3D finite element investigation of the compressive membrane action effect in reinforced concrete flat slabs

Engineering Structures 136 (2017) 233–244

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3D finite element investigation of the compressive membrane action effect in reinforced concrete flat slabs Aikaterini S. Genikomsou ⇑, Maria Anna Polak Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada

a r t i c l e

i n f o

Article history: Received 24 September 2015 Revised 22 November 2016 Accepted 10 January 2017

Keywords: Membrane action Concrete slabs Punching shear Crack pattern Finite element analysis

a b s t r a c t Design codes for punching shear resistance of flat slabs are based on test results obtained from isolated slab-column connections. However, by testing isolated slabs, the compressive membrane action of a continuous slab-column system is ignored. This can result in lower punching shear strength compared to the actual strength of the real slab system. Testing continuous slab system is very uneconomical and in most cases not possible. In this paper, finite element analyses (FEA) are performed in order to investigate the effect of the compressive membrane action in flat concrete slabs by comparing results from isolated specimens and continuous floor systems. The adopted FE formulation and the material parameters were previously calibrated on an isolated test specimen (SB1). At first, the calibrated FEA are used to investigate the boundary conditions of slab-column connection (SB1). Then, these boundary conditions are modified, and the slab is also considered to have larger in-plane dimensions in order to examine its continuity. Finally, numerical analyses of existing punching shear tests that examine the compressive membrane action effect are conducted to show the accuracy of the FEA model. All the numerical analyses indicate that the punching shear capacity of a continuous slab is higher compared to the capacity of a conventional isolated slab. The predictive capability of the FEA models can allow for future investigations on the effect of the compressive membrane action to supplement the limited testing background on this area, and to provide recommendations for future design provisions. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Punching shear failure in reinforced concrete flat slabs occurs due to the development of a three dimensional state of stresses that is created by the high transverse stresses around the column and the in-plane stresses. Inclined cracks are created inside the slab, which then propagate and form a major inclined crack. When this crack reaches the compressive zone, a punching shear cone around the column is formed leading to punching shear failure. Punching failure is brittle and sometimes can lead to a progressive collapse of the building. Many researchers performed studies to examine the punching shear failure in concrete slabs and developed methods to prevent it. Several tests have been done starting in 1950s and many theories and models have been proposed [1–3]. In these experiments, isolated slabs were considered, representing a slab-column connection limited by the line of contra-flexure for radial moments, which become zero at a distance approximately 0.22L, where L is the center-to-center span between the columns. ⇑ Corresponding author. E-mail addresses: [email protected] (A.S. Genikomsou), polak@uwaterloo. ca (M.A. Polak). http://dx.doi.org/10.1016/j.engstruct.2017.01.024 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

All these isolated tested slabs had no restraint for lateral in-plane movement and they were simply supported around the edges. These test results were the basis of design codes’ design methods for punching shear resistance of flat slabs. However, the in-plane restraining forces creating the compressive membrane action in the concrete slabs, were ignored. In continuous reinforced concrete slabs, the tensile strains at the mid-depth of a slab lead to an expansion of the slab, creating horizontal displacements. These mid-depth tensile strains are the result of concrete material nonlinearity. However, the lateral stiffness of the columns opposes this expansion by imposing compressive membrane forces (in-plane restraining forces) (see Fig. 1). The effect of this phenomenon, that is called compressive membrane action, is the increase of the flexural and the shear capacity of a slab. The first investigations on the effect of the membrane action in concrete slabs can be found in the observations done by Westergaard and Slater [4]. Later, other researchers performed tests in order to explain and examine the effect of the membrane action connected with the boundary conditions on the concrete slab’s capacity. These include work done by Ockelston [5], Elstner and Hognestad [6], Christiansen [7], Park [8], Long and Bond [9], Hop-

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Nomenclature As C 01 C 10 D1 Ec G Gf I1 I2 J el Ko Kc L Uð
Þ d 0 fc

cross-section area of reinforcement temperature-dependent parameter of neoprene temperature-dependent parameter of neoprene temperature-dependent parameter of neoprene modulus of elasticity of concrete shear modulus of neoprene fracture energy of concrete first deviatoric strain invariant of neoprene second deviatoric strain invariant of neoprene elastic volume ratio of neoprene initial bulk modulus of neoprene ratio of the second stress invariant on the tensile meridian to that on compressive meridian for concrete center-to-center span between the columns strain energy potential for neoprene effective depth of the slab compressive strength of concrete

Fig. 1. Membrane action phenomenon.

kins and Park [10], Lander et al. [11], Long et al. [12], Rankin and Long [13], Guice and Rhomberg [14], Vecchio and Collins [15], Vecchio and Tang [16], Chana and Desai [17], Alexander and Simmonds [18], Sherif [19], Choi and Kim [20]. Elstner and Hognestad [6] conducted tests by changing the boundary conditions of the slabs. They considered: (a) all edges to be simply supported; (b) only two opposite edges to be simply supported and (c) only the corners to be simply supported. The test results indicated a significant strength reduction in such cases where the edges were not continuously supported. Long and Bond [9] performed full panel tests showing an increase in the capacity of the slabs compared to the isolated test specimens. Rankin and Long [13] tested 17 specimens that ranged from the isolated specimens having their edges at the line of contra-flexure (0.22L) to the full panel specimens. All specimens were simply supported at the line of the contra-flexure. The test results showed an increased ultimate load with an increase of the slabs’ size. The lowest increase was noticed to be around 30% for full panels with reinforcement ratio 1.1% and around 50% for full panels with reinforcement ratio 0.5%. Other researchers tested slab subsystems in order to investigate the membrane action effect on slab’s capacity. Vecchio and Collins [15] examined the collapse of a four-storey warehouse building with flat slabs that happened in 1978. When the collapse took place, Vecchio and Collins found that the total load of the third floor of the building was about 4.5 times higher compared to the design load. The investigation’s results showed that high strength was created due to the effect of the membrane action. Vecchio and Tang [16] tested two slab strip specimens to

ft 0 ft fy k lc

e et ey l lo

v

qs rbo rco

w

ultimate strength of reinforcement tensile strength of concrete yield strength of reinforcement stiffness of spring elements characteristic length of element eccentricity of concrete ultimate strain of reinforcement yield strain of reinforcement viscosity parameter of concrete initial shear modulus of neoprene Poisson’s ratio reinforcement ratio initial equibiaxial compressive yield stress of concrete initial uniaxial compressive yield stress of concrete dilation angle of concrete

isolate the influence of the membrane action. The two specimens were different only in the support conditions; in the first specimen the end supports were allowed to move only horizontally, in the second specimen the edge supports were fixed. The second specimen (fixed supports) failed at a higher load compared to the first specimen. Alexander and Simmonds [18] examined three different types of boundary conditions for the slabs: (a) rotations at the edges and restrained corners, (b) rotations and restrained edges and (c) restrained rotations at the corners. The obtained test results showed that the rotational restraint increased the punching shear capacity of the slabs. Chana and Desai [17] tested isolated and continuous slabs. The side lengths of the slabs were equal to 0.4L and 1.5L for the isolated and continuous slab, respectively. The test results indicated a significant increase in the ultimate load of the continuous slab, around 52%, compared to the strength of the conventional specimen. In addition to the test programs described above, Einpaul et al. [21] presented a numerical method to calculate the capacity of continuous slabs. Comparing the results coming from their model to specific test results, they concluded that the membrane action effect increases the strength capacity of a slab. Numerical analyses of two-way slabs using finite element methods examining the membrane action effects is limited. Finite element analysis using shell elements to simulate the concrete in order to investigate the compressive membrane action against the progressive collapse of the flat slabs can be found in [22–24]. In these works, the compressive membrane action was found to be important source of enhancement for the punching shear strength of the slabs preventing a progressive collapse of the slab structures. In this paper, three dimensional nonlinear finite element analyses (FEA) are presented that are applied to simulate continuous slabs and then to compare their behavior with the isolated slab test results. FEA can be used effectively, after appropriate modeling and material calibration, to supplement the existing testing background. Parametric investigations can be performed via FEA modeling, exploring a variety of issues related to punching shear. In this work, the ABAQUS software [25] is used with the simulation of the concrete material done using the concrete damaged plasticity model. The concrete damaged plasticity model has been previously calibrated on the interior concrete slab (SB1) in order to examine its punching shear failure [26]. Herein, the calibrated model is used to investigate firstly, the boundary conditions of the isolated interior slab-column connection (SB1). Then, the isolated specimen

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SB1 is analyzed as continuous; using different boundary conditions and also different dimensions. The analysis of the whole floor system, from which the SB1 specimen has been taken, is also simulated and analyzed. Following this, two test specimens from literature [17], in which the membrane action effect was examined, are presented and analyzed to show and prove the effectiveness and the accuracy of the FEA model in continuous slabs modeling. FEA results of all these slabs are compared to the test results, and the discussion is provided. Finally, discussion based on the code provisions for punching shear and comparison with the numerical results of the continuous slabs is presented.

2. Finite element investigation of the boundary conditions of an isolated tested slab (SB1) SB1 specimen is an interior slab-column connection without shear reinforcement that has been tested at the University of Waterloo under static loading through the column [27]. It is an isolated specimen taken from a structure with four spans equal to 3.75 m between the columns in both directions. The dimensions of the tested isolated specimen were 1800  1800  120 mm with the square column (150  150 mm) to extend 150 mm at the top and the bottom of the slab. The effective depth of the slab was 90 mm. The tension flexural reinforcement consisted of 10 M bars placed at distances 100 mm and 90 mm in orthogonal directions (to allow for same reinforcement ratio in both directions). The compressive flexural reinforcement coinsisted of 10 M bars at 200 mm in both directions. During the test, simple supports were applied at the locations of the lines of the contraflexure (1500  1500 mm). Neoprene pads were installed along the supporting lines. The schematic drawing of the tested slab is shown in Fig. 2. SB1 specimen failed in punching shear. Details regarding the material properties, reinforcement and test results are presented in Table 1. The isolated slab SB1 was previously analyzed using FEA in the ABAQUS software by Genikomsou and Polak [26], where the investigation and the calibration of the adopted concrete damaged plasticity model in the FEA was performed. The material and model parameters can be found in [26]. Fig. 3 illustrates the compressive behavior of concrete for the slab SB1, as it was modeled in ABAQUS using the Hognestad type parabola. The tensile stress-strain relation of concrete is presented in Fig. 4. The bilinear tension softening response in terms of stress-displacement was defined according to [28] based on the fracture energy of concrete ðGf Þ that represents the area under the stress-displacement curve. The fracture energy was obtained from the CEB-FIP Model Code 90 [29] with maximum concrete aggregate size equal to 10 mm. The tension response of concrete was defined in terms of stress-strain by divided the displacement by the characteristic length of the element ðlc Þ. Table 2 presents all the needed material and plasticity

235

parameters of the concrete damaged plasticity model for the numerical simulations. The reinforcement was modeled with a bilinear strain hardening yield stress-strain curve as it is shown in Fig. 5. The Young’s modulus is considered 200,000 MPa and the Poisson’s ratio 0.3. Three dimensional 8-noded hexahedral elements with reduced integration (C3D8R) were used for modeling the concrete with 20 mm mesh size. For modeling the flexural reinforcement, three dimensional 2-noded linear truss (T3D2) elements were used and the embedded method was considered simulating the concretereinforcement interaction. Due to the symmetry, one quarter of the tested slab was simulated and simple supports (constraining only the loading direction) were applied in the numerical model. Quasi-static analysis in ABAQUS/Explicit performed by introducing velocity through the column stub till failure. The FEA results compared to the test results, were found to be in good agreement [26]. In the previous study, the numerical boundary conditions of the isolated interior slab-column connection SB1, were considered as simple and no further investigation was done. Herein, before analyzing the slab as continuous by modifying its boundary conditions, a more realistic approach to simulate the real supports of the slab SB1 is presented. Neoprene is simulated around the bottom edges of the slab using C3B8R elements, as it was installed in the real test. Neoprene is a hyperelastic material and it can be considered as isotropic and nonlinear. Neoprene exhibits instantaneous elastic response up to large strains and as most elastomers, it has low compressive strength compared to its shear capacity. In the FEA, when the elastomer materials are modeled with threedimensional solid elements, the numerical solution can be sensitive to the compressive behavior of the material. The Poisson’s ratio for the hyperelastic materials is close to 0.5. Hyperelastic materials are described in terms of the strain energy potential, Uð
Þ, which defines the strain energy stored in the material per unit of reference volume, as a function of the strain at that point. There are several forms of strain energy potentials available in ABAQUS to model incompressive isotropic elastomers. Among these models, the Mooney-Rivlin model is chosen for the analysis below. Mooney-Rivlin form provides the advantage that can be used by specifying the test data of the hyperelastic material or the coefficients of the strain energy potential function. This model is also available in the material libraries of many other FEA software packages, thus, the input coefficients suggested in this paper can be adopted and used in any other case of modeling neoprene in FEA. The mechanical properties of the neoprene are determined by performing uniaxial compressive test on a specimen of the neoprene pads. The testing facility consisted of a servo-controlled MTS machine equipped with a load cell with capacity of 100 kN. The grips that were used had 100 mm diameter and the specimen was attached to the upper and lower set of the grips. The lower set was fixed during the test and the upper set was moving downwards in a displacement control mode with a low speed of move-

Fig. 2. Schematic drawing of the isolated interior slab-column connection SB1.

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Table 1 Material properties and test results of specimen SB1. Specimen

f c (MPa)

f t (MPa)

f y (MPa)

As (mm2)

Tensile qs (%)

Failure load (kN)

Displacement at failure (mm)

SB1

44

2.2

455

100

1.1

253

11.9

0

0

50

800

45 600

35

stress (MPa)

stress (MPa)

40 30 25 20 15

400

fy = 455 MPa, ft = 650 MPa,

200

10

y=

0.0023 t = 0.25

5 0 0

0.001

0.002

0.003

0.004

strain (mm/mm)

0 0

0.05

0.1

0.15

0.2

0.25

0.3

strain (mm/mm)

Fig. 3. Uniaxial compressive stress-strain curve of concrete for slab SB1. Fig. 5. Stress-strain relationship of the flexural reinforcement.

2.5

Gf =0.082N/mm lc= 20 mm

stress (MPa)

2.0 1.5

Gf /lc

1.0 0.5 0.0

0

0.002

0.004

0.006

0.008

strain (mm/mm) Fig. 4. Uniaxial tensile stress-strain curve of concrete for slab SB1.

Table 2 Material properties and plasticity parameters of concrete used in the analysis of slab SB1. 0

f c [MPa] 0

f t [MPa] Gf [N/mm] Ec [MPa]

v

Dilation angle w [degrees]

e rbo =rco Kc

l

44

qffiffiffiffiffi 0 0:33 f c ¼ 2:2 0:7

Gfo ðf cm =f cmo Þ ¼ 0:082 qffiffiffiffiffi 0 5500 f c ¼ 36; 483 0.0 40 0.1 1.16 0.667 0

ment (see Fig. 6). The specimen’s dimensions were 110  50  25 mm in the undeformed configuration. The stressstrain data collected from the experiment till the 80 kN load in which no failure was observed. Fig. 3 presents the compressive stress-strain behavior of the tested neoprene. By calculating the load that the neoprene supports carried during the slab’s testing,

Fig. 6. Tested neoprene.

the level of stress that the specific specimen of neoprene reached was 1.0542 MPa at a strain level of 0.1268 mm/mm (see Fig. 7). At this stress-strain level the response of the tested neoprene can be considered as linear and the initial modulus of elasticity can be found (around 8.31 MPa). However, ABAQUS allows the input of the uniaxial compressive test data that can be used for the parametric modeling identification of the material coefficients and this procedure is followed in this analysis. By using the material evaluation option in ABAQUS, we can obtain all needed material coefficients in order to specify the Mooney-Rivlin form for the neoprene.

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300 250

Load (kN)

200

Test

150 100

Simple supports Neoprene supports

50 0 0

5

10

15

20

Displacement (mm) Fig. 8. Load-deflection curves of isolated specimen SB1 based on various boundary conditions.

Fig. 7. Compressive stress-strain behavior of neoprene.

According to ABAQUS [25] the strain energy potential in the Mooney-Rivlin form is:

U ¼ C 10 ðI1  3Þ þ C 01 ðI2  3Þ þ

2 1 el ðJ  1Þ D1

ð1Þ

where C 10 , C 01 and D1 are the temperature-dependent material parameters, I1 and I2 are the first and the second deviatoric strain invariants and Jel is the elastic volume ratio. The initial shear modulus and the bulk modulus are given according to Eqs. (2) and (3), respectively:

lo ¼ 2ðC 10 þ C 01 Þ Ko ¼

ð2Þ

2 D1

ð3Þ

The ABAQUS material evaluation gives the values for the material parameters C 10 , C 01 and D1 (C 10 ¼ 0:2292, C 01 ¼ 1:3203 and D1 ¼ 1:2994e  02). Therefore, the initial shear modulus is equal to 3.099 MPa and the initial modulus of elasticity that can be considered as three times the shear modulus is equal to 9.297 MPa. h  i.h  i 6K o o The Poisson’s ratio is equal to m ¼ 3K 2 þ 2 ¼ 0:49. l l o

o

The numerical results after the material evaluation in ABAQUS (E ¼ 9:297 MPa;v ¼ 0:49Þ are really close to the assumption that was discussed earlier by considering the neoprene as elastic material (E ¼ 8:31 MPa;v ¼ 0:5Þ. The FEA results using the neoprene to simulate the boundary conditions of the isolated specimen are in good agreement with the test results and they describe better, especially in the initial uncracked state, the load-deflection response of the isolated slab when compared to the simple supports (see Fig. 8). The simulation of the boundary conditions using the neoprene overcomes the initial stiffer response obtained from simulations by considering simple supports. Fig. 9 illustrates the cracking developed at the tension surface of the specimen SB1 at the ultimate load. By comparing the crack patterns between the FEA (Fig. 9b and c) and the test (Fig. 9a), it is obvious that in the test the flexural yield lines were not fully formed; the punching shear cone formed followed by a brittle failure. The two crack patterns from the FEA are in good agreement with the tests in terms of forming punching shear cone. The FEA crack pattern using simple supports (Fig. 9b) shows strain concentrations near the slab’s edges. Conversely, the simulation using neoprene supports (Fig. 9c) do not show strains near the supports; a result which is more consistent with the test. However, in all following analyses in this paper, simple supports are considered for simulating the simply supported isolated

