Structural Engineering and Mechanics

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Analysis of slender structural elements under unilateral contact constraints

  • Published : 2001.07.25
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Abstract

A numerical methodology is presented in this paper for the geometrically non-linear analysis of slender uni-dimensional structural elements under unilateral contact constraints. The finite element method together with an updated Lagrangian formulation is used to study the structural system. The unilateral constraints are imposed by tensionless supports or foundations. At each load step, in order to obtain the contact regions, the equilibrium equations are linearized and the contact problem is treated directly as a minimisation problem with inequality constraints, resulting in a linear complementarity problem (LCP). After the resulting LCP is solved by Lemke's pivoting algorithm, the contact regions are identified and the Newton-Raphson method is used together with path following methods to obtain the new contact forces and equilibrium configurations. The proposed methodology is illustrated by two examples and the results are compared with numerical and experimental results found in literature.

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References

  1. Adan, N., Sheinman, I., and Altus, E. (1994), "Post-buckling behaviour of beams under contact constraints",ASME J. Appl. Mech., 61, 764-772. https://doi.org/10.1115/1.2901552
  2. Alves, R.V. (1995), "Non-linear elastic instability of space frames", D.Sc. Thesis, COPPE - Federal University ofRio de Janeiro (in Portuguese).
  3. Ascione, L., and Grimaldi, A. (1984), "Unilateral contact between a plate and an elastic foundation", Meccanica,19, 223-233. https://doi.org/10.1007/BF01743736
  4. Batoz, J.L., and Dhatt, G. (1979), "Incremental displacement algorithms for nonlinear problems", Int. J. Numer.Meth. Eng., 14, 1262-1267. https://doi.org/10.1002/nme.1620140811
  5. Belytschko, T., and Neal, M.O. (1991), "Contact-impact by the pinball algorithm with penalty and Lagrangianmethods", Int. J. Numer. Meth. Eng., 31, 547-572. https://doi.org/10.1002/nme.1620310309
  6. Björkman, C.G., Klarbring, A., Sjödin, B., Larsson, T., and Rönnqvist, M. (1995), "Sequential quadraticprogramming for non-linear elastic contact problems", Int. J. Numer. Meth. Eng., 38, 137-165. https://doi.org/10.1002/nme.1620380109
  7. Chan, S.L. (1988), "Geometric and material nonlinear analysis of beam-columns and frames using the minimumresidual displacement method", Int. J. Numer. Meth. Eng., 26, 2657-2669. https://doi.org/10.1002/nme.1620261206
  8. Crisfield, M.A. (1991), Non-Linear Finite Element Analysis of Solids and Structures, 1, John Wiley & Sons.
  9. Crisfield, M.A. (1997), Non-Linear Finite Element Analysis of Solids and Structures, 2, John Wiley & Sons.
  10. Endo, T., Oden, J.T., Becker, E.B., and Miller, T. (1984), "A numerical analysis of contact and limit-pointbehavior in a class of problems of finite elastic deformation", Comput. & Struct., 18(5), 899-910. https://doi.org/10.1016/0045-7949(84)90035-X
  11. Gierlinski, J.T., and Graves Smith, T.R. (1985), "A variable load iteration procedure for thin-walled structures",Comput. & Struct., 21, 1085-1094. https://doi.org/10.1016/0045-7949(85)90221-4
  12. Givoli, D., and Doukhovni, I. (1996), "Finite element-quadratic programming approach for contact problemswith geometrical nonlinearity", Comput. & Struct., 61(1), 31-41. https://doi.org/10.1016/0045-7949(96)00012-0
  13. Holmes, P., Domokos, G., Schmitt, J., and Szberenyi, I. (1999), "Constraint Euler buckling: An interplay of computation and analysis", Comput. Meth. Appl. Mech. Eng., 170, 175-207. https://doi.org/10.1016/S0045-7825(98)00194-7
  14. Joo, J.W., and Kwak, B.M. (1986), "Analysis and applications of elasto-plastic contact problems consideringlarge deformation", Comput. & Struct., 24(6), 953-961. https://doi.org/10.1016/0045-7949(86)90304-4
  15. Kerr, A.D. (1964), "Elastic and viscoelastic foundation models", J. Appl. Mech., ASME, 31, 491-498. https://doi.org/10.1115/1.3629667
  16. Klarbring, A. (1986), "A mathematical programming approach to three-dimensional contact problems withfriction", Comput. Meth. Appl. Mech. Eng., 58, 175-200. https://doi.org/10.1016/0045-7825(86)90095-2
  17. Koo, J.S., and Kwak, B.M. (1995), "Post-buckling analysis of nonfrictional contact problems using linearcomplementarity formulation", Comput. & Struct., 57(5), 783-794. https://doi.org/10.1016/0045-7949(95)00077-T
  18. Koo, J.S., and Kwak, B.M. (1996), "Post-buckling analysis with frictional contacts combining complementarityrelations and an arc-length method", Int. J. Numer. Meth. Eng., 39, 1161-1180. https://doi.org/10.1002/(SICI)1097-0207(19960415)39:7<1161::AID-NME898>3.0.CO;2-2
  19. Lemke, C.E. (1968), "On complementary pivot theory", Mathematics of Decision Sciences, Dantzig, G.B. andYenott, A.F., eds., 95-114.
  20. Luenberger, D.G. (1973), Introduction to Linear and Nonlinear Programming, Addison-Wesley PublishingCompany, Inc.
  21. Mottershead, J.E., Pascoe, S.K., and English, R.G. (1992), "A general finite element approach for contact stressanalysis", Int. J. Numer. Meth. Eng., 33, 765-779. https://doi.org/10.1002/nme.1620330407
  22. Nour-Omid, B., and Wriggers, P. (1986), "A two-level iteration method for solution of contact problems",Comput. Meth. Appl. Mech. and Eng., (54), 131-144.
  23. Pian, T.H.H., Balmer, H., and Bucciarelli, L.L. (1967), "Dynamic buckling of a circular ring constrained in arigid circular surface", Dynamic Stability of Structures, Pergamon Press, Oxford, 285-297.
  24. Silva, A.R.D., Silveira R.A.M., and Gonçalves P.B. (2001), "Numerical methods for analysis of plates ontensionless elastic foundations", Int. J. Solids and Structures, 38(10-13), 2083-2100. https://doi.org/10.1016/S0020-7683(00)00154-2
  25. Silveira, R.A.M. (1995), "Analysis of slender structural elements under unilateral contact constraints", D.Sc.Thesis, Catholic University, PUC-Rio (in Portuguese).
  26. Silveira, R.A.M., and Gonçalves, P.B. (2000), "A modal solution for the nonlinear analysis of arches and beamsunder contact constraints", Proc. of Int. Conf. on Computational Engineering & Sciences (ICES2K), S.N.Atluri and F.W. Brust, eds., Tech Science Press, Los Angeles, USA, 1822-1827.
  27. Simo, J.C., Wriggers, P., Schweizerhof, K.H., and Taylor, R.L. (1986), "Finite deformation post-buckling analysisinvolving inelasticity and contact constraints", Int. J. Numer. Meth. Eng., 23, 779-800. https://doi.org/10.1002/nme.1620230504
  28. Stein, E., and Wriggers, P. (1984), "Stability of rods with unilateral constraints, a finite element solution",Comput. & Struct., 19, 205-211. https://doi.org/10.1016/0045-7949(84)90220-7
  29. Sun, S.M., and Natori, M.C. (1996), "Numerical solution of large deformation problems involving stability andunilateral constraints", Comput. & Struct., 58(6), 1245-1260. https://doi.org/10.1016/0045-7949(95)00081-X
  30. Wriggers, P., and Imhof, M. (1993), "On the treatment of nonlinear unilateral contact problems", Archive ofAppl. Mech., 63, 116-129. https://doi.org/10.1007/BF00788917

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