase in critical load for the
models studied in this research proje ct, because of the rigid
ends, was only 0.53 .
xperimental program
Test prog ram
An experimental program was devised to accumulate
enough d ata to check the validity of the requirements of th e
Canadian Standard CAN/CSA S16.1-M89 with regard to
the design of built-up columns that buckle in the plane of
the connectors. The model built-up columns consisted of
two bars of rectangular cross section connected intermit-
tently by plates welded to the main members. T he material
of the bars was a hot-rolled carbon steel with an average
modu lus of elasticity of 204 000 M P a and a n average yield
stress of 328.4 M Pa. These mechanical properties were
determined from tension tests performed on each of the bars
used to m ake the specimens. Th e tests were all designated
by a test num ber, such as 441-4-3. T he first number indicates
the nom inal length of the specimen, the second indicates the
nominal separation, and the third the number of connectors.
The mechanical properties for each tension test are listed
in Table 1, while the specimens made from the correspo nd-
ing bars are identified in Table 3.
Six model struts were designed and the number of con-
nectors was varied on each . Th e main members were made
from 25.4
x
6.35 mm bars. Two slenderness ratios of the
integral member, 80 and 120, were used. Thus built-up
columns with a slenderness ratio that falls in the interme-
diate and in the slender column range were tested. Each
slenderness ratio was tested with three separatio ns, namely
0, 4.02, and 7.85 mm. These separations were originally
chosen to make the slenderness ratio of the individual main
members, for each of the cases listed below, equal to
(a) on e half of the integral slenderness ratio for the built-up
member with zero separation and one connector,
(b) one third of the integral slenderness ratio fo r the built-up
member with a separation of 4.02 mm and three con-
nectors, or,
(c) one quarter of the integral slenderness ratio of the
built-up member w ith a separation of 7.85 mm and three
connectors.
Because of the length o f the en d plates these ratios were not
precisely achieved.
For the slenderness ratio of 80, these separations corre-
spond to stru t lengths of 293.0, 441.3, an d 588.0 mm ,
respectively. For the slenderness ratio of 120, the th ree sepa-
rations resulted in lengths of 440.0, 660.5, and 881.0 mm,
respectively. These lengths are measured from knife edge
to knife edge.
Each specimen was tested with fou r different num bers of
connectors in order t o vary the slenderness ratio of the indi-
vidual main members. All the specimens were tested with
zero, one, three, and seven connectors, except for the
293.0 mm specimen which, because of its shor t length, was
tested with zero, one, three, and five connectors. E nd con-
nectors were also used at each end of the specimen to mak e
sure that the two main mem bers stayed together at the ends.
As the specimens with a slenderness ratio of 120 buckled
elastically, only on e specimen was required for th e four tests.
Th e number of connec tors was simply increased after each
test on the sam e specimen. The specimens with a slenderness
rati o of 80, however, buckled in elastically, and hence a sep-
arate specimen was required for each test. Thus, for the
24 tests,
15 different specimens were required.
The connectors were welded using the Tun gsten Inert Gas
process, in two parallel planes, to the narrow width of the
main m embers. Th is type of welding was chosen, as it was
felt that the low heat generated by th e welding process would
reduce the residual stresses generated fro m welding the con-
nectors to the main members. Th e connectors all had the
same cross sectio n, 4.76 12.7 mm . Th e length of the con-
nectors was dependent u po n the separation between the main
members, b ut in general they had a length equal t o the sepa-
ration of the main members plus an overlap of about 5 mm
on each of th e main mem bers. Details of the test specimens
with three connectors are show n in Figs. 9a-9c for each of
the three separations used in the test program.
The objective of each test was to obtain an experimental
load-deflection curve from which the load-carrying capacity
of each strut could be obtained.
Test setup
Th e model st rut specimens were all tested with ends pinned
abou t the axis perpendicular t o the plane of the connectors
and with essentially fixed end conditions about the axis
parallel t o the plane of the connectors. These end conditions
were achieved by knife edges, on e at each end of the speci-
men and parallel to each other. Two types of knife edges
were used, on e when the separation was zero (see Fig. 90)
and the other when there was a separation between the spec-
imens (see Figs. 9b and 9c).
The specimens were tested either in a universal testing
machine or in a small testing frame. The universal testing
machine was used to test the sho rter specimens, that is, the
model built-up members with lengths of 293.0, 440.0, and
441.3 mm , while the longer specimens were tested in a small
testing frame. These are sho wn in Figs. 10 and 11,
respectively.
Because of th e simple buckled sh ape of the model stru ts,
one of buckling in the plane of the connectors, lateral
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C A N .
J . CIV. ENG VOL. 20 1993
FIG 9a.
Details
o f
test specimens
No. 440 0 3
and
293 0 3.
FIG
9c. Details o f test specimens No. 881 7 3 and 588 7 3.
displacements were simply measured with a dial gauge place
at the mid-height of one of the main members perpendicul
to the long side of the rectangular cross section. For th e spe
imens with zero connectors two dial gauges were used on
at mid-height of each main member.
Th e shortening of the specimens was also measured. Th
results obtained were however much larger than tho
calculated theoretically. A s a result of this discrepancy n
further attem pt was made to co mpar e the experimental an
theoretical values of the shortening of the specimens.
The dial gauge arrangement is shown in Figs. 10 and 1
The horizontal dial gauge below th e bottom knife edge
Fig. 11 was used to ensure that no significant horizont
displacement of the hydraulic jack occurred a s the load w
applied.
I 1
a?,
est procedure
I
~ ~ I
The initial out-of-straightness was determined for eac
~ :
specimen. As buckling occurred in the plane of the conne
I
I i
tors the initial out-of-straightness was measured only in th
plane. Th e out-of-straightness was determined a t mid-heig
-
t g
as well as at the quar ter points. Th e initial out-of-straightnes
varied fro m approximately zero to almost L/250 where
is taken as the length of the specimen from knife edge
knife edge. The initial out-of-straightness was used in th
iterative-incremental procedure to predict the theoretic
la81
load-deflection curves.
After the specimens were set up in the testing frame
mach ine the specimens were loaded slowly in incremen
that ranged from 2 to 8 kN depending on the predicte
FIG 96.
Details
of
test specimens
No. 660 4 3
and
441 4 3.
compressive resistance which was calculated with a resistan
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T E M P L E A N D E L M A H D Y
FI G . 10 Test setup universal testing machine.
F I G .
11.
Test setup test frame.
factor of 1 0 As the load approached the predicted com-
pressive resistance the load increm ents were gradu ally
reduced to
0.5 kN
At each load increment the dial gauge
readings were recorded once the specimen had reached equi-
librium. Each specimen was loaded until the load started
to dr op o r until a small increment in load resulted in a rela-
tively large increase in the lateral deflection.
Results and discussion
eometric properties
The g eometric dimensions of all the specimens were care-
fully measured prior to testing. The geometric properties
were calculated and are listed in Tab le
2.
The specimens have
been designated by the no minal length and nominal sepa-
ration only as the number o f connectors does not affect
these geometric properties. In the table Ai Ii and refer
to the area the momen t of inertia about an axis perpendic-
ular to the connectors and the corresponding radius of
gyration for one main member; A I and r refer to the same
properties but for the integral member; Ifi nd I are the
mom ents of inertia of one of the main members and of the
integral built-up member about the X axis; and I is the
moment of inertia of the integral cross section about the
Y
axis neglecting the mo me nt of inertia of the individual main
members abo ut their o wn centroidal axis.
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C A N . J .
C I V .
ENG. VOL. 20 1993
TABLE.
Geometric properties
Specimen
L
i i i 1 Ix I r Ti
N o . m m ) m m 2) m m 2 ) m m 4 ) m m 4 ) m m 4 mm 4 ) m m 4 ) m m ) m m )
Specimen
No.
