Experimental and parametric 3D nonlinear finite element analysis on punching of flat slabs with orthogonal reinforcement

Engineering Structures 48 (2013) 442–457

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Experimental and parametric 3D nonlinear finite element analysis on punching of flat slabs with orthogonal reinforcement Nuno F. Silva Mamede, A. Pinho Ramos, Duarte M.V. Faria ⇑ Department of Civil Engineering, Faculdade de Ciências e Tecnologia, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal

a r t i c l e

i n f o

Article history: Received 22 March 2012 Revised 4 September 2012 Accepted 13 September 2012 Available online 27 November 2012 Keywords: Punching Parametric study Reinforced concrete Numerical analysis Flat slabs

a b s t r a c t This work refers to experimental and 3D nonlinear FEA on punching. Numerical results were compared with experimental ones in order to benchmark the FE model and afterwards a parametric study was conducted, changing the reinforcement ratio, slab thickness, concrete strength and column dimensions, running a total of 360 models, where their effect on punching capacity is shown. EC2 and MC2010 provisions agreed approximately with experimental and FEA results. Based in the FEA results it is proposed an equation to predict the punching capacity with the introduction of fracture mechanics parameter, which was compared with several experimental results, giving good approximation. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Punching is one of the most complex phenomenons in reinforced concrete flat slabs, since it occurs in a geometric discontinuity zone where the stresses are highly concentrated in a small area. The influence of various parameters on the punching behaviour is still in discussion. From the few parametric studies using nonlinear analysis that have been performed, some controversial results have outcome. Parametric studies became possible with the development of computational capacities and the deepening of knowledge of nonlinear behaviour of concrete, both in compression and tension, in the last 30 years. From previously developed studies regarding parametric analysis of material factors affecting punching behaviour in RC flat slabs, Menétrey [1], Hallgren [2], Ozbolt et al. [3] and Eder et al. [4] works stand out. The first two were performed using two-dimensionally modelled systems using rotationally symmetric elements, while the last two were done using a three-dimensional finite element code. Although two-dimensionally modelled systems using rotationally symmetric elements are suitable for slabs with ring reinforcement, Menétrey [1] and Hallgren [2], used it to analyse orthogonal reinforced concrete slabs, adopting special conditions such as a fictitious fastening length [1], to simulate partial bond ⇑ Corresponding author. Address: Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade NOVA de Lisboa, Campus da Caparica, 2829516 Caparica, Portugal. Tel.: +351 962821685; fax: +351 212 948 398. E-mail addresses: [email protected] (N.F. Silva Mamede), ampr@ fct.unl.pt (A. Pinho Ramos), [email protected] (D.M.V. Faria). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.09.035

between steel and concrete, and Hallgren [2] set the bar spacing in both radial and tangential direction equal to the average spacing of bars in the upper and lower reinforcement layer of the real slab, and constant, independently of the distance from the slab centre. In both studies [1,2], the curves of the finite element analysis were stiffer than the experimental ones, and both justified it from the simplified modelling of the two-way reinforcement as an axi-symmetric mesh. The assumptions made to simulate orthogonal reinforcement with axi-symmetric elements cause the results of the analysis to be questionable, but at the time 3D analysis was difficult to perform, once computer capabilities were relatively weak. Ozbolt et al. [3] performed a 3D finite element analysis of a punching test on an interior slab–column connection, based in the microplane model for concrete and a smeared crack approach, concluding that the code is able to realistically predict the loadbearing behaviour. Nevertheless, from the presented results only peak load and corresponding displacement are well estimated. The same also happened in terms of comparison of steel strains. Later, Eder et al. [4] performed a 3D analysis showing a good approximation between predicted and measured loads. Regarding parametric analysis performed in the mentioned previous works, the main conclusions were: Ménetrey [1] concluded that punching failure is due to tensile failure of concrete along the inclined punching crack and is not due to compressive failure of concrete and that increasing the reinforcement ratio implies increasing the punching load. Size effect was also investigated, resulting in a nominal shear stress decrease with increasing thickness of the slab; Hallgren [2] showed that tensile strength has a

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Nomenclature dMN

ql w c d da dg fccm fcm

vertical displacement where the radial bending moment is zero average reinforcement ratio slab rotation around the column distance between the column face and where the bending moment is zero effective depth aggregate medium size maximum aggregate size concrete compression strength on cubes concrete compression strength on cylinders

major influence on punching capacity and compressive strength of concrete has a small influence. Showed also that the higher brittleness of higher concrete grades reduces the increase of the punching strength, and found a linear correlation between numerically obtained ultimate loads and the tensile strength multiplied with the ratio between characteristic length of concrete (lch), proposed by Hillerborg et al. [5], and the depth of compression zone; Ozbolt et al. [3] concluded that fracture energy and reinforcement ratio have a similar and relatively strong influence on the maximum load, stating that high values of fracture energy (GF) and reinforcement ratio lead to a more stable growth of the cracks in the tensile cord, and as a result, the height of the compression zone increases and leads to higher punching load and more brittle failure. The numerical results also indicate a relatively small effect of the compressive and tensile concrete strength on the punching capacity. They stated that depending on the fracture energy, the crack propagation at peak load might even become less stable if tensile strength increases, and therefore the positive effect of tensile strength remains relatively small while brittleness of the structural response tend to increase with higher tensile strengths. It was also demonstrated that increasing effective depth leads to a decrease of the nominal shear capacity of the slabs. Regarding Ozbolt et al. [3] results, Menétrey [6] stated that their conclusion, that tensile and compressive strength does not influence punching failure is not justified experimentally and this may result from the assumption of an uniaxial stress strain response according to the microplane model for which the biaxial and triaxial concrete response are neglected. Eder et al. [4] results showed that increasing concrete tensile strength affected the displacements more significantly than failure load, although the tension softening behaviour has a significant effect on punching overall performance. It was also presented that the numerical results converged as the mesh was refined and decreasing the mesh density increased the failure load. Recently, Faria et al. [7] developed a parametric study using 3D finite element software, analysing the punching shear resistance of inner slab-column connections. Faria et al. [7] analyzed the concrete parameters such as concrete tensile strength, modulus of elasticity, concrete compressive strength and fracture energy as independent, so when one changes, all the others remain constant, in order to determine their individual influence and effect in the punching phenomenon. Faria et al. [7] analyzed also, the influence of the reinforcement ratio in the punching shear resistance. Faria et al. [7] main conclusions were that tensile strength and modulus of elasticity have a small influence in punching load resistance unlike the compressive strength and fracture energy that are the main material properties that influence the punching load, since both them showed that punching load increases approximately with their cubic root. The reinforcement ratio has also an important role, as the punching load also increases approximately with its cubic root.

fctm fsu fsy lch u1 Ec Es GF VEXP VFEA VRm

concrete tensile strength steel tensile ultimate strength steel tensile yielding characteristic length of the material control perimeter around the column modulus of elasticity of concrete modulus of elasticity of steel fracture energy experimental punching load numerical punching load mean value of punching resistance

Bearing in mind some of the contradictions between the results obtained by mentioned investigators, and since they assumed certain simplifications, namely, using two-dimensionally modelled systems using rotationally symmetric elements for orthogonal reinforcement and the limitations of the microplane model, there is a need for a parametric analysis based in 3D analysis using more recent and developed material models. Thus, it will be possible to draw more reliable conclusions. The results of the parametric analysis are important to further understand the punching phenomenon and turn possible the improvement of existing provisions, namely, regarding the introduction of fracture mechanics parameters, like the characteristic length of concrete (lch), reflecting the influence of concrete brittleness, based on fracture energy, elasticity modulus and tensile strength of concrete, as first proposed previously by Hillerborg et al. [5]. Fracture mechanics parameters have already been introduced in Hallgren’s [2] proposed mechanical model and Staller’s [8] empirical model, for the determination of punching capacity.

2. Experimental models Thirteen experimental reinforced concrete flat slabs models were simulated until failure by punching. Slabs AR2, AR9, DF1, DF4 and ID1 are part of different investigation works developed by the research team, where the authors are part of. AR2 and AR9 models were tested by Ramos [9], slabs DF1 and DF4 were part of a study developed by Faria [10] and ID1 was presented by Inácio [11]. The tests arrangement are different because some of the slabs were intended to be more slender (AR and DF) while ID1 was intended to have a normal slenderness. Another reason for this was because slabs were tested in different laboratories and slabs were fixed to the ground floor in different available points. Slabs PG11, PG19 and PG20 were part of a study developed by Guidotti [12], slabs PG2-b, PG5 and PG3 were tested by Guandalini [13]. Slabs ND65-1-1 and P200 were developed by Tomaszewick [14] and Li [15], respectively. As shown in Fig. 1, slabs AR2, AR9 and DF1 have dimensions of 2300 mm  2300 mm and were 100 mm thick. Slab DF4 was similar but with a thickness of 120 mm. Slab ID1 was 1800 mm  1800 mm and 120 mm thick. Slabs PG11, PG19, PG20, PG2-b and PG5 were 3000 mm  3000 mm and 250 mm thick, while slab PG3 was double size, 6000 mm  6000 mm  500 mm. Slabs ND65-1-1 and P200 were 3000 mm  3000 mm  320 mm and 725 mm  725 mm  240 mm, respectively. They all modeled the area near a column of an interior slab panel up to the zero moment lines. In slabs AR2 and AR9 the vertical punching load was applied by two hydraulic jacks positioned under the laboratory floor. The load was transferred in eight points to the top of the slab by steel

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2300 700

150

150

500

200

A

A 200

2300

1200

200

700 2300

200

1200

500 150 500

150

500

(a)

150

(b)

150

3000 250

250

500

500

500

260 250

150

0

22.5°

260

3000

500

250

200

1000

1800

200

1000 1800

(c)

(d) 1450

3000 250

1250

1250

100

250

625

625

100

250

1250

200

200 200

3000

200

1450

1250

250

(e)

(f)

Fig. 1. Plan view of the specimens: (a) AR2 and AR9, (b) DF1 and DF4, (c) ID1, (d) PG’s (PG3 was double size), (e) ND621-1, and (f) P200 (dimensions in mm).

tendons and spreader beams. The centre of the slab was supported on a steel plate with 200 mm  200 mm, which simulated the column. In the experimental models DF1, DF4 and ID1 the load

was applied by a hydraulic jack positioned under the slab, through a steel plate with 200 mm  200 mm in the centre of the specimens. Eight points on the top of the slab were connected to the

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N.F. Silva Mamede et al. / Engineering Structures 48 (2013) 442–457 Table 1 Concrete and geometric properties.

Table 2 Steel and reinforcement bars properties.

Model

d (mm)

fccm (MPa)

fcm (MPa)

fctm (MPa)

Ec (GPa)

GF (N/ m)

dg (mm)

AR2 AR9 DF1 DF4 ID1 PG11 PG19 PG20 PG2-b PG5 PG3 ND651-1 P200

80 80 69 88 87 208 206 201 210 210 456 275

48.9 46.4 31.0 24.7 49.2 – – – – – – –

39.1 37.1 24.8 19.8 39.8 31.5 46.2 51.7 40.5 29.3 32.4 64.3

3.0 2.8 2.0 1.6 3.0 2.5 3.2 3.4 3.0 2.3 2.1 4.3

29.8 29.3 26.0 24.3 30.0 33.2 32.7 33.9 34.7 26.8 31.8 38.1

78 75 57 48 79 67 88 95 80 64 68 110

16 16 16 16 16 16 16 16 16 4 16 16

200

–

39.5

3.3

29.9

97

20

Model

strong floor of the laboratory using prestress strands and spreader beams. The test setup for slabs PG11, PG19, PG20, PG2-b and PG5 were similar to slabs AR, but in this case four hydraulic jacks were used, and the steel plate support in the centre of the slab measured 260 mm  260 mm. The slab PG3 was loaded using two hydraulic jacks positioned under the slab, and the steel plate support in the centre of the slab measured 520 mm  520 mm. All slabs were attached to the laboratory strong floor by eight strong bars along the perimeter. The load on slab ND65-1-1 was applied by a hydraulic jack with 200 mm  200 mm and supported by a square loading rig. Slab P200 was inverted tested, the reinforced concrete column had a section of 200 mm  200 mm and the load was applied through a MTS universal testing machine. The specimen’s average effective depths are shown in Table 1 along with the materials properties. The average values of concrete compression strength in cubes, fccm, were obtained experimentally for AR2, AR9, DF1, DF4 and ID1. The cylinder concrete compression strength, fcm, was considered being 80% of the fccm, while the concrete tensile strength, fctm, and the modulus of elasticity, Ec, were obtained with the relations presented on EC2 [16], referred in Eqs. (1) and (2). For the others tests the cylinder concrete compression strength presented is taken from literature. The fracture energy, GF, was obtained following the recommended on Model Code 1990 [17], shown in Eqs. (3)–(5). The reinforcement steel tensile yielding, fsy, and strength, fsu, used in models are shown in Table 2, and also the reinforcement bars layout. To clarify, slab PG11 top reinforcement was composed by alternating 18 mm and 16 mm reinforcement bars every 290 mm, slab ND65-1-1 had different reinforcement bar spacing in the different orthogonal directions and had no bottom reinforcement, slab P200 had only one reinforcement bar at the bottom close to the column, in both orthogonal directions. All the others models reinforcements were composed in both orthogonal directions.

fctm ¼ 0:30  ðfcm  8Þ2=3 Ec ¼ 9:923 

0:3 fcm

GF ¼ GF0  ðfcm =fcm0 Þ

ð1Þ ð2Þ

0:7

½N=m

ð3Þ

where

fcm0 ¼ 10 MPa

GF0

8 > < 0:025; dmax ¼ 8 mm ¼ 0:030; dmax ¼ 16 mm½N mm=mm2  > : 0:058; dmax ¼ 32 mm

ð4Þ

ð5Þ

During the tests the displacements of the top surface of the slabs were measured using LVDT’s, and the loads values were obtained

/ Bottom

/ Top

fsy (MPa)

fsu (MPa)

fsy (MPa)

fsu (MPa)

AR2 AR9 DF1 DF4 ID1 PG11

639 555 537 561 588 531

732 670 656 678 697 –

523 481 541 537 445 538

613 633 637 648 582 –

PG19 PG20 PG2-b PG5 PG3 ND651-1

500 500 500 500 500 –

– – – – – –

551 510 552 555 520 500

– – – – – –

P200

–

–

465

–

Top reinforcement

Bottom reinforcement

/10//60 mm /10//60 mm /10//60 mm /10//75 mm /10//75 mm /18 and /16 // 290 mm /16//125 mm /20//100 mm /10//150 mm /10//115 mm /16//135 mm /25//60 mm (X) /25//120 mm (Y) /12// 120.8 mm

/06//200 mm /06//200 mm /06//200 mm /06//200 mm /06//200 mm /08//145 mm /10//125 mm /10//100 mm /08//150 mm /08//115 mm /10//135 mm –

2/16

by load cells. Displacements transducers were positioned along two orthogonal lines. Nevertheless, in this work, for reasons of simplification, only one line results are presented. In Fig. 2, monitoring displacement points are shown for all the specimens, where d1 to d5 corresponds on the LVDT’s positions used in the slabs from series AR, DF and ID. The results presented in this work for the slabs in PG series were measured using the monitoring point dpg. Unfortunately, displacements data for slabs ND65-1-1 and P200 were not available. 3. Finite element modeling 3.1. Material modeling In the finite element analysis (FEA) was used ATENA 3D software [18]. It uses realistic constitutive model of concrete behaviour that enables a simulation of the real structure behaviour in service as well as in ultimate loading conditions. The redistribution of internal forces due to the nonlinear material behaviour is taken into account and the resulting stress and deformation state satisfies all three requirements of mechanics: equilibrium of forces, compatibility of deformations and material laws. The material chosen to simulate concrete was CC3DNonLinCementitious2, which consists in a combined fracture-plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. Nogueira [19] demonstrated in his study that the rotated crack model gives better approximations that the fixed crack model, so in this study it was adopted the rotated crack model. The plasticity model for concrete in compression is based on the Menétrey-William [20] failure surface. The position of the Menétrey–William [20] failure surface is not fixed but it can expand and move along the hydrostatic axis, simulating the hardening and softening stages. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The fictitious crack model developed by Hillerborg et al. [5] is considered, in which the tensile strength is controlled by the crack opening. The softening behaviour of concrete in tension adopted is the one derived experimentally by Hordjik [21], and it is used an enhanced crack model that was originally proposed by Bazant and Oh [22].

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Fig. 2. Deflection gauges positions in experimental models (dimensions in m).

Fig. 3. Example of a three dimensional model.

