1st International Conference on Modern Design, Construction and Maintenance of Structures, 10-11 December 2007, Hanoi, Vietnam

Calibration of a mixed finite element model for the monotonic analysis of continuous composite beams * Quang Huy Nguyen1, Mohammed Hijaj1 and Brian Uy2 (1. Dep of Civil Engineering, INSA of Rennes, France; 2. School of Engineering, University of Western Sydney, Australia)

Abstract: Cracking of the concrete slab in the hogging moment region decreases the global stiffness of composite steel-concrete structures, also reduces the effect of continuity and makes the structural behaviour highly nonlinear even for low stress levels. In this paper, a nonlinear analysis model using a mixed finite element formulation is presented to predict the behaviour of continuous composite beams with discrete partial shear connection in the hogging moment regions. The tension stiffening effects are introduced in the analysis by using a concrete constitutive model proposed in the CEB-FIP Model Code 1990 which incorporates steel reinforcement. Comparisons between the numerical analyses and experimental results existing in literature were undertaken to validate the accuracy of the model. Keywords: Composite beams; Negative bending; Tension stiffening;

1 Introduction For the last few decades, steel-concrete composite beams have been widely used in the construction industry, and in particular in bridge and building applications because of the benefits of combining the two construction materials. Reinforced concrete is inexpensive, massive and stiff, whilst steel is relative strong, lightweight and easy to assemble. The best use is made of the two materials when concrete is used in the compressive zone where steel may experience buckling, whilst steel is used in tensile zone where the concrete will crack. This is achieved for a simply supported steel concrete composite beam under positive bending. However in multistory buildings and bridges, continuous composite beams are often used because of the benefits at both the ultimate and serviceability limit states for long span or heavily loaded members,[1]. For these cases, there is a negative moment region, in which the concrete is cracked and the reinforcement carries the tensile forces, with the steel component being subjected to a combination of negative bending and compression. Cracking of the slab decreases the stiffness of the structure, reduces the effects of continuity and makes the structural behaviour highly nonlinear even for low stress levels, [2] and [3]. One of the main factors affecting the stiffness of cracked reinforcement in a concrete slab is the bond that develops between the reinforcement and the concrete. It allows the transfer of tensile stresses between the reinforcement and the uncracked regions of concrete. This phenomenon is called tension stiffening. In flexure, the influence of tension stiffening is most important up to service loads and should be included in the deflection calculations. This effect has been investigated in [4].

Fig. 1 Continuous composite beam system in Australia, [5]. *

Corresponding author. Tel.: +33-2 23 23 85 11 E-mail address: [email protected]

In the present paper, a finite element model is proposed to analyse the nonlinear flexural behaviour of a composite beam with discrete partial shear connection under negative bending. This model is based on the two fields mixed force-displacement formulation, [6] and [7]. The tension stiffening effect is taken into account by using the tension stiffening model proposed in the CEB-FIP model, [8]. Local buckling is not considered. Comparisons between the numerical results and experimental results existing in the literature were made to validate the accuracy of the model.

2 Structural modelling The proposed formulation in this paper consists of (a) forced-based finite element formulation for the composite beam; (b) nonlinearities of constitutive materials; (c) model for steel embedded in concrete. 2.1 Basic assumptions The following assumptions were made in the proposed model: • Preservation of the plane cross section for both the slab and the profile; • Steel beam section is class 1 or 2 according to the Eurocode 4 ; • No uplift occurs between the slab and the profile; therefore two parts of the composite section have the same rotation and the same curvature; • Slip can occur at the slab profile interface; • The axial strain of the section is linear but there is a discontinuity at the slab profile interface due to slip; • The member cross section is divided into concrete and steel layers; • When the reinforcement is in tension, all layers in the effective area Aeff are replaced by only one layer of steel embedded in the concrete; 2.2 Field equations In this section we recall the field equations for a composite beam with discrete shear connection in a small displacement setting. All variables subscripted with c belong to the concrete slab section, with sr representing the steel reinforcing bars and those with s representing the steel beam. Quantities with subscript sc are associated with the shear connectors. 2.2.1 Equilibrium conditions p0

M c + dM c

M c Tc

N c + dN c

Nc

M csc

Tc + dTc

Ts Ms

M s + dM s

x

Rsc

Rsc H2

M ssc N ssc

N s + dN s

y

Ns

N csc

H1

z Ts + dTs

dx

dx = 0

Fig. 2 Free body diagram of composite beam segment

The equilibrium conditions are derived by considering the infinitesimal beam segment without connector and the connector elements shown in Figure 1. The equilibrium conditions result in the following: (1) ∂D + P = 0

R sc = − N csc = N ssc = − M sc / H where D = {N c ( x )

N s ( x) M ( x )} ; M ( x) = M c ( x) + M c ( x) ; M sc ( x) = M csc ( x) + M ssc ( x) ;

H = H1 + H 2 ; P = {0 0

T

p0 } and the operator ∂ T

(2)

⎡d ⎢ dx ⎢ ∂=⎢0 ⎢ ⎢ ⎢⎣ 0

⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ d2 ⎥ dx 2 ⎥⎦

0 d dx 0

(3)

2.2.1 Compatibility conditions The curvature and the axial deformation at any section is related to the beam displacements through kinematic relations. Under small displacements and neglecting the relative transverse displacement between the concrete slab and the steel beam, these relationships are as follows: (4) ∂ *d − e = 0 where d = {uc ( x )

u s ( x) v( x )} is the displacement vector; e = {ε c ( x) ε s ( x ) κ ( x )} is the deformation T

T

vector and the operator ∂ are given by *

⎡d ⎢ dx ⎢ ∂* = ⎢ 0 ⎢ ⎢ ⎢⎣ 0

⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ d2 ⎥ − 2⎥ dx ⎦

0 d dx 0

(5)

2.2.1 Constitutive laws The fiber discretization is used to describe the section behaviour in the proposed model. The force-deformation relationship of the cross section is: e( x ) = f ( x ) D ( x ) (6) where f(x) is the secant flexibility matrix of the cross section derived from uniaxial constitutive relations for both steel and concrete. The constitutive relationship proposed by Ollgaard et al. [9] is considered for the stud shear connector. The analytical relationship between the shear force Rsc and the slip d sc of the generic stud is given by

Rsc = Rmax (1 − e

− β d sc

)α

(7)

where Rmax is the ultimate strength of the stud shear connector; α and β are coefficients to be determined from experimental results, [10]. By using the secant rigidity k sc may be rewritten as

Rsc = k sc .d sc

(8)

2.3 Material models 2.3.1 Concrete The stress-strain relationship suggested by the CEB-FIP model [8] is adopted in this model for both compression and tension regions (Fig.3)

σc f cm

0.5 f cm 15%

E c1

ε c1

Ec

ε c ,lim

0.9 f ctm f ctm Fig. 3 Stress-strain diagrams for concrete

εc

2.3.2 Steel In the present study, the steel was modeled as an elastic-perfectly plastic material incorporating strain hardening. Fig.4 shows the stress-strain diagrams for steel in tension. Specifically, the relationship is linearly elastic up to yielding, perfectly plastic between the elastic limit and the commencement of strain hardening, linear hardening occurs up to the ultimate tensile stress and the stress remaining constant until the tensile failure strain is reached

f su E sh

f sy

Es

εy

ε su

ε sh

Fig. 4 Stress-strain diagrams for steel

2.3.2 Steel embedded in concrete When the uncracked concrete is in tension, the tensile force is distributed between the reinforcement and concrete in proportion to their respective stiffnesses, and cracking occurs when the stress reaches a value corresponding to the tensile strength of the concrete. In a cracked cross-section all tensile forces are balanced by the steel encased in the concrete only. However, between adjacent cracks, tensile forces are transmitted from the steel to the surrounding concrete by bond forces. The contribution of the concrete may be considered to increase the stiffness of the tensile reinforcement. This effect is called tension-stiffening. To describe this effect, a number of models have been proposed. The majority of models are based on the mean axial stress and the mean axial strain of the concrete member in the reinforced concrete, [11]and [12]. To take the tension stiffening effect into account, the stress-average strain relationship of steel embedded in concrete proposed by the CEB-FIP model [8], is considered to describe the behaviour of the reinforced concrete members in tension. Fig.5 shows the stress-strain diagrams of steel embedded in concrete. σs σs

