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 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 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  π EI Z   π EI W   M o =    GJ +  ,  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  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  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- Figure  10.  Effect of strengths imperfections on buckling Figure  11.  Effect of residual stresses on buckling strength tion 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 ( ) 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 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 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 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 ... 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 I cross-sections made of high strength steel. .. 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... 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