parametric study of

Author:

Javier Pascual Ramos

Advisors:

Arcadi Sanmartín Carrillo

Nativitat Pastor Torrente

Degree in:

Construction Engineering

Barcelona, 24 of may of 2013

Department of Applied Mathematics III

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Parametric Study of Lateral-Torsional Buckling in Closed Cross-Sections


Javier Pascual Ramos Degree in Construction Engineering - UPC 2013

1

Contents

List of figures ................................................................................................................... 3

Acknowledgments ........................................................................................................ 5

Abstract ......................................................................................................................... 6

0. Introduction .................................................................................................................. 7

0.1. Motivation ............................................................................................................. 7

0.2. Outline of this document ....................................................................................... 8

1. Lateral-Torsional Buckling Overview .......................................................................... 9

1.1. History of Lateral-Torsional Buckling .................................................................. 9

1.2. Physical Vision of Lateral-Torsional Buckling and Phenomenological

Description ................................................................................................................. 12

1.3. Influencing Factors .............................................................................................. 14

1.3.1. End Restraints ............................................................................................... 14

1.3.2. Type of Loads ............................................................................................... 16

1.3.3. Load Application Point ................................................................................. 17

1.3.4. Cross-Section ................................................................................................ 18

2. Overview of Lateral-Torsional Buckling in Standards and Guides ........................... 19

2.1. Eurocode 3: Design of Steel Structures ............................................................... 19

2.1.1. Background of the European Standards ....................................................... 19

2.1.2. EN 1993 - Eurocode 3: Design of Steel Structures ...................................... 20

2.1.3. Buckling Resistance of Members According to EN 1993-1-1 ..................... 20

2.2. Specification for Structural Steel Buildings – American Institute of Steel

Construction (AISC) & American National Standard Institute (ANSI) ..................... 26

2.2.1. Studying of Buckling according to Specification for Structural Steel

Buildings 2010 (AISC/ANSI) ................................................................................ 26

2.3. British Standard (BS 5950-1:2000) ..................................................................... 31

2.4. Committee International for the Development and the Study of Tubular

Construction (CIDECT) ............................................................................................. 31

3. Parametric Study of Lateral-Torsional Buckling with Visual Basic for Applications 32

3.1. Calibrating the VBA Calculation Program .......................................................... 32

3.1.1. Variable Length ............................................................................................ 34

3.1.2. Variable Height............................................................................................. 35



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3.1.3. Variable Base ................................................................................................ 36

3.1.4. Variable Web Thickness ............................................................................... 37

3.1.5. Variable Flange Thickness ........................................................................... 38

3.1.6. Variable Web and Flange Thickness ............................................................ 39

3.1.7. Summary Table ............................................................................................ 40

3.2. Parametrical Study of Closed Cross-Sections (Rectangular and Square) ........... 41

3.2.1. Parametrical Study: Case L/b ....................................................................... 41

3.2.2. Comparison EN-1993-1-1 with CIDECT ..................................................... 52

3.2.3. Comparison EN-1993-1-1 with British Standard ......................................... 57

4. Parametric Study of Lateral-Torsional Buckling with a Finite Element Method

Program .......................................................................................................................... 66

4.1. Linear Buckling Analysis with Finite Element Method ...................................... 66

4.2. Analysis of Different Cases with SAP2000 v14 ................................................. 68

4.2.1. Boundary Conditions .................................................................................... 68

4.2.2. Particular Cases to Study .............................................................................. 70

5. Conclusions ................................................................................................................ 76

5.1. Conclusions of the calibration of the VBA model .............................................. 76

5.2. Conclusions of the parametric study ................................................................... 76

5.3. Conclusions on comparison with British Standard and CIDECT ....................... 77

5.4. Conclusions of the SAP2000 v14 study .............................................................. 77

5.5. Future Works ....................................................................................................... 78

6. References .................................................................................................................. 79



3

List of figures

Figure 1: sample of lateral-torsional buckling ................................................................ 12

Figure 2: Buckling curves. [EN-1993-1-1, 2005] .......................................................... 25

Figure 3: Sample of studied beam .................................................................................. 32

Figure 4: χ vs L/b for ratios of h/b 1 to 3 and b=0,1m ................................................... 42

Figure 5: χ vs L/b for ratios of h/b 1 to 3 and b=0,15m ................................................. 42






Figure 11: χ vs L/b for ratios of h/b 1 to 3 and b=0,45m ............................................... 45


Figure 13: χ vs L/b for b from 0,1 to 0,5 and ratio of h/b=1 .......................................... 47

Figure 14: χ vs L/b for b from 0,1 to 0,5 and ratio of h/b=1,22 ..................................... 47








Figure 22: χ vs L/b for b from 0,1 to 0,5 and ratio of h/b=3 .......................................... 51

Figure 23: χ vs L/h for b from 0,1 to 0,5 and ratio of h/b=1 .......................................... 54

Figure 24: χ vs L/h for b from 0,1 to 0,5 and ratio of h/b=1,11 ..................................... 54




Figure 28: χ vs L/h for b from 0,1 to 0,5 and ratio of h/b=2 .......................................... 56

Figure 29: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=1,25 .................................... 59


Figure 31: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=1,4 ...................................... 60






Figure 37: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=2 ......................................... 63

Figure 38: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=3 (EC-3 and BS limits

matching) ........................................................................................................................ 63

Figure 39: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=4 ......................................... 64

Figure 40: Bifurcation Graphic ...................................................................................... 66

Figure 41: square cross-section (0,2x0,2x0,15 with L=20) ............................................ 70



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Figure 42: First buckling mode for RHS 0,2x0,2x0,015 with L = 20 m. αcr = 3,13 ....... 71

Figure 43: Longitudinal view of the first buckling mode ............................................... 71

Figure 44: Front view of the first buckling mode ........................................................... 71

Figure 45: rectangular cross-section (0,2x0,1x0,15 with L=20) .................................... 72

Figure 46: First buckling mode for RHS 0,2x0,1x0,015 with L = 20 m. αcr = 0,724 ..... 73



Figure 49: rectangular cross-section (0,3x0,1x0,15 with L=20) .................................... 74

Figure 50: First buckling mode for RHS 0,2x0,1x0,015 with L = 20 m. αcr = 0,724 ..... 75





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Acknowledgments

Honestly, I want to thank my advisors, Nati Pastor and Arcadi Sanmartín for the help

and the support they have provided me in the development of this work during the last 4

months. Also, the great patience they have had with my endless questions and anything

I needed, they provided me.

Despite of their youth, they have managed me to instill their extensive knowledge in the

matter and they have made me see beyond as well as feed my curiosity to know more

and not settle with the first thing you find.

It has been a pleasure. I stand by the decision I took almost a year ago.

Now that I am about to finish my degree, I am still thinking that these kind of teachers

make students improve.

THANK YOU VERY MUCH!



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Abstract

In this work, out a Parametric Study of Lateral-Torsional Buckling in Closed Cross-

Sections has been carried. The aim is to assess the extent of this phenomenon

comparing different standards (Eurocode 3, AISC, Brisith Standard) and design guides

(CIDECT) and doing an exhaustive parametric analysis in order to assess the need of

considering the lateral-torsional buckling in these sections in some cases.

First, we present historical facts which show how the phenomenon started to be studied

centuries ago, what scientists have found out about lateral-torsional buckling and

examples of collapse of some structures because of this matter.

Secondly, the formulation of this phenomenon explaining their influencing factors, such

as end restraints, type of loads and so on is carried out.

After that, the treatment of different standards and design guides is carried out; how

they face the problem and what recommendations they give to study or not the problem,

specifically in square and rectangular cross-sections, as they are the ones studied in this

work.

Once the state of the art is analysed, a parametrical analysis with Visual Basic for

Applications (VBA) has been developed to try to assess the behaviour of the different

parameters of the cross-section and how they affect to the analysis of lateral-torsional

buckling. Also a comparison, between the parametric study carried out with VBA

(using the formulation of Eurocode 3), and other standards such as the British Standad

and Design Guide of the CIDECT is developed.

In addition, a few examples with a finite element method program (SAP2000 v14) have

been carried out, to compare with the results obtained with the formulation of Eurocode

3.

Conclusions of this work and interesting future works are exposed on.



7

0. Introduction

The lateral-torsional buckling is a phenomenon of collapse in steel construction. It is

well-known that a certain type of cross-sections, such as “I” beams tend to suffer this

problem and must be braced properly in order to limit the undesirable out-of-plane

deformation and then the collapse. This is due to their low torsional inertia and the

lateral restraint provided by the web is low and need a physical displacement to avoid

the failure. However it is not critical in closed cross-sections such as square or

rectangular which have high torsional inertia, this is because the webs are connected on

the extreme of the flanges making the cross-sections closed and that plays against

lateral-torsional buckling, so this phenomenon is unlikely to happen.

The statement above is confirmed by two main standards used for the design of steel

structures: Eurocode-3 and American Standard (AISC). Both standards state that open

cross-sections must be checked against lateral-torsional buckling, but closed-cross

sections have a special treatment. Eurocode-3 states that squared cross-sections are not

susceptible to this phenomenon but it can be checked anyhow with the general

formulation. However, the American Standard provides a table of all kind of cross-

sections, and the specific problems they can suffer, and square & rectangular cross-

sections must not be checked against the phenomenon. In addition, no guidelines are

provided to check these cross-sections against lateral-torsional buckling, contrary to

what happens in Eurocode 3.

