Shear Design of a Hollow Core Slab

Araujo, Loriggio, and Camara

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SHEAR DESIGN OF HOLLOW CORE SLABS

Carlos M. Araujo, Msc, PPGEC, Universidade Federal de Santa Catarina, Brazil Daniel D. Loriggio, Dr, Dept. Civil Eng., Universidade Federal de Santa Catarina, Brazil

Jose Camara, Dr, Dept. Engineering and Architecture, Instituto Superior Tcnico, Portugal

ABSTACT

The current design methods for the hollow core shear resistance are derived from experimental results and elastic theories that are not consistent with the behavior in the ultimate limit state.

In this paper, an analytical methodology adopted from the modified compression field theory (MCFT) and safety concepts from Eurocode 2 is properly presented and evaluated with experimental data available in the literature, proved to be accurate and simple enough for use in the design. Comparisons with the codes CSA A23.3 and Eurocode 2 are also presented.

For the validity of the process presented, the support region specific characteristics of this type of slab, as the anchorage of prestressing strands, prestressing dispersion and short support lengths, are discussed with nonlinear numerical models considering the bond between the strand and concrete.

Keywords: Prestressed concrete, Hollow Cores, Shear design.


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INTRODUCTION

The development of building techniques of hollow core slabs allowed its wide use in different types of structure, although some characteristics of prefabricated elements still cause uncertainties in the scientific community. The shear strength and failure mechanisms in the region of support are some of the issues raised including the large proportion of voids in cross sections without shear reinforcement, the lengths of support usually small, the anchorage of strands and the dispersion of prestressing force.

The traditional shear strength models are based on elastic theories and empirical results, with methods costly and not always accurate enough. Accordingly, the application of this procedure in the design of hollow cores slabs is not appropriate.

In this paper, an accurate and simple enough methodology for use in design, based on modified compression field theory MCFT [1, 2, 3, 4, 5] and safety concepts from Eurocode 2, is properly presented and verified with experimental and numerical results.

SHEAR FAILURE MECHANISMS

The possible shear failure mechanisms conceptually accepted in hollow cores slabs are shear tension failure, flexural shear, and anchorage failure of strands. The first mechanism is the most common in this type of slab, which in a non-fissured region arises a diagonal crack that propagates toward the support and the compression area, causing a brittle rupture (Fig. 1). In flexural shear failure, the crack is initiated by a vertical bending crack that develops in a diagonal crack.

Fig. 1. Shear tension failure [6]

The last mechanism, according with first ideas of JANNEY [7], occurs when bending cracks happens within the transmission length and a variation of the strand stress could cause the anchorage failure of strands. This situation is not common in hollow core slabs, where the prestressing have a good performance in the cracking control. This statement is valid, subject to compliance with the limits of initial prestressing stress to control the tensile force of bursting, splitting and spalling.


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FINITE ELEMENTS MODELS

The numerical models presented in this paper were generated with the non-linear finite element program ATENA V4 [9]. The hollow core slabs, using equivalent cross sections (Fig. 3) were modeled with plane quadrilateral isoparametric elements. The constitutive model adopted for the concrete was SBETA [9] with the fictitious crack model based on the exponential crack-opening law and fracture energy. The prestressing strands were modeled with one-dimensional geometry and bond between strand and concrete through bond-slip relationship shown in Fig. 3 [10]. The Tab. 1 shows the parameters for the definition of this relationship for concrete with 60


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Table 2 Unit VTT33.200 VTT109.265 VTT148.320 Characteristic cylinder strength of concrete (MPa) 47.5 51.8 43.5 Thickness (mm) 200 265 320 Cross sectional area (105 mm2) 1.19 1.72 2.03 Moment of inertia (108 mm4) 6.03 15.0 25.9 Centroid (mm) 103 137 164 Minimum web width (mm) 238 242 263 Top strands (-) - - 2 9.3 Bottom strands (-) 7 12.5 10 12.5 11 12.5 Top strands prestressing area (mm2) - - 104 Bottom strands prestressing area (mm2) 651 930 1023 Top strands prestressing stress (MPa) - - 900 Bottom strands prestressing stress (MPa) 1100 1000 1000 Distance between top strands and the extrados (mm) - - 49 Distance between bottom strands and the intrados (mm) 40 39 51 Specific fracture energy (10-5 MN/m) 9.56 9.44 8.40 Span of the slab (mm) 4958 4957 5945 Width of support plates (mm) 40 40 40

All models showed good agreement with the values measured in the laboratory, as shown in Tab. 3, and qualitatively, obtained similar results, from which can be drawn some important conclusions: under the load applied, the prestressing tension had a little variation between the release of the prestressing and the ultimate load, as well as the bond slips and bond stress; in the sections close to the application of the loads, the plane section hypothesis is acceptable. This analysis reaffirms the shear failure mechanisms mentioned in the previous item.

