STABILITY AD DUCTILITY OF STEEL STRUCTURES D. Camotim et al. (Eds.) Lisbon, Portugal, September 6-8, 2006 FLEXURAL BUCKLIG OF FRAES ACCORDIG TO TE EW EC3- RULES A COPARATIVE, PARAETRIC STUDY Andreas Lechner Graz Universit of Technolog e-mail: lechner@tugraz.at Kewords: frame analsis, second-order analsis, flexural buckling, GIA, steel structures. Abstract. This paper describes the application of the specific methods of the new E 1993-1-1 for the structural design of steel frames. The methods investigated are based on geometricall, materiall nonlinear analses of the imperfect structure (GIA), on second-order elasto-plastic analses and on the two methods using member design formulae. The one is related to non-swa buckling lengths, the other is the euivalent column method. 1 ITRODUCTIO The present E 1993-1-1 specifies several methods for the calculation of the flexural buckling behaviour of steel frames. Depending on the accurac of the intended results there are first-order and second-order elastic and plastic methods available. The detrimental effects of geometrical imperfections and residual stress alwas have to be accounted for. In some analsis tpes the have to be introduced directl into the model, in cases when the beam-column formulae is used member imperfections and residual stress are alread accounted for, however, swa imperfections have to be applied additionall. When the euivalent column method is used, all imperfections ma be regarded to be coved b the buckling checks based on effective column lengths. The present paper aims at comparing the different design methods with each other in order to get a thorough understanding of the behaviour of each method. The comparisons are performed numericall and are illustrated b interaction diagrams. Results presented in detail are those for GIA performed b ABAQUS [1] and those of several elasto-plastic first-order and second-order analses. Results of the method using cut-out-members with second-order end moments and non-swa buckling lengths (alternativel full length) are illustrated and compared with each other. The stud also illustrates results of the euivalent column method using appropriate buckling lengths according to the global buckling mode of the frames. Furthermore, a comparison with the Amplified Swa ethod of the British Standard BS 5950-1:2000 is included. At the end of the paper it is shown, that with a modified euivalent column method rather accurate results could be achieved. The methods were applied to a hinged swa frame with three different load cases and a clamped swa frame with two different load cases. There are pure smmetrical (non-swa) or pure antimetrical (swa) load cases. For the hinged swa frame the behaviour of a half swa and half non-swa load case is also investigated. The results are illustrated b interaction diagrams depending on the parameters, and the column slenderness λ. B these load cases the complex interaction behaviour between second-order effects due to swa and non-swa deflections has been studied. Thus, a wide range of possible frame sstems is covered.
2 REVIEW O EXISTIG RESEARC O TE CORRECT APPLICATIO OF TE C m -FACTOR FOR EQUIVALET EBERS The stabilit-check of swa frames ma be based on the buckling check of individual members, which are considered as isolated columns cut-out of the sstem and are loaded b the end forces/moments of the structure apart from internal loads along the span - (called CO). These end moments have to include the second-order and imperfection-effects of the swa-mode (P. -effect) and in some cases the effects of the local imperfections according to E 1993-1-1 (P.δ-effect), confer Fig. 1 and Fig. 7. Alternativel, buckling-checks ma be based on the euivalent column method (called ). It is applied here in its traditional form as pin-ended column (Euler-case ) of uniform cross-section and uniform axial force with a buckling length, which leads to the same critical buckling capacit cr as the structure considered [7][8]. It has been noticed in several papers [2][3][4] that there is a lack of knowledge on the correct application of bucking lengths and appropriate C m -factors, when the beam-column formulae is used for buckling checks of cut-out-members. This refers to CO and similarl. Regarding the cut-out-method (CO), there are currentl two main interpretations (called CO/1 and CO/2 in the following). A more conservative approach, CO/1, considers the effect of local imperfections b an additional buckling check based on member length and second-order end-moments. This is in accordance with E 1993-1-1. The calculation of the C m -factor for CO/1 is commonl based on the first-order shape of the bending moment diagram between the member ends. The second approach, CO/2, considers the effect of local imperfections b an additional buckling check based on the nonswa buckling length and on second-order end-moments. The calculation of the C m -factor is commonl based on the moment diagram between member ends similarl as for CO/1. owever, there is no more a phsical interrelation between the non-swa buckling length and the shape of the bending moment diagram. But, in subseuent investigations it will be shown that the method CO/2 works well compared to second-order elasto-plastic and second-order ield zone analses. Fig. 1: Definitions for cut-out-members CO/1 and CO/2. Considering the correct application of the C m -factor for the it has been shown b Lechner [5][6] that there are certain relationships between the buckling lengths and the corresponding first-order bending moment diagrams. It has been explained on the basis of numerical studies that the most critical part of the bending moment diagram is the part of the moment diagram between two consecutive inflection points of the lowest buckling mode where the maximum moment is found. Furthermore, it has been shown b interaction diagrams, that for cases where the bending deformation is coincident with the first eigenmode, exact classical euivalent members are found. In cases where the inflection points of the bending deformation (points with zero moment) are in good accordance with the inflection points of the first buckling mode good approximation with the can be found. It has been verified, that more complex cases can be solved in a similar wa. The following four figures, Fig s 2a, 2b, 3a, 3b, illustrate the proper application of the to approximate and more complex euivalent members. Although C m - factors depend on slenderness and compression force [2], the application of constant C m -factors can be verified for the investigated cases. Even in frame analsis, good results could be achieved b the.
/ pl, GIA GIA eu.memb. EC3/2 EC3/2 Cm = 0.58 λ for = 0.7L RS 200/100/10 V/V pl 1,4 1,2 V max = 1.44 V pl GIA GIA eu.memb. EC3/2 EC3/2 Cm = 0.8 λ for = 0.7L RS 200/100/10 / pl λ = / pl (2a) 1,2 (3a) 0.67L = 0.7L L e=l/1000-0.5 L*=0.7L -0.05 / pl Figure 2: Application of for approximate (2a) and more complex (2b) beam-columns. / pl max =1.33 pl e=l/1000 = L/2 L λ = GIA GIA eu.memb. EC3/2 EC3/2 Cm = 0.975 λ for = 0.5L RS 200/100/10 0.71L 0.5 0.5 L*=L/2 Figure 3: Application of for approximate beam-columns, (3a) and (3b). / pl, e=l/1000 0.7L 0.85L = 0.7L L V = 0.7L λ = V 0.43L* 0.57L* λ = L*=0.7L / pl 0.5 0.42L e=l/1000 0.125 0.125 V = L/2 L L/2 GIA 3 ε GIA eu.memb. EC3/2 EC3/2 Class 2: ε max = 3ε Cm = 0.888 λ for = L/2 RS 200/100/10 L*=L/2 (2b) (3b)
An alternative approach of the cut-out-members is also provided in BS 5950-1:2000 b the Amplified Swa ethod (BS-AS). There, the first-order moments are split into two parts. The swa moments are amplified due to the swa-effect and the non-swa moments are reduced b the C m -factor for the moment diagram along the column length. The non-swa buckling length should be taken. The application of CO/1/2, and BS-AS will be illustrated in the parametric stud in section 3. 3 FRAES The following illustrations, Fig s 4 to 6, are intended to show the behaviour of the different design methods for flexural buckling of steel frames according to E 1993-1-1. Six specific design methods are investigated b a parametric stud. It will be shown to which extent the different characteristics of swa and loading non-swa loading of swa frames will be properl accounted for b the design methods. As mentioned in section 1, two different swa frames with two or three different load-cases are investigated. All frames are built up of RS 200/100/10 sections steel grade, meaning that beam and column have the same bending stiffness. The overall frame dimensions are set to a ratio that in all critical sections pl is reached at once due to external loading. The effects of imperfections were not regarded in the choice of the external loading. All numerical simulations are based on elasto-plastic behaviour (Class 2). GIA: nonlinear Finite-Element-analses including first eigenmode-conform imperfections (L/1000), residual stress (corner +0.5*f, faces - 0.167*f ) and elasto-plastic stress-strain relationship. 2O EP-analtical: elasto-plastic second-order analses with euivalent global and euivalent local imperfections as specified in E 1993-1-1, analtical cross-section capacit. : euivalent column method according E 1993-1-1 with swa-buckling length and firstorder moments without an imperfections. The crosssection check is based on the E 1993-1-1-capacit. CO/1: cut-out-member with second-order endmoments and full member length. Euivalent imperfections are taken as specified in E 1993-1-1 with initial swa and local imperfections, see Fig. 7. C m is based on member length. CO/2: cut-out-member with second-order endmoments and non-swa buckling length. Euivalent imperfections are taken as specified in E 1993-1-1 with initial swa and local imperfections, see Fig. 7. C m is based on member length. BS-AS: Amplified Swa ethod acc. BS 5950-1, member-check with E 1993-1-1, ethod 2. resp. [F*/2] / pl,,col F / pl,col resp. [ + F/2] / pl,col L = 2* = 2.69 GIA εmax=5ε 2O EP-analtical C m=0.9 CO/2 C m=0.6,non-swa CO/1 C m=0.6 = BS-AS I,col =k amp * swa,non-swa RS 200/100/10 CO/1 λ = = CO/2 Figure 4: inged swa frame with swa load. All calculations were performed numericall either b the ABAQUS-software [1] or b atlabframe analsis tools. These atlab-tools enable parametric elasto-plastic second-order plane-frame analses and subseuent buckling checks according to E 1993-1-1, ethod 2. The cross-section capacit for the methods 2O EP-analtical, CO/1/2 and the Amplified Swa ethod is calculated with the analtic formulae. The design checks done b the are based on the section capacit of the E 1993-1-1 formulae. In all illustrations the buckling checks are limited b the cross-section capacit. Admittedl, in some cases the beam capacit is decisive. =0.92
