## 3 Steel Reinforcement Constitutive Stress-Strain Law  
The steel constitutive law of [PERSON] [16] has been incorporated in the proposed model, as it was modified by [PERSON] et al. [17] and was integrated in Opensees software with the code name Steel02 by [PERSON] et al. [18]. This model is an evolution of the well-known model of [PERSON] [19], with appropriate modifications in order to better describe the hardening of steel. In Figure 4, a full stress-strain,\(f_{c}\)=\(\epsilon_{c}\), load cycle is presented (in blue color), as it is obtained from this model, where the various stress and strain parameters are described by [PERSON] [20]. In this model, load cycles are not composed using discrete branches, as in concrete. Instead, in each loading step the stress-strain response\(f_{s}\)-\(\epsilon_{s}\) is calculated from the following equation set, using normalized values for stress and strain, \(f_{s}^{\ast}\) and \(\epsilon_{s}^{\ast}\), respectively. \[f_{s}^{\ast}=b\cdot\epsilon_{s}^{\ast}+\frac{(1-b){\epsilon_{s}}^{\ast}}{ \left[1+{(\epsilon_{s}^{\ast})}^{R}\right]^{1/R}} \tag{27}\]  
\[b=\frac{E_{sh}}{E_{o}}\text{ ( hardening ratio)} \tag{28}\]  
\[E_{o}=\frac{f_{o}-f_{rev}}{\epsilon_{o}-\epsilon_{reg}}\text{ ( tangent modulus of elasticity)} \tag{29}\]  
Figure 3: Numbering of branches. The corresponding normalized values of stress \(f_{\rm s}\) and strain \(\varepsilon_{\rm s}\) are defined as follows:  
\[{f_{s}}^{*}=\frac{f_{\rm s}-f_{\rm rev}}{f_{o}-f_{\rm rev}} \tag{30}\]  
\[{\varepsilon_{s}}^{*}=\frac{\varepsilon_{s}-\varepsilon_{\rm rev}}{\varepsilon_ {o}-\varepsilon_{\rm rev}} \tag{31}\]  
In Equations (29)-(31) (\(\varepsilon_{\rm rev},f_{\rm rev}\)) is the last load reversal point, while (\(\varepsilon_{o},f_{o}\)) is the intersection point between initial and secant stiffness. More specifically, Equation (27) describes a curve with initial tangent stiffness:  
\[E_{o}={f_{\rm s}}^{*}/{\varepsilon_{s}}^{*} \tag{32}\]  
According to experimental data, for strains where \(\varepsilon_{s}^{*}\to\infty\) the curve described by Equation (27) asymptotically tends to the following straight-envelope line with slope \(E_{\rm sh}\):  
\[{f_{\rm s}}^{*}=b\cdot{\varepsilon_{s}}^{*}+(1-b) \tag{33}\]  
In the above formulation of Equations (27)-(31), if \(\varepsilon_{\rm rev}=f_{\rm rev}=0\), then the Equation (27) describes pure monotonic loading. Moreover, if \(b=0\), then the response is elastic-perfectly plastic, while for positive or negative values of \(b\), the post-elastic branch is upward or downward, respectively. In Figure 4, the superscript of parameters \(\varepsilon_{\rm rev}\) and \(f_{\rm rev}\) denotes the serial number of each subsequent loading. For example, for the first positive loading the superscript is equal to 1, while for the subsequent negative loading the superscript is equal to 2. Figure 4: Parameters of [PERSON] and [PERSON] [16] model, according to [PERSON] [20]. Tangent stiffness, \(E_{tan}^{*}\), results from differentiating Equation (27) with respect to \(\varepsilon_{s}{{}^{*}}\) as follows:  
\[E_{\text{tan}}^{*}=\frac{\text{d}f_{s}{{}^{*}}^{*}}{\text{d}\varepsilon_{s}{{}^{ *}}^{*}}=b+\frac{1-b}{\left[1+\left(\varepsilon_{s}{{}^{*}}\right)^{R}\right]^{ 1/R}}\cdot\left[1-\frac{\left(\varepsilon_{s}{{}^{*}}\right)^{R}}{1+\left( \varepsilon_{s}{{}^{*}}\right)^{R}}\right] \tag{34}\]  
The tangent stiffness, \(E_{tan}^{*}\) of Equation (34), is used in the calculation of the section stiffness along with the corresponding Equation (8) for concrete. Parameter \(R\) is crucial for the behavior of the model and governs the curvature of the loading or unloading branch enclosed between the line with stiffness \(E_{o}\) and that with stiffness \(E_{sh}\).