Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A dual approach to perform geometrically nonlinear analysis of plane truss structures

  • Received : 2017.07.13
  • Accepted : 2018.01.21
  • Published : 2018.04.10
⟨ Previous Next ⟩

Abstract

The main objective of this study is to develop a dual approach for geometrically nonlinear finite element analysis of plane truss structures. The geometric nonlinearity is considered using the Total Lagrangian formulation. The nonlinear solution is obtained by introducing and minimizing an objective function subjected to displacement-type constraints. The proposed method can fully trace the whole equilibrium path of geometrically nonlinear plane truss structures not only before the limit point but also after it. No stiffness matrix is used in the main approach and the solution is acquired only based on the direct classical stress-strain formulations. As a result, produced errors caused by linearization and approximation of the main equilibrium equation will be eliminated. The suggested algorithm can predict both pre- and post-buckling behavior of the steel plane truss structures as well as any arbitrary point of equilibrium path. In addition, an equilibrium path with multiple limit points and snap-back phenomenon can be followed in this approach. To demonstrate the accuracy, efficiency and robustness of the proposed procedure, numerical results of the suggested approach are compared with theoretical solution, modified arc-length method, and those of reported in the literature.

Keywords

References

  1. Alnaas, W.F. and Jefferson, A.D. (2016), "A smooth unloading-reloading approach for the nonlinear finite element analysis of quasi-brittle materials", Eng. Fract. Mech., 152, 105-125. https://doi.org/10.1016/j.engfracmech.2015.04.018
  2. Arora, J.S. (2012), Introduction to Optimum Design, (Third Edition), Academic Press, Boston, MA, USA.
  3. Baguet, S. and Cochelin, B. (2003), "On the behaviour of the ANM continuation in the presence of bifurcations", Commun. Numer. Methods Eng., 19(6), 459-471. https://doi.org/10.1002/cnm.605
  4. Bathe, K.-J. (2006), Finite Element Procedures, Klaus-Jurgen Bathe.
  5. Bathe, K.-J. and Dvorkin, E.N. (1983), "On the automatic solution of nonlinear finite element equations", Comput. Struct., 17(5), 871-879. https://doi.org/10.1016/0045-7949(83)90101-3
  6. Belegundu, A.D. and Arora, J.S. (1984a), "A Computational Study of Transformation Methods for Optimal Design", AIAA J., 22(4), 535-542. https://doi.org/10.2514/3.48476
  7. Belegundu, A.D. and Arora, J.S. (1984b), "A recursive quadratic programming method with active set strategy for optimal design", Int. J. Numer. Methods Eng., 20(5), 803-816. https://doi.org/10.1002/nme.1620200503
  8. Bellini, P. and Chulya, A. (1987), "An improved automatic incremental algorithm for the efficient solution of nonlinear finite element equations", Comput. Struct., 26(1), 99-110. https://doi.org/10.1016/0045-7949(87)90240-9
  9. Bhatti, M.A. (2006), Advanced Topics in Finite Element Analysis of Structures: With Mathematica and MATLAB Computations, John Wiley & Sons, Inc.
  10. Chen, W.-F. and Lui, E.M. (1991), Stability Design of Steel Frames, CRC press.
  11. Crisfield, M.A. (1981), "A fast incremental/iterative solution procedure that handles "snap-through"", Comput. Struct., 13(1), 55-62. https://doi.org/10.1016/0045-7949(81)90108-5
  12. Crisfield, M. (1983), "An arc-length method including line searches and accelerations", Int. J. Numer. Methods Eng., 19(9), 1269-1289. https://doi.org/10.1002/nme.1620190902
  13. Crisfield, M.A. (1991), Nonlinear Finite Element Analysis of Solids and Structures. Vol. 1: Essentials, John Wiley & Sons.
  14. Damil, N. and Potier-Ferry, M. (1990), "A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures", Int. J. Eng. Sci., 28(9), 943-957. https://doi.org/10.1016/0020-7225(90)90043-I
  15. Elhage-Hussein, A., Potier-Ferry, M. and Damil, N. (2000), "A numerical continuation method based on Pade approximants", Int. J. Solids Struct., 37(46), 6981-7001. https://doi.org/10.1016/S0020-7683(99)00323-6
  16. Felippa, C.A. (2001), "Nonlinear finite element methods", Aerospace Engineering Sciences Department of the University of Colorado. Boulder.
  17. Forde, B.W. and Stiemer, S.F. (1987), "Improved arc length orthogonality methods for nonlinear finite element analysis", Comput. Struct., 27(5), 625-630. https://doi.org/10.1016/0045-7949(87)90078-2
  18. Garcea, G., Madeo, A., Zagari, G. and Casciaro, R. (2009), "Asymptotic post-buckling FEM analysis using corotational formulation", Int. J. Solids Struct., 46(2), 377-397. https://doi.org/10.1016/j.ijsolstr.2008.08.038
  19. Geers, M.-a. (1999), "Enhanced solution control for physically and geometrically non-linear problems. Part I-the subplane control approach", Int. J. Numer. Methods Eng., 46(2), 177-204. https://doi.org/10.1002/(SICI)1097-0207(19990920)46:2<177::AID-NME668>3.0.CO;2-L
  20. Hamdaoui, A., Braikat, B. and Damil, N. (2016), "Solving elastoplasticity problems by the Asymptotic Numerical Method: Influence of the parameterizations", Finite Elem. Anal. Des., 115, 33-42. https://doi.org/10.1016/j.finel.2016.03.001
  21. Hellweg, H.-B. and Crisfield, M. (1998), "A new arc-length method for handling sharp snap-backs", Comput. Struct., 66(5), 704-709. https://doi.org/10.1016/S0045-7949(97)00077-1
  22. Izadpanah, M. and Habibi, A. (2015), "Evaluating the spread plasticity model of IDARC for inelastic analysis of reinforced concrete frames", Struct. Eng. Mech., Int. J., 56(2), 169-188. https://doi.org/10.12989/sem.2015.56.2.169
  23. Jiang, G., Li, F. and Li, X. (2016), "Nonlinear vibration analysis of composite laminated trapezoidal plates", Steel Compos. Struct., Int. J., 21(2), 395-409. https://doi.org/10.12989/scs.2016.21.2.395
  24. Kondoh, K. and Atluri, S. (1985), "Influence of local buckling on global instability: Simplified, large deformation, post-buckling analyses of plane trusses", Comput. Struct., 21(4), 613-627. https://doi.org/10.1016/0045-7949(85)90140-3
  25. Liang, K., Ruess, M. and Abdalla, M. (2014), "The Koiter-Newton approach using von Karman kinematics for buckling analyses of imperfection sensitive structures", Comput. Methods Appl. Mech. Eng., 279, 440-468. https://doi.org/10.1016/j.cma.2014.07.008
  26. Liang, K., Ruess, M. and Abdalla, M. (2016), "Co-rotational finite element formulation used in the Koiter-Newton method for nonlinear buckling analyses", Finite Elem. Anal. Des., 116, 38-54. https://doi.org/10.1016/j.finel.2016.03.006
  27. Mahdavi, S.H., Razak, H.A., Shojaee, S. and Mahdavi, M.S. (2015), "A comparative study on application of Chebyshev and spline methods for geometrically non-linear analysis of truss structures", Int. J. Mech. Sci., 101, 241-251.
  28. Mansouri, I. and Saffari, H. (2012), "An efficient nonlinear analysis of 2D frames using a Newton-like technique", Arch. Civil Mech. Eng., 12(4), 485-492. https://doi.org/10.1016/j.acme.2012.07.003
  29. Papadrakakis, M. (1983), "Inelastic post-buckling analysis of trusses", J. Struct. Eng., 109(9), 2129-2147. https://doi.org/10.1061/(ASCE)0733-9445(1983)109:9(2129)
  30. Pastor, M.M., Bonada, J., Roure, F. and Casafont, M. (2013), "Residual stresses and initial imperfections in non-linear analysis", Eng. Struct., 46, 493-507. https://doi.org/10.1016/j.engstruct.2012.08.013
  31. Planinc, I. and Saje, M. (1999), "A quadratically convergent algorithm for the computation of stability points: The application of the determinant of the tangent stiffness matrix", Comput. Methods Appli. Mech. Eng., 169(1), 89-105. https://doi.org/10.1016/S0045-7825(98)00178-9
  32. Powell, M.J. (1978), "A fast algorithm for nonlinearly constrained optimization calculations", In: Numerical Analysis, Springer, pp. 144-157.
  33. Ramm, E. (1981), "Strategies for tracing the nonlinear response near limit points", In: Nonlinear Finite Element Analysis in Structural Mechanics, Springer, pp. 63-89.
  34. Rao, S.S. (2009), Engineering Optimization: Theory and Practice, John Wiley & Sons.
  35. Rezaiee-Pajand, M. and Naserian, R. (2017), "Nonlinear frame analysis by minimization technique", Iran Univ. Sci. Technol., 7(2), 291-318.
  36. Rezaiee-Pajand, M., Salehi-Ahmadabad, M. and Ghalishooyan, M. (2014), "Structural geometrical nonlinear analysis by displacement increment", Asian J. Civil Eng. (BHRC), 15(5), 633-653.
  37. Riks, E. (1972), "The application of Newton's method to the problem of elastic stability", J. Appl. Mech., 39(4), 1060-1065. https://doi.org/10.1115/1.3422829
  38. Riks, E. (1979), "An incremental approach to the solution of snapping and buckling problems", Int. J. Solids Struct., 15(7), 529-551. https://doi.org/10.1016/0020-7683(79)90081-7
  39. Rosen, A. and Schmit, L.A. (1979), "Design-oriented analysis of imperfect truss structures-part I-accurate analysis", Int. J. Numer. Methods Eng., 14(9), 1309-1321. https://doi.org/10.1002/nme.1620140905
  40. Saada, A.S. (2013), Elasticity: Theory and Applications, Elsevier.
  41. Saffari, H. and Mansouri, I. (2011), "Non-linear analysis of structures using two-point method", Int. J. Non-Linear Mech., 46(6), 834-840. https://doi.org/10.1016/j.ijnonlinmec.2011.03.008
  42. Saffari, H., Mansouri, I., Bagheripour, M.H. and Dehghani, H. (2012), "Elasto-plastic analysis of steel plane frames using Homotopy Perturbation Method", J. Constr. Steel Res., 70, 350-357. https://doi.org/10.1016/j.jcsr.2011.10.013
  43. Schweizerhof, K. and Wriggers, P. (1986), "Consistent linearization for path following methods in nonlinear FE analysis", Comput. Methods Appl. Mech. Eng., 59(3), 261-279. https://doi.org/10.1016/0045-7825(86)90001-0
  44. Thai, H.-T. and Kim, S.-E. (2009), "Large deflection inelastic analysis of space trusses using generalized displacement control method", J. Constr. Steel Res., 65(10-11), 1987-1994. https://doi.org/10.1016/j.jcsr.2009.06.012
  45. Thompson, J.M.T. and Hunt, G.W. (1973), A General Theory of Elastic Stability, John Wiley & Sons.
  46. Torkamani, M.A. and Shieh, J.-H. (2011), "Higher-order stiffness matrices in nonlinear finite element analysis of plane truss structures", Eng. Struct., 33(12), 3516-3526. https://doi.org/10.1016/j.engstruct.2011.07.015
  47. Torkamani, M.A. and Sonmez, M. (2008), "Solution techniques for nonlinear equilibrium equations", Proceedings of 18th Analysis and Computation Specialty Conference (ASCE), Vancouver, BC, Canada, April.
  48. Wan, C.-Y. and Zha, X.-X. (2016), "Nonlinear analysis and design of concrete-filled dual steel tubular columns under axial loading", Steel Compos. Struct., Int. J., 20(3), 571-597. https://doi.org/10.12989/scs.2016.20.3.571
  49. Wempner, G.A. (1971), "Discrete approximations related to nonlinear theories of solids", Int. J. Solids Struct., 7(11), 1581-1599. https://doi.org/10.1016/0020-7683(71)90038-2
  50. Yang, Y.-B. and Kuo, S.-R. (1994), "Theory and analysis of nonlinear framed structures."

Cited by

  1. Nonlinear stability of the upper chords in half-through truss bridges vol.36, pp.3, 2018, https://doi.org/10.12989/scs.2020.36.3.307
  2. Nonlinear dynamic responses of beamlike truss based on the equivalent nonlinear beam model vol.194, pp.None, 2021, https://doi.org/10.1016/j.ijmecsci.2020.106197
  3. A new procedure for post-buckling analysis of plane trusses using genetic algorithm vol.40, pp.6, 2018, https://doi.org/10.12989/scs.2021.40.6.817