sc

rb

eEffective lengthsSecond-order analysisTrusses

as a system, rather than designing members in isolation in the traditional member-based design. 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Extensive experiments have been conducted and reportedon composite columns to investigate their structural resistancesunder various loading conditions. Furlong [1] tested concrete-filledsteel tubular columns and indicated the local buckling of the steeltube was delayed by concrete infill. Circular and square concrete-filled steel tubes with different slenderness ratios were tested byKnowles and Park [2] to investigate the confinement effect onconcrete under various slenderness ratios. The results indicatedthe benefits on ultimate capacity from the confinement effect ofshort circular columns. Tomii et al. [3] carried out tests on circular,octagonal and square concrete-filled steel tubular columns, andnoted that the concrete core provided confinement effects onlyon circular and octagonal sections. Shakir-Khalil and Mouli [4]performed tests on concrete-filled steel tubular columns andconcluded that higher concrete strengths or larger steel sectionsprovided a greater structural efficiency for composite columns. Theshape effect of steel tubes on strength and behavior of concrete-filled steel tubes were studied by Schneider [5]. Fourteen circular,square and rectangular tubes were tested and the results indicatedthat the circular tubes provided greater ductility than the othertwo section shapes and the confinement effect on the concretecore was observed only on circular tubes when the yield strengthwas approached. Kilpatrick and Rangan [6] reported test results

Corresponding author. Tel.: +852 9025 6814; fax: +852 23346389.E-mail address: ceslchan@polyu.edu.hk (S.L. Chan).

on concrete-filled steel tubular columns, and indicated the effectson the behavior and strength of composite columns due to theslenderness and load eccentricity. Li et al. [7] carried out testson six high-strength concrete-filled square steel tubular columnsunder bi-axial eccentric load and indicated that both the steel arearatio and slenderness ratio influenced the capacity of the columnssignificantly. Chan and Fong [8] tested steel and composite trusseswith high-strength in-filled concrete to compare the beneficialeffect of the in-filled concrete.

From the review on the subject, most previous experimentsfocused on the behavior of isolated columns under pinned or fixedend conditions, and tests on steel and concrete-filled steel (CFS)tubes acting as members of a frame or truss structures are limited.In this paper, tests on trusses composed of the bare steel and CFStubes as chords and webs in the truss are reported and comparedwith the second-order analysis and design method. In order tocompare the behavior of the members in an isolated column andin the truss system, the columns with same properties of themembers in the truss under pinned and fixed ends are tested. Thereported work in this paper is believed to be novel in providingthe response of the members in the configuration of a structuralsystem as a truss, which should be a good reference for the futureresearch and design of composite trusses and frames.

Various international design codes provide different methodsof design for steel and composite members under different loadingconditions such as EC3 [9], EC4 [10], AISC LRFD [11], AS5100 [12]and CoPHK [13]. The accuracy of these design methods have beenverified by many researchers such as Wang [14], Al-Rodan [15],Zeghiche and Chaoui [16]. Most comparisons of experimental andpredicted results from these codes were focused on isolatedEngineering Structures

Contents lists available a

Engineering

journal homepage: www.el

Advanced design for trusses of steel andM. Fong a, S.L. Chan a,, B. Uy ba Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hb School of Engineering, University of Western Sydney, Sydney, Australia

a r t i c l e i n f o

Article history:Received 15 January 2011Received in revised form11 March 2011Accepted 2 August 2011Available online 23 September 2011

Keywords:Concrete-filled steel tubesDesign codes

a b s t r a c t

This paper presents an expeconcrete-filled steel (CFS) tuthe columns and trusses conbeneficial effects of the in-filllength and second-order ana3 (EC3), Eurocode 4 (EC4), Cproposed second-order analydesign of composite membconstituting elements in a tr0141-0296/$ see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.08.00233 (2011) 31623171

t SciVerse ScienceDirect

Structures

evier.com/locate/engstruct

oncrete-filled tubular sections

ong Kong, China

imental and analytical investigation of buckling behavior of bare steel andes used as columns and as members of trusses. The member resistances ofsisting of steel and CFS tubular members are compared to demonstrate theed concrete,with their resistances predicted using the conventional effectivelysis methods of design in various international standards such as EurocodeoPHK, AISC-LRFD and AS5100. Test results are further used to validate thesis, which skips the assumption of effective length, for accurate and reliablers. The present holistic approach of considering composite members asuss represents a piece of original work on testing and design of structures

sof second-order analysis and design method in comparison withthe first-order analysis and design method have been reportedin literature by different researchers such as Chen [17] and Chanet al. [18]. Using this design method, nonlinear effects such as theP- and P- moments, member and frame imperfections are in-cluded in the analysis process. The new design method not onlyenhances the accuracy of the design output, but it also reduces de-sign time and effort. The Pointwise Equilibrium Polynomial (PEP)element proposed by Chan and Zhou [19] is used here for its sim-

Two trusses with members composed of bare steel tubes(named as steel truss) and CFS tubes (named as composite truss)were tested with the mean dimensions of truss members arelisted in Table 1. The same cross-sections used for the isolatedcolumns tests were adopted in the construction of the trusses andthe dimensions of the trusses are provided in Fig. 2. Steel trusseswere composed of steel tubes for all members and the compositetrusses were composed of CFS tubes in the compressive membersand bare steel tubes in the tensile members. Each of these three-Pinned end boundary condition Fixed end boundary condition

Fig. 1. The setup of columns.

members with pinned or fixed end boundary conditions. However,the behaviors ofmemberswithmore realistic boundary conditionssuch as constitutive members in a truss have not been consideredin previous research. The traditional linear analysis method ofdesign assumes an effective length factor for different boundaryconditions of a chord or a web member in a truss and assessesvarious factors to compensate for buckling and second-ordereffects in the determination of buckling strength of individualmembers. Unfortunately, the accuracy on determination of theeffective length highly affects the reliability of these methods. Forsimple end conditions like pinned and fixed end, previousmethodscan give an accurate design solution as shown in the followingexample. However, for more complicated and practical structureslike trusses, the accurate determination on the effective length isdifficult and sometime impossible. This paper reports the accuracyof designmethods in several international codes for the predictionof capacities of steel and CFS tubes as composing members in atruss under pinned and fixed end conditions.

Design codes such as AISC LRFD [11], EC 4 [10] and CoPHK [13]propose the use of second-order analysis and design methods be-cause of the accuracy, reliability and convenience. The superiorities

PEP element on various forms of steel structures have been veri-fied on various structural forms such as dome structures [20], angletrusses in both elastic [21] and plastic analysis [22,23], scaffoldingsystems [24] and pre-stressed stayed column [25]. In this paper,the design method of second-order analysis with the PEP elementwould be applied to predict the resistance of steel and CFS tubesused as isolated columns and as members of trusses.

2. Experimental work

2.1. General

Tests for columns and trusses were carried out. For the columntests, four specimens of two bare steel tubes and two CFS tubeswere tested. Square hollow section steel tubes of 60 60 3 mm cross-section were used and the average width, depth andthickness are listed in Table 1. Two boundary conditions as pinnedand fixed ends were set up in the test as indicated in Fig. 1. Thespecimen length for the fixed end condition is 2 m and for thepinned end condition is 1.74 m.M. Fong et al. / Engineering Str

Table 1Material properties of steel tube.

Item Steelsection

B(mm)

D(mm)

t(mm)

Yield stre(N/mm2)

Columns 60603 60.40 60.30 3.10 407.98Trussmembers

60603 60.20 60.20 3.10 404.11

Applied force

Load Cell

2000plicity and computational stability and efficiency allowing mod-eling of one member by one element. The accuracy of using theuctures 33 (2011) 31623171 3163

s (fy) Ultimate tensile stress (fu)(N/mm2)

Youngs modulus (Es)(kN/mm2)

fu/fy

480.22 206.36 1.18473.55 205.72 1.17

Applied force

Load Cell

2000dimensional trusses consisted of 19 members which included the14main members of 2 m length and 5 tie members of 0.8 m length

Item Age at testing (days) Stress (N/mm2) E (N/mm2)

Columns 34 40.96 22.17Truss members 56 41.16 22.73

connecting the two 2-D trusses. All members were connected asmoment connections by 8 mm fillet welds.

2.2. Material properties

The steel plate slenderness ratio (D/t) was selected accordingto the guidelines provided in EC3 [9] and EC4 [10] to avoidlocal buckling. Coupon tests were carried out to determine thestressstrain curve of the steel material. Properties includingthe characteristic yield and ultimate tensile stress and Youngsmodulus of elasticity for the columns and trusses members arelisted in Table 1.

Normal strength concrete was used in filling the steel tubes.The concrete mix included 200 kg/m3 water, 408 kg/m3 OrdinaryPortland Cement, 1074 kg/m3 coarse aggregate and 716 kg/m3

concrete used in the CFS tube are given in Table 2.

2.3. Column tests

2.3.1. Test setupSteel and CFS tubular columns with pinned and fixed boundary

conditions were tested and the test setup is shown in Fig. 1. Thecolumnwas 2m in length, measured from the center of rotation ateach end. The test specimen was cast on the surface of a flat steelplate at both ends to ensure smooth surface for producing uniformloading. During the test, load was applied at the top of the columnvia the loading jack ofmaximumcapacity equal to 400 kN. The loadinterval was taken as 1 kN throughout the whole test and the loadcell was located at the top of the column to record the applied load.Rotation was only allowed in the in-plane direction for the pinnedend condition, whilst no rotation was allowed in both the in-planeand out-of-plane directions for the fixed end condition.

2.3.2. Strain gages and displacement transducersin-plane direction

out-of-plane direction

Strain gage

Fig. 3. The locations of strain gages and displacement transducers.

Table 2Material properties of concrete.

conditions as the specimens until the day of test. The characteristiccompressive cylinder strength and the modulus of elasticity of the3164 M. Fong et al. / Engineering Str

Applied forceHydraulic jackLoad Cell

2000

4000

Plan View

Fig. 2. The dimenfine aggregate. A series of standard concrete cylinders, 100 mm indiameter and 200mm in heightwas cast and cured under the sameuctures 33 (2011) 31623171

Applied force

Hydraulic jack

Load Cell

Side View

2000sions of the truss.Strain gages were mounted on the faces of the specimen asindicated in Fig. 3 to monitor the changes of strain during the

plotted in Figs. 5 and 6. For the in-plane direction as depicted inFig. 5, the deflection increased linearly with applied load at theinitial stage, and the relationship became nonlinear when the loadapproached failure load. After attaining the maximum load, thedeflection of the member increased significantly with decreasingload. The load against out-of-plane deflection curve presentedin Fig. 6 shows that the deflection was small at the maximumapplied load compared with the in-plane deflection in the pinnedend columns. However, for fixed end columns, the mid-length

2.4. Truss test

2.4.1. Test setupTwo trusses were simply supported at four points and loaded

in pairs by using the hydraulic jacks with a maximum capacity of400 kN as shown in Fig. 2. The load increment was taken to be1 kN during the test, and the two load cells were located underpoint loads. Sufficient lateral restraints were provided to ensureM. Fong et al. / Engineering Str

Table 3Column test results.

Specimen End condition Applied

Steel tubular column Pin 152.45CFS tubular column Pin 186.88Steel tubular column Fix 249.40CFS tubular column Fix 331.60

Steel tubularcolumn

(pined end)

CFS tubularcolumn

(pined end)

CFS tubularcolumn

(fixed end)

Steel tubularcolumn

(fixed end)

Fig. 4. The failure model of each column.

test. Eleven displacement transducers were used to measure thedownward and lateral movements at the top, bottom, middle andquarter points for both the in-plane and out-of-plane directionsof the columns. The locations of these transducers are shown inFig. 3.

2.3.3. Test resultsAll the specimens failed in a flexural bucklingmode as shown in

Fig. 4 and no local bucklingwas observed before reaching themax-imum applied load. Themaximum loads of all specimens are listedin Table 3. The load ratios of CFS to steel tubular columnswere 1.23and 1.33 for pinned end and fixed end conditions respectively, andthe load ratios of fixed to pinned end conditionswere 1.64 and 1.77respectively for steel and CFS tubular columns.

The axial load versus mid-length in-plane and out-of-planedeflection curves for both steel and CFS tubular columns aredeflection along the out-of-plane direction was close to the in-plane deflection,which indicated that the columndeflected in bothuctures 33 (2011) 31623171 3165

load (kN) RatioCFS column/steel column Fix/Pin

/ /1.23 // 1.641.33 1.77

CFS tubular column (fixed end)

CFS tubular column (pined end)

Steel tubular column (fixed end)

Steel tubular column (pined end)

350

300

250

200

150

Load

(kN)

100

50

00 10 20

Mid-length in-plane deflection of columns (mm)30 40 50

Fig. 5. Load against in-plane lateral deflections of columns.

CFS tubular column (fixed end)

CFS tubular column (pined end)

Steel tubular column (fixed end)

Steel tubular column (pined end)

350

300

250

200

150

Load

(kN)

100

50

05 5 15

Mid-length out-of-plane deflection of columns (mm)25 35 45

Fig. 6. Load against out-of-plane lateral deflection of columns.

directions and similar initial imperfections in both directions werepresent.

The load against strain curves at the mid-length of the columnsare also plotted in Fig. 7. These curves were noted to possess atypical and consistent pattern similar to loaddeflection curves.Signs of local buckling on the steel plate were not observed fromthese curves.

The deformed shape of each specimen is shown in Fig. 8 andtypical deformed shapes for pinned end and fixed end columnwereobserved.

For pinned end columns, owing to the friction in the pinnedend, the degree of end restraint may be varied and the maximumapplied load could be higher than the theoretically perfect pinnedend case.out-of-plane buckling at connecting nodes between members wasprevented.

L500

00 20 40Deflection (mm)

Steel tubular column(pined end)

CFS tubular column(pined end)

Steel tubular column(fixed end)

CFS tubular column(fixed end)

60 80

L

500

00 20 40Deflection (mm)

60 80

L

500

00 20 40Deflection (mm)

60 80

L

500

00 20 40

Deflection (mm)60 10080

Fig. 8. The deformed shape of the columns.

2.4.2. Strain gages and displacement transducersThe movements on the truss, and deflection of the targeted

failure members of in-plane and out-of-plane directions wererecorded by the transducers, and the strain gages were mountedon themid-length andquarter points of the target failuremembers.The detailed locations of strain gages and displacement transduc-ers are shown in Fig. 9.

2.4.3. Test resultThe maximum applied loads on both trusses are presented in

Table 4. Flexural buckling took place in the in-plane directionon the target failure member and the final deformed shapesare shown in Figs. 10 and 11. The maximum load resisted bythe member in the composite truss was 29% higher than that

deflection curves were observed for both the steel and CFS tubularmembers that, when the load was small, a linear relationshipbetween the load and deflection was noted and the relationshipbecame nonlinear approaching the maximum loads which wererespectively 250.37 kN and 323.09 kN for steel and CFS tubularmembers. The load was applied continuously to the truss afterachieving the maximum load in order to study the post-bucklingbehavior. The applied load against the out-of-plane deflection atmid-length of the failuremembers are plotted in Fig. 13. The curvesshowed that the out-of-plane deflection was small compared withthe in-plane deflection at the maximum load in both steel andcomposite trusses, and the out-of plane deflection was mainly dueto the initial imperfections.

The applied load against the strain plots at the mid-lengthSG 4

SG 4

SG 2

SG 2SG 3

SG 3

SG 1

SG 11

3

24

20000 15000 10000 5000Strain ()

0 500040000 30000 20000 10000

Strain ()0 10000 20000

150

200

250

50

0

100 150

200

300

250

50

0

100

Load

(kN)

1

3

24

Load

(kN)

Fig. 7. Load against strain values of the columns.

