10h International Conference on

Steel and Aluminium Structures (ICSAS24)

Rio de Janeiro, Brazil, 5-7, June, 2024

LATERAL-TORSIONAL BUCKLING RESISTANCE PREDICTION OF

HIGH-STRENGTH STEEL I-BEAMS WITH SINUSOIDAL WEB
OPENINGS

Douglas F. SANTOS a, Carlos H. MARTINS b, Felipe P. V. FERREIRAc, Konstantinos

D. TSAVDARIDISd and Hermano S. CARDOSO e

a
Department of Civil Engineering, Federal University of São Carlos, São Carlos, Brazil
Emails: douglassantos88@hotmail.com

b
Department of Civil Engineering, State University of Maringá, Maringá, Brazil
Emails: chmartins@uem.br

c
Faculty of Civil Engineering, Federal University of Uberlândia, Uberlândia, Brazil
Emails: fpvferreira@ufu.br

d
Department of Engineering, University of London, London, UK
Emails: Konstantinos.tsavdaridis@city.ac.uk

e
ArcelorMittal Global Research and Development, Serra, Brazil
Emails: hermano.cardoso@arcelormittal.com.br

Keywords: High-strength steel; Perforated beams; Stability; Lateral-torsional buckling; Sinusoidal web
opening; Residual stress; Finite element method.

Abstract. A new cellular beam with sinusoidal shape of web openings, aka Angelina™, shows different
behaviour in comparison with the standard beams with openings. It has its lateral-torsional buckling
(LTB) resistance affected by residual stress in a more significant way than the beams with solid webs
since the thin-walled plates are subject to the castellation process by thermal cutting and welding. The
present study aims to predict the LTB resistance of high-strength steel (HSS) I-beams with sinusoidal
web openings. For this, a finite element model is developed based on experimental tests, considering
common and HSS. Buckling and post-buckling is performed. A parametric study is conducted by varying
the strength of steel. It is expected that by increasing the strength of steel, there will be an increase in
the lateral-torsional buckling resistance, mainly in the inelastic regime. The increase of the resistance
for the different kind of steel analysed is proportional to the upgrade in the steel strength.

1 INTRODUCTION
Nowadays, it is common to use steel I-beams with web openings in structural project of
buildings. There are different kinds of geometric configuration for the web openings, which
made possible a series of benefits when compared to the use of steel beams without web
openings [1,2].
 Douglas F. SANTOS et al.

The beams with sinusoidal web openings are produced by cutting a steel I-beam or a steel
plate that will be part of the web from a welded steel I-beam. The cut in a high temperature and
with a sinusoidal shape is made to separate the element in two parts, which later will be welded
back together. This fabrication process is similar to castellated beams, which only takes one cut
in the web following a sinusoidal curve “path” [3]. Figure 1 below exemplifies the fabrication
process.

Figure 1: Fabrication scheme of beams with sinusoidal opening [1]

The possible failure modes for these beams investigated in studies such as [4,5,6]. The local
failure modes are the Vierendeel mechanism which is characterized by the occurring of plastic
hinges around the openings, the Web post buckling which is the local buckling of the web
between the openings and Tee local buckling which is the instability of the compressed tee
section on the opening part of the beam.
There are too the global failure modes, which are the Lateral-distortional buckling, Flexural
mechanism and Lateral-torsional buckling, which will be the failure mode analyzed in this study
[7].
In recent years, high strength steels (HSSs) have been increasingly used in construction and
other industries, where the main purpose is to reduce cost and size of the members structures.
Currently, 460 MPa of nominal yield stress is the number that divides the common steel to the
HSS. Steels with a higher nominal yield can be considered as HSS [8,9].
In general, there are two basic types of 𝜎 − 𝜀 curves for HSS, the curves with discontinuous
yielding, i.e., with an apparent yield plateau, and the curves demonstrating continuous yielding
without yield plateau [8,9], shown in figure 2. The occurrence of yield plateau depends on the
fabrication process and the chemical composition of the HSS during its production. HSS tends
to have to have a low ductility, and the higher the yield stress is, the ultimate strain significantly
decreases. Unsafe design may happen if it is needed for the structure to work with excessive
deformation. Therefore, it is important to avoid HSS reaching its fracture strain in any part of
the structure [8].
To reach a high variety of cross section needed in the construction industry, welding
becomes a prevailing manufacturing method for I-sections. Nevertheless, the high temperatures
of the welding provoke the transformation in the microstructure of steel materials, as well as
the residual stress within sections because of the local thermal expansion and contraction. Steel
production techniques for HSS, such as quenching and tempering (QT) process, and
thermomechanical controlled (TMCP) process, affect the mechanical behavior of steel
materials near the welding seam [10,11].

