## 3 Probabilistic Capability Model of Wall Piers
### Probabilistic in-Plane Capability Model of Wall Piers  
Based on the statistical investigation of the wall pier girder bridges, the ranges of the parameters are determined and their levels are divided, as shown in Table 4. Orthogonal experimental design is conducted by considering width-to-thickness ratio, pier width, vertical reinforcement ratio, transverse reinforcement ratio, reinforcement grade and concrete strength grade, which results in 16 combinations. A total of 256 wall piers samples were obtained by combining 16 combinations with 4-levels-all-factors combinations of axial compression ratio and shear span ratio. The finite element model of each wall pier is established by employing the layered shell element, and hysteresis analyses are carried out respectively to extract the displacement drift ratio of each sample at each performance limit point. Figure 4 describes the loading path of hysteresis analyses. Each displacement amplitude during loading, which is non-linearly increasing, is cyclically applied twice. The displacement amplitude relationship between \(i\) and \(i+1\) is \(a_{i+1}=1.4a_{i}\). After obtaining the displacement drift ratio of each performance limit point, probabilistic capability models of the wall piers can be obtained by statistical regression. Bayesian method has been widely used in the structural capability models due to its iterative ability and high efficiency in parameter stepwise screening [19; 20]. Therefore, Bayesian method is utilized to obtain the probabilistic capability models in this study. The basic formula of Bayesian method is as follows:  
\[\pi(\theta|x)=\frac{h(x,\theta)}{m(x)}=\frac{f(x|\theta)\pi(\theta)}{\int \limits_{\Theta}f(x|\theta)\pi(\theta)d\theta} \tag{2}\]  
The Bayesian statistical method not only utilizes the overall information and sample information but also utilizes some unknown parameters' information that has been known before sampling. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Level** & **1** & **2** & **3** & **4** \\ \hline Aspect ratio & 3.2 & 3.8 & 4.4 & 5 \\ Pier width (m) & 4 & 5.33 & 6.67 & 8 \\ Vertical reinforcement ratio (\%) & 0.5 & 0.63 & 0.77 & 0.9 \\ Lateral reinforcement ratio (\%) & 0.2 & 0.27 & 0.33 & 0.4 \\ Axial pressure ratio & 0.02 & 0.04 & 0.06 & 0.08 \\ Shear span ratio & 1.6 & 2.07 & 2.53 & 3 \\ Reinforcement grade & HRB400 & HRB335 & — & — \\ Concrete marking & C40 & C35 & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Factor levels of wall pier. Figure 4: Loading process of displacement control. After obtaining sample \(x\) in a sampling, the posterior distribution of the unknown parameter \(\theta\) is the conditional distribution of \(\theta\) given \(X=x\), where \(\pi(\theta)\) is the prior distribution of \(\theta\), that is, the knowledge of the possible value of the parameter \(\theta\) before obtaining the sample \(x\), and \(m(x)\) is the marginal distribution of \(X\). In this study, referred to the previous researches on capability models [21], the following probabilistic capability benchmark model is selected after considering the influence of various factors:  
\[ \begin{array}{l}\ln(\delta)=\alpha_{0}+\alpha_{1}\ln(n)+\alpha_{2}\ln(d)\\ +\alpha_{3}\ln(\lambda)+\alpha_{4}\ln(\frac{\rho_{l}f_{y}}{f_{c}})+\alpha_{5} \ln(\frac{\rho_{l}f_{y}}{f_{c}})\\ +\alpha_{6}\ln(\frac{f_{y}}{f_{c}})+\alpha_{7}\ln(\rho_{l})+\alpha_{8}\ln(\rho _{h})+\varepsilon\sigma\end{array}  \tag{3}\]  
where, \(\delta\) is the structural drift ratio, \(n\) is the axial compression ratio, \(d\) is the width-to-thickness ratio, \(\lambda\) is the shear span ratio, \(\rho_{l}\) is the vertical reinforcement ratio, \(\rho_{h}\) is the transverse reinforcement ratio, \(f_{y}\) is the yield strength of the steel, \(f_{c}\) is the compressive strength of concrete, \(\alpha\) is the unknown parameters, \(\varepsilon\) is the random variable of standard normal distribution, \(\sigma\) is the standard deviation of the model error.