extension of ec3-1-1 interaction formulae for the stability verification of tapered beam-columns
TRANSCRIPT
Journal of Constructional Steel Research 100 (2014) 122–135
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Journal of Constructional Steel Research
Extension of EC3-1-1 interaction formulae for the stability verification oftapered beam-columns
Liliana Marques ⁎, Luís Simões da Silva, Carlos Rebelo, Aldina SantiagoISISE, Department of Civil Engineering, University of Coimbra, Coimbra, Portugal
⁎ Corresponding author at: Department of Civil EngineeII, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. Tel.:239 797217.
E-mail address: [email protected] (L. Marques).
http://dx.doi.org/10.1016/j.jcsr.2014.04.0240143-974X/© 2014 Elsevier Ltd. All rights reserved.
a b s t r a c t
a r t i c l e i n f oArticle history:Received 4 September 2013Accepted 12 April 2014Available online xxxx
Keywords:StabilityEurocode 3Non-uniform membersTapered beam-columnsFEMSteel structures
EC3 provides several methodologies for the stability verification of members and frames. Regarding taperedbeam-columns, in EC3-1-1, the safety verification may be performed by the General Method. However, applica-tion of this method has been shown not to be reliable. On the other hand, the interaction formulae in EC3-1-1were specifically calibrated for stability verification of prismatic members.Recently, Ayrton–Perry based proposals for the stability verification of web-tapered columns and beams, in linewith the Eurocode principles for the stability verification of prismatic members, have shown to lead to a substan-tial increase of accuracy and to provide mechanical consistency relatively to application of the General Method.Such methodologies may be further applied to the existing interaction formulae.It is the purpose of this paper to propose a verification procedure for the stability verification of web-taperedbeam-columns under in-plane loading by adaptation of the interaction formulae in EC3-1-1, validated throughextensive FEM numerical simulations covering several combinations of bending moment about strong axis, My,and axial force, N, and levels of taper.
© 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Tapered members are used in structures mainly due to theirstructural efficiency, providing at the same time aesthetical appearance.Examples of the application of tapered steel members in variousstructures are given in Figs. 1.1 and 1.2.
Taperedmembers are commonly applied in steel frames, namely in-dustrial halls, warehouses, exhibition centers, etc. Adequate verificationprocedures are then required for these types of structures. Somestructural configurations are illustrated in Fig. 1.3.
In the scope of member design, maximum taper ratios (defined asthe ratio between the maximum and the minimum height of the ta-pered member — γh = hmax/hmin) of γh = 4 may be assumed to covera large proportion of existing structures. This issue will most likely bemore pronounced for cross sections with slender webs (class 4) forwhich a higher variation may be observed. However, in this study,since only global member failure modes were the scope of analysis,the authors assumed this limit for the proposed rules. Interaction be-tween local and global failure modes will be studied in a next step ofthe research. Figs. 1.1 to 1.3 illustrate taper ratios within that range ofγh b 4, even for the shorter members.
ring, University of Coimbra, Polo+351 239 797260; fax: +351
EC3 provides several methodologies for the stability verification ofmembers and frames. Regarding non-uniform members in general,with tapered cross-section, irregular distribution of restraints, non-linear axis, castellated, etc., several difficulties are noted regarding thestability verification. Moreover, there are yet no guidelines to overcomethese issues. As a result, to ensure safety, over-conservative verificationis likely to be performed by the designer, not accounting for the advan-tages and structural efficiency that non-uniformmembersmay provide.
In EC3-1-1 [1], the safety verification of a tapered beam-columnmaybe performed by the General Method; by a second order analysis con-sidering all relevant imperfections followed by a cross section check;or by a numerical analysis taking account of all relevant nonlineargeometrical and material effects.
The last two options involve adequate numerical modeling to incor-porate the second order effects, which is, for the time being, not the pre-ferred alternative as the definition of a wide combination of relevantimperfections is not always simple to consider. The alternative of usingthe General Method is also not very reliable for the following reasons:
• TheGeneralMethod requires that the in-plane resistance of themem-ber accounting for second order in-plane effects and imperfections isconsidered as an absolute upper bound of the member resistance(αult,k). The consideration of in-plane (local and global) imperfectionsfor the determination of the in-plane load multiplier αult,k of theGeneral Method may result in a need to perform complex numericalanalyses, as there are yet no analytical stability verification proceduresfor non-uniform members;
Nomenclature
Lowercasesa0, a, b, c, d Class indexes for buckling curves according to EC3-1-
1aγ Auxiliary term to the taper ratio for application of LTB
proposed methodologyb Cross section widthe0 Maximum amplitude of a member imperfectionfy Yield stressh Cross section heighthmax Maximum cross section heighthmin Minimum cross section heighthxcII,lim Cross section height at xc,limII
kyy, kzy,kyz, kzz Interaction factors dependent of the phenomenaof instability and plasticity involved
n Number of casestf Flange thicknesstw Web thicknessxc,limII Second order failure cross section for a high slenderness
levelxc,Ni , xc,Mi , xc,MN
i Denomination of the failure cross section (to dif-ferentiate from the type of loading it refers to): N — doto axial force only; M — due to bending moment only;MN — due to the combined action of bending momentand axial force
xcI First order failure cross sectionxcII Second order failure cross sectionxmin Location corresponding to the smallest cross sectionx-x Axis along the membery-y Cross section axis parallel to the flangesz-z Cross section axis perpendicular to the flanges
UppercasesA Cross section areaAmin Cross section area of the smallest cross section in of a ta-
pered memberCm Equivalent moment factor according to clause 6.3.3CoV Coefficient of variationE Modulus of elasticityFEM Finite Element MethodGM General MethodGMNIA Geometrical and Material Non-linear Analysis with
ImperfectionsL Member lengthLBA Linear Buckling AnalysisLTB Lateral Torsional-BucklingM Bending momentMb,Rd Design buckling resistance momentMEd Design bending momentMf,Rd Cross section resistance to bending considering the area
of the flanges onlyMNA Materially Non-linear AnalysisMpl,y,Rd Design value of the plastic resistance to bending mo-
ments about y-y axisMy Bending moments, y-y axisMy,Ed Design bending moment, y-y axisN Normal forceNcr,z Elastic critical force for out-of-plane bucklingNEd Design normal forceNpl Plastic resistance to normal force at a given cross sectionNpl,Rd Design plastic resistance to normal forces of the gross
cross sectionUDL Uniformly distributed loading
Lowercase Greek lettersα Angle of taperα, αEC3 Imperfection factor according to EC3-1-1αb(Method) Loadmultiplierwhich leads to the resistance for a given
methodαcr Load multiplier which leads to the elastic critical
resistanceαcr,op Minimum amplifier for the in-plane design loads to
reach the elastic critical resistance with regard to lateralor lateral–torsional buckling
αplM Load amplifier defined with respect to the plastic cross
section bending MomentαplN Load amplifier defined with respect to the plastic cross
section axial forceαult,k Minimum load amplifier of the design loads to reach the
characteristic resistance of the most critical crosssection
γM1 Partial safety factor for resistance of members to insta-bility assessed by member checks
δ0 General displacement of the imperfect shapeδcr General displacement of the critical modeε Utilization ratio at a given cross sectionεMI Utilization ratio regarding first order bending moment
MεMII Utilization ratio regarding the second order bending
momentεN Utilization ratio regarding the axial force Nη Generalized imperfectionλop Global non-dimensional slenderness of a structural
component for out-of-plane buckling according to thegeneral method of clause 6.3.4
λ Non-dimensional slendernessλ xð Þ Non-dimensional slenderness at a given positionλy Non-dimensional slenderness for flexural buckling, y-y
axisλz Non-dimensional slenderness for flexural buckling, z-z
axisλLT Non-dimensional slenderness for lateral–torsional
bucklingλLT;0 Plateau length of the lateral torsional buckling curves
for rolled sectionsλ0 Plateau relative slendernessφ Over-strength factorϕ Ratio between αpl
M and αplN
φy, φz, φLT Over-strength factor for in-plane buckling, out-of-plane buckling, lateral–torsional buckling
χ Reduction factorχLT Reduction factor to lateral–torsional bucklingχnum Reduction factor (numerical)χop Reduction factor for the non-dimensional slenderness
λop
χy Reduction factor due to flexural buckling, y-y axisχz Reduction factor due to flexural buckling, z-z axisχz Reduction factor to weak axis flexural bucklingψ Ratio between the maximum and minimum bending
moment, for a linear bending moment distributionψlim Auxiliary term for application of LTB proposed
methodology
123L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
• It has been proven that, for the case of columns (even prismatic),this assumption for the determination of αult,k leads to conservativeestimates of the ultimate resistance (up to 20%) [9]. In addition, itleads to also overly conservative resistance if the in-plane effects are
Fig. 1.1. Multi-sport complex — Coimbra, Portugal.
Fig. 1.2. Construction site in front of the Central Station, Europaplatz, Graz, Austria(Nahverkehrsdrehscheibe Graz-Hauptbahnhof, 02-02-2012 [19]).
124 L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
of the same magnitude as the out-of-plane effects (for example, RHSsections), see [4–6,10];
• Unconservative resultsmay also be found, mainly due to the difficultyin choosing an adequate buckling curve [7];
• For determination of the “global” reduction factor, a minimum or in-terpolated value between the reduction factors for flexural bucklingχz and lateral–torsional buckling χLT must be determined — thereare however no guidelines on how to perform this interpolation. In[17] only the first option –minimum value – is recommended. An in-terpolation between the lateral–torsional and the out-of-planemodesismechanicallymore consistent [10], although an adequate interpola-tion needs to be developed.
In summary, although a generalized slenderness procedure may ad-dress directly the true buckling behavior of themember; it still requiresa deeper study in order to establish a procedure which correctly takesinto account the weight between lateral and lateral–torsional buckling.This is even more relevant when the method is applied to verify thesafety of complex systems and there is a multiplicity of bucklingmodes that may contribute to their failures. On the other hand, as an al-ternative, the extension of the classical interaction approach applicableto the isolated member that is usually combined with global structuralanalysis has several advantages:
• The separate phenomena for flexural buckling and lateral–torsion-al buckling of the isolated member can be fairly well derived andextended to a wide range of cross sectional and member shapeswith different boundary conditions and intermediate partial re-straints, when compared to the generalized slenderness proce-dures which promise, on a “single” and “simple” expression, toverify the safety of a structural system for every possible bucklingmodes;
• At the same time, a parallelism can be kept with existing rules forprismatic columns and beams, respectively, simply by adding ade-quate factors that account for the non-uniformity of the member;
• Such factors may be given in Tables or for particular cases guide-lines may be given to the designer on how to derive such factors;
• It allows the incorporation of the local cross sectional effects byconsidering effective cross section properties.
• Many authors chose to extend the well-known interaction ap-proach in many fields, for example under fire conditions; formono-symmetric sections; etc., [11–13];
• Finally, it is of simple application as the designer is familiar with the“global second order in-plane analysis” combined with “isolatedmember verification” procedure.
As a result, in this paper, the interaction formulae for prismaticmembers in EC3-1-1 are extended to simply supported web-tapered beam-columns. In a first step of this research, Ayrton–Perry based procedures for the stability verification of isolatedweb-tapered columns and web-tapered beams were proposed bythe authors in line with EC3 basis for prismatic columns. These aresummarized in Section 1. Then, a direct adaptation of EC3-1-1 inter-action formulae is proposed for tapered members based on applica-tion of the rules for columns and beams. Although the interactionformulae was not intended for verification of non-uniformmembers,it will be shown that it leads to a similar level of reliability as of theprismatic member case, based on a numerical parametrical studyconcerning beam-columns which is carried out and compared tothe proposed methodology.
It is to be mentioned that other codes, e.g. [3], tend to providesimilar approaches for the stability verification of tapered beam-columns, i.e., to base the verification on the basis of the interactionformulae for prismatic members with the provisions for the taperedbeams and columns. The difference will then lie in a balance betweenthe simplicity and accuracy of the rules for the prismatic member. Inthis proposal the stability verification procedures for the isolatedcolumns and beam have the same analytical background of the pro-cedures for prismatic members adopted in EC3-1-1 and are thereforeconsistent with those, which are already familiar to the designer.
Fig. 1.3. Different portal frame configurations with tapered members (not to scale).
125L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
Straightforward and code conform design buckling rules are thenprovided, leading to a simple but at the same time efficient design.
In this study only global instability failure modes are analyzed, i.e.,the cross section plastic capacity may be fully attained and accordingly,the numerical models do not develop local buckling deformations.
2. Flexural and lateral–torsional buckling verification — Ayrton–Perry based procedures
The methodology for the stability verification of web-taperedbeam-columns presented in Section 2 is based on analyticallybased proposals regarding flexural buckling of tapered columnsand lateral–torsional buckling of tapered beams. More details maybe found in [2,7], respectively. For this, Ayrton–Perry analyticalmodels were built based on a linear interaction between the firstorder forces and second order bendingmoments utilizations, leadingto a maximum utilization (and, consequently, to the ultimate loadfactor) at a certain location, denoted as the second order failurelocation. The utilization ratio of a certain section is given by theratio between applied forces and resistant forces. It is also the in-verse of the load multiplier αult,k. Fig. 2.1 illustrates this aspect overthe length L of a tapered column buckling about major axis.
For the specific case of flexural buckling about minor axis, first orderutilization ratio is given by the ratio between applied and resistant axialforce, whereas second order utilization ratio is due to sinusoidal secondorder moment. The sum of both leads to a maximum stress ratio at thetip of the compressed flange, which on its turn leads to critical locationat mid span.
In the models for tapered members, similarly, eigenmode conformimperfections were considered for the second order forces' shape, lead-ing to similar equations as those presented in EC3-1-1 for the stabilityverification of prismatic columns. As a result, as long as a second order
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.2 0.4 0.6 0.8 1
Utilization
x/L
Critical location
2nd + 1st utilization
Utilization due to 2nd order forces
Utilization due to 1st order forces
Fig. 2.1. Determination of the failure location.
failure location is known and an additional imperfection factor is con-sidered to account for the non-uniformity either of the loading or ofthe cross section, the verification may be performed analogously tothe rules for prismatic columns.
Recent investigations [15] have shown that this second order fail-ure location and additional imperfection factor may be replaced insome of the terms by an “over-strength” factor φ which accountsfor the relation between the ultimate resistancemultiplier of the sec-ond order location, αult,k(xcII) and the first order location, αult,k(xcI ),such that if the φ-factor is determined, the verification is alwaysbased on xc
I . φ is given by
φ ¼FRk xIIc
� �=FEd xIIc
� �FRk xIc
� �=FEd xIc
� � ¼αult;k xIIc
� �αult;k xIc
� � ð2:1Þ
which leads to
λ xIIc� �
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiαult;k xIIc
� �αcr
vuut ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiφαult;k xIc
� �αcr
vuut ¼ ffiffiffiffiφ
pλ xIc� �
: ð2:2Þ
In Eq. (2.2), αcr is the critical load multiplier.xcI is such that the first order utilization, i.e. NEd/NRk orMy,Ed/My,Rk is
the highest along the member. It is recommended that at least 10 loca-tions along the member are considered to find xc
I , as also proposed by[8]. The verification procedures, calibrated φ-factors and second orderfailure locations, are described in Table 2.1 for web-tapered columnsand in Table 2.2 to Table 2.4 for web-tapered beams. The calibration ofsuch expressions relied on the following issues:
• φ-factors represent the level of change from first order location to sec-ond order location with increase of member length. Global utilizationis given by the sum of first and second order utilizations and, for lowslenderness, location of the maximum of this sum is prevailed by thefirst order forces, whereas, for high slenderness, it is prevailed by sec-ondorder forces, unlesswhen the first order utilization exhibits a verysteep variation near its maximum value such that the increase of non-linear effects is not enough to overcome the location of thatmaximum. In addition, if a member exhibits a constant utilizationunder first order forces, this means that αult,k is constant. The ratio φis therefore unity whichever the position xcII will be. On the otherhand, whenαult,k is not constant alongmember length andwhen sec-ond order location is different from first order location, the ratioφwillalso be higher than unity. The only exception is, as explained above,when the variation of utilization is exceptionally steep;
• As a result, for calibration of the φ-factors and/or second order loca-tions for flexural buckling and lateral–torsional buckling, both firstorder and second order utilization functions were analyzed, the latterwith the aid of numerical nonlinear analysis. Based on the range ofanalyzed taperedmembers and bendingmoment/axial force distribu-tions, trends were defined and factors were calibrated. More detailsmay be found in [2,7,15].
Table 2.1Proposed verification procedures for web-tapered I-section columns — φ approach.
Out-of-plane flexural buckling In-plane flexural buckling
αult,k(xc I) NRk(xc,N I)/NEd(xc,N I) –– for NEd = const. is the smallest cross section
αcr Numerically or from literature ≈ Ncr,z,hmin/NEd (approximately the Eulerload of an equivalent column with the smallest cross section)
Numerically or from literature
λ xIc;N� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
αult;k xIc;N� �
=αcr
r
φ1þ htw
Amin
1þ 4γhð Þ γh−1ð Þ10γh
� �1þ hmintw
Amin
γh−1γh þ 1
α Hot-rolled: 0.49Welded: 0.64
Hot-rolled: 0.34Welded: 0.45
η αzffiffiffiffiffiffiφz
pλ xIc� �
−0:2� �
If welded,ηz ≤ 0.34
αyffiffiffiffiffiffiφy
pλ xIc� �
−0:2� �
If welded,ηy ≤ 0.27
ϕ 0:5� 1þ φ� ηþ φ�λ2
xIc;N� �� �
χ(xc,NI )φ=ϕþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϕ2−φ� λ
2xIc;N
� �r≤1
Verification χ(xc,NI ) × αult,k(xc,NI )/γM1 ≥ 1 ⇒ NEd(xc,NI ) ≤ χ(xc,NI ) × NRk(xc,NI )/γM1 ⇔ NEd(xc,NI ) ≤ Nb,Rd(xc,NI )
126 L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
Finally, in Table 2.2 to Table 2.4, γh= hmax/hmin and γw =W,el,max/W,el,min are the taper ratios concerning the maximum and minimumheight or elastic bending modulus and ψ is the ratio between themaximum and minimum bending moment for linear bendingmoment distributions and UDL regards uniformly distributedloading cases. A practical limit of γh ≤ 4 and γw ≤ 6.5 isestablished for application of Table 2.2. In addition, it shouldbe noticed that new imperfection factors for welded cross sectionswere also proposed leading to more accurate levels of resistancethan the ones given in the code (i.e., even for prismatic members)[2].
Table 2.2Proposed verification procedure for web-tapered I-section beams
Lateral–torsional buck
αult,k(xc,M I) My,Rk(xc,M I)/My,Ed(xc,Mthe minimum along th
αcr Numerically e.g. or byThe multiplier αcr sha
λLT xIc;M� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
αult;k xIc;M� �
=αcr
r
xc,limII See Table 2.3φLT For ψ: A · ψ2 + B · ψ
For UDL: −0.0025aγ2
See Table 2.4 for A, B,αLT
Hot-rolled: 0:16
ffiffiffiffiffiffiffiWy;e
Wz;e
vuuut
Welded: 0:21
ffiffiffiffiffiffiffiffiffiffiffiffiWy;el x
�Wz;el x
�vuuut
ηLT αLT � λz xIIc;lim� �
−0:2�
If welded,
ηLT ≤
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWy;el xII
c; lim
� �Wz;el xII
c; lim
� �vuuut 0:1
�
λz xIIc;lim� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NRk xIIc; lim� �
=Ncr;z;hm
r
ϕLT
0:5� 1þ φ� η� λ
λ2
0@
χLTxcIÞ
φ=ϕþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϕ2−φ� λ
2LT
r
Verification χLT xIc;M� �
� αult;k xIc;�
⇔My;Ed xIc;M� �
≤Mb;R
3. Stability verification of web-tapered beam-columns
3.1. Interaction formulae in EC3-1-1 — the basic case
The interaction formulae of clause 6.3.3 for the stability verificationof prismatic beam-columns are presented in Eqs. (3.1) and (3.2) foruniaxial bending and class 1, 2 or 3 cross sections.
NEd
χy NRk=γM1þ kyy
My;Ed
χLT My;Rk=γM1≤1:0 ð3:1Þ
— xc,limII and φ combined approach.
ling
I) –e beam, e.g. 10 sectionsexpressions forMcr from the literature, see Section 5.2.4.ll afterwards be obtained with respect to the applied load.
+ C ≥1+ 0.015aγ + 1.05C and aγffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil xII
c; lim
� �l xII
c; lim
� � ≤0:49
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIIc; lim
�IIc; lim
� ≤0:64
�
2ψ2−0:23ψþ 0:35�
ffiffiffiffiin
2LT xIc;M
� �z xII
c; lim
� �þ φ� λ2LT xIc;M� �1A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffixIc;M
� �≤1
M
�=γM1≥1⇔My;Ed xIc;M
� �≤χLT xIc;M
� ��My;Rk xIc;M
� �=γM1
d xIc;M� �
Table 2.3Calculation of xc,lim,M
II /L for lateral–torsional buckling of tapered I-beams.
For ψ (0.75 − 0.18ψ − 0.07ψ2) + (0.025ψ2 − 0.006ψ + 0.06)(γh − 1) ≥ 0If ψ b 0 and |ψ| γw ≥ 1 + 1.214(γh − 1), xc,limII /L = 0.12 − 0.03(γh − 1)
For UDL 0.5 + 0.0035(γh − 1)2 − 0.03(γh − 1)2 ≤ 0.5
127L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
NEd
χz NRk=γM1þ kzy
My;Ed
χLT My;Rk=γM1≤1:0 ð3:2Þ
in which NEd and My,Ed are the design values of the compression forceand themaximummoment about the yy axis along themember. The in-teraction factors kyy and kzy may be determined either by Annex A(Method 1) or Annex B (Method 2) of the same code. Finally, besidesthe verification of Eqs. (3.1) and (3.2), an additional cross sectioncheck is required at the member ends.
The adaptation of the interaction formulae to the verification of ta-peredmember naturally leads to some questions as there is no analyticalbackground specifically developed for the tapered beam-column case asit was performed for prismatic members. Nevertheless, an adjustmentcan be fairly easily analyzed especially when considering Method 2.
In a tapered beam-column the following verifications shall beperformed:
• Out-of-plane stability verification;• In-plane stability verification;• Cross section verification at themost heavily loaded cross section, i.e.,with the highest first order utilization.
In the following, possible alternatives for each of these verificationsare discussed and results are analyzed further in Section 5.3.
3.2. Adaptation of the interaction formulae to tapered beam-columns
3.2.1. Cross section verificationThe first order failure location of tapered beam-columns varies with
varying levels of axial force relatively to the applied bending momentleading. Disregarding the fact that cross section classmay varywith vary-ing depth of the beam-column, consider the example of Fig. 3.1 which il-lustrates results of a member composed of a (hot-rolled) IPE200 crosssectionwith a taper ratio ofγh=2.5 and subject to uniformly distributedloading and constant axial force. The beam attains a maximum utiliza-tion at about 35% of the member length (xc,MI /L). With increasingaxial force the maximum utilization location, xc,MN
I , moves towardthe smallest cross section, which is the first order failure locationof the column, xc,NI , as the axial force is constant.
For illustration,My,Ed corresponds to the maximum applied bend-ing moment along the member, irrespective of the utilization — forthe example of Fig. 3.1 (uniformly distributed loading) the pairs(NEd; My,Ed(L/2)) are considered. If, for example, the utilization atxc,MNI is considered for representation at each point of the curve, no
direct information regarding the load that actually leads to firstorder failure can be obtained, as each point corresponds to a different
Table 2.4Calculation of φ for lateral–torsional buckling of tapered I-beams.
aγ − 0.0005 ⋅ (γw − 1)4 + 0.009 ⋅ (γw − 1)3 − 0.077 ⋅ (γw − 1)2 + 0.78ψlim 1þ 120 � aγ þ 600 � aγ2−210 � aγ3=1þ 123 � aγ þ 1140 � aγ2 þ 330 � aγ3φLT ψ b −ψlim −ψlim ≤ ψA −0:0665 � a6γ þ 0:718 � a5γ−2:973 � aγ4
þ 5:36 � aγ3−2:9 � aγ2−2:1 � aγ−1:09−11:37þ 1
1−10B −0:1244 � aγ6 þ 1:3185 � aγ5−5:287 � aγ4
þ 9:27aγ3−5:24aγ2−2:18aγ−2þ0:02 � aγ6
−932 � aC −0:0579 � aγ6 þ 0:6003 � aγ5−2:314 � aγ4
þ 3:911 � aγ3−2:355 � aγ2 þ 0:02 � aγ þ 0:30:02 � aγ2−
cross section location. Therefore, the first option is preferred andadopted along the examples of this paper.
As a result, and also as referred, cross section verification should beperformed in a sufficient number of locations in order to find the crosssection with the highest first order utilization. For example, for a class1 or 2 I-section at an arbitrary location of the beam-column subjectboth to major axis bending and axial force, the utilization ratio may bedetermined from (see Section 6.2.9 of EC3-1-1 for combined bendingand axial force)
ε xð Þ¼ NEd xð ÞNpl;Rd xð Þ þ 1−0:5að Þ My;Ed xð Þ
Mpl;y;Rd xð Þ ≤1:0 ð3:3Þ
where a = (A-2btf)/A ≤ 0.5. Eq. (3.3) was obtained from expression(6.36) in EC3-1-1 which gives the value of the maximum bendingmoment of a cross section when a given value of the axial force ispresent. Since this expression is valid for a constant value of theaxial force and in this study a proportional increase of the bendingmoment and axial force is considered, then expression (6.36) ofEC3-1-1 is transformed as follows at a given location along themember “x”:
MN;Rd ¼ Mpl;y;Rd
1‐NEd
Npl;Rd
1−0:5a→…→
MN;Rd
Mpl;y;Rd1‐0:5að Þ þ NEd
Npl;Rd¼ 1 ð3:4Þ
where the maximum bending moment allowed, MN,Rd, when axialforce is assumed constant may now be assumed as the maximum ap-plied bending moment My,Ed, when axial force and bending momentincrease proportionally. The result of this equation is 1 for the max-imum pair of forces allowable for the cross section and will belower than 1 otherwise (see Eq. (3.3)). Also, if NEd ≤ 0.25Npl,Rd andNEd ≤ 0.5hwtwfy/γM0, respectively, expressions (6.33) and (6.34) ofEC3-1-1, the axial force do not need to be taken into account andthe utilization is given by
ε xð Þ ¼ My;Ed xð ÞMpl;y;Rd xð Þ ≤1:0: ð3:5Þ
Another interesting aspect that can be observed from Fig. 3.1 andthat may lead to some questions is the vertical plateau around thehigh axial force zone. This can be explained because, for high axialforce relative to bending moment, the first order failure location ofthe beam-column approaches the first order location of the column(smallest cross-section) xc,MN
I → xc,NI . For the particular cases of
bending moment distributions in which there is no applied bending
⋅ (γw − 1)
≤ ψlim ψ N ψlim
2090 � aγ−8050 � aγ2 þ 1400 � aγ3
58 � aγ þ 705 � aγ2−120 � aγ3 þ 11:220:008 � aγ2−0:08 � aγ−0:157
−0:133 � aγ5 þ 0:425 � aγ4
γ3 þ 1:05 � aγ2−0:5 � aγ−0:1
−0:033 � aγ3 þ 0:04 � aγ2
þ 0:48 � aγ þ 0:370:14 � aγ þ 1:25 0:032 � aγ3−0:092 � aγ2
þ 0:06 � aγ þ 0:8
0
20
40
60
80
100
120
140
0 200 400 600 800
NEd [kN]
My,Ed [kNm]
Cross sectionresistance
b
bc1
bc2
c0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
ε
x/L
bbc_1bc_2c
a) Interaction cross section resistancecurve and analyzed cases
b) First order (plastic) utilization
Fig. 3.1. First order failure location with varying axial force relatively to the bending moment.
Table 3.1Interaction factors for web-tapered beam-columns according to Method 2.
kyy
Cmy � 1þ λy xIIc;N� �
−0:2� �|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
≤0:8≥0
NEd xIc;N� �
χy xIc;N� �
NRk xIc;N� �
=γM1
0BB@
1CCA
kzy
1−0:1λz xIIc;N
� �zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{≤0:1
Cm;LT−0:25
NEd x1c;N� �
χz x1c;N� �
NRk x1c;N� �
=γM1
for λz xIIc;N� �
b0:4 : 0:6þ λz xIIc;N� �
≤1−0:1λz xIIc;N
� �Cm;LT−0:25
NEd x1c;N� �
χz x1c;N� �
NRk x1c;N� �
=γM1
0 20 40 60 80 100
120
140
160
180
200
220
240
<1.4 ≥1.2<1.2 ≥1.0<1.0 ≥0.8<0.8 ≥0.6<0.6 ≥0.4<0.4 ≥0.2
<0.2 ≥0<0 ≥-0.2
<-0.2 ≥-0.4<-0.4 ≥-0.6
n
Dif
f (%
)
Fig. 3.2. Frequency graph— difference between alternatives for determination of kij atxc,NI and at xc,N,iII .
128 L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
moment at the smallest cross section, for high axial force, the utili-zation of the axial force at the smallest cross section is higher thanthe utilization of the combined loading at the immediate adjacentcross sections. Note that this only happens because the member istapered. As a result, the failure location is the smallest cross sectionin which only axial force is present, leading to the vertical plateau ofFig. 3.1.
Finally, it should be noticed that maximum utilization is not nec-essarily located in the same place for all I or H sections. For example,for the case of a tapered beam with varying bending moment, sincethe section modulus may exhibit higher or lower curvature depend-ing simultaneously on the cross section dimensions and taper ratio, itcannot be predicted where the utilization is maximum. First orderlocations could also be typified and expressions could be calibrated.However, unlike previously, here the ratio between applied andresistant forces can be easily computed along the member, thuseliminating increase of error naturally brought in by additionalcalibrations.
3.2.2. Adaptation of the interaction formulae to tapered membersEqs. (3.1) and (3.2) shall now be adapted to the case of tapered
beam-columns: the first question that arises is related the correctlocation to take into consideration in the given interaction formu-lae. For the case of prismatic beam-columns, this location is alwaysthe location of maximum bending moment utilization as the axialforce is constant; however, for tapered beam-columns, it may notbe the case. Nevertheless, according to the definitions of utiliza-tions for tapered beams and columns, the quantities ny = NEd/(χyNRk), nz = NEd/(χzNRk) or my = My,Ed/(χLTMy,Rk) are constantalong the member length and, as a result, it is irrelevant whichlocation is chosen and is recommended here (for simplicity) theconsideration of the first order failure location of the axial forceacting alone (xc,NI ) for the utilization term regarding axial force;and the first order failure location of the bending moment acting
alone (xc,MI ) for the utilization term regarding the bending moment.This leads to
NEd xIc;N� �
χy xIc;N� �
NRk xIc;N� �
=γM1
þ kyyMy;Ed xIc;M
� �χLT xIc;M
� �My;Rk xIc;M
� �=γM1
≤1:0 ð3:6Þ
NEd xIc;N� �
χz xIc;N� �
NRk xIc;N� �
=γM1
þ kzyMy;Ed xIc;M
� �χLT xIc;M
� �My;Rk xIc;M
� �=γM1
≤1:0: ð3:7Þ
3.2.3. Interaction factors kyy and kzyThe interaction factors given in Annex A (Method 1) and Annex B
(Method 2) of EC3-1-1 can be adapted to the tapered beam-columncase in the following. Method 1 is composed of two sets of formula
Table 3.2Adaptation of the equivalent uniform moment factors Cm for prismatic members.
Moment utilization diagram Range Cmy and CmLT
0 ≤ αs ≤ 1 −1 ≤ ψε ≤ 1 0.2 + 0.8 αs ≥ 0.4−1 ≤ αs b 0 0 ≤ ψε ≤ 1 0.1–0.8 αs ≥ 0.4
−1 ≤ ψε b 0 0.1(1−ψε) − 0.8 αs ≥ 0.4
0 ≤ αs ≤ 1 −1 ≤ ψε ≤ 1 0.95 + 0.05 αh
−1 ≤ αs b 0 0 ≤ ψε ≤ 1 0.95 + 0.05 αh
−1 ≤ ψε b 0 0.95 + 0.05 αh(1 + 2 ψε)
a) b)
Fig. 3.3. Determination of Cm factors for tapered beam-columns (plastic utilization ε).
Segment 2Segment 1
Cm,LT,2
Cm,LT,1
Cm,y
Ncr,z,1
; Mcr,1
Ncr,z,2
; Mcr,2
Ncr,z
; Mcr
Determined for the wholemember considering adequateintermediate boundaryconditions
or
6.62 S1 6.62 S2
Ncr,y
Mb,Rd
6.61
Cross section resistance (and member class)
hmin,1 h
max,1
hmin,2
hmax,2
Fig. 3.4. Verification of a member with 2 intermediate lateral restraints.
129L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
Fig. 4.1. Location of the intermediate lateral restraints in the cross section.
Table 5.1Parametric study for beam-columns failing in out-of-plane buckling.
# Smaller cs/fabrication γh Bending moment λz xh minð Þ1 IPE200 hr 2.5 UDL 1.22 IPE200 hr 3 Ψ = −0.5 2.53 IPE200 hr 3 Ψ = −0.5 1.24 IPE200 hr 1.2 Ψ = −0.25 1.25 HEB300 hr 2 Ψ = 0.25 16 HEB300 hr 1.5 Ψ = 0 1.57 HEB300 hr 1.5 Ψ = 0 0.78 HEB300 hr 2 Ψ = 1 19 IPE200 w 1.5 Ψ = −0.5 0.610 IPE200 w 1.2 Ψ = 0 0.911 IPE200 w 1.5 Ψ = −0.5 1.212 IPE200 w 1.2 Ψ = 0 1.713 IPE200 w 3 Ψ = −0.5 214 IPE200 hr 2.5 UDL 0.5
130 L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
in which the transition between the consideration of torsional defor-mations or not is implicit. In addition, a more complex transition be-tween cross-section resistance failure (for the low slendernessrange) and instability failure is also accounted for. However, thismethod contains too many parameters in a sense that it is not flexi-ble for an adaptation for tapered beam-columns. A possible applica-tion of the method to non-uniform members would bring manyquestions and probably would not lead to a satisfactory result as,from the beginning, this method was specifically developed for pris-matic members. A more detailed analysis and even calibration ofnew interaction factors would therefore be required to attainsatisfactory results. It is not the purpose of this study and, as a result,onlyMethod 2 is considered for a straightforward application/adaptationof the interaction formulae to the case of tapered beam-columns andfurther validation. Because I-sections are susceptible to torsional defor-mations, according toMethod 2, the interaction factors to be consideredare summarized in Table 3.1 – the only difference between these factorsand the ones present in EC3-1-1 is the properties of the cross section atthe respective location to be considered – which, for a prismatic mem-ber are constant. The terms NEd/(χNRk) are the same as considered inthe interaction equation. Regarding the relative slenderness, for theslenderness λy or λz , the question again arises on which location is to
be considered: (i) at xc,NI ; or (ii) at xc,N,iII , i.e. λi xIIc;N;i� �
¼ ffiffiffiffiffiφi
pλ xIc;N� �
(see Eq. (2.2)). However, provided that this slenderness seems to be re-lated to a “plateau” level (see e.g. for kyy, λy−0:2), alternative (ii) is ra-tionally more suitable. A sensitive analysis has shown that this aspecthardly influences the value of the interaction factor: for in-planebuckling cases where kyy is relevant, maximum difference of 1.3%was noticed; for out-of-plane buckling cases where kzy is relevant,maximum difference of 0.5% was noticed. For this evaluation, thecases from the parametric study to be analyzed in Section 4 (see fur-ther Table 5.1 and Table 5.2) were considered, and differences be-tween the 2 methodologies are given in Fig. 3.2 (a positive errormeans that the opted alternative gives lower value for the resistance,i.e., it is conservative). It is visible that the great majority of cases fall
Ly ≡ L
Ly / 1000
Fig. 4.2. Amplitude of the imperfections (beam
within a range of ±0.2%. As a result, no deeper study is given to thissubject and alternative (ii) is considered.
Finally, regarding the equivalent uniform moment factors Cm,y andCm,LT, Table B.3 of EC3-1-1 can be adopted provided that the diagramto be considered is the bending moment first order utilization diagraminstead of the bending moment diagram itself, see Table 3.2.
In a tapered beam subject to a linear bending moment distribution,thediagramof theutilization can be fairlywell compared to thediagramof a prismatic beam subject both to uniformly distributed loading andend moments. The Cm factor may be obtained from the respective Cmfactor due to that diagram. For the case of a tapered beam-column sub-ject to uniformly distributed loading the error would be higher. Fig. 3.3illustrates these two examples.
For application of the procedure to beam-columns with intermedi-ate lateral restraints, a similar approach as for prismatic membersshall be considered (see the flow chart of Fig. 3.4):
• Cross section resistance and determination of member class to beconsidered for application of the interaction formulae by consider-ing to the most utilized cross section along the member, see [8] formore details;
• Eq. (6.62) of EC3-1-1 or Eq. (3.7) shall be applied to each segmentof the beam-column by considering each segment as “isolated”members, i.e., treating them separately. The equivalent uniformbending moment factors Cm,LT are determined for each segment.The same applies for determination of the critical loads Mcr andNcr,z, unless numerical linear eigenvalue analysis of the globalmember is used;
• χLT (xc,MI ) that leads to the minimum resistant momentMb,Rd of theglobal member shall be kept for application of Eq. (6.61) of EC3-1-1or Eq. (3.6). This is determined as follows:
1. DetermineMb,Rd (xc,M,1I ) of segment 1, such that xc,M,1
I is the firstorder failure location of segment 1;
Lz ≡ L/2
Lz / 1000
-column with 1 intermediate restraint).
Table 5.2Parametric study for beam-columns failing in in-plane buckling.
# Smaller cs/fabrication γh Bending moment λz xhminð Þ λy xhminð Þ15 IPE200 hr 2.5 UDL 1.2 0.1816 4 0.6217 6 0.9218 HEB300 hr 1.5 Ψ = 0 0.6 0.2819 1.5 0.7020 3 1.39
Table 5.3Parametric study for beam-columns with intermediate restraints.
# Smallercs/fabrication
γh Bendingmoment
λz xhminð Þ(without restraints)
No. segments
21 hmin = 100 mmb = 300 mmtw = 10 mmtf = 20 mmhr
6 Ψ = 0 1.2 222 323 Continuous
24 hmin = 100 mmb = 100 mmtw = 10 mmtf = 20 mmhr
6 Ψ = 0 3 225 426 527 Continuous
Note: Reference case with 1 segment is not considered for analysis as, for each segment,γh N 4, and therefore the methodology of Section 3 for lateral–torsional buckling ofweb-tapered beams is not validated.
LTλ
LTχ
Cross sectioncapacity
1
Limit?
Fig. 5.2. Instability in a tapered beam-column with a high lateral–torsional slendernessplateau.
131L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
2. With the known value of Mb,Rd (xc,M,1I ), the proportional value
of Mb,Rd (xc,M,I) for the global member can be determined;
3. χLT(xc,MI ) is then obtained by applying the expression χLT(xcI) ×My,Rk(xcI)/γM1 = Mb,Rd(xcI));
0
20
40
60
80
100
120
0 200 400 600
My,Ed [kNm] GMNIA, λz=1.2
GMNIA, λz=2.5
cross section
6.62
a) Results for cases #2 and #3
0102030405060708090
0 200 400 600
NEd [kN]
NEd [kN]
My,Ed [kNm]GMNIA, λz=0.6
GMNIA, λz=1.2
cross section
6.62
c) Results for cases #9 and #11
Fig. 5.1. Interaction curve represent
4. Repeat steps 1; 2 and 3 to determineχLT(xc,MI ) but now consid-ering Segment 2;
5. The minimum χLT(xc,MI ) is to be considered for application ofEq. (6.61) of EC3-1-1;
• Eq. (6.61) of EC3-1-1 or Eq. (3.6) shall be applied to the global mem-ber. Cm,y as well as Ncr,y are also to be determined with the globallength. χLT(xc,MI ) that leads to the minimum resistant moment Mb,Rd
determined previously is to be considered here.
