Journal of Constructional Steel Research

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1 Journal of Construtional Steel Researh 7 () 6 74 Contents lists available at SiVerse SieneDiret Journal of Construtional Steel Researh Development of a onsistent bukling design proedure for tapered olumns Liliana Marques a,, Andreas Taras b, Luís Simões da Silva a, Rihard Greiner b, Carlos Rebelo a a ISISE, Department of Civil Engineering, University of Coimbra, Coimbra, Portugal b Graz University of Tehnology, Institute for Steel Strutures, Graz, Austria artile info abstrat Artile history: Reeived 5 June Aepted 6 Otober Available online 7 November Keywords: Stability Euroode 3 Non-uniform members Tapered olumns FEM Steel strutures EC3 EN provides several methodologies for the stability verifiation of members and frames. When dealing with the verifiation of non-uniform members in general, with tapered ross-setion, irregular distribution of restraints, non-linear ais, astellated, et., the ode mentions the possibility of arrying out a verifiation based on nd order theory; however, several diffiulties are noted when doing so, in partiular when the benefit of plastiity should be taken into onsideration. Other than this, there are yet no guidelines on how to apply standardized, easily reproduible rules as those ontained in Setion 6.3. to of the ode to non-uniform members. As a result, pratial safety verifiations for these members are often arried out using onservative assumptions, not aounting for the advantages non-uniform members provide. In this paper, firstly, available approahes for the stability verifiation of non-uniform members are disussed. An Ayrton Perry formulation is then derived for the ase of nonuniform olumns. Finally, and followed by a numerial parametri study overing a range of slenderness, ross-setions and fabriation proess, a design proposal is made for the relevant ase of in-plane fleural bukling of linearly tapered olumns subjet to onstant aial fore. The proposal is onsistent with urrent rules for uniform olumns provided in EC3--, i.e., lause Elsevier Ltd. All rights reserved.. Introdution The stability of uniform members in EC3-- [] is heked by the appliation of lauses 6.3. stability of olumns; lause 6.3. stability of beams and lause interation formulae for beam-olumns. Regarding the stability of a tapered member, lauses 6.3. to do not apply and verifiation should be performed either by a rosssetional verifiation based on seond-order internal fores or, with some diffiulties, aording to lause (the so-alled General Method, see [,3]). Alternatively, as the most advaned but often pratially not feasible variant, the resistane may also be heked by a numerial analysis that aounts for (geometrial and/or material) imperfetions and (material and/or geometrial) nonlinearities, heneforth denoted as GMNIA. However, for the General Method, several diffiulties are noted for the verifiation of a non-uniform member [4]. These are: (i) shape and magnitude of imperfetions (geometrial and material); (ii) hoie of an appropriate bukling urve and, as a result, of an adequate imperfetion fator; (iii) definition of ross-setion lass; (iv) determination of ross-setion properties for verifiation (or ritial design loation) this problem also eists with respet to the appliation of lauses 6.3. to For the appliation of Corresponding author at: Department of Civil Engineering, University of Coimbra, Polo II, Pinhal de Marroos, 33-9 Coimbra, Portugal. Tel.: ; fa: address: lmarques@de.u.pt (L. Marques). advaned numerial analysis (GMNIA), besides the lak of guidane onerning the shape and magnitude of imperfetions, the volume of work is still inompatible with pratial appliation and a highly eperiened engineer is required [5]. Moreover, there are no guidelines yet to overome any of these issues. In this paper, the ase of olumns subjet to in-plane bukling with varying ross-setion and with onstant aial fore is studied. It is the purpose of this paper to: (i) disuss the urrent diffiulties in performing stability verifiation of non-uniform members; (ii) present the theoretial bakground for non-uniform olumns; (iii) arry out a parametri study of FEM numerial simulations of non-uniform olumns; and (iv) develop a proposal for the stability verifiation of in-plane bukling of tapered olumns with onstant aial fore.. Available approahes for the stability verifiation of non-uniform members.. General Fig. (a) and (b) illustrates reent eamples of the use of urved and tapered members or members with polygonal entroidal ais. The evaluation of the bukling resistane of suh members lies outside the range of appliation of the interation formulae of EC3-- and raises some new problems to be solved. Firstly, taking as an eample the ase of beam-olumns (uniform or not) with varying ratios of M Ed to N Ed over the member length, the ross-setional lassifiation hanges from ross-setion to ross X/$ see front matter Elsevier Ltd. All rights reserved. doi:.6/j.jsr...8

2 6 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 Notations A Cross-setion area E Modulus of elastiity FEM Finite Element Method GMNIA Geometrial and Material Non-linear Analysis with Imperfetions I min Minimum nd moment of area I y,eq Equivalent nd moment of area, y-y ais L Member length L eq Equivalent Member length LEA Linear Eigenvalue Analysis M Ed Design bending moment M R Resistant bending moment M y,ed Design bending moment, y-y ais N Ed Design normal fore N on Conentrated aial fore N r,tapered Elasti ritial fore of the tapered olumn N pl Plasti resistane to normal fore at a given ross-setion N R Resistant normal fore Q Shear fore a, b Auiliary terms for appliation of proposed methodology a,b,,d Class indees for bukling urves b Cross setion width b ma Maimum ross setion width b min Minimum ross setion width e Maimum amplitude of a member imperfetion f y Yield strength h Cross setion height h ma Maimum ross setion height h min Minimum ross setion height n() Distributed aial fore; n Ed () Design distributed aial fore; t f Flange thikness t w Web thikness - Ais along the member First order failure ross-setion (with h=h min ) Loation of the ritial ross-setion y-y Cross-setion ais parallel to the flanges y () Displaement at a given position y () Initial imperfetion y r () Critial displaement at a given position z-z Cross-setion ais perpendiular to the flanges α, α EC3 Imperfetion fator aording to EC3-- Load multiplier whih leads to the fleural bukling resistane of the olumn α r Load multiplier whih leads to the elasti ritial resistane β Generalized imperfetion fator aounting for tapering of the member γ Taper ratio ε Utilization ratio at a given ross-setion ε M Utilization ratio regarding the bending