Solution of Contact Problems for Nonlinear Gao Beam and Obstacle

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.

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

“…Details on numerical solution of the Gao beam problem based on the control variational method including sensitivity analysis can be found in [18]. We recapitulate briefly its main principles.…”

“…Later, it was applied to solve contact problems with the Euler-Bernoulli beam [25], [5]. Recently, CVM was used for solution of contact problems with the Gao beam [18] and [19].…”

“…Unlike [18] and [19], we transform the original beam model into the optimal control problem, which is convex and smooth and the related set of admissible control variables coincides with a linear space. Numerical solution of such a problem is therefore much simpler.…”

Solution of contact problems for Gao beam and elastic foundation

“…The aim of this paper is to generalize this idea for nonlinear beam and arbitrary boundary conditions. We are also going to extend considerably the results of our previous article [11] but only the convex case will be considered here.…”

On SDP method for solving canonical dual problem in post buckling of large deformed elastic beam

“…By equation (2) we know that the axial deformation could be relatively large, while the nonconvexity of the total potential shows that this nonlinear beam model can be used for studying both pre and post-buckling problems [4,30]. Recently, the Gao beam model has been generalized for many real-world applications in engineering and sciences [1,2,3,22,23,25,26,28].…”