Structural Engineering and Mechanics

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On the local stability condition in the planar beam finite element

  • Planinc, Igor (Faculty of Civil and Geodetic Engineering, University of Ljubljana) ;
  • Saje, Miran (Faculty of Civil and Geodetic Engineering, University of Ljubljana) ;
  • Cas, Bojan (Faculty of Civil and Geodetic Engineering, University of Ljubljana)
  • Published : 2001.11.25
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Abstract

In standard finite element algorithms, the local stability conditions are not accounted for in the formulation of the tangent stiffness matrix. As a result, the loss of the local stability is not adequately related to the onset of the global instability. The phenomenon typically arises with material-type localizations, such as shear bands and plastic hinges. This paper addresses the problem in the context of the planar, finite-strain, rate-independent, materially non-linear beam theory, although the proposed technology is in principle not limited to beam structures. A weak formulation of Reissner's finite-strain beam theory is first presented, where the pseudocurvature of the deformed axis is the only unknown function. We further derive the local stability conditions for the large deformation case, and suggest various possible combinations of the interpolation and numerical integration schemes that trigger the simultaneous loss of the local and global instabilities of a statically determined beam. For practical applications, we advice on a procedure that uses a special numerical integration rule, where interpolation nodes and integration points are equal in number, but not in locations, except for the point of the local instability, where the interpolation node and the integration point coalesce. Provided that the point of instability is an end-point of the beam-a condition often met in engineering practice-the procedure simplifies substantially; one of such algorithms uses the combination of the Lagrangian interpolation and Lobatto's integration. The present paper uses the Galerkin finite element discretization, but a conceptually similar technology could be extended to other discretization methods.

Keywords

References

  1. Antman, S.S., and Rosenfeld, G. (1978), "Global behavior of buckled states of non-linearly elastic rods", SIAM Review, 20, 513-566. https://doi.org/10.1137/1020069
  2. Banovec, J. (1986), "Geometric and material nonlinear analysis of planar frames", Ph. D. Thesis (in Slovenian), University of Ljubljana, Slovenia.
  3. Belytschko, T., Fish, J., and Engelmann, B.E. (1988), "A finite element with embedded localization zone", Comput. Meth. in Appl. Mech. and Eng., 70, 59-89. https://doi.org/10.1016/0045-7825(88)90180-6
  4. Belytschko, T., and Fish, J. (1989), "Embedded hinge lines for plate elements", Comput. Meth. in Appl. Mech. and Eng., 76, 67-86. https://doi.org/10.1016/0045-7825(89)90141-2
  5. Bergan, P.G. (1984), "Some aspects of interpolation and integration in nonlinear finite element analysis of reinforced concrete structures", Computer Aided Analysis and Design of Concrete Structures, Part 1, Damjanic' , F. et al., eds., Pineridge Press, Swansea, 301-316.
  6. Cichon' , C. (1984), "Large displacements in-plane analysis of elastic-plastic frames", Comput. and Struct., 19, 737-745. https://doi.org/10.1016/0045-7949(84)90173-1
  7. Cowper, G.R. (1966), "The shear coefficient in Timoshenko's beam theory", ASME J. Appl. Mech., 33, 335-340. https://doi.org/10.1115/1.3625046
  8. Crisfield, M.A. (1991, 1997), Non-linear Finite Element Analysis of Solids and Structures, Volumes 1 and 2, John Wiley Sons, Chichester.
  9. de Borst, R., Sluys, L.J., Mühlhaus, H.-B., and Pamin, P. (1993), "Fundamental issues in finite element analysis of localization of deformation", Eng. Comput., 10, 99-121. https://doi.org/10.1108/eb023897
  10. Fujii, F., and Okazawa, S. (1997), "Bypass, homotopy path and local iteration to compute the stability point", Struct. Eng. and Mech., 5, 577-586. https://doi.org/10.12989/sem.1997.5.5.577
  11. Hohn, F.E. (1973), Elementary Matrix Algebra, 3rd edn., The Macmillan Company, New York.
  12. Hsiao, K.M., Hou, F.Y., and Spiliopoulos, K.V. (1988), "Large displacement analysis of elasto-plastic frames", Comput. Meth. in Appl. Mech. and Eng., 28, 627-633.
  13. Hughes, T.J., and Pister, K.S. (1978), "Consistent linearization in mechanics of solids and structures", Comput. and Struct., 8, 391-397. https://doi.org/10.1016/0045-7949(78)90183-9
  14. Jelenic, G., and Saje, M. (1995), "A kinematically exact space finite strain beam model-Finite element formulation by generalized virtual work principle", Comput. Meth. in Appl. Mech. and Eng., 120, 131-161. https://doi.org/10.1016/0045-7825(94)00056-S
  15. Ortiz, M., Leroy, Y., and Needleman, A. (1987), "A finite element method for localized failure analysis", Comput. Meth. in Appl. Mech. and Eng., 61, 189-214. https://doi.org/10.1016/0045-7825(87)90004-1
  16. Planinc, I. (1998), "A quadratically convergent algorithms for the computation of stability points: the application of the determinant of the tangent stiffness matrix", Ph.D. Thesis (in Slovenian), University of Ljubljana, Slovenia.
  17. Planinc, I., and Saje, M. (1998), "A quadratically convergent algorithm for the computation of stability points: the application of the determinant of the tangent stiffness matrix", Comput. Meth. in Appl. Mech. and Eng., 169, 89-105.
  18. Reissner, E. (1972), "On one-dimensional finite-strain beam theory: the plane problem", J. Appl. Math. and Physics (ZAMP), 23, 795-804. https://doi.org/10.1007/BF01602645
  19. Reese, S., and Wriggers, P. (1997), "Material instabilities of an incompressible elastic cube under triaxial tension", Int. J. Solids and Struct., 34, 3433-3454. https://doi.org/10.1016/S0020-7683(96)00205-3
  20. Saje, M., Planinc, I., Turk, G., and Vratanar, B. (1997), "A kinematically exact finite element formulation of planar elastic-plastic frames", Comput. Meth. in Appl. Mech. and Eng., 144, 125-151. https://doi.org/10.1016/S0045-7825(96)01172-3
  21. Ting, T.C.T. (1996), "Positive definiteness of anisotropic elastic constants", Mathematics and Mechanics of Solids, 1, 301-314. https://doi.org/10.1177/108128659600100302
  22. Washizu, K. (1981), Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford.
  23. Waszczyszyn, Z., Cichon , C., and Radwanska, M. (1994), Stability of Structures by Finite Element Methods, Elsevier Science B. V., Amsterdam.
  24. Wriggers, P., and Simo, J.C. (1990), "A general procedure for the direct computation of turning and bifurcation points", Int. J. Numer. Meth. in Eng., 30, 155-176. https://doi.org/10.1002/nme.1620300110
  25. Wriggers, P., Wagner, W., and Miehe, C. (1988), "A quadratically convergent procedure for the calculation of stability points in finite element analysis", Comput. Meth. in Appl. Mech. and Eng., 70, 329-347. https://doi.org/10.1016/0045-7825(88)90024-2

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