resp. [*L 2 /16 + F*/2] / pl,,col F=*L/4 (5a) λ = / pl,col resp. [ + *L/2 + F/2] / pl,col 0.56 L = 2* = 2.69 GIA εmax=3ε 2O EP-analtical C m=0.9 CO/2 C m=0.6,non-swa CO/1 C m=0.6 = BS-AS,col = I I k amp* swa +0.6* non-swa,non-swa, k,non-swa RS 200/100/10 CO/1 = CO/2 =0.92 / pl,col resp. [ + *L/2] / pl,col Figure 5: inged swa frames with half swa and half non-swa load (5a) and pure non-swa load (5b). resp. [3*F*/10] / pl,,col F (6a) λ = / pl,col resp. [ + 4/15*F] / pl,col L = 1.5* -0.67 0.67 =1.231* CO/1 GIA εmax=3ε 2O EP-analtical C m=0.9 CO/2 C m=0.4,non-swa CO/1 C m=0.4 = BS-AS =k * I E,col amp swa,non-swa, k,non-swa RS 200/100/10 CO/2 Lcr =0.65* Figure 6: Clamped swa frames with pure swa (6a) and pure non-swa load (6b). resp. [*L 2 /16] / pl,,col resp. [*L 2 /16] / pl,col L = 2* = 2.69 GIA εmax=3ε 2O EP-analtical C m=0.9 CO/2 C m=0.6,non-swa CO/1 C m=0.6 = BS-AS I,col =0.6* non-swa,non-swa, k,non-swa RS 200/100/10 CO/1 λ = = λ = CO/2 =0.92 (5b) / pl,col resp. [ + *L/2] / pl,col 0.5 L = 1.5* 0.5 =1.231 0.423 GIA εmax=3ε 2O EP-analtical C m=0.942 CO/2,col,non-swa C m=0.4 CO/1,col =, C m=0.4 BS-AS,col = I I k amp* swa +0.4* non-swa =L cr,non-swa, k,non-swa CO/1 = RS 200/100/10 CO/2 =0.65* (6b)
Formula used for cut-out-members designed b CO/1 or CO/2: + k, non swa 1 (1) χ, non swa pl pl, is the maximum second-order end moment of the critical column. Formulae which are in use for the Amplified Swa ethod based on BS 5950-1:2000: + k, non swa, C 1 m = χ, non swapl pl, (2) = kamp, swa + Cm, member, non swa (3) Λcr kamp = (second-order load amplification factor) Λ 1 (4) cr C m,member is determined from the moment distribution between the member ends.,swa considers the effects of initial swa deformation and horizontal forces.,non-swa onl considers the bending moments due to non-swa loading. All bending moments are first-order internal moments. The second-order amplification factor k amp is herein calculated b the exact values Λcr of the swa sstem. BS 5950-1 specifies if Λ is less than 4.0 second-order elastic analsis should be used to allow for swa. cr 4 EFFECT OF LOCAL IPERFECTIOS In frame analsis, it is not alwas obviousl which direction of local imperfection is most critical. Therefore, in the herein presented parametric studies two directions of local bow imperfection have been evaluated for the columns. The most critical result was taken for the interaction diagrams shown in section 3 before. In Fig. 7 the investigations which have been performed for the frame stud are shown in detail for the case of the clamped swa-frame with pure swa-load. It is shown, that for cases with high axial load the negative local imperfection e 0 is decisive. It is illustrated that local bow imperfections have to be included according to E 1993-1-1 in CO/1/2 when the following reuirement is met: Af λ > 0.5 (5) λ is the in-plane slenderness based on member length and is the compressive force. λ = 2.5 λ = 1.5 +e CO/1 = Lcr = Cm = 0.40 CO/2 = Lcr = 0.649 Cm = 0.40 e=0 λ = - e / pl,col - e +e λ = 0.5 CO/1 Figure 7: Stud of the effect of local column imperfections at clamped swa frame under swa load. CO/2 Lcr =0.65* global initial swa imperfection and relative initial local bow imperfection if reuired according E1993-1-1: L = 1.5* +e 0 =/250 φ φ=1/200*α h *α m +e 0 =/250 φ
