Issue 76

W. Hanini et alii, Fracture and Structural Integrity, 76 (2026) 183-211; DOI: 10.3221/IGF-ESIS.76.12

Structure 03 (Fig. 15): Modes 1 and 2 are two coupled dominant modes characterized by a motion that combines torsion around the Z axis and translation along the Y axis. Mode 1 has a natural frequency of 1.90 Hz, while mode 2 has a natural frequency of 1.97 Hz. The main mode in the X-axis direction is mode 10, which is a pure translation mode with a natural frequency of 2.59 Hz. Structure 04 (Fig. 16): Mode 1 is the main translation mode along the X axis, with a natural frequency of 1.62 Hz, while mode 2, a coupled translation mode, is dominant in the Y and Z-axis directions, with a frequency of 1.75 Hz. Finally, the main tortional is mode 10, with a natural frequency of 2.66Hz. Structure 05 (Fig. 17): Mode 4 is the main translation mode along the X axis, with a natural frequency of 3.28 Hz. In the Y-axis direction, mode 1 is dominant, also a translation mode, with a frequency of 1.75 Hz. Finally, the main mode along the Z axis is mode 8, a torsion mode, with a frequency of 5.10 Hz. Structure 06 (Fig. 18): Modes 3 and 2 are the two main coupled modes characterized by translation along X and Y axes. The values of the natural frequencies, for the two modes, are respectively equal to 3.13 Hz and 3.66Hz. The main mode along the Z-axis direction is mode 11, a torsional mode with a relatively high natural frequency of 17.12 Hz. Likewise, the modal analysis revealed that the first modes are dominated by horizontal translations, while the higher modes are primarily torsional, with stiffness increasing with frequency. These findings constitute a fundamental step towards better understanding: first, the dynamic distribution of forces, and second, the directional vulnerability of the structures studied.

Frequency (Hz)

Participation ratio X

Participation ratio Y

Participation ratio Z

Structure

Mode

Period (s)

1 2 5 1 2 4 1 2 1 2 1 4 8 1 2 3

1.91 2.21 3.45 2.49 2.75 4.63 1.90 1.97 2.59 1.62 1.75 2.66 1.75 3.28 5.10 3.13 3.66 7.09

0.52 0.45 0.29 0.40 0.36 0.22 0.07 0.53 0.51 0.38 0.67 0.57 0.37 0.57 0.30 0.20 0.32 0.27 0.14

0.95

0.041

0.18 0.45

Structure 01

0.056

1

0.62 0.95

0.78 0.72 0.84 0.16 0.90 0.03 0.17 0.28 1.00 0.04 0.42 0.70 0.68 0.22 1 1 1 1

1

0.007

1

0.01

Structure 02

0.19

0.002

11

11.54

0.043

1

0.23 0.07

0.92

Structure 03

1

10

1 1

0.28 0.48 0.95 0.03 0.29 1.00 1

Structure 04

0.15 0.23 0.06 1.00 0.42 0.94 0.72

10

Structure 05

0.0007 0.009

Structure 06

1

0.06 0.85

11

17.78

0.058

0.25

Table 5: Main natural vibration modes.

Mode 1, f1=1.91Hz

Mode 2, f2=2.21Hz

Mode 5, f5=3.45Hz

Figure 13: Main natural vibration modes for Structure 01.

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