Numerical validation of the General Method in EC3-1-1: Lateral buckling, Lateral-torsional buckling and bending and axial force interaction

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1 9th International Conference on Steel Concrete Composite and Hbrid Strctres. Leeds, UK, 8 10 Jl 2009 merical validation of the General ethod in EC3-1-1: Lateral bckling, Lateral-torsional bckling and bending and axial force interaction Liliana arqes; Lís Simões da Silva; Carlos Rebelo Civil Engineering Department Universidade de Coimbra, Ra Lís Reis Santos, Pólo 2, Coimbra, Portgal lmarqes@dec.c.pt; lisss@dec.c.pt; crebelo@dec.c.pt Abstract In this paper, theoretical expressions for the General ethod (EC ) are derived for colmns, beam and beam-colmns and its reslts are compared to the rles given in EC to 6.3.3, covering a combination of different load cases and member lengths. In addition, FE nmerical simlations in niform members are carried ot and compared to the reslts of the General ethod. Kewords: Steel; Erocode 3; General ethod; Stabilit; merical analsis 1. Introdction Part 1-1 of Erocode 3 incldes a general method for lateral and lateral-torsional bckling of strctral components, sch as (i) single members, bilt-p or not, with complex spport conditions or not, or (ii) plane frames or sb-frames composed of sch members which are sbject to compression and/or mono-axial bending in the plane, bt which do not contain rotative plastic hinges. The method ses a erchant-rankine tpe of empirical interaction expression to ncle the in-plane effects and the ot-of-plane effects. It is, however, qestionable that its application reslts in a lower bond estimate of the safet of the strctral component. The general method, as given in E in clase [1] states that the overall resistance to ot-of-plane bckling for an strctral component conforming to the sce defined above can be verified b ensring that: γ 1 (1) lt, k / 1 where lt,k is the minimm load amplifier of the design loads to reach the characteristic resistance of the most critical cross section of the strctral component, considering its inplane behavior withot taking lateral or lateral-torsional bckling into accont, however acconting for all effects de to in-plane geometrical deformation and imperfections, global and local, where relevant. is the redction factor for the non-dimensional slenderness to take into accont lateral and lateral-torsional bckling and γ 1 is the partial safet factor for instabilit effects (adted as 1.0 in most ational Annexes). The global non dimensional slenderness λ for the strctral component, sed to find the redction factor in the sal wa sing an apprriate bckling crve, shold be determined from

2 9th International Conference on Steel Concrete Composite and Hbrid Strctres. Leeds, UK, 8 10 Jl 2009 λ = lt, k / (2) cr, where cr, is the minimm amplifier for the in-plane design loads to reach the elastic critical resistance of the strctral component with respect to lateral or lateral-torsional bckling withot acconting for in-plane flexral bckling. In the determination of cr, and lt,k, finite element analsis ma be sed. According to [1], ma be taken either as: (i) the minimm vale of or ; or (ii) an interpolated vale between and, b sing the formla for lt,k corresponding to the critical cross section. It is noted that [2] recommends the se of the first tion onl. Defining as the ltimate load mltiplier with respect to the applied force, when the General ethod is applied to a pinned colmn, expression (3) shows that it does not exactl give the same reslt as the application of clase even whenever the exact same colmn bckling crves are sed: G = For an nrestrained beam, the General ethod exactl coincides with the application of clase 6.3.2, as (3) G = Lt = 1 (4) Derivations of expressions (3) and (4) can be fond in detail in [3]. Finall regarding bending and axial force interaction, consider a pin-ended beam-colmn, sbjected to an arbitrar axial force and a niform major axis constant bending moment. Assming prortional loading, Φ=( / max )/( / max ), the application of clase of E leads to an ltimate load mltiplier which can be defined with respect to the applied loading, given b 1 = max ; Assming the same applied loading and and evalating lt,k according to 6.3.3, the application of the General ethod leads to (5) G = lt, k = in _ pl 1 (6) Comparing expressions (5) and (6) gives: G = max in _ pl ; is determined from either of the following alternatives: (i) the minimm vale of or ; (ii) an interpolated vale between and, b sing the following prosed geometrical interpolation: (7) = (8)

