A unified approach to the Timoshenko 3D beam-column element tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations Marcos Antonio Campos Rodrigues a,⇑ , Rodrigo Bird Burgos b , Luiz Fernando Martha c a Espírito Santo Federal University, Department of Civil Engineering, Avenida Fernando Ferrari, 514, Goiabeiras, Vitória, ES 29075-910, Brazil b State University of Rio de Janeiro, Department of Structures and Foundations, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro, RJ 20550-900, Brazil c Pontifical Catholic University of Rio de Janeiro, Department of Civil Engineering, Rua Marques de São Vicente, 225, Gávea, Rio de Janeiro, RJ 22451-900, Brazil article info Article history: Received 10 July 2020 Received in revised form 14 January 2021 Accepted 16 February 2021 Available online 04 March 2021 Keywords: Tangent stiffness matrix Analytical interpolation functions Timoshenko beam theory Nonlinear geometric analysis abstract A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consid- eration of five aspects: the interpolation (shape) functions, the bending theory, the kinematic description, the strain–displacement relations, and the nonlinear solution scheme. As the FEM provides a numerical solution, the structure discretization has a great influence on the analysis response. However, when applying interpolation functions calculated from the homogenous solution of the differential equation of the problem, a numerical solution closer to the analytical response of the structure is obtained, and the level of discretization could be reduced, as in the case of linear analysis. Thus, to reduce this influence and allow a minimal discretization of the structure for a geometric nonlinearity problem, this work uses interpolation functions obtained directly from the solution of the equilibrium differential equation of a deformed infinitesimal element, which includes the influence of axial forces. These shape functions are used to develop a complete tangent stiffness matrix in an updated Lagrangian formulation, which also integrates the Timoshenko beam theory, to consider shear deformation and higher-order terms in the strain tensor. This formulation was implemented, and its results for minimal discretization were com- pared with those from conventional formulations, analytical solutions, and Mastan2 v3.5 software. The results clearly show the efficiency of the developed formulation to predict the critical load of plane and spatial structures using a minimum discretization. Ó 2021 Elsevier Ltd. All rights reserved. 1. Introduction The continuous (analytical) behavior of a solid can be approxi- mated by a discrete solution. Usually, the discrete response is obtained by nodal displacements, and an approximated continuous solution can be found by means of interpolating (shape) functions. However, the discrete solution using the FEM introduces simplifi- cations in the mathematical idealization of the structure behavior as the interpolation functions that define the deformed configura- tion of a structure are not compatible with the mathematical ide- alization of the response of a continuous medium (Martha, 2018). In a linear elastic analysis of frame models with beam elements with constant cross-sections, interpolating functions are obtained from the homogeneous solution of the equilibrium differential equation of an undeformed infinitesimal element, leading to the so-called cubic Hermitian interpolation functions (Rodrigues et al., 2019). In this case, the formulation does not consider any other approximation except those already covered in the analytical idealization of the element behavior. This explains the fact that, in linear elastic analysis, the structure response of this type of model does not depend on the level of discretization. However, for geometric nonlinear or second-order analysis, in which equilibrium should be considered in the deformed configu- ration, Hermitian interpolation functions do not represent the analytical response of the structure. To cope with this problem, high-order finite elements can be used (So and Chan, 1991; Zheng and Dong, 2011; Rodrigues et al., 2016). Burgos et al., (2005) employed classic linearization from the stability problem and used additional degrees of freedom within the elements to calculate the critical load by eigenvalue analysis. Another way of improving a second-order analysis is to use sta- bility functions (Chen and Lui, 1991; Aristizábal-Ochoa, 1997, 2007, 2008, 2012). Some authors have used the consistent field https://doi.org/10.1016/j.ijsolstr.2021.02.014 0020-7683/Ó 2021 Elsevier Ltd. All rights reserved. ⇑ Corresponding author at: Avenida Fernando Ferrari, 514, Nexem, Goiabeiras, Vitória, ES 29075-910, Brazil. E-mail address: [email protected] (M.A.C. Rodrigues). International Journal of Solids and Structures 222–223 (2021) 111003 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
A unified approach to the Timoshenko 3D beam-column element tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations
May 17, 2023
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