+ ⋅
⋅ =
⋅
0
0 1 15
4 3
ψ , (6.52)
with VRbeing the resistant punching shear force in [N], fc compressive strength in [MPa], and dg and dg0 the maximal and reference aggregate sizes in [mm], respectively, where dg0=16 mm. The model predicts well the experimentally observed fact that the punching shear strength in RC structures decreases with increasing slab rotation, explained by the presence of the critical shear crack whose opening reduces the strength of the concrete compression strut.
critical crack
Figure 6.18: Critical shear crack in flat slab, adapted from [Guandalini 2005].
The same phenomenon may be assumed for FRCs, in addition to fibre action along the crack, providing a beneficial effect on shear resistance. The force transferred by the fibres decreases with the increase in crack opening, and the relationship between a critical crack opening and the element’s capacity to sustain punching shear also seems physically justifiable in the case of FRCs. The question arises as to what extent the theory is applicable in the prediction of shear resistance in UHPFRC elements without ordinary reinforcement. The discussion will be based on existing experimental data, briefly reviewed in the following section.
6.3.2 Punching shear resistance of thin slabs: experimental results
a) b)
clamped edge
clamped edge
2 3, 1, 5 4
300 mm
measuring devices, LVDTs
loading point
350 mm
130 120
50
Figure 6.19: Punching shear test: a) test set-up; b) specimen geometry and position of measuring devices
Five UHPFRC slabs of thicknesses varying from 30 to 60 mm were tested at the EPFL Structural Laboratory. The slabs were of constant square shape, 350 mm side length, with restrained rotations along all four edges (Figure 6.19 a)) and the loading surface was 20 x 20 mm. The displacements were measured in the central and four symmetric points, as shown in Figure 6.19 b).
The force-central point deflection relationships are shown in Figure 6.20 and the related values are given in Table 6.2. Four of the five tested slabs failed in punching shear, whereas one slab, designated PP 50b, reached the maximal force in bending, and attained the punching shear failure for slightly increased deformation, in the post-peak phase. After the test, specimens were cut along two axes of symmetry in order to observe the shape of the punching surface: all the punching cones developed in a symmetrical way (Figure 6.20 c)). This could not have been concluded based on the crack pattern on the tensile side of the slab (Figure 6.20 b)), which could be associated to an asymmetric failure. Only one specimen (a 30 mm thick slab) is shown in Figure 6.20 b) and c), while the other specimens are documented in Appendix T1.
Resistance to punching shear is typically expressed by the average shear stress, v,
v = V / (b0 d), (6.53)
V being the maximal load, d the distance between the extreme compression fibre and the tensile reinforcement and b0being the perimeter of the critical section. The perimeter of the critical section can be assumed to be based on the idealised punching surface. The majority of design codes ([SIA 2003b], [ACI 2005]) suggest that the crack should be assumed to form a 45-degree angle with respect to the slab surface, leading to the relationship b0= 4 bc + π d for a square surface of punching load introduction, with bc being the side length.
a) b)
0 2 4
Δ mm
0 80 160
VkN
c)
Figure 6.20: Results of punching shear tests of slabs made of BSI: a) force-mid-point displacement relationships; b) tensile face of a specimen; c) section through punching cone
Using Equation 6.53, the results of the test are plotted as a function of elements thickness in Figure 6.21. In the same figure, results from other authors are also shown:
- slabs made of Ductal [Harris 2004], tested at the VPI; the slabs were of square shape, with a side length 900 mm and clamped edges along all sides; thickness varied between 50 and 70 mm, and the punching shear surface also varied, between 25 and 50 mm (Table 6.3);
- slabs made of BSI and Ductal [Toutlemonde et al. 2007], tested at the LCPC; the slabs were part of a prototype ribbed deck slab, restrained by the ribs at a distance of 600 mm; the applied load simulated wheel load, using a reduced introduction surface, 190x260 mm with corners cut at 40 mm (Table 6.4). The ribbed slab was prestressed, with two T15S strands per rib in one direction and external cables in the perpendicular direction providing a compressive stress of 4 MPa.
h= 60 mm 50 mm 40 mm 30 mm
a) b)
0 50 100
h mm
0 4 8 12
v
V MPa bh0
0 2 4 6
Ψ d mm
0 0.5 1
V b0d fc MPa
Figure 6.21: Punching shear resistance of UHPFRC slabs: a) average shear strength against element thickness; b) nominal strength against rotations times element depth, and failure criterion according to critical shear crack model for ordinary concrete structures
A certain dispersion of the achieved shear resistances can be noted based on the results of the above-mentioned tests (Figure 6.21 a)). Regarding only the resistances of the BSI slabs (filled squares), average shear strains vary only slightly, with a mean v value of 10 MPa. The highest resistance is attained in the thinnest specimen (11.1 MPa). However, the variation in element thicknesses is insufficient to clearly demonstrate its influence on shear strength. It is also interesting to note that a significant dispersion in shear strength exists between elements of the same thickness (e.g. h=50 mm) and made of similar materials, but with different geometries.
Tests at the VPI (filled triangles) were performed using a material with properties similar to those of BSI, as already mentioned in Section 6.2.5.2; however, significantly lower average shear strengths are attained. The resulting mean v value is 5.7 MPa.
The results for BSI slabs tested at the LCPC (voided squares in Figure 6.21 a)), show lower average shear strength than the specimens tested at the EPFL, whereas Ductal slabs tested at the LCPC (voided triangles) show higher resistance than slabs tested at the VPI. The difference between the shear strength of the BSI and that of the Ductal slabs tested at the LCPC is small (7.1 against 8 MPa mean v value). This suggests that the significant difference between the resistances of the BSI slabs tested at the EPFL and the Ductal slabs tested at the VPI is probably not due to a difference in material strengths.
Table 6.2: Results of punching shear tests on slabs made of BSI, performed at the EPFL Specimen
350 x 350 x h
nominal h
actual
h bc actual
P max δP max
[mm] [mm] [mm] [kN] [mm]
PP 30 30 31 20 61.51 1.13
PP 40a 40 38 20 76.167 1.37
PP 50a 50 51 20 117.74 1.37
PP 50b* 50 50 20 110.735 1.51
PP 60 60 60 20 162.842 0.95
* punching failure occurred after bending failure, at approx. V=102 kN and δ=1.9 mm; for a circular yield line pattern for a restrained slab [Johansen 1972] Vpl= 4π mR = 113 kN in the case of 50 mm thick slab;
EPFL, BSI
LCPC, BSI (prestressed) VPI, Ductal
LCPC, Ductal (prestressed)
Table 6.3: Results of punching shear tests on slabs made of Ductal, performed at the VPI specimen nominal
h
actual
h bc actual
P max δP max
Serie/test [mm] [mm] [mm] [kN] [mm]
1/1 50.8 (2 in.) 55.12 38.1 103.64 18.39
1/2 50.8 58.93 50.8 120.99 20.22
1/3 50.8 53.85 25.4 100.52 No data
2/1 63.5 (2.5 in.) 66.29 50.8 146.78 14.48
2/3* 63.5 64.52 38.1 135.66 32.00
3/2 76.2 (3 in.) 71.88 38.1 156.57 15.24
3/3* 76.2 71.88 25.4 178.36 27.43
* punching failure occurred after bending failure: for specimen 2/3 punching failure occurred at approx. V =115.648 and δ =32 mm; for the specimen 3/3 it occurred at approx. V =146.78 and δ =27.432.
Table 6.4: Results of punching shear tests on slabs, performed at the LCPC specimen nominal
h bc actual
P max δP max
[mm] [mm] [kN] [mm]
BSI 50 190, 260* 365 3
BSI 50 - / - 352 2.5
Ductal 50 - / - 417 2.4
Ductal 50 - / - 391 1.6
*the side dimensions of the steel plate for load introduction are 0.19 and 0.26 m with corners cut at 0.04m, giving a load-introduction surface perimeter of 0.85 m. With h=d, the accepted critical perimeter concept gives b0=1m.