PSI - Issue 26

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

425

4

(

)

     

     

( )

( ) n x n

    

   

2 2

n n

1



−

   

   

dq x

2

2   n 

n

1





( )

( )

( )

x

x

2

2 + − +

n

m x

v x

w x

=

−

−

+ +



(

)

x

dx

R

R

R

2

3

2

3

R

R

R

R

1

+



    

  

    

    

n v x R

2 

2

P R

n

1

−

( ) ( ) h n w x N  − +  2 1

( )

( )

( )

(10)



2  

m x

w x

+ −

+

+

−



x

R

R

   

    

n

n

1  +

1



( )

( )

( ) x

N

x n x 

u x

n

2

2

−

−

+ +



x

x



R

R

R R

( ) dm x n

2

2

n

n

( 1 2 1

( ) n x q x

( ) − −

( ) x

x

u x

=

+



(11)

) ( ) ( R  1

)

x

x

x



dx R

R

+

+



( )

2

dx du x

n

1

1

−



( )

( )

( ) w x

n x x

v x

=

+



+



(12)

R

R

R

( )

dv x

( ) n n x u x ( ) −

 +

2 1

=

(13)

x



dx

R

R

( )

dw x

1 =

( ) x

x 

(14)

dx R

( )

2

2

dx d x x 

n

1

−



( )

( ) R v x n  +

( )

m x x

w x

=

+



(15)

R

R

Here we used notation h  = System of eq. (8)-(15) can be easily solved using expansion in ordinary polynomials for axial coordinate 0 1 2 , , ... x x x For sake of simplicity the solution is rewritten using the method of initial parameters: ( ) 2 3 ... n x n C x C x C x = +  +  +  + 2  = 2 R E   , 2 12

x

x

0

11

12

13

...

(16)

( ) x

2 C x C x C x = +  +  +  + 3 

...



x

x

0

81

82

83

It is practical to limit our solution with fourth degree polynomials and to achieve good accuracy we can just ‘slice’ our shell in axial direction, for every sliced part solution (16) is applied. At each edge of the shell four boundary conditions must be specified. They can be generalized by the following equations:

M



x

k w  =

Q

(17)

+

x

w





(18)

x x M k   =