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

Abstract

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. Accordingly, by employing CUF, governing nonlinear equations of low-to higher-order beam theories for laminated beams are expressed in this paper as degenerated cases of the three-dimensional elasticity equilibrium via an appropriate index notation. In detail, although the provided equations are valid for any one-dimensional structural theory in a unified sense, layer-wise kinematics are employed in this paper through the use of Lagrange polynomial expansions of the primary mechanical variables. The principle of virtual work and a finite element approximation are used to formulate the governing equations in a total Lagrangian manner, 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. Several numerical assessments are proposed, including post-buckling of symmetric cross-ply beams and large displacement analysis of asymmetric laminates under flexural and compression loadings. the comprehensive review works by Carrera [2] and Kapania and Raciti [3,4], respectively. Nevertheless, among the fundamental topics in structural mechanics, the geometrical nonlinear analysis of elastic structures holds a relevant importance. It is a matter of fact that the effects of large displacements and rotations may play a primary role in the correct prediction, for example, of flexible beams, which continue to be employed for wing structures, space antennas, rotor blades, and robotic arms. The literature about this argument is large, and a detailed discussion on nonlinear formulations of composite structures falls outside the scope of this work. However, some relevant papers on nonlinear beam models are briefly outlined hereinafter for the sake of completeness.It is well known that for thin and solid cross-section beam structures, a good model for geometrically nonlinear analysis is represented by the so-called elastica [5,6,7]. The elastica beam addresses flexural problems by assuming the local curvature as proportional to the bending moment, according to the classical Euler-Bernoulli beam theory [8]. This assumption, of course, is too limiting for the analysis of composite structures, for which the shear effects may considerably alter the solution accuracy. For this reason, many works in the literature are based on the Timoshenko beam theory [9], which assumes a uniform shear distribution along the cross-section of the beam. In the domain of nonlinear analysis of metallic beams, this theory was extensively exploited; see for example the pioneering work of Reissner [10], who considered the effect of transverse force str...

Energy based collocation method to predict progressive damage behavior of imperfect composite plates under compression

“…In the field of buckling and post-buckling analyses, many researchers have extensively investigated these behaviors for composite beams and plates without considering the damage effects. Turvey and Marshall (1995) and Argyris and Tenek (1997) published excellent reviews on past studies of buckling and post-buckling of structures using different methods and recently, Pagani and Carrera (2017) have analyzed the large deflection and post-buckling behaviors of laminated beams by Carrera unified formulation.…”

Geometrically Nonlinear Analysis of Beam Structures via Hierarchical One-Dimensional Finite Elements

“…Thanks to this approach, the derivation of a family of higher-order beam theories is made formally general regardless of the through-the-thickness approximating functions and the approximation order. UF has been extended to large deflections and postbuckling analysis of beam structures in recent works by Pagani and Carrera [20,21], where the capability of such approach to investigate global and local deformations in solid and thin-walled beam structures was demonstrated by using Lagrange polynomials as approximating functions for the cross-section kinematics within a layer-wise approach. An elastoplastic analysis via UF-based one-dimensional finite elements has been carried out by Carrera et al [22], showing that such formulation can lead to a 3D-like accuracy in terms of displacements and stresses for compact and thin-walled structures subjected to localized loadings.…”

“…As far as the stress prediction is concerned, once the second Piola-Kirchoff stresses have been obtained by the TL formulation, they are transformed into the true Cauchy stresses to have a direct comparison with results coming from updated Lagrangian formulations implemented in the commercial software ANSYS. As opposed to the Lagrange layer-wise approximation [20,21], in both linear and nonlinear analyses, the use of Taylor polynomials allows an enrichment of the cross-sectional kinematics by simply increasing the order N and with no need for additional crosssectional nodes. This feature makes Taylor-based refined models particularly suitable for the analysis of multilayered structures in the framework of an equivalent single-layer approach that will be presented in a future work.…”

“…This feature makes Taylor-based refined models particularly suitable for the analysis of multilayered structures in the framework of an equivalent single-layer approach that will be presented in a future work. As further novelties with respect to [20,21], the correction of shear and membrane locking phenomena in the nonlinear regime via the MITC method has been introduced, allowing an improved convergence performance of the proposed finite elements in slender structures. Finally, a detailed description of the stress analysis under large displacements, not often encountered in the literature, has been also provided, together with the discussion of some limitations of the proposed formulation with respect to finite elements with large strains capabilities.…”

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].…”