PSI - Issue 83

Mohamed Berjal et al. / Procedia Structural Integrity 83 (2026) 295–304

298

௕௜ డ మ ௩್ ∗ ೔ ሺ௫ ೔ డ௧ ሻ మ ൅ ߦ ௕ ∗ ௜ డ௩್ ೔ ሺ௫ ೔ డ௧ ሻ ൅ ܧ ௕௜ ௕௜ ௕ ௜௩ ௜ ∗ ሺ ௜ ሻൌ0 ௜ି ଵ ൏ ௜ ൏ ௜ ௖௝ డ మ ௩ ೎ ∗ ೕ డ௧ మ ൅ ߦ ௖ ∗ ௝ డ௩ ∗ ೎ೕ డ௧ െሾ ܪ ௖௝ డ మ ௩ ೎ ∗ ೕ డ௫ ೎మ ೕ ൅ ܧ ௖௝ ௖௝ ሺ డ మ ௬ ೕ ∗ డ௫ ೎మ ೕ ൅ డ

(1)

೎మ ೕ ሻ ௝ ∗ ሺ ሻሿ ൌ 0

మ ௩ ೎ ∗ ೕ డ௫

(2)

The analysis is carried out by considering the previously defined boundary conditions of the beam. Each cable is assumed to be rigidly connected to the tower at its upper end and attached to the beam at its lower end. Accordingly, the coupled system is required to satisfy both continuity conditions and the geometric boundary constraints, as outlined in (Berjal et al., 2024a). To generalize the analysis, dimensionless variables are introduced as detailed in (Rjilatte et al., 2024a), allowing Eqs. (1) and (2) to be expressed in a non-dimensional form independent of physical units. This normalization improves result interpretation and comparison with existing studies. The corresponding boundary conditions, accounting for the elastic translational and rotational supports introduced at both beam ends, are given in Article (Berjal et al., 2026). The analysis of the previously derived equations allows the solutions to be expressed in the following form: ௕௜ ሺ ሻ ൌ ௜௕ ௕ ൅ ௜௕ ௕ ൅ ܥ ௜௕ ℎ ௕ ൅ ௜௕ ℎ ௕ , ሺ ൌ1,2,3ሻ (3) ௖௝ ሺ ሻൌ ܧ ௝௖ ௖ ൅ ௝௖ ௖ ൅ ௝௖ , ሺ݆ ൌ1,2ሻ (4) 2.2. Nonlinear formulation According to Hamilton’s principle, the dynamic behavior of the system can be formulated as follows: ቀ׬ ሺ െ ሻ ଴ ଶగఠ ቁ ൌ 0 (5) The kinetic energy of the beam is denoted by , while represents the total strain energy. The latter consists of two components: the bending strain energy, ௕ , and the strain energy associated with the axial (normal) forces, ௔ , which introduce the geometric nonlinearity of the system for each beam. By introducing ሶ ൌ ∂ / ∂ , which denotes the time derivative of the transverse displacement, the kinetic energy can be expressed as follows, in accordance with the formulation proposed in (El Kadiri et al., 2002) : ൌ ଵ ଶ ௕ ௕ ׬ ቀ డ௪್ డ ሺ ௧ ௫,௧ሻ ቁ ଶ ଴ ௅್ (6) By introducing ᇱᇱ ൌ ଶ ଶ  , which represents the second derivative of the transverse displacement with respect to the longitudinal coordinate , the bending strain energy ௕ , together with the translational and rotational spring contributions ் ௦௣௥௜௡௚ and ோ௦௣௥௜௡௚ , can be expressed as follows: ௕ ൌ ா್ ଶ ூ್ ׬ ቀ డ మ ௪್ డ௫ ሺ௫,௧ሻ మ ቁ ଶ ଴ ௅್ ൅ ா ೎ ஺ ೎ ሺ௦௜௡ఏሻ మ ଶ ௕ ሺ , ሻ ׬ ௕ ሺ , ሻ ௅ ೎ ଴ (7)

் ௦௣௥௜௡௚ ൌ ଵ ଶ ∑ ் ௡ ௜ୀଵ ௌ௥௜௡௚ ቀ ௕ ൫ ் ௌ௥௜௡௚ , ൯ቁ ଶ ோ௦௣௥௜௡௚ ൌ ଵ ଶ ∑ ௡௜ୀଵ ோ ௌ௥௜௡௚ ቀ డ௪್ ሺ௫ ೃೄೝ೔೙೒ ,௧ሻ డ௫ ቁ ଶ

(8)

(9)