# Development of Phenomenological Model for RC Circular Columns
## 1 Introduction  
A performance-based earthquake engineering study needs a relatively reliable hysteretic model. The model must be able to capture the structural seismic responses in different damage states, especially from the severe damage stage to the collapse stage; the model is also required to accurately capture the behaviors of the components' continuous deterioration in strength and stiffness under earthquake motions. The Ibarra-Medina-Krawinkler (IMK) model [1] and the Modified IMK (ModIMK) model [2], which are founded on the hysteretic energy dissipation principle, could fulfill both of the previous requirements very well.  
The hysteretic models for reinforced concrete (RC) components can be divided into \"mechanical\" and \"phenomenological\" models [3, 4, 5]. The mechanical model needs to be based on the accurate determination of the material's mechanical properties, i.e., it needs accurate constitutive laws of the material, and the typical mechanical model is the fiber model; it is often difficult to accurately describe the phenomena, such as concrete crushing and reinforcement slipping. The empirical phenomenological hysteretic model, however, can better describe these phenomena. The ModIMK model is based on phenomenology, which not only describes complex experimental phenomena but also speeds up the computation of structural analyses; therefore, it is recommended by the U.S. Code ASCE 41, which has led to the wide application of this model in earthquake engineering studies.  
The hysteretic model of a component can be determined by calibrating its individual model parameters, while empirical prediction equations for the model parameters can be obtained through statistical analyses after the calibration of the existing pseudo-static tests [6; 7]. In order to ascertain the IMK model of RC square columns, [PERSON] [8] established a database, calibrated the model parameters of each component, and created empirical equations for each IMK model parameter. For the purpose of developing the IMK model for asymmetric components, [PERSON] [9; 10] modified it and obtained the ModIMK model; he collected RC and steel components, calibrated each ModIMK model parameter, and then obtained its prediction equation. By contrasting the corroded and non-corroded RC square columns, [PERSON] [11] developed the corroded columns' empirical equations for the ModIMK model. [PERSON] [12] established a database, calibrated the ModIMK model's cyclic deterioration parameters for all columns, and created empirical prediction equations.  
The studies mentioned above are on RC square columns, while RC circular columns have seldom been studied. [PERSON] [13] developed the ModIMK hysteretic model for corroded circular columns, but this model relies on the ModIMK model for uncorroded columns, which is still incomplete. [PERSON] [14] created empirical prediction equations for some parameters of the ModIMK hysteretic model, but the equations for parameters \(\theta_{\mathrm{pc}}\) and \(M_{\mathrm{c}}/M_{\mathrm{y}}\) have not been established. Additionally, in a database of 76 components, the P-\(\Delta\) effect pattern was incorrectly chosen for 27 components, leading to inaccurate calibration results for the model parameters. Therefore, the ModIMK hysteretic model for circular columns requires further improvement and development. We collected 80 sets of pseudo-static test data for RC circular columns and conducted a calibration and prediction study of the model parameters. The peak-orientated ModIMK model was adopted, and the OpenSees (3.3.0) software was utilized to numerically simulate the test curves of each column to calibrate the hysteretic model parameters. Then, a statistical regression was performed between the components' design parameters and each model parameter, and empirical equations were established. The accuracy of these empirical prediction equations was verified through the analysis of eight columns in the database and out of the database.