Analysis of nonlinear Timoshenko–Ehrenfest beam problems with von Kármán nonlinearity using the Theory of Functional Connections

Abstract

In this paper, the Theory of Functional Connections (TFC) is used to analyze static beams, accounting for the von Kármán nonlinearity and using the Timoshenko–Ehrenfest beam theory. The authors extend their earlier framework on linear beam bending problems to nonlinear bending problems using TFC. The TFC results and performance parameters are then compared to those of the Finite Element Method (FEM) to both validate the TFC solutions and compare computational efficiencies. In addition, this paper also focuses on the benefits of using TFC over FEM for stress analysis of beam bending problems. Also, a TFC methodology to solve buckling and free vibration problems for the linearized Timoshenko–Ehrenfest beam equations is introduced and validated. The results within this paper suggest that for most static beam bending problems, TFC provides more accurate solutions in terms of the residuals of the differential equations and a faster solution time when compared to the FEM using linear or quadratic approximations. Also, TFC has the added benefit of calculating stress fields that are continuous and smooth everywhere within the domain of the beam while FEM is limited to a piece-wise continuous stress field.