Strength design of highstrength steel beams

Mechanics of Structures and Materials: Advancements and Challenges – Hao & Zhang (Eds)
© 2017 Taylor & Francis Group, London, ISBN 978-1-138-02993-4

Strength design of high-strength steel beams
M.A. Bradford & X. Liu

Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering,
The University of New South Wales, Australia

ABSTRACT:  High-Strength Steel (HSS) is advantageous in steel framed buildings because when
strength rather than stiffness predominates the design, less of it is needed by comparison to mild steel
frames and so the ensuing carbon footprint is minimised. Many major steel codes allow for members of
up to Grade 690 MPa using similar rules for mild steel, but the next step of strength increase needs careful
consideration because the residual stresses in HSS are different to mild steel, as is the stress-strain curve.
The paper looks specifically at the aspect of lateral buckling of HSS beams, because prescriptive rules for
design against lateral buckling of flexural members consider the interaction of elastic buckling, yielding,
residual stresses and geometric imperfections.
1  InTroduction

and rotation capacity, while Bradford & Ban
(2015) and Ban & Bradford (2015) considered the
buckling of tapered HSS bridge beams.
This paper develops an accurate and reliable
three dimensional Finite Element (FE) model to
investigate the lateral buckling strength of HSS
I-section beams using the ABAQUS software
package, by incorporating the stress-strain curves
and residual stresses measured experimentally
and reported in the literature. A simply supported
beam subjected to uniform bending, which represents the worst case for lateral buckling, is considered. The validated FE model is then applied to
undertake parametric studies, which include the

effects of the beam span, the steel grade, the initial
geometric imperfections, the residual stresses and
the dimensions of the cross-section. With a similar
methodology to mild steel members, the interaction of elastic buckling at the member scale with
the material characteristics at the cross-section
scale is investigated. The evaluation of current
design codes and the development of new design
rules for predicting the flexural-torsional buckling
strengths of HSS beams are presented.

The lateral (or flexural-torsional) buckling of
structural (or mild steel) prismatic I-section beams
is well-established (Trahair 1993, Trahair & Bradford 1998) and design rules in codes of practice
such as AS4100 (Standards Australia 1998) are
familiar to structural engineers. The basis of the
design rules is a so-called “beam curve”, which is
a semi-empirical reflection of the interaction of
elastic buckling, yielding and residual stresses to
express the buckling strength as a function of the
beam slenderness. It is well-known that HSS members have significantly different stress-strain characteristics and residual stress distributions to those
of mild steel, and these may potentially manifest
themselves in buckling-based strength rules for
HSS that are different from those for mild steel.
However, despite the increasing use of HSS members, surprisingly little research on their stability
appears in the open literature.
Beg & Hladnik (1996) presented an experimental and numerical analysis of the local stability of welded I-section beams made of HSS with
a yield stress of around 800 MPa while Shi et al.
(2012) investigated the overall buckling behaviour
of ultra-high strength steel I-section columns that
buckle about their major axis, and the influence of

the column end restraints on their overall buckling
behaviour was evaluated. Ban et al. (2013) undertook an experimental program to study the overall
buckling behaviour of 960  MPa HSS pin-ended
columns under axial compression. Flexural tests
on full-scale I-section beams fabricated from HSS
were undertaken by Lee et al. (2013) to study the
effect of flange slenderness on the flexural strength

2  FINITE ELEMENT MODEL
The FE model was used for a doubly symmetric
I-section beam over the span length. Figure 1 provides an overview, together with the relevant coordinate system in which the Y and Z axes define
the plan of the cross-section and the X-coordinate
defines the longitudinal beam axis. Because of
the presence of symmetry in the geometry, loading and support of the beam, only half the span

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Figure 1.  FE model of half-beam.
Figure 2.  Dimensions of cross-section.

was modelled, using the four-node shell element
with reduced integration S4R. This element has six
degrees of freedom per node and provides accurate
solutions for most applications and it allows for
transverse shear deformations and for finite strain,
being suitable for large strain analysis. It was found
that an approximate overall mesh size of 20  mm
was an appropriate balance of accuracy and computational efficiency, and was chosen for the FE
meshing. Details of cross-sectional geometric definitions of the steel beam are depicted in Figure 2.

