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   y
x
tg 2    ,
0 (3.21)
 2
yx
where  – inclination angle of the shift plane to the plane
0
where acting  .
x
Comparing the expressions (3.14) and (3.21), we find the
interrelation between the angles  and  :
0 0
1
tg 2    .
0
tg 2
0
Taking into account the relationship between the trigonometric
functions and reduction formulas, we obtain:
ctg 90  2   ctg 2  ctg  2 , so 90  2   2 ,
0 0 0 0 0
or
    45  . (3.22)
0 0
Thus, the shift planes are inclined to principal planes at the
angle 45 (fig. 3.8).
To find the values of extreme tangential stresses for initial we
take principal planes and principal stresses  and  .
max min
Substituting in the expression (3.13) values    ,    ,
x max y min
  0,  45  , we will get:
yx

     
   max min   1 3 . (3.23)
max
min 2 2

Substituting in(3.23) expressions (3.19), we find

1 2
0
        4 2 (if   ). (3.24)
max x y xy 3
min 2
The vectors of extreme tangential stresses are always directed in
the direction from  to  , as shown in fig.3.8.
min max

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