Análise paramétrica da instabilidade de estruturas reticuladas planas esbeltas com comportamento dinâmico geometricamente não linear pelo método posicional dos elementos finitos

Abstract

The demand for lightweight structural systems makes them susceptible to vibration problems that may compromise their performance and reliability. With the advancement in material technology, structures become more slender and with increasing spans, increasing the probability of loss of stability. Thus, geometrically nonlinear dynamic analysis becomes indispensable in structural design. The techniques developed over the last few years aim to assist engineers in the vibration analysis of complex structures. Thus, computational resources are essential. Software that performs dynamic analysis efficiently, allied to the accuracy and reliability of the results, becomes more necessary for engineers. The analysis and design of slender structures under vibration require the adjustment of physical and/or geometric parameters to meet a required level of performance and reliability. This research work proposes a methodology for dynamic instability analysis in plane frames using a geometrically nonlinear positional formulation of the Finite Element Method for all implementations. The study can be systematically presented as follows: (i) the evaluation of instability by dynamic snap-through in shallow arches and plane frames; (ii) calculus of the natural frequencies of vibration from the Subspace Iteration Method using the Hessian matrix; (iii) classical time-step integration methods with numerical dissipation control (Generalized-α, HHT-α, and WBZ-α), as well as recent algorithms (Truly Self-starting Two Sub-steps method and Three-parameter Single-step method) applied to nonlinear dynamic systems; (iv) classification of the systems (chaotic behavior) from the Lyapunov exponents obtained by nonlinear predictor algorithm and by Jacobian matrix analysis, as well as the Poincaré sections. Several examples from the literature were used to compare results and validate the performed implementations. Within a certain condition, the method of Iteration by Vector Subspaces using the Hessian matrix presented consistent results for the first natural frequencies of vibration. Most of the numerical integration methods proved to be efficient in the proposed analyses, with emphasis on the Generalized-α method due to its stability. The proposed algorithms for calculating the Lyapunov exponents also showed satisfactory results for the proposed examples.