Dynamic Finite Element Analysis of Bending-Torsion Coupled Beams Subjected to Combined Axial Load and End Moment

Abstract

The dynamic analysis of prestressed, bending-torsion coupled beams is revisited. The axially loaded beam is assumed to be slender, isotropic, homogeneous, and linearly elastic, exhibiting coupled flexural-torsional displacement caused by the end moment. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the vibration and stability of such beams are explored. Using the closed-form solutions of the uncoupled portions of the governing equations as the basis functions of approximation space, the dynamic, frequency-dependent, interpolation functions are developed, which are then used in conjunction with the weighted residual method to develop the Dynamic Finite Element (DFE) of the system. Having implemented the DFE in a MATLAB-based code, the resulting nonlinear eigenvalue problem is then solved to determine the coupled natural frequencies of illustrative beam examples, subjected to various boundary and load conditions. The proposed method is validated against limited available experimental and analytical data, those obtained from an in-house conventional Finite Element Method (FEM) code and FEM-based commercial software (ANSYS). In comparison with FEM, the DFE exhibits higher convergence rates and in the absence of end moment it produces exact results. Buckling analysis is also carried out to determine the critical end moment and compressive force for various load combinations.

New Frequency‐Dependent Trigonometric Interpolation Functions for the Dynamic Finite Element Analysis of Thin Rectangular Plates

“…Also, it is important to emphasize that these new DFE shape functions, which have not been presented for a thin plate element, are unique in their own right and are entirely different from the DFE shape functions for a beam [17][18][19][20] presented by Hashemi and his coworkers in the past. Although developing the QDFE method for a thin rectangular plate, currently in the final stages of its development, is beyond the scope of this paper, for the readers' interest it could be stated that having derived the frequency-dependent, trigonometric shape functions, developing the Quasi-Exact Dynamic Finite Element method now becomes a four-step process, first of which is the application of further integration by parts to the discretized form of the governing differential equation from the conventional FEM formulation.…”

“…Since its inception, the DFE method has been used in beam, beam-like, and blade vibration modelling and analysis. Hashemi and his coworkers (see, e.g., [18][19][20]) have extensively studied the free vibration of various beam configurations, such as isotropic, sandwich, composite, and thin-walled beams subjected to diverse loading configurations, using the Dynamic Finite Element (DFE) method. The results have consistently shown the DFE method to have a higher accuracy and rate of convergence compared to conventional FEM owing to the increased efficiency of the frequency-dependent, trigonometric shape functions based on the exact solutions to the governing equation which the DFE method employs.…”

“…In one study [19], application of the DFE formulation to the free vibration analysis of a sandwich beam resulted in a quasi-exact formulation. In a more recent study [20], investigating the vibration of bending-torsion coupled beams subjected to axial load and end moment, it was found that the DFE method required eight times less elements than conventional FEM to produce results with a percent difference of less than one percent. However, the DFE method has not yet been fully extended to model plate vibration problems.…”

A Finite Element Formulation for Bending‐Torsion Coupled Vibration Analysis of Delaminated Beams under Combined Axial Load and End Moment

“…Shock and Vibration e remaining terms in (Δ), corresponding to delamination edges (i.e., x � x 1 and x � x 2 ), can be resolved by applying the continuity conditions (23) and (25), with the following as a result:…”

“…In two recent publications, the authors exploited the classical [24] and frequency-dependent (dynamic) FEM [25] formulations for the free vibration of intact prestressed beams. e free vibration/buckling of intact and delaminated axially-loaded beams has also been investigated before (e.g., [17][18][19][20][21][22][23]).…”