PSI - Issue 32

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

185

6

 

  

 

 

  

  

  

  

2

2

tan

tan

1



 

 

3

   

1



4 2

4 2

( 1)

b a      

  

(19)

.

3

4 ( ) 

1

I

O

 



 





  





3

3

96 1  

  

4 2          

  

2

96 tan

1

After substituting (18) and (19) into (16), the following analytical expressions for ε and b can be obtained:

( 1) 1 96 1       



3

(20)

,

P







mean

c





3

 

1 1 (1 ) b a     ,

(21)

where

4 2         

(22)

.

2 tan mean c

mean  

Fig. 3. Contour plots of the integral I as a function of γ and b/a ( a=5m ).

The cubic equation (20) has three roots, but only one root real  , which does not contain the imaginary part. However, this root has one inconvenient property: it tends to infinity at 0 c   . For real frozen soils, the value   3 ( 1) / 96 1 c       in (21) is usually much less than value / ( 1) mean    , so we can find an approximate recursive solution of the equation (20):

2 P X c i

3

( 1)    

( 1) , P X X  

(23)

.

1  

, 1,..., i

1, n X

1

1



 







0

n

i

3

96





mean

mean

The recursion depth n in (23) can be determined based on the desired solution accuracy. Formulas (21) – (23) can be used to approximate the effect of the non-uniformity of the cohesion distribution in the FW volume on the calculated FW thickness by strength. These formulas are derived using the following main