The one-dimensional formulation relies on the approximation of the displacement field over the thin-walled cross section in order to reduce the three dimensional elasticity problem to a set of governing equations defined along the beam axis.

For a better representation of the three-dimensional effects, an enrichment of the displacement field approximation is considered both by refining the mesh and by increasing the degree of the approximation functions. An element for the cross section discretization is defined, considering linear or quadratic Lagrange functions for approximating the displacement along the beam axis and linear Lagrange functions for the transverse displacement. This beam formulation permits inclusion of the membrane shear deformation of the thin-walled beam and can be applied to a generic cross section midline profile geometry.

The beam governing equations are written in terms of the displacement coordinates, yielding a set of second order differential system of coupled equations. The corresponding homogeneous solution is sought through an exponential form, having an associated quadratic eigenvalue problem.

In order to be effective, the derived beam model must be rewritten with an appropriate set of displacement modes that reproduces uncoupled solutions of the problem and hence allows a clear interpretation of the structural phenomena. Therefore, an essential step for the successful application of the model is the definition of a consistent criterion of uncoupling.

An innovative uncoupling procedure based on the quadratic eigenvalue problem associated with the beam governing equations is derived, permitting a set of hierarquicaly uncoupled deformations modes to be obtained. The classic modes are retrieved from this set of uncoupled modes together with higher order modes. The classic modes have an associated null eigenvalue, corresponding to polynomial solutions, and the higher order eigenvalues are non-null symmetric real numbers.

Since the cross section is in-plane rigid, the eigenvalue problem can be solved efficiently through a generalised eigenvalue problem that preserves the symmetric characteristics of the beam equations and allows analytical solutions to be obtained. In fact, the equilibrium equations can be written only in terms of axial components, rendering a symmetric system that through the eigenvalue problem, yields an efficient, simple and innovative procedure for the definition of warping modes of generic thin-walled cross sections.

An application example regarding the definition and use of warping modes for the analysis of a thin-walled beam is presented. The results are compared with previous thin-walled beam theories as well as with the results obtained from a shell finite element model implemented using Abaqus.

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