Issue 58

E.S.M.M. Soliman, Frattura ed Integrità Strutturale, 58 (2021) 151-165; DOI: 10.3221/IGF-ESIS.58.11

   Y A 

  

  

  

  







B

C

D

cos

sin

cosh

sinh

(6)

1

1

1

1

1

   Y A 

  

  

  

  







B

C

D

cos

sin

cosh

sinh

(7)

2

2

2

2

2

where A i and B i , i=1, 2, 3, 4, are coefficients can be obtained from the boundary conditions. The boundary conditions for the simply supported beam at the supports are [12] At the supports no displacement and no moments:    0 1 0 Y

(8)

   0 2 1 Y

(9)

2

2 d Y d

1



0

(10)

 



0

2

d Y d

2



0

(11)

2

 



1

If ζ = µ = Lc / L is the normalized crack position, Lc is the crack distance from left hand support, the continuity conditions at the crack position are [12]:

       1 2 Y Y

(12)

Displacement:

2

2

d Y

d Y

1

2



EI

EI

Moment:

(13)

2

2





d

d

  

  

3 d Y

3 d Y

1

2



EI

EI

(14)

Shear force:

3

3





d

d

  

  

Compatibility condition i.e., jump in the slope at the crack location due to rotational flexibility can be defined as [14]:

2

dY

d Y

dY





 1

1

 2





E K T

(15)

2

d

d



d

  

  

  

K T is the bending spring constant at cracked section and it is, originally calculated by [15] and can be written as [14]:

 1 K c T

(16)

     5.346 c h EI f s h

(17)

153