Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading
Евростандарт EN 1993-1-1 описывает общий метод определения предельной нагрузки для стальных стержней при продольном изгибе с кручением. В методе учитываются кривые, описывающие потерю устойчивости при продольном изгибе. Предельная нагрузка при продольном изгибе с кручением может быть рассчитана ме...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1127272017-01-27T03:02:41Z Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading Dahmani, L. Boudjcmia, A. Научно-технический раздел Евростандарт EN 1993-1-1 описывает общий метод определения предельной нагрузки для стальных стержней при продольном изгибе с кручением. В методе учитываются кривые, описывающие потерю устойчивости при продольном изгибе. Предельная нагрузка при продольном изгибе с кручением может быть рассчитана методом конечных элементов на основе геометрического и нелинейного анализа материалов стержня с дефектами. Проведено сопоставление значений предельной нагрузки в соответствии с нормами Евростандарта EN 1993-1-1 для продольного изгиба поперечно закрепленных стержней кручения с таковыми, полученными путем моделирования методом конечных элементов на основе параметрического исследования. Євростандарт EN 1993-1-1 описує загальний метод визначення граничного навантаження для стальних стрижнів при поздовжньому згині з крутінням. У методі враховуються криві, що описують втрату cтійкості при поздовжньому згині. Граничне навантаження при поздовжньому згині з крутінням може бути розраховано методом скінченних елементів на основі геометричного і нелінійного аналізу матеріалів стрижня з дефектами. Проведено зіставлення значень граничного навантаження у відповідності з нормами Евростандарту EN 1993-1-1 для поздовжнього згину поперечно закріплених стрижнів крутіння з такими, що отримані шляхом моделювання методом скінченних елементів на основі параметричного дослідження. 2014 Article Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading / L. Dahmani, A. Boudjcmia // Проблемы прочности. — 2014. — № 3. — С. 164-168. — Бібліогр.: 8 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112727 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Dahmani, L. Boudjcmia, A. Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading Проблемы прочности |
| description |
Евростандарт EN 1993-1-1 описывает общий метод определения предельной нагрузки для
стальных стержней при продольном изгибе с кручением. В методе учитываются кривые,
описывающие потерю устойчивости при продольном изгибе. Предельная нагрузка при продольном изгибе с кручением может быть рассчитана методом конечных элементов на основе
геометрического и нелинейного анализа материалов стержня с дефектами. Проведено сопоставление значений предельной нагрузки в соответствии с нормами Евростандарта EN
1993-1-1 для продольного изгиба поперечно закрепленных стержней кручения с таковыми,
полученными путем моделирования методом конечных элементов на основе параметрического исследования. |
| format |
Article |
| author |
Dahmani, L. Boudjcmia, A. |
| author_facet |
Dahmani, L. Boudjcmia, A. |
| author_sort |
Dahmani, L. |
| title |
Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading |
| title_short |
Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading |
| title_full |
Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading |
| title_fullStr |
Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading |
| title_full_unstemmed |
Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading |
| title_sort |
lateral torsional buckling response of steel beam with different boundary conditions and loading |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2014 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112727 |
| citation_txt |
Lateral Torsional Buckling Response of Steel Beam with Different Boundary Conditions and Loading / L. Dahmani, A. Boudjcmia // Проблемы прочности. — 2014. — № 3. — С. 164-168. — Бібліогр.: 8 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT dahmanil lateraltorsionalbucklingresponseofsteelbeamwithdifferentboundaryconditionsandloading AT boudjcmiaa lateraltorsionalbucklingresponseofsteelbeamwithdifferentboundaryconditionsandloading |
| first_indexed |
2025-07-08T04:32:37Z |
| last_indexed |
2025-07-08T04:32:37Z |
| _version_ |
1837051844785864704 |
| fulltext |
UDC 539.4
Lateral Torsional Buckling Response of Steel Beam with Different Boundary
Conditions and Loading
L. Dahmani and A. Boudjemia
Mouloud Mammeri University, Tizi-Ouzou, Algeria
ÓÄÊ 539.4
Òîðñèîííàÿ ïîòåðÿ óñòîé÷èâîñòè ñòàëüíîãî ñòåðæíÿ ïðè ðàçëè÷íûõ
ãðàíè÷íûõ óñëîâèÿõ è ïðîäîëüíûõ èçãèáíûõ íàãðóçêàõ
Ë. Äàõìàíè, À. Áóäæåìèà
Óíèâåðñèòåò èì. Ìóëóäà Ìàììåðè, Òèçè-Óçó, Àëæèð
Åâðîñòàíäàðò EN 1993-1-1 îïèñûâàåò îáùèé ìåòîä îïðåäåëåíèÿ ïðåäåëüíîé íàãðóçêè äëÿ
ñòàëüíûõ ñòåðæíåé ïðè ïðîäîëüíîì èçãèáå ñ êðó÷åíèåì.  ìåòîäå ó÷èòûâàþòñÿ êðèâûå,
îïèñûâàþùèå ïîòåðþ óñòîé÷èâîñòè ïðè ïðîäîëüíîì èçãèáå. Ïðåäåëüíàÿ íàãðóçêà ïðè ïðî-
äîëüíîì èçãèáå ñ êðó÷åíèåì ìîæåò áûòü ðàññ÷èòàíà ìåòîäîì êîíå÷íûõ ýëåìåíòîâ íà îñíîâå
ãåîìåòðè÷åñêîãî è íåëèíåéíîãî àíàëèçà ìàòåðèàëîâ ñòåðæíÿ ñ äåôåêòàìè. Ïðîâåäåíî ñîïîñ-
òàâëåíèå çíà÷åíèé ïðåäåëüíîé íàãðóçêè â ñîîòâåòñòâèè ñ íîðìàìè Åâðîñòàíäàðòà EN
1993-1-1 äëÿ ïðîäîëüíîãî èçãèáà ïîïåðå÷íî çàêðåïëåííûõ ñòåðæíåé êðó÷åíèÿ ñ òàêîâûìè,
ïîëó÷åííûìè ïóòåì ìîäåëèðîâàíèÿ ìåòîäîì êîíå÷íûõ ýëåìåíòîâ íà îñíîâå ïàðàìåòðè÷åñ-
êîãî èññëåäîâàíèÿ.
Êëþ÷åâûå ñëîâà: ñòàëüíîé ñòåðæåíü, ïðîäîëüíûé èçãèá ñ êðó÷åíèåì, Åâðîñòàíäàðò
EN 1993-1-1, ìåòîä êîíå÷íûõ ýëåìåíòîâ, ïðîãðàììíàÿ ñèñòåìà ANSYS.
Introduction. Buckling and lateral stability are among the key parameters in the
design of steel structures [1–4]. Flexural members subjected to bending about their major
axis may develop buckling in the compression flange combined with lateral bending, leading
to what is known as lateral torsional buckling [3, 4]. For doubló symmetric laterally
unbraced slender beams, lateral torsional buckling can govern their ultimate limit state.
Lateral Torsional Buckling. A short beam with a compact cross section can reach its
full plastic moment capacity without any lateral instability. However, if the beam is slender
and the compression flange is not adequately braced in the lateral direction, a different
phenomenon occurs. As the beam is loaded in bending about its strong axis, it deforms in
the direction of loading, but after buckling it demonstrates an angular deformation (Fig. 1).
© L. DAHMANI, A. BOUDJEMIA, 2014
164 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
Fig. 1. Lateral torsional buckling at angle �.
The lateral torsional buckling capacity depends upon a number of material and
geometric properties, support conditions, and location of the applied load relative to the
shear centre and bending moment distribution along the length of the member. The critical
elastic lateral torsional buckling capacity for the uniform moment gradient is given by [5]:
M
EI
L
I
I
L GI
EI
cr
z w
z
t
z
� �
�
�
�
�
�
�
2
2
2
2
0 5.
, (1)
where M cr is the elastic lateral torsional buckling strength, E is the modulus of elasticity,
G is the shear modulus, I z is the moment of inertia about weak axis, I t is the Saint-
Venant torsional constant, and I w is the warping constant of the section.
