PSI - Issue 83

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

299

The strain energy associated with the axial (normal) forces, which are responsible for the geometric nonlinearity of the system, is denoted by ௔ and can be expressed for each beam as follows: ௔ ൌ ா್ ௅್ ூ଼್ ൤׬ ቀ డ௪್ డ ሺ ௫ ௫,௧ሻ ቁ ଶ ଴ ௕ ൨ ଶ (10) In this analysis, the transverse displacement is expressed in terms of a modal expansion, constructed from a series of fundamental spatial functions ௜ ሺ ሻ , where varies from 1 to , with representing the number of linear modes considered for the beam. These functions are associated with the generalized temporal coordinates ௜ ሺ ሻ , which are assumed to be harmonic in nature. Accordingly, the transverse displacement ሺ , ሻ can be written as follows: ሺ , ሻ ൌ ௜ ሺ ሻ ሺ ሻ (11) In this formulation, m ୧୨ , k ୧୨ and b ୧୨୩୪ represent the mass matrix, the linear stiffness tensor, and the nonlinear stiffness tensor, respectively. Their analytical expressions are given below: ௜௝ ൌ ܧ ௕ ௕ ׬ ቀ డ మ ௐ ೔ డ௫ మ ቁ ௅ భ ଴ ൬ డ మ ௐ ೕ డ௫ మ ൰ ൅ ܧ ௖ ௖ ሺ ሻ ଶ ௜ ׬ ௝ ௅ ೎ ଴ ൅∑ ் ௡ ௜ୀଵ ௌ௥௜௡௚ ௜ ൫ ் ௌ௥௜௡௚ ൯ ௝ ൫ ் ௌ௥௜௡௚ ൯ ൅ ∑ ௡௜ୀଵ ோ ௌ௥௜௡௚ డௐ ೔ ሺ௫ ೃೄೝ೔೙೒ ሻ డ௫ డௐ ೕ ሺ௫ ೃೄೝ೔೙೒ ሻ డ௫ (12) ௜௝௞௟ ൌ ா್ ସ௅್ ௌ್ ׬ ቀ డௐ ೔ డ௫ ቁ ଴ ௅್ ቀ డௐ ೕ డ௫ ቁ ׬ ቀ డௐ ೖ డ௫ ቁ ଴ ௅್ ቀ డௐ ೗ డ௫ ቁ (13) ௜௝ ൌ ߩ ௕ ௕ ׬ ௜ ௝ ௅ భ ଴ (14) By substituting Eqs. (12), (13), and (14) into Eq. (5), and after simplification, the following expression is obtained: 2 ௜ ௜௥ ൅3 ௜ ௝ ௞ ௜௝௞௥ െ2 ଶ ௜ ௜௥ ൌ0 (15) Prior to determining the contribution coefficient ௜ ௝ and the associated natural frequency ω , Eq. (11) is rewritten in a nondimensional form by replacing the dimensional parameters with their corresponding reduced (dimensionless) counterparts. ൌ ܮ ∗ ሺ ሻ ൌ ∗ ሺ ∗ ሻ (16) In this expression, denotes the characteristic height of the beam. Assuming that the beams illustrated in Fig. 1 are geometrically identical, the tensor expressions can therefore be written as follows: ௜ ∗ ௝ ൌ׬ ଵ ∗ ௜ ଵ ∗ ௝ ∗ ଴ ଵ (17) ௜ ∗ ௝ ൌ׬ ቀ డ మ ௐ భ ∗ ೔ డ௫ ∗మ ቁ ଴ ଵ ൬ డ మ ௐ భ ∗ ೕ డ௫ ∗మ ൰ ∗ ൅∑ ் ௡ ௜ୀଵ ௌ௥௜௡௚ ଵ ∗ ௜ ൫ ் ௌ௥௜௡௚ ൯ ଵ ∗ ௝ ൫ ் ௌ௥௜௡௚ ൯൅ ∑ ௡௜ୀଵ ோ ௌ௥௜௡௚ డௐ ೔ ሺ௫ ೃೄೝ೔೙೒ ሻ డ௫ ∗ డௐ ೕ ሺ௫ ೃೄೝ೔೙೒ ሻ డ௫ ∗ (18) ௜ ∗ ௝௞௟ ൌ ߙ ׬ ቀ డௐ భ ∗ ೔ డ௫ ∗ ቁ ଴ ଵ ൬ డௐ భ ∗ ೕ డ௫ ∗ ൰ ׬ ቀ డௐ భ ∗ ೖ డ௫ ∗ ቁ ଴ ଵ ቀ డௐ భ ∗ ೗ డ௫ ∗ ቁ ∗ (19) By substituting Eqs. (17), (18), and (19) into Eq. (15), the following system of nonlinear algebraic equations is obtained: