Closed-form solutions of non-uniform axially loaded beams using Lie symmetry analysis

Kundu, B and Ganguli, R (2020) Closed-form solutions of non-uniform axially loaded beams using Lie symmetry analysis. In: Acta Mechanica .

 PDF act_mec_2020.pdf - Published Version Restricted to Registered users only Download (1MB) | Request a copy
Official URL: https://dx.doi.org/10.1007/s00707-020-02773-w

Abstract

In this paper, the governing differential equation of a beam with axial force is studied using the Lie symmetry method. Considering the inhomogeneous beam and non-uniform axial load, the governing equation is a fourth-order linear partial differential equation with variable coefficients with no closed-form solution. We search for a favourable coordinate system where the governing equation has a simpler-form or a closed-form solution. A favourable coordinate transformation is found using the Lie transformation group method. The system of determining equations for the governing equation of a beam with non-uniform axial load is derived and then solved to find a favourable coordinate system dependent on the spatially variable stiffness, mass, and axial force. The class of non-uniform axially loaded beams which have a closed-form solution is determined. The fixed-free boundary condition is imposed to find the invariant closed-form solution. A comparison between the analytical solution derived by the Lie symmetry method and the numerical solution is presented. Lie symmetry analysis yields hitherto undiscovered closed-form solutions for non-uniform axially loaded beams. Â© 2020, Springer-Verlag GmbH Austria, part of Springer Nature.

Item Type: Journal Article Acta Mechanica Springer The copyright of this article belongs to Springer Axial flow; Axial loads; Boundary conditions; Numerical methods, Closed form solutions; Co-ordinate transformation; Free boundary conditions; Governing differential equations; Lie symmetry analysis; Lie symmetry methods; Linear partial differential equations; Variable coefficients, Algebra Division of Mechanical Sciences > Aerospace Engineering(Formerly Aeronautical Engineering) 17 Aug 2020 07:24 17 Aug 2020 07:24 http://eprints.iisc.ac.in/id/eprint/66329