# Deterioration Model for Corroded Circular Columns
## 3 Hysteretic Model and Calibration of the Model Parameters for the Corroded Columns
### Calibration of the Model Parameters  
Since the backbone curves of the ModIMK model are obtained through monotonic loading tests, but most of the components are not individually monotonically loaded, the hysteresis curves could not be used to calculate the parameters of the backbone curves directly. On the other hand, the deterioration in the strength and stiffness can only be assessed through an overall analysis of the hysteresis behavior of the components; so, the cyclic deterioration coefficients could not be obtained directly from the hysteresis curves either. Therefore, it was necessary to simulate the hysteresis curves of the test numerically. The hysteretic model parameters can be calibrated when the simulated and test hysteresis curves are well fitted.  
For reinforced-concrete structures, a value of 0 is assigned to the residual moment \(M_{r}\); four cyclic deterioration parameters, \(\Lambda_{S},\ \Lambda_{C},\ \Lambda_{A},\ and\ \Lambda_{K}\), are assumed to have an identical value, \(\Lambda\); and a value of 1 is assigned to the exponent \(c\)[1; 11]. Therefore, only six parameters are needed to determine the ModIMK hysteretic model: the five parameters  
Figure 2: The backbone curves of the corroded and uncorroded components.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Parameter** & **Physical Significance** & **Model Parameter** & **Physical Significance** \\ \hline \multirow{2}{*}{Mc/My} & Ratio of capping moment to yield & \multirow{2}{*}{\(\theta_{y}\)} & Chord rotation of elastic stage in \\  & moment in backbone curve & & backbone curve \\ \(M_{y}\) & Yield moment in backbone curve & \multirow{2}{*}{\(\theta_{p}\)} & Chord rotation of strain-hardening \\  & & & stage in backbone curve \\ \(M_{c}\) & \multirow{2}{*}{Capping moment in backbone} & \multirow{2}{*}{\(\theta_{pc}\)} & Chord rotation of softening stage in \\  & curve & & backbone curve \\ \(M_{r}\) & \multirow{2}{*}{Residual moment in backbone} & \multirow{2}{*}{\(\theta_{a}\)} & Ultimate rotation capacity \\  & curve & & \\ \(K_{0}\) & \multirow{2}{*}{Elastic stiffness} & \multirow{2}{*}{\(\Lambda_{S}\)} & Cyclic deterioration parameter for \\  & & & strength deterioration \\ \(K_{s}\) & \multirow{2}{*}{Hardening stiffness} & \multirow{2}{*}{\(\Lambda_{c}\)} & Cyclic deterioration parameter for \\  & & & post-capping strength deterioration \\ \(c\) & \multirow{2}{*}{Exponent defining the rate of} & \multirow{2}{*}{\(\Lambda_{A}\)} & Cyclic deterioration parameter for \\  & & & accelerated reloading stiffness \\ \(K_{c}\) & \multirow{2}{*}{Post-capping stiffness} & \multirow{2}{*}{\(\Lambda_{K}\)} & Cyclic deterioration parameter for \\  & & & unloading stiffness deterioration \\ \hline \hline \end{tabular}
\end{table}
Table 3: The parameters of the ModIMK model.  
associated with the skeleton curve, i.e., \(M_{y}\), \(\theta_{y}\), \(M_{c}/M_{y}\), \(\theta_{p}\), \(\theta_{pc}\), and the parameters associated with the hysteresis rule, i.e., \(\Lambda\).  
The OpenSees (3.3.0) software was utilized to establish the component analysis model, which is based on the ModIMK plastic hinge element (as illustrated in Figure 3). Since there is no plastic hinge before the component yields in a real experiment, in order to be equivalent to a fix-ended cantilever column, the stiffness of the plastic hinge and the elastic element in the analytical model must be adjusted before the yield strength is reached, i.e., the plastic hinge is tuned for high elastic stiffness, and the stiffness of the elastic element is adjusted accordingly [30; 31; 32].  
The parameters of the ModIMK model were calibrated according to the methodology provided in Refs. [11; 30]. The calibration process was conducted independently in the positive and negative loading directions. The mean values of the positive and negative calibration results are listed in Table 1.