The maximum frozen depth of surrounding rock is an important basic parameter in cold region tunnel design

such as frozen-preventing lining design

waterproof and drainage design

insulating layer design. To predict frozen depth of surrounding rock

various researches have been conducted

including field monitoring

numerical simulation

and derivation of practical estimation equations. In this study

a semi-analytical method is proposed to obtain the maximum frozen depth. First

the temperature acting on the lining surface is equivalent to a constant temperature according to the principle of equal annual accumulated temperature. Then

based on the quasi-steady state assumption

an analytical formula of the maximum frozen depth which considers the existing of lining

thermal insulation layer and the unfrozen water content in frozen rock is derived

using integration method. Next

the back analysis method is used to obtain the parameter which refers to the ratio of influence radius and frozen radius in the analytical formula

combined with the numerical simulation results of the temperature field. Finally

the semi-analytical solution of the maximum frozen depth is obtained. Through comparisons of the field monitoring data and the results of numerical simulation

the rationality of the proposed method is proved. Furthermore

the influence factors of the frozen depth are analyzed using the proposed semi-analytical solution

and the results show that the initial temperature and the annual average temperature have the most significant influence

and the rock conductivity coefficient and the porosity are less influential.