Experimental strengths from the present tests and from others reported in the literature have been compared with resistances calculated by the three methods described previ-ously. The shear reinforcement in the tests by Regan,7 Regan and Samadian,8 Beutel,9 and Birkle10 was double-headed studs made from either deformed or plain round bars. In the tests by Gomes and Regan,11 it was slices of steel I-beams with the flanges acting as anchorages. The shear reinforce-ment was positioned radially unless noted as ACI type in Table A1 in Appendix A.
The calculations of punching resistances were made using the expressions given previously, with their limits generally respected. Exceptions to this were as follows.
For ACI 318 and EC2, the limits on so/d and sr/d were given a little tolerance. Values of sr/d up to 0.8 were treated as acceptable, and for EC2 the lower limit so/d < 0.3 was waived with values going down to 0.24. EC2 does not envisage the use of plain round shear reinforcement, but this has been ignored, and lower limits on d for the use of shear reinforcement were ignored. (The least effective depth in the tests used was 124 mm [4.9 in.] in six slabs by Birkle.10)
The CSCT shear strengths were calculated using slab rotations calculated with Eq. (13), in which Vflex was calcu-lated with Eq. (14). The stresses in the shear reinforcement were calculated in accordance with the recommendations in (5). The resulting slab rotations were slightly greater than the measured slab rotations, as illustrated in Fig. 5. The predicted shear strengths typically increase by less than 5%
for the slabs tested in this program if measured rotations are used instead of calculated rotations. For the CSCT, there Fig. 13—Punching strengths according to CSCT: Slab C1.
(Note: 1 kN = 0.2248 kip.)
are only a few cases in which there were two perimeters of shear reinforcement within a distance d from the support, but there are some where a second layer was not much further out (all of the slabs by Gomes and Regan11 where the distances varied from 1.0d to 1.04d and Slabs 2, 3, 9, and 12 by Birkle10 where the distances were from 1.09d to 1.18d).
The second perimeter has been included in ΣAsw, where the distance was less than 1.05d.
Details of the individual slabs and the results obtained are given in Appendix A, while Tables 3 and 4 summarize the results of the comparisons for slabs without and with shear reinforcement.
Although there are only six results in Table 3, it is note-worthy that, for all the methods of calculation, Vu/Vcalc
decreases with increasing effective depth. This is not surprising for ACI 318, which has no size factor, or for EC2, where the size factor is constant for d ≤ 200 mm (7.9 in.).
It is surprising for the CSCT, which includes a size factor taking account of the effective depth and the maximum size of aggregate. The best correlation in the table is that for EC2*, which is the same as EC2, but without the limit on k = 1 + (200 d/ ). With this modification, however, the mean Vu/Vcalc is low, and the coefficient of 0.18 in Eq. (7) would need to be reduced.
Table 4 summarizes the results of the comparisons with the 45 slabs with shear reinforcement, and all three methods of calculation are broadly satisfactory.
The coefficient of variation of Vu/Vcalc decreases with increasing complexity in the method. The ACI method is the simplest and gives a coefficient of variation of 0.162.
The EC2 method is slightly more complex, and reduces the coefficient by 0.026, while the CSCT is considerably more complicated, but gives a further reduction of 0.015.
There are no unsafe predictions from ACI 318, but there are four from EC2 and the CSCT, with the lowest values of Vu/Vcalc being 0.88 for EC2, and 0.90 for the CSCT. An overall reduction of Vcalc by 4% would make each of these methods safe in the sense of limiting the probability of an unsafe prediction to 5%, assuming a statistically normal distribution of Vu/Vcalc.
ACI 318 and EC2 are basically empirical, but the CSCT claims a rational basis. Unfortunately, its modeling of slab deformations is incorrect. The rotation ψ is predicted satis-factorily by Eq. (13), but, as can be seen from Fig. 4, it is not divided equally into movements at the column face and in a shear crack. In addition, the surfaces at which failure occurs are not at 45 degrees to the slab plane, but have variable geometries. Refer to the section entitled, “Ultimate loads and modes of failure.”
In nine tests by Ferreira1 and five by Birkle,10 EC2 predicts outside failures for slabs that actually failed in the shear-re-inforced zones. Its predictions of failure modes in the other series are generally good. The main cause of the problem seems to be the overestimation of VR,cs. For the slabs by Ferreira,1 the mean Vu/VR,cs is 0.98, and the coefficient of variation is 0.061. For Birkle’s tests,10 the corresponding figures are 0.88 and 0.101, but would be improved if the four slabs with so/d less than 0.3 were excluded. The situ-ation could be improved by either a reduction of VR,c or by interpreting the code’s expression for the design value of the stud stress (fywd,ef) as not requiring a safety factor so long as fywd,ef is less than fyw/1.15—that is, by taking fyw,ef as (250 + 0.25d) ≤ fyw.
The EC2 predictions of VR,out for slabs with radial arrange-ments of shear reinforcement are generally satisfactory, though perhaps over-conservative for the slabs by Gomes and Regan.11 In these slabs, the 0.64d widths of the I-beam flanges reduced the clear tangential spacing of the shear reinforcement. This could be allowed for, and would make Vu/VR,out for these tests similar to those for other series.
The strength of Ferreira’s1 Slab C4 with an ACI cross arrangement of studs which failed inside is predicted very conservatively with Vu/VR,out = 1.69. For Birkle’s10 slabs with the ACI layout, which failed outside, the strengths are well predicted with Vu/VR,out = 1.21. Slab C4 was unrealistic in relation to EC2 design because it had six perimeters of studs, while the same strength would be calculated for a slab with two perimeters of studs. The performance of C4 is in marked Table 3—Comparisons with test results for slabs
without shear reinforcement
Mean 1.16 0.99 1.04 0.93
Coefficient of variation 0.13 0.15 0.12 0.11
*EC2 calculations as for EC2, but with no upper limit on k = 1+√(200/d).
Table 4—Statistics of Vu/Vcalc for slabs with shear reinforcement
No. of tests
ACI 318 EC2 CSCT
Mean COV Mean COV Mean COV
Ferreira1
11 1.47 0.148 1.20 0.160 1.23 0.108
Regan7 and Regan and Samadian8
9 1.56 0.100 1.06 0.087 1.05 0.102
Beutel9
6 1.72 0.093 1.34 0.087 1.16 0.101
Gomes and Regan11
10 1.76 0.134 1.28 0.081 1.28 0.083
Birkle10
9 1.35 0.185 1.09 0.105 1.06 0.050
Total
45 1.56 0.162 1.19 0.136 1.16 0.121
contrast to that of slabs by Mokhtar et al.,12 with up to eight perimeters of studs on stud rails. Their strengths are quite well predicted by EC2.
Influence of slab rotation on shear resistance provided by concrete
Unlike ACI 318 and EC2, the CSCT predicts that the shear resistance provided by concrete reduces with slab rotation, which is assumed to be proportional to (V/Vflex)1.5. This has significant implications for design because Vu/Vflex may be close to one for practical slabs. Consequently, the CSCT can require significantly greater areas of shear reinforcement than EC2. The influence of slab rotation on Vu/Vcalc for the CSCT is illustrated by Fig. 14, where the ratio is plotted against the normalized rotation ψd/(16 + dg), with ψ calcu-lated by Eq. (13). There is a clear tendency for the CSCT to become more conservative with increasing slab rotation, which suggests that Vu is either independent of ψ, or that the CSCT overestimates the influence of rotation. In the case of outside failure, this is to be expected as the rotation develops close to the column and not within a crack outside the shear reinforcement.
Muttoni4 plots Vu/u1d fc against ψd/(16 + dg) for 99 tests of slabs without shear reinforcement, and shows that exper-imental strengths are close to the predictions of Eq. (15), which may appear to contradict the preceding paragraph.
This, however, is not the case. Figure 15 shows Vu/u1d fc plotted against ψd/(16 + dg) for slabs similar to Ferreira’s1 C2, but without shear reinforcement, and with ρ from 0.4 to 4.0%. The values of Vu have been calculated by EC2 and the CSCT, u1 is the CSCT control perimeter, and ψ is the rotation calculated by Eq. (13). It can be seen that the effect shown in Muttoni’s figure can be accounted for by the EC2 relationship between VR,c and ρ1/3 without involving ψ in the calculations.
Failure surface and locations of shear reinforcement
ACI 318 and the CSCT assume punching surfaces to be inclined at 45 degrees, while EC2 assumes an inclination of 26.6 degrees. These are simplifications of a reality in which the angle increases with increasing shear reinforcement
(refer to Carvalho et al.13). Reasonable results can, however, be obtained in most instances with fixed angles, provided the expressions for VR,c and VR,s are constructed appropriately.
This seems to be the case with ACI 318 and EC2, with the former considering d/sr perimeters of studs acting at fyw, and the latter assuming 1.5d/sr perimeters acting at fyw,eff, which is typically approximately 0.7fyw for test slabs. The situation is more complex in the CSCT, as its numbers of perimeters depend on the exact distances of studs from the column.
There are cases where the use of different failure surfaces has a significant effect. Slabs 2, 3, 10, and 11 by Gomes and Regan11 are an example. In these slabs so = sr ≅ 0.5d, Asw = 226 mm2 (0.36 in.2) in Slabs 2 and 10, and 325 mm2 (0.5 in.2) in Slabs 3 and 11. Slabs 2 and 3 had two perime-ters of shear reinforcement, while Slabs 10 and 11 had three.
Thus, for ACI 318, two perimeters are taken into account for all the slabs, while in EC2, two perimeters are active in Slabs 2 and 3, but three are active in Slabs 10 and 11. All four slabs failed inside their shear reinforced zones. The EC2 ratios Vu/VR,cs are 1.26 and 1.21 for Slabs 2 and 3, and 1.28 and 1.31 for Slabs 10 and 11. The ACI ratios are 1.39 and 1.28 for Slabs 2 and 3, and 1.58 and 1.61 for Slabs 10 and 11, showing that the extra perimeter of shear reinforcement had an effect.
The same slabs illustrate a problem with the CSCT’s considering the active shear reinforcement to be exactly that within a distance d from a column rather than using an expression in d/sr. Because of variations of effective depths, (so + sr) = 1.05d in three cases instead of the intended 1.0d.
This discrepancy has been ignored in the calculations for Table A1, but if the CSCT were applied strictly the ratios Vu/VR,cs, which are already unusually high for Slabs 10 and 11, would be significantly increased.
CONCLUSIONS
Comparisons have been made between the punching strengths of 45 slabs with and six slabs without shear reinforce-ment and those predicted by ACI 318, EC2, and the CSCT.
ACI 318 is the simplest method, and gives only one unsafe prediction, which is for a slab without shear reinforcement.
Its mean value of Vu/Vcalc for slabs with shear reinforcement is rather high, and the coefficient of variation is 0.162. The Fig. 14—Influence of slab rotation on Vu/Vcalc CSCT.
Fig. 15—Influence of slab rotation on shear strength.
most apparent weakness is the lack of any treatment of the depth effect in the shear resistance from the concrete.
EC2 is only slightly more complicated, but reduces the coefficient of variation of Vu/Vcalc by 0.026. The mean is also reduced, and there are four unsafe predictions for slabs with and four for slabs without shear reinforcement. The simplest way to obtain a characteristic level of safety would be to reduce the constant in the expression for the concrete component of resistance and extend the range of slab depths affected by the depth factor.
The CSCT is considerably more complex, and reduces the mean and coefficient of variation of Vu/Vcalc, by a further 0.015. There are unsafe predictions for four slabs with and one without shear reinforcement.
The CSCT goes further than the other two approaches in attempting to model the slab behavior. Although its expres-sion for total slab rotation seems good, the assumption that half of this rotation occurs in the critical shear crack is incor-rect, as nearly all of it is at the column face. The assumption that all critical cracks are at 45 degrees to the slab plane is also incorrect.
The relationship assumed between the concrete compo-nent of punching resistance and slab rotation is not confirmed by the test data, and the determination of the area of active shear reinforcement as that crossed by a particular 45 degree surface seems less satisfactory than considering d/sr perimeters.
AUTHOR BIOS
ACI member Maurício P. Ferreira is a Lecturer at the Federal Univer-sity of Para, Belem, Brazil. He received his PhD from the UniverUniver-sity of Brasília, Brasília, Brazil, in 2010. His research interests include ultimate shear design, strut and tie, and nonlinear finite element modeling.
ACI member Guilherme S. Melo is an Associate Professor at the Univer-sity of Brasilia, where he was Head of the Department of Civil and Environ-mental Engineering. He is a member of ACI Committees 440, Fiber-Rein-forced Polymer Reinforcement; and Joint ACI-ASCE Committee 445, Shear and Torsion. His research interests include punching and post-punching of flat plates, the use of fiber-reinforced plastic (FRP) in concrete structures, and strengthening and rehabilitation of structures.
ACI member Paul E. Regan is a Professor Emeritus at the University of Westminster, London, UK, where he was Head of Architecture and Engi-neering. He was Chair of the European Concrete Committee (CEB) commis-sion on member design. His research interests include member design in both reinforced and prestressed concrete, with particular emphasis on problems of punching, shear, and torsion.
ACI member Robert L. Vollum is a Reader in concrete structures at Impe-rial College London, London, UK, where he also received his MSc and PhD.
His research interests include design for shear, strut-and-tie modeling, and design for the serviceability limit states of deflection and cracking.
ACKNOWLEDGMENTS
The authors are grateful to the Brazilian Research Funding Agencies CAPES (Higher Education Co-ordination Agency) and CNPq (National Council for Scientific and Technological Development) for their support throughout this research and to RFA-Tech for their permission to include test results from Reference 7.
NOTATION
Asw = area of shear reinforcement in one perimeter
c = side length of square column or diameter of circular column d = mean effective depth
dg = maximum size of aggregate
dv = depth from tension reinforcement to compression zone anchorage of shear reinforcement
Es = modulus of elasticity of reinforcement fc = cylinder compression strength of concrete fy = yield stress of flexural reinforcement
so = distance from column face to first perimeter of shear reinforcement
sr = radial spacing of shear reinforcement st = tangential spacing of shear reinforcement
st,max = maximum value of st (general in outer perimeter of shear reinforcement)
u0 = length of column perimeter
u1 = length of control perimeter for calculation of VR,c and VR,cs uout = length of control perimeter for calculation of VR,out
uout,ef = effective value of uout for calculations by EC2, where st,max > 2d V = applied shear force
Vcalc = calculated punching resistance
Vflex = flexural strength of slab calculated by yield-line theory VR,c = punching resistance of slab without shear reinforcement VR,cs = punching resistance within shear reinforced zone
VR,max = maximum punching resistance for given column size, slab effec-tive depth, and concrete strength
VR,out = punching resistance outside shear reinforced zone
VR,s = contribution of shear reinforcement to punching resistance VR,cs
Vu = experimental punching strength
ρ = ratio of flexural reinforcement ρ= ρ ρx y (calculated for width of column plus 3d to either side in EC2)
ψ = rotation of part of slab outside critical shear crack
REFERENCES
1. Ferreira, M. P., “Punção em Lajes Lisas de Concreto Armado com Armaduras de Cisalhamento e Momentos Desbalanceados,” PhD thesis, Universidade de Brasília, Brasília, Brazil, 2010, 275 pp. (in Portuguese) available at http://repositorio.bce.unb.br/handle/10482/8965.
2. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2008, 473 pp.
3. Eurocode 2, “Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings,” CEN, EN 1992-1-1, Brussels, Belgium, 2004, 225 pp.
4. Muttoni, A., “Punching Shear Strength of Reinforced Concrete Slabs without Transverse Reinforcement,” ACI Structural Journal, V. 105, No. 4, July-Aug. 2008, pp. 440-450.
5. Fernadez-Ruiz, M., and Muttoni, A., “Applications of the Critical Shear Crack Theory to Punching of R/C Slabs with Transverse Reinforce-ment,” ACI Structural Journal, V. 106, No. 4, July-Aug. 2009, pp. 485-494.
6. Fédération internationale du béton, “fib Model Code 2010, First complete draft—V. 2,” Bulletin 56, fib, Lausanne, Switzerland, Apr. 2010, 288 pp.
7. Regan, P. E., unpublished tests for RFA-TECH at Cambridge Univer-sity, 2009.
8. Regan, P. E., and Samadian, F., “Shear Reinforcement against Punching in Reinforced Concrete Flat Slabs,” The Structural Engineer, V. 79, No. 10, May 2001, pp. 24-31.
