A non-linear one-dimensional model of cross-deformable tubular beam

• Direct one-dimensional model of beam to take into account ovalization of the cross-section.
• Detection of the non-linear response function through an identification process.
• Three-dimensional refined model constituted by longitudinal fibers and transversal ribs.
• Straightforward expansion method to solve examples in statics.
• Lindstedt–Poincaré method to solve an example in free non-linear dynamics.

A direct non-linear one-dimensional model of an elastic, thin-walled, planar beam is formulated. The model accounts for changes in shape of the cross-section, in particular the ovalization (or flattening) occurring in tubular beams. The deformation of the cross-section is described in the spirit of the Generalized Beam Theory, as a linear combination of known deformation modes and unknown amplitude functions, said to be distortions. Kinematics calls for introducing distortional and bi-distortional strains, in addition to the usual strain measures of rigid cross-section beams. The balance equations are derived through the Virtual Power Principle, in which distortional and bi-distortional stresses, as well as distortional forces, are defined as conjugate quantities of distortional strain-rates and velocities, respectively. A non-linear, fully coupled, hyperelastic law is assumed. All the distortional quantities and the constitutive law are identified, via energy equalities, from a three-dimensional fiber-model of thin-walled beam where, for simplicity, just a distortion mode is considered. The model is specialized to a Euler–Bernoulli tubular beam, in which only constitutive non-linearities are retained, while kinematics is linearized. The relevant non-linear equations are solved, via a perturbation method, for several static loadings and for large-amplitude free vibrations. The interaction occurring between global bending and cross-section distortion is analyzed.