Elsevier

Thin-Walled Structures

Volume 40, Issue 9, September 2002, Pages 755-789
Thin-Walled Structures

First-order generalised beam theory for arbitrary orthotropic materials

https://doi.org/10.1016/S0263-8231(02)00025-3Get rights and content

Abstract

This paper presents the formulation of a Generalised Beam Theory (GBT) developed to analyse the structural behaviour of composite thin-walled members made of laminated plates and displaying arbitrary orthotropy. The main concepts and procedures involved in the available isotropic first-order GBT are revisited and adapted/modified to account for the specific aspects related to the member orthotropy. In particular, the orthotropic GBT fundamental equilibrium equations and corresponding boundary conditions are derived and their terms are physically interpreted, i.e., associated with the member mechanical properties. Moreover, different laminated plate material behaviours are dealt with and their influence on the GBT equations is investigated. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, a thin-walled orthotropic beam is analysed and the results obtained are thoroughly discussed.

Introduction

For a long time, the overwhelming majority of thin-walled structures were made of lightweight metals, namely cold-formed steel or aluminium. In the particular case of the construction industry, although the use of cold-formed steel can be traced back to the 1850s, thin-walled members only began to be regularly and extensively employed in buildings, with structural purposes, well into the twentieth century [1]. Moreover, since metals such as steel and aluminium are isotropic materials, it is not surprising that, up to the last couple of decades, most of the research activity dealing with the analysis and behaviour of thin-walled structures did not take into consideration the effect of orthotropy. One exception is related to members containing longitudinally and/or transversally stiffened walls, the behaviour of which has been modelled using the “equivalent orthotropic plate” concept (e.g., [2]). However, one should notice that this kind of orthotropy is due solely to the structure geometrical configuration, as all the materials involved are isotropic.

With the advent of the usage of thin-walled structures made of composite materials (materials obtained by the combination, on a macroscopic scale, of two or more basic materials), it became indispensable to account for the material orthotropy, which is due to the properties and orientation of the constituents (matrix, fibres, laminae, particles, etc.) and leads to mechanical characteristics quite different from those of isotropic materials [3]. The main advantage of composite materials, when compared with lightweight metals, lies in the fact that it is possible to achieve similar strength and/or stiffness values with considerably less weight, an aspect that makes them ideally suited for aeronautical and aerospace applications.

The structural use of composite materials started in the 1950s and has progressed steadily since then, mainly due to developments taking place in the aeronautical industry, which have led to a wide range of products extensively and routinely employed at present. However, the picture has been quite different in civil engineering, as composite material applications became significant only in the last few years, when their well known (i) structural efficiency (low weight/strength and weight/stiffness ratios) and (ii) excellent behaviour under aggressive environmental conditions were matched by (iii) sufficiently low fabrication costs. In particular, the combination of these three features is responsible for the growing demand for thin-walled composite structural members recently observed in the construction industry, namely related to offshore structures and chemical plants.

As it would seem logical to expect, the heterogeneity and orthotropic constitutive relations of the composite materials render their mechanical behaviour considerably more complex than the one displayed by metals such as steel or aluminium, thus introducing additional difficulties to the analysis of thin-walled structural members. In particular, most composite materials often exhibit (i) linear elastic stress–strain relationships (with relatively low moduli), (ii) no ductility (i.e., an elastic behaviour up to collapse) and (iii) different types of orthotropy (depending on the material constituents and fabrication procedure), leading to mechanical properties that clearly indicate a high susceptibility to (i) local and/or global instability phenomena and (ii) brittle collapse modes. Since mastering these two aspects is of paramount importance to achieve safe and economical (competitive) designs, engineers must be equipped with analytical/numerical tools that are able to model accurately their influence on the structural behaviour and load-carrying capacity of thin-walled composite members. This requires not only the access to specific and sophisticated methods of analysis but also an “unbiased mind”, in the sense that is necessary to be suspicious of intuition-based reasonings developed in the context of isotropic materials.

