Engineering Structures 30 (2008) 616–625www.elsevier.com/locate/engstruct

Second-order analysis and design of angle trussesPart I: Elastic analysis and design

S.L. Chan∗, S.H. Cho

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong

Received 9 January 2007; received in revised form 29 April 2007; accepted 11 May 2007Available online 26 June 2007

Abstract

Angle members are widely used in light-weight steel skeletons and they are commonly subject to high axial force with eccentricity. Differentdesign codes recommend varied load resistances and second-order analysis widely used in design of steel frames of doubly symmetrical sectionsis seldom reported. This paper fills the gap by proposing a practical second-order analysis and design method for trusses composed of anglessections. Realistic modeling of semi-rigid connections associated with one- and two-bolt end connections with flexible gusset plate and memberimperfections such as initial curvatures and residual stresses is made. Load eccentricity is also simulated. The proposed method can be readilyapplied to reliable, robust, efficient and effective design of angle trusses and frames without the uncertain assumption of effective length.c© 2008 Published by Elsevier Ltd

Keywords: Second-order analysis; Single angle strut; Semi-rigid connections; Buckling

1. Introduction

Single angle members have a broad range of applications,such as web members in roof trusses, members of transmissiontowers and other bracing members. Most angle members areslender and therefore relatively weak in compression resistancecompared with other steel sections, but angle sections arewidely used because of their light weight with the L-shapedsection making the angles easy for storage, transportation andfabrication. The design of angle trusses is complicated bythe structural behaviour as follows. Firstly, since angles areasymmetric or mono-symmetric, their principal axes are alwaysinclined to the plane of truss or frame. Secondly, it is notuncommon to bolt or weld an angle member to another memberdirectly or to a gusset plate at its end through their legs.Therefore, in practice, an angle member is loaded eccentricallythrough one leg. As a result, an angle member is subject toan axial force as well as a pair of end moments at it ends.Since single angle web members may be attached to the chordmembers on the same side or on alternate or opposite sides,this affects the directions of the moments. Twisting may also

∗ Corresponding author.E-mail address: ceslchan@polyu.edu.hk (S.L. Chan).

0141-0296/$ - see front matter c© 2008 Published by Elsevier Ltddoi:10.1016/j.engstruct.2007.05.010

appear simultaneously as the shear centre of the cross-sectionis located at the point of intersection of the two legs away fromthe centroid. Finally, the connection at each end provides somedegree of end fixity which is beneficial to the compressionresistance of the angle members. This further complicates theanalysis of angle members. The mentioned features are almostunique to angle sections making the design of single anglemembers controversial for some time. In a rational designprocedure, the adverse effect of the end eccentricity and thebeneficial effect due to the end restraint on the compressioncapacity should be considered. However, in most conventionaldesign methods and codes widely used today, the designprocedure is over-simplified with many assumptions not valid.For example, the load eccentricity and the end restraint may beneglected during the analysis.

In order to study the structural behaviour of eccentricallyloaded single angle struts, Trahair et al. [1] carried out aseries of tests of eccentrically loaded single angle struts. Thetested sections included 51 × 51 × 6 mm equal angle and76 × 51 × 6 mm unequal angle. The slenderness ratio rangedfrom 60 to 200, which covered the slenderness range in practice.At each end, one leg was welded to the web of a structure tee.This simulated the chord of a truss. Load was applied throughthe centre of the web of the structural tee, which contributed

S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625 617

to the load eccentricity. The load was applied to the structuralTee in three conditions as Condition A: the loading head of thetesting machine flat on the flange of the tee; Condition B: theload applied through a knife-edge in the plane of the web ofthe structural tee, i.e., in the plane of truss; Condition C: theload applied through a knife-edge parallel to the web of thetee. These three end conditions symbolized the extreme casesof a real situation. Condition B gave the lowest failure loadscorresponding to the situation that the chord buckles by twist-buckling and consequently it cannot restrain the single anglestrut from deforming out of the plane of the truss. Althoughthis situation is an extreme case and rarely occurs in reality, itrepresents the worst scenario and provides us with an aspectfor future research work. Adluri and Madugula [2] comparedresults of some experimental data on eccentrically loaded singleangle members free to rotate in any directions at the ends fromthe available literature with AISC LRFD [3] and AISC ASD [4]specifications. The experimental investigations, which covereda wide spectrum of single angle struts, were carried out byWakabayashi and Nonaka [5], Mueller and Erzurumlu [6], andIshida [7]. These results were summarized and concluded thatthe interaction formulae given in AISC LRFD [3] and AISCASD [4] are highly conservative when applied to eccentricallyloaded single angle members. It is because these interactionformulas were derived primarily for doubly symmetric sectionsand the moment ratios in these formulae are evaluated forthe case of maximum stresses about each principal axis. Thispractice does not pose a problem on doubly symmetric sectionssuch as I-sections because the four corners are critical formoments about both principal axes simultaneously. However,for angle sections, as they are monosymmetric or asymmetric,the points having maximum bending stress about both principalaxes sometimes do not coincide. As a consequence, the loadingcapacities of the sections calculated from these interactionequations are underestimated [2]. In order to eliminate theunnecessary discrepancy between the actual failure load andthe design load, Adluri and Madugula [2] suggested that themoment interaction factors given in AISC LRFD [3] shouldbe revised. Bathon et al. [8] carried out 75 full-scale testswhich covered a slenderness ratio ranging from 60 to 210.The test specimens were unrestrained against rotation at theend supports. It was noted that the ASCE Manual 52 [9]under-predicted the capacities of single angle struts. Theabove-mentioned research did not include the effect due toend connection details, which may also affect the bucklingresistances of the angle struts. Elgaaly et al. [10] conducted anexperimental program to investigate the structural behaviour ofnon-slender single angle struts as part of a three-dimensionaltruss. The specimens cover a range of slenderness ratio from60 to 120 including single-bolted and double-bolted conditions.Results show that both the ASCE Manual 52 [9] and AISCLRFD [3] are inadequate for single angle members with lowslenderness ratios. Theoretically, single angle struts whose webmembers are connected on alternate sides should have lesscompression resistance than the same struts connected on thesame side. Kitipornchai et al. [11] carried out compressiontests on two fully welded trusses which were fabricated and

tested to compare the ultimate capacity of trusses having webmembers on the same side with those having web members onalternate sides. The slenderness ratios of the failure membersranged from 120 to 160. The test confirmed that the same-sided trusses were considerably stronger than the alternate-sided trusses. However, most of the design codes seem to ignorethis consideration. Woolcock and Kitipornchai [12] proposed asimplified design method based on theoretical and experimentalobservations [13,14]. In the proposed method, not only is theeffect of load eccentricity taken into account, but cases whereweb members are all on one side or on the opposite side arealso considered. This method was considered simpler and moreeconomical than the conventional axial force-biaxial bendinginteraction approach.

Kitipornchai and Chan [15] employed a nonlinear numericalapproach to solve the problem of elastic behaviour of isolatedrestrained beam–columns. The element geometric stiffnessmatrix was derived from the principle of minimum of potentialenergy incorporating member geometrical nonlinearity. Theeffects of the load eccentricity and the shear centre notcoincident with the centroid were also taken into account. Theequilibrium paths were traced from the incremental and thetotal-force–deformation equilibrium equations using the arc-length method. The results were compared with those reportedby Trahair [16]. It was shown that when the geometry wasnot updated, with sufficient elements per member (e.g. 4elements per member), the results agreed well with thefinite integral solutions. However, when the geometry wasupdated, the influence of the pre-buckling deformations wasapparent, indicating the importance of the consideration ofgeometry update. The trusses tested by Kitipornchai et al. [11]were analyzed using both the fiber model and the lumpedplasticity model in the elasto-plastic finite element methodwith no assumption of initial curvatures of the members.Results indicated that both models could be used to predictthe nonlinear structural behaviour with moderate accuracy.To make the nonlinear method more user-friendly with fewerelements per member required to achieve satisfactory accuracy,Al-Bermani and Kitipornchai [17] proposed an approachallowing the use of least elements in a nonlinear analysisacounting for both geometric and material nonlinearity. Theprocedure is suitable for analyzing large-scaled space framessince the structures may be modelled using only a few elementsper member. It is achieved by incorporating a displacementstiffness matrix which provides the necessary coupling betweenthe axial, flexural and torsional deformations. However, thiselement does not allow for member initial imperfection, whichis mandatory in some national design codes such as Eurocode3 [18]. For simulation of member initial imperfection, at leasttwo elements per member will be required.

A review of literature shows there is a lack of research workon testing of single slender angle trusses with an objectiveof verifying a new second-order analysis design method.Comparisons with the experimental results and the predictedresults by the traditional simplified method instead of the axialforce–moment interaction method are also inadequate. Muchresearch on comparing the accuracy of a finite element package

618 S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625

against the experimental results has been conducted and it isnot the aim of this paper. The objective of this paper is topropose a new design method with verification by testing offull scale trusses and comparison with code formulae in orderto indicate the inadequacy of our current codes for designof angle trusses. In the investigation, a series of laboratorytests of angle members were conducted including single-boltedand double-bolted end conditions and web members on thesame side and on alternate sides. This paper also describesthe experimental programme and results. Comparison amongthe test results will be made between the experimental resultsagainst those predicted by the design rules. A proposed second-order analysis and design method is also introduced and themethod is validated by the test results. The suggested newmethod allowing for imperfections due to initial membercrookedness, residual stress as well as single and double boltconnections and connection on the same and on alternativesides of the trusses is envisaged to carry a great potential fordesign application to angle structures in replacement of the oldlinear elastic design approach.

2. Conventional design method

The design codes widely used today, such as AISCLRFD [19], BS5950 [20] and Eurocode 3 [18], were developedbased on nonlinear analysis of simple idealized individualmembers. However, the conventional design procedure ofstructural steelwork is based on simple linear analysis withthe nonlinear effects ignored and buckling effect approximatelyaccounted for using the effective length method. Whendesigning a steel structure, this first-order linear analysis can besummarized as follows. Firstly, a linear analysis is carried outto determine the internal forces and moments of all membersunder external loads. Afterwards, the resistance of eachindividual member is estimated according to the design rulesgiven in the codes to account for the nonlinear effects whichinvolve approximation of effective lengths for compressionmembers. A sufficiently safe structure has its resistance largerthan the factored forces and moments according to this method.

Strictly speaking, the first-order linear theory is onlyapplicable for structures with small displacements with aless severe degree of nonlinearity and for members with lowslenderness. This is in contrast with the actual non-linearstructural behaviour soon after the application of load. Takinga simply supported column subject to a point load at one endas an example, the column deflects in both axial and transversedirections once a load is applied. This type of nonlinearity isdefined as the P–δ effect. However, in a linear analysis, thecolumn will only be shortened without buckling. In this simplecase, it can be seen that the column deflects laterally as asecond-order effect which may become more severe than thefirst order effect in terms of stress induced by P–δ moment(i.e. P·δ

Z being greater than the axial stress PA in which Z is the

elastic modulus and A is the cross sectional area). If the columnis part of a structure, from a global point of view, the localdeflection and the global displacements will mutually affecteach other and the structural response of the other members

