PSI - Issue 18

Yaroslav Dubyk et al. / Procedia Structural Integrity 18 (2019) 622–629 Yaroslav Dubyk and Iryna Seliverstova / Structural Integrity Procedia 00 (2019) 000–000

624

3

2.1. Harmonic dent Assume the function of displacement in the form:

m

m

m







  

  

  

  

  

  

  cos sin u mn 

  sin cos v mn 

  cos cos w mn 

u C n 

x

v C n 

x

w C n 

x

,

,

(4)

l

l

l

General form of the equation of the dented shell:   , x    K U F

(5) Here   , x   F external load vector,   , x   U displacement vector,   , x   K stiffness matrix defined as:       , , , , , , T u x t v x t w x t         U (6)

T

   

   

   

   



N

2

2







x 



F

0 0

N

w





(7)

2 x R

2 2







The stiffness matrix K is symmetric, that is

, uv vu uw wu vw wv K K K K K K    : ,

uu K K K H K K K K K K    uv vu vv wu wv 

    

uw

K

  

(8)

vw

ww

Taking into account (8) all elements (9-11) are defined as:

2

2

2      2 2 1 1 2 R 

1

2



1 1 R x      2

K

x   

, u w w u K K ,

 



, u v K K

, v u  

(9)

, u u

R x 



  

  

2     2 1 

 

2

2

  

   

1



1



x     

 



1   

K



1 1  

, v w w v K K ,

 

 

 

(10)

  

 

, v v

2

2 2

2

2

2 2

2

2





R R  

x







R

R

2     2 1 

   

1

1

K









(11)

 

, w w

2 2

2

2

2

R

x R 







Taking into account expansion (4), we obtain an algebraic system of equations with respect to displacements:









  

   

  

   

2 n l

2

1

1

n









1 2

1 2

R

(12)

u C m   mn

v C n   mn

w   C  mn

0







l

R m 

m