moment M the at a given ross-setion ε N Utilization ratio regarding the aial fore N at a given ross-setion η Generalized imperfetion η Curvature η EC3, η uniform Generalized imperfetion for the prismati member (onsidering ross-setion properties at the ritial position) η non-uniform Generalized imperfetion for the tapered member η num Generalized imperfetion (numerial) λ ðþ Non-dimensional slenderness at a given position λ y Non-dimensional slenderness for fleural bukling, y-y ais ξ. η Retangular oordinates, longitudinal and transversal χ() Redution fator at a given position Redution fator (numerial) χ num setion, see the eample of Fig.. On the safe side, an elasti verifiation onsidering lass 3 ross-setion is likely to be performed although a qualitative analysis of the eample shows that the stresses in the interval orresponding to lass 3 ross-setion are not ritial ompared to the stresses in the remainder of the member. Seondly, the determination of an adequate bukling urve is also neessary and leads to inonsistenies, suh as: (i) The bukling urves in the ode are geared towards speifi bukling ases. That is why the interation formulae and oeffiients for uniform members have to take into aount the transitions from one failure mode to the other (fleural bukling to lateral torsional bukling, et.) The general method an only treat these transitions in a very superfiial way, by interpolation (not reommended by [7]) or, on the other hand, by a time-onsuming speifi alibration, not pratial; (ii) If the method is applied to a tapered member, the question also arises of how to ategorize the member in terms of bukling (a) Curved and tapered elements Barajas Airport, Madrid, Spain (b) Members with polygonal entroidal ais (stairs) Italy pavilion, World Epo Shanghai Fig.. Non-uniform elements. Pitures obtained from [6)

3 L. Marques et al. / Journal of Construtional Steel Researh 7 () Class 3 Class Class Table Epressions from literature for alulation of ritial aial fore of tapered members. M y,ed N Ed <<< Af y Fig.. Uniform beam-olumn with non-uniform loading. Soure Hirt and Crisinel, () [9] Lee et. al (97) [] Galambos (998) [] Petersen (98) [] Desription Epression for equivalent inertia for the tapered olumn, I eq, depending on the type of web variation. Suitable for I-shaped ross-setions. N r ¼ π EIy;eq L Epression for a modifiation fator of the tapered member length, g, i.e., alulation of the equivalent length of a prismati olumn with the smallest ross-setion whih leads to the same ritial load. Suitable for I-shaped ross-setions. N r ¼ π EI y; min ; L Leq eq ¼ g L Design harts for etration of a fator β to be applied to the ritial load of a olumn with the same length and the smallest ross-setion. Suitable for different boundary onditions and ross-setion shapes. N r ¼ β π EI min L urves as the main parameter h/b (height/width) hanges ontinuously, see Fig. 3, in whih h min (and b min ) orresponds to the properties of the olumn in the smaller etreme, and h ma (and b ma ) orresponds to the properties of the olumn in the other etreme. Beause of this, the more restritive bukling urve is most likely to be hosen, leading to over onservative results [8]. Finally, on one hand the General Method requires sophistiated global FEM models but on the other hand it ontains so many simplifiations that one must wonder if it's worth to apply it when ompared to a full non-linear seond-order analysis of the system. The latter is not really more ompliated but more preise and readable for the designer (a designer understands suh things as imperfetions and internal seond-order fores muh better than a hoie of bukling urves and terms involving y, see term η r,ma from eq. (5.9) of EC3--). Therefore it nowadays makes sense to develop simple rules for the basi ases and to inlude as muh knowledge as possible of the real behavior of members in these rules, as it will be arried out in this paper regarding in-plane fleural bukling of tapered olumns... Stability verifiation proedures for tapered olumns As mentioned in Setion., it is mainly formulae for the alulation of tapered member elasti ritial fores that are available in the literature. Some of these are summarized in Table. Nevertheless, the onsideration of a ritial position is still undefined, whih, on the safe side, requires the onsideration of the smallest ross-setion and as a result leads to over-onservative design. Regarding design rules for the verifiation, there is not muh available in the literature. A design proposal for verifiation of tapered olumns an be found in [3], in whih an additional oeffiient K, alibrated numerially and presented in the form of an abaus, is applied to the redution fator of a olumn with the smallest rosssetion (see Eq. ): N b;rd;tapered ¼ K N b;rd;min Finally, some analytial formulations are available: in [4] the equilibrium equation of a tapered olumn subjet to fleural bukling is derived, onsidering a paraboli shape for the imperfetion; in [5], the equilibrium equation is also derived, onsidering the eigenmode shape. However, these epressions are not appliable for pratial verifiation, as adequate fators for a design rule were not alibrated for this purpose. 3. Theoretial bakground for non-uniform olumns 3.. Differential equation elasti solution Fig. 4: illustrates the equilibrium of a olumn segment for arbitrary boundary onditions in its deformed onfiguration: Considering the aial fore as N ðþ¼n on þ L n ðþdξ, ξ negleting nd order terms and onsidering the internal moment given by M ðþ¼ EIðÞ d y, the differential equation is given in Eq. (): d E IðÞ y þ N ðþ y ¼ ðþ ðþ The solution of this equation leads to the elasti ritial load, see Eq. (3). As it is not the purpose of this paper to solve Eq. () analytially, numerial Linear Eigenvalue Analysis (LEA) will be arried out and used to obtain the shape of the eigenmode as well as the ritial load multiplier, α r. Curve d (h/b>) Curve (h/b ) 8 < N ðþ¼α r N Ed ðþ n ðþ¼α r n Ed ðþ : y ðþ¼y r ðþ ð3þ N Ed () is the applied aial fore and α r is the ritial load multiplier, and y r () is the ritial eigenmode. h ma bma h min h ma b min b ma Fig. 3. Change of bukling urve in a tapered member. 3.. Imperfet olumn 3... Differential equation Consider now an initial imperfetion proportional to the shape of the eigenmode (y r ()). Considering a similar approah to paragraph

4 64 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 (a) Non-uniform olumn (b) Equilibrium of fores () Detail regarding distributed fore n() N on dy dq Q + d d dm M + d dn N + d d B y n() d d ξ η dy y B n() d N A M Q A Fig. 4. Equilibrium of a olumn segment. 3. and assuming that the internal fores are independent of the imperfetion, the differential equation, Eq. (), beomes EIðÞy þ N ðþy þ N ðþy ¼ ð4þ Defining N()= N Ed (), where is the load multiplier whih leads to the fleural bukling resistane of the olumn, the solution to Eq. (4) is given by In the above, e denotes the maimum amplitude of a member imperfetion Imperfetion onsistent with European olumn bukling urves formulation. Following a similar approah as for the derivation of the European Column Bukling Curves, the imperfetion is given by y ðþ¼y r ðþe ð8þ y ðþ¼ y α r α ðþ b ð5þ The utilization ratio ε onsidering this imperfetion an now be derived This leads to a seond order bending moment of M ðþ¼ EIðÞy α ðþ¼ EI ðþ b y α r α ðþ b Defining the utilization ratio ε as the ratio between the applied fores and the orresponding, and onsidering a linear interation between moment and aial fore, the utilization ratio at eah setion of the olumn is given by εðþ¼ N Ed ðþ þ M ðþ N R ðþ M R ðþ ¼ N Ed ðþ EIðÞ þ N R ðþ h α r i y ðþ M R ðþ As a result, onsidering a first yield riterion, for a ertain load multiplier, the utilization ratio attains a maimum of ε=attheritial position of the olumn,. As only one equation is given (Eq. (7)), but two variables are unknown ( and ), an iterative proedure is needed to obtain the solution Assumptions for the magnitude of the imperfetion As already mentioned, a similar derivation was arried out in [5] appliable to fleural bukling in general, in whih, for the magnitude of the initial imperfetion, equation (5.9) of EC3-- was onsidered. It will be shown in this setion that this assumption leads to an epression mathing lause 6.3. of EC3-- for uniform olumns at the ritial position. This topi will be further disussed in this paper. Two ases are then onsidered for the proportionality fator of the eigenmode deformed shape: a) Imperfetion onsistent with the derivation of the olumn bukling urves the amplitude of this defletion is given by e ; b) Imperfetion aording to equation (5.9) of EC3-- (equivalent geometri imperfetion) the amplitude of the ritial mode is given multiplied by e and an additional fator. This derivation may be found in [5]; ð6þ ð7þ α EIðÞ 4 b ð Þ y εðþ¼ N Ed ðþ þ M ðþ N R ðþ M R ðþ ¼ α r α bn Ed ðþ r α ðþe b þ N R ðþ M R ðþ Considering λ ðþ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N R ðþ=n Ed ðþ α ; χðþ¼ b N R ðþ=n Ed ðþ α r y 3 5 ð9þ ðþ After some manipulations and reorganizing terms, the utilization ratio ε beomes (Eq. ()): 3 N εðþ¼χ ðþþχ ðþ α e R ð Þ EIðÞ y rðþ b 4 5 N RðÞ M R ð Þ M α r R ð Þ N Ed ðþα r N R ð Þ M R ðþ εð At the position =, ε( )=, Þ ¼ ¼ χð Þþ Considering χð Þ λ ð Þχ N e R ð Þ ¼ α M R ð Þ EC3 λ ð Þ : where α EC3 =α EC3 ( d ). ðþ 3 ð Þ e N R ð Þ EI ð Þ: y rð Þ 4 5 ðþ M R ð Þ α r :N Ed ð Þ ð3þ

5 L. Marques et al. / Journal of Construtional Steel Researh 7 () Eq. () beomes ¼ χð 3 EIð Þ: y rð Þ Þþχð Þ λ ð Þχð Þ α EC3ð Þ λ 4 5 ð Þ : α r :N Ed ð Þ ð4þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} η uniform ð Þ βð fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Þ η non unif orm ð Þ Eq. (4) is idential to the Ayrton Perry formulation for uniform olumns. It an be shown that, for prismati olumns with onstant aial fore, the fator β( ) is unity. This fator takes into aount the non-uniformity of the olumn and leads to a modifiation of the urrent olumn bukling urves, i.e. lause 6.3. of EC Imperfetion aording to equation (5.9) of EC3--. The equivalent imperfetion of equation (5.9) of EC3-- is given by y ðþ¼e y r ðþ N r EIy r; ma ¼ e y r ðþ N Ed:α r E:Iy r; ma ð5þ For a uniform olumn, the ritial position is at mid-span and, therefore, y r,ma =y r (L/)=y r ( ). Analogously, N Ed =onstant=n Ed ( ). Eq. (5) beomes " # N y ðþ¼y r ðþe Ed ð Þ:α r E:Ið Þ: y rð Þ ð6þ Note: The sign ( ) in Eq. (6) leads to a positive value of the imperfetion. Analogous to 3..., the utilization ratio ε beomes εðþ¼ N Ed ðþ þ MII ðþ N R ðþ M R ðþ ¼ N Ed ðþ N R ðþ ð Þ EIðÞ þ y ðþ α r M R ðþ " # χðþ εðþ¼χ ðþþ λ ðþχ ðþ e N R ð Þ EIðÞ y rðþ N Ed ð Þ N R ðþ M R ð Þ M R ð Þ EIð Þy r ð Þ N Ed ðþ N R ð Þ M R ðþ εð At the position = d, ε( )=, Þ ¼ ¼ χð Þþχð Þ λ ð Þχð Þ N e R ð Þ M R ð Þ fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ð ð Þ : α λ Þ η EC3 ð7þ ð8þ In this ase, Eq. (8) oinides eatly with equation (6.49) (lause 6.3. of EC3--) and leads to the appliation of lause 6.3. ε ()..8 εn+εm [-] εn [-] εm [-] y'' [m-] [m] Fig. 5. Curvature y and utilization ratio ε (total, due to aial fore only; due to nd order fores only). for olumns if the ross-setion properties at the position = are onsidered Interpretation of the utilization ratio ε and are obtained as follows: i ( i L) is assumed as and Eq. (4) (or (8)) is solved for ; After this, ε() in Eq. () (or (7)) is obtained for all values of ; If ε( i ) ε(),then = i. If not, the proedure is repeated for = i+. The variation of the utilization ratio ε of a non-uniform olumn (L= m) is illustrated in Fig. 5, onerning Eq. (4). For this, α r and y r are obtained numerially from a LEA analysis. The utilization ratio ε is divided into terms ε N onerns first order fores, i.e., aial fore, and ε M onerns nd order fores, i.e., bending moment due to urvature of the member. The sum of these terms leads to the total utilization ratio ε. For short members, ε N is muh higher ompared to ε M as ross-setional resistane is more signifiant, and vie-versa. It an be seen that the ritial position is loated at about % of the member length, lose to the smallest ross-setion. Note that in Fig. 5a disontinuity an be notied at about =3 m.at this position, the lass of the flanges in ompression and bending about zz (out-of-plane bukling for this eample) hanges from (plasti verifiation) to 3 (elasti verifiation), whih leads to a modifiation of the resistant moment M r and, therefore, a disontinuity in the utilization ratio due to nd order fores. Cross-setional lass was alulated aording to Clause 5.6 of EC3--, in whih there is a jump between lass and lass 3. Reent researh projet SEMI-COMP overome this issue by developing new evaluation proedures for the design resistane of lass 3 steel ross-setions []. In this projet, a smooth transation between lass and lass 4 ross-setions is provided. 4. Numerial model 4.. General L/ Fig. 6. Shape and magnitude of the imperfetion. A finite element model was implemented using the ommerial finite element pakage Abaqus, version 6. [6]. Four-node linear shell elements (S4) with si degrees of freedom per node and finite strain formulation were used. S35 steel grade was onsidered in the referene eamples, with a yield stress of 35 MPa (perfet elasti plasti), a modulus of elastiity of GPa, and a Poisson's ratio of.3. Loading is applied with referene to the plasti resistane values of the smaller ross-setion. Critial Bow Fig. 7. Critial load imperfetion vs. Bow imperfetion.