as well. To further complicate the situation with the presenceof initial curvature and limited value of yield stress, lateraldeflection occurs once the load is applied and the bucklingstrength is elasto-plastic in nature such that the concept of“elastic” effective length is strictly speaking inapplicable. Thistype of nonlinearity is regarded as the P–δ effect. In the first-order linear analysis, the column is assumed to be perfectand will only shorten with the axial load and the simplecompressive stress cannot be used to estimate the bucklingresistance; while in the design stage, in order to compensate forincreased compressive stress due to flexure from P–δ effect,the compressive strength of the column is reduced accordingto the assumed effective length of the column. The effectivelength is calculated by multiplying the effective length factoror the K-factor to the actual member length in an empiricalmanner. The effective length factor is largely dependent on theend condition of the member. So, the checking of the bucklingresistance is carried out as an independent stage instead of anintegrated part of design. The interaction with buckling andother second-order effects is always ignored. Therefore, thistype of analysis is incorrect in calculating forces and moments.Furthermore, simple columns with idealized pinned ends rarelyexist in real structures; in fact, the joints connecting two ormore members are finite in their stiffness. They would displaceand affect the compression resistance so that the accuracyof the compressive strength estimated by the design codesas the failure strength is highly dependent on the assumedeffective length. If the effective length is assumed longer thanthe actual effective length, the design may be conservative.Conversely, if the assumed effective length is taken as shorterthan the actual effective length, the design may be dangerous.Consequently, any wrong assumptions of effective length maylead to an uneconomical design or an unsafe design. This designprocedure shows an inconsistency between analysis and design.

Regarding structural design of single angle members, theactual structural behaviour is far more complicated than thatof a simple column with doubly-symmetrical section. In spiteof this, the analysis and the design processes adopted inthe conventional design approach such as BS5950 [20] andEurocode 3 [18] appear to be over-simplified. In reality, theend restraints and the end eccentricities affect the compressivestrength of the member. However, these effects are oftenignored during analysis; instead, the slenderness is modifiedand the compression resistance of the member is taken as afraction of that of a concentrically loaded member. Apart fromthe inconsistency between analysis and design, since many ofthe design rules in codes are developed based on minor axisflexural buckling, they may not be adequate for stocky memberswhich are susceptible to flexural–torsional buckling. Further,the equivalent slenderness ratio is based on a certain commonlyused detailing dimensions such as eccentricity being not toolarge which may become invalid for some new structural forms.

3. The second-order elastic analysis and design method

The conventional design procedure based on first orderlinear analysis is traditionally used because during the pre-computer age when computer time was expensive and computer

S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625 619

speeds were slow. Today, the prevalence of low-cost personalcomputers and the growing importance of environmental andeconomical concerns provide a natural choice to develop apractical second-order analysis method. This method has beenwell-researched by Chen and Chan [21], Chan and Chui [22],Chan and Cho [23] with the second-order effects includedduring the analysis through update of geometry. In other words,the member deflection (δ) and the global displacement (∆) aretaken into account so that section capacity check is adequate forstrength design as follows:

P

Ag py+

My + P (δz + ∆z)

Z y py

+Mz + P

(δy + ∆y

)Zz py

= φ ≤ 1 (1)

in which py is the design strength, P is the external forceapplied to the section, Ag is the cross-sectional area, My and Mzare the external moments about the y and z axes respectively,Z y and Zz are the section modulus about the y and z axesrespectively. P (δz + ∆z) and P

(δy + ∆y

)are the collective

moments about the y and z axes respectively due to the changeof member stiffness under load and large deflection effectsof which the consideration allows for the effect of “effectivelength” automatically. In other words, there is no need to reducethe compressive strength of the member to account for theP–δ and P–∆ effects. Moreover, the characteristics of realisticstructure (e.g. initial imperfection and residual stresses) arealso considered via the equivalent initial imperfection δ0 in theanalysis so the design is completed simultaneously with theanalysis.

To determine the design buckling load of a structure,approximately 1%–10% of the predicted failure load is appliedincrementally until the sectional capacity factor, ϕ, in Eq. (1) isgreater than 1. The Newton–Raphson method combined withthe minimum residual displacement iterative scheme [24] ofsolving the following set of nonlinear simultaneous equation asfollows is utilized with the iterative scheme illustrated in Fig. 1.[∆u

]= [KT (u, δ0, F)]−1

[∆F

](2)

[∆u] = [KT (u, δ0, F)]−1 [∆F

]. (3)

The displacement incremental is determined as,

∂[∆u]T[∆u]

∂λ=

∂ ([∆u] + λ[∆u])T ([∆u] + λ[∆u])

∂λ= 0 (4)

in which [∆u], [∆u] and [∆u] are the displacement incrementsdue to unbalanced force, applied load and summed displace-ment increment for calculation of the unbalanced force. λ is aparameter satisfying the minimum residual displacement con-dition [24].

For designing single angle struts, a method was previouslyproposed by Cho and Chan [25] based on the aforementionedsecond-order analysis and design concept. Using the softwareNIDA (structural analysis software “Nonlinear IntegratedDesign and Analysis” version 7) [26], initial curvature isimposed along the member so that bending can be triggered

Fig. 1. The minimum residual displacement method.

at the instance that the load is applied instead of only axialshortening. The values of the initial curvature are calculatedbased on the compressive strength curve given in BS5950 [20]which are 2.8 × 10−3L for equal angles and 2.0 × 10−3Lfor unequal angles. The results computed by NIDA willagree well with BS5950 [20]. The Hong Kong Steel Code2005 [27] further gives explicit values of imperfections forangles. However, this method cannot truly reflect the endcondition that a practical angle is exposed to. The methodwas modified by Chan and Cho [28]; the end condition issymbolized by a rotational spring element inserted at each endof the member. The value of the rotation spring stiffness iscalculated from the dimensions of the gusset plates and itsmaterial properties. Therefore, the rotational stiffness due to thedouble-bolted connection can be considered at the early stageof the analysis rather than at the design stage as in the linearanalysis and effective length design method. Only the rotationaldeformation of the connection spring element is consideredfor design because the effects of the axial and shear forces inthe connection deformations are small when compared withthat of bending moments. However, this modified method isstill inadequate to consider the directions of the end moments.Under some circumstances, these end moments would beadvantageous to the overall structure. In this paper, the methodis further refined. The end moments due to load eccentricity areconsidered by connecting the angle web members at each endto the chord members by rigid arms. The rigid arm will be theelement joining the centroid and the point of load application sothat the magnitude and the direction of the end moments due toload eccentricity can be taken into account immediately duringthe analysis. For single-bolted connection, the connection jointsare allowed to rotate freely. For double-bolted connection,rotational springs are inserted to the joints connecting rigidarm elements to the angle web member element in the in-plane direction so that the couples due to the double-boltedconnection can be considered. The merit of this approach overthe purely equivalent imperfection approach is that it considersthe direction of the end moments so that the aforementionedeffect can be reflected during analysis. The spring stiffness of arotational spring can be calculated as follows:

As shown in Fig. 2, the couple, M , formed by the pair ofbolts is given by:

M = F · d = kθ. (5)

620 S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625

Fig. 2. Couple formed by shear forces of two bolts.

The shear stress, τ , across the cross-section, A, of the bolt is:

τ =F

A(6)

in which A is the shear area and can be taken as 0.9 of the crosssectional area as recommended in most design codes like theHong Kong Steel Code 2005 [27].

The shear strain, γ , of the bolt shank is:

γ =2δ

l=

d · θ

l=

τ

G(7)

Rearranging terms, the rotational stiffness, k, due to the double-bolted connection will be given by:

k =G Ad2

l(8)

where F is the shear force exerted on the bolt; d is the distancebetween the centroids of the two bolts; θ is the rotation of thebolt group; δ is the displacement of the bolt; l is the length ofthe bolt shank; G is the shear modulus of elasticity.

To account for the rotation stiffness of the spring element inthe analysis, the following incremental tangent stiffness matrixis superimposed to the element stiffness matrix.[

MeMi

]=

[Sc −Sc

−Sc Sc

] [θeθi

](9)

in which Me and Mi are the incremental external and internalmoments at two ends of a connection. The external node refersto the one connected to the global node and the internal nodeis joined to the angle element. The stiffness of the connection,Sc, can be related to relative rotations at the two ends of theconnection spring as:

Sc =Me

θe − θi=

Mi

θi − θe(10)

in which θe and θi are the conjugate rotations for the momentsMe and Mi .

3.1. The incremental and iterative procedure for analysis ofangle trusses and frames

The basic equations for incorporating the end connectionstiffness are considered both in the tangent and the secantstiffness matrix equations as follows.

