PSI - Issue 32

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

184

5

k





(  

)

c

max c c

  

max

min

4 2          

(

)

f

b a 

b  a

4 2         

2

tan

1

(12)

2 tan

,

P

b

d





4 2         

2

tan



  1 , ,    

k k

       

(13)

,

f

where   1 k a b      is the coordinate of the freeze pipe contour, m; γ is the parameter characterizing the position of the freeze pipe contour in the FW volume (equal to 0.5 when the pipe contour is equidistant from the FW boundaries). For the analysis, the average cohesion value c mean and its maximum variation Δ c in the frozen soil are convenient to use, rather than c min and c max : 0.5( ) mean max min c c c   , (14)

(15)

c max min c c    .

Generally speaking, expression (14) for a cylindrical body will be the volume average only if γ=0.5 . In all other cases, it should be deemed a median. Considering (14) – (15), the integral (12) can be written as

(16)

v c P P I    ,

k



  

1 2

 

  

4 2          

(

)  

f

b a

b  a

4 2         

2

tan

1

    

    

2 tan

(17)

I

b

d

,





4 2         

2

tan



where v P is the solution (9), obtained by Vyalov. Integral I most often takes negative values, and its value is sensitive to the γ parameter (see Fig. 3). The minimum integral value is reached at 0   , which happens when soil properties with maximum strength are set at the internal FW boundary. The maximum integral value is reached in the   0.1;0.5   region, i.e., maximum strength properties of the soil are set in the outer FW area. This conclusion seems logical given that most of the external load is accepted by the external part of the FW, which follows from the radial stress field (10) and the angular stress field calculated from (2). Equation (16) indicates that if a constant average temperature is maintained in the FW, an increase in temperature variation within the FW leads to a decrease in FW bearing capacity. It is of note that the FW bearing capacity decreases linearly depending on the cohesion variation Δ c and, thus, the variation of temperature T  since ~ T c   . When 0 c   , the expression (16 ) turns into the classical Vyalov’s formula (9). The integral I in the right-hand side of (16) can be calculated analytically, but the resulting expression appears too lengthy and inconvenient for analysis. However, if one assumes that b>a and γ=0.5 , then (17) can be expanded into a Taylor series in the small neighbourhood of a point b=a . As a small parameter, it is convenient to consider the quantity:

1    

b a

(18)

.

1



  

 

Then the first non-zero term of the Taylor series for the integral (17) will be equal to: