Buckling of Built-up Compression Members in the Plane of the Connection

8/10/2019 Buckling of Built-up Compression Members in the Plane of the Connection

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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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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

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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 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


9/15


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.


10/15

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


11/15

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-


12/15

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.


13/15

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


14/15

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.


15/15

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

Buckling of Built-up Compression Members in the Plane of the Connection

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