PSI - Issue 13

Kostina A. et al. / Procedia Structural Integrity 13 (2018) 1273–1278 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

1275

3

determined in order to maximal displacement

max  is not exceeded. The three-dimensional theoretical model for

describing a stress-strain state of the ice-soil retaining structure is written as: div 0  σ

(1)





: cr   σ С ε ε

(2)

  1 grad grad 2 T       ε u u

(3)

where σ – Cauchy stress tensor [Pa], ε – full strain tensor, cr ε – creep strain tensor, u – displacement vector [m]. It is assumed that the ice-soil retaining structure is a hollow cylinder with inner radius in r and wall thickness E . The outer surface of the cylinder is loaded by the rock pressure P , the top and bottom cylinder ends are fixed in longitudinal direction by the lining and frozen soil, correspondingly, and the circle at the top end is fixed in all directions by the tubbing. On the basis of the Vyalov’s constitutive relations the creep strain cr ε of frozen soils in uniaxial stress conditions is described by the Norton-Bailey creep law. For multiaxial stress conditions with the assumption that the volumetric creep strain equals zero this law can be generalized as:

m

   

   

m   

1



  

  

m

cr eff

cr eff m eff   

A

( )





(4)

m

1/2 [2 / 3( : )] cr cr ε ε ( 1) m k m   /

1/2 [3 / 2(dev( ) : dev( ))] σ σ

cr eff  

eff  

– the effective creep strain rate,

– the effective

where stress,

1/ A      , t – time [h],  [Pa∙h λ ∙ (  C) - k ],  – temperature of the soil [  C]. By solving the problem (1-3) analytically with some assumptions, the main from which are that instantaneous elastic deformation is neglected, radial stress equals zero on the inner cylinder surface, shear stress linearly varies near the cylinder ends and equals zero on the inner surface, radial displacement on the top end equals zero and the stress-strain state is considered only at the last time moment, in (Vaylov et al. (1962)) the engineering formula for the optimal thickness E of the ice wall was obtained: ( )

     

     

1

    

    

m

1



m

1



m Ph

(1 ) 

(5)

E r

1  

1



in

m

1





m





2 3 ( ) A t 

r



pr

ma

x in

3. Numerical simulations

To estimate adequacy of (5) in this study the problem (1-4) was solved numerically. The numerical modelling was carried out for Callovian sandy loam as one of the most dangerous soil layer for a shaft sinking. The creep parameters characterizing rheological properties of the soil were estimated on the basis of experimental measurements of creep strain on time (Vaylov et al. (1962)). The data was obtained from a uniaxial compression test of a cylindrical rod with the ratio of the radius to the height equal 0.5 one of the ends of that was loaded by pressure equal 3 MPa during 12 hours. The temperature of the soil was 10     C. To identify the properties, the analytical relationship of the creep strain on time obtained by integration (4) was used. In table 1 identified values of the parameters are presented.

Table 1. Creep parameters for Callovian sandy loam at -10  C.  , MPa ∙ h λ ∙ (  C) - k k m



0.9

0.89

0.27

0.1