# Probabilistic Models for the Shear Strength of RC Deep Beams
## 4 Probabilistic Shear Strength Models
### Development of Probabilistic Models
#### 4.2.2 Calculation Results  
In the stochastic simulation process, the former 1000 values were removed to eliminate the negative effect of the initial value on chains. Then, the posterior estimate of (\(\mathbf{\theta}\), \(\sigma^{2}\)) was obtained by Equation (7). Table 3 illustrates the calculation results of the parameters,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design Provisions** & **Shear Calculation Model** & **Mean** & **SD** & **CoV** \\ \hline \multirow{4}{*}{AC1318-14 (2014) [23]} & \(V_{n}=0.85\beta_{s}f_{b_{0}w_{\mathrm{s}}}^{t}h_{0}w_{\mathrm{s}}\sin\theta_{ \mathrm{s}}\) & & & \\  & \(\tan\theta_{\mathrm{s}}=\frac{h-\frac{\gamma^{2}+\sigma^{2}}{a}}{a}\geq 0.488\) & & & \\  & \(w_{l}=0.8w_{\mathrm{t}}\) & \(1.1637\) & \(0.4148\) & \(0.3565\) \\  & \(w_{\mathrm{s}}=\frac{2.25w_{\mathrm{s}}\cos\theta_{s}+\left(\left\lfloor{ \
u}\right\rfloor_{x}+\left\{\
u}\right\rfloor_{x}\right)\sin\theta_{\mathrm{s}}}{ \tau}\) & & & \\ \hline GB 50010-10 (2010) [24] & \(V_{n}=\frac{1.75}{\lambda+1}f_{b}bh_{0}+\left(\frac{\lambda_{1}/h-2}{3}\right) f_{y\sigma}\frac{\left(\frac{5-\lambda_{1}/h}{6}\right)}{\xi_{n}}h_{0}+\left(\frac{5- \lambda_{1}/h}{6}\right)f_{y\sigma}\frac{A_{\mathrm{s}}}{\xi_{n}}h_{0}\) & \(1.6164\) & \(0.6075\) & \(0.3758\) \\ \hline CSA A23.3-04 (2004) [25] & \(V_{n}=\frac{1}{0.8+170\xi_{n}}f_{c}^{\prime}b_{w}w_{\mathrm{s}}\sin\theta_{ \mathrm{s}}\) & \(1.2327\) & \(0.6464\) & \(0.3690\) \\ \hline EC2 (2004) [26] & \(V_{n}=0.6\Big{(}1-\frac{f_{\mathrm{s}}}{250}\Big{)}f_{c}^{\prime}b_{w}w_{ \mathrm{s}}\sin\theta_{\mathrm{s}}\) & \(1.2179\) & \(0.6310\) & \(0.5181\) \\ \hline CEB-FIP (2010) [27] & \(V_{n}=0.55\Big{(}\frac{30}{f_{a}}\Big{)}^{1/3}f_{c}f_{c}^{\prime}b_{w}w_{ \mathrm{s}}\sin\theta_{\mathrm{s}}\) & \(1.3641\) & \(0.4302\) & \(0.3154\) \\ \hline [PERSON] and Gilbert (1998) [28] & \(\frac{a}{d}<2\), \(v=1.25-\frac{f_{00}^{\prime}}{500}-0.72\frac{a}{d}+0.18\big{(}\frac{a}{d}\big{)} ^{2}\leq 1\) & \(1.0900\) & \(0.4594\) & \(0.4215\) \\  & \(\frac{a}{d}<2\), \(v=0.53-\frac{f_{c}^{\prime}}{500}\) & & & \\ \hline [PERSON] and [PERSON] (1974) [29] & \(V_{n}=\frac{3.35}{\sqrt{f_{c}^{\prime}}}f_{c}^{\prime}b_{w}w_{\mathrm{s}}\sin \theta_{\mathrm{s}}\) & \(1.2440\) & \(0.4015\) & \(0.3228\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Well-known shear strength models.  
Figure 2: Flow chart of solution procedure for Bayesian–MCMC method.  
including the mean, standard deviation (SD), naive standard error (SE), time-series SE, and quintiles. Most CoVs were below 0.2, indicating that the sample averages have a relatively concentrated distribution. The naive standard error provided a measure of the potential error in this estimate that decreased as chain length increased. In turn, an appropriate chain length (i.e., 50,000) was determined by comparing it with the target value (i.e., 1/1000). Therefore, this random simulation method can accurately obtain the optimal estimation value and the corresponding reliability of model parameters.  
Figure 3 clearly shows the simulation results of parameters obtained by the iterative process of the Markov chain in Appendix A. The iterative process of two Markov chains with different initial values was shown in trace plots. After 10,000 times of iterations, the two chains were utterly mixed together and stably distributed in a minimal range near the posterior mean. Ergodic mean plots display ergodic mean statistical results of the two Markov chains. With the iterations increasing, the ergodic mean of Markov chains tended to be a horizontal line and converged to the parameter mean. It is noted that Markov chains have reached a stable state and convergence. The posterior frequency histograms were obtained from 50,000 posterior samples after removing the former 1000 values. It can be seen that the majority of the rectangle area was concentrated around the simulated result values, which is consistent with the posterior mean.