2000

1500

1000

engt

h (m

m)

2000

1500

1000

engt

h (m

m)

2000

1500

1000

engt

h (m

m)

2000

1500

1000

engt

h (m

m)3166 M. Fong et al. / Engineering Str

Steel tubular column (pined end)

Steel tubular column (fixed end)

SG 4 SG 2

SG 3SG 11

3

24

150

50

8000 6000 4000 2000Strain ()

0 2000 40000

100

Load

(kN)of the steel truss. The applied load against mid-length in-planedeflection of the failure member is plotted in Fig. 12. Similar loaductures 33 (2011) 31623171

CFS tubular column (pined end)

CFS tubular column (fixed end)

SG 4 SG 2

SG 3SG 1

6000 4000 2000Strain ()

0 2000 4000

350

150

200

50

0

100

1

3

24

Load

(kN)of the failure members are shown in Fig. 14 and the non-linearrelationship and post failure behaviors were observed. Large

Steel truss

050

10 10 30Mid-length deflection of members (mm)

50

50100150

Load

(kN)

200250

Fig. 12. Load against in-plane lateral deflection of the truss.

compressive strains were developed at the bottom fibers whichgave a consistent result with displacement transducers. The strain

using EC3 [9], EC4 [10], CoPHK [13], AISC LRFD [11] and AS5100 [12] and compared with test results. The design methods onthe compressive capacity of composite columns based on differentdesign codes are summarized below.

3.1.1. EC4 and CoPHKThe section capacity of CFS tubular members is determined by

summation of the resistances of the concrete and steel tubes andthe member capacity is reduced by multiplying a buckling reduc-tion factor , which is obtained from the effective slenderness ratioand section types, as

Pcp = (Asfyd + Ac fcd) (1)M. Fong et al. / Engineering Str

in-plane direction

strain gageout-plane direction

Fig. 9. The locations of strain gag

Fig. 10. Buckling mode of steel truss.

Fig. 11. Buckling mode of composite truss.

Composite truss

300350gages SG2 and SG4 recorded on both sides of the member werenearly identical for both steel and CFS tubular members and theuctures 33 (2011) 31623171 3167

es and displacement transducers.

Composite truss

Steel truss

050

10 10 30Mid-length deflection of members (mm)

50

50100150

Load

(kN) 200

250300350

Fig. 13. Load against out-of-plane lateral deflection of the tr.

Table 4Truss test results.

Specimen Applied load on truss (kN) Failure member force (kN)

Steel truss 250.37 216.57Composite truss 323.09 279.47

slight difference was mainly due to the out-of-straightness im-perfections. The result also implied that the out-of-plane deflec-tion was insignificant before reaching the failure load, after whichthe out-of-plane deflection increased significantly with decreasingload.

Themember forces against deflections of columnsunder pinnedand fixed end conditions and for the failuremembers of the trussesare plotted in the same figure (Fig. 15) for comparison. The testresults show that the resistances of both the steel and CFS tubularmembers were somewhere between the curves for members withpinned and fixed end conditions which implies that the boundarycondition of the trusses members is semi-rigid.

3. Comparisons between test and design capacities from codes

3.1. Design methods in EC3 and EC4, CoPHK, AISC LRFD and AS 5100

The resistances of steel and CFS tubular columnswere predictedin which As, Ac, fyd and fcd are the cross-sectional area and the de-sign cylinder strength of the steel and concrete respectively.

iandPo = AsFy + 0.85Ac f c (6)in which Fy and f c are the yield stress of steel and compressivestrength of concrete respectively and Pe is the elastic buckling load.

The design compressive resistance of CFS tubular columns canbe obtained as,Pd = 0.75Pn. (7)

3.1.3. AS 5100The ultimate section capacity can be determined by summing

the axial capacity of the steel tube and concrete as follows.

Nus = 0.9Asfy + 0.6Ac f c (8)in which fy and f c are the nominal yield strength of steel and

method in design codes and method of second-order analysis, theproperties from material tests were used in predicting the resultsand compared with test results. The effective length factors equalto 1.0 (Le = 2000) and 0.5 (Le = 1000) were assumed for thepinned and fixed end conditions respectively. Predicted capacityaccording to the different design methods are summarized andcompared in Table 5.

