Beam-Column Analysis

Effects of residual stresses and geometric imperfections

Pedro Melopedro.melo@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisboa, Portugal

July 2016

Abstract

A numerical model is developed towards the production of geometric nonlinear analysis of steelstructures, namely columns and beam-columns. The model executes second order analysis, tak-ing into account geometrical and material nonlinearities. As initial conditions the model considersboth geometrical and structural imperfections.Numerically the model consists on an application ofa distributed-plasticity Euler-Bernoulli beam displacement-based finite element method and of aNewton-Raphson nonlinear equation resolution algorithm. The developed model was validated bycomparing its solutions with the ones presented in a PhD dissertation by Ofner (1997), which wasobtained through the use of the automatic computation software Abaqus and was used as basis forthe calibration of both methods available in the EN1993-1-1:2005 for the calculation of the interac-tion factors involved in the buckling ultimate limit state safety verification of beam-columns. Thedeveloped model is then employed to evaluate the applicability of these same two methods, relyingon an extensive set of simulations.

Moreover, the model is also applied to study the effect of varying the direction of the geometricalimperfection relative to the location of the weld, i.e. relative to the orientation of the residual stressfield, in what concerns the behaviour of cold formed circular tubes with a single longitudinal weld.Keywords: steel structures, finite element method, residual stresses, second order analyses,geometrically nonlinear, physically nonlinear.

1. IntroductionStability has almost three centuries of history al-ready, starting with the Euler formula in the year1744. This formula defined the buckling load of acolumn, depending on its length and cross-sectionshape. Lagrange introduced the concept of buck-ling mode and Lamarle defined the stress associatedwith the phenomenon. Jasinsky introduced theconcept of buckling length, which allowed for theextension of the application of Euler and Lamarleformulas to elements with other support conditions.

Navier was one of the first to point out that theresults provided by the above mentioned formulasrepresented an upper-bound to the ultimate resis-tances found in real columns. It was noted that thebuckling resistance of columns is strongly influencedby its imperfections. In 1807, Young in additionto introducing the concept of modulus of elasticityalso presented the solution for the displacements ofa simply-supported column with a geometrical im-perfection with the shape of a half-sine wave.

In 1886, Ayrton and Perry, considering a imper-fection shape for the simply-supported column as a

combination of the buckling modes, i.e. a Fourierseries of sines, defined a criterion for the column re-sistance. The criterion was the yielding of the mostcompressed fiber of an axially compressed columnwith an imperfection defined through an adimen-sional factor.

In 1925, Robertson, after a set of experimen-tal measurements, suggested that the adimensionalimperfection factor would be accurately given byθ = 0.003λ. With this formula Robertson intro-duced the concept of equivalent geometric imper-fection, an imperfection deliberately overestimated,with the purpose of taking into account other im-perfections which are not as easy to simulate. Alsoworth of mention is the fact that the Robertsonformula makes the magnitude of the imperfectiondependent on the type of cross-section, (Maquoi,1980).

In 1955, the European Convention for Construc-tional Steelwork (“ECCS”) was established with theobjective of standardising the design methods ofsteel construction in Europe. With the creationof the European Community starts the project for

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the definition of the European structural norms, the“Eurocodes”, being the EN1993 (“EC3”) related tothe safety verification of steel construction.

In the sixties, the eighth technical committee(“TC8”) of the ECCS (Stability Problems) pro-duces a vast campaign of column tests, more thana thousand experiments over seven European coun-tries, together with the numerical simulation andanalytical study of the problem. As a result, in1970, the technical committee proposed three de-sign curves (a, b and c).

In 1976, Young and his team proposed the in-troduction of a plateau for extremely small slen-dernesses in the column design curves, in order totake into account the strain-hardening which hadnot been included in the numerical models. To-gether with this alteration of the shape of the col-umn curves, two new curves were introduced: a lesssevere one for high strength steel (a0) and a fifthcurve, the most severe, for cross-sections with highthicknesses (d).

