A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations

This paper deals with the formulation and implementation of the curved beam elements based on the geometrically exact curved/twisted beam theory assuming that the beam cross-section remains rigid. The summarized beam theory is used for the slender beams or rods. Along with the beam theory, some basic concepts associated with finite rotations and their parametrizations are briefly summarized. in terms of a non-vectorial parametrization of finite rotations under spatial descriptions, a formulation is given for the virtual work equations that leads to the load residual and tangential stiffness operators. Taking the advantage of the simplicity in formulation and clear classical meanings of both rotations and moments, the non-vectorial parametrization is applied to implement a four-noded 3-D curved beam element, in which the compound rotation is represented by the unit quaternion and the incremental rotation is parametrized using the incremental rotation vector. Only static problems are considered. Conventional Lagrangian interpolation functions are adopted to approximate both the reference curve and incremental rotation of the deformed beam. Reduced integration is used to overcome locking problems. The finite element equations are developed for static structural analyses, including deformations, stress resultants/couples, and linearized/nonlinear bifurcation buckling, as well as post-buckling analyses of arches subjected to different types of loads, such as self-weight, snow, and pressure (wind) loads. Several examples are used to test the formulation and its Fortran implementation.