The built-up HSS I-sections were assumed to be
fabricated from flame-cut plates and through fillet
welding with the weld size of 6 mm.
The boundary conditions are shown in Figure 3,
in which ux, uy, uz, φx, φy and φz are the displacements and the rotations about the global X, Y
and Z axes respectively. All nodes at the mid-span
section were restrained from translating in the
X direction (ux = 0) and rotating in the Y and Z
directions (φy = φz = 0). Idealised simply supported
boundary conditions that allow for major and
minor axis rotations and warping displacements,
while preventing in-plane and out-of-plane translations and twisting, were used at the support-end
section. The twist rotations of all nodes on the
section were restrained (φx = 0), while the vertical
displacement (uz = 0) of the centroid of the web
(denoted as Wc) and the lateral displacements
(uy) of all nodes of the web (all nodes located on
the Z-axis) were restrained. A uniform bending
moment about the major axis of the cross-section
was applied as a concentrated moment imposed to
node Wc at the support end. These boundary conditions and loading are the most conservative case
for lateral buckling. However, in order to avoid any
undesirable localised web deformations and stress
concentration while leaving the flange free to warp,
appropriate constraints by using the EQUATION

Figure 3.  Restraint conditions.

option were applied. For all nodes of the web, the
constraint equations were given by

Wc
i
φW
y = φy

Wc
Wi
Wi
i
and uW
x = ux + d z ⋅ φ y

(1)



i and φ Wc are the rotations of W and
where φW
i
y
y
i and uWc the displacements of
Wc respectively, uW
x
x
Wi
Wi and Wc respectively and d z the distance from
node Wi to node Wc. The node Wi denotes any
nodes located on the Z-axis. For all nodes of the
top flange, the constraint equations were expressed

by

φ zTFi = φ zTFc

and uxTFi = uxTFc + d yTFi ⋅ φ zTFi ,



(2)

where φ zTFi and φ zTFc are the rotations of nodes TFi
and TFc respectively, uxTFi and uxTFc the displacements of nodes TFi and TFc respectively and d zTFi
the distance from node TFi to node TFc. The node
TFi denotes any nodes located on the top flange

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and the node TFc is the node of the centroid of top
flange. Similar constraints equations were applied
for the bottom flange.
A plastic steel formulation with Von Mises’ yield
function, associated plastic flow and isotropic
hardening was used to model the steel beam, whose
stress-strain relationship is shown in Figure 4. The
values adopted for Grade 460 MPa (mild steel) are:
E = 200 GPa, fy = 460  MPa, fu = 550  MPa: ey =
0.22%, et = 2.0%, eu = 14%; for Grade 690 MPa, E =
200 GPa, fy = 690 MPa, fu = 770 MPa, ey = 0.33%,
et = 0.33%, eu = 8%; and for Grade 960 MPa: E =

200 GPa, fy = 960 MPa, fu = 980 MPa, ey = 0.46%,
et = 0.46%, eu = 5.5%.
The initial geometric imperfections and residual stresses are important factors that affect the
inelastic buckling strength, and should be taken
into account. To include the geometric imperfections, the first buckling mode shape derived by an
eigenvalue buckling analysis was introduced into
the FE model with the maximum magnitudes of
the initial imperfection being 1/1000 of span or
3  mm, whichever is the greater (Standards Australia 1998).
The membrane residual stresses due to the welding process were applied as the initial stresses on
the elements around the cross-section and assumed
to be uniform over the thickness of the element. It
is worth mentioning that when initial stresses were
applied, the initial stress state may not be an exact
equilibrium state for the FE model. Therefore, an
additional initial step in analysis using a statics
procedure may be necessary to be used to achieve
equilibrium. The residual stress distribution model
for HSS welded I-sections (Ban et al. 2013) based
on the relevant experimental test results is illustrated in Figure 5.
The analyses using the FE model were of two
types, viz. elastic eigenvalue buckling analysis
and non-linear load-displacement analysis. The
eigenvalue buckling analysis was conducted to
check the models and to obtain the potential buck-

