A consistent methodology for the out-of-plane buckling resistance of prismatic steel beam-columns

Journal of Constructional Steel Research 128 (2017) 839–852

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Journal of Constructional Steel Research

A consistent methodology for the out-of-plane buckling resistance of prismatic steel beam-columns Trayana Tankova a,⁎, Liliana Marques a, Anísio Andrade b, Luís Simões da Silva a a b

ISISE, Department of Civil Engineering, University of Coimbra, Coimbra, Portugal INESC Coimbra, Department of Civil Engineering, University of Coimbra, Coimbra, Portugal

a r t i c l e

i n f o

Article history: Received 7 April 2015 Accepted 12 October 2016 Available online xxxx Keywords: Beam-columns Out-of-plane buckling Design rules Steel Eurocode 3 Stability

a b s t r a c t This paper presents a design proposal for the out-of-plane buckling resistance of prismatic beam-columns subject to axial compression and uniaxial major-axis bending that was developed based on the well-known Ayrton-Perry format. Firstly, the relevant theoretical background is summarized, closely following the theoretical derivation performed by Szalai and Papp (2010). Secondly, the required transformations for the engineering application of the design procedure are detailed and extended to arbitrary bending moment distributions. Appropriate generalized initial imperfection factors for the out-of-plane buckling of beam-columns are defined so as to achieving complete consistency across the stability verifications for columns, beams and beam-columns. The proposed procedure is subsequently validated against a large set of advanced numerical simulations. A good agreement was found between the numerical results and the estimates provided by the proposed design procedure, both in terms of the overall trend and the specific quantitative results. Based on a statistical assessment, the comparison with the interaction expression of Eurocode 3 (2005) (method 2) showed that this proposal slightly outperforms the Eurocode procedure, both in terms of average values and dispersion of results. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Steel skeletal structures are often designed with individual members subject to major-axis bending and axial force (see Fig. 1). The behaviour of such members results from the combination of both action effects and varies with slenderness. At low slenderness, the load-carrying capacity is governed by cross sectional resistance. With increasing slenderness, the geometrically non-linear effects can no longer be ignored, and out-of-plane (flexural or flexural-torsional) buckling may trigger failure. For intermediate slenderness, instability usually occurs in the inelastic range of the material. In the high slenderness range, instability is essentially an elastic phenomenon. The interactions between instability and plasticity in beam-columns lead to a very complex 3D behaviour that is not easily amenable to design procedures with a consistent and transparent mechanical basis. Indeed, the resistance of beam-columns is generally checked with interaction formulae that combine the ultimate strengths of the member either as a concentrically loaded column or as a beam under uniaxial bending. Interaction formulae are typically developed either: (i) as modifications to formulae derived from an elastic analysis, with more or less empirical factors whose complexity depends on the desired accuracy and range of validity, or (ii) on a wholly

empirical basis [4]. Table 1 shows two representative examples of codified interaction formulae for beam-columns subject to axial compression and major axis bending. AISC [5] provides an interaction approach for the stability verification of beam-columns with doubly or singly symmetric cross-sections as given in Table 1. The interaction equations represent a lower bound of the resistance [6]. The verification encompasses the beam and column verifications as extreme cases and thus accounting for the limit states of yielding, flexural and/or torsional buckling, flange local buckling, and web local buckling. However, the approach has been reported to be over-conservative for members loaded with major axis bending moment and compression, which are prone to out-of-plane failure [4, 6]. Section H1.3 from AISC [5] gives an alternative equation for the verification of doubly symmetric rolled compact members subject to single axis bending and compression (AISC Commentary [6]). Focusing on the Eurocode 3 [2] implementation, the interaction factors are established on the basis of the concept of equivalent moment and the amplification of the bending effects as a function of the normalized level of applied axial force, including extensive calibration for proper account of the plasticity effects [7]. However, from the point of view of mechanical consistency and transparency, the resulting interaction formulae are hardly satisfactory, since:

⁎ Corresponding author at: Department of Civil Engineering, University of Coimbra, Polo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. E-mail address: [email protected] (T. Tankova).

▪ as a two-step procedure that depends on the buckling resistances of the member in bending only and in compression only, they require successive statistical calibrations: first, an independent calibration

http://dx.doi.org/10.1016/j.jcsr.2016.10.009 0143-974X/© 2016 Elsevier Ltd. All rights reserved.

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T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

NOTATIONS Latin upper case letters A cross-sectional area factor accounting for non-uniform bending moment C1 distributions in the elastic critical moment lateral-torsional buckling modification factor Cb factor accounting for non-uniform bending moment Cbc distributions in the elastic critical moment including compression effect; E modulus of elasticity G shear modulus St. Venant torsional constant It moment of inertia y-axis Iy warping constant Iw moment of inertia z-axis Iz L length elastic critical bending moment Mcr elastic critical bendinηg moment including the effect of Mcr,N compression force elastic critical bending moment for non-uniform bendMcr,nu ing moment distribution Mcr,N,nu elastic critical bending moment including the effect of compression force for non-uniform bending moment distribution factored lateral-torsional buckling strength Mcx maximum bending moment design value Mr. maximum bending moment design value My,Ed major axis bending moment resistance My,Rd. major axis bending moment My N compressive force elastic critical force associated with pure torsional Ncr,x buckling elastic critical force associated with pure flexural buckNcr,z ling about minor axis maximum axial design values acting on the member NEd factored buckling strength Nb,Rd. factored buckling strength Pco maximum axial design values acting on the member Pr elastic section modulus relative to y-axis Wy warping modulus Ww elastic section modulus relative to z-axis Wz

θ(x) ^θ θ0(x) ^θ0 λBC λLT λz φ χBC χLT χy χz ψ

twist rotation amplitude of twist rotation initial twist rotation amplitude of initial twist rotation normalized slenderness for beam-columns normalized slenderness for lateral-torsional buckling normalized slenderness for minor axis flexural buckling over strength factor reduction factor for flexural-torsional buckling of beamcolumns lateral-torsional buckling reduction factor major axis buckling reduction factor minor axis buckling reduction factor end moment ratio

of the imperfection factors for columns and beams and then a calibration of the interaction factors; and ▪ for class 1 and class 2 cross-sections (plastic interaction), the proposed expressions for the interaction factors (both for method 1 and for method 2) have no physical meaning.

From a practical point of view, the downside to the wide range of cases covered by the EC3-1-1 [2] interaction expressions resulted in long procedures for the determination of the interaction factors, which are especially burdensome when used for preliminary sizing of the members. The EC3-1-1 [2] design rules for columns and beams are based on the buckling curve approach. For columns, the design procedure is established on the solution of the differential equation of a pin-ended compressed member with an initial sinusoidal imperfection for the limiting condition of first yield at the critical cross-section (mid-span), cast

Fig. 1. Steel members subjected to bending and axial force [3].

