All three approaches considered herein take the punching strength of a slab with shear reinforcement as the least of
VR,cs, VR,out, and VR,max, but not less that VR,c, where VR,c is the resistance of an otherwise similar slab without shear rein- forcement; VR,cs is the combined resistance of the concrete and shear reinforcement; VR,out is the resistance from the concrete alone just outside the shear reinforcement; and
VR,max is the maximum resistance possible for a given column size, slab effective depth and concrete strength.
These resistances correspond to failures of the types shown in Fig. 9. The calculations are made for perimeters at specified distances from supports: uo is the perimeter at the
outline of the support; u1 is the perimeter used in the calcu-
lation of VR,c and VR,cs; and uout is the perimeter used in the calculation of VR,out.
Fig. 9—Types of punching failure.
Table 2—Summary of test results
Slab No. εc,max*, ‰ ry†, mm (in.)
Average stud stresses‡, MPa (ksi)
Vu§, kN (kip) Failure mode
1 2 3 C1 2.66 450 (17.7) 535 (77.6) 317 (46.0) 137 (19.9) 858 (192.9) In C2 2.81 550 (21.7) 530 (76.9) 235 (34.1) 121 (17.5) 956 (214.9) In C3 2.54 625 (24.6) 511 (74.1) 362 (52.5) 189 (27.4) 1077 (242.1) In C4 2.28|| 770 (30.3) 535 (77.6) 461 (66.8) 297 (43.1) 1122 (252.2) In C5 3.24 490 (19.3) 504 (73.1) 264 (38.3) 160 (23.2) 1117 (251.1) In C6 3.20 750 (29.5) 479 (69.5) 421 (61.0) 474 (68.7) 1078 (242.3) In C7 3.14 540 (21.3) 386 (56.0) 419 (60.8) 167 (24.2) 1110 (249.5) In C8 3.14 660 (26.0) 535 (77.6) 436 (63.2) 179 (26.0) 1059 (238.1) In S1 2.37 560 (22.0) 535 (77.6) 473 (68.6) — 1021 (229.5) Out S2 2.15 570 (22.4) 535 (77.6) 514 (74.5) 216 (31.3) 1127 (253.4) In S5 1.47 130 (5.1) — — — 779 (175.1) — S7 2.67 600 (23.6) 238 (34.5) 285 (41.3) 137 (19.9) 1197 (269.1) Out *ε
c,max is maximum tangential strain of concrete (measured 20 mm from columns in C1 to C4, S1 and S2, and 40 mm from columns in C5 to C8 and S7). For slab Type S, strains measured on centerlines.
†r
y is radius in which tangential strain > εy. ‡Averages of E
sεs ≤ fyw in Perimeters 1, 2, and 3.
§Ultimate shear force including self-weights of slabs and loading system. ||Measured at 0.85V
The locations and lengths of u1 and uout vary with the
method of calculation.
The symbols used for spacings of shear reinforcement are as follows: so is the distance from column to inner studs; sr is the radial spacing of studs; and st is the tangential spacing of studs at a perimeter. The effective depth d is taken as the average for orthogonal directions, d = (dx + dy)/2. The expressions for punching resistances are given below in SI units (N and mm) without any explicit safety factors. Those from ACI 318 are for nominal resistances, and the others are for characteristic resistances. The perimeters u1 and uout
and the detailing requirements, in relation to the spacings of shear reinforcement, are illustrated by Fig. 10, 11, and 12 for ACI 318, EC2, and the CSCT, respectively.
ACI 318-08
As double-headed studs are not considered explicitly, the equations used herein are those for studs with heads at their top ends and bottom anchorages provided by welds to struc- tural rails
VR,c = 1
3 f u dc 1 (1)
VR,cs = 0.75VR,c + VR,s (2)
VR,s = d
sr ⋅A fsw yw with fyw ≤ 414 MPa (60,000 psi) (3)
VR,out = 1 6 f u dc out (4) VR,max = 2 3 f u dc 1 if sr ≤ 0.5d (5a) VR,max = 1 2 f u dc 1 if 0.5d ≤ sr ≤ 0.75d (5b)
fc is limited to ≤ 69 MPa (10,000 psi) for calculation purposes.
EC2-04 VR,c = 0.18k(100ρfc)1/3u 1d (6) k = 1+ 200/ d ≤ 2 (7) VR.cs = 0.75VR,c + VR,s (8) VR,s = 1 5. d , sr A fsw yw ef (9)
fyw,ef = 1.15(250 + 0.25d) ≤ fyw ≤ 600 MPa (87,000 psi) (10)
VR,out = 0.18k(100ρfc)1/3u out,efd (11) VR,max = 0 3 1 250 . fc fc u d o − (12)
ρ is the ratio of flexural reinforcement calculated as ρ ρx y,
where ρx and ρy are the ratios in orthogonal directions deter- mined for widths equal to those of the column plus 3d to
Fig. 10—Detailing and control perimeters: ACI 318.
Fig. 11—Detailing and control perimeters: EC2.
either side. ρ ≤ 0.02 for calculation purposes, and the scope of EC2 is limited to fc ≤ 90 MPa (13,000 psi).
Critical shear crack theory
In the CSCT, punching resistances are related to the rota- tion ψ of the slab, outside a critical crack. Half of this rota- tion is assumed to occur in the critical shear crack and, as the slab rotates, the concrete component of shear resistance at the crack is assumed to decrease, while the component from the shear reinforcement increases up to yield. The rotation ψ is related to the ratio V/Vflex, where V is the acting shear, and
Vflex is the shear force corresponding to the flexural capacity, calculated by yield-line theory. Values of VR,c, VR,cs, VR,out, and VR,max can be determined as shown in Fig. 13, by plotting the resistances against ψ and finding their intersections with Eq. (13) ψ = 1 5 3 2 . r d f E V V s y s flex (13)
where rs is the distance from the column center to the line of radial contraflexure; and ψ is in radians. For typical punching test specimens, rs is the distance from the column center to the slab edge. The flexural failure load Vflex is approximated by5 Vflex = 2πmR r rsr q− c (14)
where mR is the moment resistance per unit length of yield line; rq is the radius at which loading is applied; and rc is the radius of the column and for square columns can be taken as 2c/π, where c is the side length of the column.
In the CSCT average method given in Reference 5
VR,c = 0 75 1 15 16 1 . / ( ) u d f d d c g + ψ + (15) VR,s = ΣAswσsi(ψ) ≤ ΣAswfyw (16) VR,cs = VR,c + VR,s (17) VR,out = 1 150 75. u ddout v/ (16 fcd ) g + ψ + (18) VR,max = 3VR,c (19)
where σsi is the stress in the i-th perimeter of shear reinforce- ment which is related to the width of the critical shear crack, where it crosses the shear reinforcement.5 The summations
ΣAsw and ΣAswσsi(ψ) are for all the shear reinforcement within a distance d from the column.
The CSCT average method is intended to give approxi- mately mean strengths. In it, the stresses in studs at different distances (≤d) from the column are calculated assuming that the width of the critical shear crack increases linearly, from zero at the slab soffit to the width corresponding to a slab rotation ψ and a crack opening angle of 0.5ψ, at the level of the tension reinforcement. The stress in a stud is then obtained by equating the vertical component of the crack opening to the elongation of the stud for a given stress at the crack.
COMPARISONS OF TESTS AND CALCULATIONS