Concrete flat slab floors provide an elegant form of construction, which simplifies and speeds up site operations, allows easy and flexible partition of space and reduces the overall height of buildings. However, flat slab construction is not ideal from the structural point of view, because of the high shear stresses around supporting columns, which can lead to abrupt punching shear failure at a load which is well below its flexural strength. The problem with this failure mode is that it is brittle and catastrophic due to the inability of the concrete to support the large tensile stresses that developed. This failure occurs with the potential diagonal crack following the surface of a truncated cone or pyramid around the column. The failure surface extends from the bottom of the slab, at the support, diagonally upward to the top surface. The angle of inclination with the horizontal depends upon the nature and amount of reinforcement in the slab. It may range between about 20 and 45° (Nilson et al., 2003). It is obvious, that in design, attention should be given to both strength and ductility when punching shear is being considered.
Although there is a great amount of experimental data available on punching shear failure of slabs, there has been little success in developing a universal theoretical model to predict the behavior of an arbitrary concrete structure (Vidosa et al., 1991). The American Society of Civil Engineers had presented an extensive review of the underlying theory and the application of the finite element method to the linear and nonlinear analysis of reinforced concrete structures in a state-of-the-art report (ASCE, 2002). It was stated that the results from the finite element analysis significantly relies on the stress-strain relationship of the materials, failure criteria chosen, simulation of the crack of concrete and the interaction of the reinforcement and concrete. The nonlinear stress distribution in the connection is not clearly understood. Several previous attempts (Malvar, 1992; Marzouk and Chen, 1993; Marzouk and Jiiang, 1996; Polak, 1998) have been focused on prediction of the slabs failure in punching shear using various finite element models. The models presented in this paper, were developed specifically to model the behavior of the experimental specimen.
The trend towards lighter and more flexible construction configurations led
to increased usage of flat plate construction in 1950s. Reinforced concrete
flat plate construction has been used as an economical structural system for
many buildings. In high seismic regions, flat plate structures are supplemented
with either a moment frame or shear wall lateral resisting system. Today, ductile
detailing for all structural connections, including for those which are gravity
load only, is a principal concept that was learned initially as a result of
the failures observed during the 1971 San Fernando earthquake (FEMA, 1997).
All the gravity load only slab column connections in a flat plate structure
must maintain their capacity at the maximum displacement of the lateral system.
During this lateral deformation, the brittle failure mode of slab punching can
occur. A punching shear failure, generated by the combination of the gravity
loading and seismically induced unbalanced moment in the slab, can occur with
little or no warning and has resulted in the progressive collapse of these types
of structures. In the 1985 Mexico City earthquake, 91 flat plate buildings collapsed
and 44 were severely damaged due to punching failure (Ghali and Megally, 2000).
Prior to the ACI (American Concrete Institute) code revisions in the 1970s,
which began to reflect ductile detailing lessons learned, reinforcing for flat
slab systems did not require continuity of top and bottom reinforcement. Top
reinforcing, used for negative bending, could be completely curtailed away from
the column supports. Bottom reinforcing was only required to extend into supports
by 150 mm (6 inches). It is now well known that positive bending can occur at
the face of supports during lateral displacements inducing a bond failure at
these short embedment locations. Due to the inadequacies of the pre-1971 design
codes, there is a need to understand the behavior of the structures designed
to these codes and to develop ways to upgrade these structures to provide an
acceptable level of seismic safety.
Previous studies have been performed on repair techniques for damaged ductile slab-column connections using steel plates and through-bolting (Farhey et al., 1995). This technique could also be used to upgrade the performance of non-ductile connections, however, the aesthetic impact to the existing structure is a potential drawback.
In recent years, use of fiber reinforced polymer composites for concrete structures has been on the rise. Despite the relatively high cost of the CFRP material, its high strength-weight ratio, high resistance to corrosion and easy handling and installation have made CFRP the material of choice in an increasingly large number of projects where increased strength or inelastic deformation capacity (ductility), or both, must be achieved for seismic retrofitting (Triantafillou, 2003).
The bulk of the work, presented in this study, focuses on numerical models developed to predict the response of the slab including the load-deflection behavior, the ultimate failure load and the crack pattern; this preliminary modeling is necessary to be considered as the basis of the main study including strengthening of slabs by using CFRP composites. Three-dimensional nonlinear finite element models were created to simulate the behavior of the flat slab by using the computer code ANSYS (ANSYS, 2004). The models are based on an eight-node isoparametric element. A parametric study was carried out to look at the variables that can mainly affect the behavior of the models.
Construction and verification of finite element static model
Experimental specimen: Internal connections having shapes of square, rectangular or circular specimens with edges simply supported and free to lift and centrally loaded through a column stub raise questions in the following respects: 1- The boundary conditions in the model are different to those of the actual structure, 2- The redistribution of moments around the column in the test specimen is different from a real structure (Desayi and Seshadri, 1997). However, work carried out by Gardner and Shao (1996) demonstrated that the isolated punching shear tests can represent the punching shear behavior of interior slab-column connections in continuous slab systems.
Slab specimen P0A tested by Kruger et al. (1998) was selected for the
numerical analysis (Fig. 1). The experimental slab-column connection
was full scale model. The boundaries of the specimen were selected at the position
of the line of contraflexure so as to contain the hogging moment region around
the column. The specimen support was considered as a simple support during the
test on knife edges fixed on steel beams so that the edges are free to lift.
The test set-up and loading arrangement are shown in Fig. 1.
The specimen was 3000x3000x150 mm3. The cylinder compressive strength
of the concrete f`c was 35 MPa. The mean ratio in each direction
of the flexural reinforcement ρ was 1.0% for the slab. The specimen was
inversely placed. The testing arrangement was convenient for both loading and
inspecting the slab for cracking during the test. The vertical load was applied
centrally through the column stub, with the slab specimen simply supported along
four edges to simulate an inverted isolated slab-column connection. The load
was applied monotonically using a hydraulic jack set by deflection control.
Readings were taken at 40 kN intervals.
||Test set-up and specimen (dimensions in mm)
||Load versus maximum vertical slab deflection
Figure 2 shows the maximum slab deflection versus the load
applied by the hydraulic jack. The specimen behaved linearly from the initial
loading up to the occurrence of the first crack. The first crack appears between
the second and the third load step at approximately 100 kN. The slab on reaching
the peak load (423 kN) failed in a brittle manner with a sudden loss of capacity
by punching, indicating very low ductility and residual strength. Failure for
this slab can be defined as punching shear failure on account of the fact that
the load quickly dropped when failure was reached as shown in Fig.
Finite element modeling of slab-column connections: In previous years some numerical tests were carried out applying the finite element method. Among these, studies with two-dimensionally modeled systems with rotationally symmetrical elements (Hallgren, 1996; Menétrey, 1994) must be distinguished from the purely three dimensional, idealized systems. A separate position is assumed by tests applying degenerated shell elements which are placed in layers (Polak, 1998). The three-dimensional idealization offers the option of defining and modeling various characteristics in any direction and the three-dimensional model is also highly suitable for observing the formation of punching cracks and their position in the slab.
In this study, it was decided to focus on modeling both the load-deflection
characteristics of the slabs and also the initial onset of cracking. The experimental
data that was obtained in a comprehensive study (Kruger et al., 1998),
described above, was used as a comparison with the numerical results. The ANSYS
finite element package was used to carry out the modeling and the Solid 65 element
used to model the concrete. The Solid 65 element is a three-dimensional isoparametric
element, capable of cracking in tension and crushing in compression. The ANSYS
element (ANSYS, 2004) is defined by eight nodal points each having three translational
degrees of freedom x, y and z (and no rotations), along with a 2x2x2 Gaussian
integration scheme which is used for the computation of the element stiffness
matrix. The model is also capable of simulating the interaction between the
two constituents, concrete and reinforcement. Thus, it can be used to describe
the behavior of the reinforced concrete material. The element has one solid
material and up to three different reinforcing bars material properties can
be defined. The most important feature of this element is that it can represent
both linear and nonlinear behavior of the concrete. For the linear stage, the
concrete is assumed to be an isotropic material up to cracking. For the nonlinear
part, the concrete may undergo plasticity and/or creep.
||Numerical model (dimensions in mm)
The load is iterated step by step using the Newton-Raphson method. As was already
stated, the slabs were simply supported along four sides, as shown in Fig.
3. Therefore the boundary conditions are quite simple to set-up. One corner
had all translational degrees of freedom fixed, while one corner was fixed with
two degrees of freedom so as to prevent the slab from moving and rotating in
its own plane. The edges of the slab were not restrained as in the experimental
set-up (Fig. 3).
