Computational & Technology Resources
an online resource for computational,
engineering & technology publications 

Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 19
TRENDS IN COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, M. Papadrakakis
Chapter 11
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", SaxeCoburg Publications, Stirlingshire, UK, Chapter 11, pp 247265, 2008. doi:10.4203/csets.19.11
Keywords: Poisson modes, linear and nonlinear constitutive models, nonlinear elasticity, finite beam elements, absolute nodal coordinate formulation, large deformations.
Summary
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
EulerBernoulli 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 NeoHookean constitutive law for compressible materials, the
NeoHookean constitutive law for incompressible materials, and the
MooneyRivlin constitutive law, will be discussed. These models, which
allow capturing coupled deformation modes, are suitable for many materials and
applications, including rubberlike 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.
purchase the fulltext of this chapter (price £20)
go to the previous chapter 
