Updated Lagrangian Curvilinear Beam Element for 2D Large Displacement Analysis

This paper presents a 2D curvilinear beam finite element model focusing the interest on its use for non-linear analysis caused by very large displacements, addressed with the Update Lagrangian strategy. The method allows using very long curvilinear beams even when high geometric nonlinearities occur. This is due inasmuch the proposed formulation does not require any pre-set shape function that would inevitably force to use a huge number of elements to achieve reliability. The lack of shape-functions is overcome using the integration of the compatibility equations, that provide the whole internal displacement field from the only knowledge of the element nodal degree of freedom. Section-slices subdivision allows to sum, not to assemble, the flexibility contribute of each slice and consequently to build up the end-to-end tangent stiffness matrix of a generic curvilinear beam element. Moreover, the flexibility feature of every slice can be deduced analytically once and for all. To validate the proposed element some comparisons are carried out with analytical and numerical solutions obtained with Runge–Kutta integration method or cubic isoparametric finite elements.