The aim of this work is to assess the extent of lateral-torsional buckling in closed cross-

sections. The point is to study commercial square & rectangular profiles and also bigger

profiles that can be prefabricated, welded or composite structures. The best way to

analyse how lateral-torsional buckling is affected by different geometric characteristics

is doing a parametrical study with the formulation given by the standards mentioned

above and besides check them with a commercial program of structures.

0.1. Motivation

First of all, this matter was of interest for the advisors, because despite of the fact that

standards state that it is not necessary to check closed sections, they have worked with

some special structures with hollow slender sections that had reduction for lateral-

torsional buckling. This is the case, for example, of a monorail with a rectangular

welded hollow section with a height about 0,7m and length of 35m.

In addition, the Design Guide of the CIDECT [CIDECT, 1996], which is not a standard,

but is widely used in the design of tubular structures, shows limit values of slenderness

of members below of which is not necessary to check against lateral-torsional buckling.

Implying beams that for the same slender ranges out of their limits necessary to check

it. Besides a design example provided in the Designers Handbook to Eurocode 3

[Gardner and Nethercot, 2005] shows the calculation of a reduction factor for lateral-

torsional buckling in a RHS is performed.



8

Therefore, our view was as that in ordinary structures it is not an important factor, but

on special structures (high slenderness) could be significant and we would like to find

out these limit values in which is necessary to take into account the lateral-torsional

buckling or not.

Furthermore, while we were working on the state of art, we found that in the British

Standard (BS 5950) [British Standard, 2001] limit values of slenderness appear (as it is

explained in chapter 3) and there is a recent published article about lateral-torsional

buckling in elliptical hollow sections [Law and Gardner, 2012]. This seems confirm the

point that for slender elements or/and special structures it could be a factor to consider

in the design of closed cross-sections, although probably, it will not decisive.

So in this way, with the objective to consider which cases we have to take into account

the lateral-torsional buckling and which was its influence on the resistance of the beams

in that cases, we developed the study described on chapters 3 and 4 of this work.

0.2. Outline of this document

This document is organised as follows. After this introduction about the aim of the

work, an overview of lateral-torsional buckling phenomena is carried out in chapter 1,

including a historical review, the formulation as well as the main influencing factors for

this phenomenon. In chapter 2, we present the treatment of this mode of failure for

closed sections in different codes and/or design guides such as Eurocode 3 (European

standard), AISC (American standard), British Standard (United Kingdom standard) or

CIDECT (Design guide). In chapter 3, a parametric study of lateral-torsional buckling

for rectangular hollow sections is carried out, including a comparison with standards

depicted in section 2. In chapter 4 a finite element buckling analysis is performed for

some selected sections and compared with the results obtained in chapter 3. Finally,

main conclusions of this work can be found in chapter 5, including a provision for

possible future works.



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1. Lateral-Torsional Buckling Overview

1.1. History of Lateral-Torsional Buckling

Several well-known scientists have noticed on their experiments the curious behaviour

of the slender beams under certain load conditions [Timoshenko, 1953]. The beginning

of lateral-torsional buckling was noticed a few centuries ago, in seventeenth century,

when the illustrated Italian mathematician Galileo (1564-1642) started to experiment

and investigate the resistance to fracture of the same bar if it

works as a cantilever with the load at the end. Assuming

that the bar has a rectangular cross-section and that material

follows Hooke’s law up to failure, we obtain the stress

distribution. Actual materials do not follow Hooke’s law

until the failure, but on this basis of his theory Galileo

draws several important conclusions. “How and in what

proportion does a prism, whose width is greater than its

thickness, offer more resistance to fracture when the force

is applied in the direction of its breadth than in the

direction of its depth” [Crew and Macmillan, 1914]. So he

arrived at the conclusion that any given prism, whose thickness exceeds its width, will

offer greater resistance. Also, he studied the cantilever-beam problem keeping a

constant cross-section and concluded that the bending moment, due to the weight of the

beam, increases as the square of the length. Finally, Galileo discusses the strength of

hollow beams and states that “such beams are seen in the bones of birds which are light

and highly resistant both to bending and breaking”.

Then arrived Euler (1707-1783), an important talented mathematician, whose sentence

“Under a large enough compressive axial load an elastic

beam will buckle.” gave rise to the modern study of the

phenomenon as it is known nowadays as buckling. It is one

of the most well-known instabilities of classical elasticity.

The critical load for buckling was first derived by Euler in

1744 and further refined for higher

modes by Lagrange in 1770. In 1744

Euler’s book “Methodus inveniendi

líneas curvas...” was the first book on

variational calculus and it also contained

the first systematic treatment of elastic

curves. Euler, as a mathematician was

interested principally in the geometrical forms of elastic curves. He

accepted Jacob Bernoulli’s theory that the curvature of an elastic beam at any point is

proportional to the bending moment at that point. In addition, he investigated the shapes

of the curves which slender elastic bar will take up under various loading conditions.

Euler considered the various cases of bending and classified the corresponding elastic

curves according to values of angle between the direction of the force P and the tangent

Galileo Galilei

Leonhard Paul Euler



10

of at the point of application of the load, “When this angle is very small, we have the

important case of columns buckling under the action of an axial compressive force”

[Timoshenko, 1953]. Euler established a formula for the compression force, modified

later by Young, obtaining:

for the buckling columns which now has such

a wide application in analyzing the elastic stability of engineering structures. Since then,

Euler buckling has played a central role in the stability and mechanical properties of

slender structures in engineering.

At the end of seventeenth and beginning of eighteenth century, Lagrange (1736-1813),

contributed to elastic curves as well. The most important

contribution made by Lagrange to the theory of elastic curves is

his memoir “Sur la figure des colonnes” [Lagrange, 1781]. He

started with a discussion of a prismatic bar having hinges at the

ends and assumed there to be a small deflection under action of

the axial compressive force P, under this configuration it is

possible to have an infinite number of buckling curves.

Lagrange did not limit himself to a calculation of critical values

of the load P but went on to investigate the deflections which

exist if the load P exceed the critical value. Larange concluded that the column of

greatest efficiency has a cylindrical form. He arrived at the same conclusion by

considering curves which pass through four points taken at equal distances from axis.

He did not succeed in getting a satisfactory solution to the problem of the shape of the

column maximum efficiency. Later on the same problem was discussed by several other

authors. He made the usual assumption that the curvature is proportional to the bending

moment and discusses several instances which might be of some interest in studying.

The shape of Lagrange’s solution is too complicated for practical application.

Finally, Prandtl (1875-1953) a German physicist, in his PhD thesis (Kipperscheinungen)

[Prandtl, 1899] wrote a paper on lateral-torsional buckling of

beams with narrow rectangular cross-section bent in the plane

of maximum flexural rigidity. He obtained solutions for several

particular cases of practical importance. In making these tests,

Prandtl developed a technique of accurately determining the

critical load which was later used by many investigators in

elastic stability. This paper on lateral buckling initiated

numerous investigations on the lateral stability of beams and

curved bars. This is the first truly scientific work aimed

exclusively towards lateral-torsional buckling of beams. Before

Prandtl presented his work, an important and an early case of failure due to lateral-

torsional buckling, was the cast iron Dee Bridge in 1847 [Delatte, 2009] designed by

Robert Stephenson. The Dee Bridge failed because torsional instability was ignored a

failure mode.

The cast iron is weakest in tension than in compression, the tension flanges of girders

were larger than compression flanges by ratio of 16:3 (see on picture below), following

Joseph Louise de Lagrange

Ludwig Prandtl



11

the ratio material strengths. The Dee Bridge had a relatively long simple span for the

day of roughly 29m. The bridge was also designed with relatively low factor of safety

for the era of 1,5 [Petroski, 1994]. The most likely cause of failure was a torsional-

buckling instability to which the bridge girders were predisposed by compressive loads

introduced by eccentric diagonal tie

roads on the girder. Indeed, the top

flange went out of plane, causing

collapse. The lengthening of cast iron

spans caused the governing mode of

failure to change from bending to

lateral-torsional buckling. Subsequently,

Stephenson pointed out the need to

study failures to learn from them.

Lateral-trosional buckling remains an

important failure mode for steel beams,

unless the compressed flanges are

braced. The problem is generally less

severe than it was with iron because steel is equally strong in tension and in

compression and thus top and bottom flanges are of equal size. This type of buckling,

however, remains an important limit state that must be checked.

Failure of Dee Bridge,1847 in Chester (UK)

Cross-section, Dee

Bridge, 1847 (UK)



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1.2. Physical Vision of Lateral-Torsional Buckling and Phenomenological

Description

As explained above, the Lateral-Torsional Buckling is an instability phenomenon which

appears in elements subjected to bending, with insufficient lateral bracing of

compression flange, and rotating around its second moment of area major axis.

Considering the elastic, perfect member (no residual stress, or geometrical

imperfections) when the maximum bending reaches a certain value, called lateral-

torsional critical moment, the element that until then has been rotating around the acting

bending moment axis, starts to twist and roll out of the vertical plane, into the

perpendicular plane. This is the reason of its name, because of its lateral deformation.

Figure 1: sample of lateral-torsional buckling

The Eurocode 3 (European Standard) and AISC (American Steel Standard) provide

different methods to study this phenomenon and what it depends on.

