## 3 Numerical Method and Verification  
### Finite Element Model  
According to the typical soil-rock composite stratum in the Jinan area, the stratum is properly simplified to fill, silty clay and limestone. Considering the structure system of the soil-rock site and pipe jacking tunnel as a plane strain problem, the finite element software MIDAS/GTS was used to establish the model and make a grid division. In order to reduce the influence of the boundary effect on the dynamic response of the tunnel structure, the calculation area of the model is taken as 50 m \(\times\) 30 m, that is the 3-5 times of structure size. In order to ensure the calculation accuracy and reduce the calculation time as much as possible, the grid division method of gradually densifying from the boundary to the vicinity of the pipe jacking is adopted. The maximum size of the grid is 1 m, which is less than 1/8 of the corresponding wavelength of the maximum wave frequency [17]. The pipe jacking structure adopts the 1D continuous beam element with equal stiffness, the soil and rock stratum adopts the 2D plane strain element and the finite element mesh division of the soil-rock-pipe jacking system is shown in Figure 3.  
When the seismic wave propagates in the soil medium, the boundary of the model absorbs energy and rebounds part of it. This phenomenon will cause some errors to the overall analysis result of the finite element calculation. Therefore, the artificial boundary on the left and right sides of the finite element model adopts a two-dimensional free infinite element boundary to eliminate the reflection effect of the boundary. The upper boundary of the model is free surface and the lower rock boundary is fixed constraint.  
The reinforced concrete pipe jacking structure adopts the linear elastic constitutive model, and the stratum adopts the Mohr-Coulomb constitutive model, without considering the separation and relative sliding between the soil and rock mass and the pipe jacking structure. The Mohr-Columb model is suitable for nonlinear, elastoplastic soil materials and has been widely used in geotechnical engineering. In the stress space, the yield surface of the Mohr-Columb condition is an irregular hexagonal cone, and the corresponding yield curve is a closed irregular hexagon in the plane \(\pi\). The stress state of any point of the soil material should meet the following [18]:  
\[\sigma_{1}\frac{(1-\sin\varphi)}{2\cos\varphi}-\sigma_{3}\frac{(1+\sin\varphi) }{2\cos\varphi}=1\]  
Figure 3: Finite element model mesh of vertical double pipe tunnel in soil rock assemblage strata.  
From [PERSON]'s circle, it can be concluded that [18]:  
\[\tau=\frac{\sigma_{1}-\sigma_{3}}{2}\cos\varphi\sigma=\frac{\sigma_{1}+\sigma_{3} }{2}+\frac{\sigma_{1}-\sigma_{3}}{2}\sin\varphi\]  
The Mohr-Columb yield criterion can be rewritten as follows [18]:  
\[\frac{\sigma_{1}-\sigma_{3}}{2}=-\frac{\sigma_{1}+\sigma_{3}}{2}\sin\varphi+ \cos\varphi\]  
The constitutive relationship of concrete material is a linear elastic model, which assumes that the stress and strain of the material are directly proportional to each other during loading and unloading and there is no residual deformation after complete unloading. The numerical calculation model considers both the self-weight and the earthquake motion. First, the stress field of the model under the dead weight is calculated, and then the response of the soil-pipe jacking structure under earthquake conditions is further calculated on the basis of the balance of the initial stress field. Rayleigh damping is used for calculation, and the empirical critical damping ratio is taken as 0.05.