Numerical validation of the general method in EC3-1 ...

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Author's Personal Copy Journal of Constructional Steel Research 66 (2010) 575–590

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Numerical validation of the general method in EC3-1-1 for prismatic members L. Simões da Silva ∗ , L. Marques, C. Rebelo ISISE, Civil Engineering Department, Universidade de Coimbra, Coimbra, Portugal

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Article history: Received 25 April 2009 Accepted 19 November 2009 Keywords: Steel structures Stability Eurocodes Safety factors Numerical analysis

abstract Part 1-1 of Eurocode 3 includes a general method for lateral and lateral–torsional buckling of structural components that uses a Merchant–Rankine type of empirical interaction expression to uncouple the in-plane effects and the out-of-plane effects. In this paper, the safety of this approach is assessed for prismatic members subjected to compression, bending and bending and axial force by comparison of both with advanced nonlinear numerical simulations (GMNIA) using beam and shell elements and the other Eurocode 3 approaches based on the European buckling curves. Analytical expressions are derived for columns, beams and beam–columns that show the variation of resistance with varying slenderness. A parametric study is carried out whereby cross-sectional shape, type of loading, slenderness and steel grade are varied. It is concluded that the method gives similar levels of safety as the methods prescribed in clauses 6.3.1 to 6.3.3 of Eurocode 3, part 1-1. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Part 1-1 of Eurocode 3 [1] includes a general method for lateral and lateral–torsional buckling of structural components such as: (i) single members, built-up or not, with complex support conditions or not; or (ii) plane frames or sub-frames composed of such members which are subject to compression and/or mono-axial bending in the plane, but which do not contain rotative plastic hinges. The method uses a Merchant–Rankine type of empirical interaction expression to uncouple the in-plane effects and the out-ofplane effects. Conceptually, the method is an interesting approach because it deals with the structural components using a unique segment length for the evaluation of the stability with respect to the various buckling modes [2]. In addition, for more sophisticated design situations that are not covered by code rules but need finite element analysis, the method simplifies this task. It is noted that EN 1993-1-6 [3] specifies a similar approach, the MNA/LBA approach, that may be seen as a generalization of the stability reduction factor approach used throughout many parts of Eurocode 3, see [4]. It is, however, questionable that the application of the method results in a lower bound estimate of the safety of the structural component for the target probability of failure that is specified in EN 1990 [4]. In addition, the method specifies two alternative criteria for the evaluation of the out-of-plane effects, leading to different levels of safety.

Apart from the doctoral thesis of Müller [2], this method was not widely validated and there is scarce published background documentation to establish its level of safety. Within Technical Committee 8 of ECCS, the need to explore deeply the field and limits of the application of the General Method was consensually recognized [5,6]. In particular, several examples have been carried out at the University of Graz [7,8], comparing advanced finite element analyses (GMNIA) using beam elements with the General Method. It is the purpose of this paper to: (i) discuss the theoretical background of this method; (ii) carry out a parametric study for a wide range of cross-section shapes, buckling lengths, loadings, production processes (welded vs. rolled profiles) and steel grades; (iii) carry out advanced numerical simulations using GMNIA for some standard cases; and (iv) assess the safety of the General Method against the numerical simulations and the procedures of clauses 6.3.1 to 6.3.3 of EC3. 2. Theoretical background 2.1. Description of the method The General Method, as given in EN 1993-1-1 in clause 6.3.4 [1] states that the overall resistance to out-of-plane buckling for any structural component conforming to the scope defined in the introduction can be verified by ensuring that:

χop αult ,k /γM1 ≥ 1, ∗ Corresponding address: Departamento de Engenharia Civil, Universidade de Coimbra, Polo 2 da Universidade, Rua Luís Reis Santos, P-3030-788 Coimbra, Portugal. Tel.: +351 239 797216; fax: +351 239 797217. E-mail address: [email protected] (L. Simões da Silva). 0143-974X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2009.11.003

(1)

where αult ,k is the minimum load amplifier of the design loads to reach the characteristic resistance of the most critical crosssection of the structural component, considering its in-plane behaviour without taking lateral or lateral–torsional buckling into

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Nomenclature A E FBzz GMNIA

Cross-section area Modulus of elasticity Flexural buckling about z–z axis Geometrical and material nonlinear analysis with imperfections L Member length LEA Linear eigenvalue analysis LTB Lateral torsional-buckling Mcr Elastic moment for lateral–torsional buckling My,max,Ed Maximum design bending moment, y–y axis Mn,y,Rd Reduced design value of the resistance to bending moments making allowance for the presence of normal forces, y–y axis Mn,z ,Rd Reduced design value of the resistance to bending moments making allowance for the presence of normal forces, z–z axis Mpl,y,Rd Design value of the plastic resistance to bending moments about y–y axis Mpl,z ,Rd Design value of the plastic resistance to bending moments about z–z axis Mw Warping moment MyMethod Maximum bending moment for the given method, ,max y–y axis Nb,Rd,y Design buckling resistance of a compression member, y–y axis Nb,Rd,z Design buckling resistance of a compression member, z–z axis Ncr ,T Elastic torsional buckling force Ncr ,y Elastic critical force for in-plane buckling Ncr ,z Elastic critical force for out-of-plane buckling NEd Design normal force Method Nmax Maximum normal force for the given method Npl,Rd Design plastic resistance to normal forces of the gross cross-section Q Factor related to Vr (see [13] for details) Ri Relation between re,i and rt ,i (see [13] for details) Rm Mean of R (see [13] for details) Rm The same as Rm , using the ‘linearization’ of the lower tail (see [13] for details) RMethod Relation between the true resistance of the member u and the theoretical resistance associated with the given method Vδ Coefficient of variation of the error term (see [13] for details) a Ratio of web area to gross area a, b, c , d Class indexes for buckling curves b Cross-section width fy Yield strength h Cross-section height kyy , kzy , kyz , kzz Interaction factors dependent of the phenomena of instability and plasticity involved in_pl kyy Interaction factor for in-plane instability n Ratio of design normal force to design plastic resistance to normal forces of the gross crosssection; r Resistance (see [13] for details) rd Design value of the resistance (see [13] for details) re Experimental resistance (see [13] for details) re,i Experimental resistance of a specimen (see [13] for details) rt ,i Theoretical resistance of a specimen (see [13] for details) tf Flange thickness

x–x y–y z–z

αcr ,op αplMy αplN αuMethod αu,pl αuMethod ,pl αult ,k γM1 γRd δi

δx δy δz ε εy ϕ Φ Φx λLt

λop λy λz χLt χop χy χz σ∆ σR σR Ψ ∆i ∆My,Ed ∆Mz ,Ed

Axis along the member Cross-section axis parallel to the flanges Cross-section axis perpendicular to the flanges Minimum amplifier for the in-plane design loads to reach the elastic critical resistance with regard to lateral or lateral–torsional buckling Load amplifier defined with respect to the plastic cross-section bending moment about y–y axis Load amplifier defined with respect to the plastic cross-section axial force Ultimate load amplifier for the given method Ultimate load amplifier defined with respect to the plastic cross-section capacity of the cross-section Ultimate load amplifier defined with respect to the plastic cross-section capacity of the cross-section, for the given method Minimum load amplifier of the design loads to reach the characteristic resistance of the most critical cross-section Partial safety factor for resistance of members to instability assessed by member checks Partial safety factor associated with the uncertainty of the resistance model Observed error term for test specimen i, obtained from the relation between Ri and Rm (see [13] for details) Longitudinal displacement Displacement about y–y axis Displacement about z–z axis Relative error Yield strain Ratio between the reduction factor for flexural buckling about z–z axis and the reduction factor for lateral–torsional buckling My N Ratio between αpl and αPL Rotation about x–x axis Non-dimensional slenderness for lateral–torsional buckling Global non-dimensional slenderness of a structural component for out-of-plane buckling Non-dimensional slenderness for flexural buckling, y–y axis Non-dimensional slenderness for flexural buckling, z–z axis Reduction factor for lateral–torsional buckling Reduction factor for the non-dimensional slenderness λop Reduction factor due to flexural buckling, y–y axis Reduction factor due to flexural buckling, z–z axis Standard deviation of the variable ∆ (see [13] for details) Standard deviation of R (see [13] for details) the same as σR , using the ‘linearization’ of the lower tail (see [13] for details) Ratio of moments in segment Value correspondent to the logarithmical transformation of δi (see [13] for details) Moments due to the shift of the centroidal y–y axis Moments due to the shift of the centroidal z–z axis

account, however, accounting for all effects due to in-plane geometrical deformation and imperfections, global and local, where relevant. χop is the reduction factor for the non-dimensional

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577

Fig. 1. Pinned column.

slenderness to take into account lateral and lateral–torsional buckling and γM1 is the partial safety factor for instability effects (adopted as 1.0 in most National Annexes). The global non-dimensional slenderness λop for the structural component, used to find the reduction factor χop in the usual way using an appropriate buckling curve, should be determined from

λop =

p

αult ,k /αcr ,op ,

(2)

where αcr ,op is the minimum amplifier for the in-plane design loads to reach the elastic critical resistance of the structural component with respect to lateral or lateral–torsional buckling without accounting for in-plane flexural buckling. In the determination of αcr ,op and αult ,k , finite element analysis may be used. According to [1], χop may be taken either as: (i) the minimum value of χ (for lateral buckling, according to clause 6.3.1 of EC3-11) or χLT (for lateral–torsional buckling, according to clause 6.3.2); or (ii) an interpolated value between χ and χLT (determined as in (i)), by using the formula for αult ,k corresponding to the critical cross-section. It is noted that [6] recommends the use of the first option only. 2.2. Application to flexural column buckling In order to illustrate the application of the method to a trivial example, consider the pinned column of Fig. 1 subjected to an arbitrary axial force NEd . Let also assume that the in-plane direction corresponds to the cross-section major axis in bending. Let also assume, for simplicity, that buckling in a torsional mode is not relevant. The application of clause 6.3.1 of EN 1993-1-1 leads, in succession, to:

s s  Afy Npl buckling curve   = −−−−−−−−−−→ χy ⇒ Nb,Rd,y λy =    Ncr ,y Ncr ,y     Npl   = χy Npl,Rd ≥ NEd  = χy γM1 s s   Afy Npl buckling curve   = λz = −−−−−−−−−−→ χz ⇒ Nb,Rd,z    N N cr ,z cr ,z      = χ Npl = χ N z z pl,Rd ≥ NEd γM1 



(3)

α

 = min

Nb,Rd,y Nb,Rd,z NEd

;



NEd

(5)

Alternatively, the application of the General Method for the same reference applied axial force yields, successively:

 Nb,y,Rd  αult ,k =  αcr ,op =

NEd Ncr ,z

→ λop =

r

αult ,k = αcr ,op

s

=

χy Npl,Rd Ncr ,z

Nb,Rd = χop αult ,k NEd = χop χy Npl,Rd ≥ NEd

(8)

or

αuGM = χop χy

Npl,Rd

≥ 1.0. (9) NEd Comparing expressions (5) and (9) for this trivial example shows that the General Method does not exactly give the same result as the application of clause 6.3.1 even whenever the same column buckling curves are used. In fact, assuming
that flexural buckling around the minor axis is critical χz ≤ χy , yields  αuGM χop χy χop = ≥ χz . αu6.3.1 χz

(10)

If advanced FEM analysis is adopted to evaluate αult ,k , different results might be obtained. This issue will be discussed later in the paper. 2.3. Application to lateral–torsional buckling For an unrestrained beam, let αult ,k denote the load level that corresponds to the attainment of the flexural resistance at the critical cross-section. Application of the General Method gives

 Mpl,y,Rd   αult ,k =   αcr ,op =

Mmax,Ed → λop = Mcr

r

αult ,k = αcr ,op

s

Mpl,y,Rd Mcr

Mmax,Ed (11)

and, in this case, the General Method exactly coincides with the application of clause 6.3.2 of EC3.

αplN =

Npl,Rd NEd

αplMy =

MN ,y,Rd = Mpl,y,Rd

My,Ed

1−n 1 − 0.5a

Φ=

αplMy αplN

.

but MN ,y,Rd ≤ Mpl,y,Rd ;

if n ≤ a;

MN ,z ,Rd = Mpl,z ,Rd 1 − (6)

Mpl,y,Rd

(12)

Clause 6.3.3 of EN 1993-1-1 states that the safety of a beam– column requires the verification of the cross-section capacity at the member ends using an appropriate interaction expression such as (for I and H cross-sections)

"

Ncr ,z

√ = λ z χy .

it follows that

MN ,z ,Rd = Mpl,z ,Rd

Nb,y,Rd

NEd

s

(7)

Consider the pin-ended beam–column of Fig. 2 subjected to an arbitrary axial force NEd and a uniform major axis bending moment My,Ed . Let (4)

≥ 1.0.

χy ≤ 1.0 ⇒ λop ≤ λz ⇒ χop ≥ χz

2.4. Application to bending and axial force interaction

or, defining αu as the ultimate load multiplier with respect to the applied axial force, 6.3.1 u

Since

= λLT → χop = χLT

and Nb,Rd = min Nb,Rd,y ; Nb,Rd,z ≥ NEd

Fig. 2. Pin-ended beam–column.



n−a 1−a

(13a) (13b)

2 #

if n > a,

(13c)

where n = NEd /Npl,Rd and a = A − 2btf /A, but a ≤ 0.5 and the verification of the two following stability interaction formulae,




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(a) IPE 200.

(b) HEB 300.

Fig. 6. Residual stresses (+ Tension and − Compression): Hot rolled. Table 1 Loading.

Fig. 3. Modelling of steel behaviour, perfect elastic–plastic behaviour.

Type of load

Steel grade

Plastic resistance HEB 300

IPE 200

Case (a) Uniform bending moment

S235 S355

420.8 kNm 635.6 kNm

49.3 kNm 74.4 kNm

Case (b) Uniform axial force

S235 S355

3356.3 kN 5070.1 kN

640.3 kN 967.3 kN

Case (c) UDL (q = 8.M /L2 (kN/m))

S235 S355

420.8 kNm 635.6 kNm

49.3 kNm 74.4 kNm

Table 2 Calibration of the mesh for HEB 300 (member in bending).

Fig. 4. Support conditions.

NEd

χy NRk /γM1

+ kyy

My,Ed + ∆My,Ed

χLT My,Rk /γM1

+ kyz

MzEd + ∆Mz ,Ed Mz ,Rk /γM1

≤ 1.0, (14a)

NEd

χz NRk /γM1

+ kzy

My,Ed + ∆My,Ed

χLT My,Rk /γM1

+ kzz

Mz ,Ed + ∆Mz ,Ed Mz ,Rk /γM1

≤ 1.0,

(a) Uniform bending moment.

My,max /Mpl,y

Error (%)

M80 × 4 × 4 M 60 × 6 × 6 M100 × 8 × 8 M120 × 12 × 12

0.8574 0.8541 0.8543 0.8538

0.42 0.04 0.06 –

Table 3 Beam–columns—L = 10.9 m; HEB 300. My

Φ=

(14b) where: NEd , My,Ed and Mz ,Ed are the calculation values of the axial force and bending moments around y and z, respectively; ∆My,Ed and ∆Mz ,Ed are the moments due to the variation of the centroid for class 4 sections; χy and χz are the reduction factors due to buckling by bending around y and z, respectively, evaluated according to clause 6.3.1 [1]; χLT is the reduction factor due to lateral buckling, evaluated according to clause 6.3.2 (χLT = 1.0 for elements that are not susceptible of buckling laterally); kyy , kyz , kzy , and kzz are interaction factors dependent on the relevant instability and plasticity phenomena, obtained according to Annex A of EC3 (Method 1) or Annex B (Method 2);

Mesh

αpl

N αpl

2 1 0.5

My

My

Ofner-beam-1/αpl

GMNIA-shell-1/αpl

Diff. (%)

0.144 0.246 0.372

0.153 0.259 0.389

5.9 5.0 4.4

Assuming proportional loading (Φ = constant) and class 1 or 2 cross-sections (NRk = Npl,Rd and My,Rk = Mpl,y,Rd ), an ultimate load multiplier can be defined with respect to the applied loading, given by   1 NEd My,Ed NEd My,Ed = max + k ; + k . zy yy αu6.3.3 χz Npl,Rd χLT Mpl,y,Rd χy Npl,Rd χLT Mpl,y,Rd (15) The application of the General Method leads to

αuGM = χop αult ,k .

(b) Axial force. Fig. 5. Modelling of loads.

(16)

(c) UDL.

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(a) HEB 300; L = 7.27 m; δy (mm) vs. N /Npl .

579

(b) HEB 300; L = 7.27 m; δz (mm) vs. My /Mpl,y .

Fig. 7. Validation of the model with cases from [11].

Comparing expressions (15) and (16) gives:

Table 4 Critical load multiplier (My,max /Mpl,y ). Case

Trahair

GMNIA

Diff. (%)

Column (λz = 1.0) Beam (λz = 1.0) Beam–column—Φ = 0.5 (λz = 1.5) Beam–column—Φ = 1.0 (λz = 1.5) Beam–column—Φ = 2.0 (λz = 1.5)

1.000a 1.880 0.595 0.381 0.212

0.976a 1.827 0.584 0.374 0.208

−2.40 −2.82 −1.85 −1.84 −1.89

a

Nmax /Npl .

1

=

χy Npl,Rd

NEd

+ kin_pl yy

My,Ed Mpl,y,Rd

.

(17)

Table 5 Parametric study (analytical evaluation).

Table 6 Parametric study (numerical evaluation)—IPE 200 and HEB 300; rolled cross-sections.

+



NEd

χz Npl,Rd

kin_pl yy

+ kzy

My,Ed

χLT Mpl,y,Rd

;



My,Ed

(18)

χLT Mpl,y,Rd

or

α α

χy Npl,Rd

= αult ,k χop max NEd



GM u 6.3.3 u

Assuming the same applied loading NEd and My,Ed and evaluating αult ,k according to 6.3.3 gives

αult ,k

α α

GM u 6.3.3 u

 = max 

1

χz

+ 1

χy



kzy

χLT Φ in_pl

+

kyy

Φ



χop ;

1

χy

+ 1

χy



kyy

χLT Φ in_pl

+

kyy

χop

  .

(19)

Φ

According to clause 6.3.4(4) of EN 1993-1-1, the reduction factor

χop may be determined from either of the following methods: (i) Method 1: the minimum value of χ (for lateral buckling, according to clause 6.3.1 of EC3-1-1) or χLT (for lateral–torsional buckling,

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Table 7 Parametric study for GMNIA numerical cases (beams and beam–columns) from Ofner [11].

Table 8 Parametric study for in-plane GMNIA numerical cases (beams and beam–columns) from Ofner [11].

Table 9 Analysis procedure. 1

GMNIA

GMNIA_s Shell elements → 1/αuGMNIA_s ↔ (Nmax ; MyGMNIA_s ,max ) ,pl GMNIA_b GMNIA_b Beam elements → 1/αu,pl ↔ (Nmax ; MyGMNIA_b ,max )

2

EC3-1-1 6.3.3

Minimum of: • In-plane resistance (YY ) or Out-of-plane resistance (ZZ ) YY or ZZ or ZZ → (Nmax ; MyYY,max ) CS • Cross-section resistance at each end of the member → (Nmax ; MyCS,max ) Leads to: M1 M1 Method 1, Annex A → 1/αuM1 ,pl ↔ (Nmax ; My,max ) M2 M2 Method 2, Annex B → 1/αuM2 ↔ ( N ; M max y,max ) ,pl

3

General Method (derived expressions in 2.4)

Minimum of: YY YY YY • αcr ,op +In-plane resistance (αult ,k ) → (NEd ; My,Ed ), where CS YY αult ,k = minimum (αult ; α ) ,k ult ,k

Leads to: GM_M1 Method 1, Annex A → 1/αuGM_M1 ↔ (Nmax ; MyGM_M1 ,max ) ,pl GM_M2 Method 2, Annex B → 1/αuGM_M2 ↔ ( N ; MyGM_M2 max ,max ) ,pl

4

General Method (numerical resistance)

Shell elements: s • GMNIA in plane—αult ,k • LEA—αcrs ,op

Beam elements: b • GMNIA in plane—αult ,k • Eq. (21) from Trahair [10]—αcr ,op Lead to: GM_s GM_s Shell elements → 1/αuGM_s ,pl ↔ (Nmax ; My,max ) GM_b GM_b Beam elements → 1/αuGM_b ↔ ( N ; M max y,max ) ,pl Table 10 Buckling curves for flexural buckling about zz. Cross-sections IPE 360–IPE 200 HEB 550–HEB 400 HEB 360–HEA 200

Table 11 All results of Eq. (19) for Φ = 1 − n = 80 for each case; χop = min(χz ; χLt ).

Rolled

Welded

b

c

c

c

according to clause 6.3.2); (ii) Method 2: an interpolated value between χ and χLT (determined as in Method 1), by using the formula for αult ,k corresponding to the critical cross-section:

 NEd My,Ed   + ≤ χop  N

M

Rk y,Rk ⇒ χop = NEd My,Ed   + ≤1  χ NRk χLt My,Rk

Φ +1 Φ χ

+

1

.

(20)

χLt

The values of χ and χLT considered for the reduction factor χop , are calculated with the global non-dimensional slenderness λop of

Fabrication process

Bending moment

EC3 Method 2

EC3 Method 1

Min.

Max.

Min.

Max.

Hot rolled

All

84.4 87.7 87.0 84.4 90.0 89.0 94.4

112.3 109.8 108.2 101.2 111.1 112.3 110.3

81.1 87.8 84.7 81.1 89.6 89.4 92.0

112.6 109.1 101.7 97.3 111.8 112.6 109.8

Ψ =1 Ψ =0 Ψ = −1 Conc. Dist.

Ψ =1

Welded

the structural component, determined from Eq. (2). According to Trahair [10], the elastic critical bending moment and axial force are given by



My,max Mcr

2

 = 1−

Nmax Ncr ,z

 1−

Nmax Ncr ,T

 (21)

Author's Personal Copy L. Simões da Silva et al. / Journal of Constructional Steel Research 66 (2010) 575–590

that αcr ,op is given by:

Table 12 Buckling curves for rolled cross-sections (General Case for LTB). Cross-sections

Buckling curve—FBzz

Buckling curve—LTB

IPE 360 IPE 200 HEB 550–HEB 400 HEB 360–HEA 200

b

b

b

a

c

a

581

αcr ,op =

Nmax NEd

=

My,max My,Ed

.

(22)

3. Numerical model 3.1. Introduction

Table 13 Sub-sets considered for the statistical evaluation. Sub-set

Level of sub-set

N Beam elements

Shell elements

Cross-section

HEB 300 IPE 200 IPE 500

227 278 140

112 112 –

Type of loading (except columns)

1-bi-triangular 0-triangular 1-constant Distributed load

158 165 245 69

– 32 128 32

Level of loading

0 (Beam) √ ]0; √ 3/3[√ [√ 3/3; 3[ [ 3; ∞[ ∞ (Column)

31 373 158

64 32 64 32 32

Slenderness (zz)

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

185 192 202 66

56 56 56 56

Steel grade

S235 S355

645 –

176 48

Total

–

645

224

83

3.2. Structural model

where Mcr is the elastic critical bending moment, Ncr ,z is elastic critical compressive buckling force in a bending mode about the z–z axis and Ncr ,T is the elastic critical compressive buckling force in a torsional mode. Eq. (21) is valid for beam–columns with constant bending moment distribution. To adapt the equation to other types of bending moment distribution, see Trahair [10]. Finally, introducing Φ =

Npl,Rd /Nmax Mpl,y,Rd /My,max

The evaluation of the safety of the General Method requires reliable estimates of the real behaviour of some reference cases. Advanced numerical simulations contemplating geometrical and material nonlinearities with imperfections (GMNIA) were adopted for this purpose, as is nowadays widely accepted [13]. Section 3.2 describes the adopted structural model and the underlying assumptions. A large number of numerical simulations were carried out, described in Section 4. In order to develop a reliable numerical model, three welldetailed reference cases were taken from the literature to allow direct comparison with independent numerical simulations. Three examples, a column, a beam and a beam–column, were selected from the Ph.D. thesis of Ofner [11], who carried out GMNIA FEM analysis using the finite element program ABAQUS and 3D-beam elements. The column (pinned–pinned), as well as the beam (fork supports), is 7.27 m long (λz = 1.0) and consists of a HEB 300 profile in steel grade S235. The beam–column (fork supports) is 10.9 m long (λz = 1.5) and also consists of a HEB 300 profile in steel grade S235. The results of these validation cases are illustrated in paragraph 3.3.

in Eq. (21) leads to (Nmax , My,max ), so

3.2.1. Finite element model The finite element model was implemented using the commercial finite element package LUSAS, version 14 [12]. Eight-noded ‘‘semiloof’’ thin shell elements (QSL8) were used. For the material nonlinearity, an elastic–plastic constitutive law based on the Von Mises yield criterion is adopted. The constitutive model is integrated by means of the explicit forward Euler algorithm. To determine the structural response of the nonlinear problem, an implicit solution strategy is used. Hence, a load stepping routine is used.

Table 14 Safety factor γRd (statistical details given in the Appendix (Tables A.2–A.6)). Sub-set

Level of sub-set

Beam elements

Shell elements

EC3-6.3.3 (M1)

EC3-6.3.4 (M1)

EC3-6.3.3 (M1)

EC3-6.3.4 (M1)

EC3-6.3.4 (num)

γRd

γRd

γRd

γRd

γRd

Cross-section

HEB 300 IPE 200 IPE 500

1.022 1.058 1.027

1.007 1.037 1.076

1.048 1.099 –

1.055 1.099 –

1.023 1.052 –

Type of loading

−1-bi-triangular 0-triangular 1-constant Distributed load

1.008 1.024 1.063 1.029

1.023 1.003 1.068 0.995

– 1.064 1.077 1.085

– 1.032 1.080 1.075

– 1.043 1.054 1.032

Level of loading (Φ )

0 (Beam) √ ]0 √; 3/3[ √ [√3/3; 3[ [ 3; ∞[ ∞ (Column)

1.098 1.048 1.009

1.098 1.061 1.055

1.036

1.015

1.104 1.046 1.052 1.100 1.012

1.104 1.050 1.038 1.060 1.054

1.067 1.084 1.044 1.033

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

1.035 1.068 1.021 0.985

1.016 1.077 1.004 0.973

1.090 1.066 1.027 1.010

1.093 1.064 1.004 0.995

0.997 1.047 1.040 0.983 0.977

S235 S355 –

1.042 – 1.042

1.057 – 1.057

1.082 1.081 1.085

1.076 1.093 1.089

1.038 1.068 1.046

Slenderness (zz)

Steel grade Total

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(a) Rolled cross-sections.

(b) Welded cross-sections. Fig. 8. Results of Eq. (10)—pin-ended columns.

(a) Method 2.

(b) Method 1. Fig. 9. Results of Eq. (19)—Constant bending moment (ψ = 1).

The increment size follows from accuracy and convergence criteria. Within each increment, the equilibrium equations are solved by means of the Newton–Raphson iteration. 3.2.2. Material properties S235 and S355 steel grade was considered in the reference examples, with a yield stress of 235 MPa and 355 MPa respectively, a modulus of elasticity of 210 GPa and a Poisson’s ratio of 0.3. Perfect elastic–plastic behaviour of the material was considered (Fig. 3). 3.2.3. Support conditions Only simply supported single span members with end fork conditions are studied in this paper. The boundary conditions are implemented in the shell model as shown in Fig. 4. The following restraints are imposed: (i) vertical (δy ) and transverse (δz ) displacements prevented at cross-sections A and B for all nodes; (ii) longitudinal displacement (δx ) prevented at point 1 only. 3.2.4. Loading The modelling of the various loads is represented in Fig. 5 and Table 1. For cases (a) and (b), the loads are distributed along the web and flanges, at the end cross-sections of the member. For case (c), the load is distributed along the web, at h/2. Cases (a), (b) and (c) correspond to uniform bending moment, uniform axial force and uniformly distributed vertical loading, respectively. The reference loading is such that, for a 1st order analysis, the plastic

resistance of the cross-section is reached (all cross-sections are class 1 or 2 for all loading cases). For the beam–column, cases (a) or (c) are combined with case (b). 3.2.5. Imperfections A geometrical imperfection of sinusoidal type relative both to the yy and zz axes is considered: y(x) = z (x) =

l 1000

sin

πx l

.

(23)

No torsional imperfection was adopted, according to EC3-1-1, clause 5.3.4 (3). Regarding material imperfections, residual stresses are constant along the thickness of the cross-section. Fig. 6 shows the adopted residual stress patterns. 3.3. Validation of the model Firstly, a mesh calibration study was carried out, leading to the following discretization for a member with an HEB 300 crosssection and a length of L = 7.27 m (λz = 1.0)—for other lengths, a linear extrapolation was performed: 6 sub-divisions in the web and flange; and 60 divisions along the axis of the member. Table 2 illustrates the relative error for 4 different meshes. For members with a IPE 200 cross-section and a length of 2.14 m (λz = 1.0), the following discretization was adopted: 6 and 4

Author's Personal Copy L. Simões da Silva et al. / Journal of Constructional Steel Research 66 (2010) 575–590

583

4. Parametric study 4.1. Scope

Fig. 10. Results of Eq. (19)—beam–columns; Eq. (10)—columns; or Eq. (11)—beams;

Ψ = 1; HEB 300; Method 2.

sub-divisions in the web and flange, respectively; and 30 divisions along the axis of the member. Subsequently, in order to validate the model, some simulations were compared with the results obtained by Ofner [11]. For the beam and column, rotations about xx axis (longitudinal) are restrained. Results are represented in Fig. 7. Agreement is excellent, especially considering that a shell model is being compared with a beam model. Table 3 summarizes some results for a beam–column with length L = 10.90 m and constant bending moment, good agreement being noted with the results from [11]. Performing an eigenvalue analysis for the three cases described in this section yields the results of Table 4 that also shows, for comparison, theoretical predictions obtained from Trahair [10].

(a) χz < χLT .

The parametric study addresses the evaluation of the safety level that is achieved with the application of the General Method to uniform members under standard boundary conditions. Two levels of comparisons are carried out. Firstly, the General Method is compared with the results that are directly obtained from the application of clauses 6.3.1 to 6.3.3. Secondly, the General Method is compared with results from advanced nonlinear numerical simulations (GMNIA) with shell elements. Table 5 summarizes the selected cases (about 1000) that are directly compared with the results of clauses 6.3.1 and 6.3.3. They comprise variations of slenderness range, loading type, production process (rolled or welded sections), cross-section shape, M /N ratio and steel grade. In the case of beam–columns, the results of the General Method are compared both with the calculation of the interaction factors using Method 1 (M1) and Method 2 (M2) from EC3-1-1 [1]. Also, for the calculation of χLT , three alternatives are also considered: General Case (GC); Special Case (SC) and General Case modified with the f-factor (GC/f) [13]. Table 6 summarizes the sub-set of cases to be compared with the advanced numerical simulations. About 224 numerical simulations with shell elements are carried out: 128 beam–columns; 64 beams; and 32 columns. They correspond to a sub-set of the parametric range described in Table 5 and cover the same parameters. For all cases, besides full 3D GMNIA numerical simulations, constrained in-plane GMNIA calculations and Linear Eigenvalue Analysis (LEA) are also carried out to provide data for the application of the General Method.

(b) χLt < χz . Fig. 11. Interaction curves regarding the value of χop .

(a) IPE 200–S235.

(b) HEB 300–S235. Fig. 12. Results for columns.

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(a) IPE 200–ψ = 1–S235.

(b) HEB 300–ψ = 1–S235. Fig. 13. Results for beams.

(a) IPE 200–ψ = 1–S235.

(b) HEB 300–ψ = 1–S235.

(c) IPE 200–ψ = 0–S235.

(d) HEB 300–ψ = 0–S235. Fig. 14. Results for beam–columns for Φ = 1.

Finally, in order to extend the range of numerical calculations for comparison and validation of the General Method, 645 and 809 cases (beams, columns and beam–columns) of steel grade S235 from Ofner [11] are considered, consisting of GMNIA (Table 7) and in-plane GMNIA (Table 8) calculations using beam elements, respectively. The GMNIA in-plane results and the out-of-plane elastic critical load using available theoretical results from Eq. (21) are then applied for the evaluation of the General Method. It is noted that other techniques could be used to further extend the range of numerical results [14].

each method, for the more general case of beam–columns. For all methods, the pair of maximum applied forces (Nmax , My,max ) is first calculated (always assuming proportional loading, Φ = Cte.). Subsequently, an ultimate load multiplier 1/αu,pl is defined with respect to the plastic cross-section capacity of the section, as shown in expression (24). 1/αu,pl provides a consistent relative indicator across the various methods.

4.2. Analysis procedure

The GMNIA simulations directly yield the maximum resistance GMNIA of the member (NEd and MyGMNIA ,Ed ). For the evaluation according to clause 6.3.3, both Method 1 and Method 2 of EC3-1-1 are considered. The minimum value of the in-plane resistance, out-ofplane resistance and cross-section resistance is calculated. For the General Method, the two following options are possible:

Given that two levels of comparisons are carried out for the General Method and that there are various options for the application of the General Method, Table 9 summarizes the various procedures for the calculation of the ultimate load factor, according to

1

αu,pl

=

Nmax Npl,Rd

=φ

My,max My,pl,Rd

.

(24)

Author's Personal Copy L. Simões da Silva et al. / Journal of Constructional Steel Research 66 (2010) 575–590

(a) IPE 200–ψ = 1.

585

(b) IPE 200–ψ = 1. Fig. 15. Results for λy = 1.0.

The ratio between the ultimate load multipliers for the General Method and clause 6.3.1 decreases as the slenderness of the column increases. It can also be noticed that the results concentrate in groups according to the chosen buckling curve (Table 10):

Fig. 16. Variation of safety factor γrd with slenderness.

(1) Analytical approach: (i) evaluation of the in-plane resistance using clause 6.3.3 (Eq. (14a) with χLT = 1 or Eq. (17), for either Method 1 or Method 2) and the cross-section interaction formulae from Eqs. (13) for the check of the end cross-sections of the member, and (ii) the evaluation of the out-of-plane elastic critical load using Eq. (21); (2) Numerical approach: (i) evaluation of the in-plane resistance from constrained in-plane GMNIA calculations, and (ii) the outof-plane elastic critical load from a numerical LEA. In order to compare the different methods, the GMNIA results are taken as the basis, assumed to represent the true behaviour of the member [13]. Since the loading is indexed to the plastic moment resistance of the cross-section, αuGMNIA ≤ 1.1 Consequently, defining Ri as the relation between the true resistance of a specimen and the theoretical resistance associated with a given method, the safety of that method can be expressed as: 1 RMethod u

=

αuMethod ,pl α

GMNIA u,pl

=

Method NEd ,max GMNIA NEd ,max

=

MyMethod ,Ed,max MyGMNIA ,Ed,max

.

(25)

If 1/RMethod < 1, that specific concretization of the resistance u according to that method is safe. 4.3. Comparison of the General Method with results from clauses 6.3.1 to 6.3.3 4.3.1. Columns Considering the case of pin-ended columns first, Fig. 8 plots expression (10) for a range of profiles and lengths, with the aim to compare the results from General Method with clause 6.3.1.

