859

BEHAVIOUR OF STEEL SINGLE ANGLE COMPRESSION MEMBERS AXIALLY LOADED THROUGH ONE LEG

Jie Sun and John W. Butterworth Department of Civil and Environmental Engineering, University of Auckland

SUMMARY A nonlinear finite element model applicable to steel single angle compression members eccentrically loaded through one leg has been developed using an existing finite element package. The model incorporates realistic initial geometric imperfections similar to a multi-wave local buckling mode, allows large inelastic deformations and predicts local and overall buckling behaviour. Calibration with both in-house and other experimentally acquired data showed that the model was able to predict behaviour up to ultimate load whilst using a relatively coarse mesh. A parameter study was undertaken to determine the ultimate axial load capacity of 121 equal leg struts and 88 unequal leg struts covering slenderness ratios from 30 to 300. Comparison of the results with the nominal loads prescribed by the relevant clauses of the New Zealand Steel Structures Design Standard, NZS 3404, revealed significant conservatism. A suggested interim measure for decreasing the conservatism by modifying the current interaction equation is suggested. 1 INTRODUCTION Steel single angle struts are of great interest in light structures as web members and are usually connected through one leg. The resulting eccentricity due to this loading arrangement introduces end moments which are most troublesome when combined with axial compression. This complicates the buckling behaviour and creates difficulty in finding a suitable design model. The applicability of the existing design models needs to be further investigated [1, 2]. Testing of crossed diagonal angles in a 3D truss by Elgaaly et al. [3] showed that different failure modes could occur, combining local, overall and torsional effects, but that residual stress had a relatively insignificant effect on the maximum loads. A buckling mode involving buckling perpendicular to the plane of the connected leg with little twisting up to the maximum load was observed by Trahair, Usami and Galambos [4] when using fixed or hinged conditions allowing out-of-plane rotations. Bathon and Mueller [5] tested a wide range of eccentrically loaded angles using a ball joint to model end conditions unrestrained against rotation. The measured ultimate strengths were compared with the American design code. Chuenmei [6] extended the finite element analysis of eccentrically loaded angles into the nonlinear range, examining the combination of torsional-flexural buckling and local plate buckling and the interaction of overall and local buckling behaviour. Beamish and Butterworth [7] used both hybrid thin-wall beam and thin shell elements to investigate the influence of local buckling on ultimate load and post-buckling response. Both elements gave results in good agreement with each other and with experimental data.

Parameter studies using a finite element numerical model present an attractive alternative to physical testing when formulating or checking design rules for members exhibiting complex behaviour. Such studies are useful only if the numerical model is carefully checked or calibrated against results based on physical testing. The purpose of this paper is to describe the development of a nonlinear finite element model for predicting the behaviour of eccentrically loaded angles. Physical testing conducted for the purpose of calibrating the model is also described. Details are given of a parameter study aimed at establishing the ultimate axial strength of a range of eccentrically loaded angles. The ultimate loads were then compared with values derived from the relevant clauses of the New Zealand Steel Structures Standard, NZS 3404 [8]. 2 PHYSICAL TESTING 2.1 Test specimens and material properties The test struts were selected from the ordinary mild steel range supplied by BHP. The section chosen, EA 90x90x6, was based on the considerations of being a fairly typical size and having a reasonably high width/thickness ratio to encourage local buckling. Four different strut lengths were selected to cover a range of slenderness ratios from 50 to 150, generating both elastic and inelastic buckling behaviour, and to suit available test equipment. The lengths were: L=892mm (λ=50), L=1298mm (λ=73), L=1704m (λ=95.7) and L=2515mm (λ=141). NOTE - refer to last page for notation.