Fig. 3. The external and internal rotations.

Tangent stiffness matrixThe incremental force is assumed in the software and

the incremental displacement is solved. The basic elementstiffness is modified by addition of the tangent stiffness of theconnection spring modeled as a dimensionless spring elementin a computer analysis as:

S1 −S1 0 0−S1 k11 + S1 k12 0

0 k21 k22 + S2 −S20 0 −S2 S2

θ1eθ1iθ2iθ2e

=

M1eM1iM2iM2e

(11)

in which S1 and S2 are the spring stiffness for simulation ofsemi-rigid connections at ends; ki j is the stiffness coefficientsof the element; θ1e, θ1i , θ2e and θ2i are respectively the rotationsat two sides for the two ends of an element shown in Fig. 3.

Assembling the element matrices, the global stiffnessmatrix for the angle frame and truss is formed and storedin a one dimensional array in the computer analysis. Theincremental displacement vector is solved and added to thelast displacement and used for determination of resistance asfollows.Resistance determination

The resistance is determined as the sum of resisting forcesfor all elements as,

[R] =

∑[ke][L S][u] (12)

in which [R] is the structural resistance of the angle frame, [ke]is the secant stiffness, [Ls] is the transformation matrix fromlocal to global coordinates and [u] is the accumulated elementdisplacement transformed to the element local axis.

The resistance of the angle frame is determined as themaximum load not violating Eq. (1). This is based on theconventional use of load causing the formation of the firstplastic hinge whereas the second part of this paper describesthe extended consideration of elasto-plastic buckling analysisof angle frames.

4. Laboratory tests of single angle struts

4.1. Experimental programme

Four single angle struts were tested as web members of atwo-dimensional truss as shown in Fig. 4. In the first set, theweb members of the truss are connected to the chord memberson the same side. In the second set, the tests are repeated withthe web members connected to the chord members on alternate

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Fig. 4. Truss before test (webs on alternate sides).

sides. The specimens are of Grade S275 and two metres longmaking the slenderness ratio around 150. The leg length-to-thickness ratio meets the BS5950 [20] requirements so thatlocal buckling can be ignored. Each end of the member isconnected to a gusset plate. The test included single and doublebolted connections. Other details of the specimens are listedin Table 1. The trusses were loaded in pairs and sufficientlateral restraints were provided to ensure out-of-plane bucklingat connecting nodes between chords and webs is fully avoided.Load was applied at the upper joint of the target failure memberthrough a hydraulic jack. As shown in Figs. 5 and 6, twelvestrain gauges were mounted evenly at the mid-length of thetargeted member and the tested truss was so designed thatthe targeted member fails first. At the targeted member, twodisplacement transducers were placed in in-plane and out-of-plane directions and transducers were also used to monitor themovements of the top and the bottom joints of the targetedmember so that its movement of the target member relativeto the truss can be measured. At the load where the targetedmember buckled or failed, the deformations of the remainingparts of the truss were small and reversible. Thus, after eachtest, the failed member was replaced by a new specimen sothat the next test could be conducted under almost the sameconditions.

Fig. 5. Locations of strain gauges.

Table 1Details of test specimens

Set Specimen Size Webarrangement

Endconditions

Gussetdimensions

1 1a 65×65×6 Same side Single bolt 240 × 180 × 81b 65×65×6 Same side Double bolt 240 × 180 × 8

2 2a 66×66×6 Alternate sides Single bolt 240×180×102b 66×66×6 Alternate sides Double bolt 240×180×10

4.2. Material properties

Four coupon tests were performed to determine the materialproperties of the steel used in the test specimens following theprocedure given in BS EN 10002-1 [29]. The test results aresummarized in Table 2. Coupons 1a, 1b, 2a and 2b were cutfrom Specimens 1a, 1b, 2a and 2b respectively.

