Procedimento incremental de dois passos associado à técnica de fluxo normal aplicado a treliças
DOI:
https://doi.org/10.5433/1679-0375.2022v43n1Espp29Palavras-chave:
Comprimento de Arco Linear, Formulação posicional, Não linearidade geométrica, Técnica de Fluxo NormalResumo
Para obter o comportamento não linear de uma estrutura, há a necessidade de traçar sua trajetória de equilíbrio no espaço de carga-deslocamento. Os sistemas treliçados são comumente empregados em vários sistemas estruturais, incluindo pontes com grandes vãos e contraventamento da estrutura po rtante de edifícios altos. Este estudo adapta um método de dois passos com convergência cúbica em um procedimento incremental-iterativo para analisar o comportamento não linear geométrico de treliças. Esse método é combinado com a técnica de continuação Comprimento de Arco Linear. Duas fórmulas são usadas para encontrar a raiz aproximada do sistema de equações não lineares. As estruturas são discretizadas por meio do Método Posicional dos Elementos Finitos e assume-se que todas as barras de treliça permaneçam elásticas lineares. A correção do vetor de subincremento de coordenadas nodais é realizada pela técnica de Fluxo Normal. Um algoritmo computacional foi implementado utilizando o programa livre Scilab. Os resultados numéricos mostram que, quando comparado com os algoritmos de Newton-Raphson Padrão e Modificado, o novo algoritmo diminui o número de iterações e o tempo de processamento nas análises não lineares de treliças. As trajetórias de equilíbrio com pontos limites de força e/ou de deslocamento são obtidas com boa precisão.Referências
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SCILAB. Version 6.1.1. France: ESI Group, 2021.
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SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I. Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems. Semina: Ciênc. Ex. Tech., Londrina, v. 42, n. 1, p. 63-74, Jan./Jun. 2021.
SOUZA, L. A. F.; SANTOS, D. F. D.; KAWAMOTO, R. Y. M.; VANALLI, L. New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities. The Journal of Strain Analysis for Engineering Design, London, v. 57, n. 2, p. 104-115, 2021.
THAI, H. T.; KIM, S. E. Large deflection inelastic analysis of space trusses using generalized displacement control method. Journal of Constructional Steel Research, London, v. 65, n. 10/11, p. 1987-1994, 2009.
TURCO, E.; BARCHIESI, E.; GIORGIO, I.; DELL’ISOLA, F. A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. International Journal of Non-Linear Mechanics, Amsterdam, v. 123, p. 103481, 2020.
YAW L. L. 3D Co-rotational Truss Formulation. Walla Walla: Walla Walla University, 2011.
WEMPNER, G. A. Discrete approximations related to nonlinear theories of solids. International Journal of Solids and Structures, New York, v. 7, n. 11, p. 1581-1599, 1971.
CODA, H. B.; GRECO, M. A simple FEM formulation for large deflection 2D frame analysis based on position description. Computer methods in applied mechanics and engineering, Amsterdam, v. 193, n. 33-35, p. 3541-3557, 2004.
CRISFIELD, M. A. Non-linear finite element analysis of solids and structures. Chichester: John Wiley \& Sons Ltda, 1991. v. 1.
DEHGHANI, H.; MANSOURI, I.; FARZAMPOUR, A.; HU, J. W. Improved homotopy perturbation method for geometrically nonlinear analysis of space trusses. Applied Sciences, [Tubingen], v. 10, n. 8, p. 2987, 2020.
FELIPE, T. R.; LEONEL, E. D.; HAACH, V. G.; BECK, A. T. A comprehensive ductile damage model for 3D truss structures. International Journal of Non-Linear Mechanics, Amsterdam, v. 112, p. 13-24, 2019.
GRECO, M.; FERREIRA, I. P. Logarithmic strain measure applied to the nonlinear positional formulation for space truss analysis. Finite elements in analysis and design, Amsterdam, v. 45, n. 10, p. 632-639, 2009.
GRECO, M.; MENIN, R. C. G.; FERREIRA, I. P.; BARROS, F. B. Comparison between two geometrical nonlinear methods for truss analyses. Structural engineering and mechanics: An international journal, New York, v. 41, n. 6, p. 735-750, 2012.
HRINDA, G. Snap-through instability patterns in truss structures. In: AIAA/ASME/ASCE/AHS/ASC STRUCTURES, STRUCTURAL DYNAMICS, AND MATERIALS CONFERENCE 51., 2010, Orlando. Proceesdings […]. Reston: AIAA, 2010. p. 2611.
KOOHESTANI, K. A hybrid method for efficient solution of geometrically nonlinear structures. International Journal of Solids and Structures, New York, v. 50, n. 1, p. 21-29, 2013.
KOU, J.; LI, Y.; WANG, X. A modification of Newton method with third-order convergence. Applied Mathematics and Computation, New York, v. 181, n. 2, p. 1106-1111, 2006.
