PSI - Issue 18

Yaroslav Dubyk et al. / Procedia Structural Integrity 18 (2019) 630–638 Yaroslav Dubyk et al. / Structural Integrity Procedia 00 (2019) 000–000

637

8













( ) C cF x dF x C cF x dF x    ( ) ( )

( ) cos( ) n 

    

(A.2)

1

1

2

2

2

1

x

  

  

2

  

   









n

2 2    

(A.3)

2 M H cd C F x C F x c d     ( ) ( )

( ) cos( ) n  

1 1 C F x C F x  2 2 ( )

1 2

2 1

x

2

R

  ( ) 3

 

  ( ) 3





   

    

2 ( ) C F x C F x cd c C F x C F x dc d    ( ) 3   2

3

1 1

2 2

2 1

1 2

(A.4)

( , ) 

cos( ) n 

Q x

D  

2

 n C cF x dF x C dF x cF x R        1 1 2 2 1 2 2 ( ) ( ) ( ) ( )

x









  2 2   2

2 2 R c d   ( n







  n 

2   ( C cd C c d F x C cd C d c F x    ) ( ) 2 ( ) ( ) sin  2 2 2 2

( , ) 

v x



(A.5)





2

1

1

1

2

2

)

 

  







  n 

( ) cos 

( , )

1 2 1 C c C d F x C d C c F x    1 2 2 ( )

u x

(A.6)

 





2 2

R c d 









1

  

  ( ) sin , n  

( , )

1 1 2 2 1 E C E C F x E C E C F x    2 1 1 2 2 ( )

L x

 

 

2 2

(

)

c d 

(A.7)

 

















  

  

1

1

dDn cDn

2 4 2 2  

  

  

2 2 2 1 1       n c d 2

4 2 n n cDn E E    ,

4 E d c d 

1

2

1

2

4

R

R













  

   

( ) C E a E b C E b E a F x   



n

1 1

2

2 1

2

1

  n 

( , ) 

cos

(A.8)

N x











x









2 2   ( ) a b R C E a E b C E b E a F x    

2 1

2

1 1

2

2

,

.

  n d x

  n d x

n c x

n c x

1 ( ) F x e  

2 ( ) F x e  

Here we have used following notations:

cos

cos

A.2. Complete set of the equation for long solution The resulting equations for the main variables of long solution ( , ,

, x u v N L ) , and additional variables written in

the explicit form:  3 3



( ) ( ) sin( ) v C F x C F x n   

(A.9)

4 4



 



(A.10)

3 ( ) ( ) cos( ) v R C a C b F x F x n n          4 3 4





















 

 

(A.11)

2

2 ( ) C ab C a b F x   2 2

2 2 C ab C a b F x  2

1

2   

( ) cos( ) n 

x N HR 





4

3

3

3

4

4

n

2



   

  ( ) 3

 

  ( ) 3



 

2 

2 3   ( ) C F x C F x ab a C F x C F x a b b    ( ) 2

3

2 1

sin( ) n 

L HR 



(A.12)

3 3

4 4

4 3

3 4

n