# Damage Analysis of Reinforced Concrete Columns Based on Fiber Beam-Column Element
## 2 Analysis Model of Reinforced Concrete Columns
### Constitutive Model of the Material  
#### 2.2.1 Constitutive Model for Concrete  
In this study, the concrete constitutive model proposed by [PERSON] [24] (as shown in Figure 1) was used. The model can simulate the continuous stiffness degradation effect and the hysteresis performance of concrete materials during unloading and reloading, as well as the constraint effect of stirrups on concrete.  
Figure 1: [PERSON]’s model for concrete.  
To consider the damage behavior in the concrete model, the tensile and compressive damage indices were used to describe the tensile and compressive damage of concrete, respectively.  
Based on the characteristics of the [PERSON] model and the basic principle of damage mechanics, the compression damage index is calculated as follows [23]:  
\[D_{\mathrm{c}}=\left\{\begin{array}{cc}0&\varepsilon_{\mathrm{cm}}^{\mathrm{c} }<\varepsilon_{\mathrm{cd0}}^{\mathrm{c}}\\ \frac{E_{\mathrm{cd0}}-E_{\mathrm{cm}}}{E_{\mathrm{cd0}}-E_{\mathrm{c0}}}& \varepsilon_{\mathrm{cd0}}^{\mathrm{c}}\leq\varepsilon_{\mathrm{cm}}^{ \mathrm{c}}\leq\varepsilon_{20}\\ 1&\varepsilon_{20}<\varepsilon_{\mathrm{cm}}^{\mathrm{c}}\end{array}\right. \tag{7}\]  
where, \(D_{\mathrm{c}}\) is the concrete compression damage index; \(E_{\mathrm{cd0}}=\left(\sigma_{\mathrm{cd0}}^{\mathrm{c}}-\sigma_{r}\right)/ \left(\varepsilon_{\mathrm{cd0}}^{\mathrm{c}}-\varepsilon_{r}\right)\) is the initial compression damage module, determined according to the compression damage starting point \(D_{0}\left(\varepsilon_{\mathrm{cd0}}^{\mathrm{c}},\sigma_{\mathrm{cd0}}^{ \mathrm{c}}\right)\) and the focus \(R(\varepsilon_{r},\sigma_{r})\); and \(E_{\mathrm{cm}}=\left(\sigma_{\mathrm{cm}}^{\mathrm{c}}-\sigma_{r}\right)/ \left(\varepsilon_{\mathrm{cm}}^{\mathrm{c}}-\varepsilon_{r}\right)\) is the module of the current unloading point, determined according to the unloading point \(D(\varepsilon_{\mathrm{cm}}^{\mathrm{c}},\sigma_{\mathrm{cm}}^{\mathrm{c}})\) and the focus \(R(\varepsilon_{r},\sigma_{r})\).  
The concrete compressive damage occurs only in the loading stage, and new compressive damage will occur after the reloading reaches the previous unloading point. According to the study by [PERSON] et al. [25], the point with a stress of \(0.3f^{\prime}_{\mathrm{c}}\) on the skeleton curve is selected as the starting point of compressive damage. The compression damage index of concrete is shown in Figure 2.  
The tensile skeleton curve in the model is shown in Figure 3. \(f_{\mathrm{t}}\) is the maximum tensile stress; \(\varepsilon_{\mathrm{cr}}\) is the strain corresponding to the maximum tensile stress; \(E_{\mathrm{c0}}\) is the initial tangent module; and \(\varepsilon_{\mathrm{ut}}\) is the ultimate tensile strain.  
The concrete tensile damage index is given as follows [23]:  
\[D_{\mathrm{t}}=\left\{\begin{array}{cc}0&\varepsilon_{\mathrm{cm}}^{\mathrm{ c}}<\varepsilon_{\mathrm{cr}}\\ \frac{E_{\mathrm{c0}}-E_{\mathrm{cm}}}{E_{\mathrm{c0}}}&\varepsilon_{\mathrm{ cr}}\leq\varepsilon_{\mathrm{cm}}^{\mathrm{c}}\leq\varepsilon_{\mathrm{ut}}\\ 1&\varepsilon_{\mathrm{ut}}<\varepsilon_{\mathrm{cm}}^{\mathrm{t}}\end{array}\right. \tag{8}\]  
Figure 2: The concrete compression damage index.  
where, \(E_{\rm tm}\) is the secant module of the current unloading point; and \(\epsilon_{\rm cm}^{\rm t}\) is the strain at the current unloading point.  
The concrete tensile damage occurs only in the loading stage, and new tensile damage occurs after the reloading reaches the previous unloading point. The tensile damage index of concrete is shown in Figure 4.  
Figure 4: The concrete tensile damage index.  
Figure 3: Tensile model of concrete.