# Probabilistic Studies on the Shear Strength of Slender Steel Fiber Reinforced Concrete Structures
## 4 Deterministic Performance Assessment of Shear Design Formulations for SFRC Beams  
The mean and design shear resistance obtained for MC-10, DAfStB and Bernat et al., models are compared to experimental observations as presented in Figures 1 and 2. The design equations for the various models are presented in Section 2 (Equations (1)-(6)). The mean or best estimate shear resistance predictions are obtained by making all the values of the safety elements or partial factors included in the design equations equal to one, and using the mean values of their material strength (e.g., concrete strength \(f_{cm}\)).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & **Range** & **Min** & **Max** & **Mean** & **S.D.** & **C.o.V.** \\ \hline \(V_{R}\) (kN) & 686.64 & 12.89 & 699.53 & 134.44 & 128.75 & 0.96 \\ \hline \(F_{sf}\) (-) & 1.90 & 0.10 & 2.00 & 0.51 & 0.31 & 0.62 \\ \hline \(d_{a}\) (mm) & 21.60 & 0.40 & 22.00 & 9.94 & 5.23 & 0.53 \\ \hline \(\rho_{l}\) & 0.053 & 0.004 & 0.057 & 0.03 & 0.01 & 0.38 \\ \hline \(f_{c}\) (MPa) & 205.23 & 9.77 & 215.00 & 49.33 & 26.94 & 0.55 \\ \hline \(a/d\) (-) & 3.50 & 2.50 & 6.00 & 3.37 & 0.66 & 0.20 \\ \hline \(b_{w}\) (mm) & 255.00 & 55.00 & 310.00 & 151.96 & 57.77 & 0.38 \\ \hline \(d\) (mm) & 837.75 & 85.25 & 923.00 & 264.15 & 160.46 & 0.61 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistical summary of the experimental evaluation database (326).  
Figure 1: (**a**) Plot of the correlation between the experimentally observed shear strength \(V_{exp}\) and mean shear resistance predictions \(V_{pral}\). (**b**) Plot of the correlation between the experimentally observed shear strength \(V_{exp}\) and mean shear resistance predictions \(V_{pral}\) (resistance up to 300 kN only).  
Figure 2: (**a**) Plot of the design shear resistance predictions against experimental observations. (**b**) Plot of the design shear resistance predictions against experimental observations (resistance up to 300 kN only).  
The plot of the correlation between the experimentally observed shear strength \(V_{exp}\) and mean or best estimate shear resistance predictions \(V_{pred}\) provided by the different procedures is presented in Figure 1. Generally, the predicted capacity by the various resistance models deviate from the 'perfect line' (defined as the point where \(V_{exp}=V_{pred}\)). This trend of results agrees with the observation reported in [21; 27]. The trend lines of DAFStB and [PERSON] et al., models are the closest to the 'perfect line'. MC-10 shear method provided more conservative predictions compared to the other methods.  
Figure 2 presents the plot of the design shear resistance predictions from the various design approaches against experimental observations. [PERSON] et al. [27] model only provides a mean estimate of shear resistance; therefore, it is not considered in the design value assessment. The trend line of MC-10 and DAFStB design capacities, as presented in Figure 2, seems to be reasonably conservative when compared to that of experimental shear capacities. The plots indicate the complexity in predicting the shear capacity of SFRC beams accurately. The resistance predictions from the various procedures differ from each other and also from the experimentally observed shear strength. The shear equations include wide-spread uncertainties due to failure in reflecting the phenomenology of shear resistance accurately. Some of the possible uncertainties include the uncertainty in the basic random variables and the uncertainty associated with the predictive model formulation itself. Given these, quantification of model uncertainties for the different shear models for SFRC beams and subsequent reliability-based performance evaluation is of high significance. This will ensure the effects of uncertainties are adequately taken into account.