PSI - Issue 26

Dubyk Yaroslav et al. / Procedia Structural Integrity 26 (2020) 422–429 Dubyk et al. / Structural Integrity Procedia 00 (2019) 000 – 000

424

3

x N L hu x R      

0

+ + =

(2)

N      L Q R x R  

hv + + + = 

0

(3)

x x R R     Q N

Q  

hw + − + = 

0

(4)



Here over dots , , u v w denotes double time derivatives of displacements. The initial stresses in the shell from the internal (external) pressure  = N PR , axial force 2  = x N N R , rotational speed 2 2 N hR    = and torque moment 2 2   = x tor N M R , form the following curvatures:

2

2

2

w

w

v

u

w

1

 









= + w

;

;

2



 x

 x

=

= − −

(5)





2 

2

x

     x R x





We have to add these terms only in equation (4):

    

       

    

Q N

  Q

2

2

P

w

w

 







2  

h w

N

+ − − −

+

−

+

x

x

2

2

 x R R R

x





 v







(6)

    

    

2

u

w

1

 





N

 + = hw

2

0

+

− −

x



x R

    x



Using the calculation procedure given in our previous work Dubyk et al (2018a), by combining the static equations with the physical and geometric, we obtain an eighth-order differential equation. Using the expansion of eight parameters in trigonometric series:

  

   





cos or n

sin   n

 = 

 

(7)

n

n

n

n

0

1

=

=

We can obtain a system of eight ordinary differential equations, in which only one equation for ( ) x dq x dx A full system of eight ordinary differential equations describing a prestressed cylindrical shell:

is changed.

( ) dn x n

( )

( ) 2 u x

x

x n x 

=

− 

(8)

dx R

(

)

     

    

( )

2

   

   

( ) n x n

n n

1 



−

   

   

    

    

dn x

2

n

n







( )

( )

( )

x



x

2

(9)

n

m x

1 + − −  + +   v x

w x

= −

−



x

(

)

dx

R

R

R

2

3

2

3

R

R

R

R

1

+ 