Two-step incremental procedure associated with the normal flow technique applied to trusses
DOI:
https://doi.org/10.5433/1679-0375.2022v43n1Espp29Keywords:
Linear Arc-Length, positional formulation, Geometric nonlinearity, Normal Flow techniqueAbstract
To achieve the nonlinear structural behavior, there is a need to trace of their equilibrium path in the space of load-displacement. Truss systems are commonly implemented in several structural systems, including high-span bridges and bracing of the supporting structure of tall buildings. Our study adapts a two-step method with cubic convergence into an incremental-iterative procedure to analyze the geometric nonlinear behavior of trusses. The solution method is combined with the Linear Arc-Length path-following technique. To find the approximate root of nonlinear equation system in the two-step method, two formulas are used. Structures are discretized using the Positional Finite Element Method and all truss bars are assumed to remain linear elastic. The correction of the nodal coordinates subincrement vector is performed using the Normal Flow technique. A computational algorithm was implemented using the free program Scilab. Our numerical results show that, when compared to the standard and modified Newton-Raphson algorithms, the new algorithm decreases the number of iterations and the computing time in the nonlinear analysis of trusses. Equilibrium paths with force and/or displacement limit points are obtained with good precision.Metrics
Metrics Loading ...
References
ALLGOWER, E. L.; GEORG, K. Homotopy methods for approximating several solutions to nonlinear systems of equations. In: FORSTER, W. (ed.). Numerical solution of highly nonlinear problems. Amsterdam: North-Holland, 1980. p. 253-270.
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.
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.
Downloads
Published
2022-06-27
How to Cite
Souza, L. A. F. de, Doná Junior, W., & Silva, E. L. C. da. (2022). Two-step incremental procedure associated with the normal flow technique applied to trusses. Semina: Ciências Exatas E Tecnológicas, 43(1Esp), 29–40. https://doi.org/10.5433/1679-0375.2022v43n1Espp29
Issue
Section
Original Article
License
The Copyright Declaration for articles published in this journal is the author’s right. Since manuscripts are published in an open access Journal, they are free to use, with their own attributions, in educational and non-commercial applications. The Journal has the right to make, in the original document, changes regarding linguistic norms, orthography, and grammar, with the purpose of ensuring the standard norms of the language and the credibility of the Journal. It will, however, respect the writing style of the authors. When necessary, conceptual changes, corrections, or suggestions will be forwarded to the authors. In such cases, the manuscript shall be subjected to a new evaluation after revision. Responsibility for the opinions expressed in the manuscripts lies entirely with the authors.
This journal is licensed with a license Creative Commons Attribution-NonCommercial 4.0 International.
