Numerical Solution of Bending of the Beam with Given Friction

Michaela Bobková and Lukáš Pospíšil
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Michaela Bobková: Department of Mathematics, Faculty of Civil Engineering, VSB-TU Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava, Czech Republic
Lukáš Pospíšil: Department of Mathematics, Faculty of Civil Engineering, VSB-TU Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava, Czech Republic

Mathematics, 2021, vol. 9, issue 8, 1-17

Abstract: We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

Keywords: bending of a beam; finite element method; sobolev spaces; linear elasticity; duality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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