## 1 Introduction  
With the construction of high-speed railways (HSRs) in China, it is inevitable to come across high-speed railway tunnels under- or over-passing road tunnels [1], railway tunnels [2], subway tunnels [3], or even another HSR tunnel. For example, the Gao Jiu Lu-Jia Hua Up-Down Cross Tunnel is a small, close distance cross tunnel project in Chongqing, with a minimum thickness of 0.9 m in the residual rock mass [4]. Another example is the Jinjiangshan Tunnel, which overpasses the existing Caomeigou Tunnel in Dandong city, Liaoning province, with a minimum distance of 16.5 m between these two tunnels. For tridimensional cross tunnels, the internal force at the tunnel lining at the intersection is usually greater than that at other positions. It may be caused by the construction of a newly built tunnel nearby [5] or other factors. Through indoor tests, [PERSON] et al. [6] found out that cross tunneling increases the initial stress, deformation, and bending moments of adjacent tunnel lining if the distance between the two tunnels is too small. For example, a significant bending moment increment of the existing tunnel lining may be induced by the construction of a new under-passing tunnel or even cause tensile cracks to the existing tunnel lining [7; 8]. Through field measurements, the maximum additional stress induced by shield tunneling below on the existing tunnel lining was about 0.7 MPa [9], and the maximum settlement of the existing upper tunnel was about 4.7 mm [10]. Other investigations have also presented that the stress level of tunnel lining and surrounding rock in an intersection is generally higher than those at other positions [11; 12]. During the long service period, the reciprocating vibration of trains is one of the key predisposing factors that cause tunnel failure. [PERSON] et al. [13] found out that dynamic metro train loads may induce a significant dynamic response in the structure of tunnel lining and the soft foundation. [PERSON] et al. [14] advocated that the dynamic train load above, indeed, has an influence on the metro line underneath through the analysis of numerical modeling and physical model experiment results. For HRS tunnels, the dynamic response of tunnel lining induced by trains would be more significant than that of tunnels for normal speed trains [15]. [PERSON] et al. [16; 17] advocated that the geometric parameters of trains have a great influence on the peak frequencies of the vibration response of track structures and that dynamic stresses of roadbed, subgrade, and subsoil are strongly dependent on train speed once the speed is higher than 150 km/h. At present, many tunnels in China have different degrees of damage, such as fatigue damage of the tunnel inverted arch [18], lining cracks [19], soil-water inrush [20], and other damage. For these tunnels, the secondary disturbance of a newly built tunnel would cause an adverse impact on the existing tunnel lining. Additionally, the displacement and stress states of tunnel lining with primary defects would be much larger than those of tunnel lining without primary defects [21]. Though the success of the above references has been achieved, the influence of long-term dynamic load on tridimensional cross high-speed railway tunnels is still unclear because very little investigation has been made on this issue. Due to the higher initial stress of tridimensional cross tunnels, the long-term dynamic train load is likely to affect the durability and service performance of tunnel lining, and its influence should be taken into consideration in the design work of similar engineering cases. The dynamic response of adjacent tunnels decreases as the distance between them becomes large and the train speed is low.