Chan_2007_Second-Order Analysis and Design of Angle Trusses_part_1

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Engineering Structures 30 (2008) 616–625

www.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 2007

Available 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. Different

design codes recommend varied load resistances and second-order analysis widely used in design of steel frames of doubly symmetrical sections

is seldom reported. This paper fills the gap by proposing a practical second-order analysis and design method for trusses composed of angles

sections. Realistic modeling of semi-rigid connections associated with one- and two-bolt end connections with flexible gusset plate and member

imperfections such as initial curvatures and residual stresses is made. Load eccentricity is also simulated. The proposed method can be readily

applied 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 transmission

towers and other bracing members. Most angle members are

slender and therefore relatively weak in compression resistance

compared with other steel sections, but angle sections are

widely used because of their light weight with the L-shaped

section making the angles easy for storage, transportation and

fabrication. The design of angle trusses is complicated by

the structural behaviour as follows. Firstly, since angles are

asymmetric or mono-symmetric, their principal axes are always

inclined to the plane of truss or frame. Secondly, it is not

uncommon to bolt or weld an angle member to another member

directly or to a gusset plate at its end through their legs.

Therefore, in practice, an angle member is loaded eccentrically

through one leg. As a result, an angle member is subject to

an axial force as well as a pair of end moments at it ends.

Since single angle web members may be attached to the chord

members 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: [email protected] (S.L. Chan).

appear simultaneously as the shear centre of the cross-section

is located at the point of intersection of the two legs away fromthe centroid. Finally, the connection at each end provides some

degree of end fixity which is beneficial to the compression

resistance of the angle members. This further complicates the

analysis of angle members. The mentioned features are almost

unique to angle sections making the design of single angle

members controversial for some time. In a rational design

procedure, the adverse effect of the end eccentricity and the

beneficial effect due to the end restraint on the compression

capacity should be considered. However, in most conventional

design methods and codes widely used today, the design

procedure 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 eccentrically

loaded single angle struts, Trahair et al. [1] carried out a

series of tests of eccentrically loaded single angle struts. The

tested sections included 51 × 51 × 6 mm equal angle and

76 × 51 × 6 mm unequal angle. The slenderness ratio ranged

from 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 through

the centre of the web of the structural tee, which contributed

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

to the load eccentricity. The load was applied to the structural

Tee in three conditions as Condition A: the loading head of the

testing machine flat on the flange of the tee; Condition B: the

load applied through a knife-edge in the plane of the web of

the structural tee, i.e., in the plane of truss; Condition C: the

load applied through a knife-edge parallel to the web of the

tee. These three end conditions symbolized the extreme casesof a real situation. Condition B gave the lowest failure loads

corresponding to the situation that the chord buckles by twist-

buckling and consequently it cannot restrain the single angle

strut from deforming out of the plane of the truss. Although

this situation is an extreme case and rarely occurs in reality, it

represents the worst scenario and provides us with an aspect

for future research work. Adluri and Madugula [2] compared

results of some experimental data on eccentrically loaded single

angle members free to rotate in any directions at the ends from

the available literature with AISC LRFD [3] and AISC ASD [4]

specifications. The experimental investigations, which covered

a wide spectrum of single angle struts, were carried out by

Wakabayashi and Nonaka [5], Mueller and Erzurumlu [6], andIshida [7]. These results were summarized and concluded that

the interaction formulae given in AISC LRFD [3] and AISC

ASD [4] are highly conservative when applied to eccentrically

loaded single angle members. It is because these interaction

formulas were derived primarily for doubly symmetric sections

and the moment ratios in these formulae are evaluated for

the case of maximum stresses about each principal axis. This

practice does not pose a problem on doubly symmetric sections

such as I-sections because the four corners are critical for

moments 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 loading

capacities of the sections calculated from these interaction

equations are underestimated [2]. In order to eliminate the

unnecessary discrepancy between the actual failure load and

the design load, Adluri and Madugula [2] suggested that the

moment interaction factors given in AISC LRFD [3] should

be revised. Bathon et al. [8] carried out 75 full-scale tests

which covered a slenderness ratio ranging from 60 to 210.

The test specimens were unrestrained against rotation at the

end supports. It was noted that the ASCE Manual 52 [9]

under-predicted the capacities of single angle struts. The

above-mentioned research did not include the effect due to

end connection details, which may also affect the buckling

resistances of the angle struts. Elgaaly et al. [10] conducted an

experimental program to investigate the structural behaviour of

non-slender single angle struts as part of a three-dimensional

truss. The specimens cover a range of slenderness ratio from

60 to 120 including single-bolted and double-bolted conditions.

Results show that both the ASCE Manual 52 [9] and AISC

LRFD [3] are inadequate for single angle members with low

slenderness ratios. Theoretically, single angle struts whose web

members are connected on alternate sides should have less

compression resistance than the same struts connected on the

same side. Kitipornchai et al. [11] carried out compression

tests on two fully welded trusses which were fabricated and

tested to compare the ultimate capacity of trusses having web

members on the same side with those having web members on

alternate sides. The slenderness ratios of the failure members

ranged 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 ignore

this consideration. Woolcock and Kitipornchai [12] proposed asimplified design method based on theoretical and experimental

observations [13,14]. In the proposed method, not only is the

effect of load eccentricity taken into account, but cases where

web members are all on one side or on the opposite side are

also considered. This method was considered simpler and more

economical than the conventional axial force-biaxial bending

interaction approach.

Kitipornchai and Chan [15] employed a nonlinear numerical

approach to solve the problem of elastic behaviour of isolated

restrained beam–columns. The element geometric stiffness

matrix was derived from the principle of minimum of potential

energy incorporating member geometrical nonlinearity. The

effects of the load eccentricity and the shear centre notcoincident with the centroid were also taken into account. The

equilibrium paths were traced from the incremental and the

total-force–deformation equilibrium equations using the arc-

length method. The results were compared with those reported

by Trahair [16]. It was shown that when the geometry was

not updated, with sufficient elements per member (e.g. 4

elements per member), the results agreed well with the

finite integral solutions. However, when the geometry was

updated, the influence of the pre-buckling deformations was

apparent, indicating the importance of the consideration of

geometry 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 method

with no assumption of initial curvatures of the members.

Results indicated that both models could be used to predict

the nonlinear structural behaviour with moderate accuracy.

To make the nonlinear method more user-friendly with fewer

elements per member required to achieve satisfactory accuracy,

Al-Bermani and Kitipornchai [17] proposed an approach

allowing the use of least elements in a nonlinear analysis

acounting for both geometric and material nonlinearity. The

procedure is suitable for analyzing large-scaled space frames

since the structures may be modelled using only a few elements

per member. It is achieved by incorporating a displacement

stiffness matrix which provides the necessary coupling between

the axial, flexural and torsional deformations. However, this

element does not allow for member initial imperfection, which

is mandatory in some national design codes such as Eurocode

3 [18]. For simulation of member initial imperfection, at least

two elements per member will be required.

A review of literature shows there is a lack of research work

on testing of single slender angle trusses with an objective

of verifying a new second-order analysis design method.

Comparisons with the experimental results and the predicted

results by the traditional simplified method instead of the axial

force–moment interaction method are also inadequate. Much

research 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 is

not the aim of this paper. The objective of this paper is to

propose a new design method with verification by testing of

full scale trusses and comparison with code formulae in order

to indicate the inadequacy of our current codes for design

of angle trusses. In the investigation, a series of laboratory

tests of angle members were conducted including single-boltedand double-bolted end conditions and web members on the

same side and on alternate sides. This paper also describes

the experimental programme and results. Comparison among

the test results will be made between the experimental results

against those predicted by the design rules. A proposed second-

order analysis and design method is also introduced and the

method is validated by the test results. The suggested new

method allowing for imperfections due to initial member

crookedness, residual stress as well as single and double bolt

connections and connection on the same and on alternative

sides of the trusses is envisaged to carry a great potential for

design application to angle structures in replacement of the old

linear elastic design approach.

2. Conventional design method

The design codes widely used today, such as AISC

LRFD [19], BS5950 [20] and Eurocode 3 [18], were developed

based on nonlinear analysis of simple idealized individual

members. However, the conventional design procedure of

structural steelwork is based on simple linear analysis with

the nonlinear effects ignored and buckling effect approximately

accounted for using the effective length method. When

designing a steel structure, this first-order linear analysis can be

summarized as follows. Firstly, a linear analysis is carried outto determine the internal forces and moments of all members

under external loads. Afterwards, the resistance of each

individual member is estimated according to the design rules

given in the codes to account for the nonlinear effects which

involve approximation of effective lengths for compression

members. A sufficiently safe structure has its resistance larger

than the factored forces and moments according to this method.

Strictly speaking, the first-order linear theory is only

applicable for structures with small displacements with a

less severe degree of nonlinearity and for members with low

slenderness. This is in contrast with the actual non-linear

structural behaviour soon after the application of load. Taking

a simply supported column subject to a point load at one endas an example, the column deflects in both axial and transverse

directions once a load is applied. This type of nonlinearity is

defined as the P–δ effect. However, in a linear analysis, the

column will only be shortened without buckling. In this simple

case, it can be seen that the column deflects laterally as a

second-order effect which may become more severe than the

first order effect in terms of stress induced by P–δ moment

(i.e. P·δ Z

being greater than the axial stress P A

in which Z is the

elastic modulus and A is the cross sectional area). If the column

is part of a structure, from a global point of view, the local

deflection and the global displacements will mutually affect

each other and the structural response of the other members

as well. To further complicate the situation with the presence

of initial curvature and limited value of yield stress, lateral

deflection occurs once the load is applied and the buckling

strength 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 simple

compressive stress cannot be used to estimate the buckling

resistance; 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 according

to the assumed effective length of the column. The effective

length is calculated by multiplying the effective length factor

or the K-factor to the actual member length in an empiricalmanner. The effective length factor is largely dependent on the

end condition of the member. So, the checking of the buckling

resistance is carried out as an independent stage instead of an

integrated part of design. The interaction with buckling and

other second-order effects is always ignored. Therefore, thistype of analysis is incorrect in calculating forces and moments.

Furthermore, simple columns with idealized pinned ends rarely

exist in real structures; in fact, the joints connecting two or

more members are finite in their stiffness. They would displace

and affect the compression resistance so that the accuracyof the compressive strength estimated by the design codes

as the failure strength is highly dependent on the assumed

effective length. If the effective length is assumed longer than

the 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 may

lead to an uneconomical design or an unsafe design. This design

procedure shows an inconsistency between analysis and design.Regarding structural design of single angle members, the

actual structural behaviour is far more complicated than that

of a simple column with doubly-symmetrical section. In spite

of this, the analysis and the design processes adopted in

the conventional design approach such as BS5950 [20] andEurocode 3 [18] appear to be over-simplified. In reality, the

end restraints and the end eccentricities affect the compressive

strength of the member. However, these effects are often

ignored during analysis; instead, the slenderness is modified

and the compression resistance of the member is taken as a

fraction of that of a concentrically loaded member. Apart fromthe inconsistency between analysis and design, since many of

the design rules in codes are developed based on minor axis

flexural buckling, they may not be adequate for stocky members

which are susceptible to flexural–torsional buckling. Further,the equivalent slenderness ratio is based on a certain commonly

used detailing dimensions such as eccentricity being not too

large 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 order

linear analysis is traditionally used because during the pre-

computer age when computer time was expensive and computer




speeds were slow. Today, the prevalence of low-cost personal

computers and the growing importance of environmental and

economical concerns provide a natural choice to develop a

practical second-order analysis method. This method has been

well-researched by Chen and Chan [21], Chan and Chui [22],

Chan and Cho [23] with the second-order effects included

during the analysis through update of geometry. In other words,the member deflection (δ) and the global displacement (∆) are

taken into account so that section capacity check is adequate for

strength design as follows:

P

Ag p y

+ M y + P (δ z +∆ z)

Z y p y

+ M z + P

δ y +∆ y

Z z p y

= φ ≤ 1 (1)

in which p y is the design strength, P is the external force

applied to the section, Ag is the cross-sectional area, M y and M zare the external moments about the y and z axes respectively,

Z y and Z z are the section modulus about the y and z axesrespectively. P (δ z +∆ z) and P

δ y +∆ y

are the collective

moments about the y and z axes respectively due to the change

of member stiffness under load and large deflection effects

of which the consideration allows for the effect of “effective

length” automatically. In other words, there is no need to reduce

the compressive strength of the member to account for the

P–δ and P–∆ effects. Moreover, the characteristics of realistic

structure (e.g. initial imperfection and residual stresses) are

also considered via the equivalent initial imperfection δ0 in the

analysis so the design is completed simultaneously with the

analysis.To determine the design buckling load of a structure,

approximately 1%–10% of the predicted failure load is applied

incrementally until the sectional capacity factor, ϕ, in Eq. (1) is

greater than 1. The Newton–Raphson method combined with

the minimum residual displacement iterative scheme [24] of

solving the following set of nonlinear simultaneous equation as

follows is utilized with the iterative scheme illustrated in Fig. 1.

