Journal of Constructional Steel Research 65 (2009) 737–748
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Journal of Constructional Steel Research
journal homepage: www.elsevier.com/locate/jcsr
Analysis of composite beams in the hogging moment regions using a mixed finiteelement formulation
Quang Huy Nguyen
a,b
, Mohammed Hjiaj
a,
∗
, Brian Uy
c
, Samy Guezouli
a
a
Department of Civil Engineering, INSA of Rennes, France
b
School of Civil, Mining and Environmental, University of Wollongong, Australia
c
School of Engineering, University of Western Sydney, Australia
a r t i c l e i n f o
Article history:
Received 19 October 2007Accepted 16 July 2008
Keywords:
Composite beamsNegative bending momentTension stiffeningMixed F.E. formulationDiscrete shear connexion
a b s t r a c t
Cracking of the concrete slab in the hogging moment region decreases the global stiffness of compositesteel–concrete structures and also reduces the effect of continuity, thus making the structural behaviourhighly nonlinear even for low stress levels. In this paper, the behaviour of continuous composite beamswith discrete shear connection is investigated using a nonlinear mixed finite element model. The modelincludesappropriatenonlinearconstitutiverelationshipsfortheconcrete,thesteelandtensionstiffeningeffect.Furthermore,thediscretenatureoftheshearconnectionisembeddedinthemodelandthetensionstiffening effects are introduced in the analysis by using a concrete constitutive model proposed in theCEBFIB Model Code 1990 which incorporates embedded steel. Special attention is paid to the hoggingmomentregions,wherecrackingoccurs.Comparisonsbetweenthenumericalanalysesandexperimentalresults in the current literature are undertaken to validate the accuracy of the model. Furthermore, aparametric study is carried out to study the influence of span length and degree of shear connection onthe strength and ductility of continuous composite beams.
©
2008 Elsevier Ltd. All rights reserved.
1. Introduction
For the last few decades, steel–concrete composite beams havebeen widely used in the construction industry (bridges and buildings) because of the benefits of combining the two constructionmaterials. Reinforced concrete is inexpensive, massive and stiff,whilst steel is relatively strong, lightweight and easy to assemble. The best use is made of the two materials when concrete isused in the compressive zone where steel may experience buckling, whilst steel is used in the tensile zone where the concretewill crack. This is the case of simply supported steelconcretecomposite beam under positive bending. However in multistoreybuildingsandbridges,continuouscompositebeamsareoftenusedbecauseofthebenefitsatboththeultimateandserviceabilitylimitstates for long spans or heavily loaded members [1]. For these
cases, there is a negative moment region, in which the concreteis cracked and the reinforcement carries the tensile forces, withthe steel component being subjected to a combination of negativebending and compression. Cracking of the slab decreases the stiffness of the structure, reduces the effects of continuity and makes
∗
Corresponding address: Department of Civil Engineering, INSA de Rennes,35043 CEDEX Rennes, France. Tel.: +33 2 23 23 87 11; fax: +33 2 23 23 84 48.
Email address:
mohammed.hjiaj@insarennes.fr (M. Hjiaj).
the structural behaviour highly nonlinear even at low stress levels(see references [2–4]). One of the main factors affecting the stiff
ness of cracked reinforced concrete slab is the bond that developsbetweenthereinforcementandtheconcrete.Itallowsthetransferof tensile stresses between the reinforcement and the uncrackedregions of concrete. This phenomenon is called tension stiffening.In flexure, the influence of tension stiffening is most important upto service loads and should be included in the deflection calculations.A good deal of research has been devoted to develop modelsto analyse the behaviour of composite beams subjected tonegative bending. Lebet [5] developed a finitedifference modelto analyse composite beams under negative bending moment.In his model, the behaviour of steel and concrete is supposedto be linear but cracking is taken into account by assuminga different bending stiffness in cracked and uncracked regions.Manfredi et al. [6] presented a nonlinear analysis technique basedon the finitedifference method and on an array of generalisedmoment–curvaturerelationshipsrelatedtointerfaceslipandbondslip. Xu [7] and Gattesco [8] proposed a displacementbased finite
elementmodelwithtwonodesandeightdegreesoffreedom.Theirnumerical procedure permits to consider nonlinear constitutivemodels for the steel, the concrete and the connexion. Recently Lohet al. [9] developed an iterativebased model using crosssectional
analysis technique. The model considered the concept of partial
0143974X/$ – see front matter
©
2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2008.07.026
738
Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748
Fig.1.
(a)Infinitesimalcompositebeamsegmentwithoutconnector;(b)Connectorelement.
interaction allowing for the occurrence of slip at the slabbeaminterface.In this article, a finite element model is proposed to analysethe nonlinear flexural behaviour of a composite beam withdiscrete partial shear connection under negative bending. Thismodel is based on the two fields mixed force–displacementformulation [10] with nonlinear constitutive relationships for the
components. The tension stiffening effect is taken into accountby using the relationship proposed in the CEBFIB Model Code90 [11]. Local buckling is not considered. Comparisons between
the numerical results and experimental results existing in theliterature are made in order to validate the accuracy of the model.Finally, a parametric study is carried out to consider the influenceof span length and degree of shear connection on the strength andductility of continuous composite beams.
2. Structural modelling
The main ingredients of the proposed formulation consistsof (a) mixed finite element formulation for the compositebeam; (b) nonlinear constitutive relationships for the componentmaterials; (c) model for steel embedded in concrete.
2.1. Basic assumptions
The following assumptions are made:
•
Preservationoftheplanecrosssectionforboththeslabandtheprofile.
•
Steel beam crosssection is Class 1 or 2 according to the EC4[12].
•
Noupliftoccursbetweentheslabandtheprofile;thereforetwoparts of the composite section have the same rotation and thesame curvature.
•
Slip can occur at the slab/profile interface.
•
Theaxialstraindistributionoverthesectiondepthislinearwitha discontinuity at the slab/profile interface due to slip.
•
Themembercrosssectionissubdividedintoconcreteandsteellayers (fibre beam element).
•
When the reinforcement is in tension, all layers in the effectivearea
A
c
,
eff
arereplacedbyasinglelayerofsteelembeddedintheconcrete (see Fig. 7).
2.2. Field equations
In this section, we recall the field equations for a compositebeam with discrete shear connection in a small displacementsetting.Allvariablessubscriptedwith
c
belongtotheconcreteslabsection and those with
s
belong to the steel beam. Quantities withsubscript
sc
are associated with the shear connectors.
2.2.1. Equilibrium conditions
Due to the discrete nature of the shear connexion, the internalforces (bending, normal force and shear force) distributions in theconcrete slab and in the steel profile are now discontinuous with jumps at each connector. To derive the equilibrium conditionsfor a composite beam with discrete shear connectors, we need toconsider first the equilibrium of an infinitesimal beam segmentwithout shear connector and the equilibrium at the crosssectioncontainingashearconnector(seeFig.1).Thefirstsetofequilibriumequations, which apply between two consecutive connectors, isreadily obtained by expressing the equilibrium of a small elementof the composite beam, of length d
x
, and subjected to internalforces(Fig.1(a)).Theequilibriumconditionsresultinthefollowingset of equations:d
N
c
(
x
)
d
x
=
d
N
s
(
x
)
d
x
=
0 (1)
T
(
x
)
= −
d
M
(
x
)
d
x
(2)d
2
M
(
x
)
d
x
2
+
p
0
=
0 (3)where
T
(
x
)
=
T
c
(
x
)
+
T
s
(
x
)
and
M
(
x
)
=
M
c
(
x
)
+
M
s
(
x
)
.The equilibrium Eqs. (1)–(3) are rewritten in the followingcompact form
∂
D
+
P
=
0 (4)where
D
=
N
c
(
x
)
N
s
(
x
)
M
(
x
)
is the internal stress resultant vector,
P
=
00
p
0
the external loading vector and
∂
a linear operator expressed as:
∂
=
dd
x
0 00 dd
x
00 0 d
2
d
x
2
.
(5)The above equations must be completed by equilibrium equationsat each crosssection containing a shear connector (Fig. 1(b)). Theresulting equation provides a relationship between the internalstress resultants and the shear force
R
sc
(Fig. 1(b)):
R
sc
= −
N
c
=
N
s
= −
M
/
H
(6)where
H
=
H
1
+
H
2
. It worth nothing that the internal stressresultants involved in Eq. (6) are acting at the crosssection level.
2.2.2. Compatibility conditions
The curvature and the axial deformation at any section arerelated to the beam displacements through kinematic relations.Under small displacements and neglecting the relative transversedisplacement between the concrete slab and the steel beam, theserelationships are as follows (see Fig. 2):
ε
c
(
x
)
=
d
u
c
(
x
)
d
x
(7)
ε
s
(
x
)
=
d
u
s
(
x
)
d
x
(8)
κ(
x
)
= −
d
2
v(
x
)
d
x
2
(9)
d
sc
(
x
)
=
u
s
(
x
)
−
u
c
(
x
)
+
H
d
v(
x
)
d
x
(10)where
u
is the longitudinal displacement,
v
the transversaldisplacement,
ε
c
the strain at the concrete section centroid,
ε
s
Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748
739
Fig. 2.