Aggregate interlock and dowel action of the reinforcement is taken in account by a reduction of the shear modulus with growing strain normal to the crack, according to the law derived by Kolmar [23]. The reinforcement can be included in two ways, either as smeared or as discrete bar elements. In this study, discrete bar elements were used, and it is represented by uniaxial ideally elastoplastic stress strain relationship. In this study, perfect bond between the concrete and reinforcement bars is assumed. In this nonlinear FEA, the load was increased deformation controlled in steps according to the standard Newton–Raphson iteration method. The convergence criteria used in the numerical analysis were displacement error, residual error, absolute residual error and energy error. 3.2. Finite element mesh The reference slabs were modeled using three dimensional isoparemetric elements with eight nodes, known as brick elements, and only a quarter of the slab was modeled using symmetry, for simplification, as shown in Fig. 3. Around the column, in the punching failure zone, the mesh was discretized with a fine mesh of brick finite elements, being the outer zone also meshed by brick finite elements with bigger size. The load on the slab was introduced by deformation control thru a steel plate in deformation increases of 0.1 mm, being the steel plates meshed using shell elements. 4. Experimental versus FEA results 4.1. Load/displacements results This section presents the results obtained for the displacements in the top surface of the slabs. Fig. 4 compares the experimental and FEA displacement obtained. In the presented results, d1 is the displacement that corresponds to the average of points 1 and 5, while d2 represents the average of the points 2 and 4 (Fig. 2).

The displacements named dPG were calculated by the slab rotation measured at the reference point. The FEA results showed a good agreement with the experimental results. For all the specimens the final failure mode was punching shear. Nevertheless, according to Guandalini [13], slabs PG2-b and PG5 reached their plastic plateau and punching failure occurred with large plastic deformations at the onset of the yield-line mechanism – flexural punching. For these specimens the FEA did not reproduce accurately the very last plastic deformations since the calculations stopped after the peak load was reached. Inconsistencies occur mainly due to differences in the concrete modulus of elasticity and in the development of concrete cracking. 4.2. Failure loads Table 3 shows the failure loads of the experimental and the numerical models, where is also shown, the ratio between them. VFEA, the maximum FEA load, was defined as the last load before a load sudden drop occurred. The obtained numerical failure loads agrees quite well with the experimental ones, although the numerical results slightly overestimate the failure load for most of the specimens. The average value of the ratio obtained between the experimental and the numerical failure loads was 0.96 with a coefficient of variation (COV) of 0.05. 5. Parametric analysis 5.1. Numerical models description For the parametric study, 360 numeric slabs models were created simulating the area near an interior column up to the zero bending moment line. In the models development, the material and geometric assumptions were the same as described in Section 3. The varying parameters were the reinforcement ratio, the thickness of the slab, the concrete strength and the column dimensions. Five different thicknesses of slabs were considered (200 mm, 225 mm, 250 mm, 300 mm and 350 mm). Based in a slenderness for the slabs of h = L/30 (where L is the slab length) and considering 0.22L as the distance where the bending moment is zero, the dimensions of the models were defined. In Table 4 are shown the dimensions of the models, according to the thickness and the distance where the bending moment is zero (0.22L) measured from the column axis. Three square column dimensions were also studied (300, 400 and 500 mm square columns). The concrete strengths considered were 20 MPa, 25 MPa, 30 MPa and 40 MPa. Table 5 presents the concrete properties

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Fig. 4. Load/displacement graphs for specimens – experimental versus FEA. MC2010 failure criterion – Exp. (8).

considered in this study, namely the compressive strength (fc), the tensile strength (fct), the modulus of elasticity (Ec) and the fracture

energy (GF). The maximum diameter of the aggregate (dg) used in the simulations was 16 mm.

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Fig. 4. (continued)

Table 3 Comparison between experimental and numerical (FEA) failure loads. Model

VEXP (kN)

VFEA (kN)

VEXP/VFEA (kN)

AR2 AR9 DF1 DF4 ID1 PG11 PG19 PG20 PG2-b PG5 PG3 ND65-1-1 P200

258.0 251.0 191.0 199.0 269.0 763.3 860.0 1094.0 438.7 550.0 2153.0 2050.0 904.0

259.1 260.6 218.9 210.0 297.4 780.8 914.4 1166.8 441.2 542.0 2117.2 2041.2 992.0

1.00 0.96 0.87 0.95 0.90 0.98 0.94 0.94 0.99 1.01 1.02 1.00 0.91

Average COV

0.96 0.05

As;min ¼ 0:26 

h (mm)

L (m)

0.22  L (m)

d (m)

200 225 250 300 350

6.00 6.75 7.50 9.00 10.50

1.320 1.485 1.650 1.980 2.310

0.165 0.190 0.215 0.265 0.315

Table 5 Concrete properties used in FEA analysis. 20 2.2 30 50

25 2.6 31 57

30 2.9 33 65

fctm bt d P 0:0013bt d fy

ð6Þ

5.2. Parametric analysis results

Table 4 Numerical models geometry.

fc (MPa) fct (MPa) Ec (GPa) GF (N/m)

The top reinforcement ratio was taken into account in this study with six different values: 0.50%, 0.75%, 1.00%, 1.25%, 1.50% and 2.00%. The top reinforcement ratio was composed with bars spaced every 100 mm in both orthogonal directions. The equivalent diameter of the bars was defined according to the respective slab reinforcement ratio. The bottom reinforcement consisted on the minimum reinforcement area for concrete elements (Eq. (6)) following the recommendations of EC2 [16], in both orthogonal directions. The reinforcement bars were spaced every 200 mm and the diameters of the bars were defined by the respective minimum reinforcement area.

40 3.5 35 80

5.2.1. Reinforcement ratio The reinforcement ratio has a significant influence on the FEA predicted punching capacity (Fig. 5). For space reasons only some graphs are shown (h = 225 mm and varying the concrete strength). The others are similar. As can be seen, with an increase of the reinforcement ratio there is also an increase of the FEA predicted punching load, approximately as a cubic root function of reinforcement ratio, similar as in the EC2 [16] prediction. The average of the exponent value, from the potential regression analysis, for all the models, was 0.35. As shown in Fig. 6, the reinforcement ratio has not only a significant influence on FEA predicted punching loads, but also on its behaviour. By increasing the reinforcement ratio, the value of the FEA predicted punching load is increased but ductility is decreased, as ultimate displacements are diminished. With increased reinforcement ratio, higher stiffness’s are recorded after cracking. This increase leads to a higher depth of the compression zone, leading to higher punching loads, accompanied by more brittle failures.

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Fig. 5. Influence of the reinforcement ratio on FEA predicted punching load.

Fig. 6. Influence of the reinforcement ratio in FEA predicted punching behaviour.

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Fig. 7. Influence of the effective depth on FEA predicted punching load.

5.2.2. Effective depth (d) The average effective depth value was defined based on the thickness of the models and the position of the reinforcement bars. It is shown in Fig. 7 that with an increase of the effective depth, there is an increase of the FEA predicted punching resistance. Fig. 8 shows that an increase on the thickness of the slab increases the FEA predicted elastic response phase, once there is an increase on the cracking moment. After this stage, as the thickness of the slab increases, the stiffness of the responses increases slightly. Ultimate displacements are practically insensitive to changes in this parameter. Also the increment of load capacity becomes smaller as the thickness of the slab increases, meaning that the normalized shear stresses decrease with the increasing of effective depth.

5.2.3. Concrete strength (fc) Increasing the concrete strength results on an increment of the FEA predicted punching load (Fig. 9). Based in a potential regression analysis, the FEA predicted punching load depends on average

on an exponent of 0.41 of the concrete strength (fc0:41 ). This value lies between the one presented by EC2 [16] (1/3) and the one recommended by the MC2010 [24] (1/2). As shown in Fig. 10, as the concrete strength increases the FEA predicted elastic response phase continues until higher loads, meaning that the cracking phenomenon is delayed, which is related to the associated increase of the tensile strength of concrete. After the first crack is opened the responses tend to become similar, with identical stiffness. Also the higher the concrete strength, higher are the FEA predicted displacements at failure.

5.2.4. Column dimensions Increasing the column dimensions leads to an increment on the FEA predicted punching capacity. Nevertheless, it is important to pay attention to situations where the column dimensions imply a stress concentration near the corners. Fig. 11 presents the load/displacement FEA predicted behaviour regarding the column dimension variation. It is shown that the FEA predicted behaviour in all response phases are similar, before and after the concrete cracking.

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Fig. 8. Influence of the effective depth in FEA predicted punching behaviour.

Fig. 9. Influence of the concrete strength on FEA predicted punching load.

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Fig. 10. Influence of the concrete strength in FEA predicted punching behaviour.

Fig. 11. Influence of the column dimensions in FEA predicted punching behaviour.

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Fig. 12. Relation between the FEA predicted and codes prediction punching loads.

Fig. 14. Relation between the numerical and proposal equation predicted punching loads.

Fig. 13. Relation between the normalized shear stress and GF/d.

For the range of column dimensions studied, the FEA predicted ultimate displacements are practically insensitive to changes in the column dimensions. However, and as was mentioned before in Section 4.1, for slabs with low reinforcement ratios, FEA did not reproduced accurately the very last plastic deformations since the calculations stopped after the peak load was reached. This means that FEA results concerning slab rotations at failure, for slabs with a reinforcement ratio of 0.5% (in Figs. 6, 8, 10 and 11), may be somehow be underestimated. 5.2.5. Comparison between predicted strength from FEA and design codes EC2 [16] uses Eq. (7) to predict the punching resistance for slabs without punching shear reinforcement, where d is the effective depth, u1 is the control perimeter around the column at a distance 2d from the support region and vmin is the minimum punching qffiffiffiffiffiffi3=2  1=2 resistance given by 0:035  1 þ 200  fcm . d The MC2010 [24] punching design recommendations is based on the critical shear crack theory, described by Muttoni [25]. Expressions (8) and (9) are for average values that can be compared with experimental results, presented by Muttoni [25].

V Rm;EC2 ¼ 0:18  1 þ

rffiffiffiffiffiffiffiffiffi! 200 ð100  ql  fcm Þ1=3  u1  d d

P v min  u1  d

In Eq. (8) dg is the maximum aggregate size, u is the control perimeter around the column at a distance 0.5d from the support region and w is the slab rotation, given by Eq. (9) where msd is the average moment per unit length for calculation of the flexural reinforcement in the support strip and it can be approximated, for inner columns without moment transfer, as V/8 (where V is the punching load); rs indicates the position where the radial bending moment is zero regarding to the column axis and may be taken as 22% of the slab length; mrd is the design average flexural strength per unit length in the support strip; Es is the modulus of elasticity of steel and fy is the yield strength of reinforcement. Those values correspond on approximation level II. Following the equations presented above the punching shear resistance was predicted using both EC2 [16] and MC2010 [24] for all the numerical models. The FEA results revealed a tendency to be higher than the predicted values using both EC2 [16] and MC2010 [24]. The average ratio between EC2 [16] punching load and numerical models predictions was 0.99 with a coefficient of variation of 0.08. The same values when compared with MC2010 [24] resulted in average ratio value of 0.89 with a coefficient of variation of 0.07. Fig. 12 presents a comparison between the numerical failure load and the punching load predictions according to EC2 [16] and MC2010 [24]. It may be observed that for this set of models the MC2010 [24] based equations for punching resistance prediction are slightly more conservative than the ones presented by EC2 [16], although presenting similar coefficients of variation.

ð7Þ 6. Influence of material/structural brittleness

V Rm;MC2010 3=4 pffiffiffiffiffiffi ¼ wd u  d  fcm 1 þ 15 16þd g  1:5 rs fy msd  w ¼ 1:5  d Es mrd

ð8Þ ð9Þ

6.1. Introduction As mentioned before, several researchers showed a relation between shear/punching strength and material/structure brittleness

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Table 6 Comparison between several experimental results and predicted failure loads. Model

VEXP (kN)

fcm (MPa)

d (mm)

q (%)

da (mm)

V EXP V MC2010

V EXP V EC2

V EXP V Prop:

IIa

IVb

16 16 16 16 16

1.17 1.16 1.22 1.10 1.21

1.09 1.07 1.09 1.01 1.00

1.24 1.23 1.28 1.15 1.25

1.07 1.06 1.14 1.09 1.09

0.77 0.78 1.56

16 16 16

1.12 1.13 1.09

0.98 0.98 1.00

0.97 0.97 0.98

1.01 0.97 0.96

210 210 456

0.25 0.33 0.33

16 4 16

1.01 1.19 1.25

– 1.05 1.02

0.74 0.94 0.92

0.74 0.99 0.92

64.3

275

1.99

16

1.07

1.07

1.04

0.98

904

39.5

200

0.83

20

1.07

1.02

1.18

1.20

Regan (1993) [30] I/1 194 I/2 176 I/3 194 I/4 194 I/5 165 I/6 165 I/7 186

26.0 23.4 27.4 32.3 28.2 22.0 30.4

77 77 77 77 79 79 79

2.40 1.20 1.40 1.20 1.50 0.80 0.80

10 10 10 10 10 10 10

0.88 1.00 0.99 0.98 0.79 1.07 1.08

– – – – – – –

1.00 1.18 1.17 1.17 0.93 1.24 1.26

0.92 1.10 1.08 1.05 0.85 1.18 1.14

Tolf (1988) [31] S2.1 S22 S2.3 S2.4

603 600 489 444

24.2 22.9 25.6 24.2

200 199 200 197

0.80 0.80 0.34 0.35

32 32 32 32

0.88 0.90 0.99 0.92

– – – –

0.90 0.92 0.96 0.90

0.86 0.89 0.91 0.86

Schaefers (1984) [32] 0 280 3 460

23.1 23.3

113 170

0.80 0.60

32 32

1.22 1.16

– –

1.25 1.12

1.09 1.04

Ladner et al. (1970/1973/1977) [32] P1 1662 M1 362 DA 6 183 DA 10 281 DA 11 324

27.6 31.4 29.6 31.6 30.0

240 109 80 80 80

1.30 1.30 1.80 1.80 1.80

32 32 16 16 16

1.47 1.21 1.39 1.43 1.46

– – – – –

1.33 1.29 1.28 1.44 1.48

1.24 1.07 1.14 1.28 1.32

Moe (1961) [32] S1-60 S2-60 S3-60 S4-60 S1-70 S5-60 S5-70 R1 R2 H1 M1A

389 356 364 334 393 343 378 312 394 372 433

23.3 22.1 22.6 23.8 24.5 22.2 23.0 26.6 27.6 26.1 20.5

114 114 114 114 114 114 114 114 114 114 114

1.10 1.50 2.00 2.60 1.10 1.10 1.10 1.40 1.40 1.10 1.50

38 38 38 38 38 38 38 10 10 38 38

1.21 1.02 0.96 0.82 1.26 1.19 1.36 1.21 1.22 1.03 1.23

– – – – – – – – – – –

1.31 1.10 1.02 0.84 1.30 1.28 1.40 1.12 1.16 1.19 1.27

1.11 0.94 0.86 0.71 1.09 1.09 1.18 1.10 1.14 0.99 1.09

Kinnunen/Nylander IA15a-5 IA15a-6 IA15c-11 IA15c-12 IA30a-24 IA30a-25 IA30c-30

(1960) [32] 255 275 334 332 430 408 491

27.6 25.4 31.0 28.4 25.6 24.3 29.2

117 118 121 122 128 124 120

0.80 0.80 1.80 1.70 1.00 1.10 2.10

32 32 32 32 32 32 32

1.14 1.25 1.06 1.09 1.22 1.19 1.15

– – – – – – –

1.11 1.21 1.01 1.04 1.24 1.22 1.18

0.95 1.05 0.85 0.89 1.09 1.07 1.00

Experimental models AR2 258 AR9 251 DF1 191 DF4 199 ID1 269

39.1 37.1 24.8 19.8 39.8

80 80 69 88 87

1.64 1.64 1.89 1.20 1.20

Guidotti [12] PG11 PG19 PG20

763 860 1094

31.5 46.2 51.7

208 206 201

Guandalini [13] PG2-b c PG5 PG3

439 550 2153

40.5 29.3 32.4

2050

Tomaszewicz [14] ND65-1-1 Li [15] P200

455

N.F. Silva Mamede et al. / Engineering Structures 48 (2013) 442–457 Table 6 (continued) Model

VEXP (kN)

IA30c-31 IA30d-32 IA30d-33 IA30e-34 IA30e-35

fcm (MPa)

540 258 258 332 332

q (%)

d (mm)

29.2 25.5 25.8 26.5 24.3

119 123 125 120 122

V EXP V MC2010

da (mm)

2.10 0.50 0.50 1.00 1.00

32 32 32 32 32

Mean Standard deviation COV

c

V EXP V Prop:

IIa

IVb

1.28 1.03 1.01 1.03 1.03

– – – – –

1.31 1.00 0.97 1.05 1.05

1.12 0.87 0.85 0.91 0.92

1.12 0.16 0.14

1.03 0.04 0.04

1.13 0.16 0.15

1.02 0.13 0.13

MC2010 approximation level II. MC2010 approximation level IV. Flexural/punching failure.