6 66 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 Table Analysis of the shape of the imperfetion. Taper ratio h ma /h min ( b ma /b min ) 4.. Geometrial imperfetions Regarding global imperfetions, a geometrial imperfetion proportional to the eigenmode defletion is onsidered with a maimum value of L/ (Fig. 6). This is onsistent with the values onsidered during the development of the European olumn bukling urves [7]: L y ðþ¼y r ðþe ¼ y r ðþ ð9þ The differene between onsidering either bow or eigenmode imperfetions (see Fig. 7) is analyzed in Table. It an be observed that the onsideration of bow imperfetions leads to an over-evaluation of resistane with the inrease of the level of taper and/or the shape of the normal fore diagram relatively to a onentrated aial fore. The Taper Ratio γ is defined as the ratio between the maimum height and the minimum height (h ma /h min ), or the maimum width and the minimum width (b ma /b min ). As for loal imperfetions, these were not onsidered, neither in the numerial models, nor in the analytial models (effetive rosssetion properties). However, this will not influene these results as, for the analyzed ases, the ritial position is always (at the most) in a lass 3 zone of the olumn Material imperfetions The material imperfetions, residual stress patterns orresponding both to stoky hot-rolled (i.e. with a magnitude of.5f y ) and welded ross-setions were onsidered. Fig. 8 shows the adopted residual stress pattern. In Fig. 9, a possible fabriation proedure for the rolled ase is illustrated (utting of the web along the length of the olumn). This hoie allowed the diret observation of the influene of the taper by omparing bukling urves for tapered members with urves for members without taper, but with otherwise the same residual stress distribution (Fig. 8(a)). (a) Hot Rolled (h/b.).5 f y Aial fore,gmnia Diff (%) Critial Bow Conentrated.55 Distributed Conentrated Distributed Conentrated Distributed f y.5 f y (b) Welded b fy.8h Fig. 8. Residual stresses: + Tension and Compression. 5 fy 5. Parametri study 5.. Definition Table 3 summarizes the sub-set of ases to be ompared with the advaned numerial simulations. The ase of in-plane fleural bukling of linearly web-tapered olumns subjet to uniform aial fore is onsidered. More than 35 numerial simulations with shell elements were arried out. Both GMNIA (Geometrial and Material Non-linear Analysis with Imperfetions) numerial simulations onstrained in-plane and LEA (Linear Eigenvalue Analysis) are arried out to provide data for appliation of the analytial formulations and for alibration of neessary parameters. Table 3 summarizes the parametri study, where the Taper Ratio is defined as γ=h ma /h min. 5.. Methodology Fig. 9. Fabriation proedure for hot-rolled tapered elements. Table 4 summarizes the alternative proedures to obtain the resistane of the tapered olumn: The first two ases (a) were already desribed in Setion Regarding the other ases (b), no iteration proedure is needed beause the ritial loation is assumed to be known from the numerial model. The proedure is implemented as follows:. Etration of from GMNIA model and of the ritial load multiplier α r from LEA model;. Calulation of λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NRðÞ=N ð Þ ¼ Ed ðþ, see Eq. (); αr 3. Calulation of the generalized imperfetion η non-uniform ( ) (when appliable) defined in Eq. (4) as η non unif orm ð Þ ¼ η uniform ð Þ βð Þ ¼ α EC3 ð Þ λ EIðÞ: ð y ð Þ : rðþþ αr:n Ed ðþ. 4. Calulation of the redution fator χ( ) and finally of, given by =χ( ).N R ( )/N Ed ( ), see see Eq. (). Table 3 Parametri study. Taper ratio γ Referene ross-setion (i.e. with h min,at= ) 8 IPE (h= mm; b= mm; t f =9 mm; t w = mm) HEB 3 (h=b=3 mm; t f =9 mm; t w = mm) (h=b= mm; t f =t w = mm) Table 4 Considered proedures for stability verifiation. Referene olumn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi slenderness λ ð Þ ¼ NRðÞ=N Ed αr Fabriation proedure 3 Welded Hot-rolled (.5 f y ) Method Desription Eq. () (a) Solution of the equation by an iterative proedure Eq. (7) (a) Solution of the equation by an iterative proedure Eq. (4) (b) Diret appliation is etrated numerially Eq. (8) (b) Diret appliation is etrated numerially Eq. (4) onsidering β( )= EC3-- Eq. (4) onsidering at the smaller ross-setion and β( )= or Eq. (8) onsidering at the smaller ross-setion GMNIA

