International Journal of Solids and Structures 49 (2012) 18021817

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International Journal of Solids and Structures

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Kinematically exact curved and twisted strain-based beam

P. Cearek, M. Saje, D. Zupan University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1115 Ljubljana, Slovenia

a r t i c l e i n f o

Article history:Received 3 July 2011Received in revised form 13 December 2011Available online 6 April 2012

Keywords:Strain measureConstant strainNon-linear beam theoryThree-dimensional beamThree-dimensional rotation

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Corresponding author. Tel.: +386 1 47 68 632; faxE-mail address: dejan.zupan@fgg.uni-lj.si (D. Zupa

a b s t r a c t

The paper presents a formulation of the geometrically exact three-dimensional beam theory where theshape functions of three-dimensional rotations are obtained from strains by the analytical solution ofkinematic equations. In general it is very demanding to obtain rotations from known rotational strains.In the paper we limit our studies to the constant strain field along the element. The relation betweenthe total three-dimensional rotations and the rotational strains is complicated even when a constantstrain field is assumed. The analytical solution for the rotation matrix is for constant rotational strainsexpressed by the matrix exponential. Despite the analytical relationship between rotations and rotationalstrains, the governing equations of the beam are in general too demanding to be solved analytically. Afinite-element strain-based formulation is presented in which numerical integration in governing equa-tions and their variations is completely omitted and replaced by analytical integrals. Some interestingconnections between quantities and non-linear expressions of the beam are revealed. These relationscan also serve as useful guidelines in the development of new finite elements, especially in the choiceof suitable shape functions.

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1. Introduction

Beam elements have played a very important role in modelingengineering structures. Their applicability is, however, stronglydependent on the accuracy, robustness and efficiency of the numer-ical formulation. This is particularly important in studying initiallycurved and twisted beams, which are well known to differ consider-ably in their behavior with respect to straight elements. That is whythe mathematical modeling of initially curved and twisted beamshas been a special subject of research both in past and at present,see, e.g. the recent publications by Atanackovic and Glavardanov(2002), Atluri et al. (2001), Gimena et al. (2008), Kapania and Li(2003), Kulikov and Plotnikova (2004), Leung (1991), Sanchez-Hu-bert and Sanchez Palencia (1999), Yu et al. (2002). Among variousexisting non-linear beam theories Reissners geometrically exactfinite-strain beam theory (Reissner, 1981) is the most widely usedone. Several finite-element formulations have been proposed forthe numerical solution of its governing equations, see, e.g. Cardonaand Gradin (1988), Ibrahimbegovic (1995), Jelenic and Saje(1995), Ritto-Corra and Camotim (2002), Schulz and Filippou(2001), Simo and Vu-Quoc (1986), to list just a few among the moreoften cited works.

Another important issue in any finite element formulation is thechoice of the primary interpolated variables. Most of the abovecited approaches use displacements and rotations or solely

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rotations as the interpolated degrees of freedom. Because thespatial rotations are elements of the multiplicative SO3 group,the configuration space of the beam is a non-linear manifold. Thatis why the way the rotations are parametrized and interpolated iscrucial. In the displacement-rotation-based formulations, the eval-uation of strains, internal forces and moments requires the differ-entiation of the assumed kinematic field which decreases theaccuracy of the differentiated quantities compared to the primaryinterpolated variables which might be very important in materiallynon-linear problems.

By contrast, if the strains are taken to be the interpolated vari-ables, the additive-type of interpolation can be used without anyrestrictions. By such an approach the determination of internalforces and moments do not require the differentiation. Instead,the fundamental problem of a strain-based formulation now be-comes the integration of rotations from the given interpolatedstrains. In the three dimensions, the derivative of the rotationswith respect to parameter equals the product of a rotation-depen-dent non-linear transformation matrix and the rotational strain. Ingeneral such a system of differential equations cannot be inte-grated in a closed form. This is probably the main reason why, inthe three-dimensional beam theories, the total strain field or evensolely the rotational strain is very rarely chosen as the primary var-iable. Some authors integrate the straindisplacement relationsand employ the results for proposing a more suitable interpolationfor the three-dimensional rotations. Tabarrok et al. (1988) as-sumed an analytically integrable curvature distribution to developa more suitable interpolation for displacements and rotations in

