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

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Nov 29, 2013 (3 years and 9 months ago)

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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
(=50), L=1298mm (=73), L=1704m (=95.7) and
L=2515mm (=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 (%)
Stress (Mpa)
341
0.16 2.18
1
2
3
4
5
6
7
Slope Strain Plastic
of curve (%) strain (%)
1 11.4 2.18 2.02
2 5.15 2.30 2.14
3 3.45 2.50 2.34
4 2.87 2.90 2.74
5 2.28 3.50 3.34
6 2.06 4.10 3.94
7 1.49 4.90 4.74
E=200 GPa
Average
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 (
x
). Displacement and load data was collected
by a data acquisition system. In-plane rotation (
y
) 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
159.1
153.4
182.9
0
50
100
150
200
250
0 5 10 15 20 25 30 35 40
Axial displacement (mm)
Load (kN)
Test 7
L=890 mm
(LT mode)
Test 4
L=1298 m m
(LT mode)
Test 5
L=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,



d
a
a



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 
d
= 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 
d
= 0.01,
however, with the smaller factors used, identical
solutions occurred between 
d
= 0.001 and 
d
= 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 geometric
imperfections for test angles EA 90x90x6
80
100
120
140
160
180
200
0 500 1000 1500 2000 2500 3000
Length (mm)
Load capacity (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)
Load (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 MPa
one leg fixed
assumed wave number of 5,7,9,13
with max. magnitude of 0.167t
disp. norm: 0.001



Fig 3.2 Comparison of analysis and test data
Fig 3.3 Convergence criteria study
(L=1704mm, P
cr
=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
0
20
40
60
80
100
120
140
160
Disp.
norm=0.1
Disp.
norm=0.01
Disp.
norm=0.001
Disp.
norm=0.0005
Load Capacity (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
Errors (%)

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
-1.0
0.0
1.0
2.0
3.0
4.0
Magnitude

= 0.333t
mm
errors

(%)
L=890mm
L=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)
Load (kN)
mesh size: 8x72
mesh size: 4x36
Test Results
L=1704 mm
Fy=341 MPa
one leg fixed
wave 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
Errors (%)
150.1
151.2
153.4
0
50
100
150
200
4x36 8x72
Test data
Load capacity (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 (
y
) and out-of-plane
(
x
) 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



0
20
40
60
80
100
120
140
160
180
200
0 2 4 6
Axial displacement (mm)
Load (kN)
fixed end
hinged end
simple end

0
50
100
150
200
0 100 200
Slenderness ratio L/r
v
Load capacity (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/r
v
) versus theoretical ultimate loads P
u
(kN)
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
L/r
v
-

P
u
EA 150x150x19
(A=5360 mm
2
,
F
y
=280 MPa)
EA 150x150x10
(A=2790mm
2
,
F
y
=320 MPa)
EA 100x100x12
(A=2260 mm
2
,
F
y
=260 MPa)
EA 100x100x6
(A=1170 mm
2
,
F
y
=260 MPa)
EA 45x45x6
(A=506mm
2
,
F
y
=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 mm
2
,
F
y
=260 MPa)
EA 90x90x6
(A=1050mm
2
,
F
y
=260 MPa)
EA 75x75x10
(A=1340mm
2
,
F
y
=260 MPa)
EA 75x75x5
(A=672mm
2
,
F
y
=260 MPa)
EA 45x45x3
(A=263mm
2
,
F
y
=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=489mm
2
,
F
y
=260 MPa)
UA 150x100x12
(A=2870mm
2
,
F
y
=300 MPa)
UA 150x100x10
(A=2300mm
2
, F
y
=320MPa)

UA 125x75x12
(A2260mm
2
,
F
y
=260 MPa)
UA125x75x6
(A1170mm
2
,
F
y
=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=1580mm
2
,
F
y
=260 MPa)
UA 100x75x6
(A=1020mm
2
,
F
y
=260 MPa)
UA 75x50x8
(A=921mm
2
,
F
y
=260MPa)
UA 75x50x6
(A=721mm
2
,
F
y
=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 20
Axial displacement (mm)
Load (kN)
imperfect column with direction 1
imperfect column with direction 2
straight column
L=890mm
L=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 N
c
for compression alone and is only
applicable to members with 

150, eq (5.1).

N N
c
*
  (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 L
e
effect is only considered in
evaluating M
bu
, and the pin-end (L
e
=L) is allowed to
determine N
ch
.

N
N
M
M
ch
h
bu
* *
cos  
  1
(5.2)


e
h
h
v
v
u
u
gusset
plate





The corresponding column strength curves using


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 50 100 150 200 250 300 350
Slenderness ratio (L/r
v
)
Load Capacity N* (kN)
Lusas predictions
Clause 6.6 Le=0.85L
Clause 6.6 Le=L
Clause 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 50 100 150 200 250 300 350
Slenderness ratio (L/r
v
)
Load Capacity N* (kN)
Lusas predictions
Clause 6.6 Le=0.85L
Clause 6.6 Le=L
Clause 8.4.6 Le=0.5L
Clause 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 1.8
Moment ratio M*/(M
bu
*cos(

))
Load ratio N*/Nch
EA 90x90x6
EA 90x90x10
EA 100x100x6
EA 100x100x12
EA 150x150x10
EA 150x150x19
EA 75x75x5
EA 75x75x10
EA 45x45x3
EA 45x45x6
EA 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 1.8
Moment ratio M*/(M
bu
*cos(

))
Load ratio N*/Nch
UA 75x50x6
UA 75x75x8
UA 100x75x6
UA 100x75x10
UA 125x75x6
UA 125x75x12
Equ.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*/N
ch
) is shown
on the Y-axis and the moment ratios (M*/M
bu
cos) 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
M
ch bu
* *
..cos0 8 17
1 

(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 =190.
A more detailed improvement would require the
incorporation of more parameters, such as the B/t ratio,
in the calculation of N
ch
and M
bu
, 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
F
y

yield stress
L
e

effective length
M
bu

nominal moment capacity
about major principal u
axis
M
h
*
end moment due to eccentricity
= eN*
N*
design axial force
N
c

nominal compression capacity
N
ch

nominal compression
capacity (about h axis)
P
cr

experimental buckling load
P
u

ultimate load
r
v

radius of gyration about minor
principal v-axis
t
thickness of angle leg

slenderness ratio about minor
principal v-axis

d

Euclidean displacement
norm (tolerance factor)


a

displacement increment

a

total displacement

strength reduction factor

x

in-plane rotation about y
axis in Fig 3.1

y

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