Abstract
A numerical procedure is presented to avoid the divergence problem during the iterative process in viscoelastic analyses. This problem is observed when the positional formulation of the finite element method is adopted in association with the finite difference method. To do this, the nonlinear positional formulation is presented considering plane frame elements with Bernoulli–Euler kinematics and viscoelastic behavior. The considered geometrical nonlinearity refers to the structural equilibrium analysis in the deformed position using the Newton–Raphson iterative method. However, the considered physical nonlinearity refers to the description of the viscoelastic behavior through the adoption of the stress-strain relation based on the Kelvin–Voigt rheological model. After the presentation of the formulation, a detailed analysis of the divergence problem in the iterative process is performed. Then, an original numerical procedure is presented to avoid the divergence problem based on the retardation time of the adopted rheological model and the penalization of the nodal position correction vector. Based on the developments and the obtained results, it is possible to conclude that the presented formulation is consistent and that the proposed procedure allows for obtaining the equilibrium positions for any time step value adopted without presenting divergence problems during the iterative process and without changing the analysis of the final results.
Highlights
IntroductionUs, structural engineering aims to address the demands for technological advances in the most diverse areas, such as infrastructure, civil construction, mechanical industry, aerospace industry, and others
Part of these studies are related to the analyses of structures with viscoelastic behavior adopting stress-strain relation deduced from rheological models, as demonstrated in the works of Akoz and Kadioǧlu [2], Mesquita and Coda [5], Panagiotopoulos et al [9], dos Santos Becho et al [11], Carniel et al [12], Oliveira and Leonel [13], Pascon and Coda [14], Rabelo et al [15], and Fernandes et al [16]. ese rheological models are schematic representations that combine elastic and viscous elements to obtain a physical interpretation of stress-strain relationships. erefore, from the interpretation of these rheological models, it is possible to deduce relations between the components of the stress and strain tensors that are appropriate to the mechanical behavior of interest and necessary for the assessment of the strain energy
It is worth mentioning that part of these studies uses the positional formulation of the finite element method to describe the viscoelastic behavior, as demonstrated in the works of dos Santos Becho et al [11], Pascon and Coda [14], Rabelo et al [15], and Fernandes et al [16]. e use of such a formulation is justified by the capacity and simplicity of its application in the analysis of nonlinear problems as highlighted in Coda and Greco [17] and Greco and Coda [18]
Summary
Us, structural engineering aims to address the demands for technological advances in the most diverse areas, such as infrastructure, civil construction, mechanical industry, aerospace industry, and others Part of these studies are related to the analyses of structures with viscoelastic behavior adopting stress-strain relation deduced from rheological models, as demonstrated in the works of Akoz and Kadioǧlu [2], Mesquita and Coda [5], Panagiotopoulos et al [9], dos Santos Becho et al [11], Carniel et al [12], Oliveira and Leonel [13], Pascon and Coda [14], Rabelo et al [15], and Fernandes et al [16]. A numerical procedure based on the retardation time and the penalization of the nodal position correction vector is presented to avoid the divergence problem
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