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

871 9

  n x h n  2 2 x

 

  n x h n 2 2





 

3 2

1

dq x

x q x





( )

  1 12 x Eh n



x

x

x





















  

dx

x

2 2 x tg h x   2

2 2 x tg h x   2

2

2 2 x tg h  

2

sin2 12 

cos 12 

cos







  w x Eh n tg n   3 4 2 2

  



   



2 2 sin

2



2 2

  

  

( ) u x Eh

h n

E

(A.3)

2

1 ( )

w x h



  

 









  

 1 12

2 x tg

2 2

 

2

2 2 x tg h  

2

2 4

2 x t

2 g h  

2

x tg





2cos 12 

2 sin x



   

   

  v x h n  2 3

  sin



 2 2 1 sin h x tg h   2 2

 

 

2 2 v x Ehn h n

2 n m x

12

x n x xtg

 







x

1  

















2



2 2 x tg h  

2

2 2cos 12 

2

2 cos 12 

2 2 x tg h  

2

x

tg  

cos 12 

  n x h n  2 x

 

dm x

3 Eh n 3 2

cos

( ) u x n







 2 cos

x

( ) w x n















   

  

2 12 sin x

3 1 2 sin x

dx

12 sin x

tg  



(A.4)

 





3 Eh x 

   

  

( ) 3 2

v x

2

  

( )

x m x

n





x

1   

1 ( ) q x





 

 

  



x

2

2

2

x

12

sin

1

x



  x du x n x  

  v x n w x ( )

  

  



( )

u x

 





(A.5)

sin

dx

H x

tg  

 

dv x

1 ( ) 

1

u x n





 

  v x n x ( ) 2

 

  

(A.6)

x



sin

dx x

Eh





 

dw x

  x x

 

(A.7)

dx

  d x m x   

  n v x w x n          sin sin tg

  



  x

x

x









(A.8)

x

2

dx

H x 

x



R

E 

2    2

.

Where we used notation

References

Dubyk, Y., Orynyak, I., Ishchenko, O., 2018. An exact series solution for free vibration of cylindrical shell with arbitrary boundary conditions. Scientific Journal of the Ternopil National Technical University 89(1), 79 – 88. Saatcioglu, M., Mitchell, D., Tinawi, R., Gardner, N., J., Gillies, A., G., Ghobarah, A., Anderson, D., L., Lau, D., 2001. The August 17, 1999, Kocaeli (Turkey) earthquake - Damage to structures. Canadian Journal of Civil Engineering 28(4), 715 – 737. Leissa, A., W., 1973. Vibration of Shells, in “America Institute of Physics”. NASA SP-288, Washington, DC: U.S. Government Printing Office. New York, pp. 425. Salmanizadeh, A., Kiani, Y., Eslami, M., R., 2022. Vibrations of functionally graded material conical panel subjected to instantaneous thermal shock using Chebyshev-Ritz route. Engineering Analysis with Boundary Elements 144, 422 – 432. Vescovini, R., Fantuzzi, N., 2023. Free vibrations of conical shells via Ritz method. International Journal of Mechanical Sciences 241. Yang, S., Hao, Y., Zhang, W., Yang, L., Liu, L., 2021. Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory. Applied Mathematics and Mechanics 42, 981 – 998. Liu, Y., Qin, Z., Chu, F., 2022. Investigation of magneto-electro-thermo-mechanical loads on nonlinear forced vibrations of composite cylindrical shells. Communications in Nonlinear Science and Numerical Simulation 107, 106 – 146.