Latin lower case letters e0 equivalent initial geometrical imperfection interaction factor kyy interaction factor kzy yield stress fy polar radius of gyration r0 v(x) transverse displacement along y axis initial transverse displacement v0 (x) ^0 amplitude of initial imperfection v ^ v amplitude of transverse displacement along y axis w(x) transverse displacement along z axis initial transverse displacement w0 (x) Greek lower case letters compression factor βN generalized initial imperfection factor for flexural-torηBC sional buckling of beam-columns generalized initial imperfection factor for lateral-torηLT sional buckling generalized initial imperfection factor for minor axis ηz flexural buckling

Table 1 Interaction formulae from representative design codes.
 Mrx þ 89 M ≤1:0 for PPcr ≥ 0:2 cx
 Pr Mrx Pr 2P c þ Mcx ≤1:0 for P c b0:2

2 Pr Pr rx þ CM ≤1:0 Pcy 1:5−0:5 Pcy bM

AISC (2010) [5]

Pr Pc

Eurocode 3-1-1 [2]

NEd χy NRd

þ kyy χ

My;Ed

NEd χz NRd

þ kzy χ

My;Ed

cx

LT My;Rd

LT My;Rd

þ ≤1:0 þ ≤1:0

Pr, NEd - the maximum axial design values acting on the member. Mr, My,Ed - the maximum bending moment design values acting on the member. Pc, Pcy - factored buckling strengths in compression. NRd – compression resistance. Mcx - factored lateral-torsional buckling strength. My,Rd. – major-axis bending moment resistance. χy, χz – flexural buckling reduction factors for major and minor axis. χLT – lateral-torsional buckling reduction factor. kyy kzy are interaction factors. Cb – lateral-torsional buckling modification factor. ⁎Alternative verification for doubly symmetric rolled compact members subject to single axis flexure and compression. ⁎⁎The terms required only to account for the shift of the centroidal axis in class 4 cross-sections have been omitted.

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

in an Ayrton-Perry format [8]. Initial geometrical and material (residual stresses) imperfections, as well as the potential plastic behaviour at ultimate load are accounted for using the concept of generalized imperfections factors, extensively calibrated and validated with experimental and numerical work [9,10,11]. For the lateral-torsional buckling of beams, the current EC3-1-1 [2] rules are given also in an Ayrton-Perry format, although they are not based on a strict mechanical background, but rather on calibration against experimental, numerical and statistical work [12]. However, recent research by Taras (2010) [13,14] resulted in a mechanically consistent Ayrton-Perry design approach for the lateraltorsional buckling of beams, also based on the solution of the differential equation of a simply-supported beam with fork supports with initial sinusoidal twist and out-of-plane bow imperfections, based on an amplification relationship that relates to the buckling mode, for the limiting condition of first yield at the critical cross-section (assessed using an “overstrength” factor), with generalized imperfection factors calibrated on the basis of an extensive numerical programme. For both columns and beams, the generalized imperfection factors reflect the most relevant parameters that affect the ultimate resistance, such as the length of the member, the cross-section properties, the buckling mode, the normalized slenderness and the fabrication procedure. In theory, beams and columns may be regarded as limiting cases of beam-columns as one of the internal forces (axial force or bending moment) approaches zero. Thus, it would be useful if a generalized AyrtonPerry formulation could be developed for beam-columns that would ensure a consistent transition between beam and column verifications. Szalai and Papp [1] have precisely developed the theoretical background for generalizing Ayrton-Perry type resistance formulas to the out-of-plane buckling behaviour of beam-columns. However, quoting directly their paper, they did “not undertake to develop formulae appropriate for design purposes in practice, as comprehensive work on deterministic and probabilistic calibration would be needed”, nor did they consider “the additional effect of partial plasticity and residual stresses – which would be needed for complete application purposes”. In this paper, an Ayrton-Perry type design procedure is proposed for the out-of-plane buckling behaviour of beam-columns subject to axial compression and uniaxial major axis bending that comprises a continuous transition between the beam and column verifications. The proposal is closely based on the theoretical derivation performed by Szalai and Papp [1]. It provides a clear mechanical background that is entirely consistent with the EC3-1-1 [2] procedures for the buckling resistance of columns and beams. The new design procedure is validated against the results of advanced numerical simulations. A statistical evaluation shows a slight improvement in accuracy when compared to the EC31-1 [2] interaction expressions. Finally, for illustrative purposes, a fully worked example is also included.

EC3-1-1 [2] convention – x denotes the centroidal longitudinal axis, while y and z correspond to the major and minor principal axes of the cross-section (see Fig. 2); • The effects of pre-buckling deflections – namely: (i) the amplification, due to the axial compressive force, of the first-order bending moments and deflections and (ii) the effect of in-plane curvature on the out-ofplane buckling resistance, are neglected; and • The unloaded state is a natural state (i.e., there are no residual stresses).

2.2. Basic equations The differential equations governing the elastic behaviour of an imperfect beam-column have the following general form: 2

EI z

d vðxÞ þ My θðxÞ þ NvðxÞ ¼ −My θ0 ðxÞ−Nv0 ðxÞ dx2

EI y

d wðxÞ þ NwðxÞ þ My ¼ −Nw0 ðxÞ dx2

ð1Þ

2

ð2Þ

3

d θðxÞ dθðxÞ dvðxÞ dθðxÞ −GIt þ My þ r 20 N dx dx dx dx3 dv0 ðxÞ dθ0 ðxÞ −r 20 N : ¼ −My dx dx

EI w

ð3Þ

In these equations, v(x) and w(x) are the transverse displacements (along y and z, respectively), θ(x) is the twist rotation (positive according to the right-hand rule) and v0(x), w0(x), θ0(x) represent the initial deviation from a perfectly straight member, as shown in Fig. 2. Moreover, EIy is the in-plane bending stiffness, EIz is the out-of-plane bending stiffness, EIw is the warping stiffness, GIt is the Saint-Venant torsional stiffness and r0 is the polar radius of gyration. The first and second equations describe the out-of-plane and inplane flexural behaviours, respectively, while the third equation represents the warping-torsion response. The stress resultants are associated with v(x), w(x) and θ(x) only and do not depend on the geometrical imperfections. 2.2.1. Flexural buckling of columns In view of its fundamental character as a guideline for subsequent developments, the out-of-plane behaviour of a column (My = 0) with an initial imperfection (see Fig. 3) ^0 sin v0 ðxÞ ¼ v

πx L

2. Analytical background The proposed design procedure is based on a second-order elastic beam theory. This section summarises the relevant results from the derivation presented in [1] for prismatic beams and beam-columns. 2.1. General assumptions and conventions The following assumptions and conventions are adopted: • The members are prismatic and simply supported, with “fork” supports; • The cross-sections are doubly symmetric I or H sections; • The material is linearly elastic until reaching a definite yield stress fy; • The applied loads consist of a centrically applied compressive force N and/or equal and opposite major axis end moments My; • The usual approximations for small displacements and rotations apply; • A fixed right-handed Cartesian reference system is associated with the idealised perfect configuration of the member, in accordance with the

841

Fig. 2. Coordinate system and cross-sectional displacements.

ð4Þ

842

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

and enforcing a first yield limit condition eventually leads to the AyrtonPerry equation [1]: χ 2LT

þ χ LT −1−

Fig. 3. Compressed column with initial imperfections.

is briefly addressed here. In these circumstances, the differential Eq. (1), together with the boundary conditions v(0) = v(L)= 0, yields vðxÞ ¼

where Ncr;z ¼

EIz π2 L2

ð5Þ

is the elastic critical load. The following amplification

relationship can thus be established: ^tot ¼ ^ vþ^ v0 ¼ v

1 ^v0 : 1−N=Ncr;z

ð6Þ

Applying a first yield criterion at mid-span (i.e., equating the normal stress at the most compressed fibre to fy) yields the well-known AyrtonPerry equation:

λ2LT

−

1 λ2LT

! ηLT

þ

1 λ2LT

¼ 0;

ð13Þ

where Wy, Wz and Ww denote elastic section moduli, χ LT

N πx πx ^0 sin ¼ v ^ sin ; v N cr;z −N L L

1

My ¼ ; λLT ¼ f yWy

sffiffiffiffiffiffiffiffiffiffiffiffi f yWy W W GI W y ^0 y þ ^θ0 y −^θ0 t ; ηLT ¼ v Mcr Ww Wz Mcr W w

ð14Þ

A similar transformation was also performed by [13,14] and further calibrated for design application, with the generalized initial imperfection factor given by: ηLT ¼

Ae0 λ2LT : W z λ2z

ð15Þ

The meaning of the geometrical parameter e0 appearing in the above definition of ηLT is indicated in Fig. 5.

factor, with Wz denoting the elastic section modulus relative to z-axis.

2.2.3. Flexural-torsional buckling of beam-columns Szalai and Papp [1] followed an identical methodology for the derivation of an Ayrton-Perry equation for the out-of-plane flexural-torsional buckling of beam-columns. Their derivation proceeds as follows. Firstly, they consider a perfect beam-column under uniform major-axis bending moment and compression (Fig. 6), for which the first buckling mode shapes are again given by Eq. (8), with the amplitudes now related according to

2.2.2. Lateral-torsional buckling of beams Considering the standard case of a simply-supported beam with fork

^ v ¼ r0 θ^

χ 2z þ χ z −1−

where λz ¼

1 λ2z

qffiffiffiffiffiffiffi Afy Ncr;z

−

1 λ2z

! ηz

þ

1 λ2z

¼ 0;

ð7Þ

is the normalized slenderness, χ z ¼ ANf is the buckling y

reduction factor and ηz ¼ ^v0 WAz is the generalized initial imperfection

2

πx L

θðxÞ ¼ ^θ sin

πx ; L

ð8Þ

where Mcr;N ¼ r 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi N cr;x −N Ncr;z −N

sffiffiffiffiffiffiffiffiffiffi Ncr;x : Ncr;z

ð9Þ

In Eq. (9), Mcr is the elastic critical bending moment and Ncr, xis the pure torsional buckling force: Mcr ¼ r0 Ncr;x ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ncr;x Ncr;z

ð10Þ

  1 EIw π 2 þ GI t : r 20 L2

ð11Þ

Assuming now that the initial imperfections have the same shape as the first buckling mode, that is, ^0 sin v0 ðxÞ ¼ v

πx L

πx ^v0 Mcr ¼ θ0 ðxÞ ¼ ^θ0 sin L ^θ0 N cr;z

sffiffiffiffiffiffiffiffiffiffiffiffi Ncr;x ¼ r0 ; N cr;z

Fig. 4. Simply supported beam subject to uniform bending.

ð17Þ

is the critical buckling moment in the presence of an axial compressive force N.

are related by ^ Mcr v ¼ r0 ¼ ^ θ Ncr;z

ð16Þ

2

d θ d θ supports (v(0) = v(L) = 0, θ(0) = θ(L) = 0 and dx 2 ð0Þ ¼ dx2 ðLÞ ¼ 0) subject to a uniform bending moment (Fig. 4), it can be verified [15] that the amplitudes of the first buckling mode shapes of the perfect system,

^ sin vðxÞ ¼ v

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ncr;x −N Mcr;N 1 ¼ ; Ncr;z 1−N=Ncr;x Ncr;z −N

Fig. 5. Amplitudes of the geometrical imperfections.

ð12Þ

Fig. 6. Simply-supported beam-column in uniform bending and compression.

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

Secondly, they address the imperfect case (i.e., v0 ≢ 0 and θ0 ≢ 0) and show that it is not possible to consider a loading history with a varying compressive force and still derive a conventional (quadratic) AyrtonPerry equation. To overcome this difficulty, instead of assuming proportional loading, the authors considered the axial compression force as “fixed”, while the bending moment is incremented up to failure. This assumption is usually considered when checking the cross-section resistance under bending and axial force [3]. Then, the first buckling mode shapes (8) and (16) are used for the initial lateral and twist imperfections, leading to the amplification relationship 3 ^0 v 6 1−N=N 7 1 6 cr;z 7 ¼ 7: 6 ^θ0 5 1−M y =Mcr;N 4 1−N=N cr;x 2



^tot v ^θtot



ð18Þ

Further imposing the first yield criterion at mid-span, it is possible to derive the Ayrton-Perry equation for beam-columns: χ 2BC þ χ BC −βN −

1 λ2BC

−

1 λ2BC

! ηBC

þ

βN λ2BC

¼ 0;

ð19Þ

χ BC ¼

λBC

ηBC

A major task is the definition of the magnitudes of the initial imperfections [14]. For beams and columns, the buckling curves incorporate the magnitude of the imperfections – initial out-of-straightness and twist, as well as residual stresses. In order to keep consistency with the rules for beams and columns, the new formulation makes use of the imperfection factors previously calibrated for these two types of members. 3.2. Uniform bending moment The derivation for the out-of-plane behaviour of beam-columns in [1] was performed assuming a simply supported member with doubly symmetric cross-section, subject to uniform major axis bending moment (My) and uniform compressive force (N). Consequently, in this section, the formulation of the new proposal is developed under the same conditions. 3.2.1. General format The Ayrton-Perry equation for beam-columns derived analytically by Szalai and Papp [1] – Eq. (19) – may be written in the alternative format χ BC ¼

where My f yWy

ð20Þ

sffiffiffiffiffiffiffiffiffiffiffiffi Wy f y ¼ Mcr;N

ð21Þ

Wy ^ Wy ^ 1 1 GIt W y ^0 ¼v þ θ0 −θ0 N Ww N Wz N Mcr;N W w 1− 1− 1− N cr;z Ncr;x Ncr;x 1

ð22Þ

843

My ¼ f yWy

ϕBC

βN ffi; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ϕ2BC −βN λ2BC

ð24Þ

with
 ϕBC ¼ 0:5 1 þ ηBC þ βN λ2BC :

ð25Þ

In Eq. (25), the factor ηBC (as given in Eq. (22)) accounts for the initial imperfect system configuration. The level of compression of the member is incorporated through the coefficient βN defined by Eq. (23). ^0 and θ^0, which are relatBoth parameters are expressed as functions of v ed by the equation

^0 N Nv − βN ¼ 1− Af y Wz f y 0 þ

GIt ^ θ0 B B Ww f y @

r2 N^θ0 1 − 0 N N Ww f y 1− 1− Ncr;z Ncr;x 1

1 N 1− N cr;x

1

C −1C A:

^0 Mcr;N v ¼ Ncr;z θ^0 ð23Þ

1 1−

N Ncr;x

ð26Þ

but are otherwise indeterminate. Moreover, member imperfections are currently considered in EC3-1-1 [2] via a modified initial local bow imperfection e0 , which also accounts for the presence of residual stresses. For doubly symmetric cross-sections, e0 is defined by (Fig. 5)

This definition of βN exactly coincides with the one for flexural-torsional buckling of columns given in [1]. Clearly, this factor is independent of the bending moment distribution and accounts for the utilization of the cross-section due to the axial force.

h ^0 þ ^θ0 : e0 ¼ v 2

3. Derivation of a Ayrton-Perry design formulation for beamcolumns

Since the proposed formulation should maintain the consistency with the design procedures for columns and beams in EC3–1-1 [2], Eqs. (22) and (23) should be expressed in terms of e0 .

3.1. Introduction In the preceding section it was shown that the Ayrton-Perry format of limit state analysis can be extended analytically from columns and beams to beam-columns [1]. In the following, the analytical derivation from Szalai and Papp [1] is used and further developed for design purposes. The proposed formulation is applicable to beam-columns and encompasses beams and columns as special cases. The result of the proposed verification expresses how much bending moment can be resisted for a given axial force. In particular, if the applied axial force is equal to the flexural buckling resistance of the member, it will not be able to resist any additional bending moment; if the axial force is equal to zero, the resistance of the member will be equal to the lateral-torsional buckling one.

ð27Þ

3.2.2. The factor ηBC Starting from Eq. (22) and bearing in mind Eqs. (26)-(27), a few straightforward transformations lead to    W z 1  2 1þ r 0 Ncr;x −N −GIt ¼ N W w M cr;N 1− Ncr;x  Iz h 1  2  1þ r N cr;x −N −GIt 2 e0 A λLT M cr Iw 2 Mcr;N 0   ¼ Mcr;N h N W z λ2 Ncr;z z þ 1− Ncr;z 2 Ncr;x

ηBC ¼ ^θ0

Wy Wz

1

ð28Þ

3.2.3. The compression factor βN In [1] it was shown that the factor βN accounts for the behaviour of a column with lateral and twist initial imperfections. Hence, it represents

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T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

the part of the cross-section that is utilized to resist the compression force, varying between zero (when the cross-section is fully utilized in compression) and unity (when the axial force vanishes). Substituting ^0 and ^θ0 into Eq. (27) and inserting the the relation (26) between v resulting expression into Eq. (23) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 Ncr;z Ncr;x −N h N cr;z C B þ r0 C Ncr;x Ncr;z −N 2 Ncr;x N B e0 A C B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ! 1 þ βN ¼ 1− C: B   C Af y B Wz N N −N h N cr;z cr;z A @ r0 þ 1− Ncr;x N cr;x −N 2 Ncr;x 0

Therefore, the final definitions for ηBC and βN are

ηBC

f int þ

λ2 Mcr ¼ ηmod LT λ2z Ncr;z

 Iz h 1  2  r Ncr;x −N −GIt f int Iw 2 Mcr;N 0   h N Mcr;N f þ 1− 2 Ncr;x int Ncr;z

ð34Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 Ncr;x −N h Ncr;z C B þ r0 f C Ncr;z −N 2 Ncr;x int N B C B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ! βN ¼ 1− C: ð35Þ B1 þ ηmod   C Af y B N N −N h N cr;z cr;z @ r0 þ f int A 1− Ncr;x Ncr;x −N 2 Ncr;x 0

ð29Þ

Ncr;z Ncr;x

3.3. Definition of the equivalent imperfection 3.4. Beams and columns as limiting cases Eqs. (28) and (29) express both factors ηBC and βN as a function of the equivalent initial geometrical imperfection e0 . Both equations feature the factor ηmod ¼

e0 A Wz

, which is called the generalized initial

imperfection. In the case of flexural buckling of columns (for which e0 ¼ ^v0 ), EC31-1 [2] defines this generalized initial imperfection as ηz ¼

e0 A ¼ α z ðλz −0:2Þ; Wz

ð30Þ

where αz depends on the cross-section type and yield stress. In the case of lateral-torsional buckling of beams with doubly symmetric cross-section (for which the full Eq. (27) applies), Taras and Greiner [13] provide

ηLT ¼

e0 A λ2LT λ2 λ2 ¼ α LT ðλz −0:2Þ LT ¼ ηLT LT : 2 2 W z λz λz λ2z

ð31Þ

ηmod =ηLT⁎ when NNb;Rd ¼ 0: ηmod

Table 2 Axial force equals zero. Let N = 0. Then,
 0 f int ¼ 1− Nb;Rd ¼1
 0 0 ¼ ηLT þ ηz Nb;Rd ηmod ¼ ηLT 1− Nb;Rd Mcr, N = Mcr and Eq. (34) reduces to

Since these are limiting cases of beam-columns, the generalized initial imperfection ηmod should range between ηzand ηLT⁎. The simplest way to account for the variation in the magnitude of ηmod is to write it as an affine function of NNb;Rd (Fig. 7), with ηmod = ηz when NNb;Rd ¼ 1 and

  N N þ ηz ¼ ηLT 1− : Nb;Rd Nb;Rd

The formulation is verified for the limiting cases of beams and columns in order to ensure its consistency. Two extreme cases are considered: i) zero axial force and ii) axial force equal to the flexural buckling resistance of the member. For ηBC, only the case of zero axial force is relevant. The other extreme case is covered by the factor βN. When the axial force is equal to the flexural buckling resistance, the factor βNbecomes zero and therefore χBCis also equal to zero (see Eq. (24)). Tables 2 and 3 illustrate these two extreme cases.

¼ ηmod

ηBC

¼ ηLT

Mcr Ncr;z

N cr;z M cr






M cr h N cr;z þ2

¼ ηLT

h Mcr þ 2 N cr;z

2 λLT 2 λz

¼

¼ ηLT ;

as required. The utilization factor βN becomes unity:
qffiffiffiffiffiffiffiffiffiffi β N ¼ 1− A0f

ð32Þ

2 λLT 2 λz

2

Iz h 1 2 λLT Mcr 1þIw 2Mcr ½r0 N cr;x −GI t  2 N h M cr λz cr;z 2þNcr;z

1 þ ηmod


y

N cr;z N cr;x

r0

N cr;x −0 h N cr;z þ N −0 2 N



cr;x cr;z 
q ffiffiffiffiffiffiffiffiffiffi

0 1−Ncr;x

N cr;z −0 h N cr;z þ Ncr;x −0 2 N cr;x

r0

! 

¼ 1:

The normalized slenderness is found to be qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi Wy f y Wy f y Mcr;N ¼ Mcr ¼ λLT :

λBC ¼

Finally, substituting the above results into Eq. (24) and Eq. (25)), one obtains

In order to achieve a seamless transition between beam and column behaviours (i.e., to ensure that χBC → 0 when N → Nb . Rd and χBC → χLT when N → 0), an additional interpolation factor is required in Eqs. (28)-(29). Indeed, all the terms in these equations stemming from ^θ0

2

2

ϕBC ¼ 0:5½1 þ ηBC þ βN λBC  ¼ 0:5½1 þ ηLT þ 1λLT  ¼ ϕLT χ BC ¼

βN

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi¼ 2

ϕBC þ

ϕ2BC −β N λBC

ϕLT þ

p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

ϕ2LT −λLT

¼ χ LT :

Therefore, when the axial force vanishes, the resistance obtained with the new beam-column proposal is exactly equal to the lateral-torsional resistance of a beam.

are multiplied by f int ¼

  N : 1− N b;Rd

ð33Þ

so that they vanish when N reaches the design buckling resistance Nb,Rd ^0 in the case of flexural buckling of columns). (as remarked above, e0 ¼ v

Table 3 The axial force equals the flexural buckling resistance. Let N = Nb,Rd. Then, N

¼0 f int ¼ 1− Nb;Rd b;Rd and the factor ηmodbecomes
 N N þ ηz Nb;Rd ¼ ηz . ηmod ¼ ηLT 1− Nb;Rd b;Rd b;Rd It follows that βN

¼

1− ANf y




1 þ ηmod





¼ 1− χ z þ

Fig. 7. Linear interpolation for ηmod.

Ncr;z N cr;x


qffiffiffiffiffiffiffiffiffiffiffi r0

N 1−Ncr;x

χz ηz 1−χ z λz

N cr;x −N N −N

cr;z 
q ffiffiffiffiffiffiffiffiffiffiffi



r0

N cr;z −N N cr;x −N



¼

¼ 1−1 ¼ 0

and, consequently, χBC = 0, which means that the member is unable to resist any bending moment.

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

Fig. 8. Bending moment distribution.

845

Fig. 9. Beam-column, IPE 180.

3.5. Generalization to other bending moment distributions Table 5 Input parameters.

The constant bending moment is the standard case for which the derivation in [1] was performed. However, a uniform bending moment distribution is seldom found in real design practice; therefore, the new proposal should be able to cope with non-uniform bending moment distributions. The normalized slenderness λBC is influenced by the critical moment including the compression effect. In [14] the proposed design verification for LTB of beams is calibrated for non-uniform bending moment cases. The verification is given in the following format: χ LT ¼

ϕLT

φ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ϕ2LT −φλ2LT

" ϕLT ¼ 0:5 1 þ φ

λ2LT λ2z

Geometry

Geometrical properties

Material Moduli properties Yield stress Safety factor Critical forces Critical moments

ð36Þ

α LT ðλz −0:2Þ þ

:

ð37Þ

3.7.1. Member geometry and loading The beam of Fig. 9 is considered. It is subject to constant axial force NEd and one end bending moment My,Ed. The magnitudes of the acting forces are specified in Fig. 9. The cross-section of the beam-column is a IPE180 in S235, all input data being shown in Table 5. Buckling curves definition →h/bN 1.2:

The factor φ may be regarded as an “overstrength” ratio, accounting for the favourable effect of non-uniform bending moment distributions. For a linearly varying bending moment diagram (Fig. 8), φ is given by φ ¼ 1:25−0:1ψ−0:15ψ2 :

ð38Þ

• Column → αz =0.34 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 qffiffiffiffiffiffiffiffi W ¼ 0:12 1:4610 • Beam → α LT ¼ 0:12 W y;el ¼ 0:31≤0:34 z;el 2:22104

For a uniformly distributed transverse load, φ = 1.05. For a transverse point load at mid-span, φ = 1.11. Since the application of the new proposal ultimately results in a reduction factor for the bending resistance (for a given axial force), it makes sense to use the “overstrength” factor φ directly in the verification formula, which is therefore rewritten as χ BC ¼

ϕBC

b = 91 mm, h = 180 mm, tf = 8 mm, tw = 5.3 mm, r=9 mm L=2.0 m A = 2390 mm2 , Iy =1.32 ×107 mm4, Wy,el = 1.46 ×105 mm3, Wy,pl = 1.66 ×105mm3, Iz = 1.01 ×106 mm4, Wz,el = 2.22 ×104 mm3, Wz, pl = 3.46 ×104 mm3, It = 4.79 ×104 mm4, Iw = 7.43 × 109 mm6 , r0 = 77 mm E = 210 GPa, G = 80.8 GPa fy = 235 MPa γM1 = 1.0 Ncr, z = 523.3 kN, Ncr, x = 1300.9 kN Mcr = 112.5 kN m, Mcr, N = 87.52 kN m

3.7. Illustrative example

!# λ2LT

Dimensions

φβN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; þ ϕ2BC −φβN λ2BC

3.7.2. Cross-section class The cross-section class is verified according to Table 5.2 from EC3-11 [2]: • Web in compression

ð39Þ

h
i ϕBC ¼ 0:5 1 þ φ ηBC þ βN λ2BC :

c h−2  t f −2  r 180−2  8−2  9 ¼ ¼ ¼ 27:5b33 t 5:3 tw

ð40Þ

3.6. Summary of the design formulation → Web in Class 1. • Out-stand part of flange in compression

Table 4 summarises the design formulation. Table 4 Summary of the design formulation. Step 1: Calculate Step 2: Verify Step 3: Obtain Step 4: Calculate

λz , ϕz and χz NEd ≤ Nb, Rd Mcr, Mcr ,N and Ncr, x qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi Wy f y Wy f y λLT ¼ Mcr and λBC ¼ Mcr;N

