Membrane Shear Strain Research Articles

New unified family of GBT deformation modes for the analysis of thin-walled cylinders

Generalised Beam Theory assumes the displacement field of a thin-walled member as a linear combination of deformation modes. The main family of deformation modes is constructed based on two simplifying assumptions: null membrane shear strain and null membrane transverse extension. If these modes are not sufficient for an accurate representation of the structural behaviour, two other families of modes are derived (from the main family) for which the two simplifying assumptions are eliminated. In this paper, for the special case of thin-walled cylinders, all three families of deformation modes are replaced by a new unified family, for which the membrane shear strain and transverse extension are not neglected. The derivation of the proposed modes is based on the stress equilibrium equations, and it involves the trigonometric series representation of the longitudinal modal amplitudes. The advantages of the proposed modes are highlighted throughout several linear buckling analyses with various loading and boundary conditions.

First-order generalised beam theory for curved thin-walled members with circular axis

This paper presents a first-order Generalised Beam Theory (GBT) formulation for naturally curved thin-walled members with deformable cross-section, whose undeformed axis is a circular arc with no pre-twist. First, the strain-displacement relations for naturally curved thin-walled members are derived and it is shown how the classic GBT assumptions concerning the strains can be incorporated, namely: (i) Kirchhoff's thin-plate assumption, (ii) Vlasov's null membrane shear strain assumption and (iii) the null membrane transverse extension assumption. The equilibrium equations are obtained in terms of GBT modal matrices and stress resultants. It is demonstrated that, for the so-called “rigid-body” deformation modes (extension, bending and torsion), the GBT equations coincide with those of the Winkler (in-plane case) and Vlasov (out-of-plane case) theories. A standard displacement-based GBT finite element is used to solve a set of representative illustrative examples involving complex local-global deformation. It is shown that the proposed GBT formulation leads to extremely accurate results with a reduced number of DOF and that the GBT modal solution provides an in-depth insight into the structural behaviour of naturally curved members.

On the shear deformation modes in the framework of Generalized Beam Theory

This paper extends previous work concerning the determination of cross-section deformation modes in thin-walled members with arbitrary polygonal cross-section, in the framework of Generalized Beam Theory (Gonçalves et al., 2010 [1]). In particular, the paper addresses the so-called “natural shear deformation modes” (i.e. the deformation modes that involve non-null membrane shear strains and are independent of the cross-section discretization employed), which are relevant for capturing the behaviour of thin-walled members with complex multi-cell cross-sections undergoing torsion and/or distortion. The contributions of the paper are (i) the derivation of fundamental properties of the shear modes, (ii) the proposal of an efficient mode extraction procedure and (iii) the development of analytical results for several particular cases. In order to illustrate the application of the proposed mode extraction procedure and demonstrate the validity of the derived properties, several cross-sections are analyzed, including complex multi-cell tubes.

A complete dynamic approach to the Generalized Beam Theory cross-section analysis including extension and shear modes

A simple and efficient ‘complete dynamic approach’ is proposed, and named GBT-D, to evaluate a suitable basis of modes for the elastic analysis of thin-walled members in the framework of Generalized Beam Theory (GBT). The basis includes conventional and non-conventional modes, the latter accounting for transverse extension and membrane shear strain of the plate elements forming the cross-section, which identically vanish in the former set. The method relies on the solution of two distinct eigenvalue problems, governing the in-plane and the out-of-plane free oscillations of a segment of a thin-walled beam. Both the eigenvalue problems, differential in origin, and defined on a one-dimensional spatial domain, are transformed into an algebraic problem by means of a discretization carried out at the cross-section middle line. Numerical examples are then presented to outline the ease of use of the proposed method considering a single plate, an open cross-section and a partially closed one. Member analyses are also performed for the simplest boundary conditions, to validate the accuracy of the proposed GBT-D approach against finite element method results and analytical solutions, highlighting the importance in including the non-conventional modes.

GBT-based buckling mode decomposition from finite element analysis of thin-walled members

This paper presents an original method based on the Generalised Beam Theory (GBT) capable to decompose the elastic buckling modes from a shell finite element analysis (FEA) of an isotropic thin-walled member, into pure buckling modes of global, distorsional or local nature. The main novelty lies in using only the GBT cross-sectional deformation modes instead of member base mode shapes. The contribution of each pure buckling mode can be calculated, allowing a better understanding of the member post-buckling behaviour and strength reserve. Following the GBT classical assumptions, the membrane shear strains and transverse extensions are neglected. The modal participations obtained from FEA are in good agreement with the values calculated via classical GBT approach.

Buckling behaviour of elliptical cylindrical shells and tubes under compression

This paper presents several issues that characterize the buckling behaviour of elliptical cylindrical shells and tubes under compression. First, a formulation of Generalised Beam Theory (GBT) developed to analyse the elastic buckling behaviour of non-circular hollow section (NCHS) members is presented. Since the radius varies along the cross-section mid-line, the main concepts involved in the determination of the deformation modes are adapted to account for the specific aspects related to elliptical cross-section geometry. After that, two independent sets of fully orthogonal deformation modes are determined: (i) local-shell modes satisfying the null membrane shear strain but exhibiting transverse extension and (ii) shell-type modes satisfying both assumptions of null membrane shear strain and null transverse extension. In order to illustrate the application, capabilities and versatility of the formulation, the local and global buckling behaviour of elliptical hollow section (EHS) members subjected to compression is analysed. In particular, in-depth studies concerning the influence of member length on the variation of the critical load and corresponding buckling mode shape are presented. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed. The results show that short to intermediate length cylinders buckle mostly in local-shell modes, exhibiting only transverse extension, while intermediate length to long cylinders buckle mostly in shell-type modes (distortional and global modes), which are characterized by transverse bending and primary warping displacements. It is also shown that the present formulation is very efficient from the computational point of view since only three deformation modes (one local-shell, one distortional and one global) are required to evaluate the buckling behaviour of EHS cylinders for a wide range of lengths.