Mathematical Problems in Engineering

Volume 2015, Article ID 940347, 12 pages

http://dx.doi.org/10.1155/2015/940347

## Higher-Order Hierarchical Models for the Free Vibration Analysis of Thin-Walled Beams

Luxembourg Institute of Science and Technology, 5 Avenue des Hauts-Fourneaux, 4362 Esch-sur-Alzette, Luxembourg

Received 30 July 2015; Accepted 6 September 2015

Academic Editor: Xiao-Qiao He

Copyright © 2015 G. Giunta and S. Belouettar. 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

This paper addresses a free vibration analysis of thin-walled isotropic beams via higher-order refined theories. The unknown kinematic variables are approximated along the beam cross section as a -order polynomial expansion, where is a free parameter of the formulation. The governing equations are derived via the dynamic version of the Principle of Virtual Displacements and are written in a unified form in terms of a “fundamental nucleus.” This latter does not depend upon order of expansion of the theory over the cross section. Analyses are carried out through a closed form, Navier-type solution. Simply supported, slender, and short beams are investigated. Besides “classical” modes (such as bending and torsion), several higher modes are investigated. Results are assessed toward three-dimensional finite element solutions. The numerical investigation shows that the proposed Unified Formulation yields accurate results as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam.

#### 1. Introduction

Many typical aeronautical and space structures involve light-weight, thin-walled, beam-like structures that operate in complex environments. The free vibration characteristics are of fundamental importance in the design of such structures. In particular, the dynamic behaviour of thin-walled beams is richer than that of solid prismatic beams since, besides classical modes (such as bending, torsion, and axial deformation), local higher modes are present. Furthermore, the number of higher modes occurring before the axial one increases as the length-to-side ratio decreases. For these reasons, the accurate modelling and analysis of thin-walled beams (as proposed in this paper) represent an interesting and up-to-date research topic.

A brief overview of recent works about the free vibration of thin-walled beams follows. Matsunaga [1] analysed the natural frequencies and buckling loads of simply supported beams subjected to initial axial forces. Thin rectangular cross sections were investigated. A bidimensional displacement field was assumed. Chen et al. [2] combined the state space method with the differential quadrature method to obtain a semianalytical method for the free vibration analysis of straight isotropic and orthotropic beams with rectangular cross sections. A discussion about properties of the natural frequencies and modes for a Timoshenko beam was presented by van Rensburg and van der Merwe [3]. Attarnejad et al. [4] introduced the basic displacement functions that are calculated solving the governing differential equations of transverse motion of Timoshenko’s beams by means of the power series method. These functions were applied to the free vibration analysis of nonprismatic beams. Gunda et al. [5] analysed the large amplitude free vibration of Timoshenko beams using a finite element formulation. Transverse shear and rotatory inertia were both accounted for. Different boundary conditions were investigated. Benamar et al. [6] presented a general model for large vibration of thin straight beams. Hamilton’s principle was used to obtain a set of nonlinear algebraic equations. Simply supported and clamped-clamped boundary conditions were investigated. Tanaka and Bercin [7] studied the natural frequencies of uniform thin-walled beams without cross sectional symmetry. A study on the free vibration of axially loaded slender thin-walled beams was presented by Jun et al. [8]. The effects of the warping stiffness and the axial force were included within Euler-Bernoulli’s beam theory. Chen and Hsiao [9] investigated these axial and torsional vibration modes of beams with a Z-shaped cross section. The governing equations were derived by the principle of virtual work. The same authors presented in [10] a finite element formulation for the coupled free vibration analysis of thin-walled beams with generic open cross sections. Duan [11] studied the nonlinear free vibrations of thin-walled curved beams with unsymmetric open cross sections via the finite element method. Vörös [12] accounted for the coupling between different vibration modes due to the eccentricity of the cross section as well as to steady-state lateral loads and internal stress resultants. The governing differential equations and boundary conditions were derived using a linearised theory of large rotations and small strains and the principle of virtual work. A finite element model with seven degrees of freedom per node was formulated. Ambrosini [13, 14] presented a numerical and experimental study on the natural frequencies of beams with unsymmetric thin-walled open cross sections. Vlasov’s theory was modified to include the effects of shear deformation, rotatory inertia, and variable cross section properties. An extension of the previous theory was proposed by de Borbón and Ambrosini [15] to investigate the natural frequencies of thin-walled beams under axial loads. Experimental tests were also carried out in order to verify the proposed theory. Arpaci et al. [16, 17] investigated the free vibrations of nonsymmetric thin-walled beams with open cross sections by solving exactly the bending and torsional dynamic equilibrium equations. The coupled bending-torsional behaviour of beams was studied by Banerjee [18] via the dynamic stiffness matrix method. S.-B. Kim and M.-Y. Kim [19] carried out a free vibration as well as a stability analysis of thin-walled tapered beams and frames by means of the finite element method. Zhou-Lian et al. [20] analysed the free vibrations in orthotropic membranes accounting for large deflections. A similar investigation was carried out by by Liu et al. [21] using the L-P perturbation method.

