Experimental Support for the CSA Shear Provisions
Extensive comparisons to experimental results have been made for the CSA shear design provisions and only a very small subset of these will be shown here. For members without transverse reinforcement, two main variables are predicted to control behaviour: the size effect and the strain effect. To demonstrate the quality of the predictions on a graph with only a single independent variable, Equation (4) will be used with в as the experimentally measured strength and both sides of the equation divided by either the strain effect term or the size effect term. Thus a 2D plot can be made showing the influence of one variable at a time. When results are normalized by the strain effect, the strain used is that predicted by the CSA provisions rather than that calculated from the experimental results.
Figure 8 shows 411 members of various depths and aggregate sizes without stirrups to demonstrate the size effect and the quality of Equation (4) at predicting this effect. The vertical axis shows the value of в in psi units which are 12 times that given by Equation (4). As seen, the CSA equations do an excellent job and generally provide a conservative estimate of shear strength across the range of depths. Recall that these provisions were not based on curve fits to experimental beam test results, but were based on the MCFT which itself resulted from more fundamental shear and material tests. The line on the plot is therefore a true prediction and not a “post-diction” or curve fit result. In addition, the experiments upon which the MCFT is based had values of the sze parameter that were generally in the range of 50 to 100 mm. The plot therefore shows a significant extrapolation from the data originally used to calibrate the MCFT. That the prediction is as good as the figure shows confirms both that the MCFT is a good model for the behaviour of cracked reinforced concrete and that the assumptions used in generating Equation (4) are appropriate.
Figure 9 shows the same data but normalized by the size term to highlight the strain effect. Again, to calculate this, the experimentally observed value of в was divided by the size effect term so that the predicted behaviour would only vary by one independent variable and a 2D plot could be generated. As with Figure 8, the overall behaviour of the strain effect is seen to be well modelled by Equation (4) for members without stirrups. As expected from the derivation in Figure 6, as the strain is increased beyond a value of about 1.0 x 10-3, the method begins to become more conservative as the crack width at failure is overestimated. The strain effect explains why FRP reinforced members have lower shear strengths than members reinforced with steel. Such members generally have a
lower stiffness of flexural reinforcement and are therefore subjected to higher strains at shear failure. Due to the strain effect, these members will therefore be weaker in shear. Members reinforced with cast in place FRP are modelled conservatively by the general method shear equations of the 2004 A23.3 concrete code.
As a final comparison, Figure 10 shows a plot of the strain effect for members with transverse reinforcement. This plot includes members of various depths, with and without axial loads and with or without prestressing. The effect of axial compression or prestress is to lower the value of the strain term ex and in this figure it can be seen that shear strength is modelled well even for negative values of ex. It may not be intuitively obvious that the strain effect should work just as well for members with stirrups, which can undergo redistribution of shear stresses as it does for the members in Figure 9. As seen in Figure 10, however, the effect applies equally well to all members when a realistic estimate of Vs is made as with Equation (5). This figure also shows why the simplified method (which assumes ex = 0.85 x 10-3) will produce poor statistical fits to databases: the
systematic effect of strain will be ignored when using the simplified method to predict member strengths.
This paper has provided a brief description of the derivation of the CSA shear design provisions of the 2004 CSA Design of Concrete Structures code. The code continues to employ a simplified and a general method, but now the general method is based on simple equations derived from the MCFT rather than tables derived from the MCFT. Because of this, the simplified method is now a clear derivation from the general method. The use of equations means that the general method is now much easier to apply, particularly with spreadsheets. Design with the general method is now a non-iterative process. Analysis with the general method requires iteration for one variable, and a process for implementing this into a spreadsheet with no programming or macros is provided. The new equations are shown to work well for experimental predictions.
In the generation of the 2004 general method, it was expected that the simplifications necessary to make practical equations would come at the cost of a significant reduction in accuracy or generality. It has been a pleasant surprise to find that the resulting simplified MCFT equations, while much easier to apply, have essentially the same ability to accurately predict shear strength for a very wide range of different parameters. It is believed that the new equations presented in this paper represent a breakthrough in the understanding and modelling of shear. It is hoped that these equations can encourage the necessary discussion to move towards international consensus on shear behaviour just as it currently exists for flexural behaviour.