Generally, consideration of the nonuniform bending moment diagram is taken into
account by means of the equivalent uniform moment factor C1 [5]. The elastic critical
moment of a simply supported beam with a uniform moment is multiplied by this factor to
obtain the elastic critical moment for any bending moment diagram,
C1
2188 140 052 2 7� � �
. . . . ,� �
where � is the ratio of the smaller factored moment to the larger one at the end points of
lateral support, �� M Ma b for M Ma b� (�
1 1� ). This ratio is positive for the
double curvature and negative for the single curvature. The moments are applied at the end
points of lateral support.
Code Requirements. For both the general and specific methods in Eurocode 3 [5] to
determine the ultimate lateral torsional buckling (LTB) load of beams in bending, the
design buckling resistance moment should be taken as
M
W f
b Rd LT
y y
M
. ,� �
� 1
(2)
in which Wy is the appropriate section modulus: W Wy pl y� . for class 1 or 2 sections.
The reduction factor � LT is a function of the imperfection factor �LT and the relative
slenderness is given by
� LT
y y
cr
W f
M
� . (3)
This relative slenderness will be used in subsequent equations to determine the
reduction factor. It should be noted that the elastic critical bending moment for LTB is not
specified by Eurocode 3 [5], but its determination is left to a designer.
General Method. This method is presented in clause 6.3.2.2 of Eurocode 3 [5] as the
“general case,” hereafter referred to as the general method (GM). According to the GM, the
reduction factor � LT for LTB of beams is similar to that for column buckling [6, 7]:
�
� � �
LT
LT LT LT
�
� �
1
1
2 2[ ]
, (4)
� � � �LT LT LT LT� � � �05 1 02 2. [ ( . ) ]. (5)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 165
Lateral Torsional Buckling Response of Steel Beam ...
When � LT
04. or the design bending moment M MEd cr
016. , then � LT � 1. The
imperfection factor �LT is selected according to the required buckling curve for the
design of the beam. The appropriate buckling curve is given in Eurocode 3 [5].
Finite Element Method. ANSYS [8], a commercial finite element software, was
used for the analysis. An eigenvalue analysis was used to get the deflected shape (mode
shape or eigenvector) and the associated load factor (eigenvalue). The resulting eigenvalues
are actually the load factors to be multiplied by the applied loading, in order to obtain the
critical buckling load.
The element used in ANSYS [8], BEAM 188, is a quadratic three-dimensional beam
element suitable for analyzing slender to moderately stocky beams. It possesses warping
degrees of freedom, in addition to the conventional six degrees of freedom (Fig.2). The
results of the buckling analysis are shown in Fig. 3, where the buckled shape and the load
factor (�) are indicated.
The above figure depicts the behavior of the lateral torsional buckling, where lateral
displacement combined with twisting can be observed.
Validation. In order to validate the finite element model developed for this
investigation, an eigenvalue buckling analysis was carried out for the model shown in
Fig. 2, and the predicted load factors (Table 1) were compared with the theoretical values
of the lateral torsional buckling capacity.
The difference between the results calculated using formula is � �
�| |
.
� �
�
ANSYS theor
ANSYS
a b
Fig. 2. Finite element model and boundary conditions for (a) simply supported and (b) cantilever
beams.
a b
Fig. 3. Buckled shape and load factor for (a) simply supported and (b) cantilever beams.
166 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
L. Dahmani and A. Boudjemia
The buckling capacity predicted using the beam element BEAM 188 of ANSYS [8] is
within 0.6% of the theoretical value.
Conclusions. This paper compares the ultimate lateral torsional buckling loads of
unrestrained beams in bending based on the design rules in Eurocode 3 for the ultimate
loads obtained via finite element simulations. For the calculations performed in the
parameter study, worrisome results have been obtained on the validity of the general
methods for lateral torsional buckling of rolled sections. It can be concluded that the
general method can lead to the underestimations of even less than 0.6% of the ultimate
lateral torsional buckling load of unrestrained beams obtained via the finite element
simulations. The general method gives good results for lateral torsional buckling of steel
beams without restraints between the supports.
For these situations, there is quite good agreement between the values given by the
Eurocode 3 design code and the numerical results of the finite element methods.
Ð å ç þ ì å
ªâðîñòàíäàðò EN 1993-1-1 îïèñóº çàãàëüíèé ìåòîä âèçíà÷åííÿ ãðàíè÷íîãî íàâàí-
òàæåííÿ äëÿ ñòàëüíèõ ñòðèæí³â ïðè ïîçäîâæíüîìó çãèí³ ç êðóò³ííÿì. Ó ìåòîä³
âðàõîâóþòüñÿ êðèâ³, ùî îïèñóþòü âòðàòó cò³éêîñò³ ïðè ïîçäîâæíüîìó çãèí³. Ãðà-
íè÷íå íàâàíòàæåííÿ ïðè ïîçäîâæíüîìó çãèí³ ç êðóò³ííÿì ìîæå áóòè ðîçðàõîâàíî
ìåòîäîì ñê³í÷åííèõ åëåìåíò³â íà îñíîâ³ ãåîìåòðè÷íîãî ³ íåë³í³éíîãî àíàë³çó ìàòå-
ð³àë³â ñòðèæíÿ ç äåôåêòàìè. Ïðîâåäåíî ç³ñòàâëåííÿ çíà÷åíü ãðàíè÷íîãî íàâàíòà-
æåííÿ ó â³äïîâ³äíîñò³ ç íîðìàìè Åâðîñòàíäàðòó EN 1993-1-1 äëÿ ïîçäîâæíüîãî çãèíó
ïîïåðå÷íî çàêð³ïëåíèõ ñòðèæí³â êðóò³ííÿ ç òàêèìè, ùî îòðèìàí³ øëÿõîì ìîäåëþ-
âàííÿ ìåòîäîì ñê³í÷åííèõ åëåìåíò³â íà îñíîâ³ ïàðàìåòðè÷íîãî äîñë³äæåííÿ.
1. N. Boissonnade, R. Greiner, J. P. Jaspart, and J. Lindner, Rules for Member Stability
in EN1993-1-1 – Background Documentation and Design Guidelines, ECCS TC8 –
Stability, ECCS Report No. 119, ISBN 92-9147-000-84, ECCS, Brussels, Belgium
(2007).
T a b l e 1
Predicted Load Factors
Section Boundary conditions loading Loading
location
Load factors
�theor �ANSYS �, %
Double symetrical
I section
h � 300 mm
b bf f1 2 15� � mm
t tf f1 2 10� � mm
tw �10 mm
L �10 m, F �10 kN
At beam fixing:
�, � , �� , �� (fixed)
Upper
flange
1.735 1.747 0.57
Shear
center
2.333 2.343 0.43
Lower
flange
2.670 2.676 0.21
Double symetrical
I section
h � 300 mm
b bf f1 2 15� � mm
t tf f1 2 10� � mm
tw �10 mm
L �10 m, F �10 kN
At beam fixing:
�, � (free), �� , �� (fixed)
Upper
flange
3.245 3.251 0.18
Shear
center
4.245 4.250 0.11
Lower
flange
4.643 4.650 0.15
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 167
Lateral Torsional Buckling Response of Steel Beam ...
2. R. H. J. Bruins, Lateral Torsional-Buckling of Laterally Restrained Steel Beams,
Master Thesis, Eindhoven University of Technology, The Netherlands (2007).
3. N. S. Trahair, “Multiple design curves for beam lateral buckling,” in: T. Usami and
Y. Itoh (Eds.), Stability and Ductility of Steel Structures, Pergamon (1998), pp. 13–
26.
4. N. S. Trahair, Flexural-Torsional Buckling of Structures, CRC Press, Boca Raton
(1993).
5. EN 1993-1-1. Eurocode 3: Design of Steel Structures – Part 1-1: General Rules and
Rules for Buildings, 2006.
6. R. Maquoi and J. Rondal, “Mise en equation des nouvelles courbes Europeennes de
flambement,” Constr. Metall., No. 1, 17–30 (1978).
7. H. H. Snijder and J. C. D. Hoenderkamp, “Buckling curves for lateral torsional
buckling of unrestrained beams,” in: Proc. of the Hommages a Rene Maquoi Birthday
Anniversary, Universite de Liege, Belgium (2007), pp. 239–248.
8. ANSYS 13.0. The General Purpose of Finite Element Software. Documentation.
Received 16. 07. 2013
168 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
L. Dahmani and A. Boudjemia
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