COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Numerical Modelling of the Lateral-Torsional Buckling of Stainless
Steel I-Beams: Comparison with Eurocode 3
N. Lopes 1*, P. M. M. Vila Real 1, L. Simões da Silva 2, E. Mirambell 3
1
University of Aveiro, Dep. Civil Engineering, 3810 Aveiro, Portugal
University of Coimbra, Dep. Civil Engineering, 3030 Coimbra, Portugal
3
Universitat Politècnica de Catalunya, Dep. Enginyeria de la Construcció, Spain
2
Email: [email protected]
Abstract This work presents a numerical study of the behaviour of stainless steel I-beams subjected to lateral torsional
buckling and compares the obtained results with the beam design curves of Eurocode 3.
The EN version of part 1-1 of Eurocode 3 [1] introduces significant changes in the evaluation of the lateral torsional
buckling resistance of unrestrained beams, that improves the too conservative approach of 1992 version [2] of
Eurocode 3 in case of non-uniform bending through the introduction of a correction factor. In fact there is an additional
beneficial effect resulting from reduced plastic zones connected with the variable bending diagram along the member
that was not considered in the earlier versions of the Eurocode 3. This beneficial effect still remains to be taken into
account in Part 1-4 of Eurocode 3 - ‘Supplementary rules for stainless steels’ [3].
The present study aims at evaluating the influence of the loading type on the lateral-torsional buckling resistance of
stainless steel I-beams and investigates the need of using a similar correction factor as the one used for carbon steel.
Key words: stainless steel, eurocode 3, numerical modelling, lateral-torsional buckling
Student Paper Competition
INTRODUCTION
The stainless steel has countless desirable characteristics for a structural material [4-5]. Although his use in the
construction is increasing, is still necessary to develop the knowledge of its behaviour in structural elements.
The stainless steels are known by its non-linear stress-strain relationships with a low proportional stress and an
extensive hardening phase. There isn’t a well defined yield strength, being usually considered the conventional limit of
elasticity to 0.2%.
The program SAFIR [6], a geometrical and material non linear finite element code, has been adapted according to the
material properties defined in prEN 1993-1-4 [3] and EN 1993-1-2 [7], to model the behaviour of stainless steel
structures. This program, widely used by several investigators, has been validated against analytical solutions,
experimental tests and numerical results from other programs, and it has been used in several studies that led to
proposals for safety evaluation of structural elements, already adopted in Eurocode 3. In the numerical simulations,
geometrical imperfections and residual stresses were considered.
The lateral-torsional buckling is an instability phenomenon induced by the compressed flange of unrestrained beams
subjected to bending around the major axis. Fig. 1 illustrates the deformed shape of a steel I-beam subjected to
lateral-torsional buckling.
Recently, during the conversion of Eurocode 3 from ENV to EN status, the project team introduced in EN 1993-1-1 [1]
significant changes in the evaluation of the lateral-torsional buckling resistance of unrestrained beams at room
temperature, compared to the over-conservative approach of ENV 1993-1-1 [2] in the case of non-uniform bending.
The lateral-torsional buckling curves proposed in the ENV version only took in consideration the loading type in the
determination of the elastic critical moment, not accounting for the additional beneficial effect resulting from the
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Figure 1: Deformed shape of a steel I-beam
reduction of the plastic zones, directly related to the fact that the bending diagrams are variable along the beam, leading
to over-conservative results in beams not subjected to uniform bending diagrams [3]. As for other international
regulations [9,10] that already considered this effect, a correction factor that considers the loading type was introduced
in the EN 1993-1-1 [1]. The EN 1993-1-1 presents two approaches for the safety evaluation of lateral-torsional
buckling of I-beams. One of them is identical to the previous procedure proposed in the version from 1992 [2], and can
be used for any kind of beams and is named “general case”. The other procedure, more specific, can only be used in hot
rolled or equivalent welded sections.
The objective of this study is to evaluate the influence of the loading type on the lateral-torsional buckling resistance of
stainless steel I-beams and investigates the need of using a similar correction factor as the one used for carbon steel.
NUMERICAL MODEL
This study is performed using the specialised finite element code SAFIR [6], which is a finite element code for
geometrical and material non-linear analysis, developed at the University of Liege.
A three-dimensional (3D) beam element has been used, based on the following formulations and hypotheses:
(1) Displacement type element in a total co-rotational description;
(2) Prismatic element;
(3) The displacement of the node line is described by the displacements of the three nodes of the element, two nodes at
each end supporting seven degrees of freedom, three translations, three rotations and the warping amplitude, plus one
node at the mid-length supporting one degree of freedom, namely the non-linear part of the longitudinal displacement;
(4) The Bernoulli hypothesis is considered, i.e., in bending, plane sections remain plane and perpendicular to the
longitudinal axis and no shear deformation is considered;
(5) No local buckling is taken into account, which is the reason why only Class 1 and Class 2 sections can be used [1],
allowing for a fully plastic stress distribution on the cross-section;
(6) The strains are small (von Kármán hypothesis), i.e.
1 ∂u
<< 1
2 ∂x
(1)
where u is the longitudinal displacement and x is the longitudinal co-ordinate;
(7) The angles between the deformed longitudinal axis and the undeformed but translated longitudinal axis are small, i.
e., sin ϕ ≅ ϕ and cos ϕ ≅ 1 where ϕ is the angle between the arc and the chord of the translated beam finite element;
(7) The longitudinal integrations are numerically calculated using Gauss’ method;
(8) The cross-section is discretised by means of triangular or quadrilateral fibres. At every longitudinal point of
integration, all variables, such as temperature, strain, stress, etc., are uniform in each fibre;
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(9) The tangent stiffness matrix is evaluated at each iteration during the convergence process (pure Newton-Raphson
method);
(10) Residual stresses are considered by means of initial and constant strains [11];
(11) The material behaviour in case of strain unloading is elastic, with the elastic modulus equal to the Young modulus
at the origin of the stress-strain curve. In the same cross-section, some fibres that have yielded may therefore exhibit a
decreased tangent modulus because they are still on the loading branch, whereas, at the same time, some other fibres
behave elastically;
(12) The collapse criterion of the structure is defined as the instant when the stiffness matrix becomes not positive
definite, becoming impossible to establish the equilibrium of the structure.
The program allows for the use of the “arc-length” method, to solve the local failure problems that sometimes appear.
In fact, in hyperstatic structures, failure in one element may not correspond to global collapse of the structure. It is
possible that, beyond local failure, part of the internal forces that can not be supported by the local element, are
redistributed to other structural elements, leading to a new equilibrium position.
CASE STUDY
A simply supported beam with fork supports, as shown in Fig. 2, was chosen to explore the validity of the beam safety
verifications. Regarding the bending moment variation along the member length, five values, (-1, -0.5, 0, 0.5 and 1), of
the ψ ratio (see fig. 2) have been investigated as well as a mid span concentrated load and a uniformly distributed
load.
Figure 2: Simply supported beam with non-uniform bending
The following welded cross-sections were used: IPE 220 steel section (representative of h/b = 2), HEA 500 steel
section (representative of h/b < 2) and IPE 500 steel section (representative of h/b > 2). It was studied the stainless steel
grade 1.4301 (also known as 304) for each cross-section.
In the numerical simulations, a lateral geometric imperfection given by the following expression was considered:
y ( x) =
l
⎛ πx ⎞
sin ⎜ ⎟
1000 ⎝ l ⎠
(2)
where l is the length of the beam.
An initial rotation around the beam axis with a maximum value of l / 1000 radians at mid span was also considered.
The adopted residual stresses follow, the typical patterns for carbon steel welded sections, considered constant across
the thickness of the web and flanges. The distribution is shown in Fig. 3, and has the maximum value of f y (yield
strength) [12].
FORMULAE IN PART 1-4 OF EUROCODE 3 FOR LATERAL-TORSIONAL BUCKLING OF
STAINLESS STEEL BEAMS
The design lateral torsional buckling resistance moment of a laterally unrestrained stainless steel beam with Class 1 or
Class 2 cross-section is determined, according to part 1-4 of Eurocode 3, by the following expression
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f
y
T
C
C
0.25 f
y
T
f
y
C
T
Figure 3: Residual stresses: C – compression; T – tension
M b , Rd = χ LT W pl , y f y
1
(3)
γ M1
where, χLT is given by
χ LT =
1
2
2
φ LT + φ LT
− λ LT
but χ LT ≤ 1
(4)
with
φ LT =
[
(
)
2
1
1 + α LT λ LT − 0.4 + λ LT
2
]
(5)
In this part 1-4 of Eurocode 3 [3] the imperfection factor αLT is 0.34 for cold formed sections and hollow sections,
whereas for welded open sections or other sections that do not possess available tests results, it should be 0.76. It was
this last value the one used in this study, but as it will be shown it must be calibrated in future works.
The non-dimensional slenderness for lateral torsional buckling is given by the expression.
λ LT =
W pl , y f y
(6)
M cr
where W pl , y is the plastic bending modulus, f y is the yield strength of steel and M cr is the elastic critical moment for
lateral torsional buckling.
If λ LT ≤ 0.4 or
M Ed
≤ 0.16 no lateral-torsional Buckling check is required.
M cr
FORMULAE IN PART 1-1 OF EUROCODE 3 FOR LATERAL-TORSIONAL BUCKLING OF
CARBON STEEL BEAMS
The design resistant moment to lateral-torsional buckling of steel profiles with class sections 1 and 2 is determined,
according to part 1-1 of Eurocode 3, by the following expression
M b , Rd = χ LT W pl , y f y
1
(7)
γ M1
This expression is similar to the one used for stainless steel (3). However to take into account the influence of the
shape of the bending diagrams, a modified reduction factor is adopted for the case of hot rolled or equivalent welded
sections, using the expression
⎯ 446 ⎯
χ LT ,mod =
χ LT
f
with
χ LT ,mod ≤ 1
(8)
where
f = 1−
(
)
2
1
(1 − kc ) ⎡⎢1 − 2 λ LT −0.8 ⎤⎥ with
⎣
⎦
2
f ≤1
(9)
The correction factor kc is given in table 1.
Table 1: Correction factor kc for the different bending diagrams
Bending diagram
kc
1
1.33 − 0.33ψ
0.86
0.94
LATERAL-TORSIONAL BUCKLING PROPOSAL FOR STAINLESS STEEL ELEMENTS
Based on the last version of the part 1-1 of Eurocode 3, it is proposed that the design lateral torsional buckling
resistance moment of a laterally unrestrained stainless steel beam with Class 1 or Class 2, can be determined with a
similar expression
M b , Rd = χ LT ,modW pl , y
fy
(10)
γ M1
where the modified reduction factor is determined as for the carbon steel, equation (8) and (9)
PARAMETRIC STUDY
Results for stainless steel beams subjected to lateral-torsional buckling are shown in Figs. 4-6 for three values, (-1, 0
and 1) of the ψ ratio, a mid span concentrated load and a uniformly distributed load. The graphics of the Figs. 4-6,
compare the curves obtained through part 1-4 of Eurocode 3, described in section 3 of this paper (denoted “prEN
1993-1-4” in the graphics), the curve obtained with the new proposal presented in the last section (denoted “prEN
1993-1-4 / f” in the graphics), and the numerical results obtained with the program SAFIR.
These figures show that as the slenderness of the cross-section increases, the beam design curve for lateral-torsional
buckling should go down to ensure a similar level of accuracy. This can be obtained by increasing the value of the
imperfection factor α LT on equation (9). For the moment works is on progress for the calibration of the value of this
imperfection factor. On the other hand the same figures show that the use of a modified reduction factor for the
lateral-torsional buckling for the case of stainless steel equal to the one that has been adopted in Eurocode 3 for carbon
steel, improves the beam design curve when compared with the numerical simulations.
CONCLUSIONS
In this paper it is presented a new proposal for the lateral-torsional buckling of stainless steel beams using the approach
already adopted in Eurocode 3 for carbon steel, considering the influence of the loading type.
Figs. 4, 5 and 6 show that the proposal made here gives results that are in good agreement with the numerical results
obtained with the program SAFIR for stainless steel beams. This study also has shown that the slenderness of the
cross-section should be taken into account as for the case of carbon steel. Further work should be done to adjust the
value of the imperfection factor α LT as a function of the ratio h/b like in pat 1.1 of the Eurocode 3.
⎯ 447 ⎯
ψ =0
M/Mpl
ψ =1
M/Mpl
prEN1993-1-4
prEN1993-1-4
prEN1993-1-4 / f
1.2
Safir
prEN1993-1-4 / f
1.2
Safir
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ψ =-1
M/Mpl
1.8
2
0
λ LT
prEN1993-1-4
0.4
0.6
0.8
1
1.2
1.4
1.6
Concentrated load
M/Mpl
prEN1993-1-4 / f
1.2
0.2
1.8
2
λ LT
prEN1993-1-4
prEN1993-1-4 / f
1.2
Safir
Safir
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λ LT
0
M/Mpl
0.2
0.4
Distributed load
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λ LT
prEN1993-1-4
prEN1993-1-4 / f
1.2
Safir
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λ LT
Figure 4: Results for the welded HEA500
ψ =0
M/Mpl
ψ =1
M/Mpl
prEN1993-1-4
prEN1993-1-4 / f
prEN1993-1-4
1.2
Safir
prEN1993-1-4 / f
1.2
Safir
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ψ =-1
M/Mpl
1.8
2
0
λ LT
prEN1993-1-4
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2
prEN1993-1-4 / f
1.2
Safir
λ LT
prEN1993-1-4
Concentrated load
M/Mpl
prEN1993-1-4 / f
1.2
1.8
Safir
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
λ LT
M/Mpl
2
0
0.2
0.4
0.6
0.8
prEN1993-1-4
Distributed load
prEN1993-1-4 / f
1.2
Safir
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
λ LT
Figure 5: Results for the welded IPE220
⎯ 448 ⎯
2
1
1.2
1.4
1.6
1.8
λ LT
2
ψ =0
M/Mpl
ψ =1
M/Mpl
prEN1993-1-4
prEN1993-1-4
prEN1993-1-4 / f
1.2
Safir
prEN1993-1-4 / f
1.2
Safir
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ψ =-1
M/Mpl
1.8
2
0
λ LT
prEN1993-1-4
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
λ LT
2
prEN1993-1-4
Concentrated load
M/Mpl
prEN1993-1-4 / f
prEN1993-1-4 / f
1.2
0.2
1.2
Safir
Safir
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
M/Mpl
λ LT
2
0
0.2
0.4
Distributed load
0.6
0.8
1
1.2
1.4
1.6
1.8
λ LT
2
prEN1993-1-4
1.2
prEN1993-1-4 / f
Safir
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
λ LT
2
Figure 6: Results for the welded IPE500
It should be pointed out that the results presented here were obtained only for the stainless steel grade 1.4301. Knowing
that the other stainless steel grades mentioned in part 1-2 of Eurocode 3, exhibit different stress-strain relationships, it
is to be expected that future work will suggest some changes in the proposal presented here, taking into account also
the influence of the steel grade.
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Rules for Buildings. EN 1993-1-1, Brussels, Belgium, 2005.
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Rules for Buildings. ENV 1993-1-1, Brussels, Belgium, 1992.
3. European Committee for Standardisation. Eurocode 3, Design of Steel Structures – part 1-4. Supplementary
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9. Trahair NS, Bradford MA, Nethercot DA. The Behaviour and Design of Steel Structures to BS5950. Spon Press,
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