σs

σ su σ sy σ srn σ sr1

Bare steel bar Embedded steel bar

ε sr1 ε srn ε sr 2

ε sry

ε sr , sh

ε sru

ε s,m

Fig. 5 Stress-strain diagrams for steel embedded in concrete in tension

According to the CEB-FIP model [8] the mean stress-strain relationship of embedded steel may be expressed as

⎛

1 ⎞

⎝

⎠

if ε s,m ≤ ε sr1

⎟E ε σ s = ⎜⎜1 + α .ρ ⎟ s s , m

σ srn − σ sr1 (ε s ,m − ε sr1 ) ε srn − ε sr1 f − σ srn (ε s,m − ε srn ) σ s = σ srn + sy ε sry − ε srn σ s = f sy

σ s = σ sr1 +

σ s = f sy +

f su − f sy

ε sru − ε sry

(ε

s ,m

− ε sry )

if ε sr1 < ε s,m ≤ ε srn if ε srn < ε s,m ≤ ε sry

(9)

if ε sry < ε s,m ≤ ε sr ,sh if ε sr ,sh < ε s,m ≤ ε sru if ε s,m > ε sru

σ s = f su

where • α and ρ are the modular ratio and the geometric ratios of reinforcing steel, respectively; • • •

σ sr1 is the steel stress in the crack, when the first crack has formed σ srn is the steel stress in the crack, when the last crack has formed ε sr1 and ε sr 2 are the steel strains at the point of zero slip and at the crack when the cracking forces reach f tm

• • • • • •

ε srn = σ srn / Es − β t (ε sr 2 − ε sr1 ) ε sry = ε sy − β t (ε sr 2 − ε sr1 ) ε sr ,sh = ε sh − β t (ε sr 2 − ε sr1 ) ε sru = ε sr ,sh + δ (1 − σ sr1 / f sy )( f su − f sy ) / Esh βt = 0.4 for instantaneous loading, and = 0.25 for long-term and repeated loading δ is a coefficient to take account of the stress ratio f su / f sy and the yield stress f sy , δ = 0.8 was proposed in CEB-FIP model [8]

2.4 The mixed finite element formulation For the composite beam element without a connector, the derivation follows the two-field mixed formulation which uses the integral form of compatibility and equilibrium equations to derive the matrix relation between element generalised forces and corresponding displacements.

q5

q1 q4

q2

(a )

q6

q8

q7

q3 p0

Q4

Q1 Q3

Q2

(b)

Q5

Q6

Fig. 6 (a) Displacement degree of freedom; (b) Force degree of freedom in reduced system

In the two-field mixed formulation, [6] and [7], both the displacement and the internal forces fields along the element are approximated by independent shape functions. The two fields are written

d ( x) = a( x).q D( x) = b( x).Q + D0 ( x)

Where D0 ( x) = {0

(10) (11)

0 p0 x( L − x) / 2} is the internal force vector including the effects of internal loading on the cross section forces; q and Q are the element displacement vector and element force vector, respectively; a(x) and b(x) are the displacement and force interpolation matrices, respectively. In the mixed method formulation the integral forms of compatibility and equilibrium equations are expressed first. These are then combined to obtain the relationship between the element forces and displacements. The weighted integral form of the compatibility equation is

∫ δD

(

)

( x) ∂ *d ( x) − e( x) dx = 0

T

(12)

L

where δD(x) is the arbitrary (virtual) force field fulfilling the equilibrium conditions and the integration extends over the element length L . By substituting (6), (10) and (11) into (12) one obtains

⎧⎪⎛ ⎪⎩⎝ L

⎞

⎛

⎞

⎠

⎝L

⎠

⎫⎪ ⎪⎭

δQT ⎨⎜⎜ ∫ bT ( x)B( x)dx ⎟⎟q − ⎜⎜ ∫ bT ( x) f ( x)b( x)dx ⎟⎟Q − ∫ bT ( x) f ( x) D0 ( x)dx ⎬ = 0 L

(13)

where B( x) = ∂ a( x) is the deformation interpolation matrix. From the arbitrariness of δQ , relation (13) reduces to the following expression G.q − F .Q = q0 (14) where *

G = ∫ b T ( x) B( x)dx

(15)

F = ∫ b T ( x) f ( x)b( x)dx

(16)

q0 = ∫ bT ( x) f ( x) D0 ( x)dx

(17)

L

L

L

The weighted integral form of the equilibrium equation (1) is derived from the virtual displacement principle and takes the form

∫ δd

T

( x)(∂D( x) + P )dx = 0

(18)

L

where δd (x) are the displacement fields fulfilling the kinematic conditions. By integrating by parts (18) and substituting (10) and (11) into (18), one obtains

⎧⎪⎛

⎞

⎪⎩⎝ L

⎠

⎫⎪ ⎪⎭

δq T ⎨⎜⎜ ∫ B T ( x)b( x)dx ⎟⎟Q − ∫ (a T ( x) P − B T ( x) D0 ( x) )dx − Q ⎬ = 0 L

(19)

where Q is the vector of applied nodal forces. Since equation (19) holds for arbitrary δq , it follows that

G T .Q = Q + Q0 where

(

(20)

)

Q0 = ∫ a T ( x) P − B T ( x) D0 ( x) dx

(21)

L

The rearrangement and combination of equation (14) and (20) results in

⎡− F ⎢ GT ⎣

G ⎤ ⎛ Q ⎞ ⎛ q0 ⎞ ⎟ ⎜ ⎟=⎜ 0 ⎥⎦⎜⎝ q ⎟⎠ ⎜⎝ Q0 + Q ⎟⎠

(22)

Equation (22) represents in matrix form the two field mixed formulation for the composite beam element. If the first equation in (22) is solved for Q and the result is substituted into the second equation, the following expression results K e q = Q + Q0 (23) −1

where K e = G F G represents the secant rigidity matrix of the composite beam element without connectors of T

−1

the two field mixed formulation and Q0 = Q0 + G F q0 represents the nodal force vector produced by internal T

element with transverse uniformed loading p0 . 2.5 Connector element The slip d sc of the connector is defined as the relative displacement between the concrete slab and the steel beam at the interface therefore (24) d sc = u ssc − u csc − Hθ sc Substituting equation (8) and (24) into equation (2) one obtains k sc (u ssc − u csc − Hθ sc ) = − N csc = N ssc = − M sc / H (25) (25) may be rewritten in matrix form as

Qsc = K sc qsc

{

where Q sc = N csc

N ssc

M

}

sc T

{

; q sc = u csc

u ssc

(26)

θ

}

sc T

−1 ⎡1 ⎢ and K sc = k sc − 1 1 ⎢ ⎢⎣ H − H

H ⎤ − H ⎥⎥ are the nodal H 2 ⎥⎦

force vector, nodal displacement vector and secant stiffness matrix of the connector element, respectively.

