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