In 1979, Maquoi and Rondal proposed the for-mulation for the “European column design curves”,based on the Ayrton-Perry formula, which is still inuse today. This formulation consists on assigning tothe adimensional imperfection factor the followingformula: θ = α(λ − 0.2), being “α” an imperfec-tion parameter, calibrated so the results from theAyrton-Perry formula would present a good agree-ment with the results of the simulations. Takinginto account residual stresses, an yield limit is nolonger reasonable, thus the result of a simulationis considered reached when no further equilibriumposition can be found.

The European column design curves were de-fined based on the specific case of a simply-supported column in a steel grade S235, with aninitial geometric imperfection with the shape of ahalf-sine wave with the amplitude of L/1000. Thisimperfection was estimated in order to take intoaccount also the other member geometric imperfec-tions. This value was already chosen by Jasinsky asa realistic approximation of the imperfections foundin construction sites. The curves can then be ex-trapolated for other support conditions and steelgrades.

The first version of “EC3” was published in1984. In 1992, the European Committee for Stan-dardization (“CEN”) publishes a new version ofEC3 with European pre-norm status, (ENV1993-1-1:1992), and in 2005, the present version as Eu-ropean norm, (EN1993-1-1:2005).

Regarding the column formulation in “EC3”, itis important to note that it is independent from thesteel grade, which implies that the initial equiva-lent geometric imperfection is proportional to theresistance of the material, for which there is no

justification, (Dwight, 1975). In fact, experimen-tal results show that with increasing material re-sistance the reduction of resistance due to imper-fections decreases. However, the diagrams of resid-ual stresses presented in ECCS(1976), used in thecolumn curve simulations, create a correspondencebetween the magnitude of the residual stresses andthe yield stress of the structural element material.No such correspondence can be identified from theexperimental measurements in the literature.

As for the beam-column elements, the methodavailable in the ENV1993-1-1:1992 for the definitionof the interaction factors to be adopted for the buck-ling safety verification of these elements, had somephysical inconsistences and was excessively conser-vative, (Greiner, 2006). For these reasons, “TC8”carried out an extensive numerical simulation study(Ofner, 1997) with which, along with a group ofexperimental results, new methods would be cal-ibrated. Two methods have resulted from thatinitiative: one from an Austrian-Germanic origin,more practical for hand calculations and one with aBelgian-Franc origin, more theoretically transpar-ent. These are the methods currently adopted inthe EN1993-1-1:2005.

A numerical model was developed in Matlab en-vironment capable of studying the details just de-scribed in the stability problem.

In this paper the developed model was appliedwith the purpose of evaluating the results of apply-ing the two new methods for the calculation of theinteraction factors. A brief evaluation of the impactof modifying the intensity of the residual stresses inhot-rolled cross-sections was performed as well.

The model was also adopted to evaluate the im-pact of varying the direction of the initial geometri-cal imperfection, with respect to the position of theweld, in what concerns the behaviour of cold-formedcircular hollow sections with a single longitudinalweld.

2. ImplementationThe developed numerical model consists on adisplacement-based Euler-Bernoulli beam finite el-ement model. The model performs second-orderanalysis including both geometrical and materialnonlinearities. With that purpose, the transversedisplacement field is approximated using cubic Her-mitian polynomials while for the axial displace-ments the linear Lagrangian polynomials are used,resulting in a six degree-of-freedom finite element.Therefore, the geometric component of the non-linear stiffness matrix is uncoupled from the elasticone.

As for material nonlinearities, a distributed-plasticity (plastic-zone) approach is used, aimingfor a more accurate simulation of the spread of

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plasticity over the section and along the element.Additionally, these approaches represent the mostgeneric and precise manner to introduce resid-ual stresses in the analysis. For integration pur-poses, the finite elements are subdivided into cross-sections, at the integration points, which are fur-ther subdivided. A stress control has to be kept foreach fibre at each cross-section because after yield-ing a fiber no longer contributes for the stiffnessesof the corresponding cross-sections, which have tobe integrated and updated. The initial values of thestress registry simply correspond to the diagram ofresidual stresses. This methodology, though muchheavier in computational terms, is simpler and moreprecise than the alternatives.