3 Comparison with experimental data

600

600

500

500

400

400 Load P [kN]

Load P[kN]

The accuracy of the proposed model is validated using experimental data derived by earlier experimental tests. Two span continuous beams tested by Ansourian [13] were considered to simulate the proposed model. The beam CTB3 was part of the experimental program planned by Ansourian [13]. Beam CTB3 was designed with two equal spans of 4500mm and loaded by concentrated loads at the centre of both spans to investigate hogging hinges. The joists in beam TCB3 was of IPBL 200 section. The slab was 1300 mm wide and 100 mm thickness, longitudinally reinforced by steel on the top and bottom. The distance from the interface to the bottom and top reinforcement steel layers was 25mm and 75 mm respectively. Stud connector diameters of 19 mm were equally spaced in triples at 350 mm along the beam except over the internal support (1050 both sides) where the pitch reduces to 300 mm.

300

200

200 Proposed model CTB3 data

100

0 0

300

10

20

30

40

50

60

Proposed model CTB3 data

100

0 0

70

10

20

600

600

500

500

400

400

300

200

10

20

30

40

50

Support curvature [1/km]

40

50

60

70

80

300

200 Proposed model CTB3 data

100

0 0

30

Midspan curvature [1/km]

P [kN]

Load P [kN]

Deflection at midspan [mm]

60

70

Proposed model Left of support - CTB3 data Right of support - CTB3 data

100

80

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Lower flange strain at support [m/m]

Fig.7 Theoretical-Experimental comparison for beam CTB3

The numerical results of the proposed model using 108 beam elements without connectors and 28 connector

elements are compared with Ansourian’s experimental results in Fig.7. It should be noted that there is reasonable agreement between the numerical curves and the experimental results.

4 Conclusions A numerical model based on a two-field mixed finite element formulation for predicting the behaviour of continuous composite beams including the nonlinearity of constitutive materials and tension stiffening effect has been presented. The theoretical-experimental comparison shown validates the model reliability and the capacity to determine the experimental behaviour. It is worth noting that the proposed model is able to perform a very good prediction of the global behaviour of continuous composite steel concrete beams subjected to monotonic loads.

References [1] Oehlers, D. and Bradford, M.A., Composite steel and concrete structural members: Fundamental behaviour, ed. Elsevier. 1995, Oxford: Pergamon. [2] Johnson, R.P. and Allison, R.W., Shrinkage and tension stiffening in negative moment regions of composite beams. Structural Engineer, 1981. 59B: p. 10-16. [3] Cosenza, E. and Pecce, M. Deflection and redistribution of moments due to the cracking in steel-concrete composite continuous beams. in ICSAS’91. 1991. New York: Elsevier Science. [4] Samy, G. and Jean-Marie, A. Numerical Investigation of Moment Redistribution in Continuous Beams of Composite Bridges. 2006: ASCE. [5] Uy, B. and Liew, J.Y.R., Composite steel-concrete structures, in Civil Engineering Handbook, W.F. Chen and Liew, J.Y.R., Editors. 2002, CRC Press, ISBN 0-8493-0958-1,. [6] Zienkiewicz, O.C. and Taylor, R.L., The finite element method. 1989, London: McGraw-Hill. [7] Ayoub, A., A two-field mixed variational principle for partially connected composite beams. Finite Elements in Analysis and Design, 2001. 37(11): p. 929-959. [8] CEB-FIP model code 1990. 1991, Lausanne, Switzerland: Comité Euro-International du Béton. [9] Ollgaard, J.G., Slutter, R.G., and Fisher, J.W., Shear strength of stud connectors in lightweight and normal weight concrete. AISC Engineering Journal, 1971. 8(2): p. 55-64. [10] Aribert, J.M. and Labib, A.G., Modèle calcul élasto-plastique de poutres mixtes à connexion partielle. Construction Metallique, 1982. 4(37-51). [11] Scanlon, A. and Murray, D.W., Time dependent deflections of reinforced concrete slab deflections. J Struct Div, 1974. 100(9): p. 1911-1924. [12] Lin, C.S. and Scordelis, A.C., Nonlinear analysis of RC shells of general form. J Struct Div, 1975. 101(3): p. 523538. [13] Ansourian, P., Experiments on continuous composite beams. Proc Inst Civil Eng, 1981. 71: p. 25-51.