Issue 60

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

  i i S x S x 4 i

     4

0

 2 i

  i x m  4 

(3)

EI

Here,  i : natural frequency of the damaged beam   x i S : Eigen functions ( i N t ) : Generalized coordinates Due to the presence of the transverse and open cracks in the structure, the beam modelling has been done as three subparts for two cracks and four beam segments for three cracks. For a simply supported beam with two cracks, the solution for Eqn. (3) can be written as below: For the first beam subdivision (   1 0 vt L )                   1 1 1 1 1 sin cos sin cosh i i i i i i i i i S vt A vt B vt C h vt D vt (4.1)

  1 2 L vt L )

For the second beam subpart (

















 









 i A vt 2 sin i

 i

 i C h vt 2 sin i

 i

2 S vt i

B

vt

D

vt

cos

cosh

(4.2)

i

i

2

2

  2 L vt L )

For the third beam part (

















 









 i A vt 3 sin i

 i

 i C h vt 3 sin i

 i

3 S vt i

B

vt

D

vt

cos

cosh

(4.3)

i

i

3

3

While the simple support prevents the deflection, it allows to rotate where it is located. The end conditions of the proposed simply supported beam is explained as follows. One end condition is geometric and the other one is dynamic. At x =0,    0, 0 y t and     0, 0 y t . At x =L,    , 0 y L t and     , 0 y L t . Applying the end states, , , i i i i A B C D constants can be determined as explained by Thomson [28]. In this investigation, crack model in Dash’s work [29] has been considered. He explained the cracks analytically in detail in his model. Eqn. (2) has been put into Eqn. (1) and then multipled by   x j S . After that the equation is integrated over the length of beam. Referring to Reis and Pala[4], The features of orthogonality and the Dirac delta function are taken into consideration, embodying some arrangements, the equation can be given as stated in Eqn. (5) and Eqn.

 0 L

 0 L

İ

    Mg N N S x S x dx S x x vt dx m M N S x S x x vt dx m M v N S S x x vt dx m M v N S S x vt dx m                   '' x       ' x x δ   2 i                         j   2 j    2 j ¨ ¨ 1 0 1 0 i i i j j L İ i j j L İ i j j L İ i j





i

1

           



(5)



   





j

1

0

117