Table 3 Slab Tests Numerical model Fcr: kN Ffail: kN Fcr: kN Fm: kN Fum/Ffail

33.200 81 108 96 112 1,04 109.265 - 178 - 188 1,06 148.320 223 238 214 233 0,98

Fig. 4 shows the bond slip, bond stress and prestressing stress along the slab for the release of the prestressing and ultimate load to the slab VTT 109.265, where also shows the values calculated with the recommendations from Eurocode 2 [12]. The shear failure mechanism in all examples was shear tension failure, clearly shown in Fig. 5 for the slab VTT 109.265. The anchorage failure of strands was discarded, for the reason that inside the transmission length did not show bending cracks and there were compression stress in the bottom side (Fig. 6). The Fig. 7 brings the longitudinal deformations along the slab to the stages of prestressing release and ultimate load.


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(a) (b)

(c) Fig. 4. Numerical results of VTT 148.320 (a) Bond slips (b) Bond stress (c) Stress in strands

Fig. 5. Crack patterns of VTT 148.320


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(a)

(b)

(c) Fig. 6. Principal stress (a) maximum (b) minimum (c) tensors

(a)

(b) Fig. 7. Numerical results of VTT 148.320

Longitudinal strain (a) on release (b) on ultimate load


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PROPOSED METHODOLOGY

The methodology presented in this paper is based on the modified compression field theory MCFT [1, 2, 3, 4, 5], also base of the CSA A23.3 code [13], and the safety concepts from Eurocode 2 [12].

BASE OF SHEAR RESISTENCE MECANISMS MODEL

The bases of the model, to the case of members without transversal reinforcement and strands with line geometry, can be seen in Fig. 8 [1]. The free body diagram of this figure cuts the longitudinal reinforcement, the flexural compression region and follows the angle of the shear crack diagonal. In this case, the shear is assumed to be carried by aggregate interlock stress (ci) and shear stress in the flexural compression region. With the equilibrium can be noted that the horizontal component of shear stresses on diagonal crack causes an increase in tension in the longitudinal reinforcement, representing one of the causes of shear-moment interaction. The bending moment is carried by forces Fc and Fs. The dowel action of longitudinal reinforcement is ignored and the aggregate interlock resistance is estimated at only in one depth.

Fig. 8. Free diagram of basic shear resisting mechanism [1]

SHEAR DESIGN MODEL

For elements without shear reinforcement, as discussed in the previous section, the shear will be resisted only by aggregate interlock component. Therefore, the shear resistance in the proposed methodology is taken as:

ctmvwc

Rd fdbV 2

= (1)

In Eq 1, the term ckf in the original equation of MCFT [4] was replaced by ctmf2 and likewise should not be taken greater than 8 MPa. According to the requirements of Eurocode 2:


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32

300 ckctm ff ,= C C50/60

+=

10122 cmctm

ff ln, C > C50/60 (2)

The coefficient models the strain effect, the first term of Eq. 3, and the size effect, the second term of the same equation. The strain effect is considered by controlling the longitudinal strain at mid-depth of the member, x, which can be taken as a good approximation equal to half the longitudinal reinforcement deformation, whereas the concrete deformation in compression is small compared to steel deformation [4].

( ) ( )xex s++= 10001300

15001400

, (3)

Figure 9 shows the deformation of the longitudinal reinforcement due to bending moment and shear. When cot is taken equal to 2, as suggested by the Canadian code [1], x deformation can be calculated by Eq. 4. In the methodology presented in this paper for hollow core slabs, the term ( ) 2vwc dbE should be added the reinforcement stiffnesses in the denominator when the design bending moment is greater than the cracking moment, as a conservative simplify. The term ( )xf p0 is calculated assuming a linear variation in the transmission lengths, whereby in the Eurocode 2. According as [4], the shear depth dv and is taken as 0.9d.

( )psp

ppsdvdx AE

xfAVdM2

0+= (4)

Fig. 9. Longitudinal strain due to bending moment and shear [3]

The size effect term is shown in Eq. 5, and crack spacing sx can be taken as the distance between layers of longitudinal reinforcing bars or equal dv, for member with longitudinal reinforcement only on the flexural tension side. The maximum coarse aggregate size ag


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should be taken as zero for high-strength concrete (fck > 70 MPa) or lightweight concrete, according as [4].

x

g

xxe s

a

ss 850

1535

,+

= (5)

where dds vx 90.=

The section which should be checked depends on the type of loading and should be where the width of the critical shear crack can be satisfactorily represented by the strain, according Muttoni;Ruiz [14]. In the methodology presented here, was found appropriate to consider a section distant dv/2 from the application point of concentrated loads and a section distant dv from the support in the case of uniformly distributed load.

COMPARISONS WITH EXPERIMENTAL DATA AND CODES

With the methodology presented in this paper comparisons were made with an experimental database, meeting in Bertagnoli; Mancini [15], where it was presented a multi-criteria design process in the company of the Eurocode 2 recommendations. The experimental campaigns have been performed by VTT Building and Transport (Finland), TNO (Netherlands), TU-Delft (Netherlands), Universit dellAquila (Italy), Istituto di Ricerche e Collaudi M.Masini (Italy).

The 129 hollow core slabs from database were analyzed using the methodology of this paper and the results presented in Tab. 4 divided into five criteria: laboratory, thickness, load scheme, type of holes and, cylinder compressive strength of concrete. The same table shows the relationship between ultimate theoretical load and ultimate experimental load, where the theoretical load obtained using the mean values and using the design values of the material properties. For all slabs was taken prestressing losses of 5%. The Tab. 5 shows the results of specific tests adopted in the numerical analyzes, compared with the CSA and the proposed methodology.

Figs. 11 and 12 show graphically relationship between ultimate theoretical load and ultimate experimental load. In the last figure can be seen that any tests has value greater than one.

With the same input data, Bertagnoli; Mancini [15] obtained a mean value of 0.89 with a coefficient of variation 25% using the mean values and mean value of 0.58 with a coefficient of variation 22% using the design values.


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Loading schemes

1

2

3

4

5

6

Fig. 10. Loading schemes [15]

L

L/7.2 L/7.2 L/7.2L/7.2P/2 P/2 P/2 P/2

L/4P/2P/2

L

L/8 L/4P/2 P/2

L/8

a300P/2P/2

L

a 300P/2 P/2

a

P

L

a 300P/2 P/2

L

CS

a

P

L


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Table 4 CSA Proposed methodology

No. Mean value

Coefficient of variation (%)

Mean value

Coefficient of variation (%)

using the mean values of the material properties

Performed by

VTT 46 0,84 17,6 0,94 15,9 TNO 39 0,85 18,4 0,98 17,1 TU-D 16 0,81 16,0 0,89 13,4 USA 14 0,68 22,5 0,77 21,1

MANSINI 14 0,94 17,9 1,00 17,9

Thickness range (mm)

155-200 13 0,86 27,9 0,92 28,2 240-260 45 0,80 17,6 0,90 17,0 300-320 35 0,88 18,3 1,00 15,6 360-400 24 0,81 18,0 0,92 16,7 420-500 12 0,90 19,6 0,93 16,4

Load scheme

(Fig. 10)

1 19 0,90 19,6 1,02 15,0 2 1 1,03 0,0 1,16 0,0 3 10 0,81 8,5 0,89 8,3 4 66 0,81 20,0 0,92 18,9 5 25 0,78 14,5 0,85 12,3 6 8 0,96 20,4 1,05 21,7

Type of holes

circular 39 0,83 23,0 0,95 21,5 irregular 90 0,83 17,9 0,93 16,8

ckf (MPa) fck 60 39 0,84 24,8 0,94 23,4

60 < fck 90 75 0,84 16,1 0,93 15,3 90 < fck 120 15 0,80 19,2 0,92 18,0

Total 129 0,83 19,0 0,93 18,0 using the design values of the material properties

Total 129 0,60 22,1 0,64 19,0

Table 5

Slab Tests CSA Proposed

methodology Numerical model

Fcr: kN Ffail: kN x (%o) Fum/Ffail x (%o) Fum/Ffail Fcr: kN x (%o) Fum/Ffail 33.200 81 108 -0,031 1,05 -0,014 1,09 96 -0,085 1,04 109.265 - 178 -0,077 0,97 -0,064 1,00 - -0,083 1,06 148.320 223 238 -0,088 0,83 -0,060 0,90 214 -0,091 0,98


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(a)

(b) Fig. 11. Relationship between ultimate theoretical load and

ultimate experimental load, where the theoretical load was obtained using the mean values material properties.


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(a)

(b) Fig. 12. Relationship between ultimate theoretical load and

ultimate experimental load, where the theoretical load was obtained using the design values of the material properties.


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USUAL CASES IN DESIGN

In the usual cases of hollow core slabs are used with uniformly distributed loads and with high values for the ratio L/h between design span and the floor thickness (Fig. 14), where the stresses verifications in the serviceability limit states are usually the critical cases. Due to the cross sections characteristics of this type of slab, the behavior in the ultimate limit state differs from the elements with solid cross section, in that the limits imposed by the shear and bending moment are closer to the hollow cores.