1 Except for the low slenderness range where strain hardening may lead to

αuGMNIA > 1.

4.3.2. Beam–columns For beam–columns, results of Eq. (19) are also analyzed for a range of profiles, loading and lengths, with the aim to compare the results from General Method with clause 6.3.3, and find any trends. The cross-sections were chosen in order to enclose a range of profiles with several depth/width ratios, and consist of class 1 or 2 cross-sections. Concerning all results, comparing the General Method with both Method 1 and Method 2 using Eq. (19) leads to a variation of results between 81% and 113%, as shown in Table 11: In the Appendix (Table A.1), more detailed results for IPE 200 and HEB 300 are presented. Fig. 9 illustrates the results for rolled cross-sections concerning a constant bending moment distribution. Results are plotted for a range of member lengths between λz = 0.5 and λz = 2.5 for Φ = 1. χLT is calculated according to the General Case from EC31-1, and χop = min(χz ; χLt ). Analyzing Fig. 9, it is seen that for more slender cross-sections (larger h/b), the General Method is less conservative. The results also tend to concentrate in groups according to the buckling curve (Table 12): Further results regarding other types of bending moment distributions are found in the Appendix (Figs. A.1 and A.2). Fig. 10 represents the results for a HEB 300 for uniform moment pl pl (Ψ = 1), with different ratios of Φ = αM /αN : An inconsistency around Φ = 0, i.e., high bending moment relatively to the axial force is observed: for Φ = 0 (lateral torsional buckling), the results are calculated using Eq. (11), that is, χop = χLT while for Φ > 0 Eq. (19) is used whereby χop = min(χz , χLT ) = χz in this case of a HEB 300 profile. In general, taking χop as the minimum value between χz or χLT , for Φ = 0, in case the reduction factor χz is smaller than χLT , the above discontinuity will be observed. For Φ = ∞, i.e., high axial force relatively to bending moment and in case reduction factor χLT is smaller than χz the same inconsistency is observed. However, if χop is calculated with Eq. (20) (interpolated value between χz and χLT ), the discontinuity in the interaction curve disappears, as

χop =

Φ +1 Φ χ

+

1

χLt

( ⇒

lim χop = χLt

φ→0

lim χop = χ.

(26)

φ→∞

To illustrate this, two cases are chosen such that: (a) χz < χLT and (b) χLt < χz . The results are plotted in the interaction

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Table A.1 All results of Eq. (19) for HEB 300 and IPE 200 (rolled)—n = 8 for each case.

Φ

Bending moment

IPE 200

HEB 300

EC3 Method 2

4

Ψ =1 Ψ =0 Ψ = −1 Conc. Dist.

2

Ψ =1 Ψ =0 Ψ = −1 Conc. Dist.

1

Ψ =1 Ψ =0 Ψ = −1 Conc. Dist.

0.5

Ψ =1 Ψ =0 Ψ = −1 Conc. Dist.

0.25

Ψ =1 Ψ =0 Ψ = −1 Conc. Dist.

EC3 Method 1

Min.

Max.

101.5 100.1 96.2 101.4 101.5

103.9 101.8 99.0 102.8 103.4

101.2 101.3 93.7 101.5 101.4

Min.

EC3 Method 2

EC3 Method 1

Max.

Min.

Max.

Min.

93.8 92.8 91.4 95.4 94.5

105.1 99.7 98.1 104.6 104.8

87.7 85.9 85.1 87.1 87.4

97.2 100.0 97.2 97.7 97.4

87.8 85.0 84.4 86.4 87.1

95.1 92.0 92.3 93.0 94.0

106.8 103.6 98.9 105.2 106.1

95.6 91.6 89.0 98.1 96.8

108.5 100.2 97.4 107.6 108.1

89.4 86.5 85.0 88.5 89.0

96.6 100.1 96.3 97.4 97.0

89.1 85.0 83.8 87.6 88.8

96.4 90.5 90.1 93.7 94.8

99.8 103.4 91.9 100.6 100.2

109.8 106.1 101.2 107.8 0.0

97.3 91.0 83.9 100.9 0.0

109.1 100.4 96.4 109.0 0.0

89.4 87.6 85.2 90.5 90.2

95.7 100.7 96.7 97.0 96.3

90.3 85.6 82.4 89.8 90.4

96.8 91.3 91.0 96.9 94.9

97.3 105.2 95.0 98.7 97.8

109.7 109.8 106.0 109.1 109.5

97.8 94.8 86.1 102.2 100.8

108.1 99.8 97.0 107.7 108.3

87.3 89.6 86.7 89.6 88.3

94.9 102.1 99.2 96.4 95.6

90.4 86.9 82.7 90.0 89.8

95.3 94.1 93.2 99.5 96.4

94.8 106.5 101.8 96.4 95.5

105.3 113.6 111.4 105.9 105.5

97.1 94.3 93.8 101.4 100.1

101.0 105.2 100.5 106.1 101.6

85.5 91.6 92.2 88.0 86.5

94.3 103.9 102.6 95.9 95.0

90.0 86.6 86.4 88.6 90.2

93.5 97.3 94.4 101.4 97.2

(a) Method 2.

Max.

(b) Method 1. Fig. A.1. Results of Eq. (26)—Triangular bending moment distribution (ψ = 0).

curves of Fig. 11, considering the results of Eq. (16)—clause 6.3.4 for beam–columns. For comparison, results of Eq. (9)—clause 6.3.4 for columns; (11)—clause 6.3.4 for beams; and (15)—clause 6.3.3, are also plotted.

4.4. Comparison of the General Method with the numerical results

4.4.1. Introduction Figs. 12–14 compare the numerical results from GMNIA and GMNIA in-plane simulations with shell elements and beam elements [11] with the results from clauses 6.3.1 to 6.3.3 and the results of the General Method. These are obtained according to the two alternative procedures described in Table 9. χLT is calculated according to the General Case from EC3-1-1 and Method 1 is adopted. χop is determined as the minimum of χz or χLT .

4.4.2. Columns For columns, the resistance of the member obtained by GMNIA simulations with shell elements is higher than for the other methods. General Method results obtained from the numerical simulations vary between results from clause 6.3.1 and clause 6.3.4 (expression (9)). Results for a HEB 300 present more scatter than results for a IPE 200. 4.4.3. Beams Analyzing Fig. 13, the results of the General Method considering in-plane GMNIA and LEA numerical simulations become more conservative with the increase of the member length. 4.4.4. Beam–columns Fig. 14 illustrates the results for a IPE 200 and a HEB 300 subjected to uniform moment and a triangular bending moment distribution (ψ = 0). Not surprisingly, the GMNIA results always give

Author's Personal Copy L. Simões da Silva et al. / Journal of Constructional Steel Research 66 (2010) 575–590

(a) Method 2.

587

(b) Method 1. Fig. A.2. Results of Eq. (26)—Uniformly distributed load.

Table A.2 Parametric study with beam elements: 6.3.3—Method 1. Sub-set

Level of sub-set

N

Rm

σ

Rm − σ

NLin

Cross-section

HEB 300 IPE 200 IPE 500

227 278 140

1.1050 1.0608 1.1086

0.0844 0.0568 0.0716

1.0206 1.0040 1.0369

−1-bi-triangular

158 165 245 69

1.0898 1.1019 1.0875 1.0502

0.0759 0.0709 0.0800 0.0324

0 (Beam) √ ]0; √ 3/3[ √ [√3/3; 3[ [ 3; ∞[ ∞ (Column)

31 373 158

1.0894 1.1125 1.0562

83

Slenderness (zz)

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

Steel grade Total

Type of loading (except columns)

Level of loading (Φ )

a

Rm

Q ≡ σ∆

(V δ )2

γRd

42 44 29

1.0049 0.9897 1.0153

0.0087 0.0152 0.0138

0.0001 0.0002 0.0002

1.022 1.058 1.027

1.0138 1.0310 1.0075 1.0179

41 31 36 9a

1.0035 1.0145 0.9884 1.0141

0.0036 0.0125 0.0161 0.0138

0.0000 0.0002 0.0003 0.0002

1.008 1.024 1.063 1.029

0.0963 0.0756 0.0464

0.9930 1.0369 1.0098

4a 68 29

1.0300 1.0111 1.0027

0.0401 0.0191 0.0040

0.0016 0.0004 0.0000

1.098 1.048 1.009

1.0282

0.0394

0.9888

10a

0.9890

0.0078

0.0001

1.036

185 192 202 66

1.0250 1.0666 1.1302 1.1859

0.0274 0.0542 0.0615 0.0649

0.9976 1.0123 1.0686 1.1209

18 31 27 8a

0.9889 0.9924 1.0449 1.1198

0.0076 0.0191 0.0211 0.0320

0.0001 0.0004 0.0004 0.0010

1.035 1.068 1.021 0.985

S235 S355

645 –

1.0868 –

0.0742 –

1.0125 –

108 –

0.9978 –

0.0128 –

0.0002 –

1.042 –

–

645

1.0868

0.0742

1.0125

108

0.9978

0.0128

0.0002

1.042

0-triangular 1-constant Distributed load

When NLin ≤ 20, a sub-set of 20 results is used.

Table A.3 Parametric study with beam elements: 6.3.4—Derived expressions from Section 2. Sub-set

Level of sub-set

N

Rm

σ

Rm − σ

NLin

Rm

Q ≡ σ∆

(Vδ )2

γRd

Cross-section

HEB 300 IPE 200 IPE 500

227 278 140

1.2244 1.0963 1.1047

0.1225 0.0798 0.0904

1.1019 1.0165 1.0144

43 34 21

1.0693 1.0038 0.9801

0.0241 0.0133 0.0175

0.0006 0.0002 0.0003

1.007 1.037 1.076

−1-bi-triangular

158 165 245 69

1.1843 1.1721 1.1229 1.0609

0.1079 0.1016 0.1270 0.0334

1.0765 1.0705 0.9960 1.0275

33 19a 19a 14a

1.0477 1.0494 0.9736 1.0235

0.0227 0.0166 0.0128 0.0059

0.0005 0.0003 0.0002 0.0000

1.023 1.003 1.068 0.995

0 (Beam) √ ]0; √ 3/3[ √ [√3/3; 3[ [ 3; ∞[ ∞ (Column)

31 373 158

1.0894 1.1684 1.1239

0.0963 0.1255 0.0966

0.9930 1.0429 1.0273

4a 58 24

1.0300 1.0154 1.0056

0.0401 0.0245 0.0193

0.0016 0.0006 0.0004

1.098 1.061 1.055

83

1.0871

0.0665

1.0206

17a

1.0125

0.0090

0.0001

1.015

Slenderness (zz)

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

185 192 202 66

1.0634 1.1327 1.1964 1.2350

0.0378 0.1042 0.1152 0.1401

1.0256 1.0285 1.0811 1.0949

38 43 45 4a

1.0130 0.9946 1.0503 1.1086

0.0094 0.0224 0.0174 0.0249

0.0001 0.0005 0.0003 0.0006

1.016 1.077 1.004 0.973

Steel grade

S235 S355

645 –

1.1432 –

0.1156 –

1.0276 –

89 –

1.0047 –

0.0197 –

0.0004 –

1.057 –

Total

–

645

1.1432

0.1156

1.0276

89

1.0047

0.0197

0.0004

1.057

Type of loading (except columns)

Level of loading (Φ )

a

0-triangular 1-constant Distributed load

When NLin ≤ 20, a sub-set of 20 results is used.

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Table A.4 Parametric study with shell elements: 6.3.3—Method 1. Sub-set

Level of sub-set

N

Rm

σ

Rm − σ

NLin

Rm

Q ≡ σ∆

(Vδ )2

γRd

Cross-section

HEB 300 IPE 200 IPE 500

112 112 –

1.1275 1.0710 –

0.0810 0.0695 –

1.046 1.001 –

21 14a –

1.0164 0.9814 –

0.0205 0.0249 –

0.0004 0.0006 –

1.048 1.099 –

−1-bi-triangular

– 32 128 32

– 1.1329 1.1014 1.1007

– 0.0835 0.0844 0.0788

– 1.049 1.017 1.022

– 5a 24 5a

– 1.0809 0.9867 1.0498

– 0.0457 0.0200 0.0424

– 0.0021 0.0004 0.0018

– 1.064 1.077 1.085

Level of loading (Φ )

0 (Beam) √ ]0; √ 3/3[ √ [√3/3; 3[ [ 3; ∞[ ∞ (Column)

64 32 64 32 32

1.0860 1.1205 1.1208 1.1051 1.0555

0.0943 0.0735 0.0802 0.0724 0.0315

0.992 1.047 1.041 1.033 1.024

11a 7a 13a 6a 6a

0.9871 1.0745 1.0278 1.0619 1.0343

0.0282 0.0383 0.0254 0.0505 0.0150

0.0008 0.0015 0.0006 0.0026 0.0002

1.104 1.046 1.052 1.100 1.012

Slenderness (zz)

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

56 56 56 56

1.0231 1.0750 1.1392 1.1596

0.0396 0.0516 0.0651 0.0771

0.984 1.023 1.074 1.083

9a 8a 10a 12a

0.9821 1.0200 1.0682 1.0757

0.0225 0.0274 0.0303 0.0272

0.0005 0.0008 0.0009 0.0007

1.090 1.066 1.027 1.010

Steel grade

S235 S355

176 48

1.1035 1.0836

0.0795 0.0829

1.024 1.001

30 7a

0.9972 1.0124

0.0249 0.0296

0.0006 0.0009

1.082 1.081

Total

–

224

1.0992

0.0805

1.019

35

0.991

0.024

0.0006

1.085

Type of loading (except columns)

a

0-triangular 1-constant Distributed load

When NLin ≤ 20, a sub-set of 20 results is used.

Table A.5 Parametric study with shell elements: 6.3.4—Derived expressions from Section 2. Sub-set

Level of sub-set

N

Rm

σ

Rm − σ

NLin

Rm

Q ≡ σ∆

(Vδ )2

γRd

Cross-section

HEB 300 IPE 200 IPE 500

112 112 –

1.1991 1.0607 –

0.1017 0.0572 –

1.097 1.004 –

27 12a –

1.0513 0.9877 –

0.0339 0.0269 –

0.0011 0.0007 –

1.055 1.099 –

−1-bi-triangular

– 32 128 32

– 1.1868 1.1227 1.1123

– 0.1010 0.1106 0.1018

– 1.086 1.012 1.011

– 4a 18a 2a

– 1.1204 0.9877 1.0472

– 0.0473 0.0213 0.0386

– 0.0022 0.0005 0.0015

– 1.032 1.080 1.075

Level of loading (Φ )

0 (Beam) √ ]0; √ 3/3[ √ [√3/3; 3[ [ 3; ∞[ ∞ (Column)

64 32 64 32 32

1.0860 1.1655 1.1605 1.1314 1.1195

0.0943 0.1115 0.1126 0.1051 0.0931

0.992 1.054 1.048 1.026 1.026

11a 5a 9a 4a 4a

0.9871 1.0894 1.0452 1.0562 1.0533

0.0282 0.0439 0.0267 0.0368 0.0341

0.0008 0.0019 0.0007 0.0014 0.0012

1.104 1.050 1.038 1.060 1.054

Slenderness (zz)

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

56 56 56 56

1.0541 1.1134 1.1637 1.1885

0.0672 0.0949 0.1026 0.1101

0.987 1.019 1.061 1.078

6a 7a 10a 13a

0.9921 1.0201 1.0595 1.0691

0.0265 0.0268 0.0203 0.0203

0.0007 0.0007 0.0004 0.0004

1.093 1.064 1.004 0.995

Steel grade

S235 S355

176 48

1.1355 1.1094

0.1079 0.1053

1.028 1.004

28 8a

1.0057 1.0186

0.0260 0.0351

0.0007 0.0012

1.076 1.093

Total

–

224

1.1299

0.1076

1.022

27

0.991

0.025

0.0006

1.089

Type of loading (except columns)

a

0-triangular 1-constant Distributed load

When NLin ≤ 20, a sub-set of 20 results is used.

higher resistance. The scatter of results from the various methods is larger for the HEB 300 and for the triangular bending moment distribution. Fig. 15 compares the resistance of two beam–columns obtained using the General Method, either with the in-plane resistance calculated numerically (GMNIA in-plane) and the elastic critical load from expression (21) or with Eqs. (16) and (17) with the results of clause 6.3.3. χLT is calculated according to the General Case from EC3-1-1 and Method 1 is adopted. χop is determined as the minimum of χz or χLT . Conservative results are noted for the General Method for the HEB 300. Results are plotted in the form of interaction curves for IPE 200 and HEB 300 cross-sections. They are compared with the results from clauses 6.3.1 to 6.3.3. 4.5. Statistical evaluation Annex D of EN 1990 (CEN, 2002) provides a methodology for the evaluation of the reliability of design models, by comparing the results obtained using the design model with experimental results.

The evaluation of the partial safety factor, γRd , is carried out according to the procedure described in detail in [9,13]. Table 13 describes the sub-sets considered for the statistical evaluation. Table 14 summarizes the results for the safety factor γRd for each sub-set and the following methodologies:

• EC3_6.3.3(M1)—resistance according to clause 6.3.3; • EC3_6.3.4(M1)—resistance according to clause 6.3.4 considering the expressions from Section 2;

• EC3_6.3.3(num)—resistance according to clause 6.3.4 considering the GMNIA in plane and LEA results (this case applies to the shell element numerical simulations only). Two sets of results are considered: GMNIA results with shell elements (results from LUSAS [12]) and GMNIA results with beam elements (results from Ofner [11]). No clear difference is observed between the chosen methods for the safety factor γRd . For most cases, the safety factor is slightly higher than unity. Regarding the slenderness sub-set, the lowest values of γRd are achieved for the low slenderness range; regarding

Author's Personal Copy L. Simões da Silva et al. / Journal of Constructional Steel Research 66 (2010) 575–590

589

Table A.6 Parametric study with shell elements: 6.3.4—Numerical results. Sub-set

Level of sub-set

N

Rm

σ

Rm − σ

NLin

Rm

Q ≡ σ∆

(Vδ )2

γRd

Cross-section

HEB 300 IPE 200 IPE 500

112 112 –

1.1989 1.0749 –

0.0990 0.0518 –

1.100 1.023 –

28 19a –

1.0647 1.0043 –

0.0280 0.0182 –

0.0008 0.0003 –

1.023 1.052 –

–

Type of loading (except columns)

−1-bi-triangular 0-triangular 1-constant Distributed load

32 128 32

– 1.1690 1.1401 1.1260

– 0.0927 0.1080 0.0977

– 1.076 1.032 1.028

– 5a 22 5a

– 1.1081 1.0064 1.0632

– 0.0474 0.0194 0.0302

– 0.0022 0.0004 0.0009

– 1.043 1.054 1.032

Level of loading (Φ )

0 (Beam) √ ]0; 3/3[ √ √ [ 3/3; 3[ √ [ 3; ∞[ ∞ (Column)

64 32 64 32 32

1.1001 1.1794 1.1647 1.1464 1.1029

0.0867 0.1185 0.1051 0.0951 0.0643

1.013 1.061 1.060 1.051 1.039

9a 6a 10a 4a 1a

1.0109 1.1015 1.0575 1.0787 1.0641

0.0247 0.0577 0.0323 0.0355 0.0194

0.0006 0.0033 0.0010 0.0013 0.0004

1.067 1.084 1.044 1.033 0.997

Slenderness (zz)

]0; 0.8] ]0.8; 1.2] ]1.2; 1.8] ]1.8; ∞[

56 56 56 56

1.0691 1.1174 1.1702 1.1909

0.0660 0.0865 0.0970 0.1023

1.003 1.031 1.073 1.089

5a 8a 7a 9a

1.0087 1.0352 1.0788 1.0846

0.0180 0.0243 0.0192 0.0190

0.0003 0.0006 0.0004 0.0004

1.047 1.040 0.983 0.977

Steel grade

S235 S355

176 48

1.1398 1.1262

0.1026 0.0922

1.037 1.034

23 8a

1.0136 1.0445

0.0168 0.0359

0.0003 0.0013

1.038 1.068

Total

–

224

1.1369

0.1004

1.037

31

1.011

0.018

0.0003

1.046

a

When NLin ≤ 20, a sub-set of 20 results is used.

the type of cross-section, the stockier cross-section HEB 300 has the lowest results. 5. Conclusions The General Method is clearly an interesting approach to deal with the verification of the safety of structural components. In particular, it avoids splitting a member into several segments for the evaluation of the relevant χ -factors. Further, it avoids using segments with different lengths for different stability phenomena. For uniform members, it is possible to directly compare the evaluation of resistance using clauses 6.3.1 to 6.3.3 and the General Method (6.3.4). For columns, when Eq. (10) is analyzed for a range of profiles and lengths, it is seen that the resistance of the member according to clause 6.3.4 when compared with clause 6.3.1 decreases with the increase of member length. For members in bending the General Method gives exactly the same results as clause 6.3.2 as χop = χLT , see Eq. (11). Concerning beam–columns, Eq. (19) leads to safe results of clause 6.3.4 (>80%) and also to unsafe results (