PSI - Issue 83

Mohamed Rjilatte et al. / Procedia Structural Integrity 83 (2026) 208–217

214

ൌ ∗ and ሺ ሻ ൌ ∗ ሺ ∗ ሻ ௜ ∗ ௝ ൌ ௜௝ ௕ ௕ ଶ ௕ ൌන ௜ ∗ ௝ ∗ ∗ ଵ ଴ ௜ ௖ ௝ ∗ ൌ ୏ ౟ౠ ௅್ య ୉ ౘ୍ ౘ ு మ ൌ׬ ቀ ப మ ௐ ೔ ∗ ப୶ ∗మ ቁ ଴ ଵ ൬ ப మ ௐ ೕ ∗ ப୶ ∗మ ൰d ∗ ௜ ∗ ௝ ൌ K ୧୨ ௖ ௖ ଶ ௕ ௖ ሺsin ሻ ଶ ൌ ଴ ቆ ௝ ∗ ∗ ቇ න ൬ ௞ ∗ ∗ ൰ ଵ ௜ ∗ න ௝ ∗ d ∗ ௅ ೎ ଴ ୠ I ୠ ଶ ൌ ߙ න ൬ ௜ ∗ ∗ ൰ ଵ

(22)

௜ ∗ ௝௞௟ ൌ ௜௝௞௟ ଷ௕ E

଴ ൬ ௟ ∗ ∗ ൰ ∗

Substituting Equations (32) into Equations (28)– (30) yields the following nonlinear algebraic system: 2 ௜ ∗ ௜ ∗ ௥ ൅3 ௜ ∗ ௝ ∗ ௞∗ ௜ ∗ ௝௞௥ െ2 ∗ଶ ௜ ∗ ௜ ∗ ௥ ൌ0 (23) The equation was solved using the single-mode approach, as previously presented (El Kadiri and Benamar, 2003). This method involves considering only a single mode, the dominant mode, while neglecting the influence of the remaining modes. ቆ ௡∗ ௟ ௟ ∗ ቇ ଶ ൌ1൅ 3 2 ௜ ଶ ௜௜௜௜ ௜௜ (24) With: ௟ ∗ ൌ ௜௜ ௜௜ (25) 3. Results and discussion 3.1 Numerical Analysis of Linear Vibrations Non-Linear In this section, we first present the results related to the linear part of the problem, before discussing those obtained from the nonlinear analysis. The linear frequencies were calculated based on the material properties provided by Cong et al. (2020) and Rjilatte et al. (2024), considering two configurations: a clamped–clamped beam and a simply supported beam. The comparison of the obtained results shows an excellent agreement with reference data, thereby confirming the reliability of the employed method. Table I provides a detailed comparison between the frequency values obtained in this study and those reported in previous works, highlighting the accuracy of the results. Table 1: The first five linear frequencies of the cable-stayed beam, expressed in hertz, for the clamped–clamped and simply supported configurations. Modes Present S-S Ref [15] C-C 1 0,1362 0,1360 0,1680 2 0,2310 0,2307 0,3370 3 0,4354 0,4349 0,5940 4 0,7849 0,7840 0,9850 5 1,2162 1,2147 1,4700 6 1,3428 1,3436 2,0440