# Probabilistic Studies on the Shear Strength of Slender Steel Fiber Reinforced Concrete Structures
## 5 Quantification of Model Uncertainties in SFRC Shear Resistance Models
### Probabilistic Model for Model Uncertainty  
In structural reliability theory, the basic variables such as actions, material properties and geometric data are considered as random variables with appropriate distribution type. One of the major objectives of this investigation is to establish the most appropriate distribution function for the model uncertainties related to the shear resistance models for SFRC beams. The most prevalently used distributions to describe actions, material properties and geometrical data, in reliability investigation, are the normal and lognormal distributions [33; 35]. In the current study, only the suitability of the normal and lognormal distribution is investigated in order to achieve an adequate representation of the actual probability distribution for the model uncertainty observations. Also, the skewness coefficients (\(\eta_{U}\)) obtained for the models, as presented in Table 3, indicate that the underlying distributions of the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **MC-10** & **DAfStB** & **[PERSON] et al.** \\ \hline \(b_{w}\) & 0.22 & 0.07 & 0.17 \\ \hline \(d\) & 0.34 & 0.10 & 0.15 \\ \hline \(\rho_{l}\) & 0.13 & 0.33 & 0.29 \\ \hline \(a/d\) & \(-\)0.38 & \(-\)0.34 & \(-\)0.29 \\ \hline \(f_{em}\) & 0.15 & 0.14 & 0.08 \\ \hline \(F_{sf}\) & 0.22 & 0.14 & \(-\)0.31 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Pearson correlation matrix between model and shear parameters.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**MC-10**} & \multicolumn{3}{c}{**DAfStB**} & \multicolumn{3}{c}{**[PERSON] et al.**} \\ \hline
**Description of** & \(\mu_{u}\) & \(\sigma_{u}\) & \(V_{u}\) & \(\mu_{u}\) & \(\sigma_{u}\) & \(V_{u}\) & \(\mu_{u}\) & \(\sigma_{u}\) & \(V_{u}\) \\ \hline Short shear span & & & & & & & & \\ length (\(\frac{a}{d}\)=2.5–3) & 1.21 & 0.32 & 0.26 & 1.13 & 0.31 & 0.28 & 1.19 & 0.32 & 0.27 \\ \(n\)=109 & & & & & & & & \\ \hline Medium Shear & & & & & & & & \\ Span Length (\(\frac{a}{d}\)=3–4) & 1.18 & 0.32 & 0.27 & 1.12 & 0.29 & 0.26 & 1.15 & 0.31 & 0.27 \\ \(\frac{a}{d}\)=180 & & & & & & & & \\ \hline Slender Shear & & & & & & & & \\ Span Length (\(\frac{a}{d}\)=4–8) & 0.78 & 0.16 & 0.21 & 0.83 & 0.18 & 0.21 & 0.99 & 0.26 & 0.26 \\ \(\frac{a}{d}\)=4–8) & \(n\)=37 & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Model uncertainty statistics for different subsets of beams.  
derived model uncertainties are closer to what is expected from a positively skewed distribution (such as lognormal).  
Figure 4: Variation of model uncertainty with shear span to effective depth ratio \(a/d\). (**a**) Plot for MC-10 model. (**b**) Plot for [PERSON] et al., model [27]. (**c**) Plot for DAßtB model.  
The type of probability distribution has been shown to play a key role in reliability performance assessment [28, 35]. The choice of the appropriate distribution for the model uncertainty variable was evaluated based on the chi-square goodness-of-fit test. The hypothesis that the model uncertainty distribution is similar to the choice of probability distribution (normal or lognormal) is rejected if the significance \(p\)-value \(<0.05\). The \(p\)-value is a measure of the goodness-of-fit, with larger values indicating a better fit. The magnitude of the \(p\)-value obtained based on the chi-square goodness of fit test suggests that the lognormal distribution is a better fit with \(p\)-value greater than 0.05. Previous studies [28, 29] and the Probabilistic Model Code by the Joint Committee on Structural Safety (JCSS) [33] have also recommended the lognormal distribution as the most suitable distribution function for model uncertainty. The model uncertainty theoretical PDF and histogram plots for the models are presented in Figure 6.  
Figure 5: Histogram of \(a/d\) for the full database.  
Figure 6: Model uncertainty distribution plots—theoretical PDF and frequency histogram plots. (**a**) PDF for MC-10 model. (**b**) PDF for DaStB model. (**c**) PDF for [PERSON] et al., model.