Abstract
In the framework of finite elements (FEs) applications, this paper proposes the use of the node-dependent kinematics (NDK) concept to the large deflection and post-buckling analysis of thin-walled metallic one-dimensional (1D) structures. Thin-walled structures could easily exhibit local phenomena which would require refinement of the kinematics in parts of them. This fact is particularly true whenever these thin structures undergo large deflection and post-buckling. FEs with kinematics uniform in each node could prove inappropriate or computationally expensive to solve these locally dependent deformations. The concept of NDK allows kinematics to be independent in each element node; therefore, the theory of structures changes continuously over the structural domain. NDK has been successfully applied to solve linear problems by the authors in previous works. It is herein extended to analyze in a computationally efficient manner nonlinear problems of beam-like structures. The unified 1D FE model in the framework of the Carrera Unified Formulation (CUF) is referred to. CUF allows introducing, at the node level, any theory/kinematics for the evaluation of the cross-sectional deformations of the thin-walled beam. A total Lagrangian formulation along with full Green–Lagrange strains and 2nd Piola Kirchhoff stresses are used. The resulting geometrical nonlinear equations are solved with the Newton–Raphson linearization and the arc-length type constraint. Thin-walled metallic structures are analyzed, with symmetric and asymmetric C-sections, subjected to transverse and compression loadings. Results show how FE models with NDK behave as well as their convenience with respect to the classical FE analysis with the same kinematics for the whole nodes. In particular, zones which undergo remarkable deformations demand high-order theories of structures, whereas a lower-order theory can be employed if no local phenomena occur: this is easily accomplished by NDK analysis. Remarkable advantages are shown in the analysis of thin-walled structures with transverse stiffeners.
Highlights
Nowadays, in many engineering fields, increasingly sophisticated structures are employed to fulfill the tasks of demanding applications
This paper is organized as follows: (i) node-dependent kinematics (NDK) approach and the Green–Lagrange relations are presented in Sect. 2, along with the geometrically nonlinear finite elements (FEs) equations ; (iii) numerical results are discussed for thinwalled beams in Sect. 3, with symmetric and asymmetric cross section and both transverse and compressive loadings; (iv) the main conclusions are drawn
B4 finite elements are employed in the analysis cases, and the notation used hereafter to identify the structural models involves the number of B4 elements approximated with LE or TE, followed by the number of FEs, so that, for example, “10LE” stands for “10B4 elements with Lagrange Expansion” and “5TE4” stands for “5B4 elements with Taylor Expansion of order 4”
Summary
In many engineering fields, increasingly sophisticated structures are employed to fulfill the tasks of demanding applications. By using variational asymptotic approaches, the 3D nature of such cases was divided into a two-dimensional (2D) analysis of the cross section and a 1D problem of the beam axis, see for instance the work by Yu et al [28,29] These solutions lack the ability to accurately catch higher-order phenomena, such as coupling or local effects, which may occur within a thin-walled structure. The post-buckling behavior of thin-walled beam structures was evaluated by an extension of GBT by Basaglia et al [30], and using the Newton–Raphson linearization method along with the Ritz approach by Machado [31] In both cases, the deformability of the structures in the large displacement field was caught by considering bending and warping effects over the cross section. This paper is organized as follows: (i) NDK approach and the Green–Lagrange relations are presented in Sect. 2, along with the geometrically nonlinear FE equations ; (iii) numerical results are discussed for thinwalled beams in Sect. 3, with symmetric and asymmetric cross section and both transverse and compressive loadings; (iv) the main conclusions are drawn
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