Strength design of highstrength steel beams

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 o

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

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 repre-sents the worst case for lateral buckling, is consid-ered 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 interac-tion 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

2 FINITE ELEMENT MODEL The FE model was used for a doubly symmetric I-section beam over the span length Figure 1 pro-vides an overview, together with the relevant coor-dinate 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, load-ing and support of the beam, only half the span

1 INTRODUCTION

The lateral (or flexural-torsional) buckling of

structural (or mild steel) prismatic I-section beams

is well-established (Trahair 1993, Trahair &

Brad-ford 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

mem-bers have significantly different stress-strain

char-acteristics 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

mem-bers, surprisingly little research on their stability

appears in the open literature

Beg & Hladnik (1996) presented an

experi-mental and numerical analysis of the local

stabil-ity 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)

under-took 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

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

com-putational efficiency, and was chosen for the FE

meshing Details of cross-sectional geometric

defi-nitions 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 u x , u y , u z , φ x , φ y and φ z are the

displace-ments 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 (u x = 0) and rotating in the Y and Z

directions (φ y = φ z = 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

trans-lations and twisting, were used at the support-end

section The twist rotations of all nodes on the

section were restrained (φ x = 0), while the vertical

displacement (u z = 0) of the centroid of the web

(denoted as W c) and the lateral displacements

(u y) 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 W c at the support end These boundary

con-ditions 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

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

y W

x W x W z W y W

i = c and u i=u c+d i⋅ i (1)

where φ W y i and φ y W c are the rotations of W i and

W c respectively, u W x i and u x W c the displacements of

W i and W c respectively and d z W i the distance from

node W i to node W c The node W i denotes any

nodes located on the Z-axis For all nodes of the

top flange, the constraint equations were expressed by

z TF

x TF x TF y TF

z TF

i= c and u i=u c+d i⋅ i, (2)

where φ z TF i and φ z TF c are the rotations of nodes TF i

and TF c respectively, u x TF i and u x TF c the

displace-ments of nodes TF i and TF c respectively and d z TF i

the distance from node TF i to node TF c The node

TF i denotes any nodes located on the top flange

Figure 1 FE model of half-beam

Figure 2 Dimensions of cross-section

Figure 3 Restraint conditions

and the node TF c 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

resid-ual stresses are important factors that affect the

inelastic buckling strength, and should be taken

into account To include the geometric

imperfec-tions, 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

Aus-tralia 1998)

The membrane residual stresses due to the

weld-ing 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

illus-trated 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-ling modes As noted, the first mode obtained from the eigenvalue buckling analysis was used to simu-late the initial geometric imperfections for the non-linear load-displacement analysis in order to trigger the lateral buckling behaviour The lowest eigen-value associated with the first eigeneigen-value buckling mode was recorded as the elastic eigenvalue

buck-ling load (Me) The non-linear load-displacement analysis took the geometric non-linearity into con-sideration and was solved by employing a modi-fied Riks method For elastic non-linear analysis, material non-linearity was not included The load-deformation 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 clas-sic 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

Figure 5 Residual stresses adopted in study

M EI

L GJ

EI L

o=   + 







2

2 2

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-sec-tional 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

numeri-cal 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-tor-sional behaviour and buckling strength of simply

sup-ported 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 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, lat-eral 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-to-depth ratios Beams having larger span-to-span-to-depth ratios exhibit more ductile pre-buckling behav-iour 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

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)

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

3.2 Effect of steel grade

Figure 8 shows the variations of the moment

ver-sus 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,

kNm, which is 39% higher than that of the steel

mem-ber, 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

sec-tion 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 s

o

= M

It can be seen that in the low and high

slender-ness regions, the influences of changes in the steel

grade are negligible However, in the region of

inter-mediate slenderness, an increase of the grade of the

steel can result in substantial increases in the

dimen-sionless 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

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 ini-tial geometric imperfections It can be seen that the changes in the magnitude of the initial

imperfec-Figure 7 Typical deformed shape for lateral buckling

failure

Figure 8 Effect of steel grade

Figure 9 Effect of steel grade on steel buckling strengths

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

3.4 Effect of residual stresses

The effects of residual stresses on the lateral

buck-ling 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

signifi-cantly and it is this ratio, rather than the magnitude

of residual stresses themselves, which governs the

reduction in strength

3.5 Effect of size of cross-section

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

intermedi-ate slendernesses, and the difference decreases with

an increase of the slenderness The strengths are

increased by an increase of the size of the

cross-section, but this effect is negligible for very slender

beams

Figure 10 Effect of imperfections on buckling

Figure 12 Effect of size of cross-section

Figure 13 Comparison of code rules with FE results

4 PRESCRIPTIVE DESIGN PROPOSAL Figure 13 compares the results from 220 FE stud-ies 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

M M

bu

=0 72 ( λ3 2 +2 18 −λ1 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 Aus-tralian Research Council by a Discovery Project (DP150100446) awarded to the first author REFERENCES

AISC 2010 ANSI/AISC 360-10 Specification for

Struc-tural 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

chal-lenges; 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

Con-ference Tampere, Finland

Lee, C.H., Han, K.H., Uang, C.M., Kim, D.K., Park, C.H & Kim, J.H 2013 Flexural strength and rota-tion capacity of I-shaped beams fabricated from

800-MPa 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

Syd-ney: SA

Trahair N.S 1993 Flexural-Torsional Buckling of

Struc-tures 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 &

Gard-ner, L 2008 The Behaviour and Design of Steel

Struc-tures to EC3. London: Taylor & Francis

Figure 14 Comparison of proposed rule with FE

results

in which the modified slenderness is given in

Equa-tion 4 Figure 14 compares the proposal with the

FE results, showing the accuracy of the

predic-tion of the new equapredic-tion is improved, but slightly

unsafe estimations can be observed in the

interme-diate slenderness regions

5 CONCLUSIONS

An accurate FE model has been developed to

inves-tigate 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

charac-teristics and initial imperfections (geometric

imper-fections and residual stresses) were incorporated

into the model The typical lateral buckling

behav-iour 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

rela-tionship 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

modifi-cations of these current design rules are needed