∆u

= [K T (u, δ0, F )]−1

∆F

(2)

[∆u] = [K T (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 increments

due to unbalanced force, applied load and summed displace-

ment increment for calculation of the unbalanced force. λ is a

parameter satisfying the minimum residual displacement con-

dition [24].

For designing single angle struts, a method was previously

proposed by Cho and Chan [25] based on the aforementioned

second-order analysis and design concept. Using the software

NIDA (structural analysis software “Nonlinear Integrated

Design and Analysis” version 7) [26], initial curvature is

imposed 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 axial

shortening. The values of the initial curvature are calculatedbased on the compressive strength curve given in BS5950 [20]

which are 2.8 × 10−3 L for equal angles and 2.0 × 10−3 L

for unequal angles. The results computed by NIDA will

agree well with BS5950 [20]. The Hong Kong Steel Code2005 [27] further gives explicit values of imperfections forangles. However, this method cannot truly reflect the end

condition that a practical angle is exposed to. The method

was modified by Chan and Cho [28]; the end condition is

symbolized by a rotational spring element inserted at each endof the member. The value of the rotation spring stiffness is

calculated from the dimensions of the gusset plates and its

material properties. Therefore, the rotational stiffness due to the

double-bolted connection can be considered at the early stage

of the analysis rather than at the design stage as in the linearanalysis and effective length design method. Only the rotational

deformation of the connection spring element is considered

for design because the effects of the axial and shear forces in

the connection deformations are small when compared withthat of bending moments. However, this modified method is

still inadequate to consider the directions of the end moments.

Under some circumstances, these end moments would be

advantageous to the overall structure. In this paper, the methodis further refined. The end moments due to load eccentricity are

considered by connecting the angle web members at each end

to the chord members by rigid arms. The rigid arm will be the

element joining the centroid and the point of load application so

that the magnitude and the direction of the end moments due toload eccentricity can be taken into account immediately during

the 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-bolted

connection can be considered. The merit of this approach over

the purely equivalent imperfection approach is that it considers

the direction of the end moments so that the aforementionedeffect can be reflected during analysis. The spring stiffness of a

rotational spring can be calculated as follows:As shown in Fig. 2, the couple, M , formed by the pair of

bolts is given by:

M = F · d = k θ. (5)




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 cross

sectional 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 Ad 2

l(8)

where F is the shear force exerted on the bolt; d is the distance

between the centroids of the two bolts; θ is the rotation of thebolt group; δ is the displacement of the bolt; l is the length of

the bolt shank; G is the shear modulus of elasticity.To account for the rotation stiffness of the spring element in

the analysis, the following incremental tangent stiffness matrixis superimposed to the element stiffness matrix.

M e M i

=

Sc −Sc

−Sc Sc

θe

θi

(9)

in which M e and M i are the incremental external and internalmoments at two ends of a connection. The external node refers

to the one connected to the global node and the internal node

is 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 = M e

θe − θi

= M i

θi − θe

(10)

in which θe and θi are the conjugate rotations for the moments M e and M i .

3.1. The incremental and iterative procedure for analysis of

angle trusses and frames

The basic equations for incorporating the end connection

stiffness are considered both in the tangent and the secant

stiffness matrix equations as follows.

Fig. 3. The external and internal rotations.

Tangent stiffness matrix

The incremental force is assumed in the software and

the incremental displacement is solved. The basic element

stiffness is modified by addition of the tangent stiffness of the

connection spring modeled as a dimensionless spring element

in a computer analysis as:

S1 −S1 0 0

−S1 k 11 + S1 k 12 0

0 k 21 k 22 + S2 −S2

0 0 −S2 S2

θ1e

θ1i

θ2i

θ2e

=

M 1e

M 1i

M 2i

M 2e

(11)

in which S1 and S2 are the spring stiffness for simulation of

semi-rigid connections at ends; k i j is the stiffness coefficients

of the element; θ1e, θ1i , θ2e and θ2i are respectively the rotations

at two sides for the two ends of an element shown in Fig. 3.

Assembling the element matrices, the global stiffness

matrix for the angle frame and truss is formed and stored

in a one dimensional array in the computer analysis. The

incremental displacement vector is solved and added to the

last displacement and used for determination of resistance as

follows.

Resistance determination

The resistance is determined as the sum of resisting forces

for all elements as,

[ R] =

[k e][ L S][u] (12)

in which [ R] is the structural resistance of the angle frame, [k e]

is the secant stiffness, [ Ls ] is the transformation matrix from

local to global coordinates and [u] is the accumulated element

displacement transformed to the element local axis.

The resistance of the angle frame is determined as the

maximum load not violating Eq. (1). This is based on the

conventional use of load causing the formation of the first

plastic hinge whereas the second part of this paper describes

the extended consideration of elasto-plastic buckling analysis

of angle frames.

4. Laboratory tests of single angle struts

4.1. Experimental programme

Four single angle struts were tested as web members of a

two-dimensional truss as shown in Fig. 4. In the first set, the

web members of the truss are connected to the chord members

on the same side. In the second set, the tests are repeated with

the web members connected to the chord members on alternate




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

sides. The specimens are of Grade S275 and two metres long

making 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 is

connected to a gusset plate. The test included single and double

bolted connections. Other details of the specimens are listed

in Table 1. The trusses were loaded in pairs and sufficient

lateral restraints were provided to ensure out-of-plane buckling

at connecting nodes between chords and webs is fully avoided.

Load was applied at the upper joint of the target failure member

through a hydraulic jack. As shown in Figs. 5 and 6, twelve

strain gauges were mounted evenly at the mid-length of the

targeted member and the tested truss was so designed that

the targeted member fails first. At the targeted member, two

displacement transducers were placed in in-plane and out-of-plane directions and transducers were also used to monitor the

movements of the top and the bottom joints of the targeted

member so that its movement of the target member relative

to the truss can be measured. At the load where the targeted

member buckled or failed, the deformations of the remaining

parts of the truss were small and reversible. Thus, after each

test, the failed member was replaced by a new specimen so

that the next test could be conducted under almost the same

conditions.

Fig. 5. Locations of strain gauges.

Table 1

Details of test specimens

Set Specimen Size Web

arrangement

End

conditions

Gusset

dimensions

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

1b 65×65×6 Same side Double bolt 240×180× 82 2a 66×66×6 Alternate sides Single bolt 240×180×10

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

4.2. Material properties

Four coupon tests were performed to determine the material

properties of the steel used in the test specimens following the

procedure given in BS EN 10002-1 [29]. The test results are

summarized in Table 2. Coupons 1a, 1b, 2a and 2b were cut

from Specimens 1a, 1b, 2a and 2b respectively.

4.3. Test results

The major failure modes are flexural buckling about the

principal minor axis as shown in the photos (Figs. 7 and 8).

Fig. 7 shows the buckled shape of Specimen 1a, of which

each 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 in

the out-of-plane direction. Fig. 8 shows the buckled shape of

Specimen 1b, of which each end is connected to the gusset plate

Fig. 6. Locations of transducers.




Table 2

Coupon test results

Set Coupon Young’s modulus,

E (kN/mm2)

Yield stress, σ y (N/mm2)

1 1a 216.9 347.0

1b 211.8 347.6

Average 214.4 347.32 2a 194.5 348.6

2b 185.6 344.9

Average 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 provides

some flexibility making it behave as if partially restrained

in the in-plane and out-of-plane directions. Figs. 9 and 10

respectively show the in-plane and out-of-plane deflections of

the four specimens. As can be seen from the curves, theirresponse pattern are similar and the out-of-plane deflections

are always more severe than the in-plane deflections. Table 3

listed the member failure loads, which are calculated using

numerical integration of the stress over the cross-sectional

area. It can be seen that the load capacities of the specimens

with double bolt end connections are 9%–15% higher than the

counterparts with single bolt end connections. For the same

end conditions, specimens with alternate side web arrangement

will have the load capacities 15%–20% lower that those with

same side web arrangement. The alternate side arrangement of

the web members will make the end moments due to eccentric

connections more severe.

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

Table 3

Experimental failure loads

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

1 1a 67.2 0.260

1b 78.4 0.303

2 2a 57.5 0.219

2b 72.1 0.275

5. Comparisons

5.1. Comparisons with BS5950

BS5950 [20] provides a simplified method for designing

struts composed of single angles. They may be treated as

axially loaded members with reduced compressive strength

with ignorance of eccentricity at end connections, provided that

the following conditions are fulfilled:

(a) by two or more bolts in standard clearance holes in line

along the angle, or by an equivalent welded connection, the

slenderness 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 taken

as 80% of the compression resistance of an axially loaded

member and the slenderness λ should be taken as the greater

of:

1.0λv but ≥ 0.7λv + 15

1.0λa but ≥ 0.7λa + 30




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 ratios

about the principal minor axis and rectangular axis respectively.

