An efficient beam element for the analysis of laminated composite beams of thin-walled open and closed cross sections

https://doi.org/10.1016/j.compscitech.2008.04.018Get rights and content

Abstract

A condensed fully coupled beam element for thin-walled laminated composite beams having open or closed cross sections is presented. An analytical technique is used to derive the cross-sectional stiffness of the beam in a systematic manner considering all the deformation effects and their mutual couplings. An efficient finite element approximation is adopted for the transverse shear deformation, which has helped to conveniently implement the C1 continuous formulation required by the torsional deformation due to incorporation of warping deformation. The performance of the element is tested through the solution of numerical examples involving open section I and channel (C) beams and closed section box beams under different loading conditions, and the obtained results are compared with model as well as experimental results available in literature.

Introduction

The problem of modelling thin-walled laminated composite beam/beam like slender structures having open or closed cross-section as one dimensional condensed beam elements has drawn a significant amount of attention of researchers over the last decades, and a number of investigations have been carried out to study the different aspects of this problem. A few representative studies relevant to the present context are given in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. One of the initial applications of composite beam theory was found in the analysis of helicopter rotor blades. It has subsequently been applied to the analysis of pultruded composite profiles and other applications, including the analysis of long and slender wind turbine blades made of composite materials.

The studies carried out so far may broadly be divided into two groups, based on the approach adopted for the evaluation of the constitutive matrix of the beam, defined as the cross-section stiffness coefficients. The first and most common approach is based on an analytical technique, while the other approach requires a two-dimensional finite element analysis to obtain the cross-section stiffness matrix. Hodges and co-workers [3] pioneered the second approach, which is referred to as the so-called variational asymptotic beam section analysis (VABS). It is based on a method known as the variational asymptotic method (VAM) [11], where the three dimensional elasticity problem is systematically divided into a two dimensional cross-sectional problem, and a one dimensional beam problem (length direction). VABS has the advantage that beams having solid or thick-walled cross sections may be analyzed, where the three dimensional stress state can be extracted in the post processing stage. Opposed to this, the fully analytical approach may be preferred specifically for the analysis of beams with thin-walled cross sections in order to avoid the additional two-dimensional finite element analysis required in VABS.

In this context it should be mentioned that Hodges and co-workers (e.g., Volovoi et al. [12], Volovoi and Hodges [13], Volovoi et al. [14], Volovoi and Hodges [15], Ye et al. [16]) further applied the concept introduced by the variational asymptotic method to two dimensional cross-sectional problem and derived closed-form expressions for the cross-sectional stiffness coefficients of thin-walled beams. As the resulting theory [12], [13], [14], [15], [16] evolved through rigorous mathematical treatments, it involves some additional terms including those for in-plane warping of the section, which is found to be important in some specific situations. Thus, the generality of the modelling has been enhanced in the asymptotic approach, but at the same time the mathematical formulation is very complex. However, the approach being a fully analytical one produced final results in the form of relatively simple closed-form expressions. The present approach proposes a fully coupled beam model which is significantly less mathematically complex, and which includes all effects except in-plane cross sectional warping.

In the present investigation, an analytical (closed-form) approach is adopted for the derivation of the cross-sectional stiffness matrix considering different effects and their coupling to yield a general formulation, which includes a torsional warping moment in addition to the classical (and well known) de St. Venant torsion contribution, an axial force, bi-axial bending moments and bi-axial transverse shear forces. In total this yields a 7 × 7 cross-sectional stiffness matrix. All the elements of this matrix are explicitly derived for open I and channel (or C) sections, and closed box section profiles. For the constitutive equation of any beam wall, defined locally, provision is kept to enable the specification of either plane stress conditions (zero normal stress along the wall profile, i.e., in the circumferential direction) or plane strain conditions (zero normal strain along the wall profile, i.e., in the circumferential direction).

For the finite element approximation of the beam element, the torsional deformation requires C1 continuity of the twisting rotation due to incorporation of the out of plane warping deformation. This requirement is satisfied with the use of a Hermetian interpolation function considering the twisting rotation and its derivative with respect to the length coordinate as the nodal unknowns. The association of the derivative of the twisting rotation helps to impose warping restraints or warping free conditions by constraining or releasing this nodal unknown. At the same time the bending deformation requires C0 continuity of the transverse displacements due to the incorporation of the transverse shear deformation of the beam walls. This requires a reduced integration technique for the evaluation of the stiffness matrix in order to avoid shear locking. As the bending deformation is not uncoupled from the other modes of deformation, including torsion, it is difficult to implement a C0 formulation with a C1 formulation having different orders of integration schemes. Lee [8] tried to solve the problem by an amended representation of the torsional deformation, so as to model it with a C0 formulation like for bending deformation, but this involved a non-physical parameter in the formulation. Moreover, the C0 formulation with reduced integration technique is susceptible to display inherent numerical disturbances like the occurrence of spurious modes.

Considering these aspects, the finite element implementation of the bending deformation in the present formulation is carried out with a different approach based on a concept proposed by the first author [17]. It does not require a reduced integration technique, and this effectively eliminates the problem described above. Based on this methodology a three node beam element, as shown in Fig. 1, has been developed, where the nodes at the two ends contain seven degrees of freedom (three translations, three rotations and the derivative of the twisting rotation), while the internal node contains five degrees of freedom (three translations and two bending rotations).

A computer program has been written in FORTRAN for the implementation of the proposed fully coupled composite beam element, which has been used to solve numerical examples of composite beams having open I and channel (C), and closed box sections. The results obtained in the form of deflections, angles of twist, and bending slopes are compared with analytical, experimental and/or other finite element analysis results available in literature. The results show a very good performance of the proposed element in terms of convergence and solution accuracy. The developed element is also utilized to derive some new results, which are presented for future references.

Section snippets

Formulation

Fig. 2 shows a portion of the composite beam shell wall, with its local coordinate system xsn, the local displacement components, the global coordinate system xyz and finally the global beam displacement components.

In Fig. 2, O is the centroid, and P is the shear centre/pole of the beam section. The displacement components at the mid-plane of the shell wall in the local coordinate system (xsn) may be expressed in terms of the global displacement components of the beam [1] asu¯=U+yθy+zθz+φθ

Results and discussions

In the following a number of numerical examples involving composite beams with I, channel (C) or box cross sections are analysed using the proposed element, and the results obtained are compared with analytical, experimental and/or numerical results available in literature for most of the cases. The analyses are usually based on plane stress conditions, unless specified otherwise. In all examples the beam walls are assumed to be constituted by identical layers of the same thickness, but the

Conclusions

A fully coupled beam element has been developed for the analysis of thin-walled laminated composite beams of open and closed cross sections including axial displacement, torsion, out of plane warping, bi-axial bending and transverse shear deformations. The constitutive equations of the beam element are derived analytically considering the coupling of all the modes of deformation, i.e., the beam model is fully coupled. The resulting composite beam theory is applied to laminated composite beams

Acknowledgements

The work presented was carried out as part the Innovation Consortium “Integrated Design and Processing of Lightweight Composite and Sandwich Structures” (abbreviated “KOMPOSAND”) funded by the Danish Ministry of Science, Technology and Innovation and the industrial partners Composhield A/S, DIAB ApS (DIAB Group), Fiberline Composites A/S, LM Glasfiber A/S and Vestas Wind Systems A/S. The support received is gratefully acknowledged. The work has been carried out during the presence of first

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