A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

• Published in: International journal of non-linear mechanics Vol. 38; no. 7; pp. 1067 - 1081
• Main Authors: Di Egidio, Angelo, Luongo, Angelo, Vestroni, Fabrizio
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01-10-2003; Elsevier Science; Elsevier
• Subjects: Beams; Engineering Sciences; Exact sciences and technology; Flexural–torsional dynamics; Fundamental areas of phenomenology (including applications); Non-linear resonances; Open cross-section; Other; Physics; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibrations and mechanical waves; Warping non-linear effects; Flexural–torsional dynamics; Beams; Non-linear resonances; Warping non-linear effects; Open cross-section; Open section beam; Vibration; Equation of motion; Warping; Flexural-torsional dynamics; Modeling; Hamiltonian mechanics; Resonance; Thin walled beam; Non linear effect; Dynamic model; Structural analysis
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/S0020-7462(02)00053-7

A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.