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Journal of Constructional Steel Research 63 (2007) 505521www.elsevier.com/locate/jcsr
Finite element modelling of composite beams with full and partial shearconnection
F.D. Queiroza,, P.C.G.S. Vellascob, D.A. Nethercot a
aDepartment of Civil and Environmental Engineering, Imperial College London, SW7 2AZ, United Kingdomb Structural Engineering Department, State University of Rio de Janeiro, RJ, Brazil
Received 25 January 2006; accepted 8 June 2006
Abstract
The present investigation focuses on the evaluation of full and partial shear connection in composite beams using the commercial finite element
(FE) software ANSYS. The proposed three-dimensional FE model is able to simulate the overall flexural behaviour of simply supported composite
beams subjected to either concentrated or uniformly distributed loads. This covers: load deflection behaviour, longitudinal slip at the steelconcrete
interface, distribution of stud shear force and failure modes. The reliability of the model is demonstrated by comparisons with experiments and
with alternative numerical analyses. This is followed by an extensive parametric study using the calibrated FE model. The paper also discusses in
detail several numerical modelling issues related to potential convergence problems, loading strategies and computer efficiency. The accuracy and
simplicity of the proposed model make it suitable to predict and/or complement experimental investigations.
c 2006 Elsevier Ltd. All rights reserved.
Keywords:Composite beams; Finite element method; Material nonlinearity; Shear connection; Parametric analysis; ANSYS
1. Introduction
Composite steelconcrete construction, particularly for
multi-storey steel frames, has achieved a high market share in
several European countries, the USA, Canada and Australia.
This is mainly due to a reduction in construction depth, to
savings in steel weight and to rapid construction programmes.
Composite action enhances structural efficiency by combin-
ing the structural elements to create a single composite sec-
tion. Composite beam designs provide a significant economy
through reduced material, more slender floor depths and faster
construction. Moreover, this system is well recognised in termsof the stiffness and strength improvements that can be achieved
when compared with non-composite solutions.
A fundamental point for the structural behaviour and design
of composite beams is the level of connection and interaction
between the steel section and the concrete slab. The term full
shear connection relates to the case in which the connection
between the components is able to fully resist the forces applied
Corresponding author. Tel.: +44 (0) 20 7594 6097.E-mail addresses:fernando.queiroz@imperial.ac.uk(F.D. Queiroz),
d.nethercot@imperial.ac.uk(D.A. Nethercot).
to it. This is possibly the most common situation; however, over
the last two decades the use of beams in building construction
has led to many instances when the interconnection cannot
resist all the forces applied (partial shear connection). In this
case, the connection may fail in shear before either of the other
components reaches its own failure state.
In the case of the serviceability limit state of composite
beams, the condition when the connection between the
components is considered as infinitely stiff is said to comprise
full interaction. Whilst this is often assumed in design, it
is theoretically impossible and cases where the connection
has more limited stiffness (partial interaction) often need
to be considered. In this case, the connection itself may
deform, resulting in relative movement along the steelconcrete
interface and the effect of increased shear deformation in the
beam as a whole. Therefore, partial interaction occurs to some
extent in all beams whether fully connected or not [1]. However,
studies [1,2] have shown that any flexibility in the connection
may be ignored for beams designed for full connection.
The use of partial connection provides the opportunity to
achieve a better match of applied and resisting moment and
some economy in the provision of connectors. Generally, the
effects of partial interaction, which are increased by the use
0143-974X/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2006.06.003
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of partial shear connection, will result in reduced strength
and stiffness, and potentially enhanced ductility of the overall
structural system [2].
It is widely known that laboratory tests require a great
amount of time, are very expensive and, in some cases, can
even be impractical. On the other hand, the finite element
method has become, in recent years, a powerful and useful
tool for the analysis of a wide range of engineering problems.
According to Abdollahi [3], a comprehensive finite element
model permits a considerable reduction in the number of
experiments. Nevertheless, in a complete investigation of any
structural system, the experimental phase is essential. Taking
into account that numerical models should be based on reliable
test results, experimental and numerical/theoretical analyses
complement each other in the investigation of a particular
structural phenomenon [4].
Previous numerical studies have been conducted to
investigate the behaviour of composite beams. Nevertheless,
most of them are based on two-dimensional analytical models
(e.g., Gattesco [5] and Pi et al. [6]), and are thus not
able to simulate more complex aspects of behaviour, which
are intrinsic for three-dimensional studies; for instance: full
distribution of stresses and strains over the entire section
of the structural components (steel beam and concrete slab),
evolution of cracks and local deformations in the concrete slab.
In addition, in the particular case of the model developed by
Pi et al. [6], it was assumed that the shear connectors were
uniformly distributed along the length of a composite member.
A three-dimensional finite element model has been developed
by El-Lobody and Lam [7] using the package ABAQUS [8],
in which the mode of failure of the beams is detected by
a manual check of the compressive concrete stress and stud
forces for each load step. Nevertheless, just two beams were
used to validate the proposed model for composite beams with
solid slabs. All these studies [57] were focused just on the
presentation and validation of their corresponding models, but
these models were not used to investigate in more detail either
the effect of particular structural parameters or other aspects
of the system behaviour. It is only very recently that papers
on finite element analyses of composite systems have started
to contain parametric studies (e.g., investigations related to the
behaviour of individual shear connectors [9,10]).In order to obtain reliable results up to failure, finite element
models must properly represent the constituent parts, adopt
adequate elements and use appropriate solution techniques.
As the behaviour of composite beams presents significant
nonlinear effects, it is fundamental that the interaction of all
different components should be properly modelled, as well as
the interface behaviour. Once suitably validated, the model can
be utilised to investigate aspects of behaviour in far more detail
than is possible in laboratory work. For instance, it permits
the study of the sensitivity of response to variability of key
component characteristics, including material properties and
shear stud layout. Consequently, different spacing in distinct
parts of the beam can be adopted, allowing the investigation ofpartial interaction effects.
