# Seismic Performance of Dam Piers Retrofitted with Reinforced Polymer Cement Mortar
## 3 Outline of Seismic Response Analysis for Dam Piers of Existing Dams
### Material Model  
In this study, the elastic-plastic material properties of the existing parts and reinforcing parts of dam piers are considered. Table 3 lists the material properties of the concrete and PCM.  
The stress-strain relationship on the compression side is shown in Equations (2)-(5) and conforms to the equation in the \"Concrete Standard Specification\" of the JSC [13]. The strain at the compression strength was defined by the inverse calculation of Equation (3). Figure 14 shows the stress-strain curves on the compression side.  
\[\sigma_{c}=E_{0}K\big{(}\varepsilon_{c}-\varepsilon_{p}\big{)} \tag{2}\]  
\[E_{0}=2f_{c}/\varepsilon_{peak} \tag{3}\]  
\[K=\exp\Bigg{\{}-0.73\frac{\varepsilon_{c}}{\varepsilon_{peak}}\Bigg{(}1-\exp \big{(}-1.25\frac{\varepsilon_{c}}{\varepsilon_{peak}}\big{)}\Bigg{)}\Bigg{\}} \tag{4}\]  
\[\varepsilon_{p}=\varepsilon_{c}-2.86\cdot\varepsilon_{peak}\Bigg{\{}1-\exp \Bigg{(}-0.35\frac{\varepsilon_{c}}{\varepsilon_{peak}}\Bigg{)}\Bigg{\}} \tag{5}\]  
where \(\sigma_{c}\) is the compressive stress (MPa), \(E_{0}\) is the initial stiffness (MPa), \(K\) is the elastic rigidity residual rate, \(\varepsilon_{c}\) is the compressive strain, \(\varepsilon_{p}\) is the plastic strain, \(f_{c}\) is the compressive strength (MPa) and \(\varepsilon_{peak}\) is the strain at compression strength. As shown in Figure 15, the stress-strain relationship on the tensile side was derived from the tensile softening curve based on the tensile stress-crack width relations from the \"Concrete Standard Specification\" of the JSC [13], as shown in Equations (6) and (7).  
\[w_{t}=\frac{5G_{f}}{f_{t}} \tag{6}\]  
\[G_{f}=10(d_{max})^{\frac{1}{3}}.f_{c}^{\frac{1}{3}} \tag{7}\]  
where \(w_{t}\) is the crack width (mm), \(G_{f}\) is the tensile fracture energy (N/mm), \(f_{t}\) is the tensile strength (MPa) and \(d_{max}\) is the maximum dimension of the coarse aggregate (mm). In addition, the von Mises yield condition with no hardening (complete elastic-plastic characteristics) was applied to the steel material, as shown in Figure 16. The post-yield handling of the rebar is thought to have some effect on the elasto-plastic response of the member, but since the main purpose of this study is a relative comparison of the four existing dam piers, this modeling will not be mentioned in depth. In addition, kinematic hardening was assumed for the steel material. Table 4 lists the material properties of the steel bars for the concrete and PCM parts.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Properties** & & **Existing Part** & **Reinforcement Part (PCM)** \\ \hline Density & kg/mm\({}^{3}\) & 2290 & 2300 \\ Young’s modulus & N/mm\({}^{2}\) & 17,500 & 27,000 \\ Poisson’s ratio & - & 0.17 & 0.2 \\ Compressive strength & N/mm\({}^{2}\) & 17.2 & 64.5 \\ Tensile strength & N/mm\({}^{2}\) & 1.8 & 4.12 \\ Maximum aggregate diameter & mm & 60 & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Physical characteristics of concrete.  
Figure 14: Stress-strain relationship of concrete (compression side).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Existing Part**} & \multicolumn{2}{c}{**Reinforcement Part**} \\
**Properties** & & **Main Rebar** & **Shear Reinforcement** & **Main Rebar** & **Shear Reinforcement** \\ \hline Material & & & SR235 & & SD345 \\ Mass density & kg/m\({}^{3}\) & & 7850 & \\ Young’s modulus & N/mm\({}^{2}\) & & 200,000 & \\ Yield modulus & N/mm\({}^{2}\) & 235 & & 345 \\ Poisson’s ratio & - & & 0.3 & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Physical characteristics of steel materials.  
Figure 15: Concrete stress\(-\)crack displacement relationship (tension side).