Estudo da estabilidade dinâmica de estruturas reticuladas planas através dos expoentes de Lyapunov considerando não-linearidades física e geométrica pela formulação posicional do método dos elementos finitos

Abstract

Nonlinear analysis of structures has been increasingly important in evaluating the behavior of structures, which can be observed in alternatives, even if simplified, present in design codes. Some types of structures are susceptible to instabilities and more detailed knowledge of the structural behavior is necessary. The present work deals with the development and application of a numerical methodology to analyze the stability of framed structures based on Lyapunov exponents. Material and geometric non-linear effects are considered. Material non-linearity is modeled using the multilinear isotropic work hardening model together with a newly developed ductile damage model called FLHB. The solution of the non-linear equilibrium equations is obtained using the positional formulation of the Finite Element Method. The time march procedures used were the Generalized-α and Newmark algorithms. To obtain the Lyapunov coefficients, two different techniques are used: maximum exponent using the non-linear predictor and the spectrum using the eigenvalues of the Jacobian matrix. Four examples are presented for validation of numerical models and two other numerical examples with dynamic actions are presented to illustrate the applicability of the methodology. In terms of the effects of material non-linearity, a great loss of stiffness of the structures under instability and a great dissipation in the dynamic response were observed. In the examples studied, the results indicated that the non-linear predictor method has a clear advantage, as it only depends on one time series. Its main disadvantage is the absence of negative exponents. Regarding the application of the calculation through the eigenvalues of the Jacobian matrix, difficulties were observed in identifying certain instabilities. It was concluded that Lyapunov exponents are good tools for identifying problems in the behavior of a structure, but, by themselves, they do not provide more information about the cause of such structural failures.