PSI - Issue 83

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

168

(c)

Fig. 3. Three-dimensional plots illustrating the relationship between normalised natural frequency, location and normalised depth of cracks for a bi-embedded beam. Sub-figures (a), (b) and (c) correspond to mode 1, mode 2 and mode 3 respectively . 4. Numerical result In this section, numerical examples are conducted on an FGM beam with dimensions: length L= 500 mm, thickness b=20 mm, and width h =20 mm. The beam possesses the following properties:

3

2 3 A E Gpa l O  70 , : m

2702 /

,

0.3

Kg m

 







m

m

(9)

3

:

380 , E Gpa 

3960 /

,

0.3

Al

Kg m

c 

c 



c

The first three natural frequencies of a clamped-clamped beam with crack (a single crack), with variations in the crack depth ratio and the crack location ratio, are determined using a numerical analysis method. These frequencies are then compared with other known results. Table 1 shows the results of this comparison. It can be seen that the values obtained from the numerical analysis agree well with those obtained from the analysis carried out by [14]. In addition, it is important to note that the natural frequencies of a beam change with crack depth and location. These frequencies are unique for each combination of crack depth and location. L h E   ) of a clamped-clamped FGM beam, for a power law exponent n =1, are obtained from 10 attempts involving different crack location and depth configurations. Table 1. Results of the first three non-dimensional frequencies ( 2 . / . / l m m

Crack

Dimensionless frequencies

No.

Loc.(mm)

Depth(mm)

1 

2 

3 

No crack [14]

9.569

26.371 26.3759 26.3635 26.3493 26.3299 26.3620 26.3460 26.3238 26.3698 26.3628 26.3531

51.685 51.7073 51.7039 51.7001 51.6948 51.6364 51.5545 51.4419 51.6440 51.5707 51.4700

No crack

9.5685 9.5559 9.5415 9.5217 9.5681 9.5678 9.5673 9.5575 9.5449 9.5274

1 2 3 4 5 6 7 8 9

40

2 3 4 2 3 4 2 3 4

100

275