3 9th International Conference on Steel Concrete Composite and Hbrid Strctres. Leeds, UK, 8 10 Jl 2009 The vales of and considered for the redction factor, are calclated with the global non dimensional slenderness λ of the strctral component, determined from eq. (2). According to Trahair [4], the elastic critical bending moment, max, and axial force, max, are given b eq. (9). cr, is then obtained from that expression.,max cr 2 = max max 1 1 (9) cr, cr, T 2. merical model Advanced nmerical simlations contemplating geometrical and material nonlinearities with imperfections (GIA) were considered. For validation of the model, some examples were selected from the PhD thesis of Ofner [5], who carried ot GIA FE analsis sing the finite element program ABAQUS and 3D-beam elements. Comparisons are illstrated in Figres 3 to 5. The finite element model was implemented sing the commercial finite element package LUSAS, version 14 [6], sing thin shell elements. S235 steel grade was considered in the reference examples, with a ield stress of 235 Pa, a modls of elasticit of 210 GPa, and a Poisson s ratio of 0.3. Perfect elastic-plastic behavior of the material was considered. evertheless, some models taking into accont strain-hardening were carried ot for members with low slenderness and differences were shown to be neglected less than 2%. Onl simpl spported single span members with end fork conditions are stdied in this paper. A geometrical imperfection of sinsoidal tpe relative both to the and axes is considered. o torsional imperfection was adted, as it does not need to be considered according to EC Regarding material imperfections, residal stresses are also considered. 3. Parametric std Two levels of comparisons are carried ot. First, the General ethod is compared with the reslts that are directl obtained from the application of clases to (eq. (3) and (7)). Secondl the General ethod is compared with reslts from advanced nonlinear nmerical simlations (GIA) with shell elements. Figre 1 smmaries the cases (abot 0) to be analsed that are directl compared to clases and In the case of beam-colmns for the calclation of the interaction factors, both ethod 1 (1) and ethod 2 (2) from EC3-1-1 [1] are considered. Also, for the calclation of, three alternatives are considered: General Case (GC); Special Case (SC) and General Case modified with the f-factor (GC/f) [6]. Figre 1: Diagram of cases to be analed

4 9th International Conference on Steel Concrete Composite and Hbrid Strctres. Leeds, UK, 8 10 Jl 2009 Table 1 smmaries the cases to be compared with the advanced nmerical simlations. Constant bending moment is adted. Besides fll 3D GIA nmerical simlations, inplane GIA and LEA (Linear Eigenvale Analsis) calclations are also carried ot to provide data for the application of the General ethod. Table 1: Parametric std for nmerical cases; rolled cross sections Cross section IPE 200 / HEB 300 λ 0.5 / 0.8 / 1 / 1.2 / 1.5 / 1.8 / 2 / 2.5 Φ 0 / 1 / For all methods, the pair of maximm applied forces ( max, max ) is first calclated (alwas assming prortional loading, Φ = constant). In order to compare the different methods, the GIA reslts are taken as the basis, assmed to represent the tre behavior of the member [7]. The methods considered in this paper are smmaried in Table 2: 4. Reslts Table 2: ethods considered for comparison ethod GIA GIA in Plane + LBA General ethod EC3-1-1, clases to General ethod theoretical approach (derived expressions in 1.) Acronm GIA G_nm / / 6.3.3_1/ 6.3.3_2 G_teo (beams and colmns) G_1 / G_2 (beam-colmns) 4.1 Comparison of the General ethod (derived expressions in 1.) with clases to G_teo / IPE 360 IPE 200 HEB 550 HEA 550 HEB 450 HEB 400 HEB 360 HEB 340 HEB 300 HEA 200 Φ= ; Hot Rolled G_2 / 6.3.3_ IPE 360 IPE 200 HEB 550 HEA 550 HEB 450 HEB 400 HEB 360 HEB 340 HEB 300 HEA 200 Φ=1; ψ=1; Hot Rolled; GC for Lt; =min(, Lt) Figre 2: Reslts of: eq. (3); eq. (7) Considering the case of pin-ended colmns, analing Figre 2, it can be seen that the ratio between the ltimate load mltipliers for the General ethod and Clase decreases as the slenderness of the colmn increases. Also, reslts concentrate in grs according to the bckling crve: for IPE profiles and HEB 400 to HEB 550 bckling crve is b and for all other profiles bckling crve is c. For beam-colmns, eq. (7) is represented in Figre 2. Reslts also tend to concentrate in grs according to the bckling crve: for IPE 360 the bckling crves are b; for IPE 200 and HEB 400 to HEB 550 the bckling crves are b and a; and for other cross sections the bckling crves are c and a; respectivel the bckling crves for flexral bckling abot the minor axis and the bckling crves for lateral-torsional bckling. ore, relativel to clase