tested slab SB1. That adoption is based on the fact that the numerical results from the neoprene and simple supports are in good agreement in terms of ultimate loads and post-cracking stiffness responses. Also, the simple supports require less computational cost compared to the analysis of the slab using neoprene supports. 3. Finite element simulation of the system continuity The increased punching shear capacity of a reinforced concrete flat slab due to the membrane action effects can be evaluated by examining the restraining in-plane forces acting in the slab. The magnitude of these restraining forces is hard to be found, because it depends on the stiffness of the flat slab structure provided by the supports and the slab itself. Also, it is difficult to test continuous slabs and for that reason, a common empirical approach is to test slabs simply supported at the outer edges. Non-linear FEA can be performed to examine all possible sources of restraint for the continuous slab specimens. In this section, the previously shown slab SB1 is examined considering the membrane action effect, and then, two slabs tested by Chana and Desai [17], where the membrane action effect was evaluated, are modeled and analyzed. In both cases the modeling and the analysis of the slab supported at the outer edges, as it is explained earlier, show the enhancement in the punching shear due to the membrane action. However, taking the advantage of the FEA, the aim of this study is to analyze also different support conditions that have not been examined in the experiments, and to simulate and analyze a real floor system. All these studies are presented on the Section 3.1, where SB1 and variations of its boundary conditions are shown. Then, Section 3.2 shows the comparison between numerical and experimental results for the slabs tested by Chana and Desai [17]; who tested both isolated and continuous slabs. 3.1. Slab-column connection SB1 tested by Adetifa and Polak [27] The previously shown isolated specimen SB1, is now examined as continuous. Fig. 10 illustrates the two continuous models: 1. Continuous Model 1: Restrained supports with both horizontal and vertical restraints (pinned supports) at the locations of the lines of the contraflexure in order to provide lateral restraints and simulate the continuous scenario (Fig. 10a). The lateral restraints are applied not only at the bottom of the slab but also at the whole height of the slab.

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Fig. 9. Crack patterns on tension side of the slab: (a) test; (b) FEA (simple supports) and (c) FEA (neoprene supports).

(L = 3750 mm). This model is based on the bending moment diagram of a floor system simulating the zero moments with the support conditions.

Fig. 10. Adopted continuous models for the slab SB1.

2. Continuous Model 2: The slab is modeled with dimensions (1.6L = 6000 mm) and simple supports are introduced at distance 0.4L = 1500 mm and at 1.5L = 5625 mm (Fig. 10b), where L denotes the center-to-center span of the slabs

The FEA results are presented in Fig. 11 in terms of loaddeflection response and are compared to the numerical and experimental results of the isolated simply supported slab. The adopted methods that represent the continuity (Fig. 10a and b), show an increase in the failure load but decreased ductility, compared to the test and the FEA results of the conventional slab. In particular, Table 3 shows the analysis results in terms of failure load and displacement for all the specimens. The load-deflection responses of the two continuous models are shown in Fig. 11. The ultimate load of the continuous model 1 is increased by approximately 68% and the ultimate load of the continuous model 2 is increased by approximately 50%, both compared to the numerical results of the isolated simply supported slab. The developed crack patterns of the analyzed continuous slabs at failure are presented in Fig. 12. The continuous slabs (Fig. 12a and b) concentrate the crack propagation around the column and it does not spread to the edges, as it happens on the isolated slab (Fig. 13). This can be explained by considering the smaller deflection of the continuous slabs, compared to the

A.S. Genikomsou, M.A. Polak / Engineering Structures 136 (2017) 233–244

400

Test isolated

Load (kN)

300

FEA isolated

200

100

Connuous Model 1

239

[23] showed that the punching shear strength of a laterally restrained slab was increased 34% compared to the strength of the isolated slab and the punching shear strength of an actual flat slab system was around 17% higher compared to the strength of the isolated slab. The crack pattern at failure of the interior slab-column connection SB1 after analyzing the floor system is shown in Fig. 18. The cracking is concentrated around the column and spreads at a distance of 300 mm from the face of the column. This development length of the crackings is smaller compared to this from the isolated simply supported slab (400 mm) and larger than the cracking development length of the continuous models (180–200 mm).

Connuous Model 2 0 0

5

10

15

20

3.2. Slabs tested by Chana and Desai [17]

Displacement (mm) Fig. 11. Load-deflection curves of the continuous slabs (comparison with the simply supported isolated).

Table 3 FEA results. Specimen

Failure load (kN)

Displacement at failure (mm)

Isolated Continuous Model 1 Continuous Model 2

234 392 352

13.9 2.7 4.5

deflection of the simply supported slab. Smaller deflection leads to lower crack widths and thus to larger punching shear capacity. Considering now that the punching shear capacity of a real slab is not the same as the capacity of the simply supported isolated slab and not such increased as this one of the isolated lateral restrained slab, the isolated slab is modeled with simple supports and axial spring elements in order to evaluate the effect of the lateral restraint. Fig. 14 shows the spring elements that are installed at the edges of the isolated simply supported slab at each element node. Different stiffness is given to the spring elements and the results in terms of load-deflection response are shown in Fig. 15. The given stiffness to the spring elements varies depending on the support conditions; low stiffness displays similar results to the simply supported slab and high stiffness shows the fully restrained case. The spring stiffness of 30,000 N/mm gives almost the same load-deflection response to the continuous model 2. Then, the whole floor system, from which the SB1 slab was taken, is simulated and analyzed using the finite element methods. Due to symmetry, one quarter of the floor flat system is considered and the adopted boundary conditions are presented in Fig. 16. A high uniformly distributed factored load of 18.5 kPa due to the high percentage of flexural reinforcement is applied to the floor system and the columns are restrained at the bottom. The punching shear load of the slab is measured as the reaction at the bottom of the column where the boundary conditions are introduced and the displacement is monitored at two points: at distance 0.2L from the column and at the middle of the slab. Fig. 17 shows the numerical results in terms of load-deflection of the slab SB1 after the analysis of the flat floor system. For comparison, in the same graph are shown the numerical responses of the isolated simply supported and continuous slabs. The ultimate punching shear load of the slab by analyzing the whole floor system is equal to 291 kN, 24% higher than the load of the simply supported isolated slab, while the analysis of the continuous model predicts an ultimate load 50–68% higher than the isolated slab. Similar results were presented in [23] where the compressive membrane action was examined in the progressive collapse of flat concrete slabs. Keyvani et al.

In order to validate the accuracy of the concrete damaged plasticity model in punching shear simulations of continuous slabs, the numerical model is applied to analyze punching shear tests reported in literature. Two tested interior slab-column connections [17], are simulated and analyzed. The purpose of these tests was to examine the effect of the membrane action on the punching shear capacity of the slabs. Both tested slabs had no shear reinforcement and were taken from a prototype structure, where the flat slab spans between the columns were equal to 6000 mm. One slab was considered as continuous and its dimensions were equal to 9000  9000 mm, while the second tested specimen was the isolated slab with dimensions 2400  2400 mm on plan. Both slabs were simply supported at the column foot. The thickness of the isolated slab was 240 mm with effective depth 200 mm and the dimensions of its square column were 300  300 mm. The continuous specimen was 250 mm thick with effective depth 210 mm and the dimensions of its column were 400  400 mm. The load was applied for both slabs through eight points placed at a radius of 1.2 m from the center. Table 4 shows the material properties, reinforcement and test results for the continuous and isolated slab. In the numerical analyses only one quarter of both slabs is simulated due to symmetry. Fig. 19 illustrates the continuous and isolated specimen with the boundary conditions and applied load. Parametric investigation is performed in order to calibrate the FEA model because these slabs were tested in a different way compared to the slab SB1 and also they were thick (250 mm-the isolated and 240 mm-the continuous) compared to the thickness of the SB1 that was 120 mm. The parametric investigation is done on the isolated specimen, where only the mesh size is examined. As it was reported in [26], the model in ABAQUS is mesh size dependent because it is a plasticity based model. For that reason as it is suggested in [25] a mesh convergence study should be always performed. Fig. 20 shows the numerical results in terms of load-deflection response for the isolated specimen by using three different mesh sizes: 20 mm, 30 mm and 40 mm. The obtained results of all different mesh sizes are in good agreement, however the mesh size of 40 mm is chosen for the analyses of both isolated and continuous slab due to less computational cost. All other material parameters are considered same with the slab SB1. Fig. 21 illustrates the comparison between the isolated and continuous slab in terms of tested and numerical failure load and deflection. The results from the FEA models are in good agreement with the test results. The numerical analysis show an ultimate load of 752 kN and 1248 kN for the isolated and continuous slab, respectively. The deflections at the loading points at failure are 16.7 mm for the isolated slab and 2.4 mm for the continuous slab. Both numerical load-deflection responses of the slabs appear stiffer compared to the tested responses. This seems to happen because of the adopted support conditions that are simplified compared to the test support conditions.

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(a)

(b) Fig. 12. Crack patterns: (a) Continuous Model 1, (b) Continuous Model 2.

Fig. 13. Crack pattern of the isolated simply supported slab SB1.

According to the test observations, both specimens failed in a brittle punching. Radial cracking was developed on the tension side of both slabs; starting from the loading column and then progating to the edges. Isolated specimen appeared the maximum crack width (0.3 mm) before failure, while the maximum crack width before failure for the continuous slab was equal to 0.15 mm. However, the radial cracks of the continuous slab stopped before the loading points, compared to the cracks of the conventional isolated slab that continued till the edges. This can

be explained due to the membrane action effect that controls the crack width and its development. Both tested crack patterns are effectively predicted by the FEA crack patterns (see Fig. 22). The continuous slab failed in a significantly increased load compared to the isolated specimen. In the tests the ultimate load of the continuous slab is increased 52% compared to the ultimate load of the isolated slab. The FEA results show that the continuous specimen has 60% higher punching shear strength compared to the isolated slab. This increase of the ultimate load (60%) is higher compared

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400

Connuous Model 1 Connuous Model 2

Load (kN)

300

200

Simply supported isolated 100

Whole floor system (Disp. @ middle) Whole floor system (Disp. @0.2L) 0 Fig. 14. Spring elements simulate the later restraint.

