PSI - Issue 32

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

181

2

The mechanical analysis of FW in the construction of underground structures includes the calculations of strength and creep (Sanger and Sayles 1979; Kostina et al. 2018). The strength calculation is aimed at obtaining the minimum FW thickness, such that the shear stresses at a given time do not exceed the maximum shear strength of the frozen rock. As a rule, the problem of the limit FW stress state is jointly solved with the differential equation of equilibrium (Vyalov 1965),

r d dr 

r r    

0  ,



(1)

and the Mohr-Coulomb condition of the limit stress state, which can be expressed in terms of primary stresses 1     and 2 r    :

(2)

r        ,

4 2         

(3)

2 tan

,

 

4 2         

(4)

2 tan c

,

 

where r  is the radial stress, Pa;   is the hoop stress, Pa; r is the radial coordinate, m; c is the cohesion, Pa; φ is the angle of internal friction, °. In equations (1) – (4), the FW is assumed to be a hollow cylinder of unlimited height (see Fig. 1). At the outer boundary r = b , a constant and uniformly distributed pressure P is set, and the inner boundary r = a is free. The influence of lithostatic pressure is implicitly taken into account when specifying the horizontal pressure P . Thus, the problem is complemented by the boundary conditions on the inner and outer FW surfaces: ( ) 0 r a   , (5) ( ) r b P   . (6) The limit FW state occurs when the plastic deformation region spreads over its entire thickness (Vyalov 1962; Jessberger 1981). In this case, not only the equilibrium equation (1) but also the Mohr-Coulomb condition (2) become true in the entire volume. This allows for the substitution of (2) into (1) to yield an equation with one unknown quantity:   1 0 r r d dr r         . (7) Considering the additional information on the limit state relating to the entire volume of frozen soil, the system (5) – (7) becomes overdetermined. To eliminate this problem, an additional unknown is introduced — location b of the external FW boundary. Solution of the problem (5) – (7) with respect to two unknowns allows us to define the FW thickness E b a   , (8) which is required to withstand an external load P . The classic solution of this problem with constant values of  and  was obtained by Vyalov(1965). As follows from this solution, the relationship between the load on the external FW boundary and the required FW dimensions ( a and b ) is determined by the formula: