PSI - Issue 47

Yaroslav Dubyk et al. / Procedia Structural Integrity 47 (2023) 863–872 Yaroslav Dubyk et al./ Structural Integrity Procedia 00 (2023) 000 – 000

865

3

N

N





2

1

1

x





0

N  

Q hv 



 

(2)

x





sin

x x

x

xtg    



1       Q xtg 

1

1

Q



0

Q  

N hw 

 

x

(3)

x



sin

x x

x



M M

    

x x   x x x  M M

1



x



0

Q   

(4)

x

sin

x

M

M





2

1

x





0

M

Q   





(5)

x



sin

x x

x



  

Displacements , , u v w with over double dots indicate displacements of double time derivatives. For physical equations, deformations are related to internal forces (Figure 1) as follows:   x N H         x x N H      x x x N N G h       (6)

H









 x



   x

1

x M M   x

 



M H  

M H 

    

x     

(7)



x

2

2 12 h

Eh

H



 

,

.

Here we used notation

2 

1



Fig. 1. Conical shell and its coordinates.

Displacements are connected with strains using geometrical equations:

1

1

1

u x

v

 



1 sin

v v     

u



u  

w







x 

x 

(8)





sin

x

x xtg

  



x x x

  

1

1 sin

w x

w    



v xtg x  









(9)



x