0

5

10

15

20

Displacement (mm) 400

Fig. 17. Load-deflection responses of the floor slab system.

Connuous Model 1 Connuous Model 2

Load (kN)

300

Simply supported isolated

200

Springs (k=30000 N/mm) Springs (k=10000 N/mm) Springs (k=5000 N/mm) Springs (k=1000 N/mm) Springs (k=50 N/mm)

100

0 0

5

10

15

20

Displacement (mm) Fig. 15. Load-deflection responses of the slabs with different stiffness of the spring elements (comparison with the isolated and two continuous slabs).

to the numerical results presented earlier for the slab SB1 (50%). That can be explained by considering that the specimens tested by Chana and Desai had lower both: the reinforcement ratio and the span to thickness ratio, compared to the slab SB1. According to tests done on one-way slabs [14] with different flexural reinforcement ratio and span to thickness ratio, it was found that the compressive forces can enhance the strength of the slabs by 30–

100%. Higher increase in the ultimate load was observed for slabs with lower reinforcement ratio and lower span to thickness ratio. Similar observations for the effect of the span-depth ratio to the punching shear strength of the slabs were made by Lovrovich and McLean [30]. They concluded after performing a test series that as the span-depth ratio decreased the punching shear strength of a slab increases due to the development of compression struts between the loading point and the supports. In this arch mechanism, the in-plane compressive forces have also concured to the increase of the punching shear strengths. Table 5 summarizes and compares the FEA results of all specimens. The analysis of the slab SB1 (Continuous Model 1) shows that the ultimate load is increased 67.5% compared to the ultimate load of the isolated slab SB1. The SB1 (Continuous Model 2) slab has an ultimate load 50.4% higher than the ultimate load of the isolated slab SB1. At this point, if we consider the numerical results of the slabs tested by Chana and Desai [17], we can see that the ultimate load of the continuous slab is 66% higher compared to the load of the isolated slab. The continuous slab tested by Chana and Desai has the same boundary conditions with the slab SB1 (Continuous Model 2). Based on these results we can conclude that the continuous models presented in this paper increase the ultimate load of the isolated slabs between 50.4 and 67.5%. However, the numerical results for the slab SB1 considered in the whole floor system shows that the ultimate load of a slab is about 24.4% higher than the observed load of the isolated slab.

Fig. 16. Boundary condition of one quarter of the whole floor system.

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Fig. 18. Crack pattern of SB1 on the floor slab system.

Table 4 Material properties, reinforcement and test results of slabs tested by Chana and Desai [17]. Specimen

f c (MPa)

f t (MPa)

f y (MPa)

As (mm2)

Tensile qs (%)

Failure load (kN)

Displacement at failure (mm)

Isolated Continuous

40.3 26.8

2.7 2.31

500 500

314 314

0.79 0.86

850 1225

18 2.49

0

0

1400

FEA Connuous slab Test Connuous slab

1200

Load (kN)

1000

FEA Isolated slab

800 600 400

Test Isolated slab Fig. 19. Boundary conditions and loads of the slabs tested by Chana and Desai [17]: (a) isolated and (b) continuous.

200 0

1000

Load (kN)

2

4

6

8

10

12

14

16

18

Displacement (mm) Fig. 21. Comparison between tested and FEA load-deflection curves of the isolated and continuous slab.

800

600

400

Test 20 mm 30 mm 40 mm

200

0

0

0

2

4

6

8

10

12

14

16

18

Displacement (mm) Fig. 20. Load-deflection response of the isolated slab [17] for different mesh sizes.

4. Comparison with the design approaches The compressive membrane action effect is ignored in the design provisions. Comparing the results of the isolated slabs (test

and FEA) with the recommendations of the design provisions (ACI318-11 [31], EC2 2004 [32]), we can conclude that the design codes underestimate the punching shear capacity of the slabs (see Table 6) introducing of course safety margins. Only for the isolated specimen tested by Chana and Desai [17] the code predictions are really close to the test results. We can state also herein that the formulae of both design codes are considered without the safety factors. However, if we compare now the numerical results of the continuous slab SB1 and the test and numerical results of the continuous slab [17] with the code predictions, we can actually conclude that the design codes are very conservative. If we compare only the SB1 (Continuous Model 2) slab with the design codes, we can observe that EC2 predicts an ultimate punching shear load about 42.6% less than the FEA load and ACI underestimates the FEA load about 46.3%. Now, if we compare the continuous slab [17] (same continuous model with slab SB1Continuous Model 2) with the design codes, we can see that EC2 predicts a punching shear load 27.8% less that the FEA load and ACI predicts an ultimate punching shear load 29.9% less than the

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Fig. 22. Crack patterns on tension side of the slabs: (a) isolated and (b) continuous.