Figure 5.  Residual stresses adopted in study.

ling modes. As noted, the first mode obtained from
the eigenvalue buckling analysis was used to simulate the initial geometric imperfections for the nonlinear load-displacement analysis in order to trigger

the lateral buckling behaviour. The lowest eigenvalue associated with the first eigenvalue buckling
mode was recorded as the elastic eigenvalue buckling load (Me). The non-linear load-displacement
analysis took the geometric non-linearity into consideration and was solved by employing a modified Riks method. For elastic non-linear analysis,
material non-linearity was not included. The loaddeformation response was determined and the
applied moment at the critical turning point on
the load-deformation curve was recorded as the
elastic non-linear buckling load (Mn). For inelastic
non-linear analysis, both material imperfections in
the form of residual stresses and material plastic
behaviour were taken into account. The peak value
of the load-deformation response was defined as
the inelastic non-linear buckling load or lateral
buckling strength (Mu).
In order to validate the FE model, the numerical
results for the flexural-torsional buckling moment
obtained from the FE analysis were compared with
the theoretical results calculated based on the classic elastic flexural-torsional buckling formulation
for simply supported beam in uniform bending
(Trahair & Bradford 1998, Trahair et  al. 2008),
given by

Figure 4.  Stress-strain relationship.

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 π 2 EI Z  
π 2 EI W  
M o = 
  GJ +

 ,
2

L2  
 L


(3)

where GJ is the torsional rigidity, EIZ the minor
axis flexural rigidity and EIW the warping rigidity.
Forty-eight slender beams were selected as
examples for analysis and comparison. Their span
lengths ranged from 4 m to 12 m and the cross-sectional dimensions are listed in Table 1. As a result,
the mean value of Me/Mo of the beams was 1.025
with a Coefficient of Variation (COV) of 0.029.
The mean values and COV of Mn/Mo are 1.006 and
0.026 respectively. It can be concluded the numerical solutions from the FE model are consistent
with the theoretical ones.
3  PARAMETRIC STUDY
Inelastic non-linear analyses were performed by using
the proposed FE model to investigate the flexural-torsional behaviour and buckling strength of simply supported doubly symmetric I-section beams in uniform
bending. Twelve beam cross-sections (Table  1) were
selected. The dimensions were chosen so that all the
plate elements are compact to eliminate the occurrence
of local buckling; the local stability criteria in AS4100
(Standards Australia 1998) were assumed to be still
applicable to high-strength steel and be adopted in
design, even though they may be conservative (Beg &
Hladnik 1996). Hence, the nominal section capacities

Ms of the beams can be calculated by Ms = Sfy, where
S is the plastic modulus of the cross-section.
3.1  Effect of span length

Figure 6.  Effect of span-to-depth ratio (L/H).

Figure 6 shows the applied moment and mid-span
deformation responses for the beams with various
span-to-depth ratios. All of the beams have the same
cross-section (410HWB) and are of Grade 960 MPa.

The curves in Figures  6(a) to (c) represent typical
applied uniform bending moment versus vertical, lateral and twist displacement curves respectively. It can
be seen that the initial stiffness and ultimate moment
capacities of the beams increase as the span-to depth
ratios decreases, and the displacements at the peak
moment increase with an increase of the span-todepth ratios. Beams having larger span-to-depth
ratios exhibit more ductile pre-buckling behaviour and stable post-buckling responses, while the
strengths of the beams with smaller span-to-depth
ratios descend dramatically after attaining the peak
moment. The deformed shapes of the beams at their
maximum moment (Mu) have been examined, and
a typical deformation at flexural torsional buckling
failure of an I-section beam is shown in Figure  7.
This figure confirms that failure of the member is
accompanied by flexural and lateral deflections and
twisting in the clockwise direction.

Table  1.  Cross-sectional dimensions of HSS I-section
beams (mm).

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Figure  7.  Typical deformed shape for lateral buckling
failure.

3.2  Effect of steel grade
Figure 8 shows the variations of the moment versus the mid-span displacements with respect to the
grade of the beam, those considered being 460, 690
and 960 MPa. For a 3 m span 410HWB member,
the maximum moment with fy = 960  MPa is 624
kNm, which is 39% higher than that of the steel
beam with fy = 460 MPa. For a 1.2 m span member, this increase rises to 68%. Increasing the steel
grade can therefore enhance the ultimate bending
capacities significantly.
In order to illustrate more comprehensively the
effect of the steel grade on the lateral buckling
strength, the buckling strengths of 410HWB section beams having three different steel grades are
conveniently illustrated in plots of the type shown
in Figure  9, in which the dimensionless inelastic
buckling resistance Mu/Mo is plotted against the
generalised slenderness defined as

λs =

Ms
.
Mo

Figure 8.  Effect of steel grade.

(4)


It can be seen that in the low and high slenderness regions, the influences of changes in the steel
grade are negligible. However, in the region of intermediate slenderness, an increase of the grade of the
steel can result in substantial increases in the dimensionless inelastic buckling resistance. Accordingly,
existing design provisions for mild steel beams may
not be applicable for HSS beams, particularly those
with yield stresses exceeding 690 MPa, and they may
underestimate the buckling strength significantly.

Figure  9.  Effect of steel grade on steel buckling
strengths.

3.3  Effect of initial geometric imperfections
A sensitivity study was conducted of a number
of 960  MPa HSS beams with initial geometric
imperfections of L/2000, L/1000, L/500. A
410HWB cross-section was chosen for the study.
Figure 10 shows the variations of the dimensionless

ultimate moment capacities of the beams against
their slendernesses for different values of the initial geometric imperfections. It can be seen that the
changes in the magnitude of the initial imperfec-

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Figure  10.  Effect of
strengths.

imperfections on buckling

Figure  11.  Effect of residual stresses on buckling
strength.

tion do not have significant impact on the buckling
strengths of beams with higher slenderness, but the
effects of initial geometric imperfections become
greater for beams with ls < 1.75. The buckling
strengths of the beams reduce as the magnitudes
of the initial imperfections increase. The codified
ones that are obtained by also considering 3  mm
as the minimum geometric imperfection value are
nearly the most conservative ones and cover lower
bounds for the other three results.
Figure 12.  Effect of size of cross-section.

3.4  Effect of residual stresses
The effects of residual stresses on the lateral buckling strengths of HSS I-section beams is illustrated
in Figure 11. It is seen that the buckling strengths
of the beams with residual stresses are significantly
smaller than those of beams without residual stresses
when the slenderness is in a relative low range (such
as ls < 1.75). On the other hand, the influence of the
residual stresses on the buckling strength becomes
less adverse for the beams with higher yield stresses.
It can be reasoned that as the grade of a steel beam

increases, the ratio of the magnitude of the residual
stresses to the steel yield strength is reduced significantly and it is this ratio, rather than the magnitude
of residual stresses themselves, which governs the
reduction in strength.

Figure 13.  Comparison of code rules with FE results.

4  Prescriptive Design Proposal

3.5  Effect of size of cross-section

Figure 13 compares the results from 220 FE studies with the results from AS4100 (SA 1998), EC3
(Trahair et al. 2008) and the AISC (2010) for Grade
960 HSS. It can be seen that the code predictions
are somewhat in error when compared with the FE
results, and so a new prediction is required.
For the AS4100 formulation, such a prediction
is proposed in the form

Figure 12 shows a comparison of ultimate moment
capacities for a group of 960  MPa HSS beams
with different sizes of their cross-sections, those
selected being 410HWB, 610HWB, 800HWB and
1000HWB (Table 1). It can be seen that the greatest
divergence of results is in the region of intermediate slendernesses, and the difference decreases with
an increase of the slenderness. The strengths are
increased by an increase of the size of the crosssection, but this effect is negligible for very slender
beams.

