PSI - Issue 83

Amal Lahrizi et al. / Procedia Structural Integrity 83 (2026) 162–170

165

2.1. Beam vibration equations To formulate the model for the cracked Functionally Graded Material (FGM) beam, we employed an approach rooted in the Euler-Bernoulli beam theory, incorporating a torsion spring to represent the presence of the crack and deriving characteristic equations. The position of the crack is denoted as lc measured from the left end of the beam. In order to describe the properties of the beam, two functions, * * 1 ( ) i w x and * * 2 ( ) i w x , were introduced to represent the left and right sides of the spring, respectively, as illustrated in Fig. 1. The ultimate expressions for the transverse vibration equations are articulated as follows:                 * * * * 1 1 1 * * * * 1 1 ( ) cosh sinh cos sin , 0 i i i i i c w x A x B x C x D x x L           (3)                 * * * * * * 2 2 2 * * * * * * * 2 2 ( ) cosh sinh cos sin , i i c i c i c i c c w x A x L B x L C x L D x L L x L               (4)

In Eq.(5), the non-dimensional parameters are introduced as :

x

D

x

w

li 

*      * 2 j j L L h i x w , , ,

,

x

c



*

2

11

(5)

c

L



4

0

where li  and j x are the linear natural frequency and the nondimensional position of the crack, respectively. 1 A 2 D are the constant coefficients that can be determined by applying the boundary conditions. The beam considered here is a clamped-clamped beam. Therefore, in equation (5), the displacement and slope are fixed at both ends of the beam:         * * 1 2 * * 1 2 * * * * 0 * * 0 0 ; 0 i i i i x x L x x L dw x dw x w x w x dx dx         (6) Additionally, the equality of displacement, bending moment, and shear force on both sides of the crack is asserted, and the discrepancy in slopes between the two segments at the crack location can be linked to the bending moment imposed by the torsional spring. These compatibility relationships are expressed in Equation (7) as follows:

  *

  *

2 d w x

2 d w x

  * x

  *

1

2

i

i

;

;

w

w x





1

2

i

i

*2

*2

*

*

dx

dx

x L 

x L 

c

c

*

*

x L 

x L 

c

c

(7)

  *

  *

  EI

3

3

d

w x

d

w x

* ( )

* ( )

2

*

1 w x ( i

)

* w x x  2 i

1 w x i







1

2

i

i

ef

f

;







*3

*3

*

*2

dx

d

x

x

K x 





*

*

*

x

L

L

L

x

x







*

*

x

L

x

L





c

c

c

c

c