Furthermore EC 3, which is very restrictive compared to other Standards [Trahair,

2010], take into account other aspects like imperfection of frames caused by lack of

verticality or straightness and any minor eccentricities present in joints of the unloaded

structure, and residual stress due to erection process e.g. welding and cooling. Those

factors reduce the buckling capacity of the beam [Martin and Purkiss, 2008]. However

AISC ignore geometrical imperfections and is more optimistic with respect to residual

stress [Trahair, 2010].

Both are developed on next chapters with its mathematical formulation and factors to

consider on different cross-sections.



13

This critical moment affects differently to the structure, and it depends on 4 important

points, hereinafter:

End restraints

Type of applied loads

Load application point with respect the centroid of the cross section

Cross-section

The general expression of lateral-torsional critical moment is as follows:

[√

]

Where,

E modulus of elasticity

Iz minor moment of inertia

Iw warping constant

It torsional constant

L member length

C1 factor depending on loads and restraints [Acces Steel, 2010 or

Timoshenko and Gere 1961]

C2 factor depending on loads and restraints

C3 factor depending on loads and restraints

zg the distance between the application load point and the centroid of the

cross-section

zj depends on the symmetry of the cross-section

This expression can be simplified depending on the cross-section we are studying. For

example in rectangular sections of width b and height h, load applied on the centre of

the cross-section (zg=0) and for a double symmetric element (zj=0). Furthermore if a

uniform moment is applied (C1=1) [Acces steel, 2010] the equation for critical moment

(Mcr) reduces to:

[√ ]



14

The reason [Seaburg and Carter, 2003] is because torsion on a closed cross-section

(hollow or solid) is resisted by shear stresses in the cross-section that vary directly with

distance from the centroid. The cross-section remains plane as it twists (without

warping) and torsional loading develops pure torsional stresses only. So warping

constant (Iw) can be take it as zero.

The analysis and design of thin-walled closed cross-sections for torsion is therefore

simplified with the assumption that the torque is absorbed by shear forces that are

uniformly distributed over the thickness of the element [Siev, 1966].

1.3. Influencing Factors

1.3.1. End Restraints

The way how a structure is supported is an important aspect to consider since it affects

directly the lateral-torsional critical moment value. The main reason is because this

problem is more likely to be found when the supports allow rotation around the strong

axis (the rotation about the longitudinal axis is usually considered restrained), so the end

restraints will modify the range of the buckling.

As in any type of structure, the supports can be restrained as follows:

TYPE OF RESTRAINT ROTATION FIXED TRANSLATION FIXED

A

B

A

B

A

-

A

B

- A

B

A

-

A

-

Table 1: End restraints

The first supports, both A and B, do not allow any rotation in the plane. This plays

against lateral buckling and particularly with these boundary conditions, it will be more

unlikely that structure suffer the phenomenon.

The second case, support A does not allow rotations, but B allows rotation in the

horizontal plane and it has to be taken into account for the calculation.

The third one, both A and B allow the rotation in the horizontal plane.



15

The fourth case, support A does not allow any movement, but B is completely free of

rotation and translation. That one is the worst situation in lateral buckling because the

restraint is lost in support B.

The first three cases [Acces Steel, 2010] are affected by two coefficients (C1 and C2)

depending on end restraints and loading (see section 1.2) that can modify (increase) the

elastic critical moment for lateral-torsional buckling.

On Standards (both Eurocode and AISC) we find typical cases, but some situations are

not currently covered. This is the case of cantilever, whose information is rather scarce

or incomplete. Some articles are focused on this issue such as [Clark and Hill, 1960]

who presented lower and upper bounds for coefficients factors applicable to cantilever

restraints. Moreover other articles published [Galéa, 1981 or Baláz and Koleková, 2004]

have looked into it more recently.



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1.3.2. Type of Loads

The lateral-torsional critical moment is function of the type of applied load on beams

and its boundary condition. Moment distribution takes part in the lateral buckling and it

has to be considered depending on the combination of both. The next table shows

different cases with loads and restraints and their moment distribution:

The values before must be taken to the coefficient C1 = 1/ ( kc2), and the Ψ=Mini/Mend

depends on the difference between both applied moments.

MOMENT DISTRIBUTION LOAD + RESTRAINT Value for kc

Uniform Bending Moment +

Simply supported

Linear Variation of Bending

Moment (same sign) + Simply

supported

Uniform Load + Simply

supported

Uniform Load + Clamped

Uniform Load + Clamped-Simply

supported

Point Load + Simply supported

Point Load + Clamped

Point Load + Clamped-Simply

supported

1,0

Ψ*

0,94

0,90

0,91

0,86

0,77

0,82

Table 2: Boundary conditions and values for kc [EN-1993-1-1]



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1.3.3. Load Application Point

The load application point with respect the centroid of the cross section has influence on

lateral-torsional critical moment. The position of the point load application is directly

related with the beam shear centre and can introduce a secondary stabilizer or

destabilize moment.

Picture 1: Possible load application points

There are 3 types of section load application:

Above Shear Centre

If it is applied above the Shear Centre the load will have a destabilizer moment and that

will play negatively on the buckling (increases the torsional moment).

Just on Shear Centre

This situation will not cause any effect on lateral buckling (does not modify the

torsional moment). That is why we tend to apply loads on beams in this position.

Beneath Shear Centre

If it is applied beneath the Shear Centre the load will have a stabilizer moment and that

will play positively on the buckling (decreases the torsional moment).

Here it makes sense the parameter zg from the critical moment (see section 1.2). It is the

distance to the shear centre for above the shear it must be taken as > 0 and beneath the

shear centre it must be taken < 0. If it is just on the shear centre, has a value of zero.



18

1.3.4. Cross-Section

The type of the cross-section is the most decisive factor on lateral buckling. There are

two main groups of cross-sections, closed cross-sections or opened cross-sections.

Closed Cross-Sections

As far as the studies are concerned, closed cross-sections normally tend not to show

lateral-torsional buckling. That is due mostly to their high torsional constant and their

warping constant equal to zero.

However slender sections, square, rectangular and round, may be subjected to lateral-

torsional buckling. According to American Standard [AISC & ANSI, 2010] this is

physically possible, but in standard design such slender sections are not used because

normally in these cases SLS deflection verification would govern the design.

Picture 2: Sample of closed-cross sections

Opened Cross-Sections

These sections are more susceptible to lateral-torsional buckling. That is due to their

low torsional constant and their warping constants different from zero.

If the bending moment is applied on major axis the beam is more susceptible to suffer

the phenomenon.

Picture 3: Sample of opened cross-sections

In addition to this other aspect related with the cross-section is the length of the beam,

as long slender beams will tend to suffer the problem.



19

2. Overview of Lateral-Torsional Buckling in Standards and Guides

2.1. Eurocode 3: Design of Steel Structures

2.1.1. Background of the European Standards

The Commission of the European Community decided on an action programme in the

field of construction. The aim of this programme came up from the idea to eliminate

some possible technical obstacles to trade and the harmonization of technical

specifications.

Within this programme, the Commission took the initiative to establish a set of uniform

technical rules for the design of construction works which, in a first stage, would serve

as an alternative to the national regulations in force the Member States and, ultimately,

would replace them.

Few years later, The Commission and the Member States decided to transfer the

preparation and publication of the Eurocodes in order to provide them with a future

status of European Standard (EN).

These Eurocodes comprise the next standards generally consisting on a number of parts:

EN 1990 Eurocode 0: Basis of structural design

EN 1991 Eurocode 1: Actions on structures

EN 1992 Eurocode 2: Design of concrete structures

EN 1993 Eurocode 3: Design of steel structures

EN 1994 Eurocode 4: Design of composite steel and concrete structures

EN 1995 Eurocode 5: Design of timber structures

EN 1996 Eurocode 6: Design of masonry structures

EN 1997 Eurocode 7: Geotechnical design

EN 1998 Eurocode 8: Design of structures for earthquake resistance

EN 1999 Eurocode 9: Design of aluminium structures

Eurocode Standard recognize the responsibility of regulatory authorities in each

Member State and have safeguarded their right to determine values related to regulatory

safety matters at national level where these continue vary from State to State.



20

2.1.2. EN 1993 - Eurocode 3: Design of Steel Structures

This part of the European Standard talks about the design of steel structures. It applies

to design of buildings and civil engineering works in steel. It complies with the

principles and requirements for the safety and serviceability of structures. It is intended

to be used with the rest of Eurocodes EN 1990, EN 1991 and EN 1992 to EN 1999.

Eurocode 3 is concerned only with requirements for resistance, serviceability, durability

and fire of resistance of steel structures. Other requirements such as concerning thermal

or sound insulation are not covered.

It is composed of twelve subparts EN 1993-1-1 to EN 1993-1-12 each addressing

specific steel components, limit states or materials.

The ones relevant to this work are EN 1993-1-1 that is about General rules and rules

for building.

2.1.3. Buckling Resistance of Members According to EN 1993-1-1

On this section it is explained how to study and the methodology to face the buckling

phenomenon according to EN-1993-1.

It is contained on EN-1993-1-1, inside section 6: Ultimate Limit State which considers

different forces on sections such as bending moment, shear, torsion, and also buckling

and so on.

2.1.3.1. Classification of Cross-Sections

The purpose of cross section classification is to identify the extent to which resistance

and rotation capacity of cross sections is limited by its local buckling resistance.

There are four different classes of cross-sections, as follows:

- Class 1 cross-sections are those which can form a plastic hinge with the rotation

capacity required from plastic analysis without reduction of the resistance.

- Class 2 cross-sections are those which can develop their plastic moment resistance, but

have limited rotation capacity because of local buckling.

- Class 3 cross-sections are those in which the stress in the extreme compression fibre of

the steel member assuming an elastic distribution of stresses can reach the yield

strength, but local buckling is liable to prevent development of the plastic moment

resistance.

- Class 4 cross-sections are those in which local buckling will occur before the

attainment of yield stress in one or more parts of the cross-section.



21

Picture 4: Classes of cross-section

A cross-section is classified according to the highest (least favourable) class of its

compression parts (see Table 3).

The classification depends on the width to thickness ratio of the parts subject to

compression, the height to thickness ratio of the parts subjected to bending and the steel

grade chosen.

On this work, Class 1, 2 and 3 are studied further on next chapters. They do not need

any reduction of their area, elastic modulus or inertia among others.

This Standard provides a table (see Table 3) to classify webs and flanges subjected to

bending or compression in closed sections. The fact is that lateral-torsional buckling

occurs when a bending moment is applied so webs are subjected to bending and one of

the flanges to compression. The other flange is subjected to tension so no classification

is applied.



22

Table 3: Classification of cross-section. [EN-1993-1-1, 2005, Table 5.2]

Both compression parts include every part of a cross-section which is either totally or

partially in compression under the load combination considered.

The various compression parts in cross-section such as web or flange can be in different

classes.



23

2.1.3.2.Uniform Members in Bending: Buckling Resistance

In order to analyse buckling, there are two main cases, according to section EN-1993-1-

1 where no lateral-torsional buckling checking is needed:

1. Beams with sufficient restraint to compression flange, which are not

susceptible to lateral-torsional buckling.

2. 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.

This last statement is developed on this work because it is interesting to know the extent

of lateral-torsional buckling in cross-sections.

The verification, given by EN-1993-1-1 to study lateral-torsional buckling on the

second case, which is a laterally unrestrained member subject to major axis bending,

should be verified as the follow:

Where,

MEd is the design value of the moment

Mb,Rd is the design buckling resistance moment

This ratio must be 1,0 because Mb,Rd is the highest value that section can reach, so

MEd cannot exceed it. The design buckling resistance moment is calculated as:

Where, Wy is the appropriate section modulus as follows:

Wy is Wpl,y for Class 1 or 2 cross-sections

Wy is Wel,y for Class 3 cross-sections

Wy is Weff,y for Class 4 cross-sections (not studied)



24

And,

is the partial factor for buckling resistance and a recommended value of

1,00

is the yield strength of the material

is the reduction factor for lateral-torsional buckling that ranges from

0 to 1 and it is non-dimensional. See below for other details of the factor.

2.1.3.3. Lateral-Torsional Buckling Parameter and Buckling Curves

For bending members of constant cross-section, the value of for the appropriate

non-dimensional slenderness , should be determined from the given formulation:

√

Where,

[ ( ) ] and ranges from 0 to 1. It is a non-

dimensional parameter.

is an imperfection factor and must be taken from Table 4. It is a non-

dimensional parameter.

is the non-dimensional slenderness, it is calculated as:

√

is the elastic critical moment for lateral-torsional buckling. It is based on

gross cross sectional properties and takes into account the loading conditions.

The Standard provides a table to determine what buckling curve we must chose and

other recommended values that belongs to each curve (see Table 4 and Table 5).

In addition, EN-1993-1-1 says that it is not necessary check the phenomenon for <

0.4. That will be analysed in a specific chapter of this work (see section 2.1.3).



25

Cross-section Limits Buckling curve

Rolled I sections h/b 2

h/b 2

a

b

Welded I sections h/b 2

h/b 2

c

d

Other cross-sections - d

Table 4: Recommended values for lateral torsional buckling curves for cross-sections using for

[EN-1993-1-1, Table 6.4

Buckling Curves

The graphical representation of these curves is the following one:

Figure 2: Buckling curves. [EN-1993-1-1, 2005]

- Curve a, represents quasi perfect shapes.

- Curve b, represents shapes with medium imperfections.

- Curve c, represents shapes with a lot of imperfections.

- Curve d, represents shapes with maximum imperfections.

These curves come from experimental tests of beams with different cross-sections and

with different values of slenderness. A probabilistic approach showed that it was

possible to draw some curves describing column strength as a function of the reference

slenderness [Batista and Rodrigues, 1994].

Buckling curve a b c d

Imperfection factor 0,21 0,34 0,49 0,76

Table 5: Recommended values for imperfection factors for lateral-torsional buckling curves

[EN-1993-1-1, Table 6.3]



26

2.2. Specification for Structural Steel Buildings – American Institute of Steel

Construction (AISC) & American National Standard Institute (ANSI)

The American Institute of Steel Construction (AISC) and American National Standard

Institute (ANSI), are a not-for-profit technical institutes and trade associations

established in 1921 and 1968 respectively, to serve the structural steel design

community and construction industry in the United States. The objective is to provide

the civil engineering specifications and code developments, research, standardization

and so on.

The Specification has been developed as a consensus document to provide a uniform

practice in the design of steel-framed buildings and other structures. The intention is to

provide design criteria for routine use and not to provide specific criteria for

infrequently encountered problems.

2.2.1. Studying of Buckling according to Specification for Structural Steel

Buildings 2010 (AISC/ANSI)

On this section is explained how to study and the methodology to face the buckling

phenomenon.

Chapter B called Design requirements has a specifically part 4, Member Properties, that

contains specifications to classify sections subjected to axial compression or flexure.

Chapter F called Design members of flexure, provides information of design members

for flexure such as I-shaped, square and rectangular, round or tees among others.

2.2.1.1.Design Requirements– Member Properties

2.2.1.1.1. Compression

For compression, sections are classified as non-slender element or as slender element

sections. For a non-slender element section, the width-to-thickness ratios of its

compression elements shall not exceed λr from the table given by this Standard. If the

width-to-thickness ratio of any compression element exceed, the section is a slender-

element section. This case is not important to this work because it is not about analysis

of compression elements.

2.2.1.1.2. Flexure

For flexure, which is our case of study because of lateral-torsional buckling, sections are

classified as compact, non-compact or slender-element sections. For a section to qualify

as compact, its flanges must be continuously connected to the web and the width-to-

thickness ratios of its compression elements shall not exceed the limiting of λp from the

Table 3. If the width-to-thickness ratio of one or more compression elements exceed,

the section is non-compact. Furthermore if the width to-thickness ratio of any

compression element exceeds λr, the section is a slender-element section.



27

There is an important difference with Eurocode 3 because cross-sections are not

classified on different classes, but we can compare them as:

This is to make the comparison with the model of classification given by the EN-1993-1

which divides the cross-section in 4 classes. But AISC does not classify in different

classes so if value of width-to-thickness ratio (see Table 6) is minor than λp the section

can be considered as being in class 1 or 2. If the value is between λp and λr is similar to

class 3. Finally, if the value is higher than λr it can be understood as class 4.

2.2.1.1.3. Stiffened or unstiffened elements

Stiffened elements are those that become rigid and are not susceptible to buckle whereas

unstiffened elements may suffer local buckle on webs or flanges.

The stiffeness plays part into the section classification as it improves the capacity of the

cross-section. All sections can be stiffened or unstiffened as I-shaped, legs or angles or

tees among others.

Class 1 and 2 λp < Class 3 < < λr < Class 4

Eurocode 3 AISC Eurocode 3 AISC Eurocode 3



28

Table 6: Classification of cross-sections subjected to flexure. [AISC, 2010]



29

2.2.1.2. Design of Members for Flexure

Chapter F of the AISC specification applies to members subject to simple bending about

one principal axis. For simple bending, the member is loaded in a parallel to a principal

axis that passes through the shear centre or is restrained against twisting at load points

supports.

The chapter is organized in 13 sections, each one for a different member type. The one

dealing with hollow (square and rectangular) sections is the F7:

F7 Square and rectangular HSS box-shaped members

This Standard is divided into each particular case as in opposition to the European

Standard which provides a general method to study any type of cross-section.

In addition to this, it is included a summary table (see Table 7) about problems that can

suffer different cross-sections.The problem of lateral-torsional buckling does not appear

on the table below on limit states column. This is an important difference from the EN-

1993-1-1.



30

Table 7: Limits states for different cross-sections. [AISC, 2010]

Since this work is focused on closed sections (square and rectangular), hereafter only

these specific cross-sections are discussed.



31

2.2.1.3.Square and Rectangular HSS and Box-Shaped Members

F7 section of AISC applies to square and rectangular hollow steel sections (HSS), and

doubly symmetric box-shaped members bent about either axis, having compact or non-

compact webs and compact or non-compact slender flanges.

The AISC Standard mentions that in case of very long rectangular HSS bent about their

major axis are subject to lateral-torsional buckling, whereas the Specification provides

no strength equation for this limit state since beam deflection will control for all

reasonable cases as it is shown on table before, that it does not appear the phenomenon

in this type of cross-section. Thus, this Standard refuses the possibility that this type of

section may fail by this phenomenon.

So there are 3 possible cases that make failure the structure

Yielding

Flange Local Buckling

Web Local Buckling

Each case is developed on the Standard with its mathematical formulation but it is not

explained here since neither of them are not relevant for lateral-torsional buckling cases

of failure. An important fact is that on the study of this type of section, it is not given

any reduced factor for slender cross-sections as it is given on European Standard.

2.3. British Standard (BS 5950-1:2000)

The British Standard [British Standard, 2001] (now is withdrawn to EN-1993) was used

in the United Kingdom to the design and analysis of steel structures. This Standard

treated the problem of lateral-torsional buckling different from the Eurocode 3 and

AISC, it provides a table (see Table 11 in section 3.2.3) with limiting values below of

which is not necessary to check the problem. The values provided on this standard are

compared to the ones obtained with EN-1993-1-1 in the parametric study developed in

section 3.2.3.

2.4. Committee International for the Development and the Study of Tubular

Construction (CIDECT)

The Design Guide 2 from the CIDECT [CIDECT, 1996], as the name define itself, it is

not a Standard that must be follow compulsory, but provides guides to the design of

steel constructions. It is widely used as a guideline for the design of structures with steel

hollow sections. We have analysed it and compared Eurocode 3 in section 3.2.2. As in

the case of British standard, they provide a table (see Table 10 in section 3.2.2) with

limiting values of slenderness to account or not for lateral-torsional buckling the case of

lateral-torsional buckling.



32

3. Parametric Study of Lateral-Torsional Buckling with Visual Basic

for Applications

In this chapter we carry out a parametric study in order to assess the extent of lateral-

torsional buckling in closed sections.

The parametric study has been developed on the basis of Eurocode’s formulation,

particularly the one provided in EN-1993-1-1.

The study has been carried out with Visual Basic for Applications (VBA), a

programming language that is determined by events and provides comprehensive

facilities to computer programmers, which allows the users to implement calculation

subroutines in junction with standard Microsoft office programs such as excel

There are 2 main steps on the VBA calculation program developed for this work:

1. Creating a calculation program that need to be calibrated justifying the obtained

results, and also allows to develop a first study of the influence of different

parameters on lateral-torsional buckling.

2. Creating a model on which is possible to obtain a large number of interesting

results from a parametric study that are useful to obtain conclusions on the need

to consider or not lateral-torsional buckling on closed sections.

3.1. Calibrating the VBA Calculation Program

The objective of this chapter is to asses in a first approximation the variation of the

different parameters affect to the lateral-torsional buckling of the beam, both taking into

account cross-sectional parameters and also the length of the beam, see figure below

Figure 3: Sample of studied beam



33

The way how the parameters influence has been studied is as follows:

1. Variable Length and remaining ones fixed

2. Variable Height and remaining ones fixed

3. Variable Base and remaining ones fixed

4. Variable Web Thickness (tw) and remaining ones fixed

5. Variable Flange Thickness (tf) and remaining ones fixed

6. Variable Web and Flange Thickness (tw = tf) remaining ones fixed

On the other hand the main that has been studied is the reduction factor for lateral-

torsional buckling ( . For this purpose, graphics have been produced with the

variation of with the parameters listed above. In a secondary way, and in order to

analyse the graphics of , the variation of the other properties such as the ones listed

below, has been assessed too.

1. Area

2. Inertia around weak axis (Iz)

3. Inertia around strong axis (Iy)

4. Elastic Modulus (Wy)

5. Torsional Inertia (It)

6. Cross-section class (c1, c2, c3 or c4)

7. Critical Moment (Mcr)

8. Non-dimensional slenderness ( )

9. , value to determine the reduction factor for lateral-torsional buckling( )

As a first idea for a good calibration of the graphics of variation of , all of them must

have the critical point (where reduction starts) at the same value. For this purpose we

have set a first geometry of a rectangular cross-section and get the critical length that

makes it buckle it. Then, once fixed the length try to get the critical height that makes it

buckle with the parameters fixed before, and the same for the rest of the geometric

parameters.

The initial parameters chosen are the follows:

PRAMETERS VALUE UNIT

Length 15 m

Height 0,745 m

Base 0,565 m

Web & Flange thickness 0,015 m

E 21x107

KN/m2

G 81x106

KN/m2

fy 355000 KN/m2

α 0,76 -

π 3,1415 -

ε 0,81 -

Table 8: Initial parameters



34

The table above shows the values of critical length, critical height and so on. In this

way, they are the inflexion point for reduction factor .

After a brief analysis of each graphic, there is a summary table that reflects the

behaviour of the main parameters.

In the following subsections the graphics for the variation of with each parameter

are shown and analyzed.

3.1.1. Variable Length

The lateral-torsional buckling is studied varying the length of the beam. The other

cross-section’s parameters are fixed. The behaviour is the following:

Graphic 1: Variable length

For lengths < 15m remains constant ( =1) and for lengths larger than 15m the

graphic follows a linear decreasing variation while the length is growing. This is logical

because the critical moment expression is inversely proportional to the length. This

makes the slenderness increase and decrease.

In addition, there are no changes on the cross-section class because the length does not

affect to the section classification.

Note that critical point occurs for 15m and this is why all the next graphics are done for

L=15m, as detailed in Table 8.

0,98

0,985

0,99

0,995

1

12 13 14 15 16 17 18 19 20

χ vs Length

χ

Limit

χ

Length (m)



35

3.1.2. Variable Height

The lateral-torsional buckling is studied varying the height of the beam. The other

cross-section’s parameters are fixed and L=15m. The behaviour is the following:

Graphic 2: Variable height

After the critical point, the graphic follows a linear decreasing variation while the height

is growing. This is logical because the height makes the slenderness increase

which reduces the critical moment. This causes the decrease for increasing heights.

In addition, there are changes on the cross-section class because the height affects to the

section classification as can be seen in the figure.

Note that the critical point is obtained for a value of height = 0,745m which corresponds

to the expected value (Table 8)

0,995

0,996

0,997

0,998

0,999

1

0,7 0,75 0,8 0,85 0,9 0,95 1

χ vs Height

χ

Limit

Class 1 χ

Height (m)

Class 2



36

3.1.3. Variable Base

The lateral-torsional buckling is studied varying the base of the beam. The other cross-

section’s parameters are fixed. The behaviour is the following:

Graphic 3: Variable base

The graphic follows a non-linear increasing variation while the base is growing until

reaching a constant value = 1 for base > 0,565m. This is logical because a larger

base makes the Iz increase which results on a larger critical moment. So this makes the

slenderness go down and the goes up for increasing values of b.

In addition, there are changes on the cross-section class because the base affects to the

classification as it is shown on the graphic.

Again, the critical point is reached for the expected value

0,965

0,97

0,975

0,98

0,985

0,99

0,995

1

0,4 0,45 0,5 0,55 0,6

χ vs Base

χ

Limit

χ

Base (m)

Class 1 Class 2



37

3.1.4. Variable Web Thickness

The lateral-torsional buckling is studied varying the web thickness of the beam. The

other cross-section’s parameters are fixed. The behaviour obtained is the following:

Graphic 4: Variable web thickness

The graphic follows a non-linear increasing variation while the web thickness increases,

until reaching the critical point. This is due to the fact that larger tw means larger inertia

and increases the critical moment. So these effects make the slenderness go down

and the increase with tw.

In addition, despite of the fact that the web thickness affects directly, there are no

changes on the class classification for the range of tw considered here

Again, the critical value for tw matches the expected value on Table 8.

0,96

0,965

0,97

0,975

0,98

0,985

0,99

0,995

1

0,005 0,007 0,009 0,011 0,013 0,015 0,017 0,019 0,021

χ vs tw

χ

Limit

χ

tw (m)



38

3.1.5. Variable Flange Thickness

The lateral-torsional buckling is studied varying the flange thickness of the beam. The

other cross-section’s parameters are fixed. The behaviour is the following:

Graphic 5: Variable flange thickness

The graphic follows a linear decreasing variation while the flange thickness is growing

after the critical point. This may seem an illogical behaviour because an increase of

thickness tf makes increase both inertia and critical moment. However, the growth of

flange thickness causes also that the slenderness increase and thus decreases,

which at the end is the governing behaviour and it goes against the expected results.

In addition, despite of the fact that the flange thickness affects directly, there are no

changes on the section classification, for the range of tf considered.

0,9986

0,9988

0,999

0,9992

0,9994

0,9996

0,9998

1

1,0002

0,0146 0,0148 0,015 0,0152 0,0154 0,0156 0,0158 0,016

χ vs tf

χ

Limit

χ

tf (m)



39

3.1.6. Variable Web and Flange Thickness

The lateral-torsional buckling is studied varying the flange thickness and web thickness

of the beam at the same time. The other cross-section’s parameters are fixed. The

behaviour is the following:

Graphic 6: Variable thickness

The graphic follows a linear decreasing variation after the critical point while the flange

thickness is growing. This is the same variation as in the last case. We notice that the

variation of flange thickness is dominant over variation of web thickness. Once again, it

is not an expected result.

In addition, despite of the thickness affects directly to it, there are no changes on the

class of the same section. This is because there is no change in the range of interest.

Note that up to 18mm, the reduction factor decreases only to 0,9997.

Again we obtain the critical thickness at the expected value.

