PSI - Issue 50

I.G. Emel’yanov et al. / Procedia Structural Integrity 50 (2023) 57–64 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

59 3

When determining the stress state of the shell, we will use the classical theory of shells based on the Kirchhoff – Love hypotheses, Grigorenko and Vasilenko (1981). The stress-strain state of an arbitrarily loaded shell of revolution is described by a system of eight partial differential equations, Grigorenko and Vasilenko (1981),

m Y

4  

s Y

 



  , 

  , , 

) ( 0 s s s L  

Am s





m f s



0



m

with boundary conditions

  1 0 1 BY s b  ,   2 . 2 B Y s L b   s r z r z Y N N S M u u , , , , , ˆ , , 

 s   ,

where N r , N z are the radial and axial forces; u r , u z are similar displacements; ŝ is the shear force; M s is the meridional bending moment; ν is the circumferential displacement; ϑ s is the rotation angle of the normal. The elements of the matrix A m depend on the geometric and mechanical characteristics of the shell. f̄ is a vector whose components depend on the loads applied to the shell. B i - given matrices and b̄ i given vectors. Since a pulsating pressure acts on the shell, the resolving system of equations should be supplemented with dynamic terms, Vasilenko and Emelyanov (1995), Vasilenko and Emelyanov (2002),

2

m N

4  

s N

N

t N 









(, ,), 

Am

B

f s t









m M

2



0



m

t 

where the elements of the matrices M and B are determined by the inertial and damping properties of the shell. The oscillatory process can be represented by a combination of the simplest oscillations. We will consider the loading of the shell by surface loads of the form , 2 cos 0 1 sin t p p p t p      where p 0 – statistical component, ω – angular frequency, t – time. Also, the shell is subjected to bending from a vertical static force P ex . In accordance with the approach for solving static one-dimensional contact problems, Emelyanov (2009), the canonical system of equations that determines the dynamic contact pressure between the shell and the circular base (for p 2 =0) will have the form, Vasilenko and Emelyanov (1995), Vasilenko and Emelyanov (2002),                2 1 2 2 0 1 sin 1 cos 1 1 N i Eh p p t R i X i DX Z            N i Eh p p t R Ni X i DX N Z N 1 2 2 2 0 1 sin cos     (1)

  N

cos 

.



вн i X i P

1

i