BUCKLING CURVES FOR HEAVY WIDE FLANGE STEEL COLUMNS

Transcription

1 BUCKLING CURVES FOR HEAVY WIDE FLANGE STEEL COLUMNS H. H. SNIJDER, L.-G. CAJOT 2, N. POPA 2, R. C. SPOORENBERG 3 Abstrct. This pper proposes existing Europen buckling curves to be used for checking the resistnce of hevy wide flnge columns mde from mild nd highstrength steel, filing by flexurl buckling. Buckling curves re not vilble in the current Eurocode3 EN 993--, for height-to-width rtios h/b >.2 nd flnge thicknesses t f > mm. The buckling curves re evluted ccording to the sttisticl procedure given in Annex D of EN 99 using finite element nlyses. Residul stress models s described in literture were used to define the initil stress stte of the column in the finite element model. A lrge dtbse ws creted contining the rtio between the elstic-plstic buckling resistnce obtined from finite element nlysis nd the buckling resistnce obtined from the proposed buckling curve for wide set of column configurtions from which prtil fctor γ Rd ws deduced. Different section types with flnge thicknesses t f > mm were investigted: the stocky HD nd more slender HL type, feturing h/b =.23 nd h/b = 2.35 respectively. The mterils investigted were: Quenched nd Self-Tempered (QST) steel vilble under the proprietry nme HISTAR 46 (High Strength ArcelorMittl) with yield stress of 46 N/mm 2 ; steel grde S46; steel grde S355; steel grde S235. For s fr s vilble, sttisticl informtion on these mterils ws used to estimte the prtil fctor for mteril properties γ m. Then the prtil (sfety) fctor γ M cn be clculted s γ M = γ Rd γ m. Bsed on the criterion tht γ M should not exceed.5, buckling curves re suggested which cn be used together with γ M =.. Buckling curves to be included in Eurocode3 EN re finlly proposed for hevy wide flnge columns in S235 to S5, with cross-sections with height-to-width rtios h/b>.2 nd flnge thicknesses t f > mm. This pper is n extended nd more complete version of n erlier pper []. Key words: buckling curves, finite element nlyses, hevy wide flnge sections, highstrength steel, mild steel, prtil fctor, sttisticl evlution.. INTRODUCTION The dvent of Quenched nd Self-Tempered (QST) steel sections which combine high strength (i.e. nominl yield stress greter thn 43 N/mm 2 ) with good ductility nd weldbility hs led to brodening of the possibilities in steel Eindhoven University of Technology, Deprtment of the Built Environment, P.O. Box 53, 56 MB Eindhoven, The Netherlnds 2 ArcelorMittl, Long Products, 66, rue de Luxembourg, L-49 Esch/Alzette, Luxembourg 3 Iv-Consult b.v., P.O. Box 55, 335 CD Ppendrecht, The Netherlnds

2 2 Buckling curves for hevy wide flnge steel columns 79 construction. This mnufcturing method cn lso be pplied to produce hevy wide flnge sections, i.e. wide flnge sections with flnges thicker thn 4 mm. At the moment, hevy wide flnge QST sections re mnufctured by ArcelorMittl under the proprietry nme of HISTAR (HIgh-STrength ARcelorMittl). Two grdes re currently produced: HISTAR 355 nd (high-strength) HISTAR 46, which possess yield stress of 355 N/mm 2 nd 46 N/mm 2 respectively, not considering reduction of yield stress with incresing mteril thickness. Hevy wide flnge HISTAR 46 sections hve lredy been pplied worldwide, with the mjority in the United Sttes where the US equivlent of HISTAR 46, Grde 65, is covered by ASTM A93 [2, 3]. Besides the high yield stress, HISTAR 46 sections hve improved mteril properties for wide flnge sections possessing thick flnges. For HISTAR 46 smller reduction in yield stress needs to be incorported for greter mteril thicknesses ccording to ETA-/56 [4] when compred to other grdes (e.g. S46M nd S5M ccording to EN 25-4 [5]) s illustrted in Fig.. For HISTAR 46 nd S46 sections with flnge thicknesses exceeding mm, the yield stress is 45 nd 385 N/mm 2 respectively. minimum yield stress [N/mm 2 ] Fig. Decrese of yield stress of HISTAR 46, S46 nd S5 with incresing mteril thickness. For S355 nd S235, depending on whether the mteril is clssified s nonlloy structurl steel, normlized fine grin structurl steel or thermomechnicl fine grin steel ccording to EN 25-2 [6], EN 25-3 [7] nd EN 25-4 [5], respectively, substntilly reduced yield stress must be used to ccount for reduction in mteril properties for thick plted prts. For the present study with sections possessing flnge thicknesses between nd 5 mm, the yield stress for S235 steel is 95 N/mm 2. For S355 the lowest yield stress vlue ccording to the three different stndrds is selected: 295 N/mm 2.

3 8 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 3 As such, hevy wide flnge HISTAR 46 sections re used to their best dvntge when the ultimte limit stte is the governing design criterion. This is the cse when pplied s grvity columns in multi-story buildings, bems in shortor medium-spn bridges or chord- nd brce members s prt of truss-like structures. In these situtions the design is most often controlled by the flexurl buckling resistnce of the member for which due llownce hs to be mde ccording to the relevnt design codes. Tble Buckling curve selection tble ccording to Eurocode3, EN Cross-section Rolled I-sections h/b >.2 Limits t f 4 mm Buckling bout xis y-y z-z Buckling curve S 235 S46 S 275 S 355 S 42 b 4 < t f mm y-y z-z b c h/b.2 t f mm y-y z-z b c t f > mm y-y z-z d d c c HISTAR 46 flls in the ctegory S46 in Tble. Smll nd medium-sized HISTAR 46 sections with flnge thickness t f smller thn or equl to 4 mm re ssigned to buckling curve or depending on the vlue of the height-to-width rtio h/b. Hevy HISTAR 46 sections which hve flnge thickness smller thn or equl to mm cn be designed ccording to buckling curve. Hevy sections possessing flnge thickness in excess of mm nd n h/b-rtio smller thn.2 re ssigned to buckling curve c. For hevy HISTAR 46 sections hving h/brtios greter thn.2 nd flnges thicker thn mm no buckling curves re vilble. The sme is true for hevy sections in S46, S355 nd S235 for h/b >.2 nd t f > mm. In order to rrive t buckling curves reflecting the buckling response for hevy sections in HISTAR 46, S46, S355 nd S235 with flnge thickness lrger thn mm nd h/b-rtios greter thn.2, combined experimentl nd numericl study ws initited by ArcelorMittl in Luxemburg nd set up nd executed by Eindhoven University of Technology in the Netherlnds. The experiments consisted of residul stress mesurements performed on two different hevy wide flnge section types mde in steel grde HISTAR 46. A residul stress model ws