For the cases that the aforementioned conditions are not

satisfied, the struts should be designed as members with

combined moment and axial force. Using the measured

material properties, the simplified method gives the following

results (Table 4). This method is not able to show the

difference between the same side arrangement and alternate

side arrangement of web members. In other words, provided

that the materials are identical in two sets of tests, the

predicted results would be identical also. As can be seen

from 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 less

conservative (10.4%–14.5%) estimates for the double-bolted

counterparts.

Table 4

BS5950 failure loads

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

1 1a 41.5 1.618

1b 68.5 1.145

2 2a 39.4 1.459

2b 65.3 1.104

5.2. Comparisons with NIDA

The failure loads are predicted by NIDA with the end

moments and end rotational restraints taken into account

during the analysis. The value of the rotational stiffness k

is approximately equal to 1000 kN m/rad. This value is

close to the rigidly fixed condition achievable for use with

preloaded bolts. In the present case of using non-preloaded




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

Table 5

NIDA failure loads

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

1 1a 55.6 1.209

1b 66.9 1.172

2 2a 50.4 1.141

2b 62.1 1.161

bolts, the value of k is taken as 10% of the calculated value. It

should be noted that the currently used section capacity check

equation in NIDA is not suitable for angle sections because the

equation was derived primarily for doubly symmetric sections

of which maximum bending stresses about each principal axis

occurs at the four corners simultaneously. However, for anglesections, as they are monosymmetric or asymmetric, the points

of maximum bending stress about both principal axes do not

necessarily coincide. As a consequence, the loading capacities

of the sections calculated are usually underestimated. It is

suggested that the axial stress is checked against every extreme

point of the section. The computed results are summarized

in Table 5. Compared with the conventional design method

using BS5950 [20], the nonlinear analysis and design method

provides less conservative estimates for the single-bolted

specimens (14.1%–20.9%) and narrowly more conservative

estimates for the double-bolted specimens (16.1%–17.2%) for

double-bolted specimens. Fig. 11 shows the experimental and

theoretical vertical deflections at the point of load application

against applied load of the trusses with webs connected on the

same side. It can be seen that the trend predicted by NIDA are

in line with the experimental results.

5.3. Discussions

Comparing the traditional code-based linear design method

with the proposed NIDA design method, the following

conclusions can be made. First, while the code method cannot

take the web arrangement (same side or alternate sides)

into consideration, the design method provides conservative

predictions of the failure loads of the truss. Second, for

compression members with single-bolted end condition, NIDA

provides more rational predictions. Third, for compressionmembers with double-bolted end condition, BS5950 provided

less conservative predictions. It is envisaged that the 0.8 factor

of calculating the compression resistance of a single-bolted

compression member is too conservative for some range of

slenderness ratio. Although the linear code method was noted

to have a more accurate prediction for the failure loads of the

double-bolted compression members using an effective length

factor of 0.85, the joint and web arrangement details cannot be

incorporated directly into the effective length method such as

the present second-order analysis approach. It is suggested that

the discrepancy between the test results and the NIDA results

can be further minimized by considering the rotational stiffnessdue to the presence of the gusset plates and fine-tuning the

rotational stiffness of the joints at the two ends of the member.

6. Conclusions

This paper illustrates the inconsistency between analysis

and design using the conventional methods and codes for

design of single angle struts. The assumption of effective

length in the design stage in fact violates the pin-ended

assumption in the analysis stage, which usually leads to an

over-conservative result and possibly unsafe results if the

displacement is large. A codified second-order analysis method

for 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 so

that an effective length is not required to be assumed. The

proposed design method is validated by laboratory tests of

trusses using single angle members of slenderness ratio about

150 as web members and the test results agreed well with

the computed results. However, this method may need to

have a checking for stocky members against flexural–torsional

buckling modes, which can be carried out by the use of a

simple empirical formula in computer programming as the

problem of determining the effective length is not so acute

here, because the torsional-flexural buckling is a local member




buckling behaviour without significant interaction with frame

system behaviour. An alternative approach is to include the

torsional stiffness matrix for prediction of the buckling. The

extension work will allow the method to cover the whole range

of slenderness ratio for compression members. Plastic design

may also be included in the analysis such that the plastic

strength 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 using

the deteriorating plastic hinge method (PolyU 5117/06E)” and

“Second-Order and Advanced Analysis of Wall-Framed Steel

Structures (PolyU 5115/05E)”.

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http://www.nida-naf.com/







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