Issue 53

A. Kostina et alii, Frattura ed Integrità Strutturale, 53 (2020) 394-405; DOI: 10.3221/IGF-ESIS.53.30 396 soil,  s ( t p ) is the ultimate strength of frozen soil to uniaxial compression, c ( t p ) is the cohesion, and  is the angle of internal friction. Eqns. (1)-(2) estimate for the cylinder wall thickness according to the criterion for the maximum radial movement of its internal wall, and Eqns. (3)-(4) - the ultimate stress criterion. We notice that Eqns. (1) and (3) are obtained on the assumption that the ice wall is a cylinder of finite height, and Eqns. (2) and (4) – on the assumption that the ice wall is a cylinder of unlimited height. It is also worth noting that formula (3) is obtained based on relationship (1) assuming that the frozen soil exhibits ideal plastic behavior described by the von Mises model. In [12], the applicability of formula (1) for calculating the frozen wall thickness was evaluated. The aim of our work is to analyze the correctness of analytical relationships (3) and (4) for calculating the frozen wall thickness according to the ultimate stress criterion. For this purpose, the ice wall thickness calculated by these formulas was compared with the numerical simulation results obtained by the formulation whose geometry is close to the real shaft sinking conditions. Using the obtained data, we propose a modification of formula (4) because it shows better agreement with the numerical simulation results. In addition, we perform a comparative analysis of the values of ice wall thickness calculated by formulas (1)-(4) and by their modifications, which provides guidelines for using each of these formulas. M ATHEMATICAL FORMULATION n order to evaluate the correctness of relations (3) and (4), we have computed the problem of deformation of ice wall exposed to the external radial load p applied to the lateral surface of the cylinder and the vertical load p’ acting on the upper end of the cylinder. The calculation scheme of shaft sinking is given in Fig.1. An ice wall is modeled as a two- piece cylinder. One part is a hollow cylinder of a height equal to that of the ice wall section without lining, and another part is a solid section. The soil inside the ice wall is assumed to be a thawed soil that experiences only elastic deformations. The load applied to the lateral surface of the cylinder is calculated by the formula for mining and hydrostatic pressure, which takes into account the influence of cohesive forces and the results of hydrogeological observations. Figure 1: Computational domain The mathematical model is based on the following hypotheses and assumptions: 1) The computational domain has radial symmetry, and therefore calculations are made in axisymmetric formulation. 2) The rocks under consideration are isotropic at macrolevel. 3) Deformations as well as strain increments at each step are small. 4) The time and temperature effects on the strength and elastic characteristics of rocks are taken into account parametrically. 5) The surrounding thawed and burden formations generate constant rock pressure. 6) A linear relation exists between the stress and elastic strain tensors. 7) The minimum value for the ice wall thickness is determined on the assumption that no plastic deformations are present on its internal wall. 8) The Mohr-Coulomb criterion is used as a yield criterion that is commonly applied to describe soil failure (especially, that of cohesionless soil). By this criterion, the rock failure begins when the maximum tangential stress    n reaches the critical value which is dependent on the normal stress  n in the same area. The mathematical statement of the problem includes equilibrium Eqn. (5), Hooke’s law (6), geometric relation for small strain tensor (7), yield condition (8), and associated plastic flow rule (9): I