It is fair to say that the first consistent study dealing with the structural behaviour of thin-walled composite (orthotropic) members, under different loading conditions, must be credited to Bauld and Tzeng [4], who developed, in 1984, a Vlassov-type beam theory (accounting for warping effects) to analyse fibre-reinforced members displaying thin-walled open cross-sections. Although the original theory assumed the cross-sections to be shear undeformable and was restricted to members formed by symmetric laminated plates, further contributions from several authors, namely Stemple and Lee [5], [6], Chandra and Chopra [7], [8], Wu and Sun [9], Maddur and Chaturvedi [10] and Fraternali and Feo [11], made it possible to remove and overcome these limitations. In fact, the theory developed by Bauld and Tzeng was extended to incorporate (i) the influence of coupling effects between membrane and bending forces (thus enabling the analysis of cross-sections formed by asymmetric laminates) and (ii) the effect of the cross-section shear deformation. However, since these approaches simply extend the classical Vlassov beam theory to account for the material orthotropy, the member cross-sections are only allowed to exhibit rigid-body motions, corresponding to extension (longitudinal translation), major and minor axis bending (transversal translations) and torsion (rotation about the member axis). In other words, the analysis is unable to predict or take into account the occurrence of in-plane cross-section deformations (e.g., local buckling effects).

More recently, investigations by Raftoyiannis [12], Godoy et al. [13], Barbero [14] and Barbero et al. [15] showed, both analytically and experimentally, that the cross-section (local) deformation may strongly affect the structural (buckling) behaviour of thin-walled composite members (e.g., the beam distortional buckling mode depicted in Fig. 1(a)). In these studies, the authors have employed the finite element method, the plates forming the member walls are discretised by using shell-type finite elements and the analyses were performed by means of commercially available codes (e.g., ABAQUS [16]). However, since the adequate (simultaneous) modelling of both local and global deformations requires the use of large numbers of shell elements and involves a considerable computational effort, the finite strip method (e.g., [17]) is likely to provide a more efficient numerical tool. In fact, Nagahama and Batista [18] employed this method to perform stability analyses of thin-walled members displaying a special type of orthotropy and showed that it yields accurate results.

A few years ago, a rather elegant and quite powerful theory to analyse the structural behaviour of isotropic elastic thin-walled members was developed by Schardt [19] and designated as Generalised Beam Theory (GBT). This theory takes into account both local (cross-section) and global (member) modes of deformation and can be applied to perform either (i) geometrically linear analyses (first-order GBT) or (ii) linear stability analyses (second-order GBT), thus providing a general and unified approach to obtain accurate and clarifying solutions for a wide range of structural problems. In the last decade, Davies and his collaborators (e.g., [20], [21], [22]) have applied GBT extensively to investigate the buckling behaviour of thin-walled cold-formed steel members and their work provided a strong contribution towards establishing this theory as a valid and often advantageous alternative to fully numerical finite element or finite strip analyses.

The research work presently undertaken by the authors is aimed at (i) formulating a GBT that can be readily applied to analyse the structural behaviour of thin-walled (folded plate) composite members made of materials which display an arbitrary type of orthotropy (arbitrary orthotropy), namely non-aligned orthotropy1, and (ii) illustrating its application and capabilities. However, given the considerable size of this task, the results obtained will be reported in two companion papers, the present one addressing issues related to geometrically linear analyses (first-order GBT) and a second one [23] focusing on aspects dealing more specifically with linear stability analyses (second-order GBT).

The objective of this paper is to present the main aspects involved in the formulation of a first-order GBT intended to be used in the analysis of prismatic composite members consisting of folded thin rectangular plates, i.e., displaying thin-walled open cross-sections (see Fig. 1(b)). The plates are formed by several layers, each of them made of a polymer plastic matrix (e.g., an epoxy matrix) reinforced with fibres (e.g., carbon or boron fibres). The particular type of orthotropy displayed by the member depends on two factors, namely (i) the plate layer configuration with respect to its mid-plane (see Fig. 1(b1)), which may be symmetric, antisymmetric or asymmetric, and (ii) the layer fibre orientation (layer natural axes) with respect the plate sides (coordinate axes)–see Fig. 1(b2). In fact, different combinations of these factors lead to members exhibiting distinct types of orthotropy, such as special, general, cross-ply or angle-ply orthotropy [3], [24], [25].

Initially, the equilibrium equations and boundary conditions for laminated plate thin-walled members displaying arbitrary orthotropy are derived and physically interpreted, i.e., their terms are associated with the member mechanical properties. The principle of virtual work is used and the equations involve three cross-section displacements, the longitudinal variation of which is described by a single function. Next, (i) the concepts and procedures required to obtain the transverse cross-section displacements from the axial ones are addressed and (ii) issues related to the influence of the laminated plate layer nature and/or configuration on the member structural behaviour are discussed. Then, the cross-section displacements are expressed as a combination of deformation modes, leading to the most convenient (powerful) form of the orthotropic GBT fundamental equations and associated boundary conditions, in the sense that they enable a rather clarifying modal analysis of the cross-section and member behaviour. Finally, in order to (i) provide a better grasp of the concepts involved in the derivation of the orthotropic GBT and (ii) illustrate its application and capabilities, the structural behaviour of a thin-walled orthotropic beam is analysed. Several laminated plate material behaviours are considered and the results obtained are thoroughly discussed.

Section snippets

Basic assumptions and kinematic relations

First, let us look at Fig. 2(a), where a prismatic member with an arbitrary thin-walled open cross-section is depicted and the coordinates x and s are associated to the member axis and cross-section mid-line directions, respectively. The member consists of a set composite thin rectangular plate (wall) elements, with constant thickness t, for which the coordinate system and displacement notations represented in Fig. 2(b) apply. The formulation of the GBT theory presented here is based on the

Determination of the transverse displacements

The a priori determination of the elementary transverse displacement functions vr(s) and wr(s), expressed in terms of the warping functions ur(s), mostly requires the use of geometrical relations, which means that the vast majority of the steps are identical to the ones included in the available isotropic GBT [19], [22], i.e., the material behaviour only affects a few of them. However, given the fundamental role played by such a procedure in the development of the theory, an overview of all of

Influence of the laminated plate characteristics on the member orthotropic behaviour

By combining the (i) plate layer (ply) configuration with (ii.1) the matrix and fibre properties and (ii.2) the fibre orientation, in each layer, it is possible to produce virtually an infinite number of laminated plates. Concerning these aspects, it is convenient to notice that [3], [24], [25]:

  • (i)

    Depending on the number, sequence and characteristics (thickness, elastic moduli, fibre orientation, etc.) of the plies, a plate may be either (layer) symmetric (S), antisymmetric (AS) or asymmetric with

Cross-section and member analysis

Once the displacement functions wk(s) are known (i.e., expressed in terms of the elementary warping functions uk(s)), it is possible to calculate the components of all the tensors (matrices) appearing in the equilibrium equations (18) and associated , , which means that the member first-order analysis can now be performed. Although this statement is technically true, the fact that all the tensors involved correspond, in general, to full matrices makes the system of equilibrium equations highly

Illustrative example

In order to enable a better grasp of the concepts and procedures just presented, the GBT is employed to analyse the structural behaviour of a thin-walled channel beam of length L=200 cm and with locally and globally simply supported free-to-warp end sections. Fig. 13(a) shows the beam cross-section geometry and the applied load, consisting of a transverse load qw(x,s)=0.01 kN/cm2, uniformly distributed over the beam web. All the beam walls are formed by identical laminated plates with thickness

Conclusion

A first-order Generalised Beam Theory (GBT) formulation was developed which enables the structural analysis of composite thin-walled members made of laminated plates and displaying arbitrary orthotropy. The GBT fundamental equilibrium equations and corresponding boundary conditions were first derived and a physical interpretation of their terms was provided. Next, the main concepts and procedures involved in obtaining the cross-section displacements were addressed, hopefully contributing to

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