5 FRAES DESIG BY ODIFIED EQUIVALET COLU ETOD During the stud of swa and non-swa second-order effects in swa frames, the application of an amended euivalent column method was discovered which achieves rather accurate results even for frames with non-swa loading. As shown in Fig. 8a, a swa frame with swa loading can be well designed with the as described in section 2. To verif this, the euivalent member was also analsed b GIA, which is in good accordance with the GIA-result of the frame. owever, when loading the swa-frame with non-swa load, the underestimation b the lead to inefficient design. I This led to a new proposal where the first-order bending moments are separated into their swa swa, I and non-swa, non swa parts in the buckling check. Imperfections need not to be accounted for, as in. The -buckling check with the formulae of E 1993-1-1, ethod 2 would be extended to euation (6). I I swa, non, swa + k, swa + k, non swa 1 (6) χ swa, pl pl, pl, k, swa = Cm, swa 1+ 0.6λ swa, Cm, swa 1+ 0.6 (7) χ, swa pl χ, swa pl k, non swa = Cm, non swa 1+ 0.6λ non, swa Cm, non swa 1+ 0.6 (8) χ, non swa pl χ, non swa pl The moment factors C m,swa and C m,non-swa are determined according to the as defined in section 2. In figures 8b and 9 results of a parametric stud with above formula (6) are illustrated. It is verified for the different frame sstems that this modified leads in general to conservative results. These results are considerabl improved in comparison to the for non-swa loading of frames. resp. [F*/2] / pl,,col (8a) λ = / pl,col resp. [ + F/2] / pl,col F L = 2* = 2.69 GIA GIA eu.memb. EC3/2 EC3/2 Cm = 0.9 λ for = 2.69* RS 200/100/10 V 0.37L* 0.63L* L* = 2.69* resp. [*L 2 /16] / pl,,col / pl,col resp. [ + *L/2] / pl,col Figure 8: works well for swa frame with swa load (8a), comparison vs. mod (8b). L = 2* limited b beam capacit + = 2.69 λ = GIA εmax=3ε EC3/2 EC3/2 mod EC3/2 Class 2: ε max = 3ε : Cm = 0.9 λ for = 2.69* mod: k-non-swa: Cm = 0.6 λ for = 0.925* RS 200/100/10 mod =0.92 (8b)
resp. [*L 2 /16 + F*/2] / pl,,col (9a) F=*L/4 / pl,col resp. [ + *L/2 + F/2] / pl,col 0.56 L = 2* Figure 9: Comparison vs. mod, hinged swa frame (9a) and clamped swa frame (9b). 6 COCLUSIO To conclude it can be stated, that current E 1993-1-1 specifications for frame design have been verified. Second-order methods are ver accurate. Based on the proper choice of the moment-factor and the buckling length the cut-out-method and the euivalent column method have been checked. It has been found out, that a modified euivalent column method achieves good results for non-swa load of frames. REFERECES = 2.69 GIA εmax=3ε EC3/2 Class 2: EC3/2 ε max = 3ε EC3/2 mod : Cm = 0.9 λ for = 2.69* mod: k-swa: Cm = 0.9 λ for = 2.69* k-non-swa: Cm = 0.6 λ for = 0.925* λ = RS 200/100/10 mod =0.92 resp. [*L 2 /16] / pl,col / pl,col resp. [ + *L/2] / pl,col [1] ABAQUS, Version 6.3, ibbit, Karlsson & Sorensen, Inc. [2] Camotim D., Gonçalves R, Comparison between Level1 / Level2 interaction formulae and FE results for beam-columns with general support conditions, Report TC8-2001-20, Technical Universit of Lisbon, 2001. [3] Gonçalves R, Camotim D., On the use of beam-column interaction formulae to design and safet check members integrated in steel frame structures, SDSS, Budapest, 2002. [4] Jaspart J.-P., Further considerations on report TC8-2001-20, Universit of Liège, 2001. [5] Lechner A., Steel Structures and Bridges 2003, ISB 80-01-02747-3, Prague 2003. [6] Lechner A., Greiner R., Eurosteel 2005, ISB 3-86130-812-6, aastricht 2005. [7] Roik K., Kindmann R., Das Ersatzstabverfahren Tragsicherheitsnachweise für Stabwerke bei einachsiger Biegung und ormalkraft, Der Stahlbau, 5/1982. [8] Roik K., Vorlesungen über Stahlbau, Ernst & Sohn, 1978. 0.5 L = 1.5* limited b beam capacit + 0.5 =1.231 0.423 λ = mod GIA εmax=3ε EC3/2 EC3/2 mod EC3/2 Class 2: ε max = 3ε : Cm = 0.942 λ for = 1.231* mod: k-non-swa: Cm =0.6 λ for = 0.649* RS 200/100/10 =0.649* (9b)