SHEAR DESIGN OF HOLLOW CORE SLABS

Transcription

1 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 Técnico, 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. Pg1

2 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. Pg2

3 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 f < 60 and 'good' quality of the bond. ck In this section are presented three hollow core slabs subjected to shear and bending moments. The experimental tests have been performed by VTT Building and Transport [11]. The main data are shown in Tab. 2 and load scheme in Fig. 10 (scheme 1). Symmetry Fig. 2. Exemple of equivalent cross section τ b Table 1 Point 1 Point 2 Point 3 Point 4 s φ τ b 0.8 f cu s Fig. 3. Bond law by BIGAJ [10] Pg3

4 Table 2 Unit VTT VTT VTT Characteristic cylinder strength of concrete (MPa) Thickness (mm) Cross sectional area (10 5 mm 2 ) Moment of inertia (10 8 mm 4 ) Centroid (mm) Minimum web width (mm) Top strands (-) Bottom strands (-) Top strands prestressing area (mm 2 ) Bottom strands prestressing area (mm 2 ) Top strands prestressing stress (MPa) Bottom strands prestressing stress (MPa) Distance between top strands and the extrados (mm) Distance between bottom strands and the intrados (mm) Specific fracture energy (10-5 MN/m) Span of the slab (mm) Width of support plates (mm) 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 F cr : kn F fail : kn F cr : kn F m : kn F um /F fail , , ,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 , 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 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. Pg4

5 (a) (b) (c) Fig. 4. Numerical results of VTT (a) Bond slips (b) Bond stress (c) Stress in strands Fig. 5. Crack patterns of VTT Pg5

6 (a) (b) (c) Fig. 6. Principal stress (a) maximum (b) minimum (c) tensors (a) (b) Fig. 7. Numerical results of VTT Longitudinal strain (a) on release (b) on ultimate load Pg6

7 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 (τ 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 F c and F s. 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: 2 VRd = β bw d v f (1) ctm γ c In Eq 1, the term f ck in the original equation of MCFT [4] was replaced by fctm 2 and likewise should not be taken greater than 8 MPa. According to the requirements of Eurocode 2: Pg7

8 f ctm 2 3 f ctm = 0, 30 f ck C C50/60 f = + cm 2, 2ln 1 C > C50/60 10 (2) The coefficient β 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, ε 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]. β = 0, ( ε ) ( ) x s xe (3) Figure 9 shows the deformation of the longitudinal reinforcement due to bending moment and shear. When cot θ is taken equal to 2, as suggested by the Canadian code [1], ε x deformation can be calculated by Eq. 4. In the methodology presented in this paper for hollow core slabs, the term ( E cbwd v ) 2 should be added the reinforcement stiffnesses in the denominator when the design bending moment is greater than the cracking moment, as a f p0 x is calculated assuming a linear variation in the conservative simplify. The term ( ) transmission lengths, whereby in the Eurocode 2. According as [4], the shear depth d v and is taken as 0.9d. M d + V A f ( x) d v d ps p0 ε x = (4) 2 E p Aps Fig. 9. Longitudinal strain due to bending moment and shear [3] The size effect term is shown in Eq. 5, and crack spacing s x can be taken as the distance between layers of longitudinal reinforcing bars or equal d v, for member with longitudinal reinforcement only on the flexural tension side. The maximum coarse aggregate size a g Pg8

9 should be taken as zero for high-strength concrete (f ck > 70 MPa) or lightweight concrete, according as [4]. where sx dv = 0. 9d 35s = s (5) x sxe 0, ag x 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 d v /2 from the application point of concentrated loads and a section distant d v 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), Università dell Aquila (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. Pg9

10 Loading schemes 1 P/2 P/2 P/2 P/2 L/7.2 L/7.2 L/7.2 L/7.2 L P/2 P/2 L/8 L/4 P/2 L/4 P/2 L/8 2 L 3 P/2 P/2 a 300 P/2 300 P/2 a L P a 4 5 P/2 P/2 a 300 L L P a 6 L CS Fig. 10. Loading schemes [15] Pg10

11 Table 4 Performed by Thickness range (mm) Load scheme (Fig. 10) Type of holes CSA Proposed methodology No. Mean Coefficient Mean Coefficient value of variation (%) value of variation (%) using the mean values of the material properties 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, ,86 27,9 0,92 28, ,80 17,6 0,90 17, ,88 18,3 1,00 15, ,81 18,0 0,92 16, ,90 19,6 0,93 16, ,90 19,6 1,02 15, ,03 0,0 1,16 0, ,81 8,5 0,89 8, ,81 20,0 0,92 18, ,78 14,5 0,85 12, ,96 20,4 1,05 21,7 circular 39 0,83 23,0 0,95 21,5 irregular 90 0,83 17,9 0,93 16,8 f ck ,84 24,8 0,94 23,4 f (MPa) ck 60 < f ck ,84 16,1 0,93 15,3 90 < f ck ,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 Proposed Tests CSA Slab methodology Numerical model F cr : kn F fail : kn ε x (%o) F um /F fail ε x (%o) F um /F fail F cr : kn ε x (%o) F um /F fail ,031 1,05-0,014 1, ,085 1, ,077 0,97-0,064 1, ,083 1, ,088 0,83-0,060 0, ,091 0,98 Pg11

12 (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. Pg12

13 (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. Pg13

14 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. q Fig. 13. Axis and load scheme x Pg14

15 (a) (b) (c) Fig. 14. Maximum uniformly distributed load due to shear (continuous line) and due to bending moment (dotted line) (a) VTT (b) VTT (c) VTT Pg15

16 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 Coordenação de Aperfeiçoamento de Pessoal de Nível 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, 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, 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 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, 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, Pg16

17 6. Jendele, L., and Cervenka, J., Finite element modelling of reinforcement with bond, Computers and Structures, 84, 2006, Janney, J. R., Nature of bond in pre-tensioned prestressedconcrete, Journal of the American Concrete Institute, May 1954, Proceedings Vol. 50, p Fusco, P. B., Técnica de armar as estruturas de concreto, Editora Pini, São Paulo, Cervenka J, Jendele L. Atena user s manual, Part 1 7. Prague: Cervenka Consl.; 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, Pajari, M. Resistance of Prestressed Hollow Core Slabs against Web Shear Failure. ESPOO, Finland, 2005, VTT Research Notes Comité Europén de Normalisation. EN :2004 Eurocode 2: Design of concrete Structures. Part 1-1: General rules and rules for buildings. CEN, Brussels, Canadian Standards Association. Design of Concrete Structures. CSA, Mississauga, Ontario, Canada, CSA Committee. A 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 Bertagnoli G., and Mancini, G., Failure analysis of hollow-core slabs tested in shear, Structural Concrete, 2009, 10, No. 3, pp Pg17