Issue 60

C. O. Bulut et al., Frattura ed Integrità Strutturale, 60 (2022) 114-133; DOI: 10.3221/IGF-ESIS.60.09

   

   

  

           

   

  

  

   

İ

İ

İ

  M N N S vt g    2 ¨ i

¨

˙





  v N S vt  ' j j

 

 

2

''





N S vt

v

j j N S vt

2

(6)

i

i

i

j

j

m

  







j

j

j

1

1

1

Due to the orthogonality of the eigenfunctions, we can write below equation:

L

L

 0 L

 0 L

 L

1

2

 0



2

2

2

2

2

 L S dx S dx S dx S dx S dx    1 2 3 i j i i i L

(7)

1

2

The equation (6) can be resolved utlizing the method proposed by Picard. In this method, basically an iteration is done by accepting the first one on the right side of the equation and neglecting the others. As per Lal M, Moffatt D [30], Michaltsos and Kounadis [27], Collins [31] , recalling the Picard’s method the following expresions are obtained; For

  1 0 t L v

  i Mg N N S vt m    2 ¨ 1 i i i

(8.1)

For

  1 2 L t L v v

  i Mg N N S vt m    2 ¨ 2 i i i

(8.2)

For

  2 L t L v v

  i Mg N N S vt m    2 ¨ 3 i i i

(8.3)

The solution of the Eqn. (6) can be given as follows for the homogeneous part: For

 1 t L v

          1 1 2 ) sin cos i h i i N t t

(9)

  Ω i i v

The particular solution of Eqn. (6) can be given as follows if

  N A t    * ) sin

  t

*

*

*

 

  )



B

C

t D

t

(

cos

sinh(

cosh(

)

(10)

i

p

i

i

i

i

i

i

i

i

1

1

1

1

1

118