Shock and Vibration

Volume 2015 (2015), Article ID 471270, 12 pages

http://dx.doi.org/10.1155/2015/471270

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

Department of Aerospace Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

Received 31 March 2015; Accepted 27 May 2015

Academic Editor: Matteo Aureli

Copyright © 2015 Mir Tahmaseb Kashani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Many terrestrial, mechanical, and aerospace structures can be modeled as beams or assemblies of beams, and, therefore, modelling and analysis of such structural elements have been the subject of numerous investigations. Depending on their applications, diverse geometries, loadings, and boundary conditions arise in the structural modeling, leading to a variety of problems. The dynamic, buckling, and vibrational analyses of diverse beam configurations, represented by different geometries and loading scenarios, governed by pertinent theories, have been investigated and reported in the literature. The vibrational analysis of prestressed beams has been the subject of several studies. Neogy and Murthy [1] carried out one of the earliest studies in this area and found first natural frequency of an axially loaded column for two different boundary conditions: pinned-pinned and clamped-clamped. Krishn et al. [2], using Rayleigh-Ritz principle, introduced an iterative approximate solution method. Gellert and Gluck [3] investigated the effect of applied axial force on the lateral natural frequencies of a clamped-free beam with transverse restraint. Pilkington and Carr [4] introduced an approximate, noniterative solution for the frequencies of beams subjected to end moment and distributed axial force. Wang et al. [5] used Galerkin’s formulation, while Tarnai [6] exploited the more generalized variational technique to investigate the lateral buckling of beams hung at both ends. Later, Jensen and Crawley [7] studied the frequency determination techniques for cases where coupling is caused by warping of composite laminate. They also compared the results of Rayleigh-Ritz and partial Ritz methods with their experimental results. Mohsin and Sadek [8] and Banerjee and Fisher [9] implemented Dynamic Stiffness Matrix (DSM) method to find natural frequencies of an axially loaded coupled beam, while Jun et al. [10] used DSM for vibrational analysis of a composite beam. The DSM method was first introduced by Kalousek [11] for an Euler-Bernoulli beam and ever since has been taken further by many researchers [9, 12–16].

With the advent of more powerful computers in recent years, there has been an increasing interest among researchers to use computational methods in structural stability and vibration analyses. This is mainly due to the fact that the experimental methods are expensive, require extensive testing and measuring techniques, and are limited in their scope of predictions. On the other hand, the analytical solutions are limited to special cases. The classical Finite Element Method (FEM), as the most popular computational technique in solid and structural mechanics, has been extensively utilized by researchers [17–20]. In FEM, fixed shape functions are used to express the field variables in terms of nodal values and to develop the element matrices. Because of their ease of manipulation, Hermite cubic shape functions are commonly used to express elements lateral displacement, resulting in an approximate solution including mass and static stiffness matrices.

In 1998, Hashemi [21] introduced a semianalytical Dynamic Finite Element (DFE) formulation, a hybrid approach that bridged the gap between DSM and FEM methods. Analogous to the conventional FEM, the DFE formulation is based on the general procedure of weighted residual method. However, in contrast to the FEM, the use of frequency-dependent trigonometric shape functions in DFE leads to a frequency-dependent (dynamic) stiffness matrix, which represents both inertia and stiffness properties of the element embedded in a single matrix. As a result, the DFE can be extended to more complex cases, for which a DSM cannot be developed. Since its inception, the DFE method has been extended to vibration analysis of various problems of beam-like structures [22, 23]. Hashemi and Richard (1999) [22] presented dynamic shape functions and a DFE for the vibration analysis of thin spinning beams. Hashemi and Richard (2000) [23] presented a DFE for the free vibration coupled bending-torsion beams and investigated the coupled bending-torsion natural frequencies of axially loaded beams and the DFE frequency results were validated against FEM and DSM [9] data and those available in the literature. When compared to the conventional FEM, the DFE generally exhibited much higher convergence rates, especially for higher modes of vibration.