In the static analysis, the stress-strain relations of concrete are modified to represent the presence of a crack. A plane of weakness in a direction normal to the crack face and a shear transfer coefficient βt (represents conditions of the crack face) are introduced in the Solid65 element. The shear strength reduction for those subsequent loads, which induce shear across the crack surface, is considered by defining the value of βt. This is important to accurately predict the loading after cracking, especially when calculating the strength of concrete member dominated by shear, such as slabs. The value βt ranges from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer) (ANSYS, 2004). The value of βt used in many studies of reinforced concrete structures, however, varied between 0.05 and 0.25 (Bangash, 1989; Hemmaty, 1998; Huyse et al., 1994). A number of preliminary analyses were attempted in this study with various values for the shear transfer coefficient within this range, but convergence problems were encountered at low loads with βt less than 0.2. Therefore, the shear transfer coefficient used in this study was equal to 0.2.
The density γ and the Poisson`s ratio v of concrete were assumed as 2400 kg m-3 and 0.2. The shear transfer coefficient for a closed crack βc was taken as 0.9. The ultimate uniaxial compressive strength f`c introduced to the program was based on the experimental results (Kruger et al., 1998).
Newton-Raphson equilibrium iterations were used for nonlinear analysis. A displacement controlled incremental loading was applied through a column stub. This was used to simulate the actual loading used in the experimental program. Small initial load steps were used for detecting the first crack in the connections. Then, automatic time stepping was used to control the load step sizes. Line search and the predictor-corrector methods were also used in the nonlinear analysis for accelerating the convergence. The failure of the connection was defined when the solution for a small displacement increment did not converge. Consequently, the finite element model was constructed following the above-mentioned assumptions and considerations.
The main focus of these analyses is to model the load-deflection behavior of
the specimen P0A that was given above. The numerical models were developed in
two different groups. One of these groups consisted of three layers of elements
in the depth of the slab with smeared reinforcement throughout (Fig.
4) and about the other, just for the bottom layer, smeared reinforcement
defined (Fig. 5).
As stated already, a parametric study is presented in the following sections.
This study was carried out to look at the effect of the volume of smeared reinforcement
on the behavior of the slab. It is possible to establish the relationship between
the volume ratio and section area of the reinforcement. However, the problem
with specifying a volume of smeared reinforcement in a three-dimensional analysis
is that, it is very difficult to relate the volume directly to the cross-sectional
area of reinforcement present in the experimental specimen. Calculating the
volume ratio to the section area ratio of the reinforcement for the shapes of
solid elements generated is difficult. If the mesh under consideration is uniform
with cubic elements throughout this relation is simple, the cross-sectional
area being exactly equal to the volume specified.
||All layers with smeared reinforcement
Bottom layer with smeared reinforcement
Uniform mesh with cubic elements was used in preliminary analyses but the results
didn`t have accuracy. In the finite element of concrete structures, it is important
to select an appropriate mesh size to meet the requirement of accuracy and computation
speed; however, considering the mesh layout used in the present models was the
same and is shown above in Fig. 3. The density of the mesh
was increased under the area of loading and gradually reduced towards the edges
of the slab. Therefore, as the mesh was not uniform, it was necessary to carry
out a parametric study and look at the effect of the volume of smeared reinforcement
present. This problem does not occur in the discrete reinforcement modeling
currently being conducted.
Verification of the finite element model against experimental data
Group I: All layers with smeared reinforcement: This group of models
consisted of three layers of elements in the depth with smeared reinforcement
throughout the slab (Fig. 4). This group was investigated to
look at the influence of the volume ratio of smeared reinforcement. The problem
with using the smeared reinforcement option was that it tended to stiffen the
element. Therefore, it was decided to carry out a parametric study to look at
the effect of different values of the steel volume ratio.
Looking at the curves for the numerical models, the first thing one should
note is that the behavior of all the models is exactly the same before cracking
and the difference in behavior only occurs after initial cracking. Cracking
of the models seems to occur at the same load as the experimental specimen,
approximately 100 kN. If the curve for the first numerical model, SVR = 0.045
(SVR = Steel Volume Ratio), is studied, one can see the effects of a high volume
of reinforcement on the behavior of the element. The element is quite stiff
and therefore, there is significantly less deflection than the experimental
specimen. The model also fails at a much greater load than the experimental
specimen. As the volume ratio of reinforcement is reduced, the approximation
of the load-deflection curve obtained using the models is improved. From the
load-deflection curves below (Fig. 6), it seems that a steel
volume ratio of 0.028 is the optimum value to achieve a good approximation of
the experimental curve.
Group II: Bottom layer with smeared reinforcement: In this set of numerical
models, the slab was divided into two parts: 1- a bottom layer consisting of
Solid 65 elements with smeared reinforcement and 2- a top and middle layer consisting
of Solid 65 elements without any smeared reinforcement (Fig. 5).
