Abstract
Contact problem for a large deformed beam with an elastic obstacle is formulated, analyzed, and numerically solved. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao, while the obstacle is considered as the elastic foundation of Winkler’s type in some distance under the beam. The problem is static without a friction and modeled either using Signorini conditions or by means of normal compliance contact conditions. The problems are then reformulated as optimal control problems which is useful both for theoretical aspects and for solution methods. Discretization is based on using the mixed finite element method with independent discretization and interpolations for foundation and beam elements. Numerical examples demonstrate usefulness of the presented solution method. Results for the nonlinear Gao beam are compared with results for the classical Euler-Bernoulli beam model.
Highlights
Contact problems belong to the most important industrial applications and contact problems for beams have their own significant position among them
This leads to the following description of the finite element space: Vh = {wh ∈ C1 ((0, L)) : VhKi ∈ P3 (Ki) ∀Ki, wh (0) (104)
We presented here quite promising methods to solve contact problem for the Gao beam and an obstacle, especially a deformable or a rigid foundation
Summary
Contact problems belong to the most important industrial applications and contact problems for beams have their own significant position among them. Nowadays it is well known that the contact of the elastic bodies is usually modeled using the Signorini conditions and followed by variational inequalities (see, e.g., [2, 3]). This approach is necessary only for the case when the obstacle is rigid as it has already been published in [4]. Using the so-called normal compliance condition (for more details see, e.g., [5] or [6]) for a deformable foundation we get description in the form of variational equation Afterwards we solve the resulting optimal control problem to obtain a solution of our initial contact problem
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