Eurocode 3 - General rules and ruled for buildings

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Eλ = π = ε93,91 f y235ε = 2(f in N/mm )yf y

NOTE B For elastic buckling of components of building structures see Annex BB.

(2) For flexural buckling the appropriate buckling curve should be determined from Table 6.2.

EN 1993-1-1: 2005 (E)6.3.1.4 Slenderness for torsional and torsional-flexural buckling

(1) For members with open cross-sections account should be taken of the possibility that the resistance of the member to either torsional or torsional-flexural buckling could be less than its resistance to flexural buckling. λ

(2) The non-dimensional slenderness for torsional or torsional-flexural buckling should be taken as:

TAf yλ = for Class 1, 2 and 3 cross-sections (6.52)

T N crA feff yλ = for Class 4 cross-sections (6.53)

T N cr= <N N but N Nwhere cr cr , TF cr cr, Tis the elastic torsional-flexural buckling force;

Ncr,TF is the elastic torsional buckling force.Ncr,T(3) For torsional or torsional-flexural buckling the appropriate buckling

curve may be determined fromTable 6.2 considering the one related to the z-axis.

6.3.2 Uniform members in bending

6.3.2.1 Buckling resistance

1. A laterally unrestrained member subject to major axis bending should be verified against lateral-torsional buckling as follows:

M ≤Ed 1,0 (6.54)
M b , Rd is the design value of the moment
where M EdM is the design buckling resistance moment.

2. Beams with sufficient restraint to the compression flange are not susceptible to lateral-torsional buckling. In addition, beams with certain types of cross-sections, such as square or circular hollow sections, fabricated circular tubes or square box sections are not susceptible to lateral-torsional buckling.
3. The design buckling resistance moment of a laterally unrestrained beam should be taken as:

f y= χM W (6.55)
b , Rd LT y γ M 1
where W is the appropriate section modulus as follows:
y W = W for Class 1 or 2 cross-sections– y pl,yW = W for Class 3 cross-sections– y el,yW = W for

Class 4 cross-sections– y eff,yχ is the reduction factor for lateral-torsional buckling.

LTNOTE 1 For determining the buckling resistance of beams with tapered sections second order analysis according to 5.3.4(3) may be performed. For out-of-plane buckling see also 6.3.4.

NOTE 2B For buckling of components of building structures see also Annex BB.60 EN 1993-1-1: 2005 (E)(4) In determining W holes for fasteners at the beam end need not to be taken into account.

y6.3.2.2 Lateral torsional buckling curves – General case(1) Unless otherwise specified, see 6.3.2.3, for bending members of constant cross-section, the value ofχ for the appropriate non-dimensional slendernessλ , should be determined from:

LT LT1χ = χ ≤but 1, 0 (6.56)

LTLT 2Φ + Φ − λ2 LTLT LT[ ]( ) 2Φ = + α λ − + λ0,5 1 0, 2

where LT LTLT LTα is an imperfection factor

LT W fy yλ =LT M crM is the elastic critical moment for

lateral-torsional buckling_cr(2) M is based on gross cross sectional properties and takes into account the loading conditions, the real_crmoment distribution and the lateral restraints.α

NOTE The imperfection factor corresponding to the appropriate buckling curve may be obtainedLT αfrom the National Annex. The recommended values are given in Table 6.3.

Imperfection factor LT

The recommendations for buckling curves are given in Table 6.4.

Values of the reduction factor for the appropriate non-dimensional slendernessλ may beLT LTobtained from Figure 6.4. M 2λλ

λ≤ ≤(4) For slendernesses (see 6.3.2.3) or for (see 6.3.2.3) lateral torsional bucklingEdLT LT,0 LT,0Mcreffects may be ignored and only cross sectional checks apply. 61EN 1993-1-1: 2005 (E)

6.3.2.3 Lateral torsional buckling curves for rolled sections or equivalent welded sectionsχ for the appropriate non-(1) For rolled or equivalent welded sections in bending the values of LTdimensional slenderness may be determined fromχ ≤ 1, 0LT1 1χ = but  (6.57)χ ≤LT 2  LTΦ + Φ − β λ2 2LTλLTLT LT[ ]( ) 2Φ = + α λ − λ + βλ0,5 1 LT LT , 0 LTLT LT λ βNOTE The parameters and and any limitation of validity concerning the beam depth or h/bLT , 0ratio may be given in the National Annex. The following values are recommended for rolled sectionsor equivalent welded sections:λ = 0, 4 (maximum value)LT , 0β = 0,75 (minimum value)The recommendations for buckling

Curves are given in Table 6.5.

(2) For taking into account the moment distribution between the lateral restraints of members the χ reduction factor may be modified as follows:

LTχχ = χ ≤LT 1but (6.58)LT , mod LT , modf

NOTE The values f may be defined in the National Annex. The following minimum values are recommended:= − − − λ − ≤2f 1 0,5(1 k )[1 2, 0( 0,8) ] but f 1,0LTck is a correction factor according to Table 6.6

6.3.2.4 Simplified assessment methods for beams with restraints in buildings

(1)B Members with discrete lateral restraint to the

compression flange are not susceptible to lateral-λtorsional buckling if the length L between restraints or the resulting slenderness of the equivalent compression flange satisfies:

Mk L c , Rdλ = ≤ λc c (6.59)

f c 0λi Mf , z 1 y , Edis the maximum design value of the bending moment within the restraint spacing

where M y,Ed f y=M Wc , Rd y γ M1W is the appropriate section modulus corresponding to the compression flange

yk is a slenderness correction factor for moment distribution between restraints, see Table 6.6

c is the radius of gyration of the equivalent compression flange composed of the compressioni f,z flange plus 1/3 of the compressed part of the web area, about the minor axis of the section

λ is a slenderness limit of the equivalent compression flange defined above

c,0 Eλ = π = ε93,91 f y235ε = 2(f in N/mm )yf y 63

EN 1993-1-1: 2005 (E)

NOTE 1B For Class 4 cross-sections i may be taken asf,zI eff , f=i f , z 1+A

A_eff, f_eff, w, c₃ is the effective second moment of area of the compression flange about the minor axis where I_eff,f of the section A is the effective area of the compression flange eff,f is the effective areas of the compressed part of the web A_eff,w,c λ may be given in the National Annex. A limit value

NOTE 2B The slenderness limit c 0λ = λ + 0,1 is recommended, see 6.3.2.3.

c 0 < T , 0 < λ(2)

NOTE B If the slenderness of the compression flange exceeds the limit given in (1)B, the design buckling fresistance moment may be taken as:

= χ ≤M k M M M but (6.60)b , Rd f c , Rd b . Rd c . Rdlχ λ is the reduction factor of the equivalent compression flange determined with where fk is the modification factor accounting for the conservatism of the equivalent compression flange fl method =k 1,10

NOTE B The modification factor may be given in the National Annex. A value is fl recommended.(3)

NOTE B The buckling curves to be used in (2)B should be taken as follows: h ≤

ε44curve d for welded sections provided that: t fcurve c for all other sections

where h is the overall depth of the cross-section

t is the thickness of the compression flange

NOTE B For lateral torsional buckling of components of building structures with restraints see also Annex BB.3.6.3.3 Uniform members in bending and axial compression

(1) Unless second order analysis is carried out using the imperfections as given in 5.3.2, the stability of uniform members with double symmetric cross sections for sections not susceptible to distortional deformations should be checked as given in the following clauses, where a distinction is made for:

members that are not susceptible to torsional deformations, e.g. circular hollow sections or sections– restraint from torsion

members that are susceptible to torsional deformations, e.g. members with open cross-sections and not– restraint from torsion.

(2) In addition, the resistance of the cross-sections at each end of the member should

satisfy the requirements given in 6.2.

NOTE 1: The interaction formulae are based on the modelling of simply supported single span members with end fork conditions and with or without continuous lateral restraints, which are subjected to compression forces, end moments and/or transverse loads.

NOTE 2: In case the conditions of application expressed in (1) and (2) are not fulfilled, see 6.3.4.

(3) For members of structural systems, the resistance check may be carried out on the basis of the individual single span members regarded as cut out of the system. Second order effects of the sway system.