440-0-7
DISPLACEMENT
mm )
FIG. 12.
Load-displacement curves, Specimen No.
440-0-7.
Load-deflection curves
Th e load-deflection curves for all specimens were plotted
from the experimental results and from the outp ut obtained
from the finite element iterative-incremental procedure.
Figure 12 shows the experimental and theoretical load-
deflection curves for Specimen No. 440-0-7, which has an
integral slenderness ratio of 120, zero separation, and seven
connectors. F or specimens with zero separation, th at is, the
two main members are in contact, th e experimental buckling
load was abo ut the same as or a little higher th an the theo-
retical buckling load as determined by the finite element
method. This is probably due to the extra stiffness that
results from one main member being in contact with the
othe r. This was probably no t correctly accounted f or in the
finite element program, even with links between the main
members.
When the separation between the main memb ers was not
zero, the theoretical buckling loads were greater than the
experimental buckling loads. Figure 13 shows the experimen-
tal and theoretical load-deflection curves for a specimen
with a separation of 4.02 m m, one connector, and a slender-
ness ratio of 120. The experimental and theoretical load-
deflections curves are, in general, in good agreemen t when
the loads are less than abou t on e half of the failure load.
These curves are in good agreem ent over the entire loading
range when the number of connectors is zero or one, as
Specimen
No.
660-4-1
DISPLACEMENT mm )
FIG.13.
Load-displacement curves, Specimen No.
660-4-1.
noted in Fig. 13. As the number of connectors increase
the agreement is not as go od. A typical set of load-deflectio
curves illustrating a case where the agreem ent is not as goo
is show n in Fig. 14. The se curves are for a specimen wit
a separation of 7.85 m m, three conn ectors, and a slenderne
ratio of 120. At least part o f this difference may be due t
residual stresses, which were not included in the fini
element program .
Experim ental results
The experimental results are summarized in Table 3
Column 2 gives the mechanical properties of the materia
used in each specimen by referring t o th e applicable tensio
test and the correspon ding p roperties listed in Table 1. Th
initial out-of-straightness, as measured at mid-height of eac
specimen, is listed in Col um n 3. Th e experimental bucklin
loads are also listed.
Equivalent slenderness ratio
Colum ns 2-4 in Table 4 list the slenderness ratio of th
integral built-up member an d the individual main member
For the individual member b etween points of connection tw
slenderness ratios are shown. In Column 3 the slendernes
ratio is based o n the clear distance between connectors. Th
is required when Clause 19.1.16 of
CAN/CSA-S16.1-M8
is considered. C olum n 4 lists the slenderness ratio of the ind
vidual main members between points of connection base
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T E M P L E A N D ELMA H D Y
T A B L E. Experimental results
Exper imental
Specimen No.
881 7 3
DISPLACEMENT mm)
FIG 14. Load-displacement curves, Specimen No. 881-7-3.
on the cen tre-to-centre distance. T hese values are used for
all equivalent slenderness ratio calculations.
Th e equivalent slenderness ratios were then calculated by
using Timoshen ko s a nd Bleich s f orm ulas, and acco rding
to the requirements of the Canadian stan dard and the AISC
LRFD specification. These are shown in Columns
5-8 of
Table
4
Timosh enko s and Bleich s form ulas, as pointed
ou t previously, differ only in the second term . Timosh enko s
form ula results in a higher equivalent slenderness ratio tha n
does Bleich s form ula. Th e maximu m difference between
these equivalent slenderness ratios is
9.3
and o ccurs in the
case of zero separation. In this case the mom ents of inertia
of the individual mem bers has a significant effect on the
overall mom ent o f inertia of the cross section abou t the axis
perpendicular to the conn ectors. In oth er cases the difference
was as low as 0.5 .
Th e equivalent slenderness ratio as calculated in accor-
dance with the requirements of the C anadian standard are
up t o 25.6 less than those obtained from Timoshenko s
formula for specimens that are sparsely connected, but as
little as 0.9 less when a specimen has seven connectors.
Calculated and experimen tal failure loads
Th e equivalent slenderness ratios were used t o calculate
the comp ressive resistance of the specimens. These a re listed
in Table 5. The compressive resistances were calculated in
accordance with the requirements of Clause 13.3.1 of the
Canadian standard, except for those shown in Column
6
which were calculated in accordance with Appendix E of
AISC L RF D specification. Column
2
lists the b uckling load
predicted by the finite element iterative-incremental
procedure.
For comparison purposes the experimental failure loads
are listed in Column
7
of Table
5.
Because of the large
variation in th e initial out-of-straightness of these specimens
and because the compressive resistance in the Canadian
stand ard is based o n a specimen with an out-of-straightness
of L/1000, it was decided to adjust the experimental
Initial Experimental
out-of- failure
Specimen Applicable
straightness load
No.
tension test
(mm) (kN)
(1) (2) (3) (4)
buckling load t o better reflect the load-carrying capacity of
the specimen if the out-of-straightness had been L/1000.
This was done as follows:
where PFEM L/lOOOs the ultimate load-carrying capacity
predicted using the finite element program when the out-
of-straightness was set at
L/1000;PEXPT
s the experimental
fai lure load; and
P F E M , M O S
is the ultimate load-carrying
capacity predicted using the finite element program and the
measured out-of-straightness. It cann ot be proven that this
procedure correctly adjusts fo r the out-of-straightness, but
it is felt that this is a reasonab le appro ach to try to minimize
the differences in the load-carrying capacity of the bu ilt-up
member d ue to the out-of-straightness. This procedure was
followed rather than t o just use the buckling load with an
out-of-straightness of L/1000, since specimens with three,
five, and seven connectors had predicted finite element
buckling loads greater than the experimental buckling load
for specimens with the same out-of-straightness. This
adjusted ex perimental failure load is listed in Co lumn
8
of
Table 5.
Comparing the finite element buckling load with the
actual experimental buckling load (Columns 2 and
7)
for
built-up mem bers with the sam e initial out-of-straightness
shows that the finite element buckling load tends to be in
good agreement when zero and on e connector are used. In
the cases where three o r seven connectors are used, the exper-
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C A N . J . C I V . ENG. VOL. 20 993
T A B L E
.
Slenderness ratios
Slenderness ratio
Individual memb er Equivalent slenderness ratio
Specimen
Built-up
between centre-to- Tim oshe nko s Bleich s
Canadian AISC LRF
No. member battens centre formula
formula standard
specification
1) 2) 3 4) 5) 6) 7) 8)
imental failure load is greater tha n the finite element buck-
ling load for the specimens with no separation, but is less
than the finite element buckling load when the se paration
is 4.02 or 7.85 mm.
It was pointed o ut previously tha t the equivalent slender-
ness ratio as calculated by T imoshenko s form ula is greater
than that calculated w ith Bleich s for mu la. T his is also
reflected in the compres sive resistance using these equivalent
slenderness ratios, a s shown in Columns
3
and 4 of Table 5.
The equivalent slenderness ratio as calculated according t o
the Can adian standard results in a compressive resistance
that is higher than tha t calculated when Timoshenko s or
Bleich s eq uivalent slenderness ratios ar e used. W hen these
compressive resistances are compared with the adjusted
experimental failure loads, it can be seen that the C anadian
standard often results in a compressive resistance that is
greater than the ad justed experimental load f or specimens
with a sepa ratio n between the main m embers. It is realized
that the slenderness ratio of a few of the main members
exceeds the allowable as established by the C anad ian stan-
dard and hence a comparison with this standard is not
applicable. It seems that using the equivalent slenderness
ratio calculated by T imoshenko s formu la results in a com-
pressive resistance that is in the best agreement with the
adjusted experimental failure load.
It should also be noted that when the compressive resis-
tance is calculated in accordance with the AI SC L RF D spec-
ification, using the equivalent slenderness ratio calculated
according to the sam e specification often results in compres
sive resistances that exceed the adjusted experimental failur
loads.
Figure 15 illustrates the difference between the compres
sive resistances calculated using th e equivalent slendernes
ratios as determined by T imosh enko s f orm ula and Bleich s
formula, that given in the Canadian standard, and the
adjusted experimental failure load. These results are for a
built-up member with an out-of-straightness of L/1000, a n
integral slenderness ratio of 120, and three connectors. I
can be seen that for the built-up members with no separa
tion, all three compressive resistances are less than the exper
imental failure load. For a separation of 4.02 mm, the
Cana dian st and ard gives a load that is high com pared with
the experimental failure load, but both Timosh enko s and
Bleich s buckling loads are very close to the experimenta
failure load. With a separation of 7.85 mm, the Canadian
standar d aga in gives a load th at is too high, while the com
pressive resistances calculated using the equivalent slender
ness ratios of T imoshe nko an d Bleich are also too high bu
are a little closer to the adjusted experimental values. I t may
be more significant in Fig. 15 to plot the compressive resis
tance versus the slenderness ratio of the main membe
between points of connection. This has been done by adding
a second horizontal axis. This graph then clearly indicate
the significant effect that this slenderness ratio has on th
equivalent slenderness ratio and hence the predicted com
pressive resistance, according t o the different m ethods.
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TEMPLE A N D ELM AHDY 907
TA BL E . Calculated and ex perimental failure loads
Calculated compressive resistance
Experimental Adjus ted experimental
Fini te e lement Cana dian AISC LRFD fai lure fai lure load
Specimen meth od Timo shenk o's Bleich's stan dard specification load
(L/
1000)
No. (kN)
(kN) (kN)
(kN) (kN) (kN) (kN)
(1) (2)
(3) (4)
(5) (6) (7) (8)
When buckling occurs about a n axis perpendicular to the
connectors, the requirement that the slenderness ratio of the
individual member between points of connection cannot
exceed the slenderness ratio o f the integral built-up member
does not ensure that a load-carrying capacity is achieved
which is equal to th e compressive resistance determ ined in
accordance with the C anadian stan dard. This conclusion has
already been reached theoretically by Libove (1985).
When buckling occurs about a n axis perpendicular t o the
connectors, the number of connectors significantly affects
the load-carrying capacity of the built-up member. The
information in Table 5 indicates that when the num ber of
connectors is increased from one to three, for built-up
mem bers with a separation between the main members that
is greater than zero, the load-carrying capacity is increased
by a factor of anywhere from 1.4 to 2.0. Figure 16 shows
the theoretical load-deflection curves of a built-up mem ber
with an integral slenderness ratio of 80, an initial out-of-
straightness of 0.1 m m, and no separation fo r the cases of
zero, one, two , three, and five connectors. The zero and on e
connector cases result in the same buckling load, but as
additional connectors are used, the load-carrying capacity
is increased. Increasing the number o f connectors fro m one
to two results in an increase in the buckling load of some
14 . Thu s it is recommended that fo r built-up members
that buckle ab out an axis perpendicular to the connectors,
at least two connectors, one at each of the third points,
should be used.
Clause 19.1.17 of the Canadian standard contains a
requirement for the length of a batten plate, a connector.
Th e length of the batten plate is the dimension of the batten
parallel to the longitudinal axis of the built-up member.
Timoshenko's equation (1961) for the critical load of a
built-up member, from which the equivalent length equation
is derived, is
where is Youn g's mo dulu s of elasticity and b is the
moment of inertia of the connector abo ut an axis perpen-
dicular to the plane of bending.