Generally speaking, for steel columns under pinned and fixedend conditions, conservative predictions were obtained fromEC3 [9], AISC LRFD [11] and CoPHK [13] with an under-estimationaround 5%23%, and a close estimation was made by AS 5100 [12](under-design by 1%3%). For CFS tubular columns under pinnedand fixed end conditions, highly conservative predictions wereobtained by AISC LRFD [11] (over-design by 29%35%) due tothe conservative resistance factor, and a reasonable predictionwas made by the other codes with the difference is less thanand

= 12

1+ ( 0.2)+ 2

(3)

in which is the imperfection factor and is the relative slender-ness.

3.1.2. AISC LRFDThe compressive strength of the CFS tubular columns is calcu-

lated as,

For Pe 0.44Po Pn = Po[0.658

PoPe

](4)

For Pe < 0.44Po Pn = 0.877Pe (5)

and

c = 1

1

[90

]2 . (10)The terms and depend on the relative slenderness ratio (r)and section constant (b).

3.2. Predicted results and comparisons

The material factors used to reduce the strengths and modulusof elasticities of steel and concrete were taken as unity incomparison with the test and predicted results according to the3168 M. Fong et al. / Engineering Str

SG 1

SG 2

1

324

15000 10000 5000Strain ()Steel truss

00

100

200

300

5000

SG 4

SG 3

Load

(kN)

Fig. 14. Load against str

Fixed end column

Truss

Pined end column

500 20 40

Mid-length deflection (mm)Steel column and truss

60 80 1000

50

100

150

Load

(kN)

200

250

300

Fig. 15. Member force against

The reduction factor is determined by

= 1 +

2 2

(2)characteristic cylinder strength of concrete. The values of 0.9 and0.6 are the capacity factors for steel and concrete respectively.uctures 33 (2011) 31623171

SG 1

SG 2

1

324

Composite truss

0

100

200

300

20000 15000 10000 5000Strain ()

0 100005000

SG 4

SG 3

Load

(kN)

ain value of the trusses.

Fixed end column

Truss

Pined end column

0 20 40Mid-length deflection (mm)

Composite column and truss

60 80 1000

50100150

Load

(kN) 200

250300350

n-plane lateral deflection of co.

The stability of CFS tubularmembers is considered by a bucklingreduction factorc in calculating themember resistance as follows,

Nuc = c(0.9Asfy + 0.6Ac f c ) (9)7%. Although the design codes such as the AS5100 [12], amongothers, over-estimate the column capacity, material factors have

reffects in analysis process.For steel members, the following section capacity equation is

adopted.

PPp+ My + P(y +y)

Mpy+ Mz + P(z +z)

Mpz= 1 (11)

in which P is the applied force, Pp is the compressive capacity ofsteel,My andMz are the external moments about the y and z axes,P(y+y) and P(z+z) are the P- and P-moments about the yand z axes,Mpy, andMpz , are the moment capacities of steel cross-section about the y and z axes. Note that Eq. (11) is symbolic inthat the P-moment need not be evaluated explicitly but includedin the analysis by updating of member curvature.

For composite members, two section capacity equations are

is equal to 1.0.The predicted results for both columns and trusses are shown

in Table 6. For the columns under pinned and fixed end conditions,the ratios of test to predicted results are 1.09 and 1.08 for steeltubular columns, and 1.14 and 1.03 for CFS tubular columns withthe initial imperfection equal to L/300, and a closer prediction wasmade with the imperfection of L/400 that the ratios are 1.02 and1.03 for steel tubular columns, and 1.08 and 0.98 for CFS tubularcolumns.

The computer model of the truss is shown in Fig. 16. The centerto center length of the members was used in the model and rigidconnection between members was assumed. The analysis resultswith two imperfection values are presented together with testresults in Table 6. A close and conservative prediction was madeM. Fong et al. / Engineering Str

Table 5Predicted result from different design codes.

Specimen Endcondition

Memberforce (kN)

Predicted capacity f

EC3 EC4 AISC

Steel tubular column Pin 152.45 124.23 141.21Steel tubular column Fix 249.40 226.59 223.63Member in steel truss / 216.57 Pin end assumed

Fix end assumedCFS tubular column Pin 186.88 184.33 145.35CFS tubular column Fix 331.60 353.21 246.39Member in composite truss / 279.47 Pin end assumed

Fix end assumed

been adopted in the codes but not considered in this paper forconsistency and therefore the final design in these codes shouldstill warrant a design with adequate factor of safety.

In the truss tests, endmovements of thememberswere allowedbut restrained partly by other members. Hence, the boundarycondition of the truss members was between the pinned and fixedend condition cases. Here, both the predicted resistances of themembers under pinned and fixed end conditions are comparedwith experimental results are reported in Table 5. As expected, avery conservative prediction was made with the difference from40% to 74% for steel tubular members and 44%92% for CFS tubularmembers if the pinned end condition was assumed which impliesan uneconomical design. If the fixed end boundary condition wasassumed, the capacity of the members would be over-estimatedfor steel tubular members in the range of 3%16%, and CFS tubularmemberswith 21% exception for AISC LRFD [11]which gives a loadbelow the tested load.

4. Comparisons between test and second-order analysis anddesign method

4.1. Design by second-order analysis

Using the second-order analysis and design method, the as-sumption of effective length is avoided and the nonlinear effectsare included automatically in analysis with true behavior of thestructure reflected. The detailed formulations of the element, tan-gent stiffness and secant stiffness matrix for steel and compositemembers have been reported by Chan and Zhou [19,26], and Fonget al. [27], and will not be repeated here.

4.2. Section capacity check equations for second-order analysis anddesign method

In the second-order analysis and design method, the sectioncapacity check equations are used and included the second-orderused for two different load conditions. In Eq. (12), the effects ofaxial force and moments are taken into account in the sectionuctures 33 (2011) 31623171 3169

om design code Ratio

CoPHK AS 5100 Test/EC3 EC4 Test/AISC Test/CoPHK Test/AS 5100

132.54 154.61 1.23 1.08 1.15 0.99236.64 256.75 1.10 1.12 1.05 0.97

1.74 1.53 1.63 1.400.96 0.97 0.92 0.84

184.33 193.96 1.01 1.29 1.01 0.96353.21 354.81 0.94 1.35 0.94 0.93

1.52 1.92 1.52 1.440.79 1.13 0.79 0.79

capacity equation, and it is used when the applied force is largerthan the capacity of concrete section (i.e. P > Ppm). And in Eq. (13),only themoment is considered since the axial force does not reducethe failure load and it is usedwhen the applied force is less than thecapacity of concrete section (i.e. P Ppm). These two sets of sectioncapacity equations are given as follows.

For P > PpmP PpmPcp Ppm +

My + P(y +y)Mcpy

+ Mz + P(z +z)Mcpz

= 1 (12)

For P Ppm My + P(y +y)Mcpy+ Mz + P(z +z)

Mcpz= 1 (13)

in which Ppm, Pcp are the compressive capacities of concrete andcomposite cross-section,Mcpy, andMcpz are the moment capacitiesof composite cross-section about the y and z axes.

As can be seen in Eqs. (11)(13), the P- and P- momentsare included in the section-capacity check equations such that theapproximation of effective length factor and computation of elasticbuckling load factor is not needed.

