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and to the expression (9.16) for the displacement the item is
added

 u    x r . (9.18)
E
In terms of practical calculations the most interesting case is
loading of thick-walled cylinder by only internal pressure p .
Then p  p and p  both expressions for stresses (9.15) are
0
1 2
simplified
2
  pr 2  r 
t 1 2
  2  1 2  . (9.19)
2
 r  r  r 1  r 
2
Diagrams  and  for this case are shown in fig. 9.6. Stresses
t r
 - tensile,  - compressing. Extreme values of these stresses
t r
occur on the inner contour of the cylinder, they are the largest and
smallest principal stresses
1 c 2
    p ,      p ,
1 t 2 3 r
1 c
where c  r r .
1 2
If stretching efforts along the axis of
the cylinder are absent, then the
0
average main tension is zero   .
2
In a closed cylinder    .
2 x
Checking the strength of the cylinder
should be done by one theory of
strength, depending on the material
properties.
Assume that the plastic material,
then the third strength theory

     ,
1 3
Figure 9.6

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