# Seismic Design and Performance Assessment of Frame Buildings Reinforced by Dual-Phase Steel
## 3 Mathematical Modelling of Cyclic Response of Columns Reinforced with B500B and DPD2  
### Description of Code-Based Mathematical Modelling of Seismic Response of RC Columns  
The code-based mathematical model of an RC column comprises an elastic element with nonlinear flexural hinges at both ends of the column. If the model is used for simulation of the cyclic test of a column, then the nonlinear flexural hinge is used only at the base of the column (see Figure 2a). Such a model is presented in the following and is used to simulate the cyclic response of columns reinforced with both B500B and DPD2 steel bars. All simulations were performed using OpenSees [30]; therefore, some features of the modelling refer to the OpenSees command language. The linear elastic element was modelled using the 'elasticBeamColumn' element available in OpenSees. The element is defined based on the height of column used in experiments and has the characteristics of a concrete cross-section (i.e., elastic modulus of concrete, and area and moments of inertia of cross-section). The four-linear moment-rotation relationship of the nonlinear flexural hinge, similar to that proposed by [PERSON] et al. [31], was defined. The cracking of the concrete cross-section is represented by the first characteristic point (CR) from the moment-rotation relationship, whereas other points are related to the reinforcement yielding (Y), the maximum moment (M) and the near-collapse limit (NC) state (see Figure 2b). Note also that an additional point (see a red dot in Figure 2b) is presented on the moment-rotation relationship. This point was added to improve the presentation of the damage of the structural elements. The data required for the calculation of the moment-rotation relationship (see Figure 2b) are elastic proprieties of structural elements, the geometry of cross-section of the structural element, longitudinal reinforcement, transverse reinforcement, the material strength, level \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Steel Grade** & \(f_{gk}\) **[MPa]** & \(f_{gm}\) **[MPa]** & \(f_{u}\) **[MPa]** & \(A_{gt}\) & \(A_{5}\) & \(k\) \\ \hline B500B & 500 & 548 & 650 & 12.3\% & 25.0\% & 1.19 \\ DPD2 & 400 & 461 & 590 & 11.9\% & 26.2\% & 1.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of B500B and DPD2 steel grades taken from the experimental results [21; 24; 27]. Figure 2: (a) mathematical model of column for simulation of cyclic response and (b) four-linear moment-rotation relationship of plastic hinge. of normalised axial force and assumed ratio \(r_{CM}\) (see below). The cracking \(M_{CR}\), yielding \(M_{\rm Y}\), and the maximum \(M_{\rm M}\) moments were calculated by performing a moment-curvature analysis of the cross-section, which was divided into many fibres. Each fibre was defined with an appropriate uniaxial material, available in OpenSees. The confined and unconfined concrete were modelled using 'Popovics Concrete Material' (or 'Concrete04 Material') [32]. In the case of unconfined cover concrete, the parameters of 'Popovics Concrete Material' were assessed according to the requirements of Eurocode 2 [13], whereas the concrete compressive strength was obtained from the compressive test performed on cubes taken from the concrete used to cast the column. The parameters of the confined concrete were estimated according to the draft of the new Eurocode 8-3 [33]. In this case, the confined concrete maximum compressive strength \(f_{cc}\) and the corresponding strain \(\varepsilon_{c0,c}\) can be determined using the following expressions  
\[f_{cc}=f_{c}(1+K) \tag{1}\]  
\[\varepsilon_{c0,c}=\varepsilon_{c0}(1+5K) \tag{2}\]  
where \(f_{c}\) is the concrete compressive strength of unconfined concrete, whereas \(\varepsilon_{c0}\) is the compressive strain in the concrete at maximum strength, which is assumed to be 0.002.