## 4 Comparative Verification  
### Comparison with Numerical Solutions  
To verify the theory, we compare and analyze the analytical solution with the numerical solution. We consider a rectangular pipe jacking excavation with a span of \(l=10\) m, height of \(h=5\) m, buried depth of \(d=5\) m, stratum loss rate of \(\eta=0.5\%\) (lateral convergence coefficient of both side boundaries and vertical subsidence coefficient of the top boundary \(\varepsilon=0.25\%\)), unit weight of the stratum of \(\gamma=20\) KN/m\({}^{3}\), elastic modulus of E = 10 MPa, and Poisson's ratio of \(v=0.33\). Referring to Figure 2, points C, D, E, and F are selected as control points (\(m=4\)) to calculate the undetermined coefficients for the mapping function. The first 10 terms of the Laurent series of the potential function are taken to solve the approximated potential function according to the method described in the previous section. Then, we calculate the stresses and displacements of the stratum and compare the result to the corresponding numerical solution calculated with the numerical software MIDAS GTS NX 2021. The finite element model mesh is shown in Figure 5, and the model size is 100 m \(\times\) 50 m. The central area of triple height and span uses a more precise grid to ensure the accuracy of the results. There are 1100 elements in the central area and 4553 elements in the outer area. The boundary conditions are longitudinal constraints on the left and right sides, fixed constraints at the bottom, given displacements at the boundary of the pipe section contour, and the excavation boundary is a free boundary. Figure 6a shows the analytical solution (AS) and numerical solution (NS) for the surface \(y\)-direction displacement \(v\) and \(x\)-direction displacement \(u\). In this figure, \(2x/l\) represents the ratio of the \(x\)-coordinate to half the tunnel span, and the ratio of the \(y\)  
Figure 5: Cross-sectional finite element model of rectangular pipe jacking excavation. coordinate to the buried depth \(d\) is 0. Figure 6b shows the analytical and finite element solutions for the \(y\)-direction displacement \(v\) and \(x\)-direction displacement \(u\) at the horizontal centerline of the pipe jacking, and the ratio of \(y\)-coordinate to buried depth \(d\) is \(-1.5\). It can be seen that the distribution characteristics of the analytical solution displacement and the finite element solution displacement are completely consistent, the analytical solution displacement values are smaller than the finite element solution displacement values, and the difference is tiny. Among them, the maximum values of the surface subsidence and the maximum difference appear at the surface of the vertical centerline of the pipe jacking excavation, and the settlement values of the analytical solution and the finite element solution at the location are 12.23 mm and 12.57 mm, respectively. The relative difference of the analytical solution is less than 3%. Figure 7 shows the horizontal normal stress \(\sigma_{x}\) and vertical normal stress \(\sigma_{y}\) of the pipe jacking excavation boundary. It can be seen that the distribution characteristics of analytical solution stress and finite element solution stress are basically the same, and the stress concentration occurs at the corner point of the excavation boundary. The analytical solution stress values are smaller than the finite element solution stress values; the difference is more significant when the stress values are larger, and the difference is more negligible when the stress values are smaller. The horizontal normal stress is close to zero at the boundary of both sides of the excavation and is larger at the top and bottom boundaries of the excavation, and the maximum values and the maximum difference appear at the top boundary of the excavation (\(\theta_{z}=33^{\circ}\), \(\theta_{z}\) is the counterclockwise angle to the positive direction of the \(x\)-axis).