4.3. Test results

The major failure modes are flexural buckling about theprincipal minor axis as shown in the photos (Figs. 7 and 8).Fig. 7 shows the buckled shape of Specimen 1a, of whicheach end is connected to the gusset plate with a single bolt,making it behave more like pin-ended in the in-plane direction.In the meantime, the gusset plate provides some flexibility inthe out-of-plane direction. Fig. 8 shows the buckled shape ofSpecimen 1b, of which each end is connected to the gusset plate

Fig. 6. Locations of transducers.

622 S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625

Table 2Coupon test results

Set Coupon Young’s modulus,E (kN/mm2)

Yield stress, σy (N/mm2)

1 1a 216.9 347.01b 211.8 347.6Average 214.4 347.3

2 2a 194.5 348.62b 185.6 344.9Average 190.0 346.7

Fig. 7. Weak-axis buckling mode (single-bolted, same side arrangement).

by two bolts. The double bolt connection at each end providessome flexibility making it behave as if partially restrainedin the in-plane and out-of-plane directions. Figs. 9 and 10respectively show the in-plane and out-of-plane deflections ofthe four specimens. As can be seen from the curves, theirresponse pattern are similar and the out-of-plane deflectionsare always more severe than the in-plane deflections. Table 3listed the member failure loads, which are calculated usingnumerical integration of the stress over the cross-sectionalarea. It can be seen that the load capacities of the specimenswith double bolt end connections are 9%–15% higher than thecounterparts with single bolt end connections. For the sameend conditions, specimens with alternate side web arrangementwill have the load capacities 15%–20% lower that those withsame side web arrangement. The alternate side arrangement ofthe web members will make the end moments due to eccentricconnections more severe.

Fig. 8. Weak-axis buckling (double-bolted, same side arrangement).

Table 3Experimental failure loads

Set Specimen Failure load (kN) Failure load/Squash load

1 1a 67.2 0.2601b 78.4 0.303

2 2a 57.5 0.2192b 72.1 0.275

5. Comparisons

5.1. Comparisons with BS5950

BS5950 [20] provides a simplified method for designingstruts composed of single angles. They may be treated asaxially loaded members with reduced compressive strengthwith ignorance of eccentricity at end connections, provided thatthe following conditions are fulfilled:

(a) by two or more bolts in standard clearance holes in linealong the angle, or by an equivalent welded connection, theslenderness λ should be taken as the greater of:

0.85λv but ≥ 0.7λv + 15

1.0λa but ≥ 0.7λa + 30

(b) by a single bolt, the compression resistance should be takenas 80% of the compression resistance of an axially loadedmember and the slenderness λ should be taken as the greaterof:

1.0λv but ≥ 0.7λv + 15

1.0λa but ≥ 0.7λa + 30

S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625 623

Fig. 9. Experimental in-plane deflections of the specimens.

Fig. 10. Experimental out-of-plane deflections of the specimens.

which λv and λa are referred to as the slenderness ratiosabout the principal minor axis and rectangular axis respectively.For the cases that the aforementioned conditions are notsatisfied, the struts should be designed as members withcombined moment and axial force. Using the measuredmaterial properties, the simplified method gives the followingresults (Table 4). This method is not able to show thedifference between the same side arrangement and alternateside arrangement of web members. In other words, providedthat the materials are identical in two sets of tests, thepredicted results would be identical also. As can be seenfrom the results, this method provides overly conservative(45.9%–61.8%) estimates for angle struts with single-bolted connection. On the other hands, it provides lessconservative (10.4%–14.5%) estimates for the double-boltedcounterparts.

Table 4BS5950 failure loads

Set Specimen BS5950 load (kN) Test load/BS5950 load

1 1a 41.5 1.6181b 68.5 1.145

2 2a 39.4 1.4592b 65.3 1.104

5.2. Comparisons with NIDA

The failure loads are predicted by NIDA with the endmoments and end rotational restraints taken into accountduring the analysis. The value of the rotational stiffness kis approximately equal to 1000 kN m/rad. This value isclose to the rigidly fixed condition achievable for use withpreloaded bolts. In the present case of using non-preloaded

624 S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625

Fig. 11. Vertical deflection vs. applied load of a pair of trusses with webs on the same side.