0,9997

0,99975

0,9998

0,99985

0,9999

0,99995

1

1,00005

0,014 0,0145 0,015 0,0155 0,016 0,0165 0,017 0,0175 0,018

χ vs t

χ

Limit

χ

t (m)



40

3.1.7. Summary Table

Hereafter it is shown a summary table which reflects on a fast way the behaviour of

each variation in relation with the main parameters studied.

PARAMETER Iy It My Mcr

Length

Height f s

Base s f

Web Thickness s f

Flange Thickness f s

Thickness f s

Table 9: Summary table

Increasing all parameters’ cross-section (h, b, tw, tf), both mechanics features and

resistance improve. However slenderness ( and reduction factor for lateral-torsional

buckling ( ) do not always follow a logical behaviour as it happens in flange

thickness. In addition we have noticed that it is dominating when we vary both

thicknesses at the same time.

The subscripts “f” and “s” mean, fast and slow, e.g. in the case of the variable height My

grows faster (f) than the Mcr (s). However in the case of the variable base My grows

slower (s) than Mcr (f).



41

3.2. Parametrical Study of Closed Cross-Sections (Rectangular and

Square)

Once we have validated the VBA calculation program, we can apply it to perform an

exhaustive parametric study to obtain interesting results that we can find in real cases.

The objective is to assess the extent of lateral-torsional buckling on this type of usual

closed cross-sections and also try to compare the results with some of the values

provided in the literature (British Standard / CIDECT).

Analyzing the graphics above we have seen that the influence of the thickness is low

( ≈ 0,9996) , so we do not take into account the thickness for the parametric study,

and we consider a simple value of thickness as 0,015m. We are going to develop 3 types

of variation graphics:

1. - χ vs L/b (see 3.2.1)

2. - χ vs L/h or CIDECT case (see 3.2.2)

3. - χ vs L/h or British Standard (BS 5950-1:2000) case (see 3.2.3)

3.2.1. Parametrical Study: Case L/b

On this section, the behaviour of closed cross-sections on the lateral-torsional buckling

related with the ratio L/b is assessed.

On the graphics obtained we see, for a fixed value b, different curves of h/b (legend).

All versus reduction factor for lateral-torsional buckling (ordinate axis) and rate L/b

(abscissa axis). It has been carried out for bases between 0,1 to 0,5m (commercial ones)

and for the ratio h/b from 1 to 2,5 (commercial ones) and up to 3 (welded sections).

The graphics obtained are the following:



42

Figure 4: χ vs L/b for ratios of h/b 1 to 3 and b=0,1m


0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,1m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,15m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b



43



0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,2m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,25m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b



44



0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,3m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,35m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b



45



0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,4m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,45m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b



46


As a general view of graphics with b fixed, we see that on the first graphic (Figure 4),

all the curves of h/b are so separated, but while the b increases, the range starts to be

narrower. So if we have a relatively narrow base the difference between higher or lower

h/b ratios is important, but as the base grows up, the size of the base starts to be less

important and the reduction values tend to get closer to each other.

Also, we see how cross-sections with low h/b (square has h/b=1) (Figure 4) need a high

L/b ratio to buckle than slender cross-sections that start to buckle before. But this

difference becomes smaller when b increases.

For this kind of representation the curves are separated and thus, we cannot obtain any

global critical point. It seems more reasonable to group the sections using its h/b ratios

regardless its base or height actual dimensions. These new arrangements can be seen in

the following figures:

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

b=0,5m

1,00

1,22

1,44

1,67

1,89

2,11

2,33

2,56

2,78

3,00

h/b



47

Figure 13: χ vs L/b for b from 0,1 to 0,5 and ratio of h/b=1

Figure 14: χ vs L/b for b from 0,1 to 0,5 and ratio of h/b=1,22

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=1

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=1,22

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base



48



0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=1,44

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base

0,91

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=1,67

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base



49



0,91

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=1,89

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base

0,9

0,91

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=2,11

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base



50



0,9

0,91

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0 10 20 30 40 50 60

χ

L/b

h/b=2,33

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

h/b=2,56

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base



51


Figure 22: χ vs L/b for b from 0,1 to 0,5 and ratio of h/b=3

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

h/b=2,78

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base

0,88

0,9

0,92

0,94

0,96

0,98

1

0 10 20 30 40 50 60

χ

L/b

h/b=3

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

base



52

Taking a look at this series of figures we can draw the following comments:

First of all, as a general view of graphics with h/b fixed, we see that on the first graphic

(h/b=1, Figure 13), all the curves of h/b are so superposed, that means that for a fixed

rate h/b=1 having a wide or narrow base (except extreme curves b=1 that correspond to

very slender elements) does not have a significant influence. But we can state a critical

point, in this case for L/b=28. And the same for the critical point for h/b=3, whose ratio

L/b is around 20. So we can conclude that for ratios h/b 1 to 3 the critical zone is 20 <

L/b < 30. This means that the lengths around which the theoretical reduction for lateral-

torsional buckling starts can be very usual, between 2 or 3 metres for small sections of

0,1m of base, up to 15m for heavy sections of 0,5m base.

Secondly, we see how cross-sections with low h/b (square has h/b=1, Figure 13) need a

high rate L/b to buckle than slender cross-sections that start to buckle before. But this

difference becomes narrow when b increases.

An important appreciation of the graphics is that there are more differences between

graphics for a fixed b and different families of h/b than fixing h/b and different families

of b. It means that the graphic with title b=0,1m (Figure 4) if we see different families

of h/b (in general) they are separated, whereas if we see the graphic with title h/b=1

(Figure 13) the families of b (in general) are very narrow. So for this reason in the

graphics of a fixed b are not appropriate to draw conclusions as h/b fixed, and we

cannot conclude an exact critical zone as before. So we will not do this analysis on next

parametric graphics below.

3.2.2. Comparison EN-1993-1-1 with CIDECT

Eurocode 3 states that members with cross-sections such as square, are not susceptible

to this phenomenon and for other cross-sections such as I beams there is no need to

check them if ≤ 0,4 (see section 2.1.3.3). On the other hand CIDECT provides with

reference values on a table to study the lateral-torsional buckling, in square and

rectangular cross-sections (see below).



53

Table 10: Limit values; below them it is not necessary to check lateral-torsional buckling [CIDECT, 1996]

The CIDECT advices with this table that for ratios below of those shown on the table, it

is not necessary to check the lateral-torsional buckling.

Therefore, in order to compare our results, we now plot our graphics with L/h in the

abscissa instead of L/b. The point is to obtain graphics for different ratios h/b (inverse

of γ on the table before) and see the difference with the Eurocode 3 and CIDECT

according to the table before.

On the graphics obtained we see, for a fixed ratio h/b, different curves of b (legend). All

versus reduction factor for lateral-torsional buckling (ordinate axis) and rate L/h

(abscissa axis). It has been carried out for the ratio h/b from 1 to 2 as it is shown on the

table above.

The straights bars represent:

EC-3 bar: from the bar to the left side the ≤ 0,4 zone is found, where there is

no need to check the lateral-torsional buckling.

CIDECT bar: from the bar to the left side the zone where no checking is needed

is found.

The graphics obtained are the following:



54

Figure 23: χ vs L/h for b from 0,1 to 0,5 and ratio of h/b=1

Figure 24: χ vs L/h for b from 0,1 to 0,5 and ratio of h/b=1,11

0,75

0,8

0,85

0,9

0,95

1

20 40 60 80 100 120

χ

L/h

h/b=1

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

CIDECT

base

0,75

0,8

0,85

0,9

0,95

1

20 40 60 80 100 120

χ

L/h

h/b=1,11

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

CIDECT

base



55



0,75

0,8

0,85

0,9

0,95

1

20 40 60 80 100 120

χ

L/h

h/b=1,25

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

CIDECT

base

0,75

0,8

0,85

0,9

0,95

1

20 40 60 80 100 120

χ

L/h

h/b=1,42

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

CIDECT

base



56


Figure 28: χ vs L/h for b from 0,1 to 0,5 and ratio of h/b=2

0,75

0,8

0,85

0,9

0,95

1

20 40 60 80 100 120

χ

L/h

h/b=1,66

0,1

0,15

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

CIDECT

base

0,75

0,8

0,85

0,9

0,95

1

20 40 60 80 100 120

χ

L/h

h/b=2

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

CIDECT

base



57

We appreciate some important aspects related with the behaviour of the graphics while

the rate h/b increases from 1 to 2. Note that for the presented range of sections h/b ratios

(from 1 to 2), the most common commercial sections can be found

First of all, as a general view, we see that on the first graphic (h/b=1) (Figure 23), the

checking point is ratio L/h=100 for Eurocode 3 and 110 for CIDECT. If we do the same

for the last case (Figure 28), the checking point is a ratio L/h=40 for Eurocode and 50

for CIDECT. So we can state a checking point for each case of h/b. In addition the ratio

L/h for which χ starts to be < 1 is between 25 ≤ L/h ≤ 35.

As an example of the explained before, low ratios of h/b, is more difficult to buckle. If

we see Figure 23, the first that shows h/b=1 (it is a square cross-section), needs a rate

L/h almost 100 to have a reduction for lateral-torsional buckling of 0,85, this means for

example, a beam with length=10m, the height will be 0,1m. But if we see the last that

shows a rate h/b=2 (it is a slender cross-section)Figure 28, needs a rate L/h about 50 to

have the same reduction for lateral-torsional buckling. It means that as the same

example before, the height will be 0,2m. That is logical, slenderness is an important

characteristic that influences this problem, so the behaviour obtained is the expected.

In spite of this we must keep in mind that ratios of L/h=100 or L/h=50 are unusual

values on normal structures, so the limits of the Eurocode are applicable for special

structures. Note if we treat the member as an I beam and a lateral-torsional buckling

check is performed we would find reduction factors of 0,85 for the limit slenderness 0,4.

This means a 15% of reduction of the member bending strength which is rather high

value.

Secondly, the recommendations from Eurocode and CIDECT look a little bit different.

We see how Eurocode 3 is more conservative than CIDECT. The Eurocode fix the

checking point for cases between 0,85 ≤ χ ≤ 0,87 (or ≤ 0,4), as it says that it is not

necessary to check the lateral-torsional buckling of square and rectangular cross-

sections. On the other hand, CIDECT fix this value in between 0,88 ≤ χ ≤ 0,83. In that

way, we can define that the Eurocode is a little more conservative than CIDECT that is

to say that CIDECT allows higher lengths with no lateral-torsional buckling checking.

But indeed, we can make a whole lecture of the graphics obtained. Really, the

difference, appreciated in both standards that range on 0,03 points, is not extremely big.

So roughly, this difference does not have an important impact on the study of lateral-

torsional buckling between designing a beam with one standard or the other.

3.2.3. Comparison EN-1993-1-1 with British Standard

This section contains a comparison between Eurocode 3 (which is currently in force)

and British Standard (now withdrawn) that was the standard used in the United

Kingdom and was superseded by Eurocode. Both of them provide technical rules to

treat lateral-torsional buckling.



58

As explained in the comparison of Eurocode 3 and CIDECT for RHS, no lateral-

torsional buckling checking is needed. According to Eurocode 3 for other cross-sections

this lateral-torsional buckling checking can be omitted for ≤ 0,4. However British

Standard provides guidance to study the lateral-torsional buckling in square and

rectangular cross-sections, as can see in the following table:

Table 11: Limit values of L/iz; below them it is no necessary to check the lateral-torsional buckling.

[British Standard, 2001]

The table before given by British Standard shows that for values below the indicated on

the table checking the lateral-torsional buckling is not necessary.

The point is to obtain graphics for different h/b ratios and see the difference of limits

between Eurocode 3 and ratios of British Standard.

On the graphics obtained we see, for a fixed rate h/b, different curves of b (legend). All

versus reduction factor for lateral-torsional buckling (ordinate axis) and rate L/iz

(abscissa axis). Where iz stands for the radius of gyration.

The straights bars represent:

EC-3 bar: from the bar to the left side means ≤ 0,4, it would not be needed

to check it.

British Standard bar: from the bar to the left side means that it is not needed to

check it.

The graphics obtained are:



59

Figure 29: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=1,25


0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,25

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,33

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base



60



0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,4

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,44

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base



61



0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,5

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,67

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base



62



0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,75

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=1,8

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base



63

Figure 37: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=2

Figure 38: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=3 (EC-3 and BS limits matching)

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=2

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=3

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base



64

Figure 39: χ vs L/iz for b from 0,1 to 0,5 and ratio of h/b=4

We appreciate some important aspects related with the behaviour of the graphics while

the rate h/b increases from 1,25 to 4. Note that for the presented ranges section h/b,

from 1 to 2 are the commercial ones, and after this (h/b=3 and 4) would be for welded

sections.

First of all, as a general view, we see that on the first graphic, all the families of b’s are

completely superposed. So in that way we can obtain a critical zone of L/iz for each h/b.

The critical zone is between 40 ≤ L/iz ≤ 80. In addition, for the first ratio h/b=1,25, the

Eurocode has the checking point in L/iz =240 and the British Standard in 400. For the

last commercial section, h/b=2 the Eurocode has the checking point in L/iz =220 and the

British Standard in 260, so the difference is narrower than for square sections. Until that

ratio of the Eurocode is more conservative since lower lengths are allowed, but for

welded sections (h/b=3 and 4) the British standard becomes more conservative and

allows lower ratios of L/iz (see Figure 38 and Figure 39).

As an example of what is explained above, for low ratios of h/b is more difficult to

buckle. If we see Figure 29 that shows h/b=1,25 (it is almost a square cross-section),

needs a L/iz ratio of almost 800 to have a reduction for lateral-torsional buckling of

0,65. This means, for example, a section with iz =0,025, the buckle length will be 20m.

But if we see the last (Figure 39) that shows a rate h/b=4 (it is a very slender cross-

section) needs a rate L/iz about 500 to have the same reduction for lateral-torsional

0,65

0,7

0,75

0,8

0,85

0,9

0,95

1

0 200 400 600 800

χ

L/iz

h/b=4

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

EC-3

BS

base



65

buckling. It means that as the same example before, the length will be 12,5m, slightly

higher than half the previous one.

Secondly, the recommendations from Eurocode and British Standard look a little bit

different. We see how Eurocode 3 in general is more conservative than British Standard.

Higher lengths are allowed for h/b ratios up to 2 in the British Standard, whereas for

higher h/b ratios the trend is reversed and the lengths allowed by Eurocode 3 are higher

than the British Standard. The Eurocode fixes the checking point for cases between

(approximate) 0,85≤ χ ≤ 0,87 (corresponding to ≤ 0,4), as he says that it is not

necessary to check the lateral-torsional buckling of square and rectangular cross-

sections.

On the other hand, British Standard fix this value in between 0,77 < χ < 0,82 (it depends

on the rate given by this standard as it is shown on graphics). In that way, we can define

that the Eurocode is more conservative than the British Standard. However for high

ratios like h/b=3 this is not absolutely true, both of them share the checking point and

then (ratios higher to h/b=3) British Standards become more conservative than the

Eurocode as happened before this rate.

But indeed, we can make a whole lecture of the graphics obtained. Really, the

difference appreciated in both standards is not extremely big for high ratios of h/b as 3

or 4. It is true that for low ratios of h/b (commercial ones) we are taking even 0,1 point

(0,85 to 0,75) of reduction factor for lateral-torsional buckling less in British Standard

than Eurocode. But if we see high ratios of h/b the difference of both standards is 0,03

points only (0,85 to 0,88). So roughly speaking, for low h/b ratios the difference

between the two standards is quite important, doubling the length of the beam for which

no checking is needed (from L/iz ≈ 300 for Eurocode 3 to 600 for British Standard). The

difference is gradually reduced while increasing the ratio h/b, becoming negligible for

high values of h/b.



66

4. Parametric Study of Lateral-Torsional Buckling with a Finite

Element Method Program

Hereafter a comparison of the results for critical moment from mathematical formulas

calculated with VBA (see chapter 3) and obtained results with a finite element method

program as carried out for some selected cases. The point is to compare the results of

different parameters such as critical moment. The program used is a commercial

program of structural analysis called SAP2000 v14, which is available on computers at

the School of Civil Engineering (UPC).

4.1. Linear Buckling Analysis with Finite Element Method

Buckling theory presumes the existence of bifurcation point. Consider the figure 1:

Figure 40: Bifurcation Graphic

Where there are two equilibrium configurations possible:

1. The beam could remain straight (primary path)

2. The beam could buckle (secondary path)

A bifurcation point exists if the beam is perfectly straight, perfectly uniform, perfectly

free of end moments and lateral loads, and forces are perfectly centred and perfectly

axial. In addition, the Mcr shown on the figure 1, represents the ideal critical moment

explained in chapter 1.

However, in reality, there are always imperfections, whose magnitude we denote by e.

If e ≠ 0, the beam displays no bifurcation point and structures in generally display limit

points. Curves corresponding to linear buckling analysis with finite elements allow

finding the critical ideal load and bifurcation points. In order to get the influence of

imperfections, residual stress and so on, we should perform a non-linear analysis, but

we are focusing in linear buckling analysis only.



67

Critical buckling loads and associate buckling modes can be obtained by performing a

linear buckling analysis. Then, second order terms of the Green-Lagrange strain are not

negligible and the geometric stiffness matrix Kσ is obtained in similar way in which

stiffness matrix K is obtained from first order terms.

So the linear stiffness matrix [Oñate, 1992] is the following one:

∫ [ ] [ ][ ]

Where:

- K only depends on the geometry of the element, only take into account first

order terms

- B is the matrix that represents state of stress

- D represents the mechanical properties (EA)

And the second order terms stiffness matrix [Cook , Malkus, Plesha, 1989] is the

following one:

∫ [ ] [ ][ ]

Where:

- Kσ is defined by an element’s geometry, displacement field, and state of stress

distribution.

- G is obtained from a shape functions by appropriate differentiation taking into

account second order terms.

- S is the matrix that represents state of stress.

So in that way, we have the next expression:

[ ]

Let’s consider a reference state previous to buckling Fref, with its corresponding Kσ,ref

and define the buckling state as:

F=αcr· Fref and Kσ = αcr· Kσ,ref

When buckling occurs, there is an increment of displacements without a change of

applied load, so combining the equilibrium equation with and equations for

buckling state, we obtain the generalized eigenvalue problem:

[ ]



68

When solving it, dV gives us the shape of the buckling mode (not the amplitude), and

the critical buckling load is given by:

Fcr= αcr · Fref

All the previous expressions are particularized for shells considering that the analysis

with SAP2000 v14 has been done with shells elements.

4.2. Analysis of Different Cases with SAP2000 v14

In this section it is developed the modelling of different cases studied that have been

compared with VBA analytical mathematical formulation. At the end, there is a resume

table with the results obtained. The commercial program SAP2000 v14 [Hernández,

2009] is an integrated software for structural analysis and design.

We want to assess that the values of critical moment obtained are similar to the critical

moment given by Eurocode 3, and also to check that the VBA program is according due

to the fact that we obtain similar values.

4.2.1. Boundary Conditions

All the cases have been studied with the following boundary conditions:

a) Simply supported as a fork. This kind of restraint allows movement into X

direction but not into Y and Z, being X the longitudinal axis and Z the vertical as

usual. For numerical stability one node in the middle of the cross-section on the

top. But one in the middle the cross section on the top flange is completely

fixed. All supports allow the rotation.

b) Uniform Bending Moment, materialized as compression forces on top flange

and tension in the low flange.

c) Shells elements (type shell-thick), due to the cross-sections are a slender

elements and shells’ behaviour is the one that give approximate results.

d) Steel S355 from the EN-1993-1-1



69

Picture 5: Sample of boundary conditions

Picture 6: Sample of one whole beam as introduced in the program completely fixed. All supports allow the rotation.



70

4.2.2. Particular Cases to Study

The cases we have carried out are a square cross-section (h/b=1) and two rectangular

cross-sections (h/b=2 and h/b=3) because they can be a commercial sections (h/b=1 to

2) or welded sections (h/b=3) and is interesting compare the behaviour with VBA and

SAP2000. All of them have the boundary conditions explained above with a uniform

bending moment applied on the extremes of the beam. In this way we could state that

the two ways drive to the same final results. The cross-sections used have the next

configuration:

Height x Base x Thickness (Length)

The different cases analyzed are the followings:

1st case: Square cross-section (h/b=1)

We want to know the results obtained with the SAP2000 v14 and compare them with

the ones obtained with VBA and get the relative error.

Cross section 0,2x0,2x0,015 (L=20m) Values Units

Med 500 KNm

α 3,13 -

Mcr,SAP 1565 KNm

Mcr,VBA 1532,94 KNm

Relative error 2,09 % Table 12: Obtained results (square cross-section, h/b=1)

The obtained results shown on the table confirm how the difference between the

formulation given by EN-1993-1-1 and calculated with VBA in section 3 is very close

to the value calculated with SAP2000. The factor α is the one explained in section 4.1

on the problem to the eigenvalues, so Mcr = Med· α. In addition, the relative error can be

taken as a lack of more discretized elements (2% of relative error), it means that if we

make a finer mesh, the relative error will decrease due to the fact that we will get more

accurate results.

Figure 41: square cross-section (0,2x0,2x0,15 with L=20)



71

The buckling mode obtained with the boundary conditions explained above can be seen

below:

Figure 42: First buckling mode for RHS 0,2x0,2x0,015 with L = 20 m. αcr = 3,13

Figure 43: Longitudinal view of the first buckling mode

Figure 44: Front view of the first buckling mode



72

2nd

case: Rectangular cross-section (h/b=2)




Med 500 KNm

α 0,724 -

Mcr,SAP 370 KNm

Mcr,VBA 350,05 KNm

Relative Error 3,41 % Table 13: Obtained results (rectangular cross-section, h/b=2)

As in the first case, the results obtained with SAP2000 are quite close to the ones

obtained with VBA.

Figure 45: rectangular cross-section (0,2x0,1x0,15 with L=20)



73


below:






74

3rd

case: Rectangular cross-section (h/b=3)




Med 770,27 KNm

α 0,766 -

Mcr,SAP 590,54 KNm

Mcr,VBA 565,02 KNm

Err_rel 4,52 % Table 14: Obtained results (rectangular cross-section, h/b=3)

Once again, the obtained results are very similar to the ones calculated with the VBA.

Figure 49: rectangular cross-section (0,3x0,1x0,15 with L=20)



75


below:






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5. Conclusions

After all the chapters above, hereafter are explained and justified the conclusions of this

work about a parametrical study of lateral-torsional buckling in closed cross-sections

(square and rectangular).

First of all, we have noticed that each standard or guide design uses their own

guidelines to check the lateral-torsional buckling on closed cross-sections, but American

Standard (AISC) does not have any guidance to face the phenomenon on these type of

sections.

Secondly, if we take the Eurocode 3 to compare with British Standard and CIDECT in a

parametrical study (done in section 3.2.2 and 3.2.3) we noticed that the difference from

the limit checking point (before which it is not necessary to check the lateral-torsional

buckling) is relatively small and remains more or less the same if compared to the

CIDECT recommendations. However if the comparison is made considering the British

Standard the difference between the two standards is quite relevant, doubling the length

of the beam for which no checking is needed for low h/b ratios. This difference is

gradually reduced while increasing the h/b ratio, becoming negligible for high values of

h/b.

Also, the boundary conditions studied are unfavourable, because the coefficient applied

C1 which depends on the load application and restraints is taken as 1 (uniform bending

moment and simply supported), so if we modify any condition (clamped, uniform load

and so on), the coefficient can take values above 1, and then the increase on the critical

moment would improve the lateral-torsional buckling checking as the slenderness is

reduced.

5.1. Conclusions of the calibration of the VBA model

After obtaining the different graphics on how affect each parameter, we have noticed

that the thickness does not have much influence, note that on Variable Flange Thickness

section, the behaviour of the χ while thickness increases is insignificant.

This is the reason why we have not focused on its influence.

5.2. Conclusions of the parametric study

First conclusion of that section is that the graphics of a fixed h/b and curves with

different bases, give more information than the ones that b is fixed and have curved

with different h/b. The reason is because if you analyze the first case, all the curves are

superposed and you can get critical limits of L/b, L/h or L/iz with accuracy. In addition

the ratios where lateral-torsional buckling starts to be found are:

Case L/b

Almost L/b=30 for low ratios of h/b as 1 and almost L/b=20 for high ratios of h/b as 3.

But it is true that the reduction factor for lateral-torsional buckling is very small. Note



77

that the case of h/b=3, the checking point is for a ratio L/b=20, but if we take a ratio

L/b=40, the χ affected is 0,94. So it is not a relevant case of failure.

Case L/h

In the case of L/h the checking point is quite far from the normal ratios on structures.

Note that critical points are between 25<L/h<35 and the checking point for lateral-

torsional buckling of Eurocode is L/h=100 and L/h=115 for the CIDECT (low ratio

h/b), so are values really unusual on structures. But if we see the case of high ratios h/b

as 2, the checking point of Eurocode and CIDECT are around L/h=40 for a χ=0,85. So it

would be possible to find some structures affected by this analysis.

Case L/iz

In this case the critical point for lateral-torsional buckling between 50 < L/iz < 80 and

the checking points for low ratios h/b for the Eurocode is L/iz 300 and L/iz 600 for the

British Standard, so it is unlikely to have this ratios in normal structures. It happens the

same for high ratios h/b as 3 or 4 that checking points for the Eurocode is L/iz 150 and

L/iz 170 for the British Standard that continues being unusual. But in case it happens,

the χ is 0,85 and should be checked.

5.3. Conclusions on comparison with British Standard and CIDECT

Firstly, the critical values are between 50 < L/iz <80 and an important conclusion with

the British Standard is that for commercial sections (h/b from 1,25 to 2) the Eurocode is

more conservative than British Standard. In fact for this commercial section, British

Standard even allows twice the ratio of L/iz (specially for h/b=1,25), and this means that

reduction factor for lateral-torsional buckling for Eurocode is 0,85 and for the British

Standard is 0,75. But when we see welded profiles (h/b=3 or 4) the difference between

both standards is very insignificant, actually both fix the checking point at χ =0,85.

Although it is true that for ratios of L/iz=100 or 200 are not common in normal

structures.

Secondly, an important conclusion with the CIDECT is that, in contrast with the British

Standard, both checking points are quite close. In addition we can assess that in the

range of h/b studied (from 1 to 2, so all commercials), Eurocode is more conservative,

while CIDECT allows higher ratios L/h until the checking point (but not too much).

Furthermore, the deference between the checking point of Eurocode 3 and CIDECT is

only 0,03 points separated.

5.4. Conclusions of the SAP2000 v14 study

The obtained results with the comparison between SAP 2000v14 and VBA are quite

similar to the critical moment calculated with the VBA code. The relative error for the

studied commercial cross-sections ranges between 2 and 5% which is a very low value,

outweighed by far by the safety factors used in the codes.



78

5.5. Future Works

As a future studies it would be interesting to know what happen in case of class 4

sections where the cross-section should be reduced to the effective area as it is

explained in EN-1993-1-5: Plated Structural Elements.

Other closed cross-sections remain to be studied, such as the cylindrical hollow sections

that are not really treated on Standards. Also the study could be extended to other

boundary conditions (e.g. clamped, continuous beams and so on), and loading

conditions (e.g. punctual load in the middle, uniform span load among others).

Also, an advanced study could be a non-linear analysis of the lateral-torsional buckling

taking into account initial imperfections, so in that way with the finite element analysis,

the load obtained would not be the ideal load, but the real buckling path.



79

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81

[Seaburg and Carter, 2003] - Seaburg P.A., Carter C. J., 2003 - Torsional

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parametric study of

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