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 multiwave local buckling mode, allows large inelastic deformations and predicts local and
overall buckling behaviour. Calibration with both inhouse 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 outofplane 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 torsionalflexural
buckling and local plate buckling and the interaction of
overall and local buckling behaviour. Beamish and
Butterworth [7] used both hybrid thinwall beam and
thin shell elements to investigate the influence of local
buckling on ultimate load and postbuckling 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 stressstrain relationship shown in Fig 2.1.
2.2 Test setup
The test rig is shown in Fig 2.2. The specimens were
orientated parallel to the test floor and the effect of the
selfweight neglected. The ends of the specimens were
bolted to pairs of backtoback 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 postbuckling
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 outofplane
rotation (
x
). Displacement and load data was collected
by a data acquisition system. Inplane 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 1.11.2 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 eightnode 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 midpoint of the connected leg through
these beam elements to achieve the desired eccentric
compression.
The nonlinear material model matched the
experimentally derived stressstrain 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 halfsine
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
bystep 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 halfwave 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 postbuckling range.
Fig 2.2 Test Rig
Fig.3.1 Lusas Eccentric Loading Model
with initial imperfection mode
(a) Viewpoint A
linear
interpolation
halfsine 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 postbuckling
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.
Loadaxial 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
postbuckling
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 inplane (
y
) and outofplane
(
x
) rotations fixed
Hinged end
outofplane 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 postbuckling 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 prebuckling
stage, but dominated the postbuckling 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 postbuckling 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 Loadaxial
displacement plot
Fig 3.9 Buckling mode under fixed
or hinged end condition
Table 3.1 Effect of end fixity
(2) postbuckling
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) postbuckling
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 postbuckling 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 postbuckling 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 postbuckling
regime [12]. A similar effect can be seen in Fig. 4.1
where the imperfection directions 1 and 2 give
contrasting postbuckling 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 beamcolumn 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 pinend (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 Yaxis and the moment ratios (M*/M
bu
cos) on
the Xaxis. 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.
Postbuckling 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, pp107112.
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 ThinWalled 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 Semiloof Element,”
Computers and Structures, Vol 12, pp 147159.
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 vaxis
t
thickness of angle leg
slenderness ratio about minor
principal vaxis
d
Euclidean displacement
norm (tolerance factor)
a
displacement increment
a
total displacement
strength reduction factor
x
inplane rotation about y
axis in Fig 3.1
y
outofplane rotation about y
axis in Fig 3.1
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