A Comparison of Methods
The failure shear predictions generated by the ACI Design Code and the SMCFT are summarized in Table 1. To account for the beam self-weight, it is appropriate, when using the ACI shear design method, to express the failure shear stress as the shear stress located at a distance d from the face of the support. When using the SMCFT, it is appropriate to calculate the failure shear at a distance dv from the face of the loading plate. While the shear due to self-weight is slightly reduced, calculating the failure shear at this location takes into account the far more dominant effect of the moment on the longitudinal strain in the web, ex. The SMCFT produced safe and accurate failure shear predictions. The average ratio of experimental to predicted failure shear was 1.17 for the small beams, with a coefficient of variation of
Figure 5: Progression of Failure Crack in
The ACI predicted failure shears for the large beams without stirrups ranged from 43% to 80% of the experimental failure shears. Clearly the ACI design method produces grossly unconservative predictions for large beams constructed without stirrups. Since the size effect is eliminated by the use of minimum stirrups, the ACI method produced an acceptable prediction of the failure shear stress of specimen SB-10- H-S. Despite the inability to account for aggregate effects, the ACI method produced otherwise excellent predictions of the failure shears of the scale model beams. This is to be expected, as the height of these beams was almost exactly the same as the average height of the beams tested to derive Equation (3).
Since the concrete strengths for all of the tested specimens varied, some thought is required to asses the true effect of the aggregate size. It is possible to normalize the experimental results by their concrete strengths and recalculate the experimental failure loads for the average f’c of all the specimens (Sherwood, 2006). The normalized and recalculated experimental failure shears of the beams without stirrups are plotted in Figure 6 as a function of the aggregate size. The averages of duplicate specimens are shown, along with the experimental range at each aggregate size. This figure clearly and explicitly demonstrates that the shear strength increases as the aggregate size increases, but ceases to increase beyond an aggregate size of 25mm. The SMCFT successfully predicts the effects of the maximum aggregate size.
Figure 6: ACI and SMCFT Prediction of the Shear Strength of the Test Specimens
Since there is no difference between the shear behaviour of beams and one-way slabs, the large specimens can be thought of as 300mm sections taken from a wide, deep slab. If this slab was a large transfer element in high-rise construction, a likely ratio of dead load to live load would be 3:1 (Sherwood et al., 2005b), resulting in a safe service load of 60% of the failure load. As such, specimens SB-10-N-2 and SB-10-H-1 failed below the safe service load predicted by the ACI code. If a one-way transfer slab similar to these specimens designed by the ACI code were in use in a real structure, there would be a risk of failure under service loads, with little to no warning of impending collapse. The ACI method, however, is perfectly acceptable when designing shallow one-way slabs.
The Size Effect Factor
A unique aspect of the SMCFT is the introduction of the effective crack spacing parameter, sze (Equation 7). It is implemented into the expression for Vc through the size effect factor, 1300/(1000+sze).
A wide range of effective crack spacings (from 214mm to 2940mm) was investigated in this study by varying only two variables: the effective depth and the maximum aggregate size. A graph showing the SMCFT predictions of the size effect factors for all of the beams tested is presented in Figure 7. Also shown are average experimental values of duplicate tests. The experimental points for the beams with stirrups are shown in white. The values of Vc, exp for these two beams were calculated by subtracting the steel contribution calculated by Equation (4) from the total experimental failure shear force. This figure shows that the size effect factor can accurately model the shear strength of d
concrete sections over a wide range of both f 0.40 ~ depths and aggregate sizes. c (1+1500E0
The effective crack spacing of 2940mm for Specimen SB-10-H-1 represents the highest value for sze ever tested. Basing the 2004 CSA shear design 1 method on a rational theory rather than empirical relationships allows it to accurately predict the shear behaviour of a reinforced concrete beam in which an experimental variable was set outside the range of previous experimental data. It is also worthwhile to note that the experimental results for the beams with stirrups follow the trend of the equivalent beams without stirrups. It is therefore appropriate to use an sze value of 300mm for beams containing at least the minimum quantity of stirrups. 1