## 1 Introduction  
Given that the high-speed railway (HSR) in China has reached 30,000 km, the number of HSR tunnels under- or over-passing road tunnels [1], subway tunnels [2; 3; 4; 5], another trunk railway tunnel [6; 7], and even another HSR tunnel has been increasing. For example, the Huofengshan Tunnel of the Chongqing comprehensive transportation hub passes over the Renhechang Tunnel of the Chongqing-Huaihua Railway, and the minimum distance of the intersection is only 5.47 m. The Anhui and Jiangxi dual-line railway and the Jiujiang-Quzhou Railway have a tunnel crossover in Jingdezhen, and the minimum distance of the intersection is 4.5 m. The Gauntouling Tunnel of the Wenzhou-Fuzhou Railway passes under the Gauntouling Expressway Tunnel, and the railway tunnel arch is approximately 2.91 m from the base of the Gauntouling Tunnel. In addition, several cross tunnels are found in the Beijing-Guangzhou and the Shanghai-Kunming HSR. The cross tunnel lining shares relatively higher stress levels compared with ordinary tunnels. Many scholars have investigated the interaction mechanism and engineering characteristics in cross tunnel construction to mitigate the adverse effect in cross-tunnelling construction. Cross-tunnelling increases the initial stress of the tunnel lining if the tunnel distance is close [8]. During the construction of an existing tunnel under a vertical tunnel, the longitudinal soil pressure of the existing surrounding rock of the tunnel has an \"arched\" distribution [9]. The stress of the upper tunnel lining changes consequently [10], the compressive failure of the shotcrete lining of the crown can occur, and the tensile forces of the rock bolts around the crown can increase substantially [11; 12] because of the maximum additional stress (approximately 0.7 MPa) induced by shield tunnelling below [13]. Other investigations have also presented that the stress level of the tunnel lining and the surrounding rock in an intersection is generally higher than that in other locations [14; 15; 16]. The long-term train loading would be another potential threat to the serviceability of cross tunnels. At present, approximately 65% of tunnels in China suffer such variable damage, such as cracking, stripping, void, seepage, and even mud pumping [17; 18]. The dynamic load from a running train could cause damage cracks in the tunnel linings and plastic deformation of surrounding rock, because of the high initial stress of the cross tunnels [19; 20; 21]. Moreover, the cross tunnel would increase the static stress level of the tunnel lining because of its complex structure [4; 5; 22]. The dynamic response of the tunnel would affect the durability and service performance of the lining because of the high static stress of the cross tunnels; moreover, the influence of a long-term dynamic load induced by high-speed trains (HSTs) should be considered in the design work of similar engineering cases [23]. This defect of the HSR cross tunnel, with a 100 year designed service duration, repeatedly occurs at an intersection, and the safe operation of the two rails would suffer an adverse effect. Thus, defining parameters, such as surrounding rock mass (\(\lambda\)), cross angle (\(\theta\)), rock mass thickness between tunnels (\(H\)), and train speed (\(v\)), is urgent and necessary in the design stage, especially for practicing engineers to directly estimate whether the lining structure is damaged because of the train load, and to what extent. In this paper, a systematic study that consists of numerical simulation and a fatigue damage experiment is conducted to develop an approximate method to enable practicing engineers to evaluate the reasonable design parameters of cross tunnels. The framework of this paper is organised as follows. Firstly, the methodology, including numerical simulation and experiment, is presented (Section 2).