Stability Analysis of Beams Subjected to Axial Loadand End Moment- A Dynamic Finite Element

A frequency-dependent, Dynamic Finite Element (DFE) formulation is developed to studythe buckling of a bending-torsion coupled, linearly elastic, slender pre-loaded beam subjectedto an axial load and end moment. The coupled governing differential equations are based onEuler-Bernoulli bending and St. Venant torsion beam theories. Exploiting the FEM procedure,and using the closed-form solutions of the uncoupled governing equations as the basisfunctions of approximation space, the relevant bending and torsion Dynamic (frequencydependent)Trigonometric Shape Function (DTSFs) are first developed. The weighted residualmethod, the principle of virtual work (PVW), and the Dynamic Shape Functions are usedto derive the Galerkin-type element Dynamic Stiffness Matrix, representing both elementstiffness and mass properties, embedded in one matrix. The assembly of element matricesand the application of boundary conditions then lead to a nonlinear Eigenproblem, which isthen solved to determine the system critical buckling loads and modes. The buckling of an illustrativeexample of a beam, with various classical boundary conditions, and subjected totensile/compressive axial loads, end moments, and combined loadings is then investigatedand the DFE results are validated against conventional Finite Element Method (FEM). Whencompared with the conventional FEM, the presented DFE is found to exhibit much higher convergence rates.

Keywords:

coupled flexural-torsional buckling , FEM , DSM , DFE