4 4 Buckling curves for hevy wide flnge steel columns 8 proposed which cn be used for hevy wide flnge QST sections. The testing procedure nd the derivtion of this so clled QST residul stress model re detiled in [9]. Since the mnufcturing process of hevy S46 sections is identicl to tht of HISTAR 46 sections, the residul stress model of the ltter ws pplied to the S46 nlyses. As no experimentl dt is vilble to model the residul stresses stte for hevy HL sections mde from S235 or S355 n ssumption ws mde on their distribution. For the S235 members the so clled ECCS residul stress model [] feturing biliner stress pttern over the web height nd flnge width ws used to define the initil stress stte. For the sections mde from S355 steel the QST residul stress model ws dopted in ddition to the ECCS residul stress model. In the present pper, existing ECCS buckling curves re proposed to check the flexurl buckling resistnce of hevy HISTAR 46 sections. The relibility of the suggested buckling curves is evluted ccording to nnex D of EN 99 []. The buckling resistnce for wide set of columns in HISTAR 46, S46, S355 nd S235 is evluted with the finite element method using the residul stress models mentioned to define the initil stress stte of the column nd with widely ccepted geometric imperfections... EARLIER APPROACHES FOR DERIVATION OF BUCKLING CURVES From the erliest experiments on pin-ended columns filing by flexurl buckling it ws observed tht the slenderness (rtio between length nd rdius of gyrtion) of the member hs profound influence on the buckling response. This led to the development of the buckling curve concept, relting the lod column cn withstnd before instbility occurs to its non-dimensionl or reltive slenderness (slenderness normlized ginst the steel properties). Importnt references include [2, 3]. The studies underlying the buckling curve concept were often bsed on two-fold pproch: to obtin the elstic-plstic buckling resistnce through fullscle column testing nd to conduct theoreticl (nd lter numericl) nlyses to replicte nd supplement the experimentl results. The theoreticl nlyses were expnded to include wide set of columns not prt of the experimentl pln from which design rules (buckling curves) were proposed. The ccurcy of the buckling curve ws often evluted through comprison with chrcteristic vlues from fullscle tests performed, where good greement between the buckling curve nd test justified the selected buckling curve..2. STATISTICAL EVALUATION OF RESISTANCE MODELS The erlier pproches to rrive t buckling curve formultions hve become obsolete s with the ppernce of EN 99 Annex D [] Design ssisted by testing it is now possible to mke sttisticl evlution for new design rules nd

5 82 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 5 existing ones nd to quntify the vribility of slient prmeters. In brief, the EN 99 sttes tht the design resistnce (R d ) my be obtined directly from the quotient of the chrcteristic (R k ) strength nd the prtil fctor γ M : where γ M cn be subdivided s follows: γ M R R k d =, () γm = γ Rd γ m, (2) where: γ M is the prtil fctor for mteril property, lso ccounting for model uncertinties nd dimensionl vritions ccording to EN 99 [] or generl prtil fctor ccording to EN [8]; γ Rd is the prtil fctor ssocited with the uncertinty of the resistnce model; γ m is the prtil fctor for mteril property. A distinction for the generl prtil fctor γ M is mde depending on the filure mode of the member under investigtion. In the present study, columns re investigted for which loss of stbility is the governing filure mode. Therefore the generl prtil fctor is in line with EN denoted Μ γ throughout this pper. The generl prtil fctor serves s reduction for the cpcity: high γ M -vlues impose lrger reduction on the buckling cpcity compred to lower γ M -vlues. One of the erliest studies concerning the sttisticl evlution of resistnce models ws crried out by Sedlcek et l. [4] t RWTH Achen, Germny. Although the investigtion ws performed prior to the finl ppernce of EN 99, it dopted the sme methodology. The study ws imed t finding new imperfection fctors for the resistnce model of Eurocode3 (EN 993--) to check the lterl-torsionl buckling resistnce of rolled nd welded bems. The relibility of the old resistnce model, originlly from the DIN, in ddition to the new resistnce model ws re-evluted. The sttisticl evlution ws bsed on 44 lterl-torsionl buckling tests. A probbilistic ssessment of the existing design rules to check the lterl torsionl buckling resistnce of bems ws performed by the University of Coimbr, Portugl for wide flnge bems. The prtil fctor ssocited with the uncertinty of the resistnce model γ Rd ws computed for different lod cses nd section types using the three different design models for lterl torsionl buckling vilble in EN 993--, Rebelo et l. [5]. The evlution of the prtil fctors ws bsed on the solution results from finite element nlyses conducted on wide set of bem configurtions. In the ccompnying pper by Simoes d Silv et l. [6], the prtil fctor for the mteril properties γ m ws determined bsed on tensile tests conducted on sections mde from different steel grdes. Using eqution (2), the fctors from [5, 6] were used to rrive t γ M -vlues for different lod cses nd steel grdes.

6 6 Buckling curves for hevy wide flnge steel columns PRESENT STUDY In the present study the methodology dopted by Rebelo et l. [5] nd in line with Annex D of EN 99 will be used to check the relibility of proposed buckling curves for hevy wide flnge sections. A lrge dtbse is creted contining the rtio between the elstic-plstic buckling resistnce obtined from non-liner finite element nlyses nd buckling resistnces obtined from the suggested buckling curve for wide set of column configurtions. It is mentioned tht the numericlly obtined elstic-plstic buckling resistnce serves s replcement of the ultimte resistnce found in column buckling test. The rtio between both buckling resistnces for specific set of columns is used to compute the prtil fctor γ Rd ssocited with the uncertinty of the resistnce model. For S46, sttisticl literture informtion concerning the relevnt mteril property, being the yield stress f y, is used to estimte γ m <.. Then the generl prtil fctor γ M is computed ccording to eqution (2). Since for HISTAR 46 such informtion is not yet vilble γ m =. cn be conservtively used to compute the generl prtil fctor γ M ccording to eqution (2). This generl prtil fctor cn hve lower vlue in the nerby future pending the vilbility of lower γ m -vlue representing the vribility in mechnicl properties of HISTAR 46 steels. For S355 nd S235 γ m =. ws lso conservtively dopted to compute the generl prtil fctor γ M ccording to eqution (2). The study is limited to hevy wide flnge sections which possess flnge thickness (t f ) greter thn mm nd for which the height-to-width rtio (h/b) is greter thn.2. The selected sections for the present study re listed in Tble 2. In the present study distinction is mde between HD en HL sections, hving n h/b-vlue of pproximtely.23 nd 2.35, respectively. Tble 2 Hevy wide flnge sections offered by ArcelorMittl with h/b >.2 nd t f > mm Section nme: Europen Americn Imperil HD 4 9 W HD 4 99 W HD 4 86 W HD 4 22 W HD W HL W HL W HL W Weight per m [kg] h [mm] b [mm] t w [mm] t f [mm] h/b [ ]

7 84 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 7 2. BUCKLING CURVE FORMULATION The theoreticl resistnce defining the mximum compressive force column cn withstnd before filing in flexurl buckling mode is determined using EN [8]. The buckling resistnce for column cn be verified s follows: N N Ed b,rd., (3) where: N E,d is the design vlue of the compression force; N b,rd is the design buckling resistnce. The design buckling resistnce is given by: Af f N, (4) b,rd = χ γ where A is the cross-sectionl re, f y is the yield stress, γ M is the generl prtil fctor for instbility nd χ is the buckling reduction fctor. This check is only vlid for sections belonging to cross-sectionl clss, 2 or 3. The product of the crosssectionl re nd the yield stress is known s the sqush lod of the cross-section or N pl. Using equtions (3, 4), the verifiction of the buckling resistnce cn be rewritten s follows: where NEd χn /γ pl M M.. (5) The buckling reduction fctor cn be computed ccording to: χ = but χ., (6) 2 2 Φ+ Φ λ 2 ( ( ) ) Φ =.5 + α λ.2 + λ. (7) The reltive slenderness λ cn be determined s follows: λ = N N, (8) where N cr is the elstic criticl force of the column. The imperfection fctor α ttins one of the vlues s listed in Tble 3, depending on the cross-section, steel grde nd buckling cse (wek-xis or strong-xis buckling) under considertion. pl cr

8 8 Buckling curves for hevy wide flnge steel columns 85 Tble 3 Imperfection fctor for buckling curves Buckling curve b c d Imperfection fctor α A grphicl representtion of the buckling curves is shown in Figure 2. Bsed on the selected buckling curve nd corresponding imperfection fctor theoreticl resistnce χ cn be computed for hevy section if the reltive slenderness is known. This vlue will be compred to the elstic-plstic buckling resistnce obtined from non-liner finite element nlysis (Section 3). Reltive resistnce χ [-] b c d Reltive slenderness λ [-] Fig. 2 Buckling curves from Eurocode3. 3. FINITE ELEMENT MODEL 3.. ELEMENTS The geometricl nd mteril non-liner nlyses on the columns contining imperfections (GMNIA) were performed in the ANSYS v.. implicit environment. The columns were modelled with bem elements. The 3D three node finite strin element (BEAM89) ws selected for the nlyses s it cn describe plsticity, lrge deformtions nd lrge strins. A user-defined cross-section ws modelled bsed on nominl dimensions (Tble 2). The cross-section is subdivided into different cells to cpture growth of plstic zones cross the cross-section. Ech

9 86 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 9 cell contins four integrtion points where the stresses re evluted (Fig. 4). Two integrtion point loctions in longitudinl direction of ech element describe progressive yielding long the length of the column. A totl of 2 elements long the length of the member were considered sufficient. Erlier reserch studies on column buckling hve shown tht this element type is ble to replicte experimentl elstic-plstic buckling tests with good ccurcy thereby tking into ccount the effects of residul stresses [7, 8] BOUNDARY CONDITIONS All selected column configurtions for the present investigtions were simply supported. The column ws pin-supported nd torsionlly restrined t the bottom. The sme boundry condition ws pplied t the top with the exception tht verticl trnsltion ws permitted. For the evlution of strong-xis buckling, the column ws restrined ginst wek-xis deflections by trnsltionl supports long the length (Fig. 3). () wek-xis buckling u y =u z =φ x = F (b) strong-xis buckling u y =u z =φ x = F L u y = x z y z y y u x =u y =u z =φ x = u x =u y =u z =φ x = Fig. 3 Boundry conditions RESIDUAL STRESSES An individul residul stress vlue ws set for ech integrtion point in the cross-section. The stress vlue specified for ech integrtion point is ssigned to the tributry re belonging to tht integrtion point, rendering step-wise initil stress pttern over the cross-section (Fig. 4b). Here the pttern for the residul stress model of [9] for HISTAR 46 is shown. The residul stresses re constnt cross the flnge thickness nd web thickness. After inserting the residul stresses into the element, first solution step ws issued to verify internl equilibrium of the residul stress model. Insignificnt differences were observed between the residul

10 Buckling curves for hevy wide flnge steel columns 87 stress model nd the stresses fter solving, indicting correct implementtion of the residul stress model (Fig. 4c). Fig. 4 Finite element discretiztion of cross-section nd implemented residul stresses (for HD 4 22). For HISTAR 46 nd for S46, the QST residul stress model of [9] ws dopted s depicted in (Fig. 5). This model hs prbolic shpe long the web height nd flnge width. The mgnitude of the residul stresses in the web is relted to the section dimensions. ) QST [9] b) ECCS [] Fig. 5 Residul stress models. Currently no experimentl dt on the residul stress distribution in mild steel HL nd HD sections is known to the uthors. In order to include the effect of residul stress on the resistnce of these sections when filing by strong- nd wekxis buckling n ssumption is mde bout their distribution. Two different residul models re selected to mke n educted guess concerning the residul stresses: the ECCS residul stress model [] commonly used for wide flnge hotrolled sections (Fig. 5b) nd the erlier derived QST residul stress model (Fig. 5).

11 88 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg As the ltter model hs residul stress vlues exceeding the (reduced) yield stress of S235 steels, which is physiclly not possible, the QST residul stress model is not used for the S235 columns. The ECCS residul stress model is fetured by biliner stress distribution long the web height nd flnge widths. The extreme vlues re t the flnge tips nd web center (compression) nd web-to-flnge junction (tension) nd set t 3 % of the (unreduced) yield stress. The unreduced yield stresses, f y = 235 N/mm 2 nd f y = 355 N/mm 2 for S235 nd S355 respectively, were used which is conservtive in view of defining the residul stress vlues. Using f y = 355 N/mm 2 my be very conservtive since normlly f y = 235 N/mm 2 is used with the ECCS residul stress model regrdless the steel grde. The residul stress vlues t the most criticl loctions for the QST nd ECCS residul stress models re listed in Tble 4 nd Tble 5, respectively. Tble 4 Residul stress vlues for QST residul stress model [9] [N/mm 2 ] Steel grde HISTAR46 S46 S355 Section σ frt σ frc σ wrt σ wrc (tension) (compression) (tension) (compression) HD HD HD HD HD HL HL HL Tble 5 Residul stress vlues for ECCS residul stress model [] [N/mm 2 ] Steel grde Section Flnge tip/web center Web-to-flnge junction (compression) (tension) S235 ll = = 7 S355 ll = = MATERIAL MODEL A bi-liner mteril model ws pplied to describe the mteril s response to loding (Fig. 6). A fixed yield stress vlue f y ws used to define the onset of yielding. This vlue is bsed on the steel properties, thereby tking into ccount reduction in yield stress due to the thickness of the flnges. A more generlly ccepted vlue for the Young s modulus of 2 N/mm 2 ws dopted to define the elstic stge of the mteril. Strin hrdening increses the ultimte lod of stocky columns but hrdly ffects columns in the intermedite nd high slenderness rnges. Neglecting strin hrdening is conservtive pproch nd is in line with wht hs previously been done by other reserchers, e.g. [5, 6]. Therefore, no strin hrdening effects were included.

12 2 Buckling curves for hevy wide flnge steel columns 89 Steel grde Yield stress f y [N/mm 2 ] HISTAR S S S E = 2 N/mm 2 Fig. 6 Mteril model GEOMETRIC IMPERFECTIONS The shpe of the geometric imperfection ws bsed on the buckling mode belonging to the lowest eigenvlue from liner buckling nlysis. This resulted in sinusoidl bow imperfection. The mplitude defining the mximum devition from the idel geometry ws L/, where L is the height of the column. This pproch is generlly ccepted for the determintion of buckling curves. A similr pproch, but then for lterl torsionl buckling of bems, ws used in [5]. The vlue L/ for the imperfection mplitude is recommended in []. This vlue is expected to be conservtive since it is very likely tht the rel imperfections of the hevy sections considered here re smller thn L/. The vlue L/ is design imperfection mplitude for use in numericl nlyses SOLUTION All elstic-plstic buckling GMNIA re lod-controlled. A force with specified mgnitude ws pplied t the top of the column. The Arc-Length method ws selected to solve the non-liner equilibrium itertions. The Arc-Length method ws selected in preference to the conventionl Newton-Rphson method s the former is ble to describe the decresing lod-deflection curve beyond the mximum lod wheres the ltter will bort the solution when the mximum resistnce hs been reched. The lod ws divided into four lod steps which in turn were further divided into substeps or lod increments. For ech lod-increment number of equilibrium itertions were performed to rrive t converged solution. The solution ws considered solved when the out-of-blnce lod vector is smller thn.5 % of the lod increment. The ultimte strength or flexurl buckling resistnce of the column (N ult;fem ) ws identified s the mximum lod on the lod-deflection curve. The elstic buckling lod (N cr;fem ) is obtined from liner buckling nlysis (LBA) using the Block-Lnczos extrction method of eigenvlues.

13 9 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg PLOTTING RESULTS IN BUCKLING CURVE For ech column configurtion for which the ultimte resistnce is evluted through non-liner finite element nlyses, the reduction fctor is obtined by normlizing the ultimte lod ginst the sqush lod of the cross-section (N pl;fem ). Nult;FEM χ FEM =, (9) N pl;fem where the sqush lod of the cross-section is computed ccording to: N = Af, () pl;fem where A is the cross-sectionl re of the element nd f y is the yield stress. The fctor χ FEM is lbeled s the experimentl resistnce for comprison with the theoreticl resistnce χ. The reltive slenderness of the column cn be computed by tking the squre root of the rtio between the sqush lod of the crosssection nd the elstic buckling lod evluted from liner buckling nlysis: y λ FEM = Npl;FEM Ncr;FEM. () Note tht eqution () is similr to eqution (8) but the sqush lod is now bsed on tht of the FEM model nd the elstic buckling lod is clculted with LBA FINITE ELEMENT RESULTS Steel HISTAR 46 Typicl lod-deflection curves s obtined from the finite element nlyses re shown in Fig. 7. N ult;fem /N cr;fem [-] HD 4x22 L=8 mm wek-xis Reduction fctor [-] b HD 92x377 L=44 mm strong-xis HD 4x22 L=8 mm wek-xis.2 HD 92x377 L=44 mm strong-xis.2 Column curve Lterl displcement [mm] 2 3 Reltive slenderness [-] Fig. 7 Finite element output: lod-deflection curves for HISTAR 46 () nd corresponding dt in buckling curve digrm (b).

14 4 Buckling curves for hevy wide flnge steel columns 9 In Fig. 7b the ultimte lods from Fig. 7 re plotted in the buckling curve digrm using the equtions (9 ) in ddition to buckling curve. Plotting the ultimte lod for specific group of columns in the buckling curve digrm in ddition to buckling curve llows first estimte to be mde s to whether tht specific buckling curve is on the conservtive or unconservtive side Steel grde S46 Typicl lod-deflection curves s obtined from finite element nlyses re shown in Fig. 8 for four HD 4x299 columns in S46 filing by wek-xis buckling. In Fig. 9 the ultimte lods from Fig. 8 re plotted in the buckling curve digrm using the equtions (9 ) in ddition to buckling curve b to show tht there is resonble fit with buckling curve b slightly on the conservtive side elstic buckling 3 elstic buckling Lod [kn] 8 elstic-plstic buckling Lod [kn] 2 elstic-plstic buckling Lterl displcement [mm] HD 4x299 S46 L=6 m elstic buckling Lterl displcement [mm] HD 4x299 S46 L=2.4 m elstic buckling 2 elstic-plstic buckling 6 elstic-plstic buckling Lod [kn] 8 Lod [kn] Lterl displcement [mm] HD 4x299 S46 L=8.8 m Lterl displcement [mm] HD 4x299 S46 L=25.2 m Fig. 8 Finite element output: lod-deflection curves for elstic nd elstic-plstic buckling nlyses of HD columns in S46.