KRENK, S.; HEDEDAL, O. A dual orthogonality procedure for non-linear finite element equations. Computer Methods in Applied Mechanics and Engineering, Amsterdam, v. 123, n. 1-4, p. 95-107, 1995.
LEON, S. E.; PAULINO, G. H.; PEREIRA, A., MENEZES, I. F.; LAGES, E. N. A unified library of nonlinear solution schemes. Applied Mechanics Reviews, New York, v. 64, n. 4, 2011.
MAHDAVI, S. H.; RAZAK, H. A.; SHOJAEE, S.; MAHDAVI, M. S. A comparative study on application of Chebyshev and spline methods for geometrically non-linear analysis of truss structures. International Journal of Mechanical Sciences, New York, v. 101, p. 241-251, 2015.
MATIAS, W. T. El control variable de los desplazamientos en el análisis no lineal elástico de estructuras de barras. Revista internacional de métodos numéricos, Barcelona, v. 18, n. 4, p. 549-572, 2002.
MAXIMIANO, D. P.; SILVA, A. R. D.; SILVEIRA, R. A. M. Iterative strategies associated with the normal flow technique on the nonlinear analysis of structural arches. Rem: Revista Escola de Minas, Ouro Preto, v. 67, n. 2, p. 143-150, 2014.
MOHIT, M.; SHARIFI, Y.; TAVAKOLI, A. Geometrically nonlinear analysis of space trusses using new iterative techniques. Asian Journal of Civil Engineering, [London], v. 21, n. 5, p. 785-795, 2020.
MUÑOZ, L. F. P.; ROEHL, D. A continuation method with combined restrictions for nonlinear structure analysis. Finite Elements in Analysis and Design, Amsterdam, v. 130, p. 53-64, 2017.
NOOR, M. A.; AHMAD, F.; JAVEED, S. Two-step iterative methods for nonlinear equations. Applied mathematics and computation, New York, v. 181, n. 2, p. 1068-1075, 2006.
RABELO, J. M.; BECHO, J. S.; GRECO, M.; CIMINI JR., C. A. Modeling the creep behavior of GRFP truss structures with Positional Finite Element Method. Latin American Journal of Solids and Structures, v. 15, n. 2, p. 1-18, 2018.
RAGON, S. A.; GÜRDAL, Z.; WATSON, L. T. A comparison of three algorithms for tracing nonlinear equilibrium paths of structural systems. International journal of solids and structures, New York, v. 39, n. 3, p. 689-698, 2002.
REZAIEE-PAJAND, M.; NASERIAN, R. Using residual areas for geometrically nonlinear structural analysis. Ocean Engineering, Elmsford, v. 105, p. 327-335, 2015.
RIKS, E. The application of newton’s method to the problem of elastic stability. Journal of Applied Mechanics, New York, v. 39, p. 1060–1065, 1972.
SAFFARI, H.; MANSOURI, I. Non-linear analysis of structures using two-point method. International Journal of Non-Linear Mechanics, Amsterdam, v. 46, n. 6, p. 834-840, 2011.
SAFFARI, H.; MIRZAI, N. M.; MANSOURI, I.; BAGHERIPOUR, M. H. Efficient numerical method in second-order inelastic analysis of space trusses. Journal of computing in civil engineering, New York, v. 27, n. 2, p. 129-138, 2013.
SCILAB. Version 6.1.1. France: ESI Group, 2021.
SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I.; ALIANO FILHO, A.; MACHADO, R. D. Trusses nonlinear problems solution with numerical methods of cubic convergence order. TEMA, São Carlos, v. 19, p. 161-179, 2018.
SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I. Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems. Semina: Ciênc. Ex. Tech., Londrina, v. 42, n. 1, p. 63-74, Jan./Jun. 2021.
SOUZA, L. A. F.; SANTOS, D. F. D.; KAWAMOTO, R. Y. M.; VANALLI, L. New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities. The Journal of Strain Analysis for Engineering Design, London, v. 57, n. 2, p. 104-115, 2021.
THAI, H. T.; KIM, S. E. Large deflection inelastic analysis of space trusses using generalized displacement control method. Journal of Constructional Steel Research, London, v. 65, n. 10/11, p. 1987-1994, 2009.
TURCO, E.; BARCHIESI, E.; GIORGIO, I.; DELL’ISOLA, F. A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. International Journal of Non-Linear Mechanics, Amsterdam, v. 123, p. 103481, 2020.
YAW L. L. 3D Co-rotational Truss Formulation. Walla Walla: Walla Walla University, 2011.
WEMPNER, G. A. Discrete approximations related to nonlinear theories of solids. International Journal of Solids and Structures, New York, v. 7, n. 11, p. 1581-1599, 1971.
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2022-06-27
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Souza, L. A. F. de, Doná Junior, W., & Silva, E. L. C. da. (2022). Procedimento incremental de dois passos associado à técnica de fluxo normal aplicado a treliças. Semina: Ciências Exatas E Tecnológicas, 43(1Esp), 29–40. https://doi.org/10.5433/1679-0375.2022v43n1Espp29
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