Table 5 Comparison of ultimate FEA load (kN) between isolated and continuous slabs. Slab specimen

FEA load VI (kN) Isolated slab

FEA load VC (kN) Continuous slab

VC/VI

SB1 (Continuous Model 1) SB1 (Continuous Model 2) SB1 (whole floor) Slab tested by Chana and Desai

234 234 234 752

392 352 291 1248

1.675 1.504 1.244 1.660

higher punching shear resistance compared to ACI for the slab SB1. However, for the isolated slab [17] the punching shear resistance from EC2 is around 9 kN higher compared to the load that ACI predicts. EC2 predicts an ultimate punching shear load about 26 kN higher than the ACI for the continuous slab [17]. Thus, the predictions from EC2 are closer to the punching shear resistance of the tested and FEA isolated slabs compared to ACI. 5. Conclusions

FEA load. Thus, the punching shear design provisions that are based on test results of isolated slabs are conservative and modifications accounting the membrane action effect can be taken into consideration. These code modifications could be proposed after analyzing many and different continuous slabs. Slabs with shear reinforcement should be also considered. The calibrated FEA models could be used for future parametric studies. Comparing now the results between the design codes, it can be said that EC2 gives higher punching shear resistance for all slabs, compared to the predictions that ACI provides. EC2 shows 13 kN

3D nonlinear finite element methods can be effectively used in punching shear simulations to examine the continuity of reinforced concrete floor systems. The existing testing database of isolated slabs can be examined using FEA by considering variety boundary conditions in order for the continuity to be adopted in the simulations. Proper calibration of FEA should be done in advance, and then parametric studies can follow. The calibrated concrete damaged plasticity model in ABAQUS predicts accurately the responses of the continuous and isolated slabs. The examples in this paper provide the following conclusions:

Table 6 Comparison of ultimate load (kN) between test, FEA and design codes for slab SB1 [27] and slabs tested by Chana and Desai [17]. Slab specimen

Test results

FEA results

ACI

EC2

[27] SB1 (isolated) SB1 (Continuous Model 1) SB1 (Continuous Model 2) SB1 (whole floor) [17] Isolated [17] Continuous

253 Not tested Not tested Not tested 850 1225

234 392 352 291 752 1248

189 189 189 189 838 875

202 202 202 202 847 901

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1. The lateral restraints combined with the simple supports at the edges in the isolated specimen SB1 (Continuous Model 1) increase the punching strength 68%, while the simple supports in the SB1 slab with dimensions 1.5L (Continuous Model 2), increase the punching strength by about 50%. Thus, the obtained numerical results of the two continuous models show quite similar increase of the punching shear load. 2. Due to the membrane action effect, the deflections of all simulated continuous specimens are smaller compared to the isolated simply supported slab. For that reason the continuous specimens show smaller crack widths and the crack patterns are concentrated around the area of the column and they are not spread to the edges of the slabs. Subsequently, the continuous slabs have higher punching shear strength. 3. Axial springs with varying stiffness, can be used in the FEA of the slabs, in order to simulate the lateral restraint leading to the formation of the compressive membrane action forces in the slab. 4. The whole floor system analysis shows that the actual slab has 24% higher punching shear resistance compared to the strength of the isolated simply supported slabs. This punching shear strength is less than the one found for the continuous slabs. Therefore, the continuous models overestimate the compressive membrane action effect and its contribution to the punching shear strength. 5. The calibrated concrete damaged plasticity model can be effectively used for analyzing slab specimens from literature. The tested slabs [17] are simulated and analyzed in ABAQUS and their numerical results are in good agreement with the test results in terms of load-deflection response and crack patterns. 6. The current design provisions of punching shear do not consider the membrane action effect. Design provisions, tests and FEA of isolated slabs give safe predictions compared to the actual flat slabs. However, the design codes accuracy could be enhanced if membrane action is considered. Finite element analyses done of isolated specimens can be extended using the appropriate dimensions and boundary conditions in order to simulate and analyze the same slabs as continuous. These results can be helpful in future code developments. 7. Full real floor systems can be analyzed using FEA even though they demand higher computational time compared to the analysis of isolated slabs. Such analyses can be very helpful in the assessment of existing structures; making them more effective and accurate and helping in rational and cost effective strengthening techniques. The aforementioned conclusions suggest further studies, both numerical and experimental, to better examine and understand the membrane action effect in reinforced concrete slabs in order to apply it for code calibration. Understanding of the membrane action effects is also needed for edge and corner slab-column connections, in slabs with low reinforcement ratios, for slabs with high strength concrete and slabs with openings. Reinforced concrete slabs with punching shear reinforcement should be also considered in the future studies. Acknowledgements The authors are grateful for the financial support that has been provided by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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