## 1 Introduction  
Tunnels are important parts of transportation networks. According to public statistics, by the end of 2024, there were 28,724 road tunnels in China, with a total mileage of 32,596.6 km [1], and 18,997 railway tunnels, with a total mileage of 24,246 km [2], with gradual expansion to high altitudes and alpine regions. However, the ambient meteorological and geological conditions faced by cold regions are extremely complex, posing a great challenge to the design and construction of tunnel projects. Under the effect of continuous low temperatures in winter, the fissure water in the surrounding rock at the entrance of a water-rich tunnel can freeze, leading to an overall volume expansion of the surrounding rock and the generation of frost heave stress. This stress is transferred to the lining structure through the surrounding rock-lining interface, resulting in freezing damage such as cracking, spalling and icing in the lining concrete [3; 4; 5; 6; 7], thus threatening traffic safety. To address this outcome, experts and scholars have conducted extensive studies on calculating frost heave mechanisms and frost heaving forces in tunnels and have proposed corresponding frost heave calculation models. Frost heave models can be divided into three main categories: accumulated water frost heave models [8], weathered layer frost heave models [9], and freeze-thaw cycle overall frost heave models [10]. Currently the most widely used is the more applicable freeze-thaw cycle frost heave model [10; 11; 12; 13; 14; 15], whereas the remaining two types of models have many assumptions, and their application is limited [16; 17]. To determine the frost heaving force, methods such as field monitoring, indoor testing, theoretical analysis, and numerical simulation are used. [PERSON] et al. [18] simultaneously measured the surrounding rock temperature and the lining pressure to carry out field measurements of frost heaving force in the Jichoushan Tunnel and analysed the variation pattern of the frost heaving force. [PERSON] et al. [19] conducted a series of unidirectional freezing experiments on clay in an open system and obtained the evolution patterns of the freezing depth, frost heaving force, and water content versus time. [PERSON] et al. [20] considered the depth of water behind the lining and carried out a three-dimensional (3D) geotechnical model test of the frost heaving force of a lining; their results showed that the model can well predict the evolution of frost heaving force caused by water accumulation. [PERSON] et al. [21] established a frost heave model based on the principles of ice-water phase change and water-heat coupling and discussed effects of factors such as geometric parameters. [PERSON] et al. [22] proposed an analytical solution for the frost heaving force in a circular tunnel based on the displacement method, performed parameter sensitivity analysis, and noted that lining thickness should be a central concern when designing tunnels for frost resistance. Based on an overall frost heave model of a broken freeze-thaw lithosphere, [PERSON] et al. [15] proposed a calculation method for the overall frost heave pressure of a horseshoe-shaped tunnel with broken surrounding rock in a seasonally frozen area. [PERSON] et al. [23] established a 3D thermo-mechanical coupling finite element model for fractured rocks to analyse the effects of fracture size, number of fractures, and fracture inclination on frost heaves. These studies used different methods and different assumptions to calculate the frost heaving force, and these methods have their own characteristics in terms of applicability, calculation accuracy, parameter requirements, and calculation efficiency, thus reflecting the lack of unified standards and specifications for frost heaving force calculations. In addition, most studies have shown that anisotropic frost heave deformations [24; 25; 26] and the multiphysics field coupling effect [27; 28] significantly affect the frost heaving force.