4.3. Analysis results

The average yield stress and Youngs Modulus of steel andconcrete from material tests were used in the computer analysis.Two initial imperfections equal to L/300 and L/400 were used forboth steel and composite members, where L is the member length.The imperfection of L/300 is recommended in CoPHK [13] for bothsteel and CFS tubular columns for second-order analysis. Anotherset of imperfections of L/400 was also used in the prediction thecolumn resistance to demonstrate the degree of conservative of therecommended magnitude of imperfections. The load incrementfactor 0.01 kNwas used in analysis until the section capacity factorwith the discrepancy within 5% for both steel and CFS tubularmembers when using the imperfection of L/400.

srwith the experimental results according to the design methods inEC3 [9], EC4 [10], CoPHK [13], AISC LRFD [11] and AS5100 [12].The comparisons show conservative predictions of steel columnsfor the design codes with the exception of AS5100 [12] givingan aggressive ultimate load prediction and a close prediction forcomposite columns was made except with AISC LRFD [11] whichunderestimates considerably the resistance of the CFS tubularcolumns. The comparisons with the test results of trusses andpredicted results according to the design codes by assuming theeffective length as member length for pinned and half of memberlength for fixed end conditions indicated that the pinned endassumption under-estimated the resistance of themembers whichled to an uneconomical design, while the fixed end assumption

sway frame classification or determination from inspection ofbuckling mode shape.

Acknowledgments

The authors acknowledge the financial support by the ResearchGrant Council of the Hong Kong SAR Government on the projectsCollapse Analysis of Steel Tower Cranes and Tower Structures(PolyU 5119/10E) and Stability and second-order analysis anddesign of re-used and new scaffolding systems (PolyU 5116/11E).

ReferencesFig. 16. Analytic truss model.

5. Conclusions

Experimental investigation on resistances of steel and CFS tubesused as isolated columns and as the members of trusses waspresented in conjunction with the numerical and codified resultsin this paper. The beneficial effects of in-filled concrete on steeltubes for both columns under pinned and fixed end conditionsand members of trusses were reported. The resistances of thesteel and CFS tubular members were determined and compared

effective length method was demonstrated in this paper that theidealized pinned and fixed end conditions did not exist in practiceand the results by either assumption were inaccurate. This paperfurther determined the resistance of isolated columns and struts intrusses by using the second-order analysis and designmethodwithtwo imperfection values. The results indicated that the second-order analysis and design method not only provided an accuratedesign solution, but also avoided uncertain approximation ofeffective lengths which involved complications in sway or non-3170 M. Fong et al. / Engineering Str

Table 6Predicted truss results from second-order analysis and de

Specimen Endcondition

Membe(kN)

Steel tubular column Pin 152.45Steel tubular column Fix 249.40Member in steel truss / 216.57CFS tubular column Pin 186.88CFS tubular column Fix 331.60Member in composite truss / 279.47over-estimated the capacities of the members and the designbecame non-conservative. The main disadvantage of using theuctures 33 (2011) 31623171

ign method.

force Predicted capacity from second-orderanalysis and design methodLoad (kN) RatioL/300 L/400 L/300 L/400

140.43 149.28 1.09 1.02229.96 241.92 1.08 1.03190.20 205.70 1.14 1.05164.11 172.30 1.14 1.08322.50 340.10 1.03 0.98254.36 272.94 1.10 1.02[1] Furlong RW. Strength of steel-encased concrete-filled beamcolumns. J StructDiv, ASCE 1967;93(5):11324.

M. Fong et al. / Engineering Str

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[10] BSI. Eurocode 4: design of composite steel and concrete structurespart 1.1:general rules and rules for buildings. London: BS EN 1994-1-1, BSI; 2004.

[11] AISC. Load and resistance factor design specification for structural steelbuildings. Chicago: American Institute of Steel Construction, AISC Inc.; 2005.

[12] Standards Australia. AS5100.6-2004 bridge design, part 6: steel and compositeconstruction. Sydney (Australia); 2004.

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Advanced design for trusses of steel and concrete-filled tubular sectionsIntroductionExperimental workGeneralMaterial propertiesColumn testsTest setupStrain gages and displacement transducersTest results

Truss testTest setupStrain gages and displacement transducersTest result

Comparisons between test and design capacities from codesDesign methods in EC3 and EC4, CoPHK, AISC LRFD and AS 5100EC4 and CoPHKAISC LRFDAS 5100

Predicted results and comparisons

Comparisons between test and second-order analysis and design methodDesign by second-order analysisSection capacity check equations for second-order analysis and design methodAnalysis results

ConclusionsAcknowledgmentsReferences