The present investigation focuses on the modelling of
composite beams with full and partial shear connection using
the software ANSYS [11]. A three-dimensional model is
proposed, in which all the main structural parameters andassociated nonlinearities are included (concrete slab, steel beam
and shear connectors). Test and numerical data available in
the literature are used to validate the model, which is able to
deal with simply supported systems with I-beams and solid flat
slabs. Other features such as steel profiled sheeting, different
types of slab (e.g., precast slabs) and distinct end-connectivities
are not included in the present study.
Based on the validated model, an extensive parametric
analysis of composite beams is performed, specifically aimed
at:
studying the effect of the continuation of shear connection
beyond the supports of simply supported composite beams; investigating the overall structural system behaviour when
different concrete compressive strengths are used in the slab
and in the associated push-out tests. This situation has often
been observed in reported laboratory studies but has not
previously received any systematic study, being important,
for instance, for the definition of the loadslip curves used
for the shear connectors;
analysing the influence of small variations in key input
parameters (i.e., concrete and steel material properties) on
the structural behaviour of uniformly loaded composite
beams. This sensitivity study can be used to identify the
variables (e.g., web and flange yield stresses, concrete
strength, etc.) which are more important in terms ofdefinition of the overall response of the system and therefore
can be helpful in terms of establishing possible differences
between numerical and test results;
assessing the influence of the effects of partial shear
connection and partial interaction not only on the overall
flexural behaviour of composite beams (represented by the
loaddeflection curve), but also on the associated failure
modes for either slab crushing or stud failure, and on the
distribution of stud shear forces along the beam length.
This paper also discusses several numerical modelling issues
related to convergence problems that arise when the concrete
material is considered, loading strategies for the simulation of
distributed loads and a comparison between the load controland the displacement control methods in terms of computer
efficiency.
2. Finite element model
2.1. Software, element types and mesh construction
Advances in computational features and software have
brought the finite element method within reach of both
academic research and engineers in practice by means of
general-purpose nonlinear finite element analysis packages,
with one of the most used nowadays being ANSYS. The
program offers a wide range of options regarding element types,material behaviour and numerical solution controls, as well as
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Fig. 1. Typical composite beam FE mesh.
graphic user interfaces (known as GUIs), auto-meshers, andsophisticated postprocessors and graphics to speed the analyses.
In this paper, the structural system modelling is based on the use
of this commercial software.
The finite element types considered in the model are as
follows: elastic-plastic shell (SHELL43) and solid (SOLID65)
elements for the steel section and the concrete slab,
respectively, and nonlinear springs (COMBIN39) to represent
the shear connectors. Both longitudinal and transverse
reinforcing bars are modelled as smeared throughout the solid
finite elements.
The element SHELL43 is defined by four nodes having
six degrees of freedom at each node. The deformation shapes
are linear in both in-plane directions. The element allows forplasticity, creep, stress stiffening, large deflections, and large
strain capabilities [11]. The element SOLID65 is used for three-
dimensional modelling of solids with or without reinforcing
bars (rebar capability). The element has eight nodes and three
degrees of freedom (translations) at each node. The concrete is
capable of cracking (in three orthogonal directions), crushing,
plastic deformation, and creep [11]. The rebars are capable of
sustaining tension and compression forces, but not shear, being
also capable of plastic deformation and creep.
The element COMBIN39 is defined by two node points
and a generalized forcedeflection curve and has longitudinal
or torsional capability. The longitudinal option is a uniaxial
tensioncompression element with up to three degrees offreedom (translations) at each node.
Symmetry of the composite beams is taken into account by
modelling only one half of the beam span. A typical FE mesh
for a composite beam is shown inFig. 1.
2.2. Material modelling
The von Mises yield criterion with isotropic hardening rule
(multilinear work-hardening material) is used to represent the
steel beam (flanges and web) behaviour. The stressstrain
relationship is linear elastic up to yielding, perfectly plastic
between the elastic limit (y ) and the beginning of strainhardening and follows the constitutive law used by Gattesco [5]
for the strain-hardening branch:
= fy+ E
h(
h)1 E
h
h
4(fu fy) (1)
where fy and fu are the yield and ultimate tensile stresses of
the steel component, respectively; Eh and h are the strain-
hardening modulus (i.e., 3500 N/mm2) and the strain at strain
hardening of the steel component, respectively.
The von Mises yield criterion with isotropic hardening rule
is also used for the reinforcing steel. An elastic-linear-work-
hardening material is considered, with tangent modulus being
equal to 1/10 000 of the elastic modulus, in order to avoid
numerical problems. The values measured in the experimental
tests for the material properties of the steel components (steel
beam and reinforcing bars) are used in the finite element
analyses.The concrete slab behaviour is modelled by a multilinear
isotropic hardening relationship, using the von Mises yield
criterion coupled with an isotropic work hardening assumption.
The uniaxial behaviour is described by a piece-wise linear
total stresstotal strain curve, starting at the origin, with
positive stress and strain values, considering the concrete
compressive strength (fc) corresponding to a compressive
strain of 0.2%. The stressstrain curve also assumes a total
increase of 0.05 N/mm2 in the compressive strength up to
the concrete strain of 0.35% to avoid numerical problems due
to an unrestricted yielding flow. The concrete element shear
transfer coefficients considered are: 0.2 (open crack) and 0.6
(closed crack). Typical values range from 0 to 1, where 0represents a smooth crack (complete loss of shear transfer) and
1 a rough crack (no loss of shear transfer). The default value
of 0.6 is used as the stress relaxation coefficient (a device that
helps accelerate convergence when cracking is imminent). The
crushing capability of the concrete element is also disabled to
improve convergence.
The concrete slab compressive strength is taken as the actual
cylinder strength test value. The concrete tensile strength and
the Poissons ratio are assumed as 1/10 of its compressive
strength and 0.2, respectively. The concrete elastic modulus is
evaluated according to Eurocode 4 [12], i.e.:
Ec = 9500(fc + 8)1/3c24
1/2 (2)
where:c is equal to 24 kN/m3.
The model allows for any pattern of stud distribution to be
considered, for instance: the conventional uniform arrangement
and a triangular spacing scheme where the stud distribution
follows the nominal elastic shear diagram [13]. In all analyses,
the number/spacing of studs adopted in the experimental
programmes is utilised. As far as the shear connector behaviour
is concerned, the loadslip curves for the studs are used
(obtained from available push-out tests) by defining a table
of force values and relative displacements (slip) as input data
for the nonlinear springs. These springs are modelled at thesteelconcrete interface, as shown inFig. 2.
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Fig. 2. Modelling of shear connectors (longitudinal view). (a) Shear studs in a typical composite beam. (b) Shear studs in a typical composite beam finite element
mesh. (c) Representation of the shear stud model.
2.3. Application of load and numerical control
Regarding application of load, concentrated loads are
incrementally applied to the model by means of an
equivalent displacement to overcome convergence problems
(displacement control). For the convergence criterion, the L2-
norm (square root sum of the squares) of displacements is
considered. Uniformly distributed loads are represented bymeans of point loads applied at all mid-section concrete
nodes. These concentrated loads are also applied to the model
incrementally using the load control strategy and the L2-
norm. The tolerance associated with this convergence criterion
(CNVTOL command of ANSYS) and the load step increment
are varied in order to solve potential numerical problems.
Whenever the solution does not converge for the set of
parameters considered, as far as load step size and converge
criterion are concerned, the RESTART command is used in
conjunction with the CNVTOL option. ANSYS allows two
different types of restart: the single-frame restart and the multi-
frame restart, which can be used for static or full transient
structural analyses. The single-frame restart only allows theuser to resume a job at the point it stopped. The multi-frame
restart can resume a job at any point in the analysis for
which information is saved. This capability enables multiple
model analyses, presenting more options for data retrieval
after an undesired aborted solution. The second approach
is used throughout the present analyses and the associated
error is controlled by comparisons between applied forces and
reactions (balance of forces) for each load step.
The load control strategy is adopted due to the fact that,in structural problems in which significant nonlinear effects
occur, it is difficult to derive a relationship between loads
and associated displacements for the case of distributed loads,
mainly for the plastic range of behaviour. For the case in
which only one point load is applied to the system, there is
a direct relationship between force and displacement, making
the displacement control method easier to be utilised. The load
control method is, however, less efficient than the displacement
control method in nonlinear analyses. This fact is observed
especially when the applied load approaches the ultimate load
of the system, as an incremental increase in the load leads
to a significant increase in the corresponding displacements,
causing difficulties in terms of numerical convergence. Forthe type and size of the finite element problem investigated,
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the load control method demanded, on average, 40% more
disk space and took 140% longer to be processed than similar
displacement control solutions.
Preliminary attempts to overcome the convergence problemsarising from the use of the load control method included the
specification of different types of equation solvers. The best
approach in terms of numerical performance was the option
in which the software ANSYS selects a solver based on the
physics of the problem. The arc-length method was also tested
but proved not to be the best option for this particular type of
analysis.
2.4. Failure criterion
Two limits are established to define the ultimate load for
each finite element investigation: a lower and an upper bound,
corresponding to concrete compressive strains of 0.2%, and0.35%, respectively. These two limits define an interval in
which the composite beam collapse load is located. A third
limit condition, hereinafter referred to as the stud failure point,
can also be reached when the composite beams most heavily
loaded stud reaches its ultimate load, as defined from the
appropriate push-out tests.
If the stud failure point is located before the lower bound
of concrete (i.e., the corresponding load of the stud failure
point is smaller than the lower bound load) then the mode
of failure of the composite beam is considered as being stud
failure. Conversely, if the stud failure point is located after the
upper bound of concrete, the mode of failure is assumed as
being concrete crushing. For the intermediate case, where thestud failure point lies between the lower and upper bounds of
concrete, than the mode of failure could be either of them.
Therefore, the proposed finite element model is able to
predict the failure modes associated with either slab crushing
or stud failure.
3. Validation of the model
The present model is validated by comparisons against
Chapman and Balakrishnan tests [13], as well as against
alternative numerical studies (Gattesco [5], Pi et al. [6] and El-
Lobody and Lam [7]).
3.1. Chapman and Balakrishnan tests
The tests performed by Chapman and Balakrishnan
successfully illustrate the behaviour of the composite system
which is being investigated. The beams spanned 5490 mm
with an I-shaped steel member 305 mm deep (12 6
44 lb/ft BSB) and a concrete slab 152 mm thick1220 mm
wide. The number and type of studs, as well as the steel and
concrete strengths, varied according to the tested composite
beam. The slab was longitudinally reinforced with four top
and four bottom 8 mm bars. The transverse reinforcement
incorporated top and bottom bars of 12.7 mm @ 152 mm
centres and 12.7 mm @ 305 mm centres, respectively. Thetensile strength, the Youngs modulus and the Poissons ratio
Fig. 3. Simply supported beam layout (dimensions in mm).
Fig. 4. Load (kN) vs. midspan deflection (mm) beams A2 and A3.
of the reinforcing steel bars were 320 N/mm2, 205000 N/mm2
and 0.3, respectively. A list of material properties for all beams
is given in [1316] and a full description of these beams is
presented inFig. 3andTable 1.Based on the composite section strength of the concrete
slab, steel components and shear connectors, the level of
shear connection could be determined. This value is defined
as the ratio between the shear connection capacity and the
weakest element capacity (concrete slab or steel beam).Table 2summarises the level of shear connection for all the composite
beams, considering two different approaches. The first one uses
the nominal values presented by [13] for the stud strength and
steel yield stress. In the second one, the material properties
are taken as the actual measured values [13]. Considerable
differences among the levels of shear connection according to
these two approaches are noticed, leading to the conclusion that,
in order to calculate the level of shear connection of composite
systems, the actual material properties of the components
(measured values), related to each experimental programme,
should be used.For all composite beams shown in Table 1, loadmidspan
deflection curves are compared to the test results. Figs. 48(beams A2 to E1) and Figs. 911 (beams U1 to U4)
depict comparisons between the FE model results and the
experimental data for the midspan concentrated and uniformly
distributed loaded composite beams, respectively. The limit
points for the concrete are represented by a full triangle (lower
bound) and by a full square (upper bound), and the stud failure
point is represented by a full circle (item 2.4). Good agreement
was obtained between test and numerical results.In order to illustrate finite element data for local results,
the numerical and test values regarding the slip at the
steelconcrete interface along the beam axis for the cases E1
and U4 are plotted inFigs. 12and13, respectively. The graphs
show that the proposed model can predict the slip distributionwith good precision. From these figures, it can be noticed that
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Table 1
Details of composite beams tested by [13]
Beam A2 A3 A4 A5 A6 B1 C1 D1 E1 U1 U3 U4
Stud diam. (mm) 19
12.7
Stud overall length (mm) 102
76
50
Number of studs 100
76
68
56
44
32
Spacing in pairs (mm) 121 a
159 a
178 a216 a
274 a
378 a
Mode of failure Slab crushing
Stud failure
Load type Midspan concentrated Uniformly
distributed
a Triangular spacing.
Table 2
Level of shear connection of the composite beams (%)
Beam A2 A3 A4 A5 A6 B1 C1 D1 E1 U1 U3 U4
Nominal values 238 213 175 138 101 138 138 313 313 175 175 101
Measured values 231 186 137 123 95 116 114 136 148 155 177 90
Fig. 5. Load (kN) vs. midspan deflection (mm) beams A4 and A5.
Fig. 6. Load (kN) vs. midspan deflection (mm) beams A6 and B1.
Fig. 7. Load (kN) vs. midspan deflection (mm) beams C1 and D1.
Fig. 8. Load (kN) vs. midspan deflection (mm) beam E1.
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Fig. 9. Load (kN/m) vs. midspan deflection (mm) beam U1.
Fig. 10. Load (kN/m) vs. midspan deflection (mm) beam U3.
Fig. 11. Load (kN/m) vs. midspan deflection (mm) beam U4.
the slip is not uniform along the beam length, even where the
externally applied shear force is uniform (midspan concentrated
load case). The maximum slip value tends to occur when X/L
equals 0.4 (point load case) and X/L equals 0.3 (uniformly
distributed load case UDL). According to [13], the large slip
which occurs near midspan is due to the high interface shear inthe plastic region.
Table 3 presents the ultimate load for each of the studied
composite beams. This load is expressed in terms of the test
results and both lower and upper bound limits (FE analysis).
This table also presents the ratio between the numerical and
test results for each limit point and their associated dispersion
values. It can be noticed that the proposed model was able to
predict the experimental ultimate load very accurately, as well
as the associated mode of failure (slab crushing or stud failure).
3.2. Comparisons with previous numerical studies
Gattesco [5] presented an analytical procedure for theinvestigation of composite beams, in which the nonlinear
Fig. 12. Slip distribution along span beam E1.
Fig. 13. Slip distribution along span beam U4.
behaviour of all materials was considered. The parametric
analysis demonstrated that the numerical program was able to
model full and partial shear connection. El-Lobody and Lam [7]
used the ABAQUS FE software to undertake a numericalanalysis of composite girders with solid and precast hollow
core slabs. Both models included the material nonlinearities,
as well as the stud nonlinear loadslip characteristics. Partial
interaction between the steel and concrete components was
also incorporated in the total Lagrangian finite element
model formulated by Pi et al. [6]. The model was validated
by comparisons against simply supported and continuous
composite beams tests.
In Figs. 14 and 15 the loadmidspan deflection curves
obtained from these previous numerical models are compared
with the present study results for the composite beams E1 and
U4. Good agreement between the curves was obtained.
3.3. Effect of the overhang region (region beyond supports)
As it can be seen in Fig. 3, the experiments conducted
by Chapman and Balakrishnan concerned composite beams in
which a number of shear connectors were used in the overhang
regions. According to [13], a more rational test procedure
would have been to limit the slab length to the distance between
supports. Nevertheless, it was also stated by [13] that, in
practice, the concrete slabs do not always necessarily terminate
at the reactions. In this section, the effect of the overhang
regions (i.e., the regions beyond supports) will be assessed in
terms of both overall and local behaviour. None of the already-
mentioned previous studies on the Chapman and Balakrishnantests [57] investigated this aspect.
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Table 3
Ultimate load results for the experimental and numerical analyses
Beam Pexp PLB(0.2%) PUB(0.35%) 1 = PLB/Pexp 2 = PUB/Pexp % = (2 1) 100
A2 448 429 469 0.96 1.05 9
A3 449 425 447 0.95 0.99 4
A4a 523 444 470 0.85 0.90 5
A5 468 462 479 0.99 1.02 3
A6 430b 449b 1.04
B1 486 459 468 (466b ) 0.94 0.96 (0.95b ) 2
C1 448 445 474 0.99 1.06 7
D1 481 457 475 0.95 0.99 4
E1 513 520 548 1.01 1.07 6
U1 191 171 178 0.90 0.93 3
U3 185 166 182 0.90 0.98 8
U4 176b 179b 1.02
Pexp, PLBand PUBare the test ultimate load, lower and upper bound loads, respectively (midspan load [kN]; uniformly distributed load [kN/m]).a The strength of the concrete used in the push-out test was much less than the concrete strength of the composite beam;b
Stud failure, so bounds not applicable.
Fig. 14. Load vs. midspan deflection Other similar studies (beam E1).
Fig. 15. Load vs. midspan deflection Other similar studies (beam U4).
In the following (Figs. 1625), the analyses in which theseregions are included in the model will be hereinafter referred to
as the extended case (results presented in item 3.1), and the
ones in which only the regions between supports are modelled
will be referred to as the standard case.
The loaddeflection curves for the composite beams A4, D1,
E1 (midspan concentrated load) and U4 (UDL) are shown in
Figs. 1619, respectively. It can be observed that an increase
in the system stiffness was present for the extended case when
compared with the standard one. The connectors placed beyond
the supports have an anchorage effect on the beam, leading to
an improvement in the composite effect of the system.
This behaviour is in accordance with the observations by
Goodman and Popov [17], who investigated the behaviour oflayered wood beam systems with interlayer slip. Results of
Fig. 16. Load (kN) vs. midspan deflection (mm) beam A4.
Fig. 17. Load (kN) vs. midspan deflection (mm) beam D1.
Fig. 18. Load (kN) vs. midspan deflection (mm) beam E1.
experiments with layered beams connected with nails, withand without glued ends, showed that there was a decrease
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Fig. 19. Load (kN/m) vs. midspan deflection (mm) beam U4.
Fig. 20. Ratio of stud forces vs. relative position of the stud beam A4.
Fig. 21. Ratio of stud forces vs. relative position of the stud beam D1.
Fig. 22. Ratio of stud forces vs. relative position of the stud beam E1.
in the beam deflection values for the former case (i.e.,
the structural system was stiffer). Although the conclusions
presented were illustrated for the case of layered systems of
wood, any mechanically-connected layered beam system (e.g.,
steelconcrete beams connected by means of shear studs) canbe analysed in the same manner. The effect of the use of
Fig. 23. Ratio of stud forces vs. relative position of the stud beam U4.
Fig. 24. Slip distribution along span beam E1.
Fig. 25. Slip distribution along span beam U4.
connectors beyond the beam supports could be linked to the
situation of glued beam ends described above.
For the case of beams subjected to uniformly distributedload (as exemplified by Fig. 19 for beam U4), the increase
in the stiffness of the system is more significant than for the
case of beams subjected to midspan point loads (Figs. 1618).
Considering the distribution of flexural moments in the regions
next to the supports, the moments for a beam subjected to a
UDL present proportionally higher values than the moments for
a beam subjected to a midspan concentrated load, making the
shear connectors in this region more important than the ones
towards the midspan. In the point load case, the studs work in
a more uniform manner along the beam. The overhang regions
lead to a more rapid development of the effective breadth of
the concrete slab (it is known that, for the standard case, the
effective breadth is theoretically zero at the support), resultingin a more accentuated increase in the stiffness.
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Table 4
Level of shear connection (%) extended vs. standard beams
Beam A2 A3 A4 A5 A6 B1 C1 D1 E1 U1 U3 U4
Extended 231 186 137 123 95 116 114 136 148 155 177 90
Standard 207 164 127 112 83 106 104 125 136 144 164 79
Table 5
Ultimate load results for the experimental and numerical analyses standard case
Beam Pexp PLB(0.2%) PUB(0.35%) 1 = PLB/Pexp 2 = PUB/Pexp % = (2 1) 100
A2 448 431 468 0.96 1.04 8
A3 449 426 445 0.95 0.99 4
A4a 523 444 469 0.85 0.90 5
A5 468 448 467c 0.96 1.00 4
A6 430b 444b 1.03
B1 486 444 (440b ) 452 0.91 (0.91b ) 0.93 2
C1 448 446 469 1.00 1.05 5
D1 481 462 472 0.96 0.98 2
E1 513 518 540 1.01 1.05 4
U1 191 162 171 0.85 0.90 5
U3 185 154 169 0.83 0.91 8
U4 176b 166b 0.94
Pexp, PLBand PUBare the test ultimate load, lower and upper bound loads, respectively (midspan load [kN]; uniformly distributed load [kN/m]).a The strength of the concrete used in the push-out test was much less than the concrete strength of the composite beam;b Stud failure, so bounds not applicable;c ANSYS analysis terminated.
In addition, it is worth pointing out that the shear connectors
in the overhang regions should be taken into account when
calculating the level of shear connection, as they have an
influence on the system behaviour. Therefore, if only the region
between supports is considered, the connection level decreases,as shown inTable 4.
In order to illustrate the results in terms of stud force
distribution, tests A4, D1, E1 and U4 are considered. In
Figs. 2023, stud force distribution graphs are plotted relating
force ratio to stud position. It can be seen that, for the standard
cases, there is a disturbance in the stud force distribution in
the region near the supports, caused by the reduction of the
effective breadth of the slab in this region. For the extended
cases, this effect is not so significant, as the effective breadth
of the slab is almost completely developed in the support
region (it can be noticed that the tendency of the curve is
maintained along the beam). In the region between the supports
and midspan, only the steel and concrete deformation patternsdefine the behaviour.
The slip at the steelconcrete interface along the beam
axis for the cases E1 and U4 is plotted in Figs. 24 and
25, respectively, for both extended and standard cases. It is
evident from these graphs that the overhang regions also have
a considerable effect on the slip values, mainly in the regions
next to the supports. The disturbance of the slip distribution
near the supports caused by the effective breadth of the slab
can be noticed in Fig. 24, as well. In addition, it can be
observed in Fig. 24 that the curve corresponding to the extended
case is above the experimental curve. This fact may be due
to a difference between the loadslip behaviour of the shear
connectors in the tested composite beam and the loadslipbehaviour of the push-out test (used in the FE analysis).
Moreover, the curves related to the standard case are also above
the ones for the extended situation. Nevertheless, the sum of the
stud forces is the same, provided that the connectors beyond the
supports are considered.
Table 5 presents the ultimate loads for all analysesconsidering the standard case (beams between supports). It can
be observed that the values predicted by this model are very
close to the ones obtained for the extended composite beam
case (Table 3).
The complete results for all standard beams are presented
and discussed in Queiroz et al. [14,15], including a detailed
study in terms of the absolute force carried by the shear studs
for three different stud overall lengths (for a fixed diameter and
spacing), and the absolute force carried by the shear studs for
five different connector spacing values (for a fixed diameter and
length).
Based on this investigation, it can be seen that the
continuation of the shear connection beyond the beam supportscan affect not only the overall response (e.g., shape of the
loaddeflection curve), but also local results (e.g., slip and
stud force distributions along the beam). Nevertheless, for
the composite beams considered, the ultimate load was not
significantly influenced.
3.4. Effects of concrete slab strength and concrete strength for
push-out specimens
This section describes a comprehensive parametric analysis
of the overall structural behaviour of steelconcrete composite
beams when different concrete compressive strengths are
present in the slab and in the associated push-out tests. Itwill always be the case, for example, for composite beams
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Table 6
Influence of the slab concrete strength
Beam Diameter
(mm)
Overall length
(mm)
Spacing in pairs
(mm)
Cylinder strength of the concrete (N/mm2)
Push-out (used in the shear connector
springs)
Slab (used in the slab finite elements)
25.0
26.9
A2 19 102 159 30.7 28.8
30.7
32.6
23.1
24.8
A5 19 102 274 28.0 26.4
28.0
29.6
Table 7
Influence of the concrete strength used in the push-out tests
Beam Diameter
(mm)
Overall length
(mm)
Spacing in pairs
(mm)
Cylinder strength of the concrete (N/mm2)
Slab (used in the slab finite
elements)
Push-out (used in the shear connector springs)
19.7 (A4)
A2 19 102 159 26.9 28.0 (A5)
30.7
19.7
A4 19 102 216 20.1 28.0 (A5)
30.7 (A2)
19.7 (A4)
A5 19 102 274 24.8 28.030.7 (A2)
with precast slabs, for which results from push-out tests with
solid concrete slabs are usually used. Two different effects
are considered, utilising the standard model discussed in item
3.3: one for which the slab concrete strength is fixed and
the concrete strength for the associated push-out specimens
is varied and another assuming the opposite situation. Results
relating to the distribution of stud forces along the beam
length and to the moment capacity of the composite system
are discussed. The cases used for the parametric study are
summarised inTables 6and7.
Table 6 presents the scope of the analysis adopted toinvestigate the effect of the slab strength on the beam structural
response. For each beam (A2 and A5) five different slab
strengths are considered. The relative differences between two
consecutive slab concrete strengths are approximately the same.
For both beams, the diameter and overall length of the studs
are also equal. The concrete strength of the associated push-out
test carried out by [13] is fixed for each beam. Nevertheless, the
shear connector spacing of each beam is distinct.Table 7 presents the range of values adopted in order
to assess the influence of the concrete strength used in the
associated push-out tests. For each beam (A2, A4 and A5)
three different push-out strengths are assumed, i.e.: the concrete
strength of the beams A2, A4 and A5. For all beams, thediameter and overall stud length are also equal. The slab
concrete strength and the shear connector spacing are fixed and
distinct for each beam.
3.4.1. Influence of the slab concrete strength on the composite
beam response
The main focus of discussion is centred on: load versus
midspan deflection curves, distribution of stud forces along the
beam lengths and absolute forces carried by the shear studs for
different slab strengths, with the same concrete strength being
used in the associated push-out specimen.
Figs. 2628andFigs. 2931present the FE model resultsfor the beams A2 and A5, respectively. By analysing the
loaddeflection curves (Figs. 26 and 29), it is possible to
observe that an increase in the slab concrete strength resulted in
a stiffer system and in an increase in the moment capacity of the
beam, represented by the ultimate bound points. Nevertheless,
the former effect seems to be less significant than the latter
one.
According to Fig. 27 (absolute force carried by the shear
studs for different slab strengths), an increase in the stud
forces is observed as the slab concrete strength increases, most
noticeably for the two beams with the highest slab concrete
strengths (30.7 N/mm2 and 32.6 N/mm2). Based onFig. 30, a
similar conclusion may be drawn for beam A5, i.e.: an increasein the stud forces occurs as the slab concrete strength increases.
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Fig. 26. Load (kN) vs. midspan deflection (mm) for different slab concrete
strengths beam A2.
Fig. 27. Stud force vs. slab concrete strength beam A2 (lower and upper
bounds).
Fig. 28. Ratio of stud forces vs. relative position of the stud beam A2.
The mode of collapse for the beams with the highest slab
concrete strengths (28.0 N/mm2 and 29.6 N/mm2) was related
to stud failure. The composite beam with slab concrete strength
of 29.6 N/mm2 presented a lower bound closer to the stud
failure point than did the composite beam with a 28.0 N/mm2-
slab concrete strength (Fig. 29).
Two concrete strengths are chosen for each beam (A2 and
A5) to illustrate the distribution of stud forces along the beam
lengths (Figs. 28 and 31, respectively). It can be noticed that, for
the beam A2, the stud forces decrease towards the beam ends.
This was not the case for the beam A5, where the stud forces
did not vary significantly along the beam length.Fig. 30showsthat the variation in the stud forces for the beam A5 is smaller
Fig. 29. Load (kN) vs. midspan deflection (mm) for different slab concrete
strengths beam A5.
Fig. 30. Stud force vs. slab concrete strength beam A5 (lower and upper
bounds).
Fig. 31. Ratio of stud forces vs. relative position of the stud beam A5.
than the variation for the beam A2 (Fig. 27), therefore resultingin a more uniform distribution of forces along the beam.
The analysis shows that, therefore, an increase in the slab
concrete strength can have an influence not only on general
aspects of behaviour (e.g., overall stiffness of the system and
ultimate moment capacity) but also on local results (e.g., stud
forces).
3.4.2. Effect of the push-out concrete strength on the composite
beam response
The main focus of discussion is centred on varying the push-
out test concrete strengths (by means of adopting different
loadslip curves for the shear connector representation)
maintaining the same slab concrete strength. Figs. 3234,3537,3840present the FE model results for the beams A2,
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Fig. 32. Load (kN) vs.midspandeflection (mm) fordifferent concrete strengths
beam A2.
Fig. 33. Stud force vs.concrete strength beam A2 (lowerand upper bounds).
Fig. 34. Ratio of stud forces vs. relative position of the stud beam A2.