http://dx.doi.org/10.1016/j.ijsolstr.2012.03.033mailto:dejan.zupan@fgg.uni-lj.sihttp://dx.doi.org/10.1016/j.ijsolstr.2012.03.033http://www.sciencedirect.com/science/journal/00207683http://www.elsevier.com/locate/ijsolstr

Fig. 1. Model of the three-dimensional beam.

P. Cearek et al. / International Journal of Solids and Structures 49 (2012) 18021817 1803

order to describe properly the rigid-body modes of arbitrarilycurved and twisted beam. Choi and Lim (1995) employed the solu-tion of the linearized straindisplacement relations to obtain thefinite-elements for constant and linear shape of varied strains.Schulz and Filippou (2001) proposed an interesting non-linearTimoshenko beam element where the displacements and boththe infinitesimal (incremental) curvatures and the infinitesimalrotations are interpolated. In Schulz and Filippou (2001) the re-duced integration has to be used to avoid shear locking. Santoset al. (2010) introduced a hybrid-mixed formulation in which thestress-resultants, the displacements and the rotations are takenas independent variables. The pure strain-based formulation wasproposed by Zupan and Saje (2003) who developed the spatialbeam finite-element formulation of the ReissnerSimo beam the-ory in which the total strain vectors are the only interpolated vari-ables. Such a formulation is locking-free, objective and a standardadditive-type of interpolation of an arbitrary order is theoreticallyconsistent and can be used for both total strains and their varia-tions. In Zupan and Saje (2003) a numerical method (the RungeKutta method) is used for the integration of the total rotations fromthe given total rotational strains, which is due to the complicatedform of the kinematical equations.

It has already been noted in the analysis of planar frames thatthe strain-based beam formulations are numerically efficient andwell applicable in various problems. In particular, applications ofthe strain-based elements to the dynamics (Gams et al., 2007),and to the statics of the reinforced concrete frame with the strainlocalization (Bratina et al., 2004) and the reinforced concrete framein fire (Bratina et al., 2007) show the advantages of both higher-order and a simple constant strain element. The constant-strainelements are especially important for the efficient numerical mod-eling of strain-softening in concrete. The same should equally ap-ply to the three-dimensional beam structures. In the paper wefollow and extend the ideas of the planar case and develop a robustand efficient three-point finite element with 24 degrees of freedombased on Reissners beam theory. In order to obtain an exact ana-lytical solution for the rotations in terms of the rotational strain,we limit our studies to the constant strain field along the element.It is important to point out that integrating the constant strain fieldresults in a non-linear, linked form of rotations and displacements.This immediately suggests that the classical additive-type of inter-polation of the rotation and displacement field, in which the dis-placements are interpolated by using only nodal displacements,and the rotations only nodal rotations, is not the most naturalchoice. This has also been observed by Borri and Bottasso (1994)by using the helicoidal approximation, and by Jelenic and Papa(2011) who studied the genuine linked interpolation functionsfor the three-dimensional linearized Timoshenko beams. In con-trast to Borri and Bottasso (1994) and Jelenic and Papa (2011),the finite-element formulation employed here is based on thestrain field rather than on the displacement-rotation field, whichresults in different types of finite elements and considerable differ-ences in the overall numerical implementation.

The analytical relationship between the rotations and the rota-tional strains is given in the exact analytical form, which enables usto perform the integration in governing equations and their varia-tions analytically. An interesting observation then follows that theanalytical approach, although based on the assumption of the con-stant strain field over the finite element, suggests the integralsmust be decomposed into the total rotational operator at theend-point of the beam and the arc-length dependent operatoralong the beam. A special study is made in searching the form ofthese operators and their similarity with respect to the Rodriguesformula. The similari