M bu

= 0.72
Ms

8

(

)

λs3.2 + 2.18 − λs1.6 ≤ 1,

(5)



Accordingly, new design proposals based on the
current design rules of AS4100 was recommended.
ACKNOWLEDGEMENT
The work in this paper was supported by the Australian Research Council by a Discovery Project
(DP150100446) awarded to the first author.
REFERENCES

Figure  14.  Comparison of proposed rule with FE
results.

AISC. 2010. ANSI/AISC 360-10 Specification for Structural Steel Buildings. Chicago: AISC.
Ban, H.Y. & Bradford, M.A. 2015. Buckling of tapered
half-through girder railway bridges using HSS. In N.
Yardimci (ed.), Steel bridges: innovation and new challenges; Proc. 8th Int. Symp. on Steel Bridges. Istanbul:
TUCSA, 585–594.

Ban, H.Y., Shi, G., Shi, Y. & Bradford, M.A. 2013.
Experimental investigation of the overall buckling
behaviour of 960  MPa high strength steel columns.
Journal of Constructional Steel Research 88: 256–266.
Beg, D. & Hladnik, L. 1996. Slenderness limit of class 3 I
cross-sections made of high strength steel. Journal of
Constructional Steel Research 38(3): 201–217.
Bradford, M.A. & Ban, H.Y. 2015. Buckling strength of
HSS steel beams. 13th Nordic Steel Construction Conference. Tampere, Finland.
Lee, C.H., Han, K.H., Uang, C.M., Kim, D.K., Park,
C.H. & Kim, J.H. 2013. Flexural strength and rotation capacity of I-shaped beams fabricated from 800MPa steel. Journal of Structural Engineering 139(6):
1043–1058.
Shi, G., Ban, H. & Bijlaard, F.S.K. 2012. Tests and
numerical study of ultra-high strength steel columns
with end restraints. Journal of Constructional Steel
Research 70: 236–247.
Standards Australia. 1998. AS4100 Steel Structures. Sydney: SA.
Trahair N.S. 1993. Flexural-Torsional Buckling of Structures. London: E&FN Spon.
Trahair, N.S. & Bradford, M.A. 1998. The Behaviour and
Design of Steel Structures to AS4100. London: Taylor
& Francis.
Trahair, N.S., Bradford, M.A., Nethercot, D.A. & Gardner, L. 2008. The Behaviour and Design of Steel Structures to EC3. London: Taylor & Francis.

in which the modified slenderness is given in Equation 4. Figure 14 compares the proposal with the
FE results, showing the accuracy of the prediction of the new equation is improved, but slightly
unsafe estimations can be observed in the intermediate slenderness regions.
5  CONCLUSIONS
An accurate FE model has been developed to investigate the lateral buckling strength of HSS I-section
beams with doubly symmetric I-sections, simply
supported boundary conditions and subjected to

uniform bending. The material non-linear characteristics and initial imperfections (geometric imperfections and residual stresses) were incorporated
into the model. The typical lateral buckling behaviour of HSS beams was elucidated in the study
and extensive parametric studies were performed.
It can be concluded that the buckling strengths
of 960 MPa HSS beams are higher than those of
beams fabricated from steel having yield stresses
that do not exceed 690  MPa on the basis of the
non-dimensional strength versus slenderness relationship. This is attributable mainly to the effects
of the residual stress being less severe for 960 MPa
HSS beams. The design formulations proposed to
predict the ultimate moment capacities of I-section
beams were assessed, showing that some modifications of these current design rules are needed.

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