Análise modal de estruturas planas com elementos finitos q4 em notação strain gradient

Abstract

The increasing advancement of technologies that enable building slender and complex structures, therefore more susceptible to deleterious effects such as dynamic loads, have increasingly demanded the development of computational tools capable of simulating the mechanical behavior of those structures. The structural analyses are mostly performed using the Finite Element Method, where the quality of the results depends on the mathematical formulation used to describe the physical model, which may contain in some cases modeling errors. Quadrilateral elements, often used in plane elasticity, are examples of elements that coutain an error known as shear locking, due to the presence of spurious terms in the expression of angular deformation. These terms can be identified using the strain gradient notation, a physically interpretable notation that enable to evaluate the modeling characteristics of the element and correctly formulate the polynomials, removing a priori the terms that do not belong to the equation. Therefore, the objective of this study is to examine the effects of shear locking on the natural frequencies and mode shapes of vibration of plane structures using 4-node quadrilaterals formulated by the strain gradient notation. For that, it is developed a computational program capable of performing modal analysis of plane structures using the strain gradient notation. The computed natural frequencies are compared to the values obtained by the isoparametric formulation and ANSYS simulation software. The analyses show the effectiveness of the strain gradient notation in the modeling of free vibration problems, where the results obtained are the same as the isoparametric formulation and ANSYS software, when there is the presence of shear locking. When comparing the results of models with and without spurious terms, it is observed that spurious terms cause errors in the values of natural frequencies. This deleterious effect has a greater preponderance for less refined meshes and in flexural vibration modes. In addition, it is observed that the errors caused by spurious terms are more relevant in slender structures and at lower values of Poisson’s ratio. Thus, it is concluded that the strain gradient notation show advantages by the numerical aspect and successfully eliminate the modeling errors in the studied problems, where it is possible to compute more accurate and compatible results according to the physical mode