The load-deflection curves for this set of models show a similar trend as obtained
before (Fig. 7). The element is still quite stiff for higher
values of smeared reinforcement. The best load-deflection curve is obtained
at a value of 0.041. Cracking occurs in all the models at the same load and
at a value slightly higher than previous models, 110 kN approximately.
Comparison of load-deflection curves: The load-deflection curves for
the best model taken from each of the groups are compared below (Fig.
8). On examination of the plots, it is obvious that the best approximation
is given by group I. For these slabs high volume ratios, such as 0.045, increase
the stiffness of the elements considerably and an optimum volume ratio of 0.028
was found to give the best approximations of the load-deflection curve for the
experimental specimen. It should be noted that this value was approximately
three times more than the percentage of main steel in the experimental specimen,
which indicates the influence of geometry of volume mesh. It was noticed that
for similar values of the reinforcement volume ratio in the bottom layer, the
deflection of the slabs decreased and the final load supported by the slab also
decreased. Similar results were noticed in the case of group I, slabs, where
the load-deflection curves for the models are not as good approximations as
those obtained using group I models.
||Load-deflection curves for slabs of group I (SVR= Steel Volume
||Load-deflection curves for slabs of group II (SVR= Steel Volume
||Comparison of load-deflection curves for experimental and
the best analytical models (SVR= Steel Volume Ratio)
Cracking pattern: Cracking may take place up to three orthogonal directions
at each integration point. A crack may develop in one plane and if subsequent
stresses tangential to the crack face are large enough, a second (or 3rd) crack
will develop. An individual crack is treated as a smeared band of cracks. Cracks
are defined by a combination of an angle θ in the xy plane and φ in
the three dimensional space. The main advantages of the smeared crack approach
compared to the discrete crack formulation is that 1- no predetermination of
the location and orientation of the crack planes are necessary and 2- the original
topology of the finite element mesh is maintained during cracking (Dyngeland
et al., 1994). This can provide the useful information for crack modeling
when discrete crack model is chosen at the later research program. Crushing
failure assumes complete deterioration of the structural integrity of the concrete,
such as spelling.
The smeared crack indicates a crack pattern similar to the experimental specimen.
The first cracks opened up on the lower surface of inverted model in the form
of flexural cracks near the column around 100 kN. After 100 kN, cracks propagated
from the middle outwards and reached the slab edges. Subsequently, with increasing
loads, more cracks occurred and advanced radially from the column faces towards
the slab edges along the four axes of symmetry of the slab (central X and Y
axes and two diagonals). Cracks parallel to the X and Y axes opened up at a
lower load, less than 240 kN (Fig. 9, 10),
whilst the cracks parallel to diagonal axes opened up at a load greater than
240 kN (Fig. 11). Figure 9, 10
and 11 shows crack pattern of modeling at first loading
increments in comparison with the figure given by Ajdukiewice and Starosolski
By the time the load increment reached approximately 320 kN, only a few new
diagonal cracks occurred and most of the cracks had already formed. After that,
with increasing load, the width of the cracks close to the column increased
substantially. In the period just prior to failure, the smeared cracks were
formed from the connection area edges at upper surface of inverted model to
the corners of the slab on the lower surface (Fig. 12). Also
a circular stress concentration was formed around the loaded area indicating
a punching shear type failure. The punching truncated pyramid is in a very close
agreement with the experiment. Figure 12 shows a crack pattern
of the model at a period just prior to ultimate load with elimination of radial
cracks in comparison with the figure given by Ajdukiewice and Starosolski (1990).
Slab strengthening with increasing steel volume ratio in the central zone:
The parametric study is carried out on the already observed control model of
the slab. In this context the influence of strengthening the slab in the central
zone with increasing steel volume ratio is investigated. In this case, the steel
volume ratio in the central zone of the model increased up to 30 % compared
to the control model. The steel volume ratio for central zone of strengthened
model was 0.0364 and for peripheral zone was 0.028. Figure 13
shows the strengthened model curve against the control model curve and slab
model curve with steel volume ratio of 0.035. On examination of the plots, it
is obvious that the curve slope for the strengthened model is higher than the
other curves and the deflection of the strengthened slab model increased and
the final load supported by the strengthened slab also increased. The curves
indicate that increasing the steel volume ratio in the central zone of the slab
improves the behavior of the slabs.
