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5.5.2 Lateral-torsional buckling of beams(1) The design buckling resistance moment of a laterally unrestrained beam shall be taken as:

(5.48)where i3 w

PwPw

1 for Class 1 or Class 2 cross-sectionsWeI.y I'NpI .v for Class 3 cross-sectionsWeff.v /W pI.v for Class 4 cross-sections

and X L T is the reduction factor for lateral-torsional buckling.(2) The value of XLT for the appropriate non-dimensional slenderness ALTmay be determined

from:

but XLT S 1 (5.49)

in which:

(3) The values of the imperfection factor aLT for lateral torsional buckling should be taken as:aLT = 0,21 for rolled sectionsaL T = 0,49 for welded sections

(4) Values of the reduction factor X L T fqf the_appropriate non-dimensional slenderness A L T maybe obtained from table 5.5.2 with A = A LT and X = X L T ' using: for rolled sections:curve a (a = 0,21)

for welded sections:curve c fa = 0,49)

(5) The value of ALT may be determined from:"LT = l8w Wp1.y fy/Merlo.s = [ALT/A,l 18wlo.s

where ", = " [E/fyJo.s = 93,9 = [235/fylo.s (fy in N/mm2)

and Mer is the elastic critical moment for lateral-torsional buckling.(6) Information for the calculation of Mer (or for the direct calculation of ALT) is given ininformative Annex F.(7) Where the non-dimensional slenderness ALT S 0,4 no allowance for lateral-torsional

buckling is necessary.(8) A beam with full restraint does not need to be checked for lateral-torsional buckling.

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5.5.3 Bending and axial tensionIII Members subject to combined bending and axial tension shall be checked for resistance to

ta t er at-tors.onat buckling, treating tn e axial force and bending moment as a vec:oriai effect,see 2.3.3.1 (4).

(2J Where the axial force and bending moment can vary independently, the design vetue of tneaxial tension should be multiplied by a reduction factor for vectorial effects:

Ii J vec = ~

(3) The net calculated stress acom.Ed (which can exceed fy) i n the extreme compression fibredue to the vectorial effects should be determined from:. ) " " r - - - (jh'r':

(5.50Jwhere Wcom is thi: etestic section modulus for the extreme compression fibreand Nt.Sd is the design value of the axial tension

. l ' i J ;

(4) The verification should be carried out using an effective design internal moment M"ff.Sdobtained from:

(5) The design buckling resistance moment Mb.Rd should be obtained using 5.5.2.5.5.4 Bending and axial compression(1) Members with Class 1 and Class 2 cross-sections subject to combined bending and axial

compression shall satisfy:

ky My.sdWp1 .y\. fy/YMl (5.51)+

in which 1 _ Jiy NSdx ; A fy but ky :S 1.5y =

-Jiy = Ay (2/3My - 4) + but Jiy :S 0,901 _ J iz NSd!~ A fy"--~ / but kz :S 1,5

but Ji z :S 0,90

Xmin is the lesser of Xy and Xz

where Xy and X z are the reduction factors from 5.5.1 for the y-y and z - z axes respectivelyand PMyand PMxare equivalent uniform moment factors for flexural buckling, see (7l.

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1 5 > , " ' y; " " ' ? -(2) Members with Class 1 and Class 2 cross-sections for which lateral-torsional buckling is a

potential failure mode shall also satisfy:

Page 1OSENV 1993-1-1:1992

k z M z .Sd'Npl z f 'f 'YM1

:$ 1

in whicn = , _ jiLT NSdx, A f y but kL T :$ 1jiLT 0,15 tlz f3 M .LT - 0,15 but jiLT S 0,90

where f 3 M.LT is an equivalent uniform moment factor for lateral-torsional buckling, see (7).(3) Members with Class 3 cross-sections shall satisfy:

+ ky M .y.SdWel.y fy lYM1 (5.53)

Jwhere ~,~ and X min are as in (1)

Jiy = tly.(2 f 3 M y - 4) but Jiy :$ 0,90and Ji z = til (2 f3 Mz - 4) but Ji z :$ 0,90

(4 ) Members with Class 3 cross-sections for which lateral-torsional buckling is a potentialfailure mode shall also satisfy:I

+ (5.54)II (S) Members with Class 4 cross-sections shall satisfy:kz(Mz.Sd + NSd eNz) :$ 1WeH .z f ylYM 1 (5.56)II

where ~,Is and Xmin are as in (1), but using A.H instead of A, see 5.3.5(9)J. ly and J .l z are as in (3). but adding NSd eN to MSd when determining 1 3

and AeH ' WeH .y , Wef f.z ' eNy and eNz are as in 5.4.8.3.(6 ) Members with Class 4 cross-sections for which lateral-torsional buckling is a potential

failure mode shall also satisfy:NSd + k L T( My.Sd + NSd eNy) + kz( Mz.Sd + NSd eNz) :s 1

X z A.ff f Y / Y M 1 XL T Welf .y f Y / Y M 1 W.H.z f Y / Y M 1 (5.57) (7 ) where k LT is as in (2). but using A .H instead of A , see 5.3.5(9)and J. lLT is as in (21. but adding NSd aNy to My.S

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I =--::Ulvaient '..:nliorm"T1orrent 'actorend moments

Ml~:/;Ml-1 ~:/; .s 1

~ , = 1,8 - 0,7 :/;M,moments due to in-planelateral loads

{3M,O = 1,3~M,O = 1,4

moments due to in-planelateral loads plus end moments

Ml~~~~MQ t

~

Me = [rnax M I due to lateral load only

for moment diagramwithout change of sign

[

,max M I6M

Imax MI-t- I min M I for moment diagramwith change of sign

Figu re 5.5.3 Equivalent uniform moment factors

I

,

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Lateral torsional buckling. ..nnex F [informative]F.1 Elastic critical moment

2 ] 0 , 5Glt+--n Z E lz

(F. 1)

F.1.1 Basis(1) The elastic critical moment for lateral-torsional buckling of a beam of uniform symmetrical

cross-section with equal flanges, under standard conditions of restraint at each end, loadedthrough its shear centre and subject to uniform moment is given by:

where G

ItIwIz

and L

E {1 + vlis the torsion constantis the warping corrstantis the second moment of area about the minor axis

s the length of the beam between points which have lateral restraint.(2) The standard conditions of restraint at each end are: estrained against lateral movementrestrained against rotation about the longitudinal axisfree to rotate in plan F.1.2 General formula for cross-sections symmetrical about the minor axis 1) In the case of a beam of uniform cross-section which is symmetrical about the minor axis,

for bending about the major axis the elastic critical moment for lateral-torsional buckling isgiven by the general formula: where C" C2 and C3 are factors depending on the loading and end restraint conditions and kw are effective length factorsZa is the coordinate of the point of load application

z, is the coordinate of the shear centreNote: See F.l.2(7) and (8) for sign conventions and F.l.4(2) for approximations for z , I/ ,

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fIIIIIIIII (8)II

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(2) The effective length factors k and kw vary from 0,5 for full fixity to 1,0 for no fixity, with0,7 for one end fixed and one end free.]) The rector < reiers to end rotation on plan. It is enetocous to me ratio i.t: for a

corroressicn r r : erno er,(4) The factor : refers to end warping. Unless special provision for warping fixity is made,

k w should be taken as t, O .(5 ) Values of C" Cz and C3 are given in tables F.l.1 and F.l.2 for various load cases, asindicated by the shape of the bending moment diagram over the length L between lateralr:~~Eair1!~_._Values are given corresponding to various values of k. . --... .~..(6) For cases with k = t , 0 the value of C 1 for any ratio of end moment loading as indicated

in table F. 1. 1, is given approximately by:C, = 1,88 -1,401.11 + 0,52t;? but C1 S 2,70 (F.3)

(7) The sign convention for determining Zj , see figure F.l .1, is:Z is positive for the compression flangeZj is positive when the flange with the larger value of Iz is in compression at thepoint of largest moment.

The sign convention for determining z~ is:for gravity loads z~ is positive for loads applied above the shear centrein the general case z'il is positive for loads acting towards the shear centre fromtheir point of application.

F.l.3 Beams with uniform doubly symmetric cross-sectionsI (1)III

For doubly symmetric cross-sections Zj = 0, thus:

(F.4)

(2) and for transverse loads applied at the shear centreor end-moment loading CzZg = 0. For these cases:

M er = C, "lEI, [ [ ~ l ' ' : : . (kL)zGlt rkU2 kw Iz rr2Elz(3) When k = kw = 1,0 (no end fixitvl:M er C, rr2Elz [,:: L'GI,r2 Iz rr2Elz

(F.S)

(F.6)

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Table F.1.1 Values of factors C" C2 and C3 corresponding :0 values ofracrcr k: E:-:d .'":1or.-:ent:oadingl..c:aclng and scccorr Bending moment diagram Vaiue I v arues O T ;ac:::rs .

i conditions o r .< II - I I ,' - , C2 I C J .!i i i = - , 1,0 1.000 1,000I I ii, I':! I Ii ; I : :: I 0,7 1,000 1,113I I : : ! : iii I i ! I ! I i \1 a , s 1,000 1, 1.to!

III = ~ ~- 1,0 1,141 0,998. l i I i ! 1 1 1 1 1 1 1 1 1 1 1 ill q 0,7 1,270 1,565a , s 1,305 2,283i i i = + Y :z 1,0 1,323 0,992I T W i ur 0,7 1,473 1.556l i\\i,l il!I!1J] 0,5 1,514 2,271i i i = ~ Y o 1,0 1,563 0.977

~ 0,7 1,739 1,531~ a , s 1,788 2,235M 1 1 1 M i i i = 0

~ - f"* '~ "'Z :< -< > "'~t \ 1,0 1,879 0,939~ 0,7 2,092 1,473. , 0,5 2,150 2,150III = - Y o

~1,0 2,281 0,8550,7 2,538 1,340---.. 0,5 2,609 1,957

i i i = - Y z~

1,0 2,704 0,6760,7 3,009 1,059~ 0,5 3,093 1,546

i i i = - Y or----___ 1,0 2,927 0,3660,7 3,009 0,575~ 0,5 3.093 0,837

III = - 1 1,0 2.752 0,000r---.._ 0,7 3.063 0,000~ a , s 3.149 0,000 1

III)

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I I Page 271ENV 1993-'-1:1992Table F .1.2 Values of factors C" Cz and C3 corresoonding to values offactor k: Transverse loading cases . l..oadl~;; .3GC succcrtccnoinons Sending momentciacr arn \lai"eso r o < C , C z C J1,0 1,132 0,~59 0,525 I0,5 0,972 0,304 0,980 w~oc z>

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z . . .l . ) (C o rnor e s s r cn l~ I I / s h e a r . .s-r - --.J...L---- -- I,I !--- - _...__ - -- - ~~ 'Centroid y

-~

I II I I (Tension)I

Zsz t/Shear centre

A 1 - -+-- - - -- (Compression)I1 I- ----I-- - ~CentrOjd y,Iu (Tension)I I

Fi9 u reF. 11 Sign convention for determining z J

.

. .

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Page 273ENV 1993-1-1: 1992F., .4 Beams with uniform monosymmetric cross-sections with unequal flanges(i) For an t -secnon wi,h unequal flanges: I...where P tIte is

1ft is

and h, is

iFf)

the second moment of area of the compression flange about the minoraxis of the sectionthe second moment of area of the tension flange about the minor axisof the sectionthe distance between the shear centres of the flanges. (2 ) The fol/owing approximations for zi can be used:when Pf > 0,5: (F.B)

~when Pf

zI

< 0,5:(F.!))

for sections with a lipped compression flange:(F. 10)Z i 0,8 (2/3f - 1)(1 + h/h) hsl2 when Pf > 0,5

1,0 (2/3f - 1)(1 + h/h) hsl2 when Pf < 0,5where hL is

II

(F. 11)

the depth of the lip

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F.2 SlendernessF 2.1 General

-i: I The sienderness ratio ,1 L T for lateral-torsional buckling is given by:

A , ~,where ,.I,

eflwflw

and flw

(F.12)T T [E/f J 0,5 = 93,9 ey

1 for Class 1 or Class 2 cross-sectionsWel.y I Wp1.y for Class 3 cross-sectionsWeff,y I Wp1,y for Class 4 cross-sections

(2) The geometrical slenderness ratio ALT for lateral-torsional buckling is given for all classesof cross-section, by:

ALT = [ n 2 EWp1,y I M,er J 0,5 (F.13)F.2.2 Beams with uniform doubly symmetric cross-sections(1) For cases with Zg = 0 (end-moment loading or transverse loads applied at the shear centre)

and k = kw 1,0 (no end fixity), the value of ALT can be obtained from:"'~"~'--'''- ''''''4'

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(4) For rolled I or H sections conforming with Reference Standard 2, the following conservativeapproximations can be used:

(C, )0.5

(F.19) A Lr = or I /'" 0,9 Lliz"l T =/ ~ ~ ~ : [ , d o [~;;J" ' 5 (F.20)

(5) For any plain lor H section with equal flanges, the following approximation is conservative: ( c + ; ~ / r ; ; ; : ] ] 0.25Cases with k < 1,0 and/or k w < 1,0 can be included by using:

(F.21 ) (6) kLlilT ;'or (F.23) (C, )0.5 tr~'?J~2"~--;kLlalT)2] 0.25/" kw 25,66~ .or for standard rolled I or H sections: AlT = kLlilT[ [~r.i, [ k L l i L T ] T 2 5(C, )0.5 kw 20 h/tfor AlT = 0,9 kLliz[ r52(C,)0.5 [':r 2~ [~~~;]

(F.24) (F.25)or for any plain I or H section with equal flanges:1 kLliz

+ _1 _20 [kLliz] 2] 0.25h/tf

(F.26)

II

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(7) Unless special provision for warping fixity is made, kw should be taken as 1 , O . . (8) Cases v irfJ transverse loading applied above the sheer centre tz : >~

centre tz _ < O.:;! C3n be inciu,dgc:LQyu~jJ1g:.,. 0,0) or below th e stieer .I 1 0.25 '~'.kL Wpl./ -"""'-I I z 1 . , 0 ; \- - - - - - - - - - - - - - - - - - - - - - - - . _ - - - - ~ - - - - - - - - - - - - - - - - - - - - ~ ~k r l " \ (F.27) .IC,~05~H [ k~]' - (~- :L :-)2-E l-~-" IC, Z,)' , ' : 1 . 5 - c, z, -r alternatively: -L T = IC,)05 {[[ k ~ ] ' (F.28) -or for standard rolled I or H sections:

[k L i i L T ] 2 +h/t,

[ 2C~. z g ] ' ] 0.5 _ 2C~. Z g I 05(F.29) -

120 -0,9 k L i i z (F.30)or alternatively: ! [ [ ~ J + _1 [ k L l i Z ] 2 + [2C2 z g ] 2 ] 0 , 5 _ 2C2 Z g 1 ' 5kw 20 h/tf h, h,

or for any plain I or H section with equal flanges: [ k L l i Z ] 2 + [2C2 -l ] , 5 _ 2C2 z g l o . 5h/tf h, h, (F.31 )