Analysis of Timoshenko–Ehrenfest beam problems using the Theory of Functional Connections

Abstract

The numerical solution of boundary value problems in solid mechanics is dominated by the Finite Element Method (FEM). The present study provides an alternative numerical approach using the Theory of Functional Connections (TFC) for solving problems of this type. Here TFC is used to solve multiple beam problems using the Timoshenko–Ehrenfest beam theory which accounts for transverse shear strain. The results obtained are then compared to those obtained with both the FEM and the analytical approaches. For cases where the analytical solution is continuous and smooth, TFC out performs FEM both in terms of the L2 norm of the residuals and the solution time. When the analytical solution is continuous and smooth, TFC results in an L2 norm on the order of machine-error level precision. However, when the higher order derivatives of the analytical solution are discontinuous (i.e. when the applied load is discontinuous), TFC under performs compared to FEM. This is due to TFC approximating the solution by continuous functions when the true solution is piece-wise continuous. In terms of computational expense, TFC consistently solved the differential equations faster than FEM across each type of problem analyzed in this paper.