Since the analysis to be performed are nonlin-ear in nature, the usual linear procedure associ-ated with displacement-based finite element meth-ods will not work. For dealing with the nonlinearaspect of the problem, the Newton-Raphson algo-rithm was adapted to find the load bearing capac-ity of the structural elements. This algorithm isan incremental iterative procedure, the most tradi-tional and most simple to understand and to imple-ment from the available ones, with load incrementsin which the tangent stiffness matrix is updated atthe beginning of each iteration and increment.

Regarding the numerical integration process,a variation of the Gauss-Legendre quadrature isused, the Gauss-Lobatto one. The advantage ofthe Gauss-Lobatto quadrature is the imposition thattwo of the integration point are located exactly atthe extremities of the domain, despite the numberof integration points being adopted. Since plastic-ity tends to concentrate at the extremities of theelements and start at the tips of the cross-sections,this choice of alternate quadrature allows a moreprecise monitoring of the spread of plasticity. Onthe other hand the abscissa of two of the integrationpoints are pre-established, it is then generally nec-essary one more point in this quadrature, comparedto the number of points needed in the standard one,to integrate exactly the same polynomial.

The developed numerical model is written inMatlab environment, being prepared to obtain in-put from an Excel file; the residual stresses di-agrams for hot-rolled I/H sections and rectangu-lar hollow sections, and cold-formed circular hollowsections are already built-in. The developed modelis also capable to distinguish two different compo-nents of loading, a uniform one and a incrementalone.

Given the volume of results planned to be ob-tained, the process of acquiring them is automatic,pre-establishing the simulations to be run, havingthe program alter the input itself and record theresult in an appropriate manner.

3. Validation and Application3.1. ValidationThe developed numerical model (“DNM”) is vali-dated comparing its solutions with the results pre-sented by Ofner (1997). These results were cho-sen for validation, not only because of their quan-tity and quality, but also because these results wereadopted for the calibration of both methods avail-able in the EN1993-1-1:2005 for the verification ofbeam-column buckling. In fact Ofner, performedmore than 20000 second-order geometrically andmaterially non-linear simulations of steel members,taking into account initial residual stresses, througha user-defined subroutine introduced in the Abaqussoftware, and geometrical imperfections.

Ofner simulated simply-supported steel mem-bers with four different sections — IPE200,IPE500, HEB300, RHS200 × 100 × 10 — in steelof grade S235. For the analysis on the plane, Ofnerpresents analysis for both principal planes for mem-bers with three different values of normalized slen-derness — 0, 5 1, 0 1, 5 — under six different trans-verse load configurations:

A two equal concentrated moments on the beamends, making a uniform moment diagram;

B a single concentrated moment on an extremity,making a linear moment diagram with zero onone extremity;

C two symmetrical concentrated moments on theextremities, making a linear moment diagram;

D concentrated load at mid-span;

E constant distributed load;

F combination of configurations “A” and “D”,making a bilinear moment diagram where themid-span value is symmetrical with the onesat the extremities.

In these plane analysis the geometrical imper-fections implemented have the form of a half-sinewave with an amplitude of L/1000. Regarding theresidual stresses diagrams, the widely accepted onesfrom the ECCS(1976), used in the definition of theEuropean column curves, were applied.

The results in Ofner (1997) are presented ingraphics and the curves of the graphics corre-sponding to plane analysis were reproduced witheight simulations in the developed model, makinga subtotal of approximately 1150 simulations. Theresults regarding buckling along the strong axes ofan IPE200 from both models are presented in figure1 the remaining in Melo (2016). As can be seen inthese figures, and in the ones in the reference, theresults from the developed model show an excellent

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Figure 1: IPE200 strong axes buckling DNM vs. Ofner

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Strong axes Weak axes

AverageMaximum

AverageMaximum

Non-conservative Conservative Non-conservative Conservative

Method 1 4,8% 5,3% 14,9% 4,5% 5,3% 19,7%Method 2 5,1% 5,3% 26,0% 9,1% 5,3% 28,2%

Table 1: Whole results.

agreement with the results by Ofner (1997). There-fore the developed numerical model is consideredvalidated.

3.2. ResultsThe developed numerical model is adopted toevaluate the applicability of the two methods inthe EN1993-1-1:2005 regarding the verification ofbeam-column buckling, namely: method 1, pre-sented in annex “A” and described in Boisson-nade(2004) and method 2, presented in annex “B”and explained in Greiner(2006). The model wasalso applied to study the impact of varying the di-rection of the initial geometrical imperfection, withrespect to the position of the fillet weld in the be-haviour of cold-formed circular hollow sections witha single longitudinal weld.

3.2.1 Results hot-rolledFirstly, the set of gathered results is expanded toHEA sections, since these are the sections betterdesigned to be used as columns. The analysed sec-tions are a HEA200 in S235 steel and a HEA500 inS355, these were simulated for buckling about bothprincipal axis, as were the other sections, with thesame three values of normalized slenderness (0, 51, 0 1, 5), under the same six different transverseload configurations, described in the previous sec-tion, with the same initial conditions. The set of re-sults now amounts to more than 1700 simulations,which cannot all be reproduced in this document.The results for both axes of the IPE200 section aredisplayed in figures 2 and 3, along with the resultsobtained from both methods available in EN1993-1-1:2005. The remaining results are available in Melo(2016), in the same format.

Considering the examples analysed (e.g. figures2 and 3), one can safely say that, for I/H type sec-tions, method 2 is the most conservative of the two,when analysing buckling about the weaker axes.As for the strong axis it is not possible to deter-mine which of the two methods is more conserva-tive. This is because the result from method 2 givesa clearly better approximation for buckling aboutthe strong axes than along the weak axes.

Regarding the differences in behaviour underthe six transverse load configurations, it is possi-ble to notice that the quality and consistency ofthe answer of both methods, when analysing buck-ling about the strong axis, gradually worsens from

configuration “A” to “C”, i.e. worsens with the pro-gressive inversion of the concentrated moments atthe beam ends. Under configuration “A” the qual-ity of the answers is approximately constant andindependent of both method and slenderness, whileunder configuration “C” the quality of the answervaries with the intensity of the transverse load, andit is also no longer independent with respect to theslenderness and method chosen. Similarly, underconfigurations “D” and “E” the answer behaves asunder configuration “A”, and under configuration“F” behaves as under configuration “C”, i.e. whilethere is only transverse loads the methods have agood behaviour, when there is a combination oftransverse loads and concentrated end moments thebehaviour of the answers is not as good.

Regarding buckling about the weak axis, forI/H type sections, no distinction can be made be-tween the behaviour under the different transverseload configurations, all six of them being most simi-lar to the behaviour under configuration “C” aboutthe strong axis.

Also relevant to note that for all analysed sec-tions the case where method 2 is most conservativeis for the element with normalized slenderness of1, 5 and a transverse load configuration “C”, aboutthe strong axis this is an isolated case while aboutthe weak axis there are other close simulations.Such conservative results are partly due to the pur-posefully conservative lower limit introduced in theAustin formula (0.6+0.4ψ ≥ 0.4), which is used forthe calculation of the equivalent uniform momentfactor in method 2.

Specifically for the RHS200 × 100 × 10 section,as expected, it features a lower level of distinctionbetween the performance about the two axes, andthe results obtained for this section are most similarwith the ones from the I/H type sections about thestrong axis, particularly in what concerns the be-haviour under the different transverse load configu-rations. Hence, method 2 no longer exhibits such ahigher level of conservatism about the weaker axisin comparison with method 1.

An analysis of the whole results, evaluating theerror of the obtained solution relatively to bothmethods, is summarized in table 1. The resultsfound for method 1 are in accordance with theones reported by Boissonnade(2004) and clearlymore satisfactory than the ones presented in the

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Figure 2: IPE200 strong axes buckling DNM vs. EC3

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Figure 3: IPE200 weak axes buckling DNM vs. EC3

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same document for the previous version of EC3,ENV1993-1-1:1992, — average error 24, 4%, max-imum conservative error 87, 0%. The presentednumbers imply that the errors are bigger about theweak axis and using method 2, however, it is impor-tant to underscore that the set of results obtainedis limited to plane analysis and these methods havea domain which includes spatial problems with pos-sibility of lateral-torsional buckling.

Furthermore, a brief analysis is made about theinfluence of the magnitude of the residual stresseson the ultimate resistance of the steel elements. Intheoretical terms as well as in experimental mea-surements, the intensity of the residual stressesdoes not depend on the steel grade. However, thewidely used formulas presented in ECCS(1976) es-tablish a relation between the intensity of the resid-ual stresses and the yield stress of the steel used.With this objective the beam-column results of theHEA500 in S355 are repeated, but with the resid-ual stresses scaled as considering the element madeof S235 steel. In addition, a series of columns aresimulated with several values of slenderness andwith both magnitudes of residual stresses, analysedabout both principal inertia axes.

It was found that by introducing residualstresses proportional to S235 in an S355 steel el-ement, one obtains a slightly higher ultimate resis-tance: maximum of 3, 5% in columns (essentially inthe weak axes for normalized slenderness close to 1),and 5% in beam-columns (concentrated in the weakaxes in the area with transversal loads with lowerintensity). The significative difference between thereduction in the intensity of residual stresses andthe gain in resistance comes from the fact that, al-though the reduction in residual stresses increasesthe elastic load bearing capacity of the element, italso reduces the difference between the elastic andinelastic load bearing capacities.

3.2.2 Results CHSThe model was also adopted for the analysis ofcolumns with a CHS section. The effect of vary-ing the direction of the geometrical imperfection,relatively to the location of the weld, with respectto the behaviour of cold formed circular tubes witha single longitudinal weld was analysed.

Since there is no standard residual stressesdiagram for cold-formed circular hollow sections(“CHS”), a diagram was defined by considering thatthe components of residual stresses due to the pro-duction process of the original steel plate and bend-ing in the circular shape are negligible when com-pared with the component due to the welding. Inso being, the applied residual stresses diagram isderived from the formula presented in ECCS(1976)for the residual stresses diagram of a plate with a

single weld at one edge. the formula is adapted fora half-circumference, being written as:

σc = fy×2 (cos (ψ) + 1)ψ + sin (ψ) (π − ψ)

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2 (cos (ψ) + 1) (π − ψ) − sin (ψ) (π − ψ)2 ;

(1)

σt = σc−2fy×ψ

π − ψ; ψ = π× c× b

2; (2)

where “c” is the width of the tension block at eachside of the weld, given by:

c =12000 × p×Aw

fy ×∑t

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where “p” is the process efficiency factor, assumed0, 9 for submerged arc; “Aw” is the cross-sectionarea of added weld metal in mm2; “

∑t” is the sum

of the plate thickness at the weld in mm and “b” isthe perimeter in mm. In addition, the yield stressshould be introduced in MPa. The resulting dia-gram is presented in figure 4, linear with respect tothe centre angle.

Figure 4: Residual stresses diagram in a circularhollow section due to a single weld.

In what concerns the residual stresses in thistype of section it is only left to specify how “Aw” isobtained. For a single curve — curve c — to be ableto suitably adapt to all cold-formed circular hollowsections the same geometrical layout for the filletweld must be applicable to all sections. The layoutapplied in this study is the one illustrated in figure5.

Figure 5: Geometrical layout for the weld cross-section.

The studied section is a CHS457 × 8 in S235steel grade, of resistance class 2 and to which cor-respond the following data for the definition of theresidual stresses diagram:

ψ = 0, 41rad; σc = 142, 2MPa; σt = 71, 768MPa.

