Wire element: a kinematic compatibility-based curvilinear beam element for geometrically nonlinear analysis of highly flexible slender structures

Abstract This work presents a planar curvilinear beam finite element, referred to as the Wire Element, for the geometrically nonlinear analysis of slender one-dimensional structures undergoing large displacements and finite rotations. The proposed formulation differs from conventional displacement-interpolation approaches for it reconstructs the internal kinematics through direct integration of compatibility equations along the beam axis. Within a flexibility framework, the equilibrium relations, cross-sectional constitutive behaviour and kinematic compatibility altogether combine to derive the element response in the current configuration. The element accommodates the evolving geometry through internal section-slice discretization. At every converged configuration of the incremental–iterative procedure, each section-slice updates its orientation forming the one-dimensional curvilinear beam. The flexibility contribution of each slice is evaluated analytically and accumulated to construct the end-to-end element flexibility matrix; this allows to obtain the tangent stiffness matrix directly referred to ending nodes. As a result, the formulation enables the modelling of highly curved beams using a minimal degrees-of-freedom set, whatever curvature complexity is. The proposed element is assessed through comparisons with analytical Elastica solutions and with conventional geometrically nonlinear finite element analysis. The results demonstrate accurate prediction of large displacements and rotations equilibrium paths accompanied with a significant reduction of the required degrees-of-freedom and therefore computational effort. Owing to its compatibility-driven kinematic reconstruction, the formulation is particularly suited for applications where geometric fidelity and computational efficiency must be simultaneously addressed.