Unilateral elastic subsoil of Winkler’s type: Semi-coercive beam problem. (English) Zbl 1199.49051

Sysala, Stanislav

Appl. Math., Praha 53, No. 4, 347-379 (2008).

Summary: 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 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.

MSC:

49K15	Optimality conditions for problems involving ordinary differential equations
49M25	Discrete approximations in optimal control
65D30	Numerical integration
74B20	Nonlinear elasticity
74K10	Rods (beams, columns, shafts, arches, rings, etc.)
74S05	Finite element methods applied to problems in solid mechanics

Keywords:

non-linear subsoil of Winkler’s type; semi-coercive beam problem; existence; uniqueness

References:

[1]	R. A. Adams: Sobolev Spaces. Academic Press, New York, 1975.
[2]	P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.
[3]	K. Frydrýšek: Beams and Frames on Elastic Foundation 1. VŠB-TU, Ostrava, 2006. (In Czech.)
[4]	S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier, Amsterdam, 1980.
[5]	I. Hlaváček, J. Lovíšek: Optimal control of semi-coercive variational inequalities with application to optimal design of beams and plates. Zeitschr. Angew. Math. Mech. 78 (1998), 405–417. · Zbl 0910.49008 · doi:10.1002/(SICI)1521-4001(199806)78:6<405::AID-ZAMM405>3.0.CO;2-Z
[6]	J. V. Horák, H. Netuka: Mathematical models of non-linear subsoils of Winkler’s type. In: Proc. 21st Conference Computational Mechanics 2005. ZČU, Plzeň, pp. 235–242, 431–438. (In Czech.)
[7]	J. V. Horák, S. Sysala: Bending of plate on unilateral elastic subsoil: Mathematical modelling. Proceedings 13th Seminar Modern Mathematical Methods in Engineering. JČMF, VŠB-TU, Ostrava, 2004, pp. 51–55. (In Czech.)
[8]	N. Kikuchi, J. T. Oden: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, 1988. · Zbl 0685.73002
[9]	J. Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague, 1967. (In French.)
[10]	J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam, 1981.
[11]	K. Rektorys: Variationsmethoden in Mathematik, Physik und Technik. Carl Hanser Verlag, München-Wien, 1984. (In German.) · Zbl 0568.49001
[12]	I. Svobodová: Cone decomposition of Hilbert space: Existence of weak solution for 1D problem. http://home1.vsb.cz/\(\sim\)svo19 (2005).
[13]	S. Sysala: 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, pp. 85–86. (In Czech.)
[14]	S. Sysala: 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, pp. 95–100.