Lateral Buckling in Steel-concrete Composite Beams

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Lateral buckling in steel-concrete composite beams

Renato José Vacas Guedes

Abstract

This dissertation presents results that concerns to a study about composite steel-concrete

beams, where the steel area is a metallic I section, however the stability problems of themetallic section are the main subject of this work, since this is one of the major problem when

concerning to metallic sections.

The Lateral Buckling problem is quite complex phenomena which evolutes in the hogging

moment area of the continue composite beams, to do an exact analysis of this problem would

be too hard and would be such a complex process, so to setting the value of Critical Moment

of the section it was used the formula present in the EN 1994-1-1, which is less complex

than the exact solution, and a posteriori used the Critical Moment to set the value of the

Lateral Buckling Moment . It will be developed a toolkit which enables us to set the value

of the Lateral Buckling Moment in a prompt way following the methodology used in the

EN 1994-1-1.

Keywords

Steel-concrete composite beams; Lateral torsional buckling; Critical moment; Uniform Lateral

Slenderness; continue composite beams; lateral buckling moment

1- Introduction

A composite structure is the one where two or more materials work together. The goal of this

connection is to get the best of the materials and make the new material better than the ones

that had originated it. The composite steel-concrete beam has been used with more regularity

lately in construction because they provide a fast, economic and structural solution.

In sagging moments the concrete slab is working on compression and the steel beam is working

on traction, when that happens the composite beam fulfills is potential because both materials

are tensioned in a way that their performance is better. On the other hand, if the beam is in a

hogging moment zone and the concrete are on traction, which isn’t good because the concrete

has a very low resistance to traction and will crack. The steel will be compressed and instability



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problems will developed in the beam. The instability problems develop on steel sections

because they are really slender. The concrete slab will increase the steel section resistance to

that phenomenon, because the concrete doesn’t enable the upper flange deformation and

consequently the web deformation is restrained to.

It will be developed an excel toolkit with the purpose of calculate de Lateral Buckling ResistantMoment (Mb,Rd) in a prompt way, following the methodology of EN 1994-1-1.

There are some authors that had developed this matter in the last years. Bradford (1981)

developed a method to calculate the critical moment (Mcr ). Dekker, Kemp and Trinchero (1995)

proposed a method to predict the buckling strength of the composite plain webbed beams.

Bradford (1998) analyzed the distortional buckling in I beams and developed a method

considering that the upper flange have the translation totally restrained and the lower is partially

restrained. Bradford and Kemp (2000) defined the buckling phenomenon of composite beams

as exclusive of the negative moments. Hanswille (2000) developed a method to calculate the

critical moment; this methodology is based on the U-frame inverted model where the steel beam

is laterally and elastically restrained by the concrete slab. Vrcelj and Bradford (2007) studied a

mode of restrained distorcional buckling (RDB) of composite beams, which involve a distortional

deformation of the web, the authors developed a finite element model to study the problem

(bubble augmented spline finite strip). Vrcelj and Bradford (2009) developed a method to

determinate the elastoplastic bifurcation load of composite beams based on the bubble

augmented spline finite strip, developed by the authors. Salah and Gizewjowski (2008)

developed a finite element model with the purpose of analyze the instability in the hogging

moment zone, they concluded that when the beam becomes longer the collapsing mode tend to

have lateral buckling collapse and not plastic.

2- Critical Moment evaluation methods

2.1- Hanswille method

The Hanswille method (2000) is based on the invert U-frame model (fig. 1), in which the steel

section has is rotation and translational movements restrained because the presence of the

concrete slab. The restrain of the slab is simulated by an analogy with the buckling of a

compression member on elastic foundation.

The method developed by Hanswille is based in the theory of Vlasov, where the warping

restrictions are included, plus the rotation restriction conferred by the concrete slab, and the

following differential equation represents the solution for the problem presented to Hanswille.

(1)

The previous expression origins the critical moment equations that Hanswille developed, where

equation (2) is for uniform moment and equation (3) is for non-uniform moment.



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(2)

(3)

Fig.2: U-frame inverted model[2]

2.2- Dekker et al method

The restrain that the web provides to the compression flange will influence the type of

deformation of the steel beam, introducing a distortional component when the beam deforms,

which doesn’t existed when the steel beam acted alone.

To simulate the restrain imposed by the web to the lower flange, Dekker developed an

approximated method based on the rigid web theory, with some differences to make possible

restrain the deformation of the upper flange and simulate the web flexion stiffness. The author

developed a method based on the critical moment of isolated I beams, using coefficients to

simulated the presence of the concrete slab.

(4)

The previous formula is the definition of the critical moment of an isolated steel beam, and with

the Dekker’s approximation some of the parameters are multiplied by coefficients (C1; C2; C3)

which simulate the presence of the slab.

(5)

2.3- Eurocode 4 method

In order to elaborate this study of the behavior of composite steel-concrete beams was

developed an Excel toolkit, based in the Eurocode 4 methodology.

The Eurocode 4 deals with the lateral buckling of continuous composite beams by reducing the

section moment resistance at the internal support to a lower value referred to the beam buckling

strength. Because the composite beam is one of several parallel members attached to the same

concrete slab, design is based on the inverted U-frame model. When the slab deforms under



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applied load, there is a tendency for the steel beams to rotate as a result of slab deformations

and additionally distort laterally. The slab may always be regarded as providing the full lateral

restraint to the upper flange of steel I-section.

Acording to the Anex B of ENV 1994-1-1 (CEN, 2000), the critical o moment is calculated based

on the inverted U-frame model, resulting in the following expression.

(6)

(7)

The Dekker’s formula is the most conservative method because of his assumption of the rigid

steel section, when he tries to simulate the beam behavior. So the rigidity of the section will

diminish the value of the critical moment. Hanswille considers constant the flexional transverse

stiffness and the torsional St. Venant constant, unlike Dekker. In the EC4 the approach is less

complex than the one developed by Hanswille, so it’s normal if the EC4 method is more

conservative.

3- Presentation of the model

3.1- Eurocode 4 model

The study object of this work is the composite steel-concrete beams and their tendency to have

lateral instability problems. With that in mind it was elaborated an Excel toolkit based on the

norm EN 1994-1-1 to imitate behavior of 2 and 3 span beams.

There are several parameters that control the lateral buckling in this study, see fig (2). The

parameters are provided to the toolkit and it turn back the value of critical moment (Mcr ),

normalized slenderness ( ), the reduction coefficient and the reduced resistant moment

(Mb,Rd). The program receives the data of the parameters defined above and generates de

areas, moments of inertia and flexure modules of the concrete, steel and composite sections.

After the program generates all this information, it’s operated a macro which will provide the

shear and resistant moment’s diagrams. Those diagrams will allow an analysis of the cracking

of the concrete slab and permit the decision of making a cracked or un-cracked analysis.

Those diagrams will allow an analysis of the cracking of the concrete slab and permit the

decision of making a cracked or un-cracked analysis.

A macro is created to develop a program of finite elements with the SAP2000. The concrete

slab effective width doesn’t remains constant along the span of the beam, so to mimic this

behavior the program generated two different width, one in the hogging zone and one other at

the sagging zone (fig. 3). The program uses a frame element to represent the beam, the frame

element is represented by a generic I section and 15 finite elements are use to simulate the

sagging zone at the outer span and 20 to simulate the other zones (fig. 4), to differentiate



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between the different beam width the program changes the moment of inertia of the section

generated. In fig. 5 it can be seen the out-put of the program.

Fig. 2: Composite steel-concrete beam transversal section.

Fig. 3: Determination of the effective width.

Fig. 5: Finite element model, frames.

Fig. 5: Sap2000 out-put.

Bc,eff

Bc

h c

bf

tw h

w

t f

h s , t

h s , l

As,l As,t

e

L AB LBC LCD

A B C D

0,75 L AB 0,25 L AB 0,25 LBC 0,50 LBC 0,25 LBC 0,25 LCD 0,75 LCD

e f f



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3.2- Models Comparison

3.2.1- 2-span beam

The three models used to calculate the critical moment will be compared in this chapter as can

be seen in fig. 6. The Hanswille method, excluding the fig. 6 (a), is the less conservative

method and with the increase of the web height is value is more far away then the other

methods. The Dekker method is more the conservative and the EC4 is in the middle. Even with

different values de graphics develop in the same way. So in a 2-span beam the method in study

is between the other to methods and presents similar results to the more conservative method,

therefore is concluded that the this method is good to predict the critical moment of a composite

2-span beam.

(a) (b)

(c) (d)Fig.6: Influence of thickness of the web in the value of the critical moment in a 2-span beams

for : (a) ; (b) ; (c) ; (d) .

4.2.2- 3-span beam

In fig. 7 the method of Eurocode 4 is compared to the Hanswille and the critical moments of

both methods are smaller when compared to the 2-span beam. Hanswille still are less

conservative then the Eurocode 4 approach and when the web height increases the critical

3500

4500

5500

6500

7500

8500

10 11 12 13 14

M o m e n t o C r í t i c o ,

M c r ( K N . m

)

Espessura da alma, tw (mm)

EC4 hw=420

Hanswille hw=420

Dekker hw=420

3500

5500

7500

9500

11500

13500

10 11 12 13 14 M o m e n t o C r í t i c o ,

M c r ( K N . m

)


EC4 hw=515

Hanswille hw=515

Dekker hw=515

3500

4500

5500

6500

7500

8500

9500

10500

10 11 12 13 14


M c r ( K N . M

)


EC4 hw=470

Hanswille hw=470

Dekker hw=470

3500

5500

7500

9500

11500

13500

15500

17500

10 11 12 13 14


M c r ( K N . m

)


EC4 hw=560Hanswille hw=560Dekker hw=560



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moment values diverge on both methods, because in the Hanswille method critical moment had

a higher increase.

In all of the graphics critical value adopt the same development, increasing with the increase of

the web thickness. So in a 3-span beam the method in study as the same behavior than in the

2-span beam therefore is concluded that this method is good to predict the critical moment of acomposite 3-span beam.

(a) (b)

(c) (d)Fig. 7: Influence of web height in the value of critical moment of a 3-span composite beam, for:

(a) ; (b) ; (c) ; (d)

4- Parametrical Study

4.1- Influence of the parameters in normalized slenderness

In this chapter several parameters are analyzed and their influence in the process of lateral

buckling. And the analyzed parameters are web height and thickness, the flange width and

thickness, the concrete slab width, the span length, the overload, the concrete rupture strength

and the steel yield strength.

2000

2500

3000

3500

4000

4500

5000

10 11 12 13 14


M c r ( K N . m

)


EC4 hw=420

Hanswille hw=420

2500

3500

4500

5500

6500

7500

8500

10 11 12 13 14

M o m e n t o C

r í t i c o ,

M c r ( K N . m

)


EC4 hw=515

Hanswille hw=515

25003000

3500

4000

4500

5000

5500

6000

6500

10 11 12 13 14


M c r ( K N . m

)


EC4 hw=470

Hanswille hw=470

2000

3500

5000

6500

8000

9500

11000

10 11 12 13 14

M o m e n t o C

r í t i c o ,

M c r ( K N . m

)


EC4 hw=560

Hanswille hw=560



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Fig. 8: Web thickness influence in normalized slenderness Fig. 9: Overload influence in normalized slenderness

Fig. 10: Span length width influence in normalized Fig. 11: Steel yield strength influence in normalized

4.2- Proposed formulas to the normalized slenderness

4.2.1- 2-span beam

Fig. 12: hw/tw influence in normalized slenderness Fig. 13: tf /bf influence in normalized slenderness

0,20

0,30

0,40

0,50

0,60

0,70

0,80

10 11 12 13 14

N o r m a l i z

e d S l e n d e r n e s s ,

λ L T

Web thickness, tw (mm)

hw=420-2span hw=470-2spanhw=515-2span hw=560-2spanhw=420-3span hw=470-3spanhw=515-3span hw=560-3span

0,20

0,30

0,40

0,50

0,60

0,70

5 10 15 20 25 30 35 40 N o r m a l i z e d

S l e n d e r n e s s ,

λ L T

Overload, sc (KN/m2)

hw=420hw=470hw=515hw=560

0,20

0,30

0,40

0,50

0,60

3 4 5 6 N o r m a l i z e d S l e n d e r n e s s ,

λ L T

Span length , L (m)

hw=420-2spanhw=470-2spanhw=515-2spanhw=560-2span

0,20

0,30

0,40

0,50

0,60

0,70

0,80

235 275 315 355 N o r m a l i z e d S l e n d e r n

e s s ,

λ L T

Steel yield strength, f yd (MPa)

hw=420hw=470hw=515hw=560

0,30

0,35

0,40

0,45

0,50

30,0 35,0 40,0 45,0 50,0 55,0

N o r m a l i z e d S l e n d e r n e s s ,

λ L T

hw/tw

hw=420

hw=470

hw=515

hw=560Proposta

0,30

0,35

0,40

0,45

0,50

0,055 0,06 0,065 0,07 0,075 0,08 0,085 0,09 N o r m a l i z e

d S l e n d e r n e s s ,

λ L T

tf /bf

hw=420hw=470hw=515Proposta



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Fig. 14: 1/C4 influence in normalized slenderness Fig. 15: L influence in normalized slenderness

(13)

4.2.2- 3-span beam

Fig. 16: hw/tw influence in normalized slenderness Fig. 17: tf /bf influence in normalized slenderness

Fig. 18: f yd influence in normalized slenderness Fig. 19: L influence in normalized slenderness

(5.14)

0,30

0,35

0,40

0,45

0,50

3 4 5 6 N o r m a l i z e d

S l e n d e r n e s s ,

λ L T

L

hw=420

hw=470

hw=515

Proposta

0,40

0,50

0,60

0,70

30,0 35,0 40,0 45,0 50,0 55,0

E s b e l t e z a n o r m a l i z a d a ,

λ L T

hw/tw

hw=420

hw=470

hw=515

hw=560

0,30

0,40

0,50

0,60

0,70

0,055 0,06 0,065 0,07 0,075 0,08 0,085


λ L T

tf/bf

hw=420

hw=470

hw=515

Proposta

0,30

0,40

0,50

0,60

0,70

3 4 5 6

E s b e l t e z a n o r m

a l i z a d a ,

λ L T

L

hw=420

hw=470

hw=515

Proposta0,40

0,50

0,60

0,70

235 275 315 355


λ L T

f yd

hw=420

hw=470

hw=515

Proposta

0,30

0,40

0,50

0,60

0,70

0,04 0,045 0,05 0,055 0,06 0,065 0,07

N o r m a l i z e d S l e n d e r n e s s ,

λ L T

1/C4

hw=420

hw=470

hw=515

Proposta



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5- Conclusion future developments

The parameters with more influence in the process of lateral buckling are steel yield strength,

the web thickness and height and the length of the span. When the value of those parameters

increases they make a large variation in the value of the normalized slenderness. On the other

hand the flange doesn’t have much influence on the problem, although the width is more

important than the thickness of the flange. The slab is important to restrain the upper flange and

the web, a variation on the rupture strength of the concrete slab only make the slenderness

change when an elastic analysis is made. The overload it’s only important if it is applied in a

non-symmetrical way, which will change the moment diagram and change the C4 parameter.

In future developments should be developed a method to define the value of the normalized

slenderness of class 3 and 4 sections and of beams with 4 or more spans.

6- Bibliography

[1] EN 1993-1-1, Eurocode 3 Part 1-1: Design of steel structures – Part 1-1: General rules and

rules for buildings, CEN, Brussels 2005

[2] EN 1994-1-1, Eurocode 4: Design of composite steel and concrete structures, Part 1-1:

General rules for buildings, CEN, Brussels 2005

[3] Salah, W; Gizewjowski, M; M. A. 2008. Restrained distortional buckling of composite beams-

FE modeling of the behavior of steel-concrete beams in the hogging moment region, in Proc.

Eurosteel Conference, 2008, Graz, Austria, 1629-1634.

[4] Hanswille, G. 2000. Torsional buckling of composite beams, comparison of more accurate

methods with Eurocode 4, in Proc. Composite Construction in Steel and Concrete IV, 2000,

Banff, Alberta, Canada, 105-116. doi:10,1061/40616(281)10

[5] Dekker, N.; Kemp, A. R.; Trinchero P. 1995. Factors influencing the strength of continuous

composite beams in negative bending, Journal of Constructional Steel Research 34:262-185

[6] Gizewjowski, M.; Salah, W. 2010. Restrained distortional buckling strength of steel-concrete

composite beams – a review of current practice and new developments, in Proc, International

Conference, 2010, Vilnius, Lithuania, 604-612.

[7] Vrcelj, Z. e Bradford, M. A., 2006. Elastic Distortional Buckling of Continuous Restrained I-

section Beam-columns. Journal of Constructional Steel Research. 62:223-230.

[8] Vrcelj, Z. e Bradford, M. A., 2007. Elastic Bubble Augmented Spline Finite Stripe Method in

Analysis of Continuous Composite Beams. Australian Journal of Structural Engineering.

7(2):75-84.

[9] Vrcelj, Z. e Bradford, M. A., 2009. Inelastic Restrained Distortional Buckling of Continuous

Composite T-beams. Journal of Constructional Steel Research. 65:850-859.

[10] Bradford, M. A. e Gao, Z., 1992. Distortional Buckling Solutions for Continuous Composite

Beams. Journal of Structural Engineering, vol.118:1144.

[11] Bradford, M. A. e Kemp, A. R., 2000. Buckling in Continuous Composite Beams. Progress



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in Structural Engineering and Materials, 2:169:178.

[12] Calado, L. e Santos, J., 2010. Estruturas Mistas de Aço e Betão. IST Press, Lisboa.

[13] Bradford, M. A., 1998. Inelastic buckling of I-beam with continuous elastic tension flange

restraint. Journal of Constructional Steel Research, 48:63-77.

Lateral Buckling in Steel-concrete Composite Beams

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