## 2 Modeling of Structural Materials  
### Reinforcing Steel  
The elastic-plastic model with regard to material hardening was applied for the reinforcing steel, which is shown in Figure 1. The reduced plane stress state (compression/tension with shear) was considered, because of the nature of the reinforcing slender bars in the reinforced concrete element. The stress state in subsequent load steps is described by the equations below. \[\sigma_{11}^{n}=\frac{\sigma_{11}^{n-1}+E_{s}\Delta e_{11}^{n}}{1+\frac{2}{3} E_{s}\Delta\Lambda^{n}}\text{ and }\sigma_{12}^{n}=\frac{\sigma_{12}^{n-1}+2\mu_{s}\Delta e_{12}^{n}}{1+2\mu_{s} \Delta\Lambda^{n}} \tag{1}\]  
where \(E_{s}\) is the deformation modulus of steel, \(\mu_{s}=\frac{E_{s}}{2(1+\
u_{s})}\) is the shear modulus, \(\
u_{s}\) is the initial Poisson's ratio \(\varepsilon_{11}^{n}\), \(\varepsilon_{12}^{n}\), \(\Delta\varepsilon_{11}^{n}=\varepsilon_{11}^{n}-\varepsilon_{11}^{n-1}\), and \(\Delta\varepsilon_{12}^{n}=\varepsilon_{12}^{n}-\varepsilon_{12}^{n-1}\) are the known strains and strain increments. \(\Delta\Lambda^{n}\) is the scalar multiplier that defines the state of stress according to the associated plastic flow rule for the Huber-Mises-Hencky yield criterion, \(n\) is the step number of the instantaneous strain-stress state in a subsequent time instant, and \(t_{n}=t_{n-1}+\Delta t\), and \(\Delta t\) is the time step increment. To determine the scalar multiplier \(\Delta\Lambda^{n}\), the value of the plasticity function must be checked for consistency with the Huber-Mises-Hencky yield criterion:  
\[F^{n}=\frac{\sigma_{i}^{n}}{f(\varepsilon_{ef}^{pn})}-1 \tag{2}\]  
where \(\sigma_{i}^{n}=\sqrt{\left(\sigma_{11}^{n}\right)^{2}+3\left(\sigma_{12}^{n} \right)^{2}}\) is the stress intensity, and \(f(\varepsilon_{ef}^{pn})\) is the hardening function of the steel yield limit. The variation of the steel yield limit function describes the law of isotropic hardening:  
\[f(\varepsilon_{ef}^{pt})=f_{y}+H_{s}\varepsilon_{ef}^{pn} \tag{3}\]  
where \(\varepsilon_{ef}^{pn}=\int\limits_{l_{y}}^{t_{n}}\Delta\varepsilon_{ef}^{p}dt\) is the effective plastic strain, \(\Delta\varepsilon_{ef}^{p}=\sqrt{\left(\Delta\varepsilon_{11}^{p}\right)^{2} +\frac{4}{3}\left(\Delta\varepsilon_{12}^{p}\right)^{2}}\) is the increment of the effective plastic strain, \(f_{y}\) is the steel yield limit for tension/compression, \(H_{\text{S}}\) is the steel plastic hardening modulus, and \(t_{y}\) is the time instant when the steel yield limit is achieved. Figure 1: Strength and stress parameters of the uniaxial tensile/compressive test for reinforcing steel. The uniaxial idealization for the tensile/compressive test is shown in Figure 1. After applying the assumption of elastic and plastic strain increments \(\Delta\varepsilon_{11}=\Delta\varepsilon_{11}^{e}+\Delta\varepsilon_{11}^{p}\), where \(\Delta\varepsilon_{11}=\frac{\Delta\sigma_{11}}{E_{T}}\), \(\Delta\varepsilon_{11}^{e}=\frac{\Delta\sigma_{11}}{E_{s}}\) and \(\Delta\varepsilon_{11}^{p}=\frac{\Delta\sigma_{11}}{H_{s}}\), the steel plastic hardening modulus is shown in the equation below. \[H_{S}=\frac{E_{S}E_{T}}{E_{S}-E_{T}} \tag{4}\]  
The tangent deformation modulus of steel is \(E_{T}=\frac{f_{t}-f_{y}}{\varepsilon_{t}-\varepsilon_{y}}\), where \(f_{t}\) is the steel tensile/compressive strength, \(\varepsilon_{y}=\frac{f_{y}}{E_{s}}\) is the maximum elastic strain, and \(\varepsilon_{t}\) is the strain that is related to the steel strength. In the initial assumption \(\Delta\Lambda^{n}=0\), if \(F^{n}<0\), or \(F^{n}=0\) and \(\Delta F^{n}<0\), we have a purely elastic stress state (or unloading). \(F^{n}>0\) represents an elastic-plastic state for which the multiplier value \(\Delta\Lambda^{n}\) is determined from the conditions below. \[F^{n}=0\text{ and }\Delta F^{n}=0 \tag{5}\]  
These equations can be numerically solved using [PERSON]'s method. The solution at the \(i\)-th iteration is shown below.