Unilateral elastic subsoil of Winkler's type: Semi-coercive beam problem

Stanislav Sysala

Applications of Mathematics (2008)

  • Volume: 53, Issue: 4, page 347-379
  • ISSN: 0862-7940

Abstract

top
The mathematical model of a beam on a unilateral elastic subsoil of Winkler's type and with free ends is considered. Such a problem is non-linear and semi-coercive. The additional assumptions on the beam load ensuring the problem solvability are formulated and the existence, the uniqueness of the solution and the continuous dependence on the data are proved. The cases for which the solutions need not be stable with respect to the small changes of the load are described. The problem is approximated by the finite element method and the relation between the original problem and the family of approximated problems is analyzed. The error estimates are derived in dependence on the smoothness of the solution, the load and the discretization parameter of the partition.

How to cite

top

Sysala, Stanislav. "Unilateral elastic subsoil of Winkler's type: Semi-coercive beam problem." Applications of Mathematics 53.4 (2008): 347-379. <http://eudml.org/doc/37788>.

@article{Sysala2008,
abstract = {The mathematical model of a beam on a unilateral elastic subsoil of Winkler's type and with free ends is considered. Such a problem is non-linear and semi-coercive. The additional assumptions on the beam load ensuring the problem solvability are formulated and the existence, the uniqueness of the solution and the continuous dependence on the data are proved. The cases for which the solutions need not be stable with respect to the small changes of the load are described. The problem is approximated by the finite element method and the relation between the original problem and the family of approximated problems is analyzed. The error estimates are derived in dependence on the smoothness of the solution, the load and the discretization parameter of the partition.},
author = {Sysala, Stanislav},
journal = {Applications of Mathematics},
keywords = {non-linear subsoil of Winkler's type; semi-coercive beam problem; existence; uniqueness; continuous dependence on data; finite element method; numerical quadrature; non-linear subsoil of Winkler's type; semi-coercive beam problem; existence; uniqueness},
language = {eng},
number = {4},
pages = {347-379},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Unilateral elastic subsoil of Winkler's type: Semi-coercive beam problem},
url = {http://eudml.org/doc/37788},
volume = {53},
year = {2008},
}

TY - JOUR
AU - Sysala, Stanislav
TI - Unilateral elastic subsoil of Winkler's type: Semi-coercive beam problem
JO - Applications of Mathematics
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 4
SP - 347
EP - 379
AB - The mathematical model of a beam on a unilateral elastic subsoil of Winkler's type and with free ends is considered. Such a problem is non-linear and semi-coercive. The additional assumptions on the beam load ensuring the problem solvability are formulated and the existence, the uniqueness of the solution and the continuous dependence on the data are proved. The cases for which the solutions need not be stable with respect to the small changes of the load are described. The problem is approximated by the finite element method and the relation between the original problem and the family of approximated problems is analyzed. The error estimates are derived in dependence on the smoothness of the solution, the load and the discretization parameter of the partition.
LA - eng
KW - non-linear subsoil of Winkler's type; semi-coercive beam problem; existence; uniqueness; continuous dependence on data; finite element method; numerical quadrature; non-linear subsoil of Winkler's type; semi-coercive beam problem; existence; uniqueness
UR - http://eudml.org/doc/37788
ER -

References

top
  1. Adams, R. A., Sobolev Spaces, Academic Press New York (1975). (1975) Zbl0314.46030MR0450957
  2. Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland Amsterdam (1978). (1978) Zbl0383.65058MR0520174
  3. Frydrýšek, K., Beams and Frames on Elastic Foundation 1, VŠB-TU Ostrava (2006), Czech. (2006) 
  4. Fučík, S., Kufner, A., Nonlinear Differential Equations, Elsevier Amsterdam (1980). (1980) MR0558764
  5. Hlaváček, I., Lovíšek, J., 10.1002/(SICI)1521-4001(199806)78:6<405::AID-ZAMM405>3.0.CO;2-Z, Zeitschr. Angew. Math. Mech. 78 (1998), 405-417. (1998) MR1632693DOI10.1002/(SICI)1521-4001(199806)78:6<405::AID-ZAMM405>3.0.CO;2-Z
  6. Horák, J. V., Netuka, H., Mathematical models of non-linear subsoils of Winkler's type, In: Proc. 21st Conference Computational Mechanics 2005 ZČU Plzeň 235-242, 431-438 Czech. 
  7. Horák, J. V., Sysala, S., Bending of plate on unilateral elastic subsoil: Mathematical modelling, Proceedings 13th Seminar Modern Mathematical Methods in Engineering JČMF, VŠB-TU Ostrava (2004), 51-55 Czech. (2004) 
  8. Kikuchi, N., Oden, J. T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Philadelphia (1988). (1988) Zbl0685.73002MR0961258
  9. Nečas, J., Les Méthodes Directes en Théorie des Équations Elliptiques, Academia Prague (1967), French. (1967) MR0227584
  10. Nečas, J., Hlaváček, I., Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier Amsterdam (1981). (1981) MR0600655
  11. Rektorys, K., Variationsmethoden in Mathematik, Physik und Technik, Carl Hanser Verlag München-Wien (1984), German. (1984) Zbl0568.49001MR0799323
  12. Svobodová, I., Cone decomposition of Hilbert space: Existence of weak solution for 1D problem, http://home1.vsb.cz/ svo19 (2005). (2005) 
  13. Sysala, S., Problem with unilateral elastic subsoil of Winkler's type: Numerical methods, In: Proc 8th International Scientific Conference Applied Mechanics 2006, FAV ZČU Plzeň, Srní (2006), 85-86 Czech. (2006) MR2433726
  14. Sysala, S., On a dual method to a beam problem with a unilateral elastic subsoil of Winkler's type, In: Proc. Seminar on Numerical Analysis---SNA'07 Institute of Geonics AS CR Ostrava (2007), 95-100. (2007) MR2433726

NotesEmbed ?

top

You must be logged in to post comments.