Most of the above-mentioned investigations focused on either uncoupled lateral or geometric/materially coupled stability/vibration of beams. The presence of prestress in structural components can also significantly change the system’s stability, dynamic behavior, and response. Helicopter, propeller, compressor and turbine blades, aircraft wing, and rockets internal structure subjected to axial acceleration are some examples of such situations where, at the preliminary design stage, an axially loaded beam model is often used for the dynamic and stability analyses of the system. Also, a beam column with two planes of symmetry, connected through semirigid connections, loaded in the plane of greater bending rigidity by end moments, exhibits coupled torsional-lateral displacements (in the plane of smaller bending rigidity). The former configurations have been thoroughly investigated and reported in open literature. However, studies on the latter cases and the cases of coupled buckling and dynamic behavior of beams subjected to combined axial load and end moment as well as the coupling effects caused by end moments are scarce. Joshi and Suryanarayan [24] developed a closed-form analytical solution for vibrational analysis of a simple uniform beam subjected to both constant end moment and axial load. Later, they unified their solution for different boundary conditions [25] and subsequently developed a general iterative method for coupled flexural-torsional vibration of initially stressed beams [26]. More recently, the authors presented a comprehensive study of the coupled bending-torsion stability and vibration analysis of such elements using the conventional FEM, where the flexural and torsional displacements were expressed using cubic Hermite and linear interpolation functions, respectively [27].

In what follows, a Dynamic Finite Element (DFE) for the coupled flexural-torsional stability and vibration analyses of slender beams, subjected to combined axial force and end moment, is presented. Based on the previously developed applicable governing differential equations of motion, the weighted residual method and integration by parts are exploited to develop the weak integral form of the governing equations. The closed-form solutions of the differential equations governing uncoupled bending and torsional vibrations of an axially loaded beam are used as the basis functions of approximation space to derive the pertinent Dynamic (frequency-dependent) Trigonometric Shape Functions (DTSFs). Introducing the field variables, expressed in terms of the DTSFs and the nodal displacements, into the weak integral form of the governing equation followed by extensive mathematical manipulation leads to the element Dynamic Stiffness Matrix (DSM). The element matrices are then assembled and the boundary conditions are applied to form the system’s nonlinear eigenvalue problem, which is finally solved to extract natural frequencies and modes of the system or to evaluate the critical axial load/end moment. It is worth noting that the present DFE is applicable to the members composed of closed sections, with the torsional rigidity, , being very large compared to the warping rigidity, , and with ends free to warp, that is, state of uniform torsion, where the twist rate is constant along the span. However, the presented DFE formulation can also be extended to more complex configurations, such as thin-walled beams with closed or open cross sections, where torsion-related warping effects cannot be neglected.

#### 2. Theory

Consider a linearly elastic, homogeneous, isotropic slender beam subjected to two equal and opposite end moments, , about -axis and an axial load, (i.e., loaded in the plane of greater bending rigidity), undergoing linear coupled torsion and lateral vibrations along -axis. Figure 1 depicts the schematic of the problem, where , , and stand for the beam’s length, width, and height, respectively. Governing differential equations of motion can be developed by defining an infinitesimal element (refer to the authors’ earlier work [27]) and by using the following assumptions:(1)The beam is made of linearly elastic material.(2)The displacements are small.(3)The stresses induced are within the limit of proportionality.(4)The cross section of the beam has two axes of symmetry.(5)The cross-sectional dimensions of the beam are small compared to the span.(6)The transverse cross sections of the beam remain plane and normal to the neutral axis during bending, and the beam’s torsional rigidity () is assumed to be very large compared with its warping rigidity (), and the ends are free to warp, that is, state of uniform torsion.The governing differential equations for a prismatic Euler-Bernoulli beam ( and constant) subjected to static constant axial force () and end moment (), undergoing coupled flexural-torsional vibrations caused by end moment, is written as follows [27]:where stands for derivative with respect to and denotes derivative with respect to (time). The internal shear force, , bending moment, , and torsional torque, , are defined asAs can be observed from (1) and (2), the lateral and torsional displacements of the system are coupled by the end moments, . Exploiting the simple harmonic motion assumption, displacements, and , are written aswhere denotes the frequency and and are the amplitudes of flexural and torsional displacements, respectively. Substituting (4) into (1) and (2) leads to havingApplication of the Galerkin weighted residual formulation [19, 20] leads to the integral form of the differential equations (5) and (6), written aswhere and are the weighting functions associated with flexure and torsion, respectively. Performing integrations by parts twice on (7) and once on (8) leads to the following weak integral forms:The above expressions also satisfy the principle of virtual work (PVW),where is the total virtual work, is the internal virtual work, and is the external virtual work. For free vibrations, , and where and denote the components of the virtual work associated with flexure and torsion, respectively. It is worth noting that the bracketed boundary terms in (6) and (7) will vanish regardless of the type of boundary conditions applied, for example, zero displacements and slope, , and virtual displacements and slope, , at the fixed end (i.e., where the field variables are imposed) and zero resultant internal shear force, , bending moment, , and torsional torque, , at the free end (see expressions (3)).