Kinematic of composite beam.
the strain at the steel section centroid,
κ
the curvature and
d
sc
the relative slip between the concrete slab and the steel beam.The compatibility relationships (7)–(9) can also be rewritten in acompact form as
∂
∗
d
−
e
=
0 (11)where
d
=
u
c
(
x
)
u
s
(
x
)v(
x
)
is the displacement vector;
e
=
ε
c
(
x
)ε
s
(
x
)κ(
x
)
is thesection deformation vector and the operator
∂
∗
is given by
∂
∗
=
dd
x
0 00 dd
x
00 0
−
d
2
d
x
2
.
(12)
2.3. Material models
Thematerialbehaviourisdescribedusingexplicitrelationshipsbetween the total stress and the total strain with appropriateloading/unloading conditions.
2.3.1. Concrete
The stress–strain relationship suggested by the CEBFIB ModelCode 90 [11] is adopted in this paper for both compression andtension regions (Fig. 3). In compression regions, the stress–straincurve suggested by the CEBFIB Model Code 90 [11] includes a
monotonically increasing branch up to a peak value, followed bya descending part that gradually flattens to a constant value equalto zero. The initial portion of the ascending branch is linearlyelastic, but at about 30% of the ultimate strength, the presenceof microcracks leads to a nonlinear behaviour, with a reductionin tangent modulus. In the subsequent descending branch, theconcrete is severely damaged with prominent cracks.The
σ
c
–
ε
c
relationship is approximated by the followingfunctions:
•
For
ε
c
< ε
c
,
lim
:
σ
c
= −
ε
c
1
ε
c
,
lim
ξ
−
2
ε
c
1
ε
c
,
lim
2
ε
c
ε
c
1
2
+
4
ε
c
1
ε
c
,
lim
−
ξ
ε
c
ε
c
1
−
1
f
cm
Fig. 3.
Stress–strain diagram for concrete.
•
For
ε
c
,
lim
≤
ε
c
≤
0:
σ
c
= −
1
+
E
ci
E
c
1
−
2
ε
c
ε
c
1
−
1
E
ci
E
c
1
ε
c
ε
c
1
−
ε
c
ε
c
1
2
f
cm
(13)
•
For 0
< ε
c
≤
0
.
9
f
ctm
/
E
ci
:
σ
c
=
E
ci
ε
c
•
For 0
.
9
f
ctm
/
E
ci
< ε
c
≤
0
.
00015:
σ
c
=
f
ctm
−
0
.
1
f
ctm
0
.
00015
−
0
.
9
f
ctm
/
E
ci
(
0
.
00015
−
ε
c
)
•
For
ε
c
>
0
.
00015:
σ
c
=
0where
ε
c
1
= −
0
.
0022
;
ξ
=
4
1
+
E
ci
E
c
1
−
2
ε
c
,
lim
ε
c
1
−
2
×
E
ci
E
c
1
−
2
ε
c
,
lim
ε
c
1
2
+
2
ε
c
,
lim
ε
c
1
−
E
ci
E
c
1
.
In the above relationships, the symbols have the followingmeaning:–
f
cm
is the mean compressive concrete strength;–
f
ctm
is the mean tension concrete strength;–
ε
c
1
is the strain at the peak stress;–
ε
c
,
lim
is the strain at half the peak stress;–
E
ci
is the initial tangent modulus;–
E
c
1
is the secant modulus at the peak stress.
2.3.2. Steel
In the present study, the steel is modelled as an elasticperfectly plastic material incorporating strain hardening. Fig. 4shows the stress–strain diagram for steel in tension. Specifically,the relationship is linearly elastic up to yielding, perfectly plasticbetween the elastic limit and the commencement of strainhardening,linearhardeningoccursuptotheultimatetensilestressand the stress remains constant until the tensile failure strain isreached.
2.3.3. Steel embedded in concrete
When uncracked concrete is in tension, the tensile force is distributedbetweenthereinforcementandtheconcreteinproportionto their respective stiffness, and cracking occurs when the stressreaches a value corresponding to the tensile strength of the concrete. In a cracked crosssection all tensile forces are balanced by
740
Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748
Fig. 4.
Stress–strain diagram for steel.
Fig. 5.
