lateral torsional stability of timber beams€¦ · proceedings of the 6th international conference...

Proceedings of the 6th International Conference on Mechanics and Materials in Design, Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015

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PAPER REF: 5742

LATERAL TORSIONAL STABILITY OF TIMBER BEAMS Ivan Baláž1(*), Yvona Koleková2 1Department of Metal and Timber Structures, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Bratislava, Slovak Republic, European Union 2Department of Structural Mechanics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Bratislava, Slovak Republic, European Union (*)

Email: [email protected] ABSTRACT

Lateral torsional stability of timber beams with monosymmetric cross-sections. Proposals are given (Baláž, Koleková, 2000) for approximate formulae enabling to compute the values of elastic critical moments crM of beams under different loadings and various boundary

conditions. These formulae were accepted by prEN 1999-1-1: May 2004 and EN 1999-1-1: May 2007 for design of aluminium structures and are used also for design of steel structures. It is shown that they may be used for design of timber structures too. The aim is to unify different Eurocodes procedures.

Keywords: timber, critical bending moment, lateral torsional stability.

INTRODUCTION

A beam bent about major axis in its stiffer principal plane may buckle out of that plane by deflecting laterally and twisting. Eurocode EN 1995-1-1: 2004 requires to verify the resistance of timber beam as follows

dmcritdm fk ,, ≤σ , (1)

where

dm,σ is the design bending stress,

d,mf is the corresponding design bending strength,

critk is a factor which takes into account the reduced strength due to lateral buckling due to

beam imperfections. For beams with an initial lateral deviation from straightness within the limits defined in EN 1995-1-1: 2004, section 10, critk may be determined as follows:

1kcrit = for 75.0, ≤mrelλ

mrelcritk ,75.056.1 λ−= for 4.175.0 , ≤< mrelλ

2,

1

mrel

critkλ

= for mrel ,4.1 λ<

The factor critk may be put equal to 1.0 for a beam where lateral displacement of the

compression side is prevented throughout its length and where rotation is prevented at the supports.

The relative slenderness for bending is defined by

Symposium_29 Safety in Wood Materials

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critm

km

mrel

f

,

,, σ

λ = (2)

where

kmf , is corresponding characteristic bending strength,

critm,σ is the critical bending stress calculated according to the classical theory of stability,

with 5- percentile stiffness values.

The critical stress crit,mσ at which buckling of an ideal beam takes place is

y

cr

critmW

M=,σ (3)

The same concept of lateral torsional verification was used in various publications, national standards, prestandard Eurocodes and their National Application Documents or in Eurocodes and their National Annexes. These procedures differ in critical moment crM calculations.

ELASTIC CRITICAL MOMENT crM

Calculation of critical moment crM is the classical problem of theory of elasticity. There are

many more or less exact approximate formulae for calculation of crM value. It was shown by

(Hooley and Madsen, 1964), that formulae for elastic critical moment may be used also for timber beams. Today there are also many computer programs which enable to compute crM

value for any boundary conditions and any loading. Standards contain: a) no formulae for computing of crM value (EN 1993-1-1:2005), or b) limited number of approximate formulae

valid only for some basic boundary conditions of beams loaded by only basic action types, e.g. EN 1995-1-1:2004, DIN 1052:2004), or c) general formulae for a lot of loading types and many combinations of boundary conditions (e.g. EN 1999-1-1:2007, which is based on works (Baláž, 1999), (Koleková, 1999), (Baláž, Koleková, 1999, 2000a, 2000b, 2002a, 2002b).

Formulae defining crM value are valid for members made from any structural material. It is

necessary, of course, to use corresponding material properties G,E (e.g. for steel, aluminium alloy, concrete or timber). Specialists designing timber beams in bending may use the general formulae for crM given in EN 1999-1-1:2007, when they will use material properties of

timber E = E0.05 and G = G0.05.

The value of elastic critical moment crM of the reference beam, which is characterized by

− uniform non-warping and doubly-symmetrical cross-section, − standard boundary conditions (the beam is simply supported on both ends in all

three cases: bending in xz-plane, bending in xy-plane and in torsion about axis x, which may be expressed through the factors ky=1.0, kz=1.0, kw=1.0, see e.g. (Baláž, Koleková, 2000a, 2000b, 2002a, 2002b) or prEN 1999-1-1: May 2004; EN 1999-1-1: May 2007,

− and it is loaded by equal end moments M and ψ.M, ψ = 1.0, (moment line distribution is uniform),

may be calculated from simple formula


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L

GIEIM

tzcr

π= (4)

where: It is the torsion constant,

Iz is the second moment of area about the minor axis,

L is the length of the beam between points that have lateral restraint.

All other cases may be transformed in above defined equivalent reference beam by replacing the length L by the effective length Lef which can take into account the following influences

− mono-symmetry of the cross-section (through mono-symmetry parameter ζj) − warping of the cross-section (through the warping constant Iw, or wtκ ),

− any combination of boundary conditions, − any loading type, − the location of point of load application in cross-section (through gz or az).

The general formula may be written in the form

efL

GIEIM tz

cr

π= (5)

The effective length Lef can be calculated according to various standards and publications.

ACCORDING TO EN 1999-1-1: MAY 2007 or (Baláž, Koleková, 2000a, 2000b, 2002a, 2002b) it is possible to calculate the effective length may be computed from the formula

cref LL µ/= (6)

a) Elastic critical moment crM of beams with uniform monosymmetric cross-sections

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 is given by the general formula

L

GIEIM tz

crcr

πµ= (7)

where relative non-dimensional critical moment crµ is

[ ])()(1 j3g22

j3g22wt

z

1cr ζζζζκµ CCCC

k

C−−−++= (8)

non-dimensional torsion parameter is

t

w

wwt

GI

EI

Lk

πκ = (9)

relative non-dimensional coordinate of the point of load application related to shear center

t

z

z

gg

GI

EI

Lk

zπζ = (10)


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relative non-dimensional cross-section mono-symmetry parameter

t

z

z

jj

GI

EI

Lk

zπζ = (11)

where: C1, C2 , C3 are factors depending mainly on the loading and end restraint conditions (see Tables 1 and 2), kz , kw are buckling length factors,

sag zzz −= (12)

az is the coordinate of the point of load application related to centroid,

sz is the coordinate of the shear center related to centroid,

gz is the coordinate of the point of load application related to shear center.

The sign convention for determining zg is for: − gravity loads zg is positive for loads applied above the shear centre,

– general case zg is positive for loads acting towards the shear centre from their point of application

dAzzyI

zzA

∫ +−= )(5.0 22

ysj (13)

0j =z ( 0j =y ) for cross sections with y-axis (z-axis) being axis of symmetry.

The following approximation for zj can be used:

+=

f

sfjh

chz

2145.0 ψ (14)

where c is the depth of a lip hf is the distance between centerlines of the flanges.

ftfc

ftfc

fII

II

+

−=ψ (15)

Ifc is the second moment of area of the compression flange about minor axis of the section,

Ift is the second moment of area of the tension flange about the minor axis of the section,

hs is the distance between the shear centre of the upper flange and shear centre of the bottom

flange (Su and Sb).

For an I-section with unequal flanges without lips and as an approximation also with lips

2

z2fw 2)1(

−= shII ψ (16)

The sign convention for determining z and zj is as follows − coordinate z is positive for the compression flange. When determining zj from formula in

(14), positive coordinate z goes upwards for beams under gravity loads or for cantilevers


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under uplift loads, and goes downwards for beams under uplift loads or cantilevers under gravity loads

− sign of zj is the same as the sign of cross-section mono-symmetry factor ψf in (15). Take the cross section located at the M-side in the case of moment loading, Table 1, and the cross-section located in the middle of the beam span in the case of transverse loading, Table 2.

b) Beam with uniform cross-section symmetrical about major axis, centrally symmetric and doubly symmetric cross-section

Fig. 1 - Beams with uniform cross-sections symmetrical about major axis, centrally symmetric and doubly

symmetric cross-sections

For these beams loaded perpendicular to the mayor axis in the plane going through the shear centre, zj = 0, thus

[ ]ggwt CCk

Cζζκµ 2

22

2

z

1cr )(1 −++= (17)

crµ values and factors C1 and C2 may be found in (Baláž, Koleková, 2000b).

For end-moment loading C2 = 0 and for transverse loads applied at the shear centre zg = 0. For these cases

2

z

1cr 1 wt

k

Cκµ += (18)

If also 0=wtκ

z

1cr

k

C=µ (19)

Values of C1, C2 and C3 are given in Tables 1 and 2 for various loading cases, as indicated by the shape of the bending moment diagram over the length L between lateral restraints. The values given in Table 1 correspond to various values of kz and in Table 2 correspond to various combinations of values kz and kw .

For cases with kz = 1,0 the values of C1 for any ratio of end moment loading as indicated in Table 1, are given approximately by formula

5.021 )262.0428.0310.0( −++= ψψC (20)


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Table 1 - Values of factors 1C and 3C corresponding to various end moment ratios ψ , values of buckling length

factor zk and cross-section parameters fψ and wtκ . End moment loading of the simply supported beam with buckling length factors 0.1y =k for major axis bending and 0.1w =k for torsion

1) For beams with rectangular cross-section the values in bold may be used. They may be replaced by approximate values obtained from the formula (17). 2) ( ) 1,1wt0,11,10,11 CCCCC ≤−+= κ , ( 0,11 CC = for 0wt =κ , 1,11 CC = for 1wt ≥κ ).

3) =L7,0 left end fixed, R7,0 = right end fixed. 4) In EN 1999-1-1: May 2007 there are incorrectly: f75,09,0 ψ− instead of correct value f77,09,0 ψ− taken from (Baláž, Koleková,2000b), f9,023,0 ψ− instead of correct value f9,0225,0 ψ− taken from (Baláž, Koleková,2000b).


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Table 2 - Values of factors C1, C2 and C3 corresponding to various transverse loading cases, values of buckling length factors yk , zk , wk , cross-section monosymmetry factor fψ and torsion parameter wtκ

1) For beams with rectangular cross-section the values in bold may be used. 2) ( ) 1,1wt0,11,10,11 CCCCC ≤−+= κ , ( 0,11 CC = for 0wt =κ , 1,11 CC = for 1wt ≥κ ).

3) Parameter fψ refers to the middle of the span. 4) Values of critical moments Mcr refer to the cross-section, where Mmax is located.


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c) Beam with rectangular cross-section

The formulae (17-19) given in previous paragraph are valid also for rectangular cross-section. Parameter wtκ in (17, 18) may be calculated with the help of the following formulae.

Torsion constant

35

052.063.013

1hb

h

b

h

bIt

+−= (21)

Warping constant

0=wtI m6, 33532

067.197.4884.41144

10 hb

h

b

h

b

h

bIII wnwtw

−

+

−+=+= (22)

For the timber beams with ratios

,16

1,0.1,15,5

05.0

05.0 ≈==<>E

G

E

Gk

b

h

h

Lw (23)

we obtain that 1853.0≤wtκ . In many cases the value of wtκ may be neglected and formula

(17) can be simplified by inserting 0=wtκ into the form

[ ]gg CCk

Cζζµ 2

22

z

1cr )(1 −+= (24)

For end-moment loading C2 = 0 and for transverse loads applied at the shear centre zg = 0 and from formula (24) we obtain simple formula(19).

For beams supported on both ends ( 1=yk , 1=zk , 15,0 ≤≤ wk ) or for beam segments

laterally restrained on both ends, which are under any loading (e.g. different end moments combined with any transverse loading), the following value of factor 1C may be used in the formula (18) to obtain approximate value of the critical moment

5,27,1

275,0

25,0

225,0

max1 ≤

++=

MMM

MC (25)

where

maxM is maximum design bending moment,

75,025,0 , MM are design bending moments at the quarter points and

5,0M is design bending moment at the midpoint of the beam or beam segment with length

equal to the distance between adjacent cross-sections which are laterally restrained.

Factor C1 defined by (25) may be used also in formula (17), but only in combination with relevant value of factor C2 valid for given loading and boundary conditions. This means that for the six cases in Table 2 with boundary condition 15,0,1,1 ≤≤== wzy kkk , as defined

above, the value 5,02 =C may be used in (17) together with factor C1 calculated from (25) as an approximation. Similar approximation is used in German DIN 18 800: 1990, Part 2.


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d) Beam with rectangular cross-section under uniform bending moment

For all above given crM formulae the influence of curvature due to major axis bending was ignored. It has been shown in (Broude, 1953, p.85), see also (Baláž, Koleková, 1999, p.61), that the influence of pre-buckling displacements is positive one. For special case of doubly symmetric cross-section of the beam under uniform moment with 0.1=wk , 0.1=zk it leads

to the increase of the critical moment described by the factor 0.1≥ξ

( )2

2

, 1

111

1wt

wt

z

t

y

z

y

z

crcr

EI

GI

I

I

I

I

κ

κ

µξµ ξ +

+−

−

== (26)

Formula (26) is valid only for above mentioned special case. It is often used as an evidence that no lateral torsional buckling occurs if zy II ≤ . Such rules may be found in many codes

including German ones. It is necessary to stress, that generally it is not true. No lateral torsional buckling will occur if zy II ≤ only if point of load application in to cross-section is

not too high above the centre of shear. Examples of factor ξ calculation

(i) IPE 360: m5.4=L , 064.0=yz II , 029.1=K , 118.0=zt EIGI , 035.1=ξ .

(ii) HD 360x162: m5.4=L 360.0=yz II 526.1=K , 078.0=zt EIGI , 255.1=ξ .

The factor ξ increasing the critical moment due to pre-buckling displacements is for doubly

symmetric beam under uniform bending with 0.1,0.1 == wz kk given in Table 3.

Table 3 - The coefficient ξ increasing the critical moment due to pre-buckling displacements

In the national and international standards coefficient ξ = 1 being on the safe side.

e) Cantilever with uniform cross-sections symmetrical about the minor axis

For bending about the major axis the elastic critical moment for lateral-torsional buckling is given by the formula (7), where relative non-dimensional critical moment crµ is given in Table 4 and 5. In Table 4 and 5 non-linear interpolation shall be used (see Fig. 3 and 4).

The sign convention for determining jz is below formula (16) and zg is below formula (12).


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Table 4 - Relative non-dimensional critical moment crµ for cantilever )2( wzy === kkk loaded by

concentrated end load F

1) For beams with rectangular cross-section the values in bold may be used. See Fig. 3. 2) For 0j =z , 0g =z and 8wt0 ≤κ : 2

wt0wt0cr 017,014,127,1 κκµ ++= .

For beams with rectangular cross-section 27,1cr =µ .

3) For 0j =z , 44 g ≤≤− ζ and 4wt ≤κ , crµ may be calculated also from formulae (17-19),

where the following approximate values of the factors 21, CC should be used for the cantilever under tip load F:

3wt

2wtwt1 5,062,2675,456,2 κκκ +−+=C , if 2wt ≤κ or 55,51 =C , if 2wt >κ .

4wt

3wt

2wtwt2 024,0245,0931,0566,1255,1 κκκκ −+−+=C , if 0g ≥ζ or

g2wtwt

2wtwt2 )013,0102,0032,0(054,0585,0192,0 ζκκκκ −+−−+=C , if 0g <ζ .

For beams with rectangular cross-section 56,21 =C and

255,12 =C , if 0g ≥ζ or g2 032,0192,0 ζ−=C , if 0g <ζ .


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Table 5 - Relative non-dimensional critical moment crµ for cantilever )2( wzy === kkk loaded by uniformly

distributed load q

1) For beams with rectangular cross-section the values in bold may be used. see Fig. 4. 2) For 0j =z , 0=gz and 80 ≤wtκ : 2

00 021,068,204,2 wtwtcr κκµ ++= .

For beams with rectangular cross-section 04,2cr =µ .

3) For 0j =z , 44 g ≤≤− ζ and 4wt ≤κ , crµ may be calculated also from formulae (14-16),

where the following approximate values of the factors 21, CC should be used for the cantilever under uniform load q:

3wt

2wtwt1 975,065,52,1111,4 κκκ +−+=C , if 2wt ≤κ , or 121 =C , if 2wt >κ .

4wt

3wt

2wtwt2 014,0153,0609,0068,1661,1 κκκκ −+−+=C , if 0g ≥ζ or

g2wtwt

2wtwt2 )0085,0074,0061,0(029,0426,0535,0 ζκκκκ −+−−+=C , if 0g <ζ .

For beams with rectangular cross-section 11,41 =C and

661,12 =C , if 0g ≥ζ or g2 061,0535,0 ζ−=C , if 0g <ζ .


