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TRENDS IN COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, M. Papadrakakis
On the Large Deformation Finite Element Formulations of Beam Elements
A.A. Shabana, L.G. Maqueda and B.A. Hussein
Department of Mechanical Engineering, University of Illinois at Chicago, United States of America
A.A. Shabana, L.G. Maqueda, B.A. Hussein, "On the Large Deformation Finite Element Formulations of Beam Elements", in B.H.V. Topping, M. Papadrakakis, (Editors), "Trends in Computational Structures Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 11, pp 247-265, 2008. doi:10.4203/csets.19.11
Keywords: Poisson modes, linear and non-linear constitutive models, non-linear elasticity, finite beam elements, absolute nodal coordinate formulation, large deformations.
Existing large deformation finite element beam formulations do not capture significant modes that couple the cross section deformations with the stretch and bending of the beam. These deficiencies in beam formulations, including Euler-Bernoulli and Timoshenko beam theories, are mainly attributed to the kinematic assumption that the cross section of the beam remains rigid when the beam deforms. For this reason, existing finite element beam formulations do not fully capture Poisson effects, that is, a stretch of the beam for example does not lead to a reduction in the dimensions of the cross section. In many applications, the assumption of a rigid cross section can lead to unrealistic models, particularly in the case of very flexible structures. The coupling between the deformation of the cross section and other beam deformations can be significant and can be the source of geometric stiffness. The neglect of this geometric stiffness effect can lead to wrong dynamic and stability results. In applications where the cross section deformation is important, beam models are developed in the literature using solid finite elements. Solid elements, however, do not impose continuity on the rotations and gradients and are known to perform poorly in beam bending problems. The use of the new large deformation finite element absolute nodal coordinate formulation (ANCF) allows capturing the coupled deformation modes of beams and other structural elements. For this reason, new beam models, in which the coupling between the deformation of the cross section and bending and stretch of the beam is accounted for, can be developed. The goal of this work is to shed light on the beam coupled deformation modes, and discuss the use of linear and nonlinear constitutive models in the case of large deformation of beams. The absolute nodal coordinate formulation, unlike most of the existing formulations for structural elements such as beams, plates and shells, allows for the use of general nonlinear constitutive models in a straightforward manner. Three nonlinear constitutive models, which are based on the Neo-Hookean constitutive law for compressible materials, the Neo-Hookean constitutive law for incompressible materials, and the Mooney-Rivlin constitutive law, will be discussed. These models, which allow capturing coupled deformation modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behaviour and are considered incompressible or nearly incompressible. Numerical results are presented in order to show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson's formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.
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