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2
d  m   x
  . (5.15)
dx 2 GJ

The equations (5.7) and (5.15) are the basic equations of torsion
problems. Integrating them under given boundary conditions we
can find the relative angle of twist   x or twisting moment M
x
and the angle of twist   x .
To determine the angle of twist of an arbitrary cross-section we
will use the equation (5.7), integrating it we find
x M
   x dx  0 , (5.16)
0 GJ 
where  – the angle of twist at x  .
0
0
In case when M  const , GJ  const , the distance between
x 
the cross sections x  , and   we obtain
0
l
0
M l
  x . (5.17)
GJ

If the rod has several areas where the laws of changes of the
twisting moment M and the polar moment of inertia J (or one
x 
of these values) are different, full twisting angle of the rod:
n M dx
   i x ,
i 1 l GJ
i i 
where l – length of the section within which the laws of
i
changes M and J are constant.
x 
The angle of twist per unit length
 M
   x . (5.18)
l GJ

The condition of hardness in torsion has the form

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