860

0

50

100

150

200

250

300

350

400

450

500

0 1 2 3 4 5 6 7 8 9

Strain (%)

Str

ess

(Mp

a)

341

0.16 2.18

12

3 45

67

Slope Strain Plastic of curve (%) strain (%)

1 11.4 2.18 2.022 5.15 2.30 2.143 3.45 2.50 2.344 2.87 2.90 2.745 2.28 3.50 3.346 2.06 4.10 3.947 1.49 4.90 4.74

E=200 GPaAverage yield stress = 341 MPa

Standard tensile tests on a number of coupons gave an average yield stress of 339MPa and an average ultimate stress of 493MPa. For comparison, the manufacturer’s nominal values were 260MPa and 480MPa respectively. The test data was idealised a little to give the typical stress-strain relationship shown in Fig 2.1. 2.2 Test set-up The test rig is shown in Fig 2.2. The specimens were orientated parallel to the test floor and the effect of the self-weight neglected. The ends of the specimens were bolted to pairs of back-to-back angles which were simulating the truss chord. A portion of box section strut (130x130x6) with two pairs of guides was used to apply jack loads parallel to the line joining the ends of the angle. The purpose was to prevent the loading face of the jack from rotating when the angle under test underwent large lateral deflection in the post-buckling range. Friction and play at all the interfaces was minimised by the use of close fitting greased plates and shims. The measuring system included three measurements taken at mid span of the angle specimen with one displacement transducer for the resultant vertical movement and two horizontal displacement transducers to measure lateral displacement and rotation. Another four displacement transducers were used, two at the loading face to measure the axial shortening and to check the loading face rotations, with the other two at each end of the angle to check the relative out-of-plane rotation (θx). Displacement and load data was collected by a data acquisition system. In-plane rotation (θy) was measured manually using a micrometer bubble level at selected points in the loading cycle. 2.3 Test Results A total of seven struts were tested, including tests 1 and 7 with L=890mm, tests 2, 3 and 4 with L=1298mm, test 5 with L=1704mm and test 6 with L=2515mm. The experimental curves of load and axial displacement from tests 7, 4, 5 and 6 (representing the four different lengths) are shown in Fig. 2.3. It was found that the

load increase after initial buckling up to the point of maximum load represented a modest but useful ‘strength reserve’. Maximum loads were typically in the range of 1.1-1.2 times the initial buckling loads. The failure mode in seven of the test specimens involved predominant local buckling in the connected leg. This local buckling then coupled with either torsional buckling or flexural buckling about an axis parallel to the unbuckled angle leg. Most of the local buckling occurred near the end connection (tests 1, 3, 4, 5 and 7, while in tests 2 and 6 it occurred away from the connection near the mid span. Fig. 2.4 shows photographs of some failure modes.

179.7

209.6

172.21

190.1

145.9

159.1153.4

182.9

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40

Axial displacement (mm)

Lo

ad (k

N)

Test 7 L=890 mm (LT mode)

Test 4L=1298 m m (LT mode)

Test 5L=1704 m m (LT mode)

Test 6 L=2515 mm (LG mode)

Buckling

Maximum load

Note: LG: Local buckling of the connected leg followed by flexural geometric axis buckling LT: Local buckling of the connected leg followed by torsional buckling

Fig 2.1 Material model for test specimens

Fig 2.3 Load-axial displacement plots for EA 90x90x6 from tests 4,5,6 and 7

Fig 2.4 Typical local-overall buckling modes

(a)

(b)

861

3 NUMERICAL MODELLING 3.1 Lusas nonlinear model A 3D eight-node thin shell Semiloof element (QLS8 in the Lusas [9] element library) was selected as the primary element. The general idealisation of the steel angle is shown in Fig.3.1. In order to achieve the eccentric loading through the attached leg, stiff beam elements simulating the gusset plates were connected to the shell elements at each end. The axial load was applied at the mid-point of the connected leg through these beam elements to achieve the desired eccentric compression. The nonlinear material model matched the experimentally derived stress-strain curve of Fig. 2.1 and used a Von Mise yield criterion, an associated flow rule and isotropic hardening, giving three distinct regimes - elastic, perfectly plastic and multilinear strain hardening respectively. A large displacement small strain Total Lagrangian formulation took account of the significant geometric nonlinearity. The Total Lagrangian formulation was preferred to the equivalent Updated Lagrangian formulation as it avoided the lengthy evaluations of shape function derivatives for the Semiloof elements at each load step [9,10]. Initial imperfections in the shape of several half-sine waves (resembling a typical local buckling mode) were adopted in the longitudinal direction with linear interpolation in the transverse direction as shown in Fig 3.1. The effect of residual stress was not considered in the analysis due to its insignificant effect on the maximum loads as reported by Elgaaly [3]. The solution strategies adopted for the nonlinear step-by-step response analyses involved full Newton-Raphson iteration combined with load incrementation. A restepping option was selected to accelerate the convergence. Displacement control was introduced in place of load control to avoid convergence difficulties when the solution approached a limit point [9]. A variety of other strategies including arc length control were also tried but found to be less satisfactory.

3.2 Convergence studies Mesh Density is usually an important factor influencing both the accuracy and cost of the numerical solution. Analyses to assess the effect of mesh density were performed on a typical test angle having a length of 1704mm and with both ends fixed. Initial Imperfections - In matching the mesh to the initial imperfection mode, two cases consisting of four elements per half-wave and eight elements per half-wave were considered. For the strut with 9 half waves the resulting mesh densities became 4 x 36 (coarse mesh) and 8 x 72 (fine mesh), where 4 and 8 were the transverse divisions and 36 and 72 were the longitudinal divisions. The numerical solutions and the physical test results (from Section 2) are shown in Fig 3.2. Nearly identical solutions were obtained from the coarse and fine mesh models, with the ultimate load capacities matching the test buckling loads with reasonable accuracy, having errors of 2.1% and 1.4% respectively. However, as can be seen there was considerable difference in the post-buckling range.

Fig 2.2 Test Rig

Fig.3.1 Lusas Eccentric Loading Model with initial imperfection mode

(a) Viewpoint A

linear interpolation

half-sine waves twisted about the shear centre

shear centre

(b) viewpoint B

(c) 3D view

A

B

first half wave

862

Convergence criteria - tight tolerances were required to maintain control of the analysis in the presence of significant geometric nonlinearity. The Euclidean displacement norm,

γδ

d

aa

=r

r 2

2

x 100 (5.1),

one of a number of convergence measures available in Lusas, was generally used. The results obtained by using criteria of γd = 0.1 ~ 0.0005 are presented in Fig 3.3. Solutions for the ultimate loads varied little when using different tolerance factors. The maximum error of 2.8% compared to the test data was found at γd = 0.01, however, with the smaller factors used, identical solutions occurred between γd = 0.001 and γd = 0.0005. 3.3 Initial imperfection effect Amplitude of the initial imperfection wave - Numerical predictions for the test specimens using initial imperfection amplitudes of 0.167t, 0.333t, 0.5t and 0.667t are summarised in Fig 3.4, together with the existing physical test results. The greatest difference in ultimate loads occurred between a straight column and the corresponding imperfect column with smaller differences resulting from the different imperfection amplitudes. The differences were largest for the shortest column (L=890mm), with the sensitivity decreasing with increasing length of angle.

Apart from the apparently anomalous result for the 2515mm specimen (discussed later), numerical predictions using an initial imperfection amplitude of e=0.333t gave the best agreement with the test buckling loads, with errors in the range ±2.5%.

Effect of the magnitude of the initial geometricimperfections for test angles EA 90x90x6

80

100

120

140

160

180

200

0 500 1000 1500 2000 2500 3000

Length (mm)

Load

cap

acity

(kN

)

straight column

Magnitude = 0.167t

Magnitude = 0.333t

Magnitude = 0.5t

Magnitude = 0.667t

Test results

Direction of the initial wave - Analyses to assess the effect of the direction of the assumed imperfection wave were performed by imposing waves with opposite directions, 1 and 2, as defined in Fig 3.5. Higher capacities were obtained when using direction 2 for angles of L=890 ~ 1704mm. In the post-buckling region, significant difference was observed for the shortest angle with the difference larger than the effect on the ultimate loads, but the influence decreasing with member length. For the most slender member, identical solutions were obtained for both directions.

Load-axial displacement plot for EA 90x90x6 (Lusas analysis)

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6

Axial displacement (mm)

Lo

ad (

kN)

L=890mm (direction 2), Pu=188.2kN

L=890mm (direction 1), Pu=184.4kN

L=1298mm (direction 2), Pu=175.4kN

L=1298mm (direction 1), Pu=170.9kN

L=1704mm (direction 2), Pu=153.8kN

L=1704mm (direction 1), Pu=150.1kN

L=2515mm (direction 2), Pu=97.6kN

L=2515mm (direction 1), Pu=99.84kN

Fy=341 MPaone leg fixedassumed wave number of 5,7,9,13with max. magnitude of 0.167tdisp. norm: 0.001

Fig 3.2 Comparison of analysis and test data

Fig 3.3 Convergence criteria study (L=1704mm, Pcr=153.4 kN)

Fig 3.6 Effect of imperfection direction

(a) Load capacity (b) Errors

Fig 3.4 Comparison of Lusas imperfect models with test data

149.8 149.1 150.1 150.1

020406080

100120140160

Disp.norm=0.1

Disp.norm=0.01

Disp.norm=0.001

Disp.norm=0.0005

Lo

ad C

apac

ity (

kN)

2.35

2.80

2.15 2.15

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Disp.norm=0.1

Disp.norm=0.01

Disp.norm=0.001

Disp.norm=0.0005

Err

ors

(%)

first half sine wave twisted about the shear centre in anti-clockwise direction

first half sine wave twisted about the shear centre in clockwise direction

Fig 3.5 assumed wave directions

2.5

-2.45 -2.09

-4.0-3.0-2.0-1.00.01.02.03.04.0

Magnitude = 0.333tmm

erro

rs (

%)

L=890mmL=1298mm

L=1704mm

(a) direction 1 (b) direction 2

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6 7 8 9 10

Axial displacement (mm)

Lo

ad (

kN)

mesh size: 8x72

mesh size: 4x36

Test Results

L=1704 mmFy=341 MPaone leg fixedwave number: 9 (direction1)with magnitude of 0.333t disp. norm: 0.001

post-buckling strength reserve

buckling load

2.15

1.43

0.00

0.50

1.00

1.50

2.00

2.50

4x36 8x72

Err

ors

(%)

150.1 151.2 153.4

0

50

100

150

200

4x36 8x72 Test data

Lo

ad c

apac

ity

(kN

)

863

3.4 Numerical buckling behaviour Ultimate load capacity and buckling response - the effect of end fixity was evaluated by applying three end conditions to the established model - Fixed end both the in-plane (θy) and out-of-plane

(θx) rotations fixed Hinged end out-of-plane rotation released Simple end both rotations released. The results (using direction 1 imperfections in all cases) are summarised in Table 3.1. Fixed end provided load capacities of 20% ~ 42% higher than the simple end and the hinged end gave ultimate loads of 7.5% ~ 13% lower than the fixed end. The significant stiffening effects due to end fixity on both ultimate loads and post-buckling response is shown in Fig 3.7. The effect on ultimate load capacity decreased with increasing slenderness as shown in Fig 3.8.

Ultimate load (kN) strut length

(mm) fixed end

hinged end

simple end

890 184.4 159.8 130 1298 170.9 149.5 123 1704 150.1 133 112 2515 99.8 92.4 83.3

020406080

100120140160180200

0 2 4 6Axial displacement (mm)

Load

(kN

)

fixed end

hinged end

simple end

0

50

100

150

200

0 100 200

Slenderness ratio L/rv

Lo

ad c

apac

ity

(kN

)

fixed end

hinged end

simple end

Buckling modes - The finite element results showed predominant vertical rather than horizontal deformation with the angle buckling about an axis parallel to the connected leg as shown in Fig 3.9. NOTE: the connected leg was horizontal in the finite element model, but vertical in the physical tests. Twist was almost imperceptible in the pre-buckling stage, but dominated the post-buckling region. The local waves were amplified and interacted with the overall bending, with the effect more pronounced in the stockier members, as shown in Fig 3.10. However, under simple end conditions the angles exhibited quite different modes. Horizontal bending increased rapidly and became dominant at the ultimate loading stage in the post-buckling region. The angle buckled about a major principal axis first and was prone to bend about the axis perpendicular to the connected leg at the ultimate loading stage. Twist was insignificant throughout the response and local waves were

amplified at the connected leg rather near mid span, as shown in Fig 3.11.

Parameter Study - Analyses were conducted on eleven equal leg angle sections and eight unequal leg angle sections using properties taken from the BHP Specification [11]. Eleven slenderness ratios were considered for each different angle section. For the first nine, an imperfect model with imperfection wave amplitude of 0.333t was adopted. For the two highest slenderness ratios imperfections were set to zero as

Fig 3.8 Effect of end fixity

Fig 3.7 Typical Load-axial displacement plot

Fig 3.9 Buckling mode under fixed or hinged end condition

Table 3.1 Effect of end fixity

(2) post-buckling stage

(3) ultimate stage

C

B

A

(b) viewpoint B

(d) 3D view (a) viewpoint A

(c) viewpoint C

vertical bending

amplified local waves

twist effect

(1) pre-buckling stage

(2) post-buckling stage

(3) ultimate stage

C

B

A

(b) viewpoint B

(d) 3D view (a) viewpoint A

(c) viewpoint C

horizontal bending

amplified local waves

Fig 3.11 Buckling mode, simple end condition

Fig 3.10 Interaction in a slender member

(1) pre-buckling stage

864

their effect was known to be negligible. The load was applied at the mid point of the connected leg. The hinged boundary condition was used, allowing out of plane rotations. The results are summarised in Table 3.2. 4 COMPARISON Compared with the physical test results, the analyses based on a relatively coarse mesh predicted response up to the ultimate load level reasonably closely and permitted appropriate choices for initial imperfection parameters to be made. Some disagreement was apparent in the post-buckling region and the theoretical buckling modes did not give satisfactory coincidence with experimental observations. It was thought that this disagreement was due to the assumed imperfection modes affecting the post-buckling responses to a

greater extent than the ultimate loads. The experimental specimens had a range of (undetermined) imperfections causing these behaviour variations. Another item of interest was the disparity in the experimental buckling load of the most slender angle, which exceeded the analytical value by 35.2%. A possible explanation was that the initial imperfection resulted in the angle bowing away from its preferred buckling direction causing it to follow a different equilibrium path leading to a higher limit point from which it buckled and reverted to its preferred path. This type of behaviour would also be expected to show the observed steeper fall off in the immediate post-buckling regime [12]. A similar effect can be seen in Fig. 4.1 where the imperfection directions 1 and 2 give contrasting post-buckling slopes, although relatively small differences in buckling load.

Slenderness ratio (L/rv) versus theoretical ultimate loads Pu (kN)

L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu L/rv - Pu

EA 150x150x19 (A=5360 mm2, Fy=280 MPa)

EA 150x150x10 (A=2790mm2, Fy=320 MPa)

EA 100x100x12 (A=2260 mm2, Fy=260 MPa)

EA 100x100x6 (A=1170 mm2, Fy=260 MPa)

EA 45x45x6 (A=506mm2, Fy=260 MPa)

30 - 837 150 - 504 30 - 442 150 - 236 30 - 326 150 - 206 30 - 142 150 - 86 30 - 74 150 - 47 50 - 822 170 - 443 50 - 422 170 - 202 50 - 322 170 - 179 50 - 140 170 - 75 50 - 73 170 - 41 70 - 807 190 - 385 70 - 400 190 - 174 70 - 317 190 - 156 70 - 134 190 - 65 70 - 72 190 - 36 90 - 784 240 - 303 90 - 372 240 - 129 90 - 308 240 - 125 90 - 125 240 - 50 90 - 70 240 - 29

110 - 633 300 - 210 110 - 314 300 - 90 110 - 250 300 - 87 110 - 110 300 - 35 110 - 57 300 - 20 130 - 556 130 - 273 130 - 229 130 - 98 130 - 51

EA 90x90x10 (A=1620 mm2, Fy=260 MPa)

EA 90x90x6 (A=1050mm2, Fy=260 MPa)

EA 75x75x10 (A=1340mm2, Fy=260 MPa)

EA 75x75x5 (A=672mm2, Fy=260 MPa)

EA 45x45x3 (A=263mm2, Fy=260 MPa)

30 - 243 150 - 151 30 - 134 150 - 81 30 - 206 150 - 131 30 - 95 150 - 56 30 - 34 150 - 20 50 - 240 170 - 133 50 - 131 170 - 70 50 - 204 170 - 113 50 - 91 170 - 49 50 - 33 170 - 18 70 - 235 190 - 117 70 - 125 190 - 61 70 - 201 190 - 101 70 - 87 190 - 43 70 - 31 190 - 15 90 - 227 240 - 92 90 - 118 240 - 47 90 - 196 240 - 80 90 - 83 240 - 33 90 - 30 240 - 12

110 - 188 300 - 64 110 - 102 300 - 33 110 - 159 300 - 56 110 - 72 300 - 19 110 - 26 300 - 8.2 130 - 167 130 - 92 130 - 142 130 - 64 130 - 23

EA 55x55x5 (A=489mm2, Fy=260 MPa)

UA 150x100x12 (A=2870mm2, Fy=300 MPa)

UA 150x100x10 (A=2300mm2, Fy=320MPa)

UA 125x75x12 (A2260mm2, Fy=260 MPa)

UA125x75x6 (A1170mm2, Fy=260 MPa)

30 - 73 150 - 43 30 - 381 150 - 341 30 - 329 150 - 283 30 - 257 150 - 246 30 - 120 150 - 110 50 - 71 170 - 39 50 - 375 170 - 316 50 - 326 170 - 260 50 - 255 170 - 242 50 - 119 170 - 106 70 - 68 190 - 34 70 - 372 190 - 283 70 - 322 190 - 232 70 - 253 190 - 229 70 - 118 190 - 100 90 - 67 240 - 27 90 - 367 240 - 230 90 - 315 240 - 184 90 - 251 240 - 198 90 - 116 240 - 88

110 - 56 300 - 19 110 - 362 300 - 163 110 - 310 300 - 129 110 - 248 300 - 149 110 - 115 300 - 64 130 - 50 130 - 357 130 - 301 130 - 248 130 - 113

UA 100x75x10 (A=1580mm2, Fy=260 MPa)

UA 100x75x6 (A=1020mm2, Fy=260 MPa)

UA 75x50x8 (A=921mm2, Fy=260MPa)

UA 75x50x6 (A=721mm2, Fy=260 MPa)

30 - 208 150 - 183 30 - 118 150 - 98 30 - 113 150 - 107 30 - 83 150 - 77 50 - 205 170 - 167 50 - 116 170 - 90 50 - 112 170 - 101 50 - 82 170 - 73 70 - 203 190 - 148 70 - 114 190 - 80 70 - 111 190 - 91 70 - 81 190 - 67 90 - 201 240 - 120 90 - 112 240 - 63 90 - 110 240 - 78 90 - 81 240 - 56

110 - 199 300 - 86 110 - 109 300 - 44 110 - 110 300 - 58 110 - 79 300 - 41 130 - 194 130 - 105 130 - 109 130 - 79

Table 3.2 Parameter study - ultimate axial loads for equal and unequal angles

865

0

50

100

150

200

250

0 5 10 15 20Axial displacement (mm)

Load

(kN

)

imperfect column with direction 1

imperfect column with direction 2

straight column

L=890mmL=1298mm L=1704mm

L=2515mm

5 RELEVANCE TO NZS 3404 Two design models are currently recommended in the NZ Steel Design Standard, NZS 3404 [8]. Clause 6.6 determines ϕN

c for compression alone and is only

applicable to members with λ ≥ 150, eq (5.1).

N Nc

* ≤ ϕ (5.1) In clause 8.4.6, the beam-column model is allowed for designing for a combination of moment and axial load (eq 5.2 and Fig 5.1). The Le effect is only considered in evaluating Mbu, and the pin-end (Le=L) is allowed to determine Nch.

N

N

M

Mch

h

bu

* *

cosϕ ϕ α+ ≤ 1 (5.2)

e

h h

v

vu

u

gussetplate α

The corresponding column strength curves using ϕ = 10. with effective length factors of 1.0 and 0.85 for clause 6.6 and of 1.0 and 0.5 for clause 8.4.6 are displayed together with the analytical ultimate loads in Table 3.2. Typical results are plotted in Fig 5.2 for an equal leg angle (EA) and in Fig 5.3 for an unequal leg angle (UA). Clauses 6.6 underestimated capacities in the high slenderness ratio range. Similarly, clause 8.4.6 underestimated capacities in the low slenderness ratio range. Clause 8.4.6 was also relatively insensitive to effective length and consequently unable to reflect the real effect of the end conditions even though the end restraint may exert a strong influence on the load capacity for struts of low to medium slenderness.

Column Strength Curve for EA 90x90x10

0

50

100

150

200

250

300

350

400

450

0 50 100 150 200 250 300 350Slenderness ratio (L/r v)

Load

Cap

acity

N*

(kN

)

Lusas predictions

Clause 6.6 Le=0.85L

Clause 6.6 Le=LClause 8.4.6 Le=0.5L

Clause 8.4.6 Le=L

Column Strength Curve for UA 75x50x6

0

20

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250 300 350Slenderness ratio (L/r v)

Lo

ad C

apac

ity

N*

(kN

)

Lusas predictions

Clause 6.6 Le=0.85LClause 6.6 Le=L

Clause 8.4.6 Le=0.5LClause 8.4.6 Le=L

Load ratio vs. moment ratio for Lusas prediction as per NZ 3404 provisions (clause 8.4.6, Le=0.5L)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Moment ratio M*/(Mbu*cos(α ))

Lo

ad r

atio

N*/

Nch

EA 90x90x6EA 90x90x10EA 100x100x6EA 100x100x12EA 150x150x10EA 150x150x19EA 75x75x5EA 75x75x10EA 45x45x3EA 45x45x6EA 55x55x5

Equ. 5.2

Modified Equ. 5.3

Load ratio vs. moment ratio for Lusas prediction as per NZ 3404 provisions (clause 8.4.6, Le=0.5L)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Moment ratio M*/(Mbu*cos(α))

Lo

ad r

atio

N*/

Nch

UA 75x50x6

UA 75x75x8

UA 100x75x6

UA 100x75x10

UA 125x75x6

UA 125x75x12Equ.5.2

Modified Equ. 5.3

The combination of axial loads and bending moments on the single angle struts for use with clause 8.4.6 are shown in Figs 5.4 and Fig 5.5 for equal and unequal angles respectively. Axial load ratio (N*/Nch) is shown on the Y-axis and the moment ratios (M*/Mbucosα) on the X-axis. The majority of the numerical results are scattered above the interaction equation, with different

Fig 4.1 Effect of imperfection modes Fig 5.2 Typical column strength curve (EA)

Fig 5.3 Typical column strength curve (UA)

Fig 5.4 Load - moment interaction (EA)

Fig 5.5 Load ratio- moment interaction (UA)

Fig 5.1 Design model in clause 8.4.6

866

scatter ‘patterns’ in the EA and UA groups. Significant conservatism in the interaction equation is apparent. Because of the scatter it is clear that the present interaction equation can not be simply altered to produce a good fit. However, limited improvement could be achieved by applying clause 8.4.6 over the full range of slenderness ratios, retaining the linear interaction formula in its present form and imposing different scaling factors for interaction. This was done by rotating the interaction line and moving it so that it formed a closer lower bound to the groups of calculated capacities. The resulting modified equation became:

N

N

M

Mch bu

* *

. . cos0 8 171+ =

α (5.3)

This gave increases in axial load capacity in the range 4.1% to 15.4% for the groups of angles used in the parameter study when compared with the ‘old’ interaction equation. The range of slenderness ratios considered ranged up to λ=190. A more detailed improvement would require the incorporation of more parameters, such as the B/t ratio, in the calculation of Nch and Mbu, together with a nonlinear interaction equation. 6 CONCLUSIONS A nonlinear finite element model applicable to steel single angle compression members was able to predict local and overall buckling behaviour under eccentric axial loading through one leg up to ultimate load whilst using a relatively coarse mesh. Calibration against physical tests enabled suitable waveforms and amplitudes of initial imperfections to be chosen in order to predict initial buckling and ultimate loads with good accuracy. An amplitude of 0.333t for the initial imperfection waves combined with a mode ‘1’ form generally gave the best match to test results. Post-buckling behaviour, appeared to be particularly sensitive to both magnitude and direction of initial imperfections making it more difficult to predict. Comparison of the results of a parameter study giving the axial strengths of over 200 equal and unequal angles with the nominal axial strength prescribed by the relevant clauses of the New Zealand Steel Structures Design Standard, showed significant conservatism. The inclusion of two additional scaling factors in the current code interaction equation resulted in increases in axial capacity of the order of 10%. 7 REFERENCES 1. Temple, M.C. and Sakla, S.S., “Consideration for

the Design of Single Angle Compression Members Attached by One Leg,” Proceedings, International Conference on Structural Stability and Design, Kitipornchai, Hancock, Bradford (Eds), Balkema, Rotterdam, 1995, pp107-112.

2. NZ Heavy Engineering Research Association, “Steel Design and Construction Bulletin”, No. 17, December 1995.

3. Elgaaly, H., Dagher, and Davids, W., “Behaviour of Single Angle Compression Members,” Journal of Structural Engineering, Vol.117 No12, December, 1991, ASCE pp3720~3741

4. Trahair, N.S., Usami, T. and Galambos, T.V., “Eccentrically Loaded Single Angle Columns”, Research Report No. 11, Dept. of Civil and Environmental Engineering, Washington Univ., St Louis, Mo., Aug., 1969.

5. Bathon, L, Mueller, W.H. and Kempner, L., “Ultimate Load Capacity of Single Steel Angles,” Journal of Structural Engineering, Vol No1, ASCE 1993, pp 229~300

6. Chuenmei, G., “Elastoplastic Buckling of Single Angle Columns,” Journal of Structural Engineering, Vol. 110, No6, June, 1984, ASCE, pp 1391~1395

7. Beamish, M.J., and Butterworth, J.W., “Inelastic local and lateral Buckling of Thin-Walled Steel Members,” Report No 494, Dept. of Civil Engineering, University of Auckland, 1991.

8. Steel Structures Standard, NZS 3404:Part 1:1997, Standards New Zealand, Wellington.

9. Lusas Finite Element Analysis System - Theory Manual, Version 10.0, 1990 Finite Element Analysis LTD, UK

10. Javaherian, H., Dowling, P.J., Lyons, L.P.R., “Nonlinear Finite Element Analysis of Shell Structures Using the Semi-loof Element,” Computers and Structures, Vol 12, pp 147-159.

11. “Hot Rolled and Structural Steel Products”, 1994 Edition, BHP Co. Pty Ltd, Melbourne.

12. Thompson, J.M.T. and Hunt, G.W., A General Theory of Elastic Stability, Wiley, 1973.

8 NOTATION A cross section area E elastic modulus

Fy yield stress Le effective length

Mbu nominal moment capacity about major principal u axis

Mh* end moment due to eccentricity = eN*

N* design axial force Nc nominal compression capacity

Nch nominal compression capacity (about h axis) Pcr experimental buckling load

Pu ultimate load rv radius of gyration about minor principal v-axis

t thickness of angle leg λ slenderness ratio about minor principal v-axis

γd Euclidean displacement norm (tolerance factor) δ

ra displacement increment

ra total displacement ϕ strength reduction factor

θx in-plane rotation about y axis in Fig 3.1 θy

out-of-plane rotation about y axis in Fig 3.1