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ACCORDING TO NATIONAL APPLICATION DOCUMENTS TO STANDARDS ČSN P ENV 1995-1-1: 1996; STN P ENV 1995-1-1: 2002

The effective length is defined as follows. The m values are given in Table 6.

mLLef = (27)

ACCORDING TO EN 1995-1-1: 2004

Lef = lef may be calculated using values from Table 6, see also Table 7.

Table 6 - Effective length as a ratio of the span (EN 1995-1-1: 2004)

Table 7 - Effective length efl (Colling, 2014)


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ACCORDING TO DIN 1052: 2004, ANNEX E.3

The effective length is given by

−

==

t

zz

eflef

GI

EI

L

aaa

LLkL

21

,

1

(28)

where

gz za = (29)

is the coordinate of the point of load application related to shear center.

Table 8 - Factors 1a and 2a (Neuhaus, 2009)

COMPARISONS OF CORRESPONDENT VALUES Lef/L ≈ 1/µcr , lef/L, kl,ef, m

The comparisons of corresponding values of Lef/L ≈ 1/µcr , lef/L, kl,ef, m of the beams under various loading types and boundary conditions taken from various publications are given in Table 9. From Table 9 it is clear that the procedure proposed in (Baláž, Koleková, 2002b), which was accepted by prEN 1999-1-1: May 2004 and EN 1999-1-1: May 2007, is the most general and the most exact one.


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The procedure given in EN 1995-1-1: 2004 is limited only to few cases and gives very rough values of Lef/L ≈ lef/L. The procedure from DIN 1052: 2004, which is offered to designers also in German National Annex to DIN EN 1995-1-1: December 2005 is less limited but contains incorrect value.

Authors recommend not to use the procedure given in prestandard Eurocodes ENV 1993-1-1:1992 Design of steel structures and ENV 1999-1-1:1998 Design of aluminium structures which contain a lot of problems and incorrect values C1, C2, C3. Authors criticized the draft of prEN 1993 which originally contained the same tables as above prestandards ENV. As a consequence of the criticism these tables were removed from the final version of EN 1993-1-1: March 2005. Eurocode for design of steel structures EN 1993 therefore does not contain any procedure for calculation of critical moment Mcr. Designers should avoid to use that part of publications containing ENV procedure for Mcr calculation, e.g. very good books (Hirt, Bez, 1998) and (Hirt, Bez, Nussbaumer, 2007).

Eurocodes prEN 1999-1-1: May 2004 and EN 1999-1-1: May 2007 accepted without change the procedure proposed by (Baláž, Koleková, 2000b). It may be found in Annex I. Prof. Sedlacek accepted this procedure in draft of DASt Richlinie 023, Stabilität von Tragwerken aus der Haupttragebene Biegeknicken und Biegedrillknicken.

The tables created in (Baláž, Koleková, 2000b) are used after modification in handbook (ECCS Technical Committee 8 – Stability Rules for Member Stability EN 1993-1-1, No. 119. 2006) without any reference to the original source. But the ECCS handbook No. 119 contains incorrect rules and some incorrect Ci values relating to Mcr calculation. Details see in (Baláž, Koleková, 2014a).

Authors of another ECCS publication (Dubina, Ungureanu, Landolfo, 2012) during copying the tables from handbook No.119 produced so many print errors in Ci values that using of these tables is dangerous.

Consequence of the facts that: (i) Eurocode EN 1993-1-1: March 2005 for design of steel structures contains no rules for Mcr calculation and (ii) Eurocode EN 1999-1-1: May 2007 for design of aluminium structures is not known to many designers is, that incorrect procedures given in prestandard ENV 1993-1-1:1992 or in two above ECCS publications are used in many books, National Annexes and in design practice.

In (Živner, 2010) there is a large collection of many procedures for Mcr calculation. The procedure (Baláž, Koleková, 2000b), which is in EN 1999-1-1: May 2007, was found as the best one. This was confirmed also in the Brazilian PhD thesis (Fruchtengarten, 2010). See also (Kulmann et al., 2014, p. III-144 and III-146).

The formulae taken from the DIN 1052: 2004 seems to be only simplifications of the more exact (Baláž, Koleková, 2000b) formula because zkCa /11 = , zkCa /22 = , gz za = . The error

as a consequence of such simplification is described by the ratio of the less exact DIN 1052: 2004 formula to more exact (Baláž, Koleková, 2000b) and EN 1999-1-1:2007 formula

x

xx

xx

xl

L

M

M

cr

DINef

ENKolekovaBalazcr

DINcr

−−−

=

−−

−=

=−− 1

1

1

11

12

2

1052

111999,,,

1052,

µ (30)

The graphical interpretation of (30) is shown in Fig. 2. Fig. 2 shows that the less exact DIN 1052: 2004 formula gives smaller Mcr values comparing with the more exact (Baláž,


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Koleková, 2000b) and EN 1999-1-1:2007 formula. The error on the safe side (Fig. 2) is less than 7 % for the values

3.02 ≤−=t

zz

GI

EI

L

aax (31)

Fig. 2 - Graphical interpretation of the ratio Mcr,DIN 1052 / Mcr.Baláž, Koleková, EN 1999-1-1

NOTE: it is necessary to stress that in Table 3 and 4 non-linear interpolation shall be applied. For beams with rectangular cross-section diagrams in Fig. 3 and 4 may be used.

Fig. 3 - Interpolation in Table 4 for 0,0 == wtjz κ Fig. 4 - Interpolation in Table 5 for 0,0 == wtjz κ


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SOLUTIONS BASED ON THE BESSEL FUNCTIONS

They may found in (Baláž, Koleková, 2000c) and in the very good book (Voľmir, 1967). For the force F acting on the cantilever end ( 2=zk ) in the shear center, we can obtain the value

crµ by applying Bessel functions )(xJν and gamma function )1( +Γ ν for solution of

differential equation (Kamke, 1965) of lateral torsional buckling (Voľmir, 1967) as follows

k

cr

n

k

kk

cr

n

k

k

cr

g

kkkk

2

0

2

0 8)125.0(!

)1()75,0(

8)125.0(!

)1()25.0(

4

++−Γ

−Γ−=

++Γ

−Γ ∑∑

−−

µπ

µπ

µζ

(32)

For the smaller values of parameter gζ the following approximate formula may be used

)2

1(2

013.4 gcr ζππ

µ −= (33)

The error in approximate formula (33) is lesser than 9.8% for 4.0≤gζ . The comparable

value with the value 1 / 1.28 = 0.781 taken from the Table 10 is the value π / 4.013 = 0.783.

CONCLUSION

This study shows that for verification of lateral torsional stability of timber beams it is possible to use procedure proposed in (Baláž, Koleková, 2000b), which is used also in the Annex I of Eurocode EN 1999-1-1: May 2007 for design of aluminium structures. It is shown that this procedure is convenient after considerable simplification also for timber beams with rectangular cross-section or cross-sections with other shape.

The comparisons in Table 10 illustrate that this procedure gives the best results and show disadvantages of all other investigated procedures including ones used in Eurocode EN 1995-1-1: 2004, German standard DIN 1052: 2004 and German National Annex to DIN EN 1995-1-1: December 2005.

ACKNOWLEDGMENTS

Project No. 1/0748/13 was supported by the Slovak Grant Agency VEGA.

REFERENCES

[1]-Baláž, I.: Comparison of Different Design Curves for Lateral Torsional Buckling of Rolled and Welded Steel Beams, Proc. of 18th Czecho-Slovak International Conference on Steel Structures and Bridges `97 . Brno, 1997, p. 2 / 3 – 2 / 11.

[2]-Baláž, I.: Buckling of monosymmetric beams – conjured problem, in: Proc. Eurosteel 2nd

European Conference on Steel Structures, Praha, 1999, pp.701-704.

[3]-Baláž, I.: Klopenie drevených nosníkov. Zborník prednášok z konferencie „Výstavba a ob-nova budov“, výstava PRO DOMO. Košice 12. marec 2001. Dom techniky Košice, s. 27-32.


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[4]-Baláž, I.: Lateral torsional buckling of timber beams. Wood Research 50(1): 2005, p.51-58.

[5]-Baláž, I., Jahelka, K., Poledňák, P.: Ověření skutečného působení lepených žebrových bloků jako nosné konstrukce zatimních mostů. Zborník referátov z II. medzinárodného sympózia "Drevo v stavebných konštrukciách". Bratislava - Kočovce, 1980, s. 315-324.

[6]-Baláž, I., Koleková, Y.: Stability of Monosymmetric Beams. Proceedings of the 6th International Colloquium Stability and Ductility of Steel Structures. Timişoara, Romania, 9-11 September 1999., p.57-64.

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