## 2 Performance of IWP with Steel Anchor Beam
### FE Analysis and Results
#### 2.3.1 FE Model of the IWP Test Model  
General FE software ABAQUS 2020 was utilized to analyze the test model. To simulate the stress distribution of the structure accurately, the steel plate, concrete, and plate-perforating rebar were modeled by C3D8R, 8-node linear hexahedral solid elements with reduced integration, with at least four layers of element in the thickness direction, as shown in Figure 14. As Figure 14a shows, two rigid plates were established under the test model to simulate the supporting beam, the reference points of which are constrained in all degrees of freedom (DOFs). The translational DOFs along the X- and Z-axes of the nodes inside six  
Figure 13: Interfacial deformation of PBL specimen and IWP specimen: (**a**) slip; (**b**) separation. square regions (shown in red) are constrained to simulate the ground anchors. The end of the test model was constrained in the Y-axis to simulate the effect of the reaction beam. The nodes on the loading surface are coupled to a reference point RP-1, on which an imposed displacement was introduced to apply the load. The loading speed was controlled to be 0.02 mm/s, and the ratio of kinematic to internal energy was controlled under 5%. Hard contact with penalty frictional formulation was adopted to simulate the interaction between the supporting beam and the test model and all steel-concrete interfaces, with the friction coefficient assumed as 0.5. The mesh sizes of the concrete and steel parts were determined based on mesh sensitivity checks shown in Table 3 and Figure 15, and the regular mesh with less than 5% of error compared to fine mesh is utilized in the following analysis. Figure 14b,c shows the finite element models of the steel part. Besides the perforating bars, the rest of the rebars were modeled by TRUSS elements. Figure 14: FE model: (**a**) overall model; (**b**) side view of steel part; (**c**) schematic view of steel part. The stress-strain relationships for steel plates and rebars were assumed as a perfectly elastic-plastic model, with the yield strength being 430 MPa and 335 MPa and the modulus of elasticity being 200 GPa and 195 GPa, respectively. The material properties for concrete were determined based on material tests of three cubic and six cylindrical specimens. A concrete damaged plasticity model was assumed for concrete. The elastic modulus, Poisson ratio, dilation angle, eccentricity, \(f_{\rm bo}/f_{\rm co}\), \(K\), and the viscosity parameter were taken to be 34.4 GPa, 0.2, 30\({}^{\circ}\), 0.1, 1.16, 0.67, and 0.0, respectively. The material nonlinearity of the concrete was considered in the stress-strain relationship under compression with the following relationship [32]:  
\[\sigma_{c}/f_{c}=\frac{\gamma(\epsilon_{c}/\epsilon_{c}^{\prime})}{\gamma-1+( \epsilon_{c}/\epsilon_{c}^{\prime})^{\gamma}} \tag{4}\]  
\[\gamma=(f_{c}/32.4)^{3}+1.55 \tag{5}\]  
where \(\sigma_{c}\) stands for the compressive stress of concrete (MPa); \(\epsilon_{c}\) stands for the compressive strain; \(\epsilon_{c}\) is the peak strain of concrete; \(f_{c}\) is the compressive strength of concrete (MPa), taken as 44.3 MPa based on material tests; and \(\gamma\) is an undermined coefficient. The stress-strain relationship of concrete under tension before cracking was assumed to be linear, and the relationship between the tensile stress and crack width after cracking was calculated based on the following [33]:  
\[\sigma_{t}/f_{t}=f(\omega)-(\omega/\omega_{c})f(\omega_{c}) \tag{6}\]  
\[f(\omega)=\left[1+(c_{1}\omega/\omega_{c})^{3}\right]\exp(-c_{2}\omega/\omega _{c}) \tag{7}\]  
\[G_{f}=73f_{c}^{0.18} \tag{8}\] \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & **Steel** & & **Concrete** & \\ \cline{2-4}  & **Global Size (mm)** & **Layer Through** & **Global Size (mm)** & **Local Size at** \\  & **Thickness** & **Global Size (mm)** & **Steel–Concrete** \\  & & & & **Interface (mm)** \\ \hline Coarse mesh & 30 & 3 & 60 & 30 \\ Regular mesh & 20 & 4 & 40 & 20 \\ Fine mesh & 15 & 6 & 25 & 15 \\ \hline \hline \end{tabular}
\end{table}