This behavior can be better understood with the aid of curves of maximum load that each section allows until to reach the ultimate limit state due to bending moment or due to shear. Fig. 15 shows these curves for three different cross sections and various ratios L/h. The cross sections used are the same of the examples presented in the numerical analysis of this paper. Continuous lines in the charts represent the maximum load in each section limited by the shear and the dotted line, the maximum load limited by the bending moment. The results presented are shown for half the structure, taking advantage of its symmetry.

Fig. 13. Axis and load scheme

q

x


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(a)

(b)

(c) Fig. 14. Maximum uniformly distributed load due to shear (continuous line) and due to bending moment (dotted line)

(a) VTT 33.200 (b) VTT 109.265 (c) VTT 148.320


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CONCLUSIONS

The shear failures of hollow core slabs, it always been sudden and brittle, are difficult to be characterized and measured in conventional laboratory tests. Therefore, the main models verification applied to these slabs does not match with nonlinear mechanisms present in the behavior under shear. However, such behavior can be adequately explained with theory for the shear strength of reinforced concrete presented in Collins et al. [4]. Using this theory, numerical modeling nonlinear, experimental results and the safety concepts from Eurocode 2 [12], in this paper was possible to propose a methodology for shear verification of hollow core slabs, bringing as main advantages: it is based on a general theory of shear, applied to elements of reinforced and prestressed concrete; compared to experimental results and codes, gives very good results; it has use simple and appropriate to the design; easy to understand.

The nonlinear analyses presented in this paper represent well the behavior of hollow core slabs. In these analyses, stress variations in the prestressing steel, between the stages of release and ultimate limit state, are very small and the stresses close to support on the bottom side of the slab usually are compression. This indicates that an anchorage failure of strands could not occur and that the stress concentration in the web produces a tension shear failure.

ACKNOWLEDGEMENTS

The authors wish to express their sincere thanks to Coordenao de Aperfeioamento de Pessoal de Nvel Superior (CAPES) for scholarship granted to the first author of this paper.

REFERENCES

1. Bentz E. C. and Collins M. P. Development of the 2004 CSA A23.3 shear provisions for reinforced concrete. Canadian Journal of Civil Engineering, 2006, 33, No. 5, 521534.

2. Bentz E. C., Vecchio F. J. and Collins M. P. The simplified MCFT for calculating the shear strength of reinforced concrete elements. ACI Structural Journal, 2006, 103, No. 4, 614624.

3. Collins, M. P., Mitchell D., Adebar, P., and Vecchio, F. J. A general shear design method, ACI Structural Journal, V. 93, No. 1, January-February 1996, pp. 36-45.

4. Collins, M. P., Bentz, E. C., Sherwood, E. G., and Xie, L., An adequate theory for the shear strength of reinforced concrete structures, Magazine of Concrete Research, V. 60, No. 9, 2008, 635650.

5. Vecchio F. J. and Collins M. P. The modified compression field theory for reinforced concrete elements subjected to shear. ACI Journal, Proceedings, 1986, 83, No. 2, 219231.


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6. Jendele, L., and Cervenka, J., Finite element modelling of reinforcement with bond, Computers and Structures, 84, 2006, 17801791.

7. Janney , J. R., Nature of bond in pre-tensioned prestressedconcrete, Journal of the American Concrete Institute, May 1954, Proceedings Vol. 50, p. 717.

8. Fusco, P. B., Tcnica de armar as estruturas de concreto, Editora Pini, So Paulo, 1995.

9. Cervenka J, Jendele L. Atena users manual, Part 17. Prague: Cervenka Consl.; 20002002.

10. Bigaj AJ. Structural dependence of rotation capacity of plastic hinges in RC beams and slabs. Ph.D. Thesis, Department of Civil Engineering, Delft University of Technology, 1999.

11. Pajari, M. Resistance of Prestressed Hollow Core Slabs against Web Shear Failure. ESPOO, Finland, 2005, VTT Research Notes 2292.

12. Comit Europn de Normalisation. EN 1992-1-1:2004 Eurocode 2: Design of concrete Structures. Part 1-1: General rules and rules for buildings. CEN, Brussels, 2004.

13. Canadian Standards Association. Design of Concrete Structures. CSA, Mississauga, Ontario, Canada, 2004. CSA Committee. A23.3

14. Muttoni, A., Ruiz, M. F., Shear Strength of Members without Transverse Reinforcement as Function of Critical Shear Crack Width, ACI Structural Journal, V. 105, No. 2, March-April 2008, pp. 163-172.

15. Bertagnoli G., and Mancini, G., Failure analysis of hollow-core slabs tested in shear, Structural Concrete, 2009, 10, No. 3, pp. 139152.

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Shear Design of a Hollow Core Slab