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 Douglas F. SANTOS et al.

Figure 2: Comparison between stress-strain curves of different types of steel [8]

Residual stress models for hot-rolled and welded regular strength I-section steel members
have been widely reported in the literature [12,13], but it is less common to find formulations
for welded HSS I-sections. In a more recent studies [14,15] proposed models and formulations
for welded HSS I-section. Its model was validated for steel grades S235 to S890. Most recent
measurements of residual stress in welded I-section available in the literature such as [14,16-
19].
Therefore, this paper aims to analyse with finite element method using ABAQUS, the LTB
resistance of HSS I-section beam with sinusoidal web opening.

2 NUMERICAL MODEL

2.1 FE Model
The FE models of the I-section beams with sinusoidal web openings were developed by the
commercial software ABAQUS as illustrated in figure 3. As reference, it was also simulated a
beam with web opening with common strength steel (𝑓𝑦 ≤ 460 MPa), in this study, it was used
S355 as reference.

Figure 3: One of the numerical models of the HSS I-section steel beam with sinusoidal web opening

It was adopted the quadrilateral shell element with reduced integration (S4R) with mesh size
of 10 mm.
A total number of 36 beams was analysed. One of the variables considered was different
boundary conditions and different load conditions. It was modelled beams for 3 points and 4
points bending test. And with and without lateral restrain where the load was applied.

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 Douglas F. SANTOS et al.

There were three cross-sections analysed in this study. The first two were defined to match
class 1 and 2 of Eurocode 3 [20] (beams EC1 and EC2 from table 1), and the third cross section
was defined following the Brazilian Standard [21] (beam NBR from table 1) to avoid local
buckling of the plates from the I-section beam. The size of the cross-sections is presented at
figure 4.
In addition to the variation of the boundary conditions and cross-sections, the beams were
analysed following three different steel grades, S355, S500 and Q690. The combination of the
variables of the 36 beams analysed is shown at table 1.
For the geometries of the openings in the web, it was used the equation (1), with the variables
explained in figure 4, provided in [22].

𝑙𝑠 𝑥
𝑦= {𝑠𝑖𝑛 [𝜋 ( + 1.5)] + 1} (1)
4 𝑎0

Figure 4: Cross-sections

2.2 Material characteristics

To characterize the three different steel grades, it was used the stress-strain curves provided
by [8] figure 5. Table 2 summarizes the steel grades and its mechanical characteristics. For the
S355 steel, it was used the values presented by [23], for S500 it was used the values provided
by ArcelorMittal, which were found by experimental tests, and for the Q690 the values used
was the same cited by [10]. The data of the experimental test for S500 showed it fits better in
the discontinuous model. Since, it is known that S355 has the same behaviour, this model was
adopted for the three steel grades.

2.3 FE Analysis
The simulations of the 36 beams were made in a two-step analysis being the first one a
buckling analysis, which was used mainly to find the first positive eigenvalue. Second analysis
were made using Static-Riks, so it was possible to understand through every increment, the
behaviour of each beam and, using the data from the analysis to plot load-displacement curves.
It was in this second phase where it was applied in the models de geometric imperfections
and residual stress, according to what was proposed by [10] and illustrated in the figure 6.

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 Douglas F. SANTOS et al.

Table 1: Combination of the variables from the models analysed

EC1- EC1- EC1- EC2- EC2- EC2- NBR- NBR- NBR-

Tested Beam S355 S500 Q690 S355 S500 Q690 S355 S500 Q690
Yield Limit
355 500 690 355 500 690 355 500 690
(MPa)
Parent Profile
322.5 322.5 322.5 370.5 370.5 370.5 202.5 202.5 202.5
Height: h (mm)
Profile height: d
430.5 430.5 430.5 491.5 491.5 491.5 278.5 278.5 278.5
(mm)
Flange Width: bf
122 122 122 136 136 136 218 218 218
(mm)
Flange
Thickness: tf 9.5 9.5 9.5 9.5 9.5 9.5 9.5 9.5 9.5
(mm)
Web Thickness:
6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3
tw (mm)
Opening height:
180 180 180 205 205 205 114 114 114
a0 (mm)
Sinusoid length:
270 270 270 310 310 310 175 175 175
ls (mm)
Web-post length:
140 140 140 180 180 180 110 110 110
w (mm)
Beam openings
9 9 9 7 7 7 13 13 13
number (4PBT)
Beam openings
8 8 8 6 6 6 12 12 12
number (3PBT)
Opening length:
680 680 680 800 800 800 460 460 460
L0 (mm)
Span: P (mm)
Between 7500 7500 7500 7500 7500 7500 7500 7500 7500
Supports
Span: PL (mm)
2460 2460 2460 2940 2940 2940 2850 2850 2850
between loads
Total length of
the beam: L 8000 8000 8000 8000 8000 8000 8000 8000 8000
(mm)

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 Douglas F. SANTOS et al.

Figure 5: HSS stress-strain curves proposed by [8]

Table 2: Mechanical properties for steel grades

Elastic Ultimate
Steel Limit fy Limit fu
Grade (MPa) (MPa) εsh (%) εu (%) E (MPa) Reference
S355 355 490 1.74 16.53 210000 [23]
S500 560 642.85 2.8 9.553 200000 ARCELORMITTAL
Q690 788.6 834.5 2.42 6.92 217200 [10]

Figure 6: HSS residual stress model

3 RESULTS
After the analysis was concluded, it was possible to plot 36 curves load-displacement, as
shown in figure 7 and 8, where the displacement was measured at the middle of the span in the
vertical axis. The curves are classified by the boundary conditions, cross-sections and steel
grades and they are illustrated in the figures 10 and 11. Figures 12 presents some deformed
shapes in the analysis.

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 Douglas F. SANTOS et al.

100 160

90 140
80
120
70
60 100
P (kN)

P (kN)
50 80
40
60
30
S500
20 40
Q690 S500
10 S355 20 Q690
0 S355
0 50 100 150 200 0
0 50 100 150 200
U(mm)
U(mm)
(a) (b)
180
120
160
100
140

120 80
100
P (kN)

P (kN)

60
80

60 40
40 S500 S500
20
20 Q690 Q690
S355 S355
0 0
0 100 200 300 0 50 100 150 200
U(mm) U(mm)
(c) (d)
160 250

140
200
120

100
150
P (kN)

P (kN)

80

60 100

40
S500 50 S500
20 Q690 Q690
S355 S355
0 0
0 50 100 150 200 0 100 200 300
U(mm) U(mm)
(e) (f)
Figure 7: 4-point (a, b, c) and 3 point-bending (d, e, f) with lateral restrain tests curves for beams
EC1, EC2 and NBR respectively

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 Douglas F. SANTOS et al.

25 35

30
20
25
15
P (kN)

20

P (kN)
10 15
S500
10
5 Q690 S500
S355 5 Q690
0 S355
0 50 100 150 200 0
U(mm) 0 50 100 150 200
U(mm)
(a) (b)
60 30

50 25

40 20
P (kN)
P (kN)

30 15

20 10

S500
S500 5 Q690
10
Q690 S355
S355 0
0
0 50 100 150 200
0 100 200 300
U(mm)
U(mm)
(d)
(c)
40 80

35 70

30 60

25 50
P (kN)

P (kN)

20 40

15 30

10 20
S500 S500
5 Q690 10 Q690
S355 S355
0 0
0 50 100 150 200 0 50 100 150 200 250
U(mm) U(mm)
(e) (f)
Figure 8: 4-point (a, b, c) and 3 point-bending (d, e, f) without lateral restrain tests curves for beams
EC1, EC2 and NBR respectively

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 Douglas F. SANTOS et al.

(a) (b)

(c) (d)

Figure 9: Some examples of deformed shapes from the analysis

4 CONCLUSION
Studying the LTB for HSS of beams with sinusoidal opening using FEM, made possible to
observe through the curves of load-displacement that the yield stress impact in the resistance of
the beams in the ultimate load of them. It also has an impact in the displacement measured in
the middle of the span. Steel grades with higher yield stress made possible a higher
displacement.
The beams with the cross-section based on the class 1 and 2 from Eurocode presented a very
similar behaviour while the cross-section based on the Brazilian standard showed a different
curve shape, since it was the cross-section with the smaller depth.

ACKNOWLEDGMENTS
This study was financed by the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq).

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