## 3 FE Analysis
### Material Modeling
#### 3.2.1 Compression Behavior  
The compressive behavior of concrete under uniaxial loading conditions can be characterized by three distinct phases, as illustrated in Figure 3a. Each of these phases is described in detail below:  
Phase 1: During the first phase, the stress is linearly proportional to strain, as described by Equation (1):  
\[\sigma_{\mathrm{(c1)}}=\mathrm{E_{0}}\varepsilon_{c} \tag{1}\]  
With \(\mathrm{E_{0}}\), \(\varepsilon_{\mathrm{c}}\) represents the concrete's initial elastic stiffness and strain, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Concrete** & \(\psi\) & \(\varepsilon\) & \(\sigma_{\mathrm{b0}}/\sigma_{\mathrm{c0}}\) & \(\mathbf{K_{c}}\) & **Viscosity** \\ \hline NC & 30 [18] & 0.1 [18] & 1.16 [18] & 0.6667 [18] & 0.0001 [18] \\ \hline \end{tabular}
\end{table}
Table 2: Parameters of CDP model for NSC and UHPS materials.  
Figure 3: Concrete response to uniaxial loading: (**a**) compression; (**b**) tension.  
Phase 2: After the linear stage, cracks begin to appear, and the stress-strain relationship of concrete becomes nonlinear, as described by Equation (2):  
\[\sigma_{\left(\zeta 2\right)}=\frac{\mathrm{E_{ci}}\frac{\varepsilon_{c}}{\mathrm{f_{cm} }}-\left(\frac{\varepsilon_{c}}{\varepsilon_{\mathrm{c1}}}\right)^{2}}{1+ \left(\mathrm{E_{ci}}\frac{\varepsilon_{c}}{\mathrm{f_{cm}}}-2\right)\frac{ \varepsilon_{c}}{\varepsilon_{\mathrm{c1}}}}\mathrm{f_{cm}} \tag{2}\]  
\(\mathrm{E_{ci}}\) represents the modulus of elasticity of deformation of concrete for zero stress, while \(\mathrm{f_{cm}}\) and \(\varepsilon_{\mathrm{c1}}\) denote the strength and strain at peak, respectively.  
Phase 3: This softening phase assumes that the stress-strain relationship follows the theory proposed by [PERSON] et al. [19], which takes into account the mesh length (\(\mathrm{l_{eq}}\)). The constitutive law of this phase is presented in Equation (3):  
\[\mathrm{s_{\left(\zeta 3\right)}}=\left(\frac{2+\mathrm{g_{c}}\mathrm{f_{cm} }\mathrm{e_{cl}}}{2\mathrm{f_{cm}}}-\mathrm{g_{c}}\mathrm{e_{c}}+\frac{ \mathrm{g_{c}}\mathrm{e_{c}^{2}}}{2\mathrm{e_{c1}}}\right)^{-1} \tag{3}\]  
With \(\mathrm{\gamma_{c}}=\frac{\pi^{2}\mathrm{f_{cm}}\varepsilon_{\mathrm{c1}}}{ 2\left[\frac{\mathrm{c_{ch}}}{\mathrm{v_{eq}}-0.5\mathrm{f_{cm}}\left( \varepsilon_{\mathrm{c1}}(1-\mathrm{b})+\mathrm{b}\frac{\mathrm{f_{cm}}}{ \mathrm{f_{0}}}\right)}\right]}\); \(\mathrm{b}=\frac{\varepsilon_{\mathrm{c1}}^{\mathrm{pl}}}{\varepsilon_{ \mathrm{c}}^{\mathrm{pl}}}\).  
Where \(\varepsilon_{\mathrm{c}}^{\mathrm{pl}}\), \(\varepsilon_{\mathrm{c}}^{\mathrm{pl}}\) denote plastic strain and inelastic strain, respectively. \(\mathrm{l_{eq}}\) is the characteristic length, which depends on the mesh size.  
\(\mathrm{G_{ch}}\) is the crushing energy, which is determined by \(\mathrm{G_{ch}}=\left(\frac{\mathrm{f_{cm}}}{\mathrm{f_{cm}}}\right)^{2}0.073 \mathrm{f_{cm}^{0.18}}\).  
The compression damage variable, \(\mathrm{d_{c}}\), is determined by the portion of normalized energy dissipated by damage [20] in the following Equation:  
\[\mathrm{d_{c}}=1-\frac{1}{2+\mathrm{a_{c}}}\left[2(1+\mathrm{a_{c}})\exp(- \mathrm{b_{c}}\mathrm{\varepsilon_{c}^{ch}})-\mathrm{a_{c}}\mathrm{exp}(-2 \mathrm{b_{c}}\mathrm{\varepsilon_{c}^{ch}})\right] \tag{4}\]  
\(\mathrm{\varepsilon_{c}^{ch}}\) is the crushing strain.  
Here, the \(\mathrm{a_{c}}\) and \(\mathrm{b_{c}}\) dimensionless coefficients are determined from Equations (5) and (6):  
\[\mathrm{a_{c}}=2(\mathrm{f_{cm}}/\mathrm{f_{c0}})-1+2\sqrt{\left(\mathrm{f_{ cm}}/\mathrm{f_{c0}}\right)^{2}-\mathrm{f_{cm}}/\mathrm{f_{c0}}} \tag{5}\]  
\[\mathrm{b_{c}}=\frac{\mathrm{f_{c0}}\mathrm{l_{eq}}}{\mathrm{G_{ch}}}\left(1+ \frac{\mathrm{a_{c}}}{2}\right) \tag{6}\]  
\(\mathrm{f_{c0}}\) is the compressive strength at zero crushing strain, and \(\mathrm{f_{cm}}\) is determined by \(\mathrm{f_{cm}}=\frac{\mathrm{f_{c0}}\left(1+\mathrm{a_{c}}\right)^{2}}{ \mathrm{a_{c}}}\).