Page 117 - 4560

du u
  ,   . (9.10)
r t
dr r
The physical aspect of the problem is described by equations of
Hooke's law, which, after substituting in them the values of the
relative deformation (9.10) solve relatively stress  and  :
r t
E  du u  E  u du 
    ,     . (9.11)
r 2   t 2  
1   dr r  1   r dr 
Substituting the expressions for stresses (9.11) in the static
equation (9.9), after transformations we get a differential equation
with the unknown u

2
d u 1 du u
   0. (9.12)
dr 2 r dr r 2
The solution of this second order differential equation with
variable coefficients (Euler’s equation) has the form
C
u  C r  2 . (9.13)
1
r
Substituting the expression for u in equation (9.11) leads to the
following expressions for stresses:

E  1  
  C 1   C ,
r 2  1 2 2 
1   r 

E  1  
  C 1   C .
t 2  1 2 2 
1   r 
Steels integration C and C define the boundary conditions on
1 2
the inner and outer contours of the ring
r
   r   p ,     p .

r 1 1 r 2 2
These conditions give two equations:

117