Stress–strain diagram for steel embedded in concrete in tension.
the steel encased in the concrete only. However, between adjacentcracks,tensileforcesaretransmittedfromthesteeltothesurroundingconcretebybondforces.Thecontributionoftheconcretemaybeconsideredtoincreasethestiffnessofthetensilereinforcement.Thiseffectiscalledtensionstiffening.Todescribethiseffect,anumberofmodelshavebeenproposed.Themajorityofthemodels are based on the mean axial stress and the mean axial strain of the concrete member in the reinforced concrete, [13–16].
To take the tension stiffening effect into account, the stressaveragestrainrelationshipofsteelembeddedinconcreteproposedby the CEBFIB model [11] is considered to describe the behaviour
of the reinforced concrete members in tension. Fig. 5 shows thestress–strain diagram of steel embedded in concrete.According to the CEBFIB Model Code 90 [11] the mean
stress–strain relationship of embedded steel may be expressed as
•
For
ε
s
,
m
≤
ε
sr
1
:
σ
s
=
1
+
1
αρ
E
s
ε
s
,
m
•
For
ε
sr
1
< ε
s
,
m
≤
ε
srn
:
σ
s
=
σ
sr
1
+
σ
srn
−
σ
sr
1
ε
srn
−
ε
sr
1
ε
s
,
m
−
ε
sr
1
•
For
ε
srn
< ε
s
,
m
≤
ε
sry
:
σ
s
=
σ
srn
+
f
sy
−
σ
srn
ε
sry
−
ε
srn
ε
s
,
m
−
ε
srn
(14)
•
For
ε
sry
< ε
s
,
m
≤
ε
sr
,
sh
:
σ
s
=
f
sy
Fig. 6.
Load–slip diagram for stud shear connector.
•
For
ε
sr
,
sh
< ε
s
,
m
≤
ε
sru
:
σ
s
=
f
sy
+
f
su
−
f
sy
ε
sru
−
ε
sry
ε
s
,
m
−
ε
sry
•
For
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
sr
1
is the steel stress in the crack, when the first crack hasformed–
σ
srn
is the steel stress in the crack, when the last crack hasformed–
ε
sr
1
and
ε
sr
2
are the steel strains at the point of zero slip andat the crack when the cracking forces reach
f
tm
–
ε
srn
=
σ
srn
/
E
s
−
β
t
(ε
sr
2
−
ε
sr
1
)
–
ε
sry
=
ε
sy
−
β
t
(ε
sr
2
−
ε
sr
1
)
–
ε
sr
,
sh
=
ε
sh
−
β
t
(ε
sr
2
−
ε
sr
1
)
–
ε
sru
=
ε
sr
,
sh
+
δ
1
−
σ
sr
1
/
f
sy
f
su
−
f
sy
/
E
sh
–
β
t
=
0
.
4 for instantaneous loading, and
β
t
=
0
.
25 for longterm and repeated loading–
δ
isacoefficienttotakeaccountthestressratio
f
su
/
f
sy
andtheyield stress
f
sy
(
δ
=
0
.
8 was proposed in CEBFIB Model Code90 [11]).
2.3.4. Stud shear connectors
The constitutive relationship proposed by Ollgaard et al. [17] is
considered for the stud shear connector. The analytical relationship between the shear force
R
sc
and the slip
d
sc
of a generic studis given by
R
sc
=
R
max
(
1
−
e
−
β

d
sc

)
α
(15)where
R
max
is the ultimate strength of the stud shear connector;
α
and
β
are coefficients to be determined from experimental results [18]. By using the secant stiffness
k
sc
(see Fig. 6) may berewritten as
R
sc
=
k
sc
d
sc
.
(16)
2.4. Cross sectional analysis
The procedure followed to predict the flexural behaviourof steel–concrete composite beam sections is an incrementaliterative technique with secant stiffness formulation. For thepresent numerical procedure, the cross section is divided into afinite number of discrete layers (fiber model). The axial strain in
Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748
741
Fig. 7.
Crosssectional analysis by finite layer approach.
any layer of the concrete slab section and the steel beam sectioncan be evaluated as
ε
cz
=
ε
c
+
κ
z
c
ε
sz
=
ε
s
+
κ
z
s
(17)To take into account the tension stiffening occurring when thereinforced layer is in tension, a single layer of embedded steel isconsidered in replacement of all layers in the effective concretearea
A
c
,
eff
(see Fig. 7).
A
c
,
eff
is the area of concrete surrounding thetensile reinforcement. According to the CEBFIB Model Code [11],
A
c
,
eff
is calculated as
A
c
,
ef
=
2
.