Slab strengthening by CFRP
Carbon fiber reinforced polymers: Several techniques are currently available to retrofit and strengthen buildings with insufficient stiffness, strength and/or ductility. While these techniques are effective in improving the earthquake resistance of a building, they may add significant weight to the structure and thus alter the magnitude and distribution of the seismic loads. Also, the existing techniques are generally very labor intensive. Fiber reinforced polymers are composite materials consisting of high strength fibers immersed in a polymer matrix. The fibers in an FRP (Fiber Reinforced Polymer) composite are the main load-carrying element and exhibit very high strength and stiffness when pulled in tension. An FRP laminate will typically consists of several million of these thin, thread-like fibers. The polymer matrix protects the fibers from damage, ensures that the fibers remain aligned and allows loads to be distributed among many of the individual fibers in the composite (Kheyroddin and Naderpour, 2006).
Finite element modeling of strengthened connection: Here, the effectiveness
of slab strengthening by CFRP in the central zone would be investigated. A layered
solid element, Solid 46, was used to model the FRP composites. The element allows
for up to 100 different material layers with different orientations and orthotropic
material properties in each layer. The element has three degrees of freedom
at each node and translations in the nodal x, y and z directions. Nodes of the
FRP layered solid elements were connected to those of adjacent concrete solid
elements in order to satisfy the perfect bond assumption. The material properties
for FRP composites are available at Table 1. By using Solid
46 element, one layer of CFRP with the thickness of 2 mm in two sides of base
model in central zone was defined (Fig. 14).
||Crack pattern at first loading increments
||Crack pattern at first loading increments
||Crack pattern at middle loading increments
||Crack pattern at period just prior to ultimate load with eliminating
||Comparison of load-deflection curves (SVR= Steel Volume Ratio)
The various thicknesses of the FRP composites create discontinuities, which
are not desirable for the finite element analysis. These may develop high stress
concentrations at local areas on the models; consequently, when the model is
run, the solution may have difficulties in convergence. Therefore, a consistent
overall thickness of FRP composite was used in the models to avoid discontinuities.
The equivalent overall stiffness of the FRP materials was maintained by making
compensating changes in the elastic and shear moduli assigned to each FRP layer.
For example, if the thickness of an FRP laminate was artificially doubled to
maintain a constant overall thickness, the elastic and shear moduli in that
material were reduced by 50% to compensate. Note that the relationship between
elastic and shear moduli is linear (ANSYS, 2004).
||Summary of material properties for CFRP composites
||Slab strengthening model in central zone by using Solid 46
||Comparison of load-deflection curves between Control Model
(CM) and Strengthened Slab Model with CFRP (SSMCFRP)
Figure 15 shows the load-deflection curve of the strengthened
model using CFRP in comparison with the same curve of control model. According
to the curves, it could be derived that curve slope for strengthened model using
CFRP is lower than the other curve. Also, the deflection of strengthened model
has been increased by 36 percent; while the slab strength has been increased
in a low rate. The deflection of strengthened model was 34 mm and the ultimate
load reached to 456 kN; while the deflection of basic model was 25 mm and the
ultimate load had reached to 449 kN. By considering that in order to increase
the ductility and shear strength, it is necessary that the lines of shear cracks
should be cut by shear reinforcement, it seems that by applying the shear studs
made of CFRP in the central zone between the CFRP plates, the shear strength
and ductility could be increased properly.
From the discussion of results obtained by testing and numerical modeling,
the following conclusions can be drawn:
||The numerical investigations provided good agreement between
the predicted and the available test results of the ultimate load and associated
||It is important to select an appropriate mesh size to meet the requirement
of accuracy and computation speed.
||It is necessary to carry out a parametric study and look at the effect
of the volume of smeared reinforcement, if the mesh is not uniform. This
problem does not occur in the discrete reinforcement modeling.
||The slab model of group I which consisted of the smeared reinforcement
throughout the entire slab provides the closest results compared with the
||Although the smeared models do give accurate approximations, their behavior
is slightly different. The precracking branch of the different curves follows
the experimental results very closely. Beyond cracking, the models of group
II appear stiffer.
||The smeared crack indicated a crack pattern similar to the experimental
||3D finite element model simulated punching shear behavior of the experimental
||The deflection of strengthened model by using CFRP has been increased
by 36%; while the slab strength has been increased in a low rate. The deflection
of strengthened model was 34 mm and the ultimate load reached to 456 kN;
while the deflection of basic model was 25 mm and the ultimate load had
reached to 449 kN.
||Considering the cracking pattern, it seems that by applying the shear
studs made of CFRP in the central zone between the CFRP plates, the shear
strength and ductility could be increased properly.
||Strengthened model with increasing steel volume ratio in the central zone
increased both, the final load and deflection.