PSI - Issue 83

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

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Fig. 1 Configuration of the multi-cable-stayed beam model By simplifying the differential equation governing the cable motion and neglecting the higher-order and nonlinear squared terms in the integral of the uniform dynamic elongation of the cable, ௖௝ ሺ ሻ , the equations for the undamped free vibrations of the cable and beam are obtained as follows [10]: ௖௝ ଶ ² ௖௜ ² െ ² ௖௜ ௖ ଶ ௝ െ ߤ ̂ ௝ ² ௖௜ ௖ ଶ ௝ ൌ0 ௕௜ ସ ଶ ௕௜ ଶ ൅ ଶ ௕௜ ௜ ସ ൌ0 (1) In the equation above: ̂ ௝ ൌ ௖௝ ሺ1, ሻ ௝ ൅׬ ൬ డ௬ ೎ೕ డ௫ ೎ೕ డ௩ ೎ೕ డ௫ ೎ೕ ൰ ௝ ଴ ଵ for ሺ ൌ 1, 2, … , ሻ (2) By adopting a similar approach, the mechanical boundary conditions can be represented in a linear form, as illustrated below: As a result, this work utilizes a model of a symmetric double-cable beam. Based on the theoretical principles previously discussed and considering a symmetrical configuration of cables anchored at one-third of the beam length with an angle of 30 degrees, the following variable substitutions will be made: ௖௝ ൌ ௖ , ௝ ൌ , ௝ ൌ , ௖௝ ൌ ௖ , ௖௝ ൌ ௖ , ௖௝ ௖௝ ൌ ௖ ௖ , ௖௝ ൌ ௖ , ௕௜ ൌ ௕ , ௕௜ ௕௜ ൌ ௕ ௕ (3) Within this equation, ௝ denotes the dimensionless deflection of the -th cable. By employing the method of separation of variables, we formulate the following relationship: