# Probabilistic Models for the Shear Strength of RC Deep Beams
## 6 Conclusions  
Probabilistic models of the shear strength of deep beams were developed with an extensive database by using the Bayesian-MCMC method. The following conclusions can be drawn:  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{2}{c}{**Level of Approximation**} & \multirow{2}{*}{\(n\)} & \multicolumn{2}{c}{**Proposed Model**} & \multicolumn{2}{c}{**ACI**} & \multicolumn{2}{c}{**ACI.MCMC**} \\ \hline \multicolumn{2}{c}{**Description of Sample**} & & \multicolumn{1}{c}{**Mean**} & \multicolumn{1}{c}{**CoV**} & \multicolumn{1}{c}{**Mean**} & \multicolumn{1}{c}{**CoV**} & \multicolumn{1}{c}{**Mean**} & \multicolumn{1}{c}{**CoV**} \\ \hline \multicolumn{2}{c}{Whole Databases} & 645 & 1.0357 & 0.2263 & 1.1637 & 0.3565 & 1.0387 & 0.2798 \\ \hline \multirow{3}{*}{Concrete compressive strength} & Normal strength 10–40 MPa & 457 & 1.0402 & 0.2169 & 1.2491 & 0.3437 & 1.0733 & 0.2812 \\  & High strength 40–70 MPa & 129 & 1.0201 & 0.2156 & 1.0210 & 0.2865 & 0.9941 & 0.2570 \\  & Ultra strength 70–120 MPa & 59 & 1.0356 & 0.3101 & 0.8140 & 0.2704 & 0.8686 & 0.2173 \\ \hline \multirow{3}{*}{Reinforcement ratio} & 0 \(<\rho\leq\) 1.0 & 96 & 1.0558 & 0.2494 & 1.3640 & 0.4725 & 1.0827 & 0.3050 \\  & 1.0 \(<\rho\leq\) 2.0 & 273 & 1.0177 & 0.2346 & 1.1250 & 0.3442 & 1.0441 & 0.2694 \\  & 2.0 \(<\rho\leq\) 5.0 & 276 & 1.0466 & 0.2092 & 1.1325 & 0.2728 & 1.0182 & 0.2795 \\ \hline \multirow{3}{*}{Component section size} & 0 \(<\) \(b/h\)\(\leq\) 0.2 & 156 & 1.0408 & 0.2752 & 1.5299 & 0.3259 & 1.1142 & 0.2662 \\  & 0.2 \(<\) \(b/h\)\(\leq\) 0.4 & 296 & 1.0537 & 0.2082 & 1.0485 & 0.2769 & 0.9683 & 0.2565 \\  & 0.4 \(<\) \(b/h\)\(\leq\) 0.7 & 193 & 1.0029 & 0.2106 & 1.0630 & 0.3184 & 1.0935 & 0.2945 \\ \hline \multirow{3}{*}{Shear span ratio} & 0 \(<\) \(a/d\)\(\leq\) 1.0 & 246 & 1.0367 & 0.2320 & 1.3359 & 0.3608 & 1.0558 & 0.2746 \\  & 1.0 \(<\) \(a/d\)\(\leq\) 2.0 & 211 & 1.0991 & 0.1838 & 1.0971 & 0.2688 & 0.9950 & 0.2337 \\  & 2.0 \(<\) \(a/d\)\(\leq\) 5.0 & 188 & 0.9633 & 0.2495 & 1.0129 & 0.3477 & 1.0654 & 0.3209 \\ \hline \end{tabular}
\end{table}
Table 7: Sample characteristics for different ranges of the key explanatory terms.  
Figure 7: Variation of model uncertainty factor with target reliability. (a) \(f_{\rm{c}}\)\({}^{\prime}\)/MPa. (b) \(\rho\)/%.  
* The deterministic methods improved by the Bayesian-MCMC method showed more accurate and robust predictions with experiment-to-prediction ratios between 1.0322 and 1.0744 and CoVs between 0.2540 and 0.3879.
* The proposed model developed without prior models had higher accuracy (the mean was 1.0357) and lower discreteness (the CoV was 0.2312) than other modified probabilistic models. Uncertainties related to the proposed model were described by the following statistical characteristics and partial factors: whole database (\(\beta\) = 0.38, \(\mu\) = 1.0357, \(V\) = 0.2263, and \(\gamma_{\text{Rd}}\approx 1.25\)).
* Gibbs sampling method was introduced to solve high-dimensional and complex integration problems by which the sampling progress was simplified, improving the optimization and reliability of model parameters.
* Explanatory functions such as \(f_{\text{c}}^{\prime}/f_{\text{y}}\), \(l_{0}/h\), \(a/l_{0}\), \(b/h\), \(\rho_{\text{v}}\), \(\rho_{\text{h}}\), and \(\rho\) were identified by different removal processes, which improves the modified probabilistic models.  
Investigation, D.K., Y.H. and Q.Y.; Data curation, Q.Z.; Writing--original draft, Z.L.; Writing--review & editing, X.L. All authors have read and agreed to the published version of the manuscript.  
The authors gratefully acknowledge the financial support provided by the Natural Science Foundation of Shaanxi Province (No. 2022 KJ)X-03).  
The findings and opinions expressed in this paper are those of the authors and do not necessarily reflect those of the sponsors.  
The authors declare there are no conflicts of interest regarding the publication of this paper.