## 6 Case Study Validation
### Applicability Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(R_{T}\)} & \multirow{2}{*}{**Seismic Wave**} & \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**A section Axial**} & \multicolumn{1}{c}{**E Section Shear**} & \multicolumn{1}{c}{**F Section Bending**} & \multicolumn{1}{c}{**Deformation of Central**} \\  & & & **Force N (\&N)** & **Force Q (kN)** & **Moment \(M\) (kN-m)** & **Columns (mm)** \\ \hline \multirow{4}{*}{0.5} & \multirow{4}{*}{Loma Prieta} & Dynamic time history method & \(-31.54\) & \(-11.51\) & \(-32.91\) & \(-0.6801\) \\  & & Traditional & \(-21.52\) & \(-7.82\) & \(-23.42\) & \(-0.5482\) \\  & & IRDM & \((-31.78\%)\) & \((-32.07\%)\) & \((-28.82\%)\) & \((-19.39\%)\) \\  & & New method & \(-32.48\) & \(-10.51\) & \(-30.44\) & \((-0.6403\) \\  & & (\(29\%\)) & \((-86.6\%)\) & \((-7.50\%)\) & \((-5.85\%)\) \\ \hline \multirow{4}{*}{0.5} & \multirow{4}{*}{Kobe} & Dynamic time history method & \(-32.54\) & \(-10.85\) & \(-31.87\) & \(-0.6587\) \\  & & Traditional & \(-21.92\) & \(-7.88\) & \(-24.24\) & \(-0.5616\) \\  & & IRDM & \((-32.64\%)\) & \((-27.35\%)\) & \((-23.93\%)\) & \((-14.74\%)\) \\  & & New method & \((-33.09\%)\) & \((-10.62\) & \(-31.38\) & \(-0.6542\) \\  & & New method & \((1.71\%)\) & \((-2.12\%)\) & \((-1.54\%)\) & \((-0.69\%)\) \\ \hline \multirow{4}{*}{0.5} & \multirow{4}{*}{Northridge} & Dynamic time history method & \(-31.88\) & \(-11.32\) & \(-32.66\) & \(-0.6703\) \\  & & Traditional & \(-21.96\) & \(-8.06\) & \(-23.94\) & \(-0.5603\) \\  & & IRDM & \((-31.11\%)\) & \((-28.84\%)\) & \((-26.70\%)\) & \((-16.40\%)\) \\  & & New method & \((-33.19\)) & \(-10.82\) & \(-31.14\) & \((-0.6555\) \\  & & (\(41\%\)) & \((-4.41\%)\) & \((-4.64\%)\) & \((-2.20\%)\) \\ \hline \multirow{4}{*}{0.1} & \multirow{4}{*}{Loma Prieta} & Dynamic time history method & \(-31.54\) & \(-11.51\) & \(-32.91\) & \(-0.6801\) \\  & & Traditional & \(-25.27\) & \(-9.06\) & \(-26.64\) & \(-0.5923\) \\  & & IRDM & \((-18.31\%)\) & \((-21.27\%)\) & \((-19.03\%)\) & \((-12.91\%)\) \\  & & New method & \((-33.37\%)\) & \((-11.02\) & \(-31.68\) & \(-0.6512\) \\  & & (\(5.80\%)\) & \((-4.23\%)\) & \((-3.72\%)\) & \((-4.25\%)\) \\ \hline \multirow{4}{*}{1.0} & \multirow{4}{*}{Kobe} & Dynamic time history method & \(-32.54\) & \(-10.85\) & \(-31.87\) & \(-0.6587\) \\  & & Traditional & \(-26.26\) & \(-9.15\) & \(-27.51\) & \(-0.6058\) \\  & & IRDM & \((-19.30\%)\) & \((-15.73\%)\) & \((-13.67\%)\) & \((-8.03\%)\) \\  & & New method & \((-34.01\)) & \((-11 As shown in Table 3, the errors in the axial force, shear force, bending moment, and deformation calculated via the proposed method under varying medium stiffnesses are all within 6%. Moreover, as the shear wave velocity of the surrounding soil medium decreases, the errors tend to diminish. The applicability of the proposed method is further examined under varying structural stiffness conditions. Let the elastic modulus of the structural material in Section 3 be denoted as \(E^{*}\). In this analysis, the elastic modulus \(E_{\rm s}\) is set to \(0.5E^{*}\), \(1.5E^{*}\), and \(2.0E^{*}\), while all other parameters remain consistent with those in Section 3. The dimensionless boundary distance ratio is fixed at \(R=0.5\), and the artificial boundary parameters are set to \(\alpha_{T}=0.5\) and \(\alpha_{N}=1.0\). Seismic response analyses of the underground structure under Kobe wave excitation are performed via both the proposed method and the dynamic time history method. A comparative study is conducted, and the results are presented in Table 4.  
The results indicate that the errors in the internal force and deformation calculated via the proposed method remain within 8% across all structural stiffness scenarios. Furthermore, as the structural stiffness increases, the calculation errors of the proposed method show minimal variation. This demonstrates that the proposed method offers high computational accuracy and robust applicability, with the influence of structural stiffness on the method's accuracy being relatively insignificant.