5
(
c
+
φ/
2
)
(18)where
c
isthecoatinglengthoftheconcreteand
φ
thediameterof the reinforcement.By using the secant stiffness of the constitutive relationships of materials for each layer, the axial resultant and bending resultantof the slab section can be expressed as
N
c
=
(
EA
)
c
ε
c
+
(
EB
)
c
κ
(19)
M
c
=
(
EB
)
c
ε
c
+
(
EI
)
c
κ
(20)where
(
EA
)
c
=
nlc
i
=
1
E
ic
+
E
i
+
1
c
2
b
ic
t
ic
+
E
sr
A
sr
(21)
(
EB
)
c
=
nlc
i
=
1
E
c
,
i
(
z
i
+
1
+
2
z
i
)
+
E
c
,
i
+
1
(
2
z
i
+
1
+
z
i
)
6
b
c
,
i
t
c
,
i
+
E
sr
A
sr
z
sr
(22)
(
EI
)
c
=
nlc
i
=
1
E
c
,
i
˜
z
i
+
E
c
,
i
+
1
˜
z
i
+
1
12
b
c
,
i
t
c
,
i
+
E
sr
A
sr
z
2
sr
(23)with
˜
z
i
=
z
2
i
+
1
+
2
z
i
z
i
+
1
+
3
z
2
i
and
˜
z
i
+
1
=
3
z
2
i
+
1
+
2
z
i
z
i
+
1
+
z
2
i
.
E
c
,
i
and
E
c
,
i
+
1
are the secant concrete modulus at the layer ends(trapezoidal integration scheme),
b
c
,
i
is the section width,
t
c
,
i
thethickness of the concrete layer
i
and
nlc
is number of concretelayers in the slab section.Inthesamewaytheaxialresultantandbendingresultantofthesteel beam sections are given by
N
s
=
(
EA
)
s
ε
s
+
(
EB
)
s
κ
(24)
M
s
=
(
EB
)
s
ε
s
+
(
EI
)
s
κ.
(25)Finally the force–deformation relationship of the cross section canbe written in matrix form as
e
(
x
)
=
f
(
x
)
D
(
x
)
(26)where
f
(
x
)
=
(
EA
)
c
0
(
EB
)
c
0
(
EA
)
s
(
EB
)
s
(
EB
)
c
(
EB
)
s
(
EI
)
c
+
(
EI
)
s
−
1
is the secant flexibilitymatrix of the cross section.
2.5. The mixed finite element formulation
Several finite element formulations for composite beams withcontinuous shear interaction have been already proposed in theliterature (see for example [19–21]). However, little attention has
been paid to composite beams with discrete shear connection asit will result in large number of degree of freedom. Nevertheless,discrete connection is more representative of the actual behaviourand provides accurate estimation on the load carried by the studsand the slip distribution along the beam. A displacementbasedformulation of composite beam with discrete shear connexionhas been proposed by Aribert et al. [22] and used to analysethe bending moment redistribution in continuous compositebeams [23].
In this paper, a new mixed finite element formulation [24] for
composite steel–concrete beams with discrete shear connectionis used to investigate the nonlinear behaviour of continuouscomposite beams. A similar formulation has been proposed by[25] for composite beams with continuous shear connexion. The
stiffness matrix is derived by combining the stiffness matrix of a single shear connector element and the stiffness matrix of acomposite beam element without connector.For the composite beam element without connector, thederivationfollowsthetwofieldmixedformulationwhichusestheintegral form of compatibility and equilibrium equations to derivethe matrix relation between the element generalized forces andthe corresponding displacements.In the twofield mixed formulation [10], both the displacement
and the internal forces fields along the element are approximatedby independent shape functions. The displacement is assumed tobe continuous along the whole beam:
d
(
x
)
=
a
(
x
)
q
(27)where
a
(
x
)
is amatrix of 3
×
n
d
shape functionswith
n
d
=
8 beingthe total number of displacement degrees of freedom and
q
thevector of element displacement (Fig. 8).
Theinternalforcesfieldisassumedtobecontinuousalongeachbeam element (but not across adjacent elements):
D
(
x
)
=
b
(
x
)
Q
+
D
0
(
x
)
(28)where
b
(
x
)
is a matrix of 3
×
n
s
force interpolation functionswith
n
s
=
4 being the total number of force degrees of freedom,
Q
the vector of element forces (Fig. 8) and
D
0
(
x
)
=[
0 0 0
.
5
p
0
x
(
L
−
x
)
]
T
isa internal forcevector accountingforthe effects of internal loading on the cross section forces.In the mixed formulation, the integral forms of compatibilityand equilibrium equations are enforced. These are then combinedto obtain the relationship between the element forces and thedisplacements.