# Probabilistic Models for the Shear Strength of RC Deep Beams
## 5 Discussion
### Continuity and Uncertainty  
The consistent performance and model uncertainty concerning essential variables were investigated by comparing test and model results. The boxplots of the experiment-to-prediction ratios were made for different ranges of each of the following key explanatory terms: \(f_{\mathrm{c}}{}^{\prime}\) (Figure 6a), \(a/d\) (Figure 6b), \(b/h\) (Figure 6c), and \(\rho\) (Figure 6d). The boxplots of the bias of the proposed model, ACI 318-14, and the modified probabilistic model were made for comparison. As shown in Figure 6a, the ACI 318-14 tended to underestimate the shear strengths of normal-strength members and overestimate ultra-strength members. The experiment-prediction ratio decreased as concrete strength increased. Similarly, the ACI 318-14 model tended to underestimate the shear strengths for \(0<\rho\leq 1.0\) members and had great discreteness. Figure 6c depicts the ACI 318-14, which had a very large scatter of prediction for all ranges of \(b/h\), and the bias was not uniform. The modified probabilistic model based on ACI 318-14 effectively corrected these biases and allowed for more consistent performance over the considered ranges. This may be due to the fact that the bias correction term indirectly considers the effect of \(f_{\mathrm{c}}{}^{\prime}\), \(b/h\), and \(\rho\), etc. As for ultra-strength and \(0<b/h<0.2\) members, the proposed model still had great variability (i.e., CoVs > 0.25), which probably results from uniform distributions of parameters in the database. It is concluded that these boxplots confirmed that the proposed  
Figure 5: Performance of the probabilistic models at each experimental data. **(a)** The Proposed Model. **(b)** ACI 318-14. **(c)** GB 50010-10. **(d)** CSA 23.3-04. **(e)** CEB-FIP. **(f)** EC2. **(g)** Foster and Gilbert. **(h)** Mitchell and Collins. model performed uniformly well over the entire ranges of the selected key parameters. The Bayesian-MCMC method further improves the consistency of the performance. An adequate description of the uncertainties in shear prediction models was recognized as one of the critical issues in reliability investigations of RC structures [79; 80; 81; 82]. The partial factor method was used to evaluate model uncertainty. Assuming a lognormal distribution and unity characteristic (nominal) value of model uncertainty, the partial factor \(\gamma_{\text{Rd}}\) can be obtained from the following equation:  
\[\gamma_{\text{Rd}}=\frac{1}{\mu\exp(-\alpha_{\text{R}}\beta v)}, \tag{20}\]  
where \(\alpha_{\text{R}}\) is the FORM (First Order Reliability Method) sensitivity factor; \(\beta\) is the selected target reliability index; \(\mu\) and \(v\) are the mean and coefficient of the experiment-to-prediction ratios, respectively. Considering the model uncertainty characteristics shown in Table 7, the variation of model uncertainty factor with target reliability \(\beta\) is displayed in Figure 7. \(\alpha_{\text{R}}\) = 0.4 \(\times\) 0.8 = 0.32, that is, \"non-dominant resistance variable\" according to ISO 2394 (2015) [83] and fib Bulletin 80 (2016) [84]. The range of \(\beta\) represented the target lifetime levels commonly associated with the Ultimate Limit States for new and existing structures. As shown in Figure 7, the partial factor \(\gamma_{\rm{Rd}}\) increased with an increasing target reliability index \(\beta\). The value of \(\beta\) is 3.8 for ordinary structures with 50 years of design service life.