## 1 Introduction  
With China's growing population and increasing traffic congestion, the development of underground space has become a mainstream trend, and the emergence of underground highway and railway tunnels has greatly eased traffic congestion [1, 2, 3]. However, due to the increase in the number of tunnels, it is inevitable that there will be spatial overlapping of tunnels, in which the lower part of new tunnels passes through the existing tunnels, which is a commonly used method in the project [4, 5, 6, 7]. The excavation of new tunnels not only induces stress redistribution in the surrounding rock strata but also generates additional stresses, settlement deformations, and additional vibrations on the existing tunnels, resulting in the deterioration of the mechanical properties of the existing tunnel structures within a certain range of the underpass area [8, 9, 10]. When the clear distance between the cross-section tunnels is small, there is an overlapping influence area between the excavation of the new tunnel and the train operation in the existing tunnel, which can lead to further deterioration of the mechanical properties of the surrounding rock [11, 12, 13]. Currently, there are fewer studies on the overlapping impact areas of tunnel crossing sections, and research on cases of double-line highway tunnels passing under existing railway tunnels at oblique angles is lacking. Therefore, it is of great significance to study the area affected by the excavation of road tunnels under existing railway tunnels. At present, scholars have certain research results on the effects of static loads on excavation under existing tunnels for new tunnels [14; 15; 16]. [PERSON] et al. [17] investigated the effect of tunnel depth, overburden pressure, tunnel dimensions, and palisade pressure on tunnel displacements in a single-lane tunnel. [PERSON] et al. [18] investigated the bending moments and deformation patterns of a newly constructed two-lane tunnel vertically crossing an existing tunnel at different clear distances using centrifuge modeling tests. [PERSON] et al. [19] constructed a model of a newly constructed tunnel vertically under an existing tunnel by means of three-dimensional finite element (ICFEP 1999) software and analyzed the circumferential forces, bending moments, and lining deformations due to excavation. [PERSON] et al. [20] systematically monitored the deformation characteristics and stress changes of new tunnels using a proximity earth pressure balance shield (EPBS) on existing tunnels based on a double tunnel project in a sandy soil formation in Changsha City. It is proposed that the settlement development of the existing tunnel undergoes four stages: settlement, bulge, secondary settlement, and reaching a steady state, and it is found that the settlement caused by the second shield tunnel penetration is significantly larger than the settlement caused by the first tunnel penetration. Vertical tunnel crossing is typical in the theoretical model, but in practical engineering, angled crossings with more complex disturbances are often adopted. [PERSON] et al. [21] investigated the response of underground pipes to tunnel boring by centrifuge tests. The results showed that the pipe tunnel settlement and strain were related to the tunnel intersection angle, and the change was 20% higher at an intersection angle of 60\({}^{\circ}\) compared to a 90\({}^{\circ}\) intersection angle. [PERSON] et al. [22] studied the lateral deformation, transverse internal force, and torsional deformation of new two-lane tunnels diagonally angled down through the existing tunnels based on the Changsha Metro Line 2 and Line 4 projects in China.