Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23

(1)

(2)

(3)

(4)

(3)

(4)

where the coefficients are determined from the recurrence relations available on E resource (www. icmm.ru/compcoeff) and can be calculated using the suggested options. For this purposes, it is necessary to input the number of terms of a series m , Poisson's ratio  , the number of harmonics k and the desired  , which can be either complex or real. Substituting (35) into (33), we obtain partial solutions (1) (2) (3) (4) 0 0 0 0 , , , v v v v for the function 0 v , , , m m m m m m A A A A B B , ,

¥ å

( x x S x x S x x S x x - - - ] ( ] ( ( ] ( ( 1 ) 1 2 1 1 2 1 1 2 1 1 2 S ( - - + - - - + ) ) ) ) ) - - + )

é ê ë

ù ú û

(1) v x

m

(1) P x m

=

,

( )

)

0

[

2

(

a

a a

= ¥

m

0

é ê ë

ù ú û

å

(2) v x

m

(2) P x m

=

,

( )

)

0

[

2

(

a

a a

=

m

0

ì ï ï í ï î ì ï ï í ï ï î

ü ï ï ý ï ï þ

¥ å

(

)

+

S

1

(

)

é ê ë

ù ú û

(36)

(3) v x

m

(3)

(3)

=

+

( ) P D x x + ln m m

,

( )

)

0

[

2

x

(

) - - + ï a a

a

=

m

0

ü ï ï ý ï ï þ

¥ å

(

) ( ) 1 m -

é ê ë

ù ú û

(4) v x

(4)

(4)

=

( ) P D x x + ⋅ ln m m

,

( )

] (

)

0

[

2

(

a

a a

=

m

0

(1) , , , , m m m m m m P P P P D D (2) (3) (4) (3) (4) ,

where the coefficients can be calculated ibidem. A general solution for 0

are available on E-resource (www. icmm.ru/compcoeff) and

u and 0

v can be written as

    4 0 , u x C u x C u x C u x C u x v x C v x C v x C v x C v x                                 (1) (2) (3) (4) 0 1 0 2 0 3 0 4 0 (1) (2) (3) (4) 0 1 0 2 0 3 0

(37)

where 1 2 3 4 , , , C C C C are the constants defined by the prescribed combination of boundary conditions (3) - (5).

C ONSTRUCTION OF PARTIAL SOLUTION FOR NONZEROTH HARMONICS OF THE FOURIER SERIES

T

he construction of partial solutions to Eqns. (10)-(12) involves some transformations [21], which yield a system of two differential equations with respect to , k k w v

 

 

 

 

4 d w x

3 d w x

2 d w x

dw x

 

 

 

 

   

k

k

k

k











4 f x

3 f x

2 f x

1 f x

f x w x

0

k

0

4

3

2

dx

dx

dx

dx

(38)

 

 

 

 

2 d v x

3 d w x

2 d w x

k dw x

  x

    x v x

  x

  x

  x

    x w x

k

k

k





3 

2 

1 

0 











k

k

2

0

2

3

2

dx

dx

dx

dx

,   

2 0 1 2 3 , , , ,  

 are given by

where 0 1 2 3 4 , , , , , f f f f f

0

1 2 1 2

1

  f x x 

  

 1 1 2 x  



  2 1 x             2 k k 2 3 1 1 k  



2

 

 

x

1



0

16

1

  f x x 

  

 1 1 2 x  



  2 1 x             2 k k 2 3 1 1 k  



2

 

 

x

1



0

16

2 1 1 2 2 k

  x  

  

 

  f x x x x    



2



3 1 x      

1 2 1 4 

1



233