5 9th International Conference on Steel Concrete Composite and Hbrid Strctres. Leeds, UK, 8 10 Jl , the General ethod leads to nsafe reslts for some cases. For other load cases, the respective crves are similar to the crves of Figre 2. Reslts are smmaried in Table 3: Table 3: Reslts of eq. (10) for: Φ=1; GC for Lt ; =min(, Lt ) Fabrication Process Bending oment EC3 ethod 2 EC3 ethod 1 in. ax. in. ax. All Ψ= Hot Rolled Ψ= Ψ= Conc Dist Welded Ψ= Regarding the inflence of, if all the cases of the parametric std of beam-colmns are compared, differences between SC or GC/f and GC are, at the most, 13% lower. Also, for the calclation of, when eq. (7) is considered interpolated vale between and, the ratio between the ltimate load mltipliers for the General ethod and Clase increases, at the most, of 11%. 4.2 merical reslts In Figres 3 to 5 nmerical reslts of the GIA calclations are compared to the different methods described in 3.2: reslts from clases to 6.3.3; reslts for the General ethod obtained from the derived expressions in eq. (3) and eq. (7); and reslts for the General ethod with the nmerical calclations (GIA in plane and LEA). merical reslts from literatre [5] are also plotted for comparison. Analing Figres 3 to 5, for some cases, neither the derived expressions for the General ethod nor the nmerical reslts of the General ethod are alwas conservative, especiall for the low slenderness range. ore, HEB300 presents safer reslts than IPE200, also evident in Figre 2. ethod / GIA 6.3.1/GIA G_teo/GIA G_nm/GIA Lit./GIA ethod / GIA 6.3.1/GIA G_teo/GIA G_nm/GIA Lit./GIA IPE200 Colmn HEB300 Colmn Figre 3: Reslts for colmns IPE 200; HEB 300 ethod / GIA 6.3.2/GIA G_nm/GIA Lit./GIA IPE200 Beam Ψ=1 ethod / GIA 6.3.2/GIA G_nm/GIA Lit./GIA HEB300 Beam Ψ=1 Figre 4: Reslts for beams IPE 200; HEB 300

6 9th International Conference on Steel Concrete Composite and Hbrid Strctres. Leeds, UK, 8 10 Jl 2009 ethod / GIA 6.3.3_2/GIA G_teo/GIA G_nm/GIA Lit./GIA IPE200 Beam-Colmn Ψ=1 / Φ=1 ethod / GIA 6.3.3_2/GIA G_teo/GIA G_nm/GIA Lit./GIA HEB300 Beam-Colmn Ψ=1 / Φ=1 5. Conclding remarks Figre 5: Reslts for beam-colmns IPE 200; HEB 300 For flexral bckling, when eq. (3) is applied, General ethod leads to lower resistance than clase Also, with the increase of the member slenderness, General ethod becomes more conservative. For beam-colmns, when eq. (7) is applied, for some cases General ethod leads to higher resistance than clase Both for colmns and beam-colmns reslts of eq. (3) and eq. (7) are gred according to the respective bckling crves. Regarding the nmerical reslts, for colmns, the General ethod alwas leads to conservative reslts relativel to GIA reslts. However, for some cases of beams and beam-colmns, for low member slenderness, the General ethod leads to nsafe reslts, becoming more conservative with the decrease of the member length. When comparing the nmerical reslts for HEB 300 and IPE 200, it is also noticed that HEB 300 leads to safer reslts than IPE 200, which shold be related to the bckling crves of each cross section. A parametric std for beam-colmns with other tpes of loading and also considering nonniform members is crrentl being carried ot. It is hed that it will frther clarif the application of the General ethod to strctral components. 6. References [1] CE, Erean Committee for Standardiation, (2005), E :2005, Erocode 3: Design of steel Strctres Part 1-1: General Rles and Rles for Bildings, Brssels, Belgim. [2] ECCS TC8, (2008), Resoltion of ECCS/TC8 with respect to the general method in E , TC8 Stabilit. [3] arqes L., Simões da Silva L., and Rebelo C., (2008), merical validation of the General ethod in EC3-1-1: Lateral and lateral-torsional bckling of non-niform members, Erosteel 2008 Conference, pp. 3-8, Astria. [4] Trahair. S., (1993), Flexral-Torsional Bckling of Strctres, E & F SPO, London, UK. [5] Ofner R., (1997), Traglasten von Stäben as Stahl bei Drck and Biegng PhD thesis, Fakltät für Baingenierwesen der Technischen Universität Gra, Gra, Astria. [6] LUSAS 14.1, (2007), Theor manal, Finite Element Analsis Ltd, Version 14, Srre UK. [7] Rebelo C., Les., Simões da Silva L., ethercot D., and Vila Real P., (2008), Statistical evalation of the lateral-torsional bckling resistance of steel I-beams Part 1: Variabilit of the Erocode 3 design model, Jornal of Constrctional Steel Research, doi: /j.jcsr