Applications of Mathematics, Vol. 62, No. 6, pp. 661-677, 2017

Control variational method approach to bending and contact problems for Gao beam

Jitka Machalová, Horymír Netuka

Received June 23, 2017.   First published November 30, 2017.

Abstract:  This paper deals with a nonlinear beam model which was published by D. Y. Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.
Keywords:  nonlinear beam; elastic foundation; contact problem; normal compliance condition; control variational method; finite element method
Classification MSC:  49J15, 49S05, 65K10, 74K10, 74M15
DOI:  10.21136/AM.2017.0168-17

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Affiliations:   Jitka Machalová, Horymír Netuka, Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail:,

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