7 L. Marques et al. / Journal of Construtional Steel Researh 7 () Results Fig.. Critial position aording to GMNIA analysis. Finally, onerning nonlinear numerial alulations, the maimum load fator of GMNIA analysis orresponds to load multiplier. The ritial position is also etrated from the numerial model orresponding to the element with the maimum strain at the maimum load fator,, see Fig.. Results are represented relatively to the loation of the smallest ross-setion,. Beause N R ( )=N Ed, Eq. () beomes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N λ ð Þ ¼ R ð Þ=N Ed ð Þ ¼ p ffiffiffiffiffiffiffi ; χð α r α Þ r α ¼ b ðþ N R ð Þ=N Ed ð Þ ¼ χð Þ Auray of the analytial model Fig. illustrates the numerial results from GMNIA analyses against results from Eq. () for different Taper Ratios γ, regarding the maimum load fator and the relative ritial position /L. A olumn onsisting of the hot-rolled ross-setion defined in Table 3 is hosen for illustration. Although differenes of 5% (unsafe) to 7% (safe) are notied, it an be seen that the analytial model haraterizes the behavior of the tapered olumn well when ompared to the numerial model. This was also found for other ases of Table 3 (not shown in Fig. ). It is also notieable an inrease of up to % in terms of resistane with the inrease of tapering at a slenderness range of λ ð Þ ¼ :5toλ ð Þ ¼, whih shows the relevane of Eq. (). Eq. () was solved onsidering an imperfetion fator α EC3 =.34 (urve b of EC3--), in agreement with the adopted residual stresses of.5f y. Finally, in Fig. (a), the Taper Ratio of γ=, i.e., prismati olumn, is shown for omparison. It is epeted that the relative ritial loation is loated at mid-span where the urvature is maimum and therefore /L=.5 (see Fig. (a.)). Moreover, Eq. () should give the same results as the EC3 urve b. This is visible in Fig. (a.). (a.) γ= (a.) γ= γ= GMNIA. γ= Model (b.) γ= λ y ( ) EC3 Curve b γ= GMNIA γ= Model EC3 Curve b γ=.5 GMNIA γ=.5 Model EC3 Curve b γ=5. GMNIA γ=5. Model.5 /L.3 λ y ( ) λ y ( ) λ y ( ) (b.) γ=.5.5 /L.3 (.) γ=5 (.) γ=5..5 /L.3. λ y ( ) γ=.5 GMNIA γ=.5 Model γ=5. GMNIA γ=5. Model λ y ( ) Fig.. Analytial derivation Eq. () against numerial alulations GMNIA. (.) Resistane χ( ) against slenderness λ ð Þ; (.) Critial position /L against slenderness λ ð Þ.

8 68 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 (a) Hot-Rolled. (b) Welded..8.8 γ= γ=.5 γ= γ=3 γ=6 EC3-b.5.5 λ y ( ) ()IPE Welded. (d) HEB3 Welded. γ= γ= γ=3 γ=5 γ=8 EC3-b.5.5 λ y ( ).8.8 γ= γ=.5 γ=.5 γ= γ=3 γ=5 EC3-b.5.5 λ y ( ) γ = γ=.33 γ = γ=3 γ =5 EC3-b.5.5 λ y ( ) Fig.. Numerial alulations GMNIA organized by Taper Ratio. Resistane χ( ) against slenderness λ ð Þ Influene of the taper ratio, γ=h ma /h min Fig. illustrates GMNIA results of resistane against slenderness (based on N pl of the smallest ross-setion) organized by Taper Ratio. Curve b of EC3-- is shown for omparison. Note that, for the welded ross-setion ases, the numerial urve orresponding to the uniform element (γ =) shows deviations that fall below the ode urve results for the relevant slenderness range up to. This will be disussed in Setion 6 Verifiation proedure. A smooth inrease in the resistane with the inrease of Taper Ratio γ along all slenderness ranges an be observed in all ases of Fig.. It also shows to be less signifiant for higher levels of Taper Ratio Analysis of the ritial position and of an additional fator β( ) to the generalized imperfetion η uniform The importane of identifiation of the ritial loation has already been disussed. Nowadays, there is no straight-forward proedure to obtain this loation. Therefore, most designers in pratie will use the smallest ross-setion properties for verifiation aording to lause 6.3. of EC3--. Moreover, an additional fator β( ) derived in Setion 3... and given in Eq. (7) haraterizes the inrease of resistane of the tapered member relatively to the uniform member. This fator attains a limit for the ase of uniform members, reahing unity for those ases. When assoiated to the generalized imperfetion of the uniform member η uniform to give a generalized imperfetion of the tapered member η non-uniform, see Eq. (4), the latter beomes lower and, as a onsequene the resistane of the tapered member beomes higher. βð Þ ¼ EIð Þ: y rð Þ α r :N Ed ð Þ ðþ Fig. 3 illustrates the influene of these two parameters for a member with an initial ross-setion of (hot-rolled) and γ=4 (h ma =4 mm). Table 5 shows results for the ase of λ y ð Þ ¼ :74. In..8 No No β No β β EC3-b GMNIA.5.5 λ y ( ) Fig. 3. Influene of the ritial position and of the imperfetion in the resistane of the tapered olumn.

9 L. Marques et al. / Journal of Construtional Steel Researh 7 () Table 5 Influene of the ritial position and of the imperfetion in the resistane of the tapered olumn ( λ y ð Þ ¼ :74; α r =.85). Case /L β( ) λy ð Þ χ( ) =χ( )N Rk /N Ed Diff (%) No Noβ Noβ. (GMNIA) β. (GMNIA) GMNIA..96 order to obtain resistane for the ases onsidering,thenumerialposition was onsidered. Moreover, to alulate β( ), y r is etrated from LEA analysis. It an be seen that fator β has a great influene in the resistane of the olumn. Regarding the ase in whih is onsidered with urrent EC3 imperfetion (no β), note that the first 3 ases oinide with urrent EC3 bukling urve b. This happens beause at this slenderness range and regarding this taper ratio, the nd order failure ross-setion (or ) is the same as the st order failure rosssetion (smaller end). Table 5 shows an imperfetion derease of more than 5% (β( )=8) for the analyzed ase of λ y ð Þ ¼ :74. Finally, it an also be observed that the relative ritial loation /L and the additional imperfetion fator β( ) are independent of the fabriation proess or of the initial ross-setion proportions, see Figs. 4 and 5, omputed for all the analyzed ases Influene of the funtion for the magnitude of the imperfetion In Setion 3.. Assumptions for the magnitude of the imperfetion, two ases are onsidered. Results have been shown regarding the amplitude of the imperfetion given by e (Setion 3...), i.e., onsistent with the derivation of the olumn bukling urves of EC3--. Fig. 6 ompares the solution of Eqs. () and (7), in whih for the latter, Eq. (6) (equation (5.9) of EC3--) is onsidered for the imperfetion. This derivation is also given in [5]. Two representations of resistane are onsidered and illustrated onerning a Taper Ratio of γ=4 and the referene ross-setion (hot-rolled): Fig. 6(a) illustrates the redution fator λ ð Þas a funtion of the redution fator χ( ), and, therefore, resistane an be diretly ompared; Fig. 6(b) illustrates the redution fatorλ ð Þas a funtion of the redution fator χ( ) it is stated in [5] that when equation (5.9) of EC3-- (Eq.(6)) is onsidered for the imperfetion, results of the redution fator χ( ) oinide with urrent bukling urves for olumns (see also Fig. 6(b)) and that good agreement is ahieved with numerial models. This is to be epeted if the imperfetions onsidered in the numerial models are also obtained from Eq. (6). However, and as already mentioned in Setion., the magnitude of the geometrial /L λ y ( ) Fig. 4. Relative ritial position /L against the relative slenderness λ y ð Þ, all ases. γ= β( ) γ= λ y ( ) Fig. 5. Additional imperfetion fator β( ) against the relative slendernessλ y ð Þ, all ases. imperfetion should only be dependent on the member length [8]. Moreover, for the alibration of EC3 imperfetion fators for olumns, this magnitude was given by e =L/(and additional residual stresses for the material imperfetions). The same approah is onsidered in this study. Both Fig. 6(a) and (b) show a better agreement with the EC3 onsistent approah regarding Eq. (). Note that onerning Fig. 6(b), GMNIA is also illustrated in terms of the redution fator χ( ), in whih is obtained from the numerial model. Finally, Fig. 7(a) and (b) respetively illustrate resistane and relative ritial loation /L regarding all Taper Ratios of the analyzed ross-setion (hot-rolled). A higher spread is notied for Eq. (7). 6. Design methodology 6.. Introdution Considering the developed analytial formulation and the numerial alulations, a verifiation proedure for the stability of tapered olumns subjet to in-plane bending is now proposed. In a first step, regarding the imperfetion fator for uniform welded ross-setions it was notied that, for I-setions, the imperfetion fator α=.34 provides unsafe results for slenderness up to approimately (differenes of 8% were observed, see Fig. (b), () and (d)). Beause this proposal has, as a referene limit, the ase of uniform members (γ=), it was deided to alibrate new imperfetion fators for welded ross-setions for that purpose. In a seond step, the development of a verifiation proedure for tapered olumns is done. Here, epressions for the ritial loation and the additional imperfetion fator β( ) are alibrated against numerial results shown in Figs. 4 and Imperfetion fators for fleural bukling of uniform welded olumns Ayrton Perry formulation for uniform olumns is given by ¼ χ þ χ λ χ e N R α λ M ð Þ : ηec3 ð R ¼ α λ ð Þ : ¼ λ χ χ Þ η EC3 ðþ The generalized imperfetion η EC3 for in-plane bukling of uniform olumns is given by Eq. (3), for a flange thikness t f mm η EC3 ¼ α λ ð Þ : ¼ :34 λ ð Þ : ð3þ Fig. 8(a) illustrates the generalized imperfetion of EC3-- η EC3 ompared to the generalized imperfetion η num of about

10 7 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 (a) Resistane χ( ) against slenderness λ ( ) (b) Resistane χ( ) against slenderness λ( )..9.7 χ( ).5 EC3 -b. GMNIA.9 Eq. () Eq. (7).7 χ( ).5 EC3 -b GMNIA Eq. () Eq. (7) λ y ( ) λ y ( ) Fig. 6. Influene of imperfetion magnitude bukling urve representation (a) Resistane (b) Critial loation /L. Eq. (8) Eq. ().5 Eq. (7) Eq. () (Equation).8 /L(Equation) α E (GMNIA)..3.5 /L (GMNIA) Fig. 7. Influene of imperfetion magnitude (hot rolled), all Taper Ratios. numerial alulations overing a range of uniform olumns with different h/b ratios varying from.95 (HEA) to.5 (IPE5) and slenderness varying from λ y ¼ : to λ y ¼ :. The value η num is alulated aording to Eq. (4), see also [9], in whih χ is etrated numerially and orresponds to the maimum load fator of GMNIA alulation, : η num ¼ λ χ num χ num ð4þ Fig. 8(a) shows the differene, on the unsafe side, in onsidering for the imperfetion fator α the value of.34. A value of α=5 was shown to fit the redution fator χ y very aurately up to slenderness of. However, in order not to get too onservative for slenderness above and to take into aount the bukling behavior of olumns with a welded residual stress pattern for that slenderness range, a ut-off of η ¼ α λ : :7was also shown to be adequate. If the ut-off of 7 is applied, for higher slenderness of about λ y ¼ :5, imperfetion beomes unsafe again. However, for high (a) Generalized imperfetion η against slenderness λ y η y 5.5. α=.34 (EC3).5 α=5 (Best fit) Cut-off..5. λ y.5. (b) Resistane χ y against slenderness λ y χ y.5 α=.34 (EC3) α=5 (Best fit) α=5 + Cut-off λ y Fig. 8. Generalized imperfetion of in-plane fleural bukling of welded olumns.

11 L. Marques et al. / Journal of Construtional Steel Researh 7 () slenderness range, the olumn is not so sensitive to the imperfetion level and resistane onverges to the load. (b) illustrates the redution fator χ y against the relative slenderness λ y Proedure Eq. () was shown to follow aurately the bukling behavior of a tapered olumn. However, the appliation of this epression is not straight forward: The ritial loation is needed throughout the appliation of Eq. () for this an iterative proedure is needed; One is known, the additional imperfetion fator β( ) an be alulated. However, to obtain it, the funtion for the ritial urvature is needed this is not a diret proedure γ= β( ) λ y ( ) (Equ).5 (Equ). (Equ).5 (Equ) 3. (Equ) 4. (Equ) 5. (Equ) (Equ) Assuming that the ritial load multiplier, α r, is obtained from a numerial analysis, LEA, epressions regarding and β( ) are still needed for the diret alulation of resistane. Elliptial epressions were shown to give good approimation for both these parameters. Fitting equations for and β( ) are illustrated in Figs. 9 and respetively. Corresponding epressions are shown in Fig., whih illustrates the omplete proedure for in-plane stability verifiation of tapered olumns. Note that, for higher Taper Ratios, β ould be higher. However, for safety reasons onerning the resistane multiplier, the limit of β= for all tapered ratios was hosen. Finally, Fig. illustrates the steps to be followed. Firstly, the ritial position is determined based on the referene relative slenderness of the smallest ross-setion. α r shall be alulated numerially. Note that from this step, geometrial properties of are onsidered, inluding slenderness alulation for the determined position. Imperfetion an now be alulated by ombining the imperfetion effets of the uniform member (η uniform ) and of the tapered member (β). With this, the redution fator at is determined and the verifiation is finally made Comparison with numerial results Fig. illustrates the resistane of the numerial results χ( ) as a funtion of the relative slenderness λ ð Þ, onerning GMNIA analysis as well as the proposed formulation. The urrent EC3 urve for uniform members that would be applied is also illustrated (i.e. α=.34; β=; and onsidering as the minimum ross-setion as no guidelines eist at the moment). Good agreement is noted with the proposed methodology. Fig.. Fitting elliptial epression for the additional imperfetion fator β( ). 7. Eample 7.. Introdution A tapered olumn omposed of a IPE welded ross-setion in the smallest end with a linearly varying height and a Taper Ratio of γ=h ma / h min =3 is now analyzed (Fig. 3). The applied load is given by the plasti aial fore at the smallest end, i.e. N Ed =5 kn for a yield stress of f y =35 MPa. The olumn has a length of L=.9 m. In-plane bukling resistane is alulated using several methods. 7.. Elasti ritial analysis A numerial linear eigenvalue analysis LEA attains a ritial load multiplier of α r =.85. For omparison, the ritial load is also alulated by some of the methods desribed in Table. A negative differene illustrates a higher value of the ritial load obtained in the literature relatively to the numerial value. Results are summarized in Table 6: 7.3. Verifiation of stability In Setion 7.3. the proposed verifiation proedure is applied. The appliation of other methodologies is summarized in Setion A numerial GMNIA analysis leads to a maimum load fator of = Appliation of the proposed method.5 5 γ= /L (Equ).5.7 (Equ).33 (Equ)..5 (Equ). (Equ).5.5 (Equ) 3. (Equ) 4. (Equ) 5. (Equ) (Equ) 8. (Equ) λ y () Fig. 9. Fitting elliptial epression for the ritial position, Calulation of slenderness at = (smaller ross-setion). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N λ ð Þ ¼ Pl ð Þ=N Ed 64:3=5 ¼ ¼ :83 :85 α r Critial ross-setion relative position, /L. a ¼ :5 þ :6 :6 ¼ :5 þ ¼ :8 and b ¼ γ 3 γ þ ¼ 3 þ ¼ :5 a ¼ : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λð Þ ¼ :83 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ð =L ¼ b Þ ¼ :5 ð:83 Þ ¼ :9 a :8

12 7 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 Required data N Rk ( ) / N Ed λ( ) = ; γ = hma hmin ; α r a =.5 + ; γ b = γ + Critial Position, / L =, b b, ( λ( ) ) a, λ( ) < a a λ( ) λ( ) > or λ( ) Slenderness at λ( ) = N Rk ( ) / N α r Ed Aount for Imperfetion β ( ) =,, ( λ( ) ) a, λ( ) < a a λ( ) λ( ) > or λ( ) η Uniform = α where Uniform α η ( λ( ) ) Uniform Uniform Hot Rolled.34 Welded 5 7 Redution fator χ( ) = φ( ) + φ ( ) λ ( ), with φ( ) =.5 η( ) = η ( + η( ) + λ ( )) Uniform β ( ) Verifiation N Ed χ( ) N Rk ( ) N Rk ( ) Fig.. Design proposal Calulation of slenderness at =. N Pl ð λ ð Þ ¼ Þ ¼ A ð Þf y ¼ 74:5 kn sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N Pl ð Þ=N Ed 74:5=5 ¼ ¼ :895 :85 α r Determination of imperfetion, η. Additional Imperfetion fator β( ) a ¼ :5 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ð Þ ¼ :895 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ð βð Þ ¼ Þ ¼ ð:895 Þ ¼ :789 a :8 η non unif orm ¼ η uniform βð Þ ¼ :7 :789 ¼ : Redution fator at =. ϕð χð Þ ¼ :5 þ η þ λ ð Þ Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕð Þþ ϕ ð Þ λ ¼ ð Þ ¼ :68 ¼ :5 þ :3 þ :895 ¼ :7 p :7 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :7 : Verifiation. N b;rd ¼ χð ÞN Pl ð Þ ¼ :68 74:5 ¼ 54:kN N Pl ð Þ N b;rd N Ed 54: > 5 Design hek verified! ¼ N b;rd =N Ed ¼ 54:=5 ¼ :8ðGMNIA; ¼ :4Þ Summary of results Results are summarized in Tables 7 and 8. Firstly, from results of Setion 7.3., the resistane alulated aording to the proposed methodology of Setion 6.3 is pratially oinident with the GMNIA resistane (% of differene). Table 7 summarizes results onsidering the smallest ross-setion for verifiation. Note that, in this ase, the ase orresponding to α=.34 leads to a smaller differene (%) beause urrent bukling urves for welded olumns lead to unsafe results in this slenderness range. Therefore, the omparable orret differene is 5%, whih orresponds to proposed bukling urve with α=5 onsidering the ut-off of η 7. Finally, onsidering [] or [9] for α r alulation leads to different relative slenderness λ y ð Þ. Considering [9] with the proposed verifiation proedure leads to unsafe level of resistane. 8. Conlusions In this paper an analytial derivation of non-prismati olumns was arried out and ompared against numerial simulations. It was shown

13 L. Marques et al. / Journal of Construtional Steel Researh 7 () (a) Hot-Rolled γ =6 (b) Welded γ= EC3-b EC3-b GMNIA GMNIA Proposal Proposal λ y ( ) λ y ( ) () IPE Welded γ=3 (d) HEB3 Welded γ=..8 Eample Setion 7..8 EC3-b GMNIA Proposal EC3-b GMNIA Proposal λ y ( ) λ y ( ) Fig.. Resistane against λ y ð Þ. Evaluation of the proposed method. (a) In-plane stress distribution (perspetive view) (b) In-plane bukling shape (front view) Fig. 3. Analyzed tapered olumn. that, onerning non-uniform olumns, Euroode rules needed to be adapted in the following aspets: A pratial approah for the determination of the design position needed to be developed; Table 6 Calulation of ritial aial fore of the tapered olumn. Method Approah α r =N r,tapered /N Ed Diff (%) Hirt and Crisinel [9] I y,eq ( )=33% I y,ma Galambos [] L eq ( )=5% L LEA Numerial.85 Table 7 Results = (minimum ross-setion), urrent proedure EC3 (β=). α (imperfetion) Cutt-off: α λ ð Þ : :7 αb Diff (%) GMNIA.4 The olumn design formula had to be amended by an additional fator β, whih speifially takes into the seond-order behavior of tapered olumns. The urrent imperfetion fator of lause 6.3. for welded setions needed to be modified and re-alibrated. For this purpose, a wide parametri study of more than 35 LEA and in-plane GMNIA simulations was arried out regarding linearly web tapered olumns with onstant aial fore subjet to in-plane bukling. After that, a proposal for the stability verifiation of these tapered olumns was presented. It was notied that, most of all, the onsideration of the most stressed position is neessary in order not to ahieve over- Table 8 Results proposed method applied with other formulae for α r (from literature). Method for α r λ p ð Þ ¼ = ffiffiffiffiffiffiffi α r Diff (%) Hirt and Crisinel [9] Galambos [] LEA.83.8 GMNIA.4

14 74 L. Marques et al. / Journal of Construtional Steel Researh 7 () 6 74 onservative levels of resistane. The above-mentioned fator β was developed based on the prior analytial formulation and alibrated with numerial results. Finally, a new generalized imperfetion for welded uniform olumns was also alibrated in order to obtain improved results for the tapered ases. The net step is to perform a reliability analysis of the proposal and determine γ M [,]. Moreover,thesametopi will be analyzed for other tapering shapes and loading. Aknowledgment Finanial support from the Portuguese Ministry of Siene and Higher Eduation (Ministério da Ciênia e Ensino Superior) under ontrat grant SFRH/BD/37866/7 is gratefully aknowledged. Referenes [] CEN. Euroode 3, EN-993--:5, Euroode 3: design of steel strutures Part : general rules and rules for buildings. Brussels, Belgium: European Committee for Standardization; 5. [] Müller C (3). Zum Nahweis ebener Tragwerke aus Stahl gegen seitlihes Ausweihen, PhD Thesis, RWTH Aahen, Germany. [3] Simões da Silva L, Marques L, Rebelo C. Numerial validation of the General Method in EC3-- for prismati members. J Constr Steel Res ;66(4): [4] Simões da Silva L, Gervásio H, Simões R. Design of steel strutures. ECCS Euroode Design Manuals. ECCS Press and Ernst & Sohn;. [5] Simões da Silva L, Gervásio H. Manual de Dimensionamento de Estruturas Metálias: Métodos Avançados. Coimbra, Portugal: mm Press; 7. [6] in June 6th,. [7] ECCS TC8. Resolution of ECCS/TC8 with respet to the general method in EN 993--, ECCS TC8 Stability; 6. [8] Marques L, Simões da Silva L, Rebelo C. Appliation of the general method for the evaluation of the stability resistane of non-uniform members. Proeedings of ICASS, Hong Kong, 6 8 Deember; 9. [9] Hirt MA, Crisinel M. Charpentes Métaliques Coneption et Dimensionnement des Halles et Bâtiments. Traité de Génie Civil, vol.. Lausanne: Press Polytehniques et Universitaires Romandes;. [] Lee GC, Morrell ML, Ketter RL. Design of tapered members. Weld Res Coun Bull June 97(73): 3. [] Galambos TV, editor. Guide to Stability Design Criteria for Metal Strutures. Fifth Edition. John Wiley & Sons In.; 998 [] Petersen C. Stahlbau. Wiesbaden: Vieweg Verlag; 993. [3] Baptista AM, Muzeau JP. Design of tapered ompression members aording to Euroode 3. J Constr Steel Res 998;46( 3):46 8. [4] Raftoyiannis I, Ermopoulos J. Stability of tapered and stepped steel olumns with initial imperfetions. Eng Strut 5;7(5): [5] Naumes (9). Biegekniken und Biegedrillkniken von Stäben und Stabsystemen auf einheitliher Grundlage, PhD thesis, RWTH Aahen, Germany. [6] Abaqus. v.6., Dassault Systems/Simulia, Providene, RI, USA;. [7] Beer H, Shulz G. Die Traglast des planmä βig mittig gedrükten Stabes mit Imperfektionen. VDI-Zeitshrift 969;: , [8] Greiner R, Taras A. New design urves for LT and TF bukling with onsistent derivation and ode-onform formulation. Steel Constr ;3(3): [9] Taras A, Greiner R. New design urves for lateral-torsional bukling proposal based on a onsistent derivation. J Constr Steel Res ;66: [] Rebelo C, Lopes N, Simões da Silva L, Netherot D, Vila Real P. Statistial evaluation of the lateral torsional bukling resistane of steel I-beams Part : Variability of the Euroode 3 design model. J Constr Steel Res 8;65(4):88 3. [] Simões da Silva L, Rebelo C, Netherot D, Marques L, Simões R, Vila Real P. Statistial evaluation of the lateral torsional bukling resistane of steel I-beams Part : variability of steel properties. J Constr Steel Res 8;65(4): [] Greiner R, Kettler M, Lehner A, Jaspart J-P, Weynand K, Ziller C, Örder R. SEMI- COMP+: valorisation ation of plasti member apaity of semi-ompat steel setions a more eonomi design, RFS-CT--3. Bakground Doumentation, Researh Programme of the Researh Fund for Coal and Steel RTD;.