15 92 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 5.8 HD 4x299 S46 wek-xis Reduction fctor [-] Buckling curve b FEM 2 3 Reltive slenderness [-] Fig. 9 Finite element results plotted in buckling curve digrm for buckling of HD columns in S Steel grdes S355 nd S235 Typicl lod-deflection curves for strong xis buckling s obtined from finite element nlyses re shown in Fig. for columns with different lengths with cross-section HL in S355 nd S235 using the QST nd ECCS residul stress models S355 QST model 4 Lod [kn] 3 2 S235 ECCS model Lod [kn] 3 2 S355 QST model HL 92 x 377 L = 3 m S235 ECCS model HL 92 x 377 L = 5 m Lterl displcement [mm] Lterl displcement [mm] Fig. Finite element output: lod-deflection curves for buckling nlyses of HL columns.

16 6 Buckling curves for hevy wide flnge steel columns 93 In Fig. the ultimte lods from Fig. re plotted in the buckling curve digrm using the equtions (9 ) in ddition to buckling curve to show tht there is resonble fit..8 L=3 m S235 ECCS Model Reduction fctor [-].6.4 L=5 m S355 QST Model buckling curve Reltive slenderness [-] Fig. Finite element results plotted in buckling curve digrm for buckling of HL columns. 4. STATISTICAL EVALUATION AND SUGGESTED BUCKLING CURVES 4.. PARTIAL FACTOR EVALUATION PROCEDURE The prtil fctor evlution procedure follows Annex D of EN 99 nd is pplied here in similr wy s in [5]. For ny hevy QST column i comprison cn be mde between its experimentl resistnce (r e,i ) nd its theoreticl resistnce (r t,i ): R r e,i i =. (2) rt,i In the present study the experimentl resistnce refers to χ FEM from eqution (9) for column i filing by flexurl buckling s obtined from non-liner finite element nlysis, so r e,i = χ FEM. The theoreticl resistnce of column i refers to the buckling reduction fctor χ ccording to the buckling curve formultion from EN (eqution (6)), so r t,i = χ. It is noted tht in order to rrive t theoreticl resistnce selection for buckling curve (imperfection fctor from Tble 3) must lredy be mde. A vlue of R i smller thn. or lrger thn. reflects n unconservtive or conservtive theoreticl resistnce model, respectively. For ny

17 94 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 7 group of column configurtions belonging to set with smple size n, the men vlue correction fctor R m nd corresponding vrince cn be determined: n n 2 m i R i n n i = i = ( ) 2 R = R, σ = R R. (3) When plotting the experimentl resistnce on the y-xis nd corresponding theoreticl resistnce on the x-xis for ll column configurtions belonging to subset n, the points will be distributed round the so-clled estimtor line: r e = R m r t. For ech column configurtion belonging to subset n n error term δ i is introduced: m δ i = r r t, i ei, R M. (4) A logrithmic trnsformtion is performed: ( ) = n δ. (5) i For the logrithmic error terms belonging to smple size n, the men vlue nd corresponding vrince re determined s follows: n 2 i n i = i= i n ( ) 2 i =, =. n σ (6) The vrince cn be used to compute the coefficient of vrition s follows: V δ ( σ 2 ) = exp. (7) When using subset with smple size n > the prtil fctor ssocited with the uncertinty of the resistnce model cn be determined s follows: for which: γ =., ( d,n ) Rd 2 Rmexp k Q.5Q 2 ( Vδ ) (8) Q = n + (9) nd where k d,n is the chrcteristic frctile fctor: = 3.4.

18 8 Buckling curves for hevy wide flnge steel columns 95 So, finlly eqution (8) gives the prtil fctor γ Rd belonging to suggested buckling curve bsed on set of column configurtions PARTIAL FACTOR FOR SUGGESTED BUCKLING CURVES Non-liner finite element nlyses were crried out for the hevy wide flnge cross-sections of Tble 2. Per steel grde, cross-section, buckling xis nd where pproprite residul stress model, t lest nlyses were performed indicted by the smple size n Steel HISTAR 46 For HISTAR 46 ll eight cross-sections were considered, (Tble 2). For ech cross-section the wek-xis nd strong-xis buckling response ws evluted. The reltive slenderness of the investigted columns ws in the rnge between.3 nd 3.3. Plotting the finite element results in buckling curve digrm permits first judgment on the suitbility of the buckling curve to represent the column strength for hevy HISTAR 46 sections. In cse the chosen buckling curve is positioned below the finite element results, it will provide conservtive column strength vlues. The buckling curve cn be regrded s unconservtive when the finite element dt is below the buckling curve. Figure 2 (left) shows the finite elements results for HL 4 22 section in HISTAR 46 buckling bout its wek xis in buckling curve digrm in ddition to buckling curve b. Similr trends re found when plotting the theoreticl column strength r t,i = χ ginst its numericl counterprt r e,i = χ FEM such s shown in Fig. 2 (right). In cse the buckling curve produces column strengths similr to the finite element results, the dt is positioned on the line r e = r t. Dt distributed bove the line r e = r t indictes tht the buckling curve provides conservtive vlues for the column strength. Unconservtive columns strengths re found when the dt points re below r e = r t. When the buckling curve formultion represents column strengths different from those obtined with finite element nlyses, the dt points will be distributed round the line r e = R m r t, where R m is men vlue correction fctor ccording to eqution (3), [, 9]. This line will give better description of the correltion between the theoreticl nd numericl vlues in comprison to r e = r t. The prtil fctor ssocited with the uncertinty of the resistnce model is evluted for ech buckling curve. The corresponding γ Rd -vlues for ech section type nd buckling xis re presented in Tble 6. For unfvorble buckling curves the γ Rd -vlue is lower in comprison to more fvorble buckling curves for mjority of the investigted cses. Hence, relting the elstic-plstic buckling response of hevy HISTAR 46 section to more fvorble buckling curve is t

