Unified formulation of geometrically nonlinear refined beam theories

Abstract

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, low-to 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 Green-Lagrange 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.

Large-deflection and post-buckling analyses of laminated composite beams by Carrera Unified Formulation

“…According to those fundamental works, the constraint relationship corresponds to a multi-dimensional sphere with radius equal to the given arch-length value ∆l 0 , which varies at each load step depending on the ratio of convergence at the previous iteration. A detailed discussion about the numerical iterative scheme employed for solving the gemetric nonlinear problem is not given in this paper, but it can be found in [1]. Nevertheless, it is important to clarify that we employ a full Newton-Raphson method that, as opposed to a modified scheme, utilizes an updated tangent stiffness matrix at each iteration.…”

“…For this reason, in the past and recent literature, some authors have explored new possibilities of formulating symmetric forms of the secant stiffness matrix, see for example [45,46,47,48,49]. In the present work, and according to Carrera [50,51], a symmetric form of the secant stiffness matrix is devised by expressing the virtual variation of the internal strain energy due by the contribution K ijτ s nll as follows (see [1]):…”

“…In this context, the present research wants to introduce a unified beam formulation able to deal with large displacement/rotation analysis of composite laminated beams accounting for higher-order effects, which include (but are not limited to) complex shear deformations, bending-torsion coupling, accurate post-buckling, and nonlinear three-dimensional stress/strain state analysis. The proposed geometrical nonlinear formulation is based on the Carrera Unified Formulation (CUF) [32,33], which was recently extended to the nonlinear analysis of metallic beams by the same authors [1]. According to CUF, which assumes that any theory of structures can degenerate into a generalized kinematics by using an appropriate arbitrary expansion of the generalized variables, the nonlinear governing equations and the related finite element arrays of the generic, and eventually hierarchical, geometricallyexact composite beam theory are written in terms of fundamental nuclei.…”

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The Carrera Unified Formulation (CUF) was recently extended to deal with the geometric nonlinear analysis of solid cross-section and thin-walled metallic beams [1]. The promising results provided enough confidence for exploring the capabilities of that methodology when dealing with large displacements and post-buckling response of composite laminated beams, which is the subject of the present work.

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Computationally efficient, high-fidelity micromechanics framework using refined 1D models

“…The efficiency of the framework is derived from the ability of CUF models to provide accurate 3D displacement and stress fields at a reduced computational cost (approximately one order of magnitude of degrees of freedom less as compared to standard 3D brick elements) [24,25,27]. Over the last couple of decades, CUF models have been extensively used for wide range of structural simulations such as static analysis of laminated beams [28], dynamic response for aerospace structures [29], vibration characteristics of rotating structures [30], evaluation of failure indices in composite structures [26], buckling and post-buckling analysis of compact and composite structures [31,32]. Carrera et al reported an extended review of recent developments in refined theories for beam based on CUF with particular focus on diverse applications [33].…”