9. Beutel, R., “Punching of Flat Slabs with Shear Reinforcement at Inner Columns,” Rheinisch-Westfälischen Technischen Hochschule Aachen, Aachen, Germany, 2002, 267 pp. (in German)
10. Birkle, G., “Punching of Flat Slabs: The Influence of Slab Thickness and Stud Layout,” PhD thesis, Department of Civil Engineering, University of Calgary, Calgary, AB, Canada, Mar. 2004, 152 pp.
11. Gomes, R., and Regan, P. E., “Punching Strength of Slabs Rein-forced for Shear with Offcuts of Rolled Steel I-Section Beams,” Magazine of Concrete Research, V. 51, No. 2, 1999, pp. 121-129.
12. Mokhtar, A. S.; Ghali, A.; and Dilger, W., “Stud Shear Reinforce-ment for Flat Concrete Plates,” ACI Journal, V. 82, No. 5, Sept.-Oct. 1985, pp. 676-683.
13. Carvalho, A. L.; Melo, G. S.; Gomes, R. B.; and Regan, P. E.,
“Punching Shear in Post-Tensioned Flat Slabs with Stud Rail Shear Reinforcement,” ACI Structural Journal, V. 108, No. 5, Sept.-Oct. 2011, pp. 523-531.
APPENDIX A
Table A1—Comparison between theoretical and experimental results
Author
Vu/Vcalc and critical strength Studs
Notes: Vu includes self-weight; Vflex is approximate yield-line capacity from Eq. (14).
Shear reinforcement: In References 1 and 7: deformed studs, 3φ heads, so as given for all lines; in Reference 8, slabs R, plain studs, 2.5φ heads, Slabs A deformed studs, 2.5φ heads, so as given for orthogonal lines, so = 40 mm for diagonal lines; in Reference 9, deformed studs, 3φ heads, so as given for all lines; in Reference 11, I-beam slices, flange breath 102 mm, web breath 4.7 mm. φ values in the table are equivalent diameters giving the same areas as the actual web sections. so as given for orthogonal lines, so = 40 mm for diagonal lines; in Reference 10, plain studs with 3.2φ heads, so as given for all lines. Birkle’s Slabs 5 and 6 had 7 perimeters of studs. The outer two, with sr = d, have been ignored.
Aggregate (maximum size and type): In Reference 1, 9.5 mm crushed limestone. In References, 7, 8, 9, and 11, 20 mm gravel. In Reference 10, Slabs 1-6—14 mm; Slabs 7-12—20 mm, type unknown.
Failure modes: P is punching of slabs without shear reinforcement, In = failure inside shear reinforced zone (VR,cs), Out = failure outside shear reinforced zone (VR,out); Max = inclined compression failure of concrete close to column (VR,max); in Reference 7 and Slab 9 of Reference 10, the concrete soffit around the column crushed and spalled due to tangential compression, the spalling extended and at failure there was inclined cracking starting at the end of the spalled area.
1 mm = 0.03937 in.; 1 kN = 0.225 kip; 1 MPa = 145 psi.
ACI STRUCTURAL JOURNAL TECHNICAL PAPER
Fiber-reinforced polymer (FRP) materials have proven their effec-tiveness as an alternative reinforcement for concrete structures in severe environmental conditions. Many studies have investigated the flexural and shear behaviors of FRP-reinforced concrete beams and slabs. Limited research, however, has gone into investigating the behavior of internally reinforced FRP concrete columns. This paper reports the experimental investigation of the compressive performance of concrete columns reinforced longitudinally with FRP or steel bars and with FRP as transverse reinforcement.
Twenty concrete columns measuring 350 x 350 x 1400 mm (13.8 x 13.8 x 55.1 in.) were constructed and tested under concentric compressive load. The parametric study included variables such as transverse reinforcement configuration, material type and spacing, longitudinal reinforcement ratio, and confining volumetric stiff-ness. Results showed that FRP bars have contribution as
Twenty concrete columns measuring 350 x 350 x 1400 mm (13.8 x 13.8 x 55.1 in.) were constructed and tested under concentric compressive load. The parametric study included variables such as transverse reinforcement configuration, material type and spacing, longitudinal reinforcement ratio, and confining volumetric stiff-ness. Results showed that FRP bars have contribution as