A free vibration analysis of thin-walled isotropic beams is addressed within this paper. The relevance of the proposed analysis is due to the fact that the thinness of the cross section elements results (besides classical bending, torsion, and axial deformation) in local higher modes and corresponding frequencies. The dynamics of thin-walled beams is richer than that of solid prismatic beams and the number of higher modes before the axial one increases as the length-to-side ratio decreases. Therefore, accurate higher-order theories are required. Furthermore, the considered analysis represents a severe test for the proposed models in order to outline both their merits and limitations. In this work, several hierarchical models as well as the classical theories are derived via a Unified Formulation (UF) that has been previously formulated for plates and shells (see Carrera [22] and Carrera and Giunta [23, 24]) and lately extended to beams with solid and thin-walled cross sections; see Carrera and Giunta [25], Carrera et al. [26], and Giunta et al. [27]. Within this UF, the a priori approximation of the displacement field is written in a compact form. The governing equations are derived through the Principle of Virtual Displacements in terms of a “fundamental nucleus.” This nucleus is an invariant of the formulation in the sense that it does not depend upon the theory order of expansion. As a result, a very broad set of beam models that account for transverse shear deformation and cross section in- and out-of-plane warping (although a warping function is not explicitly assumed) can be obtained. Governing differential equations are solved via a Navier-type closed form solution. Open and closed thin-walled simply supported beams are investigated. Slender as well as short simply supported beams are investigated. Open and closed thin-walled cross sections are considered. The modes up to the axial one (which is not included for the sake of brevity) are all considered. The axial mode is disregarded since it is accurately predicted by classical Euler-Bernoulli’s and Timoshenko’s models. Results are compared with three-dimensional finite element models showing that the dynamic behaviour of thin-walled beam structures can be accurately predicted.

#### 2. Preliminaries

A beam is a structure whose axial extension () is predominant if compared to any other dimension orthogonal to it. The cross section () is identified by intersecting the beam with planes that are orthogonal to its axis. A Cartesian reference system is adopted: - and -axes are two orthogonal directions laying on . The coordinate is coincident to the axis of the beam. Cross sections that are obtained by the union of rectangular subdomains withare considered; see Figure 1. Points are the coordinates of the corner points of a subdomain. Through the paper, superscript “” represents a cross section subdomain index. The cross section is considered to be constant along . The displacement field is in which , , and are the displacement components along -, -, and -axis, respectively, and is the time variable. Superscript “” represents the transposition operator. Stress, , and strain, , vectors are grouped into vectors , that lay on the cross section: and , laying on planes orthogonal to : The strain-displacement geometrical relations are Subscripts “,” “,” and “,” when preceded by comma, represent derivation versus the corresponding spatial coordinate. A compact vectorial notation can be adopted for (6): where , , and are the following differential matrix operators: being the unit matrix.