Global–Local-Distortional Vibration of Thin-Walled Rectangular Multi-Cell Beams

Abstract

This paper presents the results of an investigation concerning the free vibration behavior (undamped natural frequencies and vibration mode shapes) of thin-walled beams with rectangular multi-cell cross-section (assemblies of parallel rectangular cells in a single direction). Besides local (plate-type) and global (flexural, torsional and extensional) vibration modes, attention is paid to the relatively less-known distortional vibration modes, which involve cross-section out-of-plane (warping) and in-plane deformation, including displacements of the wall intersections. A computationally efficient semi-analytical Generalized Beam Theory (GBT) approach is employed to obtain insight into the mechanics of the problem. In particular, the intrinsic modal decomposition features of GBT — the fact that the beam is described using a hierarchical set of relevant cross-section deformation modes — are exploited to identify and categorize the most relevant vibration modes and deformation mode couplings.

A new approach to the identification of distortion modes of thin-walled structures based on plate/shell theory

“…where H denotes the Hamiltonian, L a =T-U-U p is the Lagrangian, and t 1 and t 2 are boundary values of time. Substituting Equations (5), (6) and 7into (8) yields…”

“…To support this, several trends of enhancing beam theories have been developed in the last decades: asymptotical methods, such as variational asymptotical sectional analyses by Yu and Hodges [3], expansion of the beam displacement field through Taylor series [4] and refinement of the classic thin-walled theories by Vlassov (in particular the theory of Kim et al [5]). In engineering, other formulations are deduced considering specific structural behaviors, such as the multi-cell distortion (Gonçalves et al [6]) and warping due to shear-lag effects (Chen et al [7]). Moreover, some studies are focused on physically meaningful section modes [8].…”

A Simplified Approach to Identify Sectional Deformation Modes of Thin-Walled Beams with Prismatic Cross-Sections

“…where N and X represent the shape function matrix and the nodal generalized displacement vector, respectively. The submatrices in Equation (12) are written as follows:…”

GBT-Based Buckling Analysis Using the Exact Element Method

“…For the discretisation considered (involving 15 nodes), a total of = 3 × 15 = 45 deformation modes are obtained  the in-plane or out-of-plane configurations of the most relevant ones are depicted in Figure 3. They comprise: (i) the four classical rigid-body (or "global") modes (axial extension (1), major-and minor-axis bending (2-3) and torsion (4)), (ii) two distortional modes, associated with quasi-rigid body flange-lip motions (5-6), (iii) a sequence of local modes, involving transverse plate bending with increasing curvature (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), (iv) five global shear modes (18)(19)(20)(21)(22), consisting of the warping components of the Vlasov modes 2-6, (v) a set of local shear modes, (23)(24)(25)(26)(27)(28)(29)(30)(31), (vi) five global transverse extension modes (32)(33)(34)(35)(36) and (vii) the local transverse extension modes (37-45).…”

“…Recent progress concerning the cross-section analysis has made GBT applicable in the context of members exhibiting arbitrary flat-walled cross-sections [6][7] or circular/elliptical tubular cross-sections [8][9][10]. Concerning the member analysis, formulations/studies have been reported for various types of structural analysis, namely first-order [11,12], buckling [3,[13][14][15][16], vibration [17][18][19], post-buckling [20][21][22] and dynamic [23] analyses involving elastic members (mostly), frames and trusses. Recently, the second version of GBTUL [24], a GBT-based freeware code which performs linear buckling and vibration analyses of general thin-walled bars, has been released online [25].…”