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

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4

Displacements are connected with bending strains as follows:

x x x         

2

  

2 v w w       1

1

1 sin

w









cos

x

x 



x 









x

(10)

 





2

2 2 sin

2  

x

x

x x 



  

 





x N in physical equations (6):

We obtain the equation for N  , eliminating   from forces expressions N  and

1 sin

  

  

v u w   x xtg



N N Eh   

(11)

x



x

  



x M in physical equations (7) and

The expression for M  take, considering the moments expressions M  and

  (10):

bending strain

  

  

3

2

cos

1

Eh

v

w

x 







M M 

 





x

(12)

x



2 2 12 sin x

2 2 sin

2   

x

  

x M  will be by substituting the expression for x   (10) into the moment expressions

x M  in physical equations

(7):

 2 1



  





3

2

cos

2

Eh

w

v

u

  







M

N











x

(13)

 

 24 1



x

x





2

2 x tg x 

2 2 sin

sin

sin       x

x

xEhtg





  



The fifth equilibrium equation (5) is used to determine Q  , with respect to defined values of M  (12 ), but it won’t be presented for more concise definitions. The differential equation of the eighth order can be obtained using the calculation procedure given in the work of Dubyk et al (2018), combining static equations with physical and geometric ones. Such system of eight ordinary differential equations is obtained by using the expansion of eight parameters in trigonometric series: x M  eq. (13) and

 

 





cos  

sin    n n

n or

(14)

 

n

0

1

n

n





A complete system of eight ordinary differential equations describing a conical shell dynamic behavior showed in Appendix A. 2.2. Method of solution To solve this system of ordinary differential equations, we use ordinary polynomials. As a result, we take an approximate solution, and as the degree of polynomial increases n n x C  , accuracy will increase, but the difficulty of obtaining coefficients n C will also increase. Such a simplification in the construction of analytical expressions somewhat complicates the numerical scheme, but it is more effective than searching for large powers, as, for example, it is done in works Xie et al. (2015) and Xie et al. (2017). Based on the authors' experience with cylindrical shells Dubyk et al (2018), limiting the solution with 3rd or 4th polynomial degree is enough. For conical shells this issue is investigated below by means of a numerical experiment. This solution is applicable in numerical scheme for a conical shell using the transfer matrix (TM) method. For this purpose, the shell is divided into small equal segments, for each the solution is known (see fig. 2a). We can create a