It can be seen from this equation that the greater the
moment of inertia of the batten, the greater is the critical
load. According to Bleich (1952), however, the term con-
taining the moment of inertia of the batten is small com-
pared with the other terms in the denomin ator and can be
neglected for a properly designed batten. T hus, once a batten
with sufficient flexural rigidity is selected, any further
increase in the moment of inertia of the batten is of insig-
nificant importance. The origin of Clause 19.1.17 is not
known and will be the subject of further research.
The connector used in Specimen 881-7-3 does not meet
the length requirements of Clause 19.1.17 of S16.1. Thus
it was decided to check, theoretically, to see what effect it
would have on the compressive resistance if the length was
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908
CAN.
J . C I V . ENG.
VOL.
20
1993
Adjusted Expt.
Canadian Std.
~ imoshenko sEq.
* Bleich s Eq.
*
Specimens with
15 th ree connecto rs
SEPARATION
(mrn)
L / r
OF
MAIN MEMBER
FIG. 15. Buckling load vs. separation an d slenderness ratio of
main member .
90
i No. of connectors
zero
and I
DISPLACEMENT
(rnrn)
FIG. 16 Theor e t ica l load- def lec t ion cur ves f or va r ious
numbers of connectors .
changed to that required by the Stan dar d. For the specimens
with a separation of 7.85 mm between the m ain members,
the battens should have a length parallel to the main mem-
bers of abo ut 17.9 mm. The actua l length of the batten was
only 12.7 mm. T he moment of inertia of the ba tten, Ib s
F I G . 17
Predicted critical load
vs.
moment of inertia of the
bat ten.
813 mm4, while the Canadian standard would require an
I
of 2256 mm 4. Equation [13] was used to calculate a n
equivalent length factor. The compressive resistance was
then calculated in accordance with Clause 13.3.1 of S16.1
The length, which should probably be called a depth in the
Standard, was varied to see what effect the moment of
inertia of the batten h as o n the compressive resistance o
the built-up member. The results are shown in Fig. 17
Changing the length of the batten from 12.7 to 17.9 mm
(a 40 increase) changes the mom ent of inertia fro m 813
to 2256 mm 4 (a 175 increase) an d the predicted compres
sive resistance fro m 23.08 to 23.34 kN (an increase of only
1 ). Thu s it seems tha t the requirement in Clause 19.1.17
of 516.1 requiring the battens to have a length of not less
than the distan ce between the lines of welds may be unneces
sarily restrictive.
Conclusions
The following conclusions may be stated from this
research for built-up members that buckle about an axis
perpendicular to the connectors.
1. The slenderness ratio of the main member between
points of connec tion has a significant effect on the compres
sive resistance of the built-up member.
2. A minimum of two intermediate connectors, one a
each of the third points, should be used.
3 Clause 19.1.4 of the Canadian standard should be
changed to the equivalent length formula derived by
Timo shenko . This equivalent length formula together with
the compressive resistance calculated in accordance with
Clause 13.3.1 gives the best agreement with the experimenta
loads.
4. Th e requirement t hat the slenderness ratio of the indi
vidual main memb ers between points of connection be equa
to or less than the slenderness ratio of the integral membe
is not app licable to these membe rs and does not en sure tha
the required load-carrying capacity is achieved.
8/10/2019 Buckling of Built-up Compression Members in the Plane of the Connection
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T E M P L E A N D
E L M A H D Y
AISC. 1986. Load and resistance factor design specification for
structural steel buildings. American Institute of Steel Construc-
tion, Chicago, Ill.
AISC. 1989. Specification for structural steel buildings, allowable
stress design and plastic design. American Institute of Steel
Construction, Chicago, Ill.
Bleich, H. H. 1952. Buckling strength of metal structure s. McGraw-
Hill Book Company, Inc., New York, N.Y.
BSI. 1985. Structu ral use of steelwork in building. BS 5950, Par t 1,
British Standards Institution, London, England.
CSA. 1989. Limit states design of steel structures. CAN/CSA
S16.1-M89, Canadian Standards Association, Rexdale, Ont.
DIN . 1952. German buckling specification. Deutscher Normenauss-
chuss, Beuth Vertrieb G.M .b.H., Berlin and Colog ne, Germany.
Translated by J . Jones and T.V. Galambos. C olumn Research
Council (now Structural Stability Research Council), Lehigh
University, Bethlehem, PA.
Hibbitt, Karlsson and Sorenson, Inc. 1989. ABAQUS,ersion 4-8,
Vol. : Theory manual; Vol. 2: Verification manual; Vol. 3:
User s m anual; Vol. 4: Example problem ma nual.
Libove, M. 1985. Sparsely connected built-up columns. ASCE
Journal of Structural Engineering, lll(3): 609-627.
Livesley, R.K., and C handler, D.B. 1956. Stability functions for
structural frameworks. Manchester University Press, Manchester,
England.
Timosh enko, S. , and Gere, J.M . 1961. Theory of elastic stability.
3rd ed. McGraw-Hill Book Company, New York, N.Y.
ist of symbols
total cross-sect ional area
cross -sect ional area of o ne ma in mem ber
centre-to-centre dis tance between connectors
c lear d i s tance between ad jacen t connectors
compressive resis tance
d i s tance f rom cen t ro id to cen t ro id of the main
members
Young s mod ulus of elast ici ty
ax ia l fo rce in one main member in panel r
length of r igid end plates
mom ent of inert ia of th e integral cross sect ion
abou t the ax i s perpendicu lar to th e p lane of
the connectors
m o m en t o f i n e rt i a o f a co n n ec t o r ab o u t t h e
horizontal centroidal axis perpendicular to th e
p lane of the connectors
m o m en t o f i n e r ti a o f o n e m a i n m em b er ab o u t
i t s cen tro idal ax i s perpendicu lar to the p lane
of the connectors
mom ent of inert ia of the integral cross section
abo ut the axis perpendicular to the connectors ,
the axis abou t which buckl ing occ urs , neglect-
ing the moment o f iner t i a o f the ind iv idual
main members abo ut the i r ow n cen t ro idal ax i s
~ ; d ~ / 2
Subscripts
moment o f iner t i a o f the in tegra l bu i l t -up
m em b er ab o u t t h e axi s
m o m en t o f i n e r ti a o f o n e m a i n m em b er ab o u t
its axis
effect ive length factor
leng th of member
length of cen t re par t o f co lumn between the
rigid ends
end mo ments app l ied a t ends a an d b , respec-
t ive ly , t o p roduce d i s tu rbance of mem ber
moment in connector r
moment in main member in panel r
n u m b er o f m a i n m em b er s
ex ternal load
cri t ical load
exper imenta l fa i lu re load
adjus ted exper imenta l buckl ing load
ultimate load-carryin g capacity predicted using
the f in i t e e lement p rogram when the ou t -of-
s t raightness was set at
L 1000
ultimate load-c arrying capacity predicted using
the f in i te e lement p rogram and th e measured
out-of-s t raightness
la tera l shear fo rce
shear fo rce in connector in panel r
shear fo rce in panel r
radius of gyrat ion of the integral bui l t -up
m em b er ab o u t t h e ax i s p e rp en d icu l ar t o t h e
p lane of the connectors
min imum rad ius o f gyra t ion for one of the
m a i n m em b er s
coord inate axes
d i s p la c e m e n t i n a n d
Z
d i r e c t i o n s
respectively
lateral deflect ion caused by bending of the
co n n ec t o r s
la tera l def lec t ion caused by bending of the
m a i n m em b er s
ro ta t ional degree of f reedom
dis turbances appl ied a t ends a a nd b , respec-
t ively, of a member
equivalent
ind iv idual main member
modi f i ed
in tegra l bu il t -up mem ber
a b o u t o r Y axis, respectively