# Implementation of PMDL and DRM in OpenSees for Soil-Structure Interaction Analysis
## 4 Application  
Following the successful incorporation and validation of both DRM and PMDL in OpenSees, this section illustrates an application that employs the integrated DRM-PMDL system for examining the influence of the angle of incidence (\(\theta\)). As documented in [47; 48], the angle of incidence has been demonstrated to exert a considerable influence on the response of tunnels, particularly those with a vertical orientation. This section presents an examination of a single-cell tunnel and a double-cell tunnel in homogeneous media subjected to SV waves at varying angles of incidence. The numerical models are illustrated in Figure 15. The DRM interface is employed to define the input motion, while the PMDL is utilized to truncate the domain and mitigate the effects of scattering waves. The input motion is identical to that utilized for validating the DRM implementation (see Figure 11). Four-node quadrilateral plane-strain elements are employed for the domain of interest, while two-node frame elements are utilized for the tunnel structures, with the latter having a thickness of 0.50 m. The contact between the soil and the tunnel is assumed to be bonded. The material properties for the soil and the tunnels are given in Table 4.  
\[ \begin{split} R_{\alpha}=\tan^{-1}\left(\frac{X_{2}-X_{1}}{h_{T}} \right)-\tan^{-1}\left(\frac{Z_{1}-Z_{4}}{w_{T}}\right)\\ R_{\beta}=\tan^{-1}\left(\frac{Z_{1}-Z_{4}}{w_{T}}\right)\end{split}  \tag{33}\]  
In Equation (33), \(R_{\alpha},~{}R_{\beta}\) refer to the drift ratio and rocking rotation, respectively. The shear and rocking deformation (see Figure 16) of the tunnel structure are depicted in Figures 17 and 18. In Figure 16, \(w_{T}\) and \(h_{T}\) denote the width and height of the tunnel. The \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Young’s** & **Shear Modulus** & **Poisson’s Ratio** & **Shear Wave** & **Density (kg/m\({}^{3}\))** \\  & **Modulus (kPa)** & **(kPa)** & **Poisson’s Ratio** & **Velocity (m/s)** & **Density (kg/m\({}^{3}\))** \\ \hline Soil & - & 80,000 & 0.3 & 200 & 2000 \\ Tunnels & 32,000,000 & - & 0.2 & - & 2500 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material properties of the soil and tunnels. Figure 15: Illustration of (**a**) single-cell, and (**b**) double-cell tunnels for studying the effects of incidence angle. Figure 20 illustrates the maximum spectral amplification (as percentages) at the ground surface, caused by the presence of a subsurface tunnel structure. The said percentages are obtained by first calculating the difference between the spectral values when considered with and without tunnel structures, and subsequently dividing these differences by the corresponding spectral values for the case where there is no tunnel structure. As illustrated in  
Figure 19: Variation in peak: (**a**) drift ratios and (**b**) rocking rotations. Figure 18: Illustration of rocking rotations: (**a**) single-cell tunnel, (**b**) its Fourier amplitudes, (**c**) double-cell tunnel, and (**d**) its Fourier amplitudes. Figure 20, the amplification percentages of spectral values for the single-cell tunnel decrease as the angle of incidence increases. For the double-cell tunnel, the amplification percentages of Sd and PSV remain nearly constant. Conversely, the amplification percentages of PSA for the double-cell tunnel increase with the angle of incidence. At a specific angle of incidence, the highest amplification percentages are observed in the single-cell tunnel. Figures 21 and 22 present the deformation plots at the moment of maximum bending moment for the single-cell and double-cell tunnels, respectively, under various incidence angles. It can be observed that greater bending deformation occurs at smaller incidence angles, while more rotation is evident at larger incidence angles. It should be noted that all figures show the deformations magnified by a factor of 10,000. Figures 23 and 24 display the maximum bending moment diagrams for the single-cell and double-cell tunnels, respectively.