19 96 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 9 the expense of higher prtil fctor γ Rd. The most fvorble buckling curve selected for hevy HISTAR 46 sections filing by flexurl buckling is bsed on the criterion γ Rd <.5, s denoted in bold in Tble 6. This is ssocited with trget vlue γ Rd =. so tht vlues greter thn.5 cnnot be ccepted. The.5 boundry is rbitrry but it is believed to be resonble..8 HD 4x22 HISTAR 46 wek-xis.8 HD 4x22 HISTAR 46 wek-xis Reduction fctor [-] Buckling curve b FEM re,i [-] FEM re=rmxrt re=rt 2 3 Reltive slenderness [-] rt,i [-] Fig. 2 Finite element dt in buckling curve (left) nd compred ginst theoreticl solutions for buckling curve b (right) wek-xis buckling of HD 4x22 in HISTAR 46. Tble 6 Prtil fctors γ Rd per buckling curve for HISTAR 46 Hevy section HD 4 9 HD 4 99 HD 4 86 HD 4 22 HD HL HL HL Buckling curve Buckling Smple xis size n b c d Wek Strong Wek Strong Wek Strong Wek Strong Wek Strong Wek Strong Wek Strong Wek Strong

20 2 Buckling curves for hevy wide flnge steel columns 97 The differences between sections belonging to the sme type (HD or HL) nd buckling round the sme xis (wek or strong) re reltively smll, indicting tht section geometry for the sme section type hs little influence on the prtil fctor. In generl the prtil fctors for n identicl buckling curve re greter for the wek-xis buckling cse thn those for the strong-xis buckling cse. This reflects the more detrimentl influence of residul stresses for columns filing by wekxis buckling. Assuming tht γ Rd -vlue smller thn.5 llows γ Rd =. to be used, HD nd HL sections filing by wek-xis buckling should be ssigned to buckling curve b. Buckling curve is ssigned to HD sections filing by strong xis buckling. HL sections buckling bout the strong xis should be checked by buckling curve. These results re summrized in Tble 7. Tble 7 Proposed buckling curves for HISTAR 46 sections bsed on γ Rd Cross-section Limits Buckling bout xis Buckling curve Rolled I-sections HD section: h/b.23 HL section: h/b 2.35 y-y z-z y-y z-z b b Steel grde S46 For S46 four different cross-sections were considered, nmely HD 4 9, HD 4 299, HL nd HL (Tble 2). For ech cross-section the wek-xis nd strong-xis buckling response ws evluted. The reltive slenderness of the investigted columns ws in the rnge between.23 nd Figure 3 (left) shows the finite elements results for HL section in S46 buckling bout its strong xis in buckling curve digrm in ddition to buckling curve. In Fig. 3 (right) the sme results re plotted with the theoreticl column strength r t,i = χ on the horizontl xis ginst its numericl counterprt r e,i = χ FEM on the verticl xis.

21 98 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 2.8 HL 92x377 S46 strong-xis.8 HL 92x377 S46 strong-xis Reduction fctor [-].6.4 re,i [-].6.4 FEM.2 Buckling curve.2 re=rmxrt FEM re=rt 2 3 Reltive slenderness [-] rt,i [-] Fig. 3 Finite element dt in buckling curve (left) nd compred ginst theoreticl solutions for buckling curve b (right) strong-xis buckling of HL in S46. The prtil fctor ssocited with the uncertinty of the resistnce model is evluted for ech buckling curve. The corresponding γ Rd -vlues for ech section type nd buckling xis re presented in Tble 8. Agin the prtil fctor should be smller thn.5 to llow γ Rd =. to be used. The ssocited vlues re denoted in bold in Tble 8. Agin the differences between sections belonging to the sme type (HD or HL) nd buckling round the sme xis (wek or strong) re reltively smll nd the prtil fctors for n identicl buckling curve re greter for the wek-xis buckling cse thn those for the strong-xis buckling cse. The proposed buckling curves for S46 sections re the sme s for HISTAR 46 (Tble 7) except for HL sections buckling round the wek xis where buckling curve c seems to be more pproprite. However, it should be noted tht for tht cse the prtil fctors belonging to buckling curve b (γ Rd =.66 nd γ Rd =.58 for the sections HL nd HL , respectively) just slightly exceed γ Rd =.5. Hevy section HD 4 9 HD HL HL Tble 8 Prtil fctors γ Rd per buckling curve for S46 Buckling curve Buckling xis Smple size n b c d Wek Strong Wek Strong Wek Strong Wek Strong

22 22 Buckling curves for hevy wide flnge steel columns Steel grdes S355 nd S235 Since HL cross-sections pprently result in less fvorble buckling curves thn HD cross-sections, only HL cross-sections were considered for 355 nd S235. For two cross-sections in S355 nd S235 strong-xis buckling ws considered: HL nd HL (Tble 2). For cross-section HL mde from steel grde S355 buckling bout the wek-xis ws considered. The reltive slenderness of the investigted columns ws in the rnge between.2 nd HL 92x377 S355 ECCS strong-xis.8 HL 92x377 S355 ECCS strong-xis Reduction fctor [-].6.4 re,i [-].6.4 FEM.2 Buckling curve FEM.2 re=rmxrt re=rt 2 3 Reltive slenderness [-] rt,i [-] Fig. 4 Finite element dt in buckling curve (left) nd compred ginst theoreticl solutions for buckling curve b (right) strong-xis buckling of HL in S355 with ECCS residul stress model. Figure 4 (left) shows the finite elements results for HL section in S355 with ECCS residul stress model, buckling bout its strong xis in buckling curve digrm in ddition to buckling curve. In Fig. 4 (right) the sme results re plotted with the theoreticl column strength r t,i = χ on the horizontl xis ginst its numericl counterprt r e,i = χ FEM on the verticl xis. The prtil fctors ssocited with the uncertinty of the resistnce model re evluted for ech buckling curve nd shown in Tble 9. Also the residul stress model used is indicted. Agin the prtil fctor should be smller thn.5 to llow γ Rd =. to be used. The ssocited vlues re denoted in bold in Tble 9. For S355 nd S235 Tble 9 suggests buckling curve for strong-xis buckling nd buckling curve c for wek-xis buckling. It should be noted tht the strong-xis prtil fctors ssocited with buckling curve obtined with residul stress model QST re quite close to the.5 boundry vlue. It should lso be noted tht no knowledge is vilble with respect to the rel residul stress models nd levels for these hevy sections in S355 nd S235.

23 2 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 23 Hevy section HL S355 HL S355 Tble 9 Prtil fctors γ Rd per buckling curve for S355 nd S235 Buckling curve Res. str. Buckl. Smple mod. xis size n b c d Steel gr. S235 ECCS Strong ECCS Strong QST Strong Wek S235 ECCS Strong ECCS Strong QST Strong STATISTICAL INFORMATION ON MATERIALS The presented nlyses so fr were limited to the computtion of γ Rd -vlues. Using vilble sttisticl informtion on the prtil fctor for the mteril γ m nd using eqution (2) my produce lower γ M -vlues thn the γ Rd -vlues, since γ m -vlues re generlly smller thn.. The relevnt mteril property is the yield stress. So, if sttisticl informtion on the yield stress is vilble such tht γ m -vlue cn be evluted, then the generl prtil fctor γ M cn be clculted with eqution (2). As the yield stress of single coupon cn never be lower thn the nominl yield stress, s this would led to the member being rejected, it cn be resonbly ssumed tht γ m -vlue of. is conservtive vlue to ccount for the vribility of the mteril properties. So, if sttisticl informtion on the yield stress is not vilble, the prtil fctor for the mteril γ m cn be sfely set equl to γ m =. nd the generl prtil fctor bsed on eqution (2) is equl to γ M = γ Rd Steel HISTAR 46 Since for HISTAR 46 published sttisticl informtion on the yield stress is not yet vilble γ m =. is used to compute the generl prtil fctor γ M ccording to eqution (2): i.e. γ M = γ Rd. Assuming tht γ M =. is the trget vlue for the generl prtil fctor, then the (rbitrry but resonble) criterion for choosing buckling curve is tht γ M <.5. So, in fct for HISTAR 46 with γ M = γ Rd, lso the prtil fctor should fulfil the requirement γ Rd <.5. This mens tht for HISTAR 46 the buckling curves of Tble 7 cn be used with γ M =.. As soon s dtbse becomes vilble contining the yield stress from wide set of coupon tests on HISTAR 46 sections, vlue of γ m lower thn. cn be obtined resulting in either lower generl prtil fctor or more fvorble buckling curve Steel grde S46 For S46, sttisticl literture informtion concerning the yield stress f y is used to estimte γ m <.. Two RFCS (Reserch Fund for Col nd Steel) projects provide sttisticl dt: OPUS [2] nd PROQUA [2].

24 24 Buckling curves for hevy wide flnge steel columns 2 OPUS [2] provides sttisticl dt for the yield stress of S46M in the flnge thickness rnge of 6 < t f < 4mm: the men vlue is f y,m = 52. N/mm 2, the stndrd devition is f y,σ = N/mm 2 nd the coefficient of vrition then is V fy = f y,m / f y,σ =.5. The rtio between men nd nominl yield stress cn be clculted s R = f y,m / f y,nom = 52./46 =.3. If it is ssumed tht this rtio lso pplies to S46 cross-sections with flnge thicknesses t f > mm, then with reduced nominl yield stress for thickness f y,nom = 385 N/mm 2, the men vlue becomes: f y,m = f y,nom R = = 435 N/mm 2. Keeping the coefficient of vrition the sme, the mteril prtil fctor cn be clculted using: f y,nom γ m =. f (.64V ) y,m fy This results in γ m = 385/(435(.64.5)) =.966. If this mteril prtil fctor is used with the prtil fctors γ Rd of Tble 8 in eqution (2), more fvorble generl prtil fctors γ M re obtined. This does not ffect the buckling curves for HD sections nor does it ffect the buckling curve for strong-xis buckling of HL columns but it does ffect the buckling curve for wek-xis buckling of HL columns. For HL columns in wek-xis buckling nd for buckling curve b the γ Rd - vlues re.66 nd.58 for HL nd HL respectively. Multiplied (eqution (2)) by γ m =.966 the γ M -vlues become.3 nd.22 respectively. So γ M <.5 mening tht γ M =. my be used in this cse together with buckling curve b. This mens tht lso for S46 the buckling curves of Tble 7 cn be used with γ M =.. Sttisticl dt given in PROQUA [2] together with plnt mesurements support the vlue γ m =.966 used Steel grdes S355 nd S235 Though sttisticl informtion on the yield stress of S355 nd S235 is redily vilble, there re other resons (to be mentioned herefter) not to use this informtion. (2) 5. BUCKLING CURVES FOR EUROCODE3 Though for HISTAR 46 nd S46 the buckling curves of Tble 7 my be used for hevy sections with h/b >.2 nd t f > mm in combintion with the generl prtil fctor γ M =., these buckling curves cnnot be incorported in Eurocode3 EN [8] if the formt of the buckling curve selection tble (Tble ) is to be kept, since no distinction is mde between HD nd HL crosssections. For tht reson, the most unfvorble buckling curves for HD nd HL

25 22 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg 25 cross-sections of Tble 7 re used in the proposed new Eurocode3 buckling curve selection tble (Tble ). Also the proprietry nme HISTAR cn obviously not be mentioned in the proposed Eurocode3 buckling curve selection tble. For S355 nd S235 it ws shown in section tht for hevy sections with h/b >.2 nd t f > mm the buckling curves nd c my be used for strong nd wek-xis buckling respectively, together with γ Rd =.. This result my even be improved using sttisticl dt on the yield stress resulting in γ m <. nd thus in either lower vlue of the generl prtil fctor γ M or in more fvorble buckling curves thn mentioned. However, this does not mke sense s long s no further nd better informtion is vilble on residul stress distributions in hevy sections with h/b >.2 nd t f > mm in S355 nd S235. In fct, informtion on residul stresses is not vilble t this moment t ll. For tht reson it is proposed to conservtively use the buckling curves b nd c, which fit nicely in the tble when compring with buckling curves for other cses. Often, moving from S46 to lower steel grdes mens shift of one buckling curve nd the current proposl is in line with tht. The QST residul stress model [9] s used in the present nlyses is representtive for ny hevy wide flnge section hving similr cross-sectionl dimensions nd mde with the Quenched nd Self-Tempered process. As such, the residul stress model cn be used to define the initil stress stte in hevy sections mde from grde S46 but lso from S5 s these grdes re mnufctured with identicl methods by ArcelorMittl s HISTAR 46 steel. Grde S5 lso hs the sme nominl yield stress fter reduction to ccount for mteril thickness effects (i.e. 45 N/mm 2, see Fig. ). For these resons S5 cn be dded in the lst column of Tble. Cross-section Rolled I-sections Tble Proposed buckling curve selection tble for Eurocode3, EN Limits t f 4 mm Buckling bout xis y-y z-z Buckling curve S235 S46 S275 S5 S355 S42 b h/b >.2 4 < t f mm y-y z-z b c h/b.2 t f > mm t f mm t f > mm y-y z-z y-y z-z y-y z-z b c b c d d b c c

26 26 Buckling curves for hevy wide flnge steel columns CONCLUSIONS In this pper buckling curves re proposed to check the flexurl buckling resistnce of hevy wide flnge columns, which hve flnge thickness t f > mm nd height-to-width rtio h/b >.2. These sections re currently not covered by Eurocode3 (EN 993--). A dtbse ws creted contining the elstic-plstic buckling resistnce for wide set of hevy HISTAR 46, S46, S355 nd S235 columns (both the stocky HD-type nd slender HL-type) filing by wek-xis nd strong-xis buckling. The buckling resistnce ws evluted using non-liner finite element nlyses using n erlier proposed residul stress model [9] nd the ECCS residul stress model [] to define the initil stress stte. The numericl buckling lods were compred ginst theoreticl vlues, where the ltter correspond to the buckling resistnces for selected buckling curve ccording to EN Bsed on the rtio between both vlues, prtil fctor γ Rd ssocited with the uncertinty of the resistnce model ws evluted ccording to Annex D of EN 99 for ech of the five buckling curves. Aiming t trget vlue for the generl prtil fctor of γ M =., mening tht the resulting γ M -vlues should not exceed.5, buckling curves re proposed. For cross-sections with h/b >.2 nd t f > mm in steel grdes S46 nd S5 the buckling curves nd b re proposed for strong nd wek-xis buckling respectively, while for these cross-sections in steel grdes S235 up to nd including S42 the buckling curves b nd c re proposed for strong nd wek-xis buckling respectively. Quenched nd Self-Tempered (QST) steel cross-sections re currently mnufctured under the proprietry nme HISTAR (HIgh STrength ARcelorMittl) by ArcelorMittl. For stocky HD cross-sections in HISTAR 46 with h/b.23 nd t f > mm it ws shown tht the buckling curves nd b cn be used for strong nd wek-xis buckling respectively. For slender HL cross-sections in HISTAR 46 with h/b 2.35 nd t f > mm it ws shown tht the buckling curves nd b cn be used for strong nd wek-xis buckling respectively. Received on July 6, 24 REFERENCES. SPOORENBERG, R.C., SNIJDER, H.H., CAJOT, L.-G., POPA, N., Buckling curves for hevy wide flnge QST columns bsed on sttisticl evlution, Journl of Constructionl Steel Reserch,, pp , AXMANN, G., Steel going strong, Modern Steel Construction, 43, pp. 56 6, POLLAK, B.S., Designing with Grde 65, Modern Steel Construction, 44, Europen Orgnistion for Technicl Approvls, Europen Technicl Approvl ETA-/56, Long Products mde of HISTAR 355/355L nd HISTAR 46/46L, Deutsches Institut für Butechnik, 2.

27 24 H. H. Snijder, L.-G. Cjot, N. Pop, R. C. Spoorenberg Europen Committee for Stndrdiztion, Hot rolled products of structurl steels Prt 4: Technicl delivery conditions for themo-mechnicl rolled weldble fine grin structurl steels, EN-25-4, Europen Committee for Stndrdiztion, Hot rolled products of structurl steels Prt 2: Technicl delivery conditions for non-lloy structurl steel, EN 25-2, Europen Committee for Stndrdiztion, Hot rolled products of structurl steels Prt 3: Technicl delivery conditions for normlized/ normlized rolled weldble fine grin structurl steels, EN 25-3, Europen Committee for Stndrdiztion, Eurocode 3. Design of steel structures, generl rules nd rules for buildings, Brussels: EN 993--, SPOORENBERG, R.C., SNIJDER, H.H., CAJOT, L.-G., MAY, M.S., Experimentl investigtion on residul stresses in hevy wide flnge QST steel sections, Journl of Constructionl Steel Reserch, 89, pp , 23.. VOGEL, U., et l., Ultimte Limit Stte Clcultion of Swy Frmes with Rigid Joints, ECCS Publiction No 33, Europen Committee for Stndrdiztion, Eurocode Bsis for structurl design, Brussels, EN 99, BEER, H., SCHULZ, G., Bses théoriques des courbes européennes de flmbement, Construction Métllique, 3, RONDAL, J., MAQUOI, R., Single Eqution for SSRC Colomn-Strength Curves, Journl of the Structurl Division, Proc. of the ASCE, 5, ST, pp , SEDLACEK, G., UNGERMANN, D., KUCK, J., MAQUOI, R., JANSS, J., Bckground documenttion Document 5.3 (prtim) Evlution of test results on bems with crosssectionl clss -3 in order to obtin strength functions nd suitble model fctors, Eurocode 3, Editoril Group, REBELO, C., LOPES, N., SIMOES DA SILVA, L., NETHERCOT, D., VILA REAL, P.M.M., Sttisticl evlution of the lterl-torsionl buckling resistnce of steel I-bems. Prt : Vribility of the Eurocode 3 resistnce model, Journl of Constructionl Steel Reserch, 65, pp , SIMOES DA SILVA, L., REBELO, C., NETHERCOT, D., MARQUES, L., SIMOES, R., VILA REAL, P.M.M., Sttisticl evlution of the lterl-torsionl buckling resistnce of steel I- bems, Prt 2: Vribility of steel properties, Journl of Constructionl Steel Reserch, 65, pp , BAN, H., SHI, G., SHI, Y., WANG Y., Overll buckling behvior of 46 MP high strength steel columns: Experimentl investigtion nd design method, Journl of Constructionl Steel Reserch, 74, pp 4 5, SHI G., BAN, H., BIJLAARD, F.S.K., Tests nd numericl study of ultr-high strength steel columns with end restrints, Journl of Constructionl Steel Reserch, 7, pp , BIJLAARD, F.S.K., Eurocode 3 Design of steel structures Present sttus nd further developments, Steel Construction Design nd Reserch,,, pp. 6 23, BRACONI, A., et l., OPUS, Optimizing the seismic performnce of steel nd steel-concrete structures by stndrdizing mteril qulity control, Europen Commission, Technicl Steel Reserch, Steel products nd pplictions for building construction nd industry, Finl report, Directorte-Generl for Reserch, EUR 25893, CAJOT, L.-G., et l., PROQUA, Probbilistic quntifiction of sfety of steel structure highlighting the potentil of steel versus other mterils, Europen Commission, Technicl Steel Reserch, Steel products nd pplictions for building construction nd industry, Finl report, Directorte-Generl for Reserch, EUR 2695 EN, Luxembourg, Office for Officil Publictions of the Europen Communities, 25.