Table 5NIDA failure loads

Set Specimen NIDA load (kN) Test load/NIDA load

1 1a 55.6 1.2091b 66.9 1.172

2 2a 50.4 1.1412b 62.1 1.161

bolts, the value of k is taken as 10% of the calculated value. Itshould be noted that the currently used section capacity checkequation in NIDA is not suitable for angle sections because theequation was derived primarily for doubly symmetric sectionsof which maximum bending stresses about each principal axisoccurs at the four corners simultaneously. However, for anglesections, as they are monosymmetric or asymmetric, the pointsof maximum bending stress about both principal axes do notnecessarily coincide. As a consequence, the loading capacitiesof the sections calculated are usually underestimated. It issuggested that the axial stress is checked against every extremepoint of the section. The computed results are summarizedin Table 5. Compared with the conventional design methodusing BS5950 [20], the nonlinear analysis and design methodprovides less conservative estimates for the single-boltedspecimens (14.1%–20.9%) and narrowly more conservativeestimates for the double-bolted specimens (16.1%–17.2%) fordouble-bolted specimens. Fig. 11 shows the experimental andtheoretical vertical deflections at the point of load applicationagainst applied load of the trusses with webs connected on thesame side. It can be seen that the trend predicted by NIDA arein line with the experimental results.

5.3. Discussions

Comparing the traditional code-based linear design methodwith the proposed NIDA design method, the followingconclusions can be made. First, while the code method cannottake the web arrangement (same side or alternate sides)into consideration, the design method provides conservativepredictions of the failure loads of the truss. Second, for

compression members with single-bolted end condition, NIDAprovides more rational predictions. Third, for compressionmembers with double-bolted end condition, BS5950 providedless conservative predictions. It is envisaged that the 0.8 factorof calculating the compression resistance of a single-boltedcompression member is too conservative for some range ofslenderness ratio. Although the linear code method was notedto have a more accurate prediction for the failure loads of thedouble-bolted compression members using an effective lengthfactor of 0.85, the joint and web arrangement details cannot beincorporated directly into the effective length method such asthe present second-order analysis approach. It is suggested thatthe discrepancy between the test results and the NIDA resultscan be further minimized by considering the rotational stiffnessdue to the presence of the gusset plates and fine-tuning therotational stiffness of the joints at the two ends of the member.

6. Conclusions

This paper illustrates the inconsistency between analysisand design using the conventional methods and codes fordesign of single angle struts. The assumption of effectivelength in the design stage in fact violates the pin-endedassumption in the analysis stage, which usually leads to anover-conservative result and possibly unsafe results if thedisplacement is large. A codified second-order analysis methodfor design of angle trusses is proposed. Using this method, somesecond-order effects (e.g. initial curvatures, residual stresses,P–δ and P–∆) are explicitly included in the analysis sothat an effective length is not required to be assumed. Theproposed design method is validated by laboratory tests oftrusses using single angle members of slenderness ratio about150 as web members and the test results agreed well withthe computed results. However, this method may need tohave a checking for stocky members against flexural–torsionalbuckling modes, which can be carried out by the use of asimple empirical formula in computer programming as theproblem of determining the effective length is not so acutehere, because the torsional-flexural buckling is a local member

S.L. Chan, S.H. Cho / Engineering Structures 30 (2008) 616–625 625

buckling behaviour without significant interaction with framesystem behaviour. An alternative approach is to include thetorsional stiffness matrix for prediction of the buckling. Theextension work will allow the method to cover the whole rangeof slenderness ratio for compression members. Plastic designmay also be included in the analysis such that the plasticstrength reserve in the structure can be determined. This partis reported in the accompanying or part two of this paper.

Acknowledgements

The authors acknowledge the financial support by the Re-search Grant Council of the Hong Kong Special Administra-tive Region Government under the projects “Advanced analy-sis of steel frames and trusses of non-compact sections usingthe deteriorating plastic hinge method (PolyU 5117/06E)” and“Second-Order and Advanced Analysis of Wall-Framed SteelStructures (PolyU 5115/05E)”.

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