## 3 FEA Model and Validation
### Material Properties and Modelling Procedure  
The Mohr-Coulomb model is widely used in numerical studies as a relatively simple constitutive model, the same applies when the internal friction angle (\(\varphi\)) is less than 22\({}^{\circ}\), but relatively speaking the DP model converges better [43]. In order to reduce the difficulty of computation convergence for soils with the internal friction angle \(\varphi\ <\ 22\), Drucker-Prager (D-P) criterion can be used for modeling the plasticity behavior instead of the Mohr-Coulomb assumption. Herein, the D-P criterion together with the linear elasticity assumption is employed for the soils in the FEA model. The D-P criterion is summarized as follows.  
Figure 5: Geometry of the finite element analysis (FEA) model (redrawn after [34]).  
The yield surface of D-P model can be expressed as Equations (2)-(4):  
\[\mathrm{p}=-\frac{1}{3}\sigma\mathbf{:I} \tag{2}\]  
\[\mathrm{q}=\sqrt{\frac{3}{2}}\mathbf{S}\mathbf{:S} \tag{3}\]  
\[\mathbf{r}^{3}=\frac{9}{2}\mathbf{S}\mathbf{:S}\mathbf{:S} \tag{4}\]  
where \(\mathrm{p}\) is the average principal stress, \(\mathrm{q}\) is equivalent shear stress, \(\mathbf{r}\) is the third stress tensor invariant, and \(\mathbf{S}\) is the deviator stress tensor as:  
\[\mathbf{S}=\sigma+\mathrm{pI} \tag{5}\]  
The linear D-P criterion can be expressed as Equations (6) and (7).  
\[\mathrm{F}=\mathrm{t}-\mathrm{p}\mathrm{t}\mathrm{a}\mathrm{n}\beta-\mathrm{d}=0 \tag{6}\]  
\[\mathrm{t}=\frac{1}{2}\mathrm{q}\left[1+\frac{1}{\mathrm{K}}-\left(1-\frac{1 }{\mathrm{K}}\right)\left(\frac{\mathbf{r}}{\mathrm{q}}\right)^{3}\right] \tag{7}\]  
where \(\mathrm{\beta}\) is the inclination of the yield surface in the \(p-t\) stress space, \(d\) is the intercept of the \(\mathrm{t}\) axis and the yield surface in the \(p-t\) stress space. \(\mathrm{\beta}\) and \(k\) can be converted and calculated according to Equations (8) and (9).  
\[\mathrm{t}\mathrm{a}\mathrm{n}\beta=\frac{6\mathrm{a}\mathrm{n}\mathrm{p}}{3- \mathrm{a}\mathrm{n}\mathrm{p}} \tag{8}\]  
\[\mathrm{d}=2c\frac{\mathrm{a}\mathrm{n}\mathrm{p}}{1-\mathrm{a}\mathrm{n} \mathrm{p}} \tag{9}\]  
\[\mathrm{K}=\frac{3-\mathrm{a}\mathrm{n}\mathrm{p}}{3+\mathrm{a}\mathrm{n} \mathrm{p}} \tag{10}\]  
Considering isotropic elastic damage, isotropic tensile, and compressive plasticity theory, the concrete damage plasticity (CDP) model is a continuous, plastic-based concrete constitutive model, which can describe the non-linear damage behavior of concrete [44]. The CDP model assumes that, when the concrete under tension or compression, the strain contains isotropic elastic damage (\(\varepsilon_{t}^{el},\varepsilon_{c}^{el}\)) and plastic damage (\(\varepsilon_{t}^{pl},\varepsilon_{c}^{pl}\)). Comparing with elastic materials, strain can also be regarded as the elastic strain (\(\varepsilon_{0t}^{el},\varepsilon_{0c}^{el}\)) with the same \(E_{0}\) plus inelastic strain (\(\overline{\varepsilon}_{t}^{nk},\overline{\varepsilon}_{c}^{ln}\)), where \(d_{t}\) and \(d_{c}\) are both damage factors.  
The concrete strain in tension can be obtained as:  
\[\sigma_{t}=(1-d_{t})E_{0}\left(\varepsilon_{t}-\overline{\varepsilon}_{t}^{ pl}\right) \tag{11}\]  
\[\sigma_{t}=E_{0}\left(\varepsilon_{t}-\overline{\varepsilon}_{t}^{nk}\right)\  
According to previous research [45], the recommended values of \(b_{t}\) and \(b_{c}\) are 0.1 and 0.7, respectively. The CDP parameters of C55 (i.e., 55 MPa cube strength) concrete is employed in this model. Parameters of each component of this model are shown in Table 1.