Finally, it should be mentioned that such adaptation of theinteraction formulae leads to a similar level of reliability as for prismaticmembers for the analyzed cases – restrained and unrestrained beam-columns subject to axial force and major axis bending. This occurs notonly because the utilization ratios for axial force and for bending mo-ment determined from the rules provided in Section 2 present a closerlevel of safety when compared to the prismatic cases; but also becausefor determination of Cm factors, a direct equivalency of the tapered toa prismaticmember is done by considering the utilization of the appliedbending moment with the resistant bending moment. In Section 4,unrestrained beam-columns that fail under Eq. (3.7) (out-of-planebuckling) are analyzed and for some of the cases, the number of
0
100
200
300
400
500
600
700
800
0 1000 2000 3000 4000
My,Ed [kNm] GMNIA, λz=0.7
GMNIA, λz=1.5
cross section
6.62
b) Results for cases #6 and #7
NEd [kN]
NEd [kN]
0
10
20
30
40
50
60
70
0 200 400 600 800
My,Ed [kNm] GMNIA, λz=0.9
GMNIA, λz=1.7
cross section
6.62
d) Results for cases #10 and #12
ation — out-of-plane buckling.
0
20
40
60
80
100
120
0 200 400 600 800
NEd [kN] NEd [kN]
My,Ed [kNm] CsectionGMNIA λy (xc,I)=0.18GMNIA λy (xc,I)=0.62GMNIA λy (xc,I)=0926.61
0
100
200
300
400
500
600
700
800
0 1000 2000 3000 4000
My,Ed [kNm] cross sectionGMNIA λy (xc,I)=0.28GMNIA λy (xc,I)=0.7GMNIA λy (xc,I)=1.396.61
a) Results for cases #15, #16 and #17 b) Results for cases #18, #19 and #20
Fig. 5.3. Interaction curve representation — in-plane bucking.
132 L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
intermediate lateral restraints is increased such that Eq. (3.6) maybecome more restrictive.
4. Numerical Model
A finite element model was implemented using the commercial fi-nite element package Abaqus, version 6.10 [14]. Four-node linear shellelements (S4)with six degrees of freedomper node andfinite strain for-mulation were used.
S235 steel grade was considered in the reference examples, with ayield stress of 235 MPa (perfect elastic–plastic), a modulus of elasticityof 210 GPa, and a Poisson's ratio of 0.3.
Loadingwas appliedwith reference to theplastic resistance values ofthe smaller cross-section.
Only members with end fork conditions were considered for thisstudy, and modeled to be supported against vertical displacement atthe centroid of the end cross sections. Intermediate lateral restraintsequally spaced may be applied at the extremes of the flanges accordingto Fig. 4.1.
Regarding global imperfections, a geometrical imperfection propor-tional to the eigenmode deflection was considered with a maximumvalue of δ0 = L/1000:
δ0 xð Þ ¼ δcr xð Þe0 ¼ δcr xð Þ L1000
ð4:1Þ
050
100150200250300350400450
0 200 400 600 800 1000 1200
NEd [kN]
My,Ed [kNm]CSGMNIA6.616.62 S16.62 S2
a) All equations
Fig. 5.4. Results for case
• For the continuously restrainedmodels about y-y axis, only imperfec-tions with respect to the in-plane mode (buckling about y-y axis)were considered;
• For the models with intermediate restraints, both imperfectionsabout major and minor axis with an amplitude of Ly = L/1000and ez = Lz/1000 were considered. This was done in accordancewith the models used for calibration of the imperfection factorsof the interaction formulae EC3-1-1, as the influence of any of thebuckling modes may be relevant. The shape of the buckling modeis here considered, see Fig. 4.2;
• For the unrestrained models, only the imperfection of the firstbuckling mode were considered as, for these cases, it was noticedthat the failure of the beam-column about the major axis is unlikelyto occur, especially for the cases of web-tapered members. In addi-tion, for the analyzed cases, the first member in-plane bucklingmode was found to be considerably higher than the first out-of-plane buckling mode.
Local imperfections are not considered provided that only global in-stability failure modes are analyzed. Therefore, the numerical modelsare prevented from developing local buckling deformations in thecross sections.
Regarding material imperfections, residual stress patterns corre-sponding both to stocky hot-rolled (i.e. with a magnitude of 0.5fy) andwelded cross-section were considered. A discussion concerning theadopted residual stress patterns and magnitudes may be found in [2].
0
50
100
150
200
250
0 200 400 600
NEd [kN]
My,Ed [kNm]GMNIA
6.62 S1
b) Failure condition eq.; elastic buckling modes
#24 — 2 segments.
NEd [kN] NEd [kN]
050
100150200250300350400450
0 200 400 600 800 1000 1200
My,Ed [kNm] cross sectionGMNIA6.616.62 S16.62 S26.62 S36.62 S4
0
50
100
150
200
250
300
350
0 200 400 600 800 1000
My,Ed [kNm] GMNIA
6.62 S1
6.61
c) All equations d) Failure condition eq.; elastic buckling modes
Fig. 5.5. Results for case #25 — 4 segments.
133L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
5. Parametric Study
5.1. Definition
Theparametric study comprises 520 simply supported symmetricallyweb tapered beam-columns of which:
• 275will fail in out-of-plane flexural buckling (with orwithout lateral–torsional buckling) — application of Eq. (2), summarized in Table 5.1;
• 140 will fail in in-plane flexural buckling (without lateral–torsionalbuckling) — application of Eq. (1) with χLT = 1, summarized inTable 5.2;
• 105 with intermediate restraints which will fail in in-plane and/orout-of-plane buckling (with or without lateral–torsional buckling) —application of both Eqs. (3.6) and (3.7), summarized in Table 5.3.Here, with the increase of restraints, the influence of major axis buck-ling increases relatively to minor axis buckling.
It is noted that whenever major axis buckling is the critical mode, ahigh level a taper is not possible, as this would lead to unpracticalslenderness ranges due to the buckling resistance provided by a web-tapered I-section about the major axis.
5.2. Methodology
For evaluation of results, the followingmethodologies are considered:
• GMNIA — Results given by the numerical models (geometrically andmaterially nonlinear analysis with imperfections);
• Application of proposed procedure:
050
100150200250300350400450500
0 500 1000 1500
NEd [kN]
My,Ed [kNm] cross sectionGMNIA6.616.62 S16.62 S26.62 S36.62 S46.62 S5
1
1
2
2
3
3
4M
a) All equations b
Fig. 5.6. Results for case
o 6.61 — results given by adapted Eq. (6.61) of EC3-1-1 (Eq. (3.6) ofthis paper);
o 6.62 — results given by adapted Eq. (6.62) of EC3-1-1 (Eq. (3.7) ofthis paper), and:
o 6.62 Si— results of Eq. (6.62) regarding segment nº i (i is consideredfrom the smallest cross section to the largest cross section)
o Cross section — cross section resistance.
Results are presented in interactionN-My curves. In the xx axisNEd isrepresented and in the yy axisMy,Ed corresponding to themaximumap-plied bending moment along the member, as discussed in Section 3.2.1.
5.3. Results and discussion
Fig. 5.1 compares the analytical results from the procedure present-ed in Section 3.1, regarding Eq. (3.7) with the numerical results of cases#2 and #3; cases #6 and #7; cases #9 and #11 and cases #10 and #12.Cross section interaction curve is also shown for comparison. In generalthe interaction formula only gives conservative results and neverexceeds 20% of conservativeness on the safe side. The highest differenceis noticed for cases #3 and #9. These correspond to a beam column suchthat χLT(xc,MI ) = 1. Because web-tapered sections present higher ratiosh/b, these present low torsional rigidity and fail mostly in flexural buck-ling. On the other hand, these specific examples are mainly beam-columns such that the reduction factor is χLT = 1 and with very lowslenderness, λLT , sufficiently smaller than the plateau slenderness —i.e. even with less probability to present torsional deformations, seeFig. 5.2. If, in addition the bending moment is much higher relativelyto the axial force, flexural buckling is negligible relatively to the bending
0
50
00
50
00
50
00
50
00
0 200 400 600 800 1000 1200
NEd [kN]
y,Ed [kNm] GMNIA
6.62 S1
6.61
b
a
cd
) Failure condition eq.; elasticbuckling modes
#26 — 5 segments.
(a.1) (a.2)
(b.1) (b.2)
(c.1) (c.2)
(d.1)(d.2)
Fig. 5.7. Deformed shape of “orange” points in Fig. 5.6(b). 1 — Front view; 2 — top view.
134 L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
moment effect and, as a result, cross section capacity prevails. The inter-action factor kzy of Table B.2 of EC3-1-1whichwas developed specifical-ly for members susceptible to torsional deformations does not properlytake advantage of this behavior for the case of tapered members whichoften present a higher slenderness plateau.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
χMethod
χGMNIA
6.62
20%
a) Unrestrained beam-columns
0
0.2
0.4
0.6
0.8
1
0 0.2 0.
χMethod
χ
6.61 +20%-6%
c) Intermediate
Fig. 5.8. Results given by th
Fig. 5.3 illustrates results of Eq. (1) for the two sets of cases presentedin Table 5.2. It is also clear that results given by Eq. (1) follow sufficientlywell the numerical models.
Finally, the influence of intermediate restraints in the failure mode isnowanalyzed. The set of cases #24; #25 and#26 are shown, respectively,
χGMNIA
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
χMethod
6.61
+15%
-5%
b) Continuously restrained beam-columns
4 0.6 0.8 1GMNIA
or 6.62
restrained models
e interaction approach.
Table 5.4Statistical evaluation concerning the ratio χov
Method/χovGMNIA for the interaction formulae.
Case Methodology n Mean St. Dev. CoV (%) Min. Max. % cases b0.9 % cases N1.03
Table 5.1 6.62 273 0.93 0.053 5.66 0.78 1.07 26.7 2.2Table 5.2 6.61 140 0.93 0.049 5.22 0.85 1.03 26.4 0.0Table 5.3 6.61 or 6.62 Si 105 0.90 0.050 5.54 0.81 1.06 47 1.9
135L. Marques et al. / Journal of Constructional Steel Research 100 (2014) 122–135
in Figs. 5.4; 5.5 and 5.6, in which (a) illustrates results for all equationsand (b) illustrates only the equation that leads to the least resistance. Inaddition, (b) also illustrates the dominance of the buckling mode givenby a linear bifurcation analysis, LBA. It is seen that, in what concernsapplication of the interaction formulae, with the increase of the numberof intermediate restraints, Eq. (3.6) becomes more relevant andrestricting than Eq. (3.7) for out-of-plane buckling, even when it is no-ticed that the first bucklingmode is not the in-planemode. Nevertheless,this only shows that the formulae are also safe-sided for cases that theslenderness λLT and λLT are of the same or similar order of magnitude.
Finally, Fig. 5.7 illustrates the deformed shape of the points inorange from Fig. 5.6(b). Firstly, if one notices the first bucklingmode LBA, it is seen that, for high axial force, it is out-of-plane;with increasing bending moment relatively to axial force, in-planemode becomes higher; and finally, with high level of bendingmoment relatively to axial force, out-of-plane mode (with lateral–torsional buckling) is dominant again. The same is visible inFig. 5.7, where such defined frontiers are not as visible since bothtypes of relevant imperfections (in and out) were applied to themodels. For example, Point c and b yield some level of in-plane deforma-tion even if the level of bending moment is not high. On the other hand,lateral–torsional buckling deformations increase relatively to lateraldeformations with the increase of the bending moment.
5.4. Statistical evaluation
Regarding the interaction formulae, in general, the consideration ofthe analyzed adapted interaction approach for tapered beam-columnsleads to a resistance level between 80% and 106% of the GMNIAresistance.
In Fig. 5.8, to have a common basis, the generalized reduction factorsare compared: χov
GMNIA = αbGMNIA/αult,k and χov
interaction = αbinteraction/αult,k,
in which αb is the resistance multiplier obtained numerically or by theinteraction approach and αult,k is the cross section resistance multiplier.Fig. 5.8(a); (b) and (c) gives results, respectively, for the parametricstudy of Table 5.1 (unrestrained models); Table 5.2 (continuouslyrestrained models) and Table 5.3 (intermediate restrained models).
It should be mentioned that the General Method would lead to ascatter or results between 50% up to 120%, depending on the bucklingcurve, see [15].
Finally, statistical indicators are also shown in Table 5.4 and indicategood average and low CoV for all sets when considering the proposedformulation.
It is clear that the interaction approach leads to a very good approx-imation to the numerical results, identical results are found in [16] forthe case of prismatic members.
In summary, it can be concluded that a straightforward adaptation ofthe interaction formulae in EC3-1-1 always gives safe results, and theconservatism of this methodology was not shown to be greater than20%.
6. Conclusions
In this paper, the stability verification of tapered beam-columnswas discussed. The interaction formulae in EC3-1-1 for prismatic
members were adapted for tapered members, validated throughextensive FEM numerical simulations covering several combinationsof bending moment about strong axis, My, and axial force, N, andlevels of taper. For the time being, a parametric study of 520 beam-columns with or without intermediate lateral restraints indicatedthat the interaction approach leads to results that are mostly on thesafe side. Maximum differences of 20% relatively to the numericalresults were achieved for any of the possible failure modes of thebeam-column.
In a next step, the partial safety factor γM1will be determined [18]. Inaddition, the procedure shall be extended for bending about the minoraxis and, more importantly, for cross sections with class 3 or 4.
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