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Control variational method approach to bending and contact problems for Gao beam. (English) Zbl 1458.74088
Summary: This paper deals with a nonlinear beam model which was published by D. Y. Gao [Mech. Res. Commun. 23, No. 1, 11–17 (1996; Zbl 0843.73042)]. 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.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74M15 Contact in solid mechanics
49J15 Existence theories for optimal control problems involving ordinary differential equations
49S05 Variational principles of physics
65K10 Numerical optimization and variational techniques
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