PSI - Issue 32

Mikhail Semin et al. / Procedia Structural Integrity 32 (2021) 180–186 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

182

3

  

   

1



1           b a

1

P

(9)





.

This is the most commonly used formula for calculating the FW strength, despite having a safety margin due to some assumptions (Levin et al. 2018; Zhang et al. 2018; Kostina et al. 2020):

 Infinitely large height of the unlined section of the shaft.  Negligible deformation of the surrounding unfrozen soils.  Homogeneous strength properties in the FW volume.

Fig. 1. Structural model.

In practice, the alternative formula for FW strength calculation obtained by Vyalov (1965) for the condition of the finite height of the unlined mine shaft wall appears to be less applicable, owing to more significant simplifications. In later works (Yang et al. 2013; Zhang and Wang 2018), modifications of this formula were proposed. These works additionally consider the elastic-plastic deformation of the surrounding unfrozen soil and the change in FW thickness caused by the displacements at the internal and external boundaries of the FW. According to numerous studies (Tsytovich 1960; Vyalov 1965; Akagawa and Nishisato 2019), the cohesion of frozen soil strongly depends on temperature, while the angle of internal friction changes insignificantly. The temperature throughout the volume of frozen soil is non-uniform, with a minimum in the FW center and maximums at its external and internal boundaries. This indicates that ultimately  and  are functions of the radial coordinate for a given temperature field in the frozen soil volume. At the same time, the variation of  in the FW is the most significant, and the dependence of  on the radial coordinate can be ignored in most cases. This paper provides a simplified theoretical analysis of the effect of temperature non-uniformity and the resulting non-uniformity of strength properties inside the FW volume on the FW thickness determined from the limit stress state problem of a frozen soil cylinder. 2. Temperature non-uniformity effect Equation (7) with boundary condition (5) assumes an accurate analytical solution for the case when  is a constant, and  is an arbitrary function of radial coordinate r . This solution is given by

r

( )



d 

a 

(10)

,

1



r

r 









where ρ is the integration variable.