# 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
### Numerical Simulation of Cyclic Test of Column Reinforced with B500B  
The proposed model was used to simulate the cyclic response of a column reinforced with reinforcing steel B500B and tested within a research project sponsored by the Slovenian Research Agency [24]. The specimen height was 1.70 m with a cross-section b/h = 30/30 cm. The square column was reinforced with 8 bars of diameter \(\phi\) 16 mm in the longitudinal direction and \(\phi\)8 mm/7.5 cm \(n=2+\sqrt{2}\) in the transverse direction (see Figure 4). The mean compressive strength of concrete was obtained from the experimental test on the cylinder, and it was equal to 41.2 MPa, whereas for the reinforcing steel, the nominal strength of the B500B was assumed. The column was loaded with an axial force that corresponded to the normalised axial force 0.15. The distance between the column base and the centre of the horizontal hydraulic jack, which was used to impose the displacements to the column, was 1.50 m.  
The simulation of the response of the column was performed by imposing on the model the displacements that were used during the cyclic test. The model of the column was developed according to the rules defined in Section 3.1. The elastic part of the model was based on the concrete cross-section dimensions and the elastic modulus of concrete (\(E_{c}\) = 36,000 MPa, see Table 2). The moment-rotation relationship was assessed using the characteristics of the reinforced concrete cross-section at the base of the column. For the reinforcing steel, the stress-strain curve for B500B, presented in Figure 1, was used. The material properties of confined and unconfined concrete are presented in Table 2. Note that the axial force (N) was not applied to the columns, but its effect was taken into account in the calculation of the moment-rotation relationship (see Section 3.1). \begin{table}
\begin{tabular}{c c c} \hline  & **Unconfined Concrete** & **Confined Concrete** \\ \hline \(f_{c}\) [MPa] & 41.2 & 56.8 \\ \(\epsilon_{c0}\) [\%] & 2.22 & 6.41 \\ \(\epsilon_{c0}\) [\%] & 3.50 & 19.66 \\ \(f_{ct}\) [MPa] & 3.1 & 3.1 \\ \(\epsilon_{t}\) [\%] & 0.086 & 0.086 \\ \(E_{c}\) [MPa] & 3.6\(\cdot 10^{4}\) & 3.6\(\cdot 10^{4}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Compressive strength (\(f_{c}\)), corresponding compressive strain at \(f_{c}\) (\(\epsilon_{c0}\)), ultimate strain (\(\epsilon_{c0}\)), axial tensile strength (\(f_{ct}\)), tensile strain at \(f_{ct}\) (\(\epsilon_{t}\)) and elastic modulus (\(E_{c}\)) of confined and unconfined concrete. Figure 4: Elevation view of specimen with reinforcement, and cross-sections of column of specimen reinforced with B500B. The characteristic points of the moment-rotation relationship were calculated as defined in Section 3.1 by assuming \(r_{CM}=3.5\). The parameter \(\beta\) was considered, as defined in Section 3.1 (i.e., 0.75 and 0.69 for, respectively, uniaxial material _Hysteretic_ corresponding to zero-length element with smaller and higher initial stiffness (see zero-length element 2 in Figure 3)). The pinching and damage parameters were set as 0. The model for the moment-rotation envelope of the column was further calibrated to improve the match between the simulated and measured cyclic response of the column. For this purpose, only the yielding rotation from Equation (7) was multiplied by the correction factor \(\gamma=0.60\), and the \(r_{CM}\) was set to 7. The parameter \(\beta\) and the damage parameter (_damage1_) of the column were also estimated based on the experimental results. For the zero-length element with smaller initial stiffness, the parameter \(\beta\) was set to 0.75, and _damage1_ was taken to equal 0.009, whereas, for another zero-length element (Figure 3), the corresponding values were considered to equal 0.55 and 0.001. The cyclic test results and the corresponding simulations using the basic and the calibrated model are presented in Figure 5. From Figure 5a, it can be observed that the basic model overestimates the yield rotation and rotation at the maximum moment.