PSI - Issue 59

Andrii Babii et al. / Procedia Structural Integrity 59 (2024) 609–616 Andrii Babii et al. / Structural Integrity Procedia 00 (2019) 000 – 000

613 5

According to the Fig. 2, we refer the tank shell to the coordinate system ). We assume that the tank is filled with liquid having specific gravity  , and in addition, the tank is subjected to the excessive internal pressure 0 p . The external loading on the shell can be described by the following function (

      .

1 1 2 (1 cos ) р ( α ,α ) p R      , 0 0

(6)

Here  is dynamism coefficient, which takes into account the increase in the effect of the liquid weight when the sprayer moves over irregularities and possible tank vibrations (Vikovych (2003)). The occurrence of field irregularities is closely related to the specifics of technological operations during sowing, care or harvesting of crops (Nanka et al. (2019)). Similarly, let us develop dependence (6) in series by the system of functions 1 1 sin cos . k m l          Applying the above mentioned approaches, for we obtain

  

cos      

  

k

m

1 

2 

1



 

,

(7)

( , )

sin

p

p



1 1 2  

1

km

l

l

1,3,...

0

k

m





1

2





8

p R 



4

R



0

0

p

p

0





where . The same algorithm is used to model the contact interaction of the supports. Let us take the most general law of contact pressure distribution - according to the hyperbolic cosine in both directions for rectangular support     0 0 1 2 1 1 1 2 2 2 ( , ) ( ) ( ) q q Ach a ch a          , (8) 1 0 k k  , 1 1 k k  sides of the rectangular support; 1 2 , , q A a a are constant values. Considering half of the cylindrical shell due to its symmetry (Fig. 2)   l R   , and denoting the areas of interaction between the support and the cylindrical shell by 0 0 1 2 l b              . Then the model of the action of the supports on the shell is written in the form of single expression  1 2 ( , ):0 1 1,     , D D , we have: 2 2 0 l l     , 2 0   0 0 1 1 2 ( , ): D b             , 1 1 1 0 ,   0 0 2 1 2 ( , ): 1 1 1 1 0 , D where: 0 1 1 b     , 1 0 2 2 b     ; 2 0 0 1 2 ( , )   are coordinates of the support center; 1 2 2 , 2 b b are lengths of the

    

A ch a  

   

0 1 2 ) , ( , )       2

0  

0

( (

)  

(

,

ch a

D







1 1

1

2 2

1

q

(9)

1 1 1 A ch a l  q

   

0 1 2 ) , ( , )       2

0    )

0

1 2 ( , )  

(

,

q

ch a

D

 



 





1

2 2

2

0 ( , ) , ( , ) . D D       0 1 2 1 1 2 2

0,



Let us find the development of function (9) into the Fourier series

  

cos      

  

4

k

m





   

1 2 ( , )  

( , , )   

sin

q

k m A



1 

2 

,

(10)

m km

l l

l

l

1,3,...

0

k

m





1 2

1

2

16

q Ab

0 l     1 sin k

  

1

where

1 ( 1) , k    

(22) (23)

cos(

0  ) m I I  k

A







km

m

l





1

1