0.70 0.60

6.00

VNorm,MC2010

a b

V EXP V EC2

y = 1.1285x

y = 0.5412x-0.372 y = 0,5472x-0,520

FEA

4.00

MC2010

0.50

EXP

0.40

5.00

3.00 2.00

0.30

VR 0.20 1.00

2.00

3.00

4.00

5.00

6.00

1.00 1.00

2.00

3.00

(a)

4.00

5.00

6.00

(b)

Fig. 15. (a) Relation between slab rotation around the support region, normalized shear stress; (b) relation between the computed and MC2010 [24] slab rotation.

[26,2,8]. The brittleness of the material may be measured using a characteristic length of the material lch (Eq. (10)), while the brittleness of the structure may be defined as the ratio between structural size, usually defined as the effective depth d, and lch, according to Eq. (11). It is the ratio between the stored elastic energy in the structure and the dissipated energy during fracture process [27]. In other words, the size effect is due to the release of strain energy from the structure into the fracture zone as the cracking extends. The larger the structures, the greater is the energy release and the more fracture energy is needed to avoid a sudden failure. When eventually, the tensile strength is exceeded and a critical diagonal crack form, the residual stresses acting across the crack have to resist the impulse load that is connected to the energy release. Hence, the requirements on the tensile properties of concrete will increase as the size increases.

Ec  GF 2 fctm 2 d d  fctm ¼ lch Ec  GF

lch ¼

role in the punching load, contrary to tensile strength and modulus of elasticity, that by themselves has practically no influence. In this case, the punching capacity should be proportional to Eq. (12), meaning that elastic properties of concrete in tension are not important regarding ultimate punching load.

d Ec d  2 ¼ lch fctm Gf

ð12Þ

According to Faria et al. [28], EC2 [16] equation does not take into account, at least, in an accurate way, the influence of material/structure brittleness, since in its size effect parameter, the only variable is the effective depth, d. Based on their results, and in the evaluation of several experimental results, it is proposed a change to the size qffiffiffiffiffiffi  effect parameter, 1 þ 200 ; of EC2 [16], based on fracture d

ð10Þ

mechanics parameters, making it possible to attain more accurate predictions and less scatter in the results.

ð11Þ

6.2. Fracture mechanics parameters/size effect replacement

A higher value of d/lch corresponds to a more brittle structure, in structures sensitive to tensile stress-induced fracture. Gustafsson et al. [24] suggested that in equations regarding shear strength, where the size effect depends on effective depth d, this d should be replaced by d/lch. Nevertheless, based on the results obtained in the parametric analysis, developed by Faria et al. [28], regarding tensile properties of concrete, it was clear that only fracture energy has an important

To develop the modified equation it is necessary to determine an accurate relation between the numerical punching loads, VFEA, and GF/d. The normalized shear stress values, Eq. (13), based in EC2’s equations, were plotted against GF/d, as shown in Fig. 13.

V Norm;EC2 ¼

V FEA 0:18  ð100  ql  fc Þ1=3  u1  d

ð13Þ

456

N.F. Silva Mamede et al. / Engineering Structures 48 (2013) 442–457

The relation between normalized shear stress and GF/d is shown in Fig. 13, based in a potential regression function, and for reasons of simplification is assumed as 2.47(GF/d)0.18. It was applied a factor of 0.96 to the proposed equation, this value results from the average ratio between VFEA and VEXP, presented in Table 3. The proposed equation to predict the punching capacity for slabs without punching shear reinforcement according to the results presented above, with the fracture energy (GF) given in N/m, is as follows:

V Prop: ¼ 0:42 

 0:18 GF  ð100  ql  fc Þ1=3  u1  d d

ð14Þ

Comparing the FEA predicted punching load that results from the numerical analysis and the punching load calculated using the proposed equation a good agreement was obtained. The average ratio between the numerical and calculated values by the proposed equation punching loads is of 1.02 (Fig. 14), with a coefficient of variation 0.08. 6.3. Comparison with others experimental results In order to see the validity of the proposed equation a comparison between the punching loads of different experimental campaigns, from different investigators, with the predicted punching loads computed using MC2010 [24], EC2 [16] and the proposed equation (Eq. (14)) was done. The MC2010 approximation level IV predictions were computed for all models with a predicted failure mode of punching in Section 2. It was obtained applying the Eq. (8) where the rotations, w, were converted to vertical displacements, on the top of the specimens, as shown in Fig. 4. All concrete properties were computed based on compressive strength, using equations presented in Section 2. The comparison results are shown in Table 6. From the results presented, it may be stated that the proposed equation to predict the punching load gives a good approximation with the experimental results. In average, the EC2 [16] and MC2010 [24] predictions are conservative, underestimating the punching resistance. As presented in Table 6, the experimental models loads predictions using MC2010-IV results in better values than the MC2010 [24] prediction using the approximation level II, as expected. 7. Slab rotations The evaluation of the rotation of the slab around the support region outside the punching zone, thru wd, was carried out, comparing the numerical and the predicted results. Using Eq. (15), it is computed wd, where dMN is the vertical displacement where the radial bending moment is zero and c is the distance between the column face and the point where the bending moment is zero.

wd ¼

dMN d c

ð15Þ

To determine the rotation of the slabs, the numerical punching loads were normalized with basis on MC2010 [24], following Eq. (16), and plotted against wd. It was also plotted the normalized punching load, VR, versus wd, as proposed by Muttoni [25], Eq. (17). They can be observed in Fig. 15a. It was also introduced in Fig. 15a the rotation related to the experimental models (green dots1). The maximum aggregate size assumed in normalized punching load was 16 mm.

1 For interpretation of colour in Fig. 15, the reader is referred to the web version of this article.

V Norm;MC2010 ¼ VR ¼

V FEA pffiffiffiffi fc

d  b0 

3=4 wd 1 þ 15 16þd g

ð16Þ ð17Þ

The obtained rotation around the support region using the numerical analysis, wdFEA, was also plotted against the one obtained using the MC2010 [24], wdMC2010, as shown in Fig. 15b. By observation of the graphs, it is evident that both the potential regression and the MC2010 functions show a similar behaviour. The smaller the slab rotation, the greater is the numerical and the MC2010 [24] equation agreement. In both graphs, it may be seen that MC2010 [24] results are slightly higher than the numerical ones, in terms of slab rotation, and the difference is bigger for higher values of wd. The graph shows that the predicted shear resistance reduces as the slab rotation increases, but the FEA predictions shows a less steep decrease than it is assumed in MC2010. The experimental slab rotations presented in Fig. 15a (green dots), agrees well with the FEA predictions. 8. Conclusions The results of the non-linear 3D numerical analyses of the thirteen reference slabs models gives a satisfactory agreement with the experimental results, comparing deflections and punching strength. The numerical analysis shows that it is possible to effectively predict the real behaviour of column/slab connections, with a certain degree of accuracy. It can be assumed that this finite element software can be used for further research of reinforced concrete slab/column connections behaviour. Based on the finite element model capable of simulating 3D nonlinear behaviour of reinforced concrete elements, previously verified, it was developed a parametric analysis in order to determine the influence of some parameters on the punching load capacity in reinforced concrete flat slabs. The parameters studied were: the reinforcement ratio, the thickness of the slab, the concrete strength and the transverse column dimension. The main conclusions are:  Increasing the reinforcement ratio lead to an increment of the FEA predicted punching load approximately with their cube root, matching with EC2 [16] provisions.  The increment of reinforcement ratio, resulted in as increase of the FEA predicted punching capacity, but with less ductility. With higher reinforcement ratios the stiffness after cracking are bigger, with higher depths of the compression zone leading to more brittle failures.  The FEA predicted punching load was proportional on average to fc0:41 . That value is in between the one presented in EC2 [16] (1/3) and the one proposed by the MC2010 [24] (1/2).  Higher the concrete strength, higher the cracking loads. After cracking the responses tend to be similar with identical stiffness.  The increase of the thickness of the slab and column dimensions also increased the FEA predicted punching strength. It is also proposed an equation to predict the punching capacity, based on EC2 equation but with a change in the size effect parameter, where fracture mechanics parameters are introduced, namely, structure brittleness (d/lch). This parameter allows taking directly into account the type and size of aggregates. The results of the proposed equation match with the numerical analysis performed, and gives a good approximation with the several experimental results presented.

N.F. Silva Mamede et al. / Engineering Structures 48 (2013) 442–457

According to the results presented in Fig. 15 it is possible to conclude that results from FEA further support that punching strength reduces with slab rotation as described in MC2010, and as already been observed by Kinnunen and Nylander [29] and Muttoni [25]. The rotation of the slabs around the support region outside the punching zone was slightly smaller in the numerical models when compared with the MC2010 [24] predictions, and more so for higher values of wd. However, and accordingly to the results presented in Section 4.1, it is important to keep in mind that FEA results concerning slab rotations at failure, for slabs with low reinforcement ratios, may be somehow underestimated, due to the fact that the calculations stopped after the peak load was reached. Acknowledgements This work received support from the Fundação para a Ciência e Tecnologia – Ministério da Ciência, Tecnologia e Ensino Superior through Project PTDC/ECM/114492/2009. We would also like to thank CONCREMAT for making the slab specimens. References [1] Menétrey P. Numerical analysis of punching failure in reinforced concrete structures. PhD thesis no 1279. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland; 1994. 177 p. [2] Hallgren M. Punching shear capacity of reinforced high strength concrete slabs. PhD thesis. Bulletin 23. Department of Structural Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden; 1996. [3] Ozbolt J, Vocke H, Eligehausen R. Three-dimensional numerical analysis of punching failure. In: Proceedings of the international workshop on punching shear capacity of RC slabs. Royal Institute of Technology, Department of Structural Engineering, Stockholm; June, 2000. p. 65–74. [4] Eder M, Vollum R, Elghazouli A, Abdel-Fattah T. Modelling and experimental assessment of punching shear in flat slabs with shearheads. Eng Struct 2010;32(October):3911–24. [5] Hillerborg A, Modéer M, Petersson P-E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6:773–82. [6] Menétrey P. Synthesis of punching failure in reinforced concrete. Cem Concr Compos 2002;24:497–507. [7] Faria D, Biscaia H, Lúcio V, Ramos A. Punching of reinforced concrete slabs – numerical and experimental analysis and comparison with codes. In: Joint IABSE-fib conference, Dubrovnik, Croatia; May, 2010. [8] Staller M. Analytical studies and numerical analysis of punching shear failure in reinforced concrete slabs. In: Proceedings of the international workshop on punching shear capacity of RC slabs. Royal Institute of Technology, Departament of Structural Engineering, Stockholm; June, 2000. p. 374–6. [9] Ramos A. Punching in prestressed concrete flat slabs. Ph.D thesis. Lisbon: Technical University of Lisbon; 2003.

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