PSI - Issue 83

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

164

( , ) P z T Effective ownership of the material ( , ), ( , ) u x t w x t   * * w x Shape of the nth linear mode of the beam * x Non-dimensional coordinate , , x y z Structure axis system

Movements in the x and z directions, respectively

2. Modal Analysis of Cracked FGM Beams: Modelling and Vibratory Behaviour Comprehending the vibrational behavior of structures with cracks holds immense significance in the proactive detection of damage. When conceptualizing a structure as a linear system, the changes in its characteristics, characterized by the stiffness and mass matrices, have a direct impact on alterations in its frequencies and eigenvectors [10]. In the scope of our investigation, our attention is directed towards a bi-encased Functionally Graded Material (FGM) beam characterized by length L , height h , and width b , featuring an open transverse crack with a depth denoted as dc (refer to Fig. 1).

Fig. 1. Representative model of an FGM beam with a crack.

In order to characterize the vibrational behavior of the cracked beam, we partition it into two uniform segments connected by a torsion spring positioned at the crack location (refer to Fig. 1). The stiffness of this torsion spring, influenced by both the depth of the crack and the physical and geometric attributes of the beam, encapsulates the impact of the crack on the natural frequencies. The coefficient of the torsion spring, represented as K  , is computed using the equation [11]:       2 1 / 6 1 eff c EI K h f d h     (1) where   eff EI ,  and c d represent the effective bending stiffness, Poisson's ratio, and crack depth of the FGM beam, respectively. The geometric parameter   / c f d h is determined through the following calculation [12]:

2

3

4

5

6

c d       h

c d       h

c d       h

c d       h

c d       h





/ f d h

1.8224

3.95

16.375

37.226

76.81











c

(2)

7

8

9

10

c d       h

c d       h

c d       h

c d       h

126.9

172

143

66.56









This model allows us to explore the subtle changes in the vibratory properties of the beam as a function of crack depth. Indeed, the effect of the crack is localised, which justifies the representation in distinct segments linked by the rotating spring. By combining the concepts of modal analysis and dynamic behavior, we are able to decipher how the presence of cracks affects the vibratory properties of the beam.