A nonlinear one-dimensional model of inextensional, shear undeformable, thin-walled beam with open cross-section is developed. Nonlinear warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, the nonlinear warping is described in terms of the flexural and torsional curvatures, the sectorial area law, and three new warping functions. Due to the internal constrains, the displacement field depends on three components only, two transversal translation of the shear center and the torsional rotation. Three nonlinear ordinary differential equations, governing the motion of the beam, corrected up to the third order, are derived through the Hamilton principle. After having analyzed the order of magnitude of the various terms, the equations are simplified and the influence of each contribution is discussed with reference to its kinematical nature. Symmetries properties are also taken into account to analyze the effects on the dynamic of the beam due to the presence of one or two symmetry axis. By a Galerkin procedure a discreet form of the equations is given by taking into account only of the first three vibration modes. Through multiple scale method the modulation and phase equations are obtained. Stationary solutions and their stability are studied for a beam with a symmetry axis.

### Nonlinear Nonplanar Vibrations of Open Cross-Section Thin-Walled Beams

#### Abstract

A nonlinear one-dimensional model of inextensional, shear undeformable, thin-walled beam with open cross-section is developed. Nonlinear warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, the nonlinear warping is described in terms of the flexural and torsional curvatures, the sectorial area law, and three new warping functions. Due to the internal constrains, the displacement field depends on three components only, two transversal translation of the shear center and the torsional rotation. Three nonlinear ordinary differential equations, governing the motion of the beam, corrected up to the third order, are derived through the Hamilton principle. After having analyzed the order of magnitude of the various terms, the equations are simplified and the influence of each contribution is discussed with reference to its kinematical nature. Symmetries properties are also taken into account to analyze the effects on the dynamic of the beam due to the presence of one or two symmetry axis. By a Galerkin procedure a discreet form of the equations is given by taking into account only of the first three vibration modes. Through multiple scale method the modulation and phase equations are obtained. Stationary solutions and their stability are studied for a beam with a symmetry axis.
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1999
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11697/40175`
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