Structural Engineering and Mechanics

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Effective modeling of beams with shear deformations on elastic foundation

  • Gendy, A.S. (Structural Engineering Department, Cairo University) ;
  • Saleeb, A.F. (Civil Engineering Department, The University of Akron)
  • Published : 1999.12.25
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Abstract

Being a significant mode of deformation, shear effect in addition to the other modes of stretching and bending have been considered to develop two finite element models for the analysis of beams on elastic foundation. The first beam model is developed utilizing the differential-equation approach; in which the complex variables obtained from the solution of the differential equations are used as interpolation functions for the displacement field in this beam element. A single element is sufficient to exactly represent a continuous part of a beam on Winkler foundation for cases involving end-loadings, thus providing a benchmark solution to validate the other model developed. The second beam model is developed utilizing the hybrid-mixed formulation, i.e., Hellinger-Reissner variational principle; in which both displacement and stress fields for the beam as well as the foundation are approxmated separately in order to eliminate the well-known phenomenon of shear locking, as well as the newly-identified problem of "foundation-locking" that can arise in cases involving foundations with extreme rigidities. This latter model is versatile and indented for utilization in general applications; i.e., for thin-thick beams, general loadings, and a wide variation of the underlying foundation rigidity with respect to beam stiffness. A set of numerical examples are given to demonstrate and assess the performance of the developed beam models in practical applications involving shear deformation effect.

Keywords

References

  1. Bathe, K.J. (1982), Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ.
  2. Bowles, J.E. (1974), Analysis and Computer Methods in Foundation Engineering, McGraw-Hill Co., Inc., New York, N.Y., 147-186, 1974.
  3. Bowles, J.E. (1977), Foundation Analysis and Design, McGraw-Hill Co., Inc., New York, 2nd. Edn., 276-285.
  4. Cheung, Y.K. and Nag, D.K. (1968), "Plates and beams on elastic foundation-linear and nonlinear behavior" , Geotechnique, 18, 250-260. https://doi.org/10.1680/geot.1968.18.2.250
  5. Cook, R.D. (1982), Concepts and Applications of Finite Element Analysis, Wiley, New York, 2nd. Edn.
  6. Eisenberger, M. and Yankelevsky, D.Z. (1985), "Exact stiffness matrix for beams on elastic foundation", Comput. & Struct., 21(6), 1355-1359. https://doi.org/10.1016/0045-7949(85)90189-0
  7. Gendy, A.S., Saleeb, A.F. and Chang, T.Y.P. (1992), "Generalized thin-walled beam models for flexural-torsional analysis", Comput. & struct., 42(4), 531-550. https://doi.org/10.1016/0045-7949(92)90120-O
  8. Gendy, A.S. and Saleeb, A.F. (1994), "Generalized mixed finite element model for pre- and post-quasistatic buckling response of framed structures" , Int. J. Num. Meth. Eng., 37, 297-322. https://doi.org/10.1002/nme.1620370208
  9. Hughes, T.J.R, Cohen, M. and Haroun, M. (1978), "Reduced and selective integration techniques in the finite element analysis of plates" , Nucl. Engng. Design, 46, 303-322. https://doi.org/10.1016/0029-5493(78)90017-1
  10. Malkus, D.S. and Hughes, T.J.R (1978), "Mixed finite element method - reduced and selective integration techniques: A unification of concepts" , Comput. Meths. Appl. Mech. Engng., 15, 63-81. https://doi.org/10.1016/0045-7825(78)90005-1
  11. Miranda, C. and Nair, K. (1966), "Finite elements on elastic foundation" , J. Struct. Div., ASCE 92(ST 2), 131-142.
  12. Noor, A.K. and Peters, J.M. (1981), "Mixed model and redeced/selective integration displacement models for nonlinear analysis of curved beams", Int. J. Numer. Meths. Engng., 17, 615-631. https://doi.org/10.1002/nme.1620170409
  13. Pian, T.H.H. (1982), "Recent advances in hybrid/mixed finite elements", Proceedings of the International Conference of Finite Element Methods, Shanghai, China, 1-19.
  14. Pian, T.H.H. and Chen, D.P. (1982), "Alternative ways for formulation of hybrid stress elements" , Int. J. Numer. Meths. Engng., 18, 1679-1684. https://doi.org/10.1002/nme.1620181107
  15. Pian, T.H.H. and Chen, D.P. (1983), "On the suppression of zero energy deformation modes", Int. J. Numer. Meths. Engng., 19, 1741-1752. https://doi.org/10.1002/nme.1620191202
  16. Pian, T.H.H. (1985), "Finite element based on consistently assumed stresses and displacements", J. of Finite Elements in Analysis and Design, 1, 131-140. https://doi.org/10.1016/0168-874X(85)90023-X
  17. Stolarski, H. and Belytschko, T. (1983), "Shear and membrane locking in 0 curved C-elements", Comput. Meths. Appl. Mech. Engng., 41, 172-176.
  18. Timoshenko, S. and Goodier, J.N. (1970), Theory of Elasticity, McGraw-Hill Co., Inc., New York, 3rd. Edn., 97-104.
  19. Timoshenko, S.P. (1956), Strength of Material, Part 2, 3 rd Edn., Van Nostrand Reinhold, New York.
  20. Ting, B.Y. (1982), "Finite beam on elastic foundation with restraints" , J. Struct. Div., ASCE, 108(ST3), 611-621.
  21. Ting, B.Y. and Mockry, E.F. (1984), "Beam on elastic foundation finite element", J. Struct. Engng., ASCE, 110(10), 2324-2339. https://doi.org/10.1061/(ASCE)0733-9445(1984)110:10(2324)
  22. Tong, P. and Rossettos, J.N. (1977), Finite Element Method: Basic Techniques and Implementation, Massachusetts Institute of Technology Press, Cambridge, Mass., 212-214.
  23. Washizu, K. (1982), Variational Method in Elasticity and Plasticity, Perganon Press, Oxford, 3rd. Edn..

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