A4 and A5, respectively. Two concrete strengths are chosen foreach beam to illustrate the distribution of stud forces along the
beam lengths (Figs. 34,37and40).
The concrete strength used in the push-out test appears to
have a small influence on both the overall behaviour of the
composite beam and the stud force distribution, as can be
seen in the graphs of applied load versus midspan deflection
and absolute force carried by the shear studs versus concrete
strengths used in the push-out tests (beam A2:Figs. 32and33;
beam A4:Figs. 35and36; beam A5:Figs. 38and39).
4. Sensitivity study
In this section, an investigation is performed aimed atassessing the sensitivity of the overall response of composite
Fig. 35. Load (kN) vs.midspandeflection (mm) fordifferent concrete strengths
beam A4.
Fig. 36. Stud force vs.concrete strength beam A4 (lowerand upper bounds).
Fig. 37. Ratio of stud forces vs. relative position of the stud beam A4.
Fig. 38. Load (kN) vs.midspandeflection (mm) fordifferent concrete strengths
beam A5.
beams (represented by loaddeflection curves and ultimate
moment capacity) to likely variations in material strengths.
Beams U1, U3 and U4 described in item 3 are used forthis analysis, as well as the standard model discussed in item
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Fig. 39. Stud force vs.concrete strength beam A5 (lowerand upper bounds).
Fig. 40. Ratio of stud forces vs. relative position of the stud beam A5.
3.3. The numerical modelling of composite beams subjected
to uniformly distributed loads is not a straightforward process,
particularly due to their highly nonlinear behaviour. Hence, by
identifying the key structural variables which can affect thesystem response, possible differences between numerical and
tests results could be further comprehended.The range of steel and concrete strengths is assumed based
on observed variations for nominally identical samples. The
material properties studied are: (a) steel: web and flanges yield
stresses and ratios between the strain at strain hardening and the
yield strain; (b) concrete: compressive strength.The main results are as follows (a comprehensive description
of the material properties adopted and the case studies
considered can be found in Queiroz et al. [16]).
4.1. Influence of the slab concrete strength
It was noticed that, increasing the concrete strength, for afixed combination of yield stresses for the flanges and web:
the lower and upper bounds increase in terms of both the
ultimate load and the associated deflection. In some cases,
the failure mode of the composite beam can change from
slab crushing to stud failure; the corresponding loaddeflection curve becomes stiffer, but
this effect is not very significant when compared with the
effect previously mentioned.
These results confirm the outcomes discussed in item3.4.1.
4.2. Influence of the yield stress of the flanges and web
It could be observed that, for a fixed slab concrete strengthand increasing either the flanges or the web yield stresses:
the lower and upper bounds increase in terms of the ultimate
load;
the corresponding loaddeflection curve becomes stiffer. For
the range of material properties considered in the parametricanalysis, the change in the loaddeflection curve was more
significant for the cases in which the web yield stress was
modified.
4.3. Influence of the ratio of strains at strain hardening and at
yield
For the range of values considered, the steel ratio between
the strain at strain hardening and the yield strain did not
significantly affect the overall response of the composite beam.
Based on the outcomes of the sensitivity analysis, it can
be concluded that the web yield stress is the key structural
parameter in the definition of the overall shape of the
loaddeflection curve of the composite beams analysed. An
increase in this property makes the curve become stiffer. In
addition, the failure mode of the composite beams can be
influenced by the concrete strength.
5. Effect of partial shear connection
In this section, the proposed finite element model is used
to assess the influence of the effects of partial shear connection
and partial interaction not only on the overall flexural behaviour
of the structural system (represented by the loaddeflection
curve), but also on the associated failure modes for either slabcrushing or stud failure, and on the distribution of stud shear
forces along the beam length. In order to isolate the effect of
the level of shear connection, the material properties (steel,
concrete and shear connectors) and dimensions of the beam
E1 tested by [13] are used for all analyses. Two load cases
are considered: midspan point load and uniformly distributed
load. The levels of shear connection are calculated using the
actual material properties measured during the experimental
procedure. In the present study, levels ranging from 47% to
136% (by means of varying the number of shear connectors)
are analysed using the standard finite element model
(item 3.3).
As the present model of composite beams adopts a solidelement to represent the concrete material and a nonlinear
spring to model the shear connectors, the interaction between
the slab and the studs cannot be investigated explicitly. A
more refined model which adopts three-dimensional solid
elements for the shear connectors would be the ideal option
in order to capture this effect. Nevertheless, such an approach
would certainly lead to complications as far as the total size
of the finite element mesh (which can lead to a significant
increase in running times) and numerical convergence problems
are concerned, especially if the crushing capability of the
concrete material is not disabled. Based on comparisons with
experimental tests, the use of springs to model the studs has
proved to be, in spite of its simplicity, very efficient in terms offinite element analyses.
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Fig. 41. Load-deflection curve Midspan concentrated load.
Fig. 42. Load-deflection curve Midspan concentrated load Initial
stiffness.
5.1. Midspan point load case
The loadmidspan deflection curves for all cases (each one
having a different level of shear connection) are shown inFig. 41.Fig. 42presents these curves just for the initial branch,
in order to better illustrate the differences between the curves
in terms of initial stiffness. It can be observed that, as expected,
on decreasing the level of shear connection the system became
more flexible, with reduced strength and stiffness. However, the
decrease in stiffness did not seem to be as significant as the
decrease in the ultimate load.
The three higher levels (118%, 130% and 136%) resulted
in very similar curves, in terms of both stiffness and ultimate
load, represented by the lower and upper bound values, with
the mode of failure being slab crushing at the midspan of
the beam (location of maximum bending moment). Regarding
level 100%, its mode of failure could be either slab crushingor stud failure, as a stud failure point was obtained between
the lower and upper bounds. As the three lower levels (47%,
71% and 89%) resulted in stud failure, the level 100% was an
intermediate level between the two possible modes of failure.
The ratio between the ultimate load and the load at the end
of the elastic behaviour was not much affected by the distinct
levels of shear connection, being in the range from 1.5 to 2
for all cases, which is a common value as far as steelconcrete
composite beams are concerned.
The stud force distribution related to the composite beam
ultimate load is plotted relating force ratio to stud position in
Fig. 43. It can be seen that the variation of the stud force ratio
of all curves is approximately 10%. Moreover, if the 136% levelis not considered, this variation is only 5%. This means that, at
Fig. 43. Ratio of stud forces vs. relative position of the stud.
Fig. 44. Load-deflection curve UDL.
Fig. 45. Load-deflection curve UDL Initial stiffness.
ultimate load and regardless of the level of shear connection, the
connectors were able to redistribute the load almost uniformly
among them.
The results demonstrate that, therefore, the effects of partial
interaction, which are increased by the use of partial shearconnection, can be neglected for levels of shear connection
above 100%, as no significant improvement in terms of either
strength or stiffness of the beam was observed.
5.2. Uniformly distributed load case
The loadmidspan deflection curves for all cases are shown
inFig. 44. It can be observed that, as occurred for the midspan
load case, decreasing the level of shear connection makes
the system become more flexible, with reduced strength and
stiffness. Once more, the decrease in stiffness did not seem to
be as significant as the decrease in the ultimate load (Fig. 45).
The two higher levels (130% and 136%) resulted in verysimilar curves, in terms of both stiffness and ultimate load,
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Fig. 46. Ratio of stud forces vs. relative position of the stud.
represented by the lower and upper bound values, with the
mode of failure being slab crushing at the midspan of the
beam (location of maximum bending moment). Regarding level
118%, its mode of failure could be either slab crushing or studfailure, as a stud failure point was obtained between the lower
and upper bounds. As the other four levels (47%, 71%, 89%
and 100%) resulted in stud failure, the level 118% was an
intermediate level between the two possible modes of failure.
The ratio between the ultimate load and the load at the end
of the elastic behaviour was similar to the one observed for the
midspan point load case (from 1.5 to 2 for all cases).
The stud force distribution graph (Fig. 46) is plotted relating
force ratio to stud position. It can be noticed that the connectors
were able to plastify and redistribute the load almost uniformly
among them for a length of approximately 30% of the span
from each of the supports (0 X/L 0.3). In the central
region of the beam (0.3 X/L 0.5), the studs were lessheavily loaded. This behaviour occurred for all levels of shear
connection, except the two highest ones (130% and 136%), for
which the length of the central region increased by 10%.
The results show that, for the UDL case, the effects of
partial shear connection can be important for levels of shear
connection above 100%. It was observed that the connectors
were not loaded uniformly along the beam length (Fig. 46), with
the stud forces decreasing towards the midspan of the beam.
In addition, this behaviour is augmented by local deformations
of the concrete slab. Consequently, more shear connectors
are needed in order to obtain a total interaction level, for
which the effects of partial interaction may be neglected.
In spite of the fact that current design codes assume thesame strength for all shear connectors, as well as a uniform
distribution of load among them for the ultimate limit state,
an 18% difference between the total-interaction levels for the
concentrated load (100%) and UDL (118%) cases was obtained.
Nevertheless, this difference is not sufficiently significant to
propose modifications to well accepted design methods.
6. Conclusions
A three-dimensional finite element model of composite
beams is proposed based on the use of the commercial
software ANSYS. It has proved to be effective in terms of
predicting the loaddeflection response for beams subjected to
concentrated or uniformly distributed loads, longitudinal slipat the steelconcrete interface, shear force carried by the studs
and the mode of failure (stud failure or concrete crushing). It is
also able to investigate beams with either full or partial shear
connection. Comparisons against experimental results and
against alternative numerical analyses indicate that the presentmodel can be used to perform extensive parametric studies.
Based on the use of this model, it was shown that
the continuation of the shear connection beyond the beam
supports of simply supported beams can affect not only the
overall system response, but also the slip and the stud force
distributions along the beam.
Results from a parametric analysis, designed to investigate
the overall structural behaviour of composite beams when
different concrete compressive strengths are used in the slab
and in the associated push-out tests, revealed that the slab
concrete strength (for a fixed concrete strength for the push-out
specimen) can have an influence not only on the overall stiffness
of the system and on its ultimate moment capacity, but also onthe stud forces. In addition, for a fixed slab concrete strength,
it was observed that the concrete strength of push-out tests
appears to have a small influence on both the overall behaviour
of the composite beam and the stud force distribution.A sensitivity study was also undertaken, focused on the
assessment of the influence of small variations in key input
parameters (i.e., concrete and steel material properties) on the
overall structural behaviour of composite beams. For the range
of material properties considered, it was noticed that the web
yield stress was the main structural parameter influencing the
definition of the overall shape of the loaddeflection curve of
the composite beams analysed. In addition, it could be observed
that the slab concrete strength can affect the mode of failure ofthe composite beam, confirming the outcomes obtained in the
previous parametric study.Finally, a study was carried out on the effects of partial
shear connection/partial interaction. It was demonstrated that,
by decreasing the level of shear connection, the composite sys-
tem becomes more flexible, with reduced strength and stiffness,
mainly for beams with less than 100%, for which the partial in-
teraction effects are significant and must be taken into account.The proposed three-dimensional model provides the
opportunity to develop insights that would be virtually
impossible using experimental tests, due to costs and,
especially, the dispersion of material properties that inevitably
occurs in laboratory work. For instance, in the investigation
of the influence of the level of shear connection, the concrete
strength effect could be easily disregarded by the consideration
of a fixed value for this property. Despite the fact that the
proposed three-dimensional model is able to accurately provide
a wide range of results, including the detection of local
aspects of behaviour (e.g., local deformations of the concrete
slab), a two-dimensional model could be the solution for
more complex structural systems (e.g., beams with different
degrees of continuity), due to numerical convergence aspects
and processing times.
Acknowledgment
The authors would like to acknowledge the support providedby the Brazilian Foundation CAPES.
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