Step 5: Obtain

N N N ηmod ¼ ηLT ð1− Nb;Rd Þ þ ηz Nb;Rd , where ηz ¼ α z ðλz −0:2Þ and ηLT ¼ α LT ðλz −0:2Þ; f int ¼ 1− Nb;Rd and the “overstrength” ratio φ

Step 6: Calculate



ηBC ¼ ηmod

2 λLT 2 λz

f þ Iz h 1 ½r2 ðN −NÞ−GI t Mcr int Iw 2Mcr;N 0 cr;x Mcr;N N cr;z h N 2ð1−N cr;x Þ f int þ N cr;z

βN ¼ 1− ANf ð1 þ ηmod y

Step 7: Calculate Step 8: Verify

N cr;z N cr;x ðr 0

N cr;x −N hN cr;z Ncr;z −N þ2N cr;x f int Þ

qffiffiffiffiffiffiffiffiffiffiffi

N cr;z −N hN cr;z N cr;x −N þ2Ncr;x f int Þ

ϕBC ¼ 0:5½1 þ φðηBC þ β N λBC Þ and χ BC ¼ MEd ≤χBCfyWy

and

qffiffiffiffiffiffiffiffiffiffiffi

N Þðr ð1−Ncr;x 0

2

f int 

Þ φβN pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

ϕBC þ

ϕ2BC −φβN λBC

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T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

interaction formula in Eurocode 3 [2] for out-of-plane buckling with LTB (method 2). In this section, firstly, the sources and scope of the numerical studies are summarized; then the assessment methodology is explained and finally the results are compared and discussed.

Table 6 Design verification. λz ¼

qffiffiffiffiffiffiffi Afy Ncr;z

¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2390mm2 235N=mm2 523:3kN

¼ 1:04

ηz ¼ α z ðλz −0:2Þ ¼ 0:34ð1:04−0:2Þ ¼ 0:28 2

ϕz ¼ 0:5ð1 þ ηz þ λz Þ ¼ 0:5ð1 þ 0:28 þ 1:042 Þ ¼ 1:18 1 p1ffiffiffiffiffiffiffiffiffiffiffi2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ 0:57

χz ¼

ϕ2z −λz

ϕz þ

4.2. Sources of numerical results

1:182 −1:042

1:18þ

The flexural buckling resistance becomes: Nb,Rd = χzAfy = 0.57 × 2390mm2 × 235N/mm2 = 322.6kN ≥ NEd = 200kN Calculate the normalized slenderness for a beam: ffi qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wy f y 1:66105 mm3 235N=mm2 ¼ 0:59 λLT ¼ Mcr ¼ 112:5kNm q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffi Wy f y 1:66105 mm3 235N=mm2 ¼ 0:67 λBC ¼ Mcr;N ¼ 87:52kNm

The main source of numerical results is a parametric study performed at the University of Coimbra in the scope of a MSc thesis by Anwar [16]. Some results by Ofner [17] are used for initial validation of the proposed design method, as well as for validation of the results by Anwar [16]. As shown in Fig. 10, the comparison is performed for two cross-sections IPE200 and HEB300, considering three normalized slenderness ratios λz = 0.5; 1.0 and 1.5. Although the number of simulations is not the same, it is clear that the numerical estimations by Anwar [16] closely follow the results by Ofner [17], allowing to consider that the numerical model provides results with satisfactory accuracy. The main parameters included in the parametric study [16] are summarized in Table 7. In order to cover the use of different buckling curves for beams and columns, various flange thicknesses tfand h/b ratios were considered within both h/b groups (Nor ≤1.2), thus allowing a thorough assessment of the proposed interpolation of Eq. (32). Besides, various levels of major axis bending moment My and axial force N were considered, which were defined on the basis of the parameter Φ (see Eq. (41)).

ηLT ¼ α LT ðλz −0:2Þ ¼ 0:31ð1:04−0:2Þ ¼ 0:26 200kN Ed f int ¼ 1− NNb;Rd ¼ 1− 322:6kN ¼ 0:38 N N ηmod ¼ ηLT ð1− Nb;Rd Þ þ ηz Nb;Rd ¼ 0:26  0:38 þ 0:28  0:62 ¼ 0:27

The factor φ = 1.25, because the bending moment ratio ψ = 0. ηBC ¼ ηmod

Iz h 1 2 2 λLT Mcr f int þIw 2Mcr;N ½r0 ðNcr;x −NEd Þ−GI t f int  M cr;N 2 N N Ed h λz cr;z ð1− N cr;x Þ f int þ N 2

¼ 0:11

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi cr;z

βN ¼ 1− ANfEd ð1 þ ηmod y

N cr;z N cr;x ðr 0 N

N cr;x −N Ed hN cr;z N cr;z −N þ2N cr;x f int Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ed Þðr ð1−Ncr;x 0

Þ ¼ 0:51

N cr;z −N hN cr;z N cr;x −N Ed þ2N cr;x f int Þ

2

ϕBC ¼ 0:5½1 þ φðηBC þ β N λBC Þ ¼ 0:5½1 þ 1:25ð0:11 þ 0:51  0:672 Þ ¼ 0:71 χ BC ¼

ϕBC þ

φβ N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 ϕ2BC −φβ N λBC

0:71þ

1:250:51 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:55 0:712 −1:250:510:672

Final verification: MyN, b, Rd = χBCWy, plfy = 21.3kNm ≥ My, Ed = 10kNm →Satisfactory.

M

α pl y

M pl;y;Rd My;Ed

N pl;Rd : NEd

c 0:5ðb−t w Þ−r 0:5ð91−5:3Þ−9 ¼ ¼ ¼ 4:23b9 t tf 8

Φ¼

→ Flanges in Class 1.

All sections considered are compact, and their plastic resistance is used in the verifications. In total, 4419 cases were considered.

The cross-section is Class 1 in compression and therefore it is Class 1 in combined bending in compression.

α Npl

M

; with α pl y ¼

ð41Þ

4.3. Input parameters 4.3.1. Material properties The material properties are adopted in accordance with the recommendations of EN 10025 (2004) [18], taking into account the reduction of the yield stress fywith increasing flange thickness tf.

3.7.3. Stability verification Table 6 summarises the various steps of the design procedure. 4. Validation and statistical assessment of the new formulation 4.1. Introduction The proposed method is validated in this section against advanced numerical simulations (GMNIA) and a comparative statistical assessment is performed that compares the proposed methodology and the

4.3.2. Imperfection factors In the application of the new formulation, the imperfection factors for the flexural buckling of columns αz are used from clause 6.3.1 in Eurocode 3 [2]; for the lateral-torsional buckling of beams the recently calibrated αLT from [13,14] are adopted.

HEB 300

IPE 200 1.0

1.0 0.9 0.8

Anwar z=0.5

0.9

Anwar z=0.5

Ofner z=0.5

0.8

Ofner z=0.5

Anwar z=1.0

0.7 0.6

Anwar z=1.5

0.5

Ofner z=1.5

0.4

0.2

0.1

0.1

0.4

0.6

0.8

1.0

Ofner z=1.5

0.4

0.3

0.2

Anwar z=1.5

0.5

0.2 0.0

Ofner z=1.0

0.6

0.3

0.0

Anwar z=1.0

0.7

Ofner z=1.0

M/Mpl

M/Mpl

α Npl ¼

0.0 0.0

0.2

N/Npl Fig. 10. Validation of the numerical model by Anwar [16].

0.4

0.6

N/Npl

0.8

1.0

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852 Table 7 Scope of the parametric study. Steel Bending moment

λz

Φ

Limit h/b

tf

Section

S235

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

∞ 5.67 2.74 1.73 1.19 0.83 0.57 0.36 0.17

h/b N 1.2

5.7 7.4 8.0 12.7 19 28 46 31 46 14 19 43.7 29 21.5

IPE100 IPE160 IPE180 IPE360 HE400A HE500B HE600x337 HE650B HE650x342 HE300A HE300B HD400x347 HE300C HE340B

h/b ≤ 1.2

1.82 1.95 1.98 2.12 1.30 1.67 2.04 2.17 2.20 0.97 1.00 1.01 1.05 1.13

4.3.3. Critical moment The elastic critical moment Mcr and the critical moment in the presence of a compressive force Mcr,N for uniform bending moment are calculated using the analytical expressions (10) and (17) respectively. Although it is possible to easily obtain the critical moments numerically [19] for non-uniform bending moment distributions, the elastic critical moment Mcr , nuis calculated using the approximate Eq. (42) [7], while the critical moment in the presence of a compressive force Mcr,N,nuis determined using Eq. (43) from Trahair [20].

Mcr;nu ¼ C 1

π2 EI z L2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Iw L2 GIt þ ¼ C 1 Mcr I z π2 EIz

ð42Þ

Mcr;N;nu ¼ C bc Mcr;N

ð43Þ

initial geometrical imperfection e0 ; λBC can be calculated analytically; and χBC , num is obtained from the advanced numerical simulations. Therefore, Eq. (44) is solved in order to find e0;num , which is further used to calculate “numerical” estimates of ηBC and βN (henceforth referred as ηBC , num and βN , num). Furthermore, for the same level of compression N/Npl, the “theoretical” values of ηBC and βN can be calculated using the proposed interpolation in Section 3.3, thus allowing to assess the accuracy of the new formulation. 4.4.2. Theoretical and experimental resistance The difference between the theoretical and experimental resistance is used as a measure of the accuracy of the new formulation. The “theoretical” result obtained using the new formulation is the reduction factor χBC, which is calculated for a given axial force, i.e. for every ratio of the applied axial force divided by the compression resistance, a ratio of applied bending moment divided by the bending resistance is found. Likewise, the numerical results can be represented as pairs of M/Mpl and N/Npl. The following notation is adopted: n = N/Npl and m = M/Mpl. Fig. 11a shows an example of theoretical and experimental plots of m against n representing the resistance of a given beam-column. In general, a perfect match between the two curves is barely achieved. Moreover, for the same abscissa ne = nt, the corresponding ordinates me and mt may be considerably different, even though the curves are close to each other. Since deviations between the numerical and theoretical estimates always exist for both beams (bending moments) and columns (axial forces), in this section, the “resistance” is calculated as shown in Fig. 11b: re ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2e þ n2e ; r t ¼ m2t þ n2t ;

ð45Þ

where the theoretical values are found using an iterative procedure in order to obtain the same ratio Φ as defined in Eq. (41)

4.4. Methodology 4.4.1. Numerical estimates for ηBC and βN In order to understand the nature of the parameters ηBC and βN, the new formulation is applied in “reverse order”, meaning that, using Eq. (44), it is possible to “extract” the required geometrical imperfection e0;num such that the numerical estimate χBC,numis reached. χ 2BC;num

847

þ χ BC;num −βN −

1 λ2BC

−

1 λ2BC

! ηBC

þ

βN λ2BC

¼ 0:

ð44Þ

The parameters ηBC and βN, given by Eqs. (34) and (35), are both functions of the generalized initial imperfection ηmod, or the equivalent

a-

theoretical and experimental plots of m against n

5. Results and discussion 5.1.1. Comparison of ηBC and βN Firstly, a comparison is performed for the parameters ηBCand βN. The assessment is based on the numerical simulations performed by Ofner [17], considering uniform bending moment, since the analytical derivation was performed using the same assumption. Two cross-sections are chosen, IPE200 and HEB300, representing two different column buckling curves. Additionally, for each cross-section, two slenderness ratios are considered λz = 0.5 and λz = 1.5, allowing the verification of the performance in different slenderness ranges. The comparison is based on ηBC , num and βN , num evaluated

b-

Experimental and theoretical resistance

Fig. 11. Numerical vs theoretical results. a. Theoretical and experimental plots of m against n. b. Experimental and theoretical resistance.

848

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

Fig. 12. Parameters ηBCand βN for IPE200 λz = 0.5.

Fig. 13. Parameters ηBCand βN for IPE200 λz = 1.5.

Fig. 14. Parameters ηBCand βN for HEB300 λz = 0.5.

Fig. 15. Parameters ηBCand βN for HEB300 λz = 1.5.

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

using the procedure of Section 4.4.1 and the corresponding theoretical estimates ηBCand βNusing the equations provided in Section 3.6. Fig. 12 and Fig. 13 present the variation with axial force of the parameters ηBC and βN for a IPE 200 and Fig. 14 and Fig. 15 for a HEB300, for the two slenderness ratios λz = 0.5 and λz = 1.5, respectively. The “theoretical” parameter βN follows the numerical results very closely varying between 0 and 1. In contrast, this is not the case for the parameter ηBC, for λz = 0.5. In the low slenderness range, for both IPE200 and HEB300, the theoretical estimate is a safe-sided approximation, while in the medium-to-high slenderness range good agreement is observed between the numerical and theoretical values. In order to assess the influence of the variation of the generalized initial imperfection factor, the bending moment vs axial force interaction plots are presented for both λz = 0.5 and λz = 1.5 (Fig. 16 and Fig. 17). Although the theoretical estimate for ηBC reaches more than twice the numerical value for certain load ratios, the overall behaviour is not strongly influenced. This can be explained by the fact that the imperfections do not have a high influence in the low slenderness range [3]. It is also observed that the new model is consistent when applied to different buckling curves. Plotting the results for the same slenderness but different cross-sections (IPE200 or HEB300) shows similar behaviour, even though different buckling curves were used. More importantly, the new formulation strictly follows the numerical curves. 5.1.2. Comparison between the theoretical and the experimental resistances In this sub-section, the assessment follows the methodology presented in sub-section 4.4.2. The results from the parametric study

849

Table 8 Statistical parameters – proposed method. Bending moment distribution

Total

Nb1

Nb0.97

Mean value

C.o.v Min value

Max value

N

1.05

4.8%

0.953

1.192

1120 200 21

1.06

4.6%

0.953

1.217

1110 99

10

1.10

6.4%

0.953

1.301

1090 42

5

1.05

4.8%

0.948

1.181

1099 229 24

1.07

5.5%

0.948

1.301

4419 570 60

described in Section 4.2 are used. The accuracy of the proposed formulation is compared with the existing design rules. Additionally, the performance of the new formulation for non-uniform bending moment distributions is studied. The comparison is performed based on statistical parameters, obtained for the ratios re/rt, where re is the result from the numerical simulations as shown in Section 4.3.2 and rt is either the new method or the interaction formula given in clause 6.3.3 in EC3 [2]. The ratio re/rt is higher (respectively lower) than unity when the “theoretical” result is a safe-sided (respectively non-conservative) estimate. Table 8 and Table 9 list the statistical parameters (mean value, coefficient of variation, minimum and maximum values, the total number of cases per load case; the number of cases b 1 and b0.97) for the proposed

Fig. 16. Bending moment - axial force interaction IPE200.

Fig. 17. Bending moment - axial force interaction HEB300.

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T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

Table 9 Statistical parameters – interaction formula Bending moment distribution

Total

Nb1

Nb0.97

Mean value

C.o.v Min value

Max value

N

1.07

5.2%

0.953

1.224

1120 121 5

1.12

6.9%

0.953

1.328

1110 64

6

1.10

6.6%

0.953

1.330

1090 41

5

1.08

5.5%

0.953

1.231

1099 145 8

1.09

6.4%

0.953

1.330

4419 371 24

method and the interaction formula, respectively. The results are separated by load case in order to distinguish between the accuracy of the new formulation when applied to different bending moment distributions. In a similar assessment performed in [7], the cases that were lower than 0.97 were disregarded as statistical outliers. However, in this

study all such cases were retained. A difference in the total number of cases per load case can be found because some cases were disregarded due to convergence criteria. Figs. 18 to 21 show the scatter-plots obtained for each bending moment distribution for the new model and the interaction formula, respectively. It is observed that the mean value of re/rt. for the proposed model is never higher than 1.1 and so the new model is not overly conservative; the coefficients of variation are approximately 5% which shows that the proposed model is stable with a small scatter. The minimum values for all bending moment distribution are near 0.95, however the majority of the “unsafe” results lie between 0.97 and 1.0. Analysing the results of the interaction formula, similar conclusions can be drawn. The results do not exhibit high scatter; they are neither over-conservative nor very unsafe-sided. When both methods are compared, the new formulation has a lower mean value for all load cases and the corresponding c.o.v. is slightly lower. The load cases with linearly varying bending moment ψ= − 1.0 and ψ = 0.0 are the most conservative for both methods and also exhibit the highest scatter.

New Method

Interaction formula 1.0

0.8

0.8

0.6

0.6

rt

rt

1.0

0.4

0.4 New Method

Interaction formula

20% safe

0.2

0.2

10% unsafe

20% safe 10% unsafe

0.0

0.0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

re

0.6

0.8

1

re Fig. 18. Constant bending moment.

New Method

Interaction formula 1.0

0.8

0.8

0.6

0.6

rt

rt

1.0

0.4

0.4

0.2

0.2

New Method

Interaction formula 20% safe

20% safe

10% unsafe

10% unsafe

0.0

0.0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

re

re Fig. 19. Bending moment distribution ψ = 0.

0.8

1

T. Tankova et al. / Journal of Constructional Steel Research 128 (2017) 839–852

New Method

851

Interaction formula

0.8

0.8

0.6

0.6

rt

1.0

rt

1.0

0.4

0.4

0.2

New Method

Interaction formula

0.2

20% safe

20% safe

10% unsafe

10% unsafe

0.0

0.0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

re

0.6

0.8

1

re Fig. 20. Bending moment distribution ψ = −1.

6. Conclusion This paper presented a design proposal for the out-of-plane buckling resistance of prismatic beam-columns subject to axial compression and uniaxial major-axis bending that was developed based on the wellknown Ayrton-Perry format. It is therefore grounded on a “model with a clear mechanical background that defines the appropriate meaning of the model parameters” [1]. Firstly, the relevant theoretical background was summarized, closely following the theoretical derivation performed by Szalai and Papp [1]. Secondly, the required transformations for the engineering application of the design procedure were detailed and extended to arbitrary bending moment distributions. Appropriate generalized initial imperfection factors for the out-of-plane buckling of beam-columns were defined so as to achieving complete consistency across the stability verifications for columns, beams and beam-columns. The proposed procedure was subsequently validated against a large set of advanced numerical simulations. A good agreement was found between the numerical results and the estimates provided by the proposed design procedure, both in terms of the overall trend and the specific quantitative results. Based on a statistical assessment, the comparison with the interaction expression of EC3–1-1 [2] (method 2) showed that this proposal slightly outperforms the Eurocode procedure, both in terms of average values and dispersion of results.

The validation was carried out only for class 1 and class 2 cross-sections. Although the formulation is general, specific validation covering class 3 and class 4 should be implemented. Additionally, further validation studies covering boundary conditions other than “standard”, mono-symmetric cross-sections and transverse loads offset from the shear centre are forthcoming. A straightforward generalization of the proposal to web-tapered beam-columns can be envisaged, following the results obtained in [21] for columns and [22] for beams. For complete practical application it remains to derive a complementary proposal for in-plane buckling of beam-columns, including the challenging task of covering the combined effect of in-plane and lateral-torsional buckling modes, which requires a solution involving the three differential Eqs. (1) to (3). Acknowledgments The research leading to these results has received funding from: • the European Community's Research Fund for Coal and Steel (RFCS) under grant agreement SAFEBRICTILE RFS-PR-12103 – SEP no. 601596; • the Portuguese Foundation for Science and Technology (FCT) under grant agreement TAPERSTEEL PTDC/ECM-EST/1970/2012.

New Method

Interaction formula 1.0

0.8

0.8

0.6

0.6

rt

rt

1.0

0.4

0.4

0.2

0.2

New Method

Interaction formula

20% safe

20% safe

10% unsafe

10% unsafe

0.0

0.0 0

0.2

0.4

0.6

0.8

1

0

0.2

re

0.4

0.6

re Fig. 21. Bending moment distribution under distributed load.

0.8

1

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