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Three combinations of imperfections are stud-ied. Combination “A” considers the residualstresses and the initial geometric imperfection inopposite directions by placing the fillet weld at theinner side of the initial curvature, illustrated in fig-ure 6a. Combination “B” considers both imperfec-tions in orthogonal directions, by placing the filletweld at 90o with the initial geometrical imperfec-tion, represented in figure 6b. Combination “C”is the opposite of combination “A”, both imperfec-tions have the same direction placing the fillet weldat the outer side of the initial curvature, presentedin figure 6c. Also illustrated on figures 6 is a qual-itative stress diagram due to loading and elementgeometry, termed “σcarreg”.

Combination “A” is instinctively more con-straining since it overlaps the area on the resid-ual stresses diagram with the higher intensities,both in compression and in tension, with the areamost compressed in “σcarreg”. Consequently, re-duces both the elastic and inelastic load bearingcapacities. By opposition, combination “C” isinstinctively the least constraining as it overlapsthe area on the residual stresses diagram with thehigher intensities with the area least compressed in“σcarreg”. In what concerns combination of imper-fections “B”, although it overlaps the area on theresidual stresses diagram with the higher intensitieswith the area with mean compression in “σcarreg”,it concentrates the tensioned areas of the residualstresses diagram around the centre of gravity of thesection. In this way, as the compressed areas are thefirst to yield, the section rapidly loses the greaterpart of its bending moment resistance. For thisreason, for small values of slenderness this secondcombination of imperfections yields lower values ofultimate element resistance than combination “A”.

It is important to note that the results obtainedfor combination “B” have an inherent error due tothe fact that the column structural behaviour isbiaxial bending as a result of the combination ofimperfections considered. The developed numericmodel only has the ability to analyse a single plane,the out-of-plane component is then neglected.

With respect to curve “c” (α = 0, 49), the col-umn curve defined in EC3 for cold-formed hollowsections, the results attained with combination ofimperfections “A” are in a good agreement: curve“c” is on the safe side throughout the slendernessrange. As well as combination “B” for slendernessesgreater than 1; for smaller slendernesses though,curve “c” is not on the safe side with respect tocombination “B”, the maximum non-conservativeerror found is 8%. As predicted, for slendernessessmaller than 2, combination “C” is not the con-straining one, corresponding to values of ultimateresistance significantly greater than the other two

(a) Combination “A”

(b) Combination “B”

(c) Combination “C”

Figure 6: Qualitative diagrams of the residualstresses and stresses due to the loading and elementgeometry.

combinations. For values of slenderness greaterthan 2, combination “C” results in ultimate resis-tances lower than the ones obtained by the othertwo combinations and by curve “c”, though withonly slight differences.

The effect of the residual stresses in the deflec-tions of the element, as obtained through the de-veloped numeric model, is materialized through anadditional bending moment in the inelastic sections.The stress diagram in a given section may be seen

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Comprimento L = 4m L = 14m L = 21mCombinacao A B C A B C A B C

Gomes (KN) 4231 3863 4306 2376 2105 2663 1325 1334 1542DNM (KN) 4275 3974 4443 2416 2216 2704 1332 1399 1553

Erro 1,0% 2,9% 3,2% 1,7% 5,3% 1,5% 0,5% 4,9% 0,7%

Table 2: Agreement between the two models.

as the sum of the two components represented infigures 6. In the first stages of inelasticity one canconsider that the component due to loading remainsintact and that it is the residual stresses componentwhich is “truncated”. A truncated residual stressesdiagram no longer possesses the auto-equilibratedcharacteristic, both in terms of axial force and mo-ments, as illustrated in figure 7b.

(a) (b)

Figure 7: (a) Imperfect column; (b) Qualitative di-agram of truncated cross-sectional residual stresses,lateral view combination “C”.

For small values of slenderness, in combina-tion of imperfections “A” the moment from thetruncated residual stresses is opposed to the oneproduced by the axial force, illustrated in figure7a, therefore the need arises for bigger deflectionsin order to establish equilibrium. In combination“B” two moments arise from the truncated residualstresses: one has an effect similar to the one de-scribed for combination “A”, the other one wouldgive rise to out-of-plane deflections, thus bi-axialbending. In combination “C”, once again, the ef-fect is opposed to the one in combination “A”, amoment arises with the same direction as the oneproduced by the axial force, making the equilibriumconfiguration less deformed.

As the effect of the geometric imperfection isproportional to the element slenderness and the ef-fect of the residual stresses is not, for small valuesof slenderness, in combination “C” there must bea value of slenderness for which both effects bal-ance each other out. In so being, along the resultsfor this combination there are actually two differ-

ent behaviours. For greater values of slendernessthe effect from the geometric imperfection is pre-dominant and the equilibrium path presents the ex-pected form, progressive increase of the deflectionsas in the other combinations. For smaller values ofslenderness there is a point along the equilibriumpath in which the effects of the residual stressesovercome the effect of the geometrical imperfectionarising a reduction or even inversion of the deflec-tions.

The combination “C” becomes determining forslendernesses greater than 2, since for these val-ues of slenderness, in terms of stresses, the effectof the moment produced by the axial force intro-duced surpasses its compressive effect and, in com-bination “C”, the first areas to yield are no longerthe compressive ones but the tensioned ones. Thismeans that the moment due to the truncated resid-ual stresses does not have the same direction as theone produced by the axial force anymore.

A comparison with the results obtained byGomes (2014) was performed so as to evaluate themodel accuracy. The results of Gomes are obtainedthrough the use of shell elements in the computa-tional software Abaqus. The section CHS508×8 inS355 grade steel is the one chosen in the document,having the following data for the definition of theresidual stresses diagram:

ψ = 0.49; σc = 272MPa; σt = 141MPa,

these possess some differences with the ones pre-sented in Gomes (2014). The differences arise frompractical considerations related with the dimensionsof the shell elements and from the way by which thesoftware produces the residual stresses. The valuespresented in the document are:

ψ = 0.47; σc = 240MPa; σt = 95MPa.

Three columns were simulated with the threedifferent combinations of imperfections: L = 21mλ ≈ 1, 7; L = 14m λ ≈ 1, 15; L = 4m λ ≈0, 33. The results for these simulation are pre-sented in table 2, both the ones from Gomes (2014)and the ones from the developed numerical model(“DNM”), as well as the error between the two mod-els.

Taking into account that the Abaqus model istri-dimensional and composed of shell elements, the

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results obtained through the developed model areconsidered in satisfactory agreement. The greatererror values found for combination “B” can be jus-tified by the fact that the developed model neglectsthe out-of-plane deformations.

Finally, it is important to underscore that, tothe best knowledge of the author, this is a subjectnot very well covered in the literature, and also thatthe residual stresses diagram, though similar to theone used by Gomes (2014) and to the one used inrectangular hollow sections, lacks experimental ba-sis.

Additionally, the geometric layout used for thecross-section of the fillet weld may be a variablewhich influences the attained results and aboutwhich no study was performed.

4. ConclusionsAn accurate and efficient numerical model capableof producing second-order analysis of steel mem-bers was successfully developed. The model consistson a distributed-plasticity Euler-Bernoulli beamdisplacement-based finite element model, resortingto the Newton-Raphson procedure to solve the non-linear component of the analysis. The results ob-tained with this model are generally in good agree-ment with the ones provided by the “EC3” for-mulations. Of particular note: the conservativeresults from method 2 for “I/H” type sections inbuckling about the weak axes, specially when theend-moments are of opposite sign; and the non-conservative results in CHS cold-formed columnswhen the the fillet weld is positioned at a 90o anglewith the geometrical imperfection direction.

AcknowledgementsI would like to express my sincere gratitude to myadvisors Prof. Ricardo Vieira and Prof. FranciscoVirtuoso for the continuous support of my studyand knowledge. And I would like to thank my fam-ily for the support throughout writing this paper.

ReferencesN. Boissonnade, J-P. Jaspart, J-P. Muzeau, and

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