Unified formulation of geometrically nonlinear refined beam theories

By using the Carrera Unified Formulation (CUF) and a total Lagrangian approach, the unified theory of beams including geometrical nonlinearities is introduced in this paper. According to CUF, kinematics of one-dimensional structures are formulated by employing an index notation and a generalized expansion of the primary variables by arbitrary cross-section functions. Namely, in this work, lowto higher-order beam models with only pure displacement variables are implemented by utilizing Lagrange polynomials expansions of the unknowns on the cross-section. The principle of virtual work and a finite element approximation are used to formulate the governing equations, whereas a Newton-Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. By using CUF and three-dimensional GreenLagrange strain components, the explicit forms of the secant and tangent stiffness matrices of the unified beam element are provided in terms of fundamental nuclei, which are invariants of the theory approximation order. A symmetric form of the secant matrix is provided as well by exploiting the linearization of the geometric stiffness terms. Various numerical assessments are proposed, including large deflection analysis, buckling and post-buckling of slender solid cross-section beams. Thin-walled structures are also analysed in order to show the enhanced capabilities of the present formulation. Whenever possible, the results are compared to those from the literature and finite element commercial software tools.