On a geometrically exact curved/twisted beam theory under rigid cross-section assumption R. K. Kapania, J. Li Abstract A geometrically exact curved/ twisted beam theory, that assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal frames of reference starting from a 3-D beam theory. The relevant engineering strain measures with an initial cur- vature correction term at any material point on the current beam cross-section, that are conjugate to the first Piola- Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The stress re- sultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimen- sional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are con- sidered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constit- utive relations are obtained with the initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also summa- rized in a manner that is easy to understand. Keywords Curved beam theory, Geometrically exact, Rigid cross-section, Finite strain, Finite rotation, Curved beam element, Orthonormal frame 1 Introduction Beams have found many applications in civil, mechanical and aerospace engineering. Exact and efficient nonlinear analysis of structures, built up from beam components, using robust numerical methods, e.g. finite element meth- ods, should be based on proper nonlinear beam theories. Reissner’s finite strain beam theory (1972, 1973, 1981) is one of the simplest and most important ones. This theory is based on Timoshenko’s plane cross-section assumption, which has been extended, given in detail, and used by many other authors (see e.g. Simo 1985; Simo and Vu-Quoc 1986; Cardona and Geradin 1988; Iura and Atluri 1988, 1989; Saje 1991; Simo and Vu-Quoc 1991; Pimenta and Yojo 1993; Jelenic ´ and Saje 1995; Simo et al. 1995; Ibrahimbegovic ´ 1995; Pimenta 1996; etc.) for 2-D and 3-D cases for both static and dynamic problems. Most authors dealt with straight beams and used orthonormal reference frames. For curved/twisted beams, the slender beam or rod assumption is hidden in the beam theories (see e.g. Simo et al. 1995; Ibrahimbegovic ´ 1995; etc.). For truly geometrically exact curved/twisted beam theory for moderate thick beams, Iura and Atluri (1988, 1989), etc., kept the rigid cross-section assumption and used curvilinear reference frames for the formulation. Certain authors considered shear and torsion warping by the introduction of a simple warping function (see e.g. Simo and Vu-Quoc 1991; etc.) for straight beams. For computationally analyzing curved beams or arches, many authors prefer using straight beam elements based on straight beam theories (see e.g. Simo and Vu-Quoc 1986; Cardona and Geradin 1988; Ibrahimbegovic ´ et al. 1995; Franchi and Montelaghi 1996; etc.). This is a simple and good approximation for slender curved beams or flexible curved beams although more elements will be used to get a satisfactory accuracy. Others prefer using curved beam/arch elements to analyze curved beams or arches based on slender beam theories to reduce the number of elements used (see e.g. Sandhu, Stevens and Davies 1990; Saje 1991; Simo et al. 1995; Jelenic ´ and Saje 1995; Ibrahimbegovic ´ 1995; Pimenta 1996). However, for thick and moderately thick curved beams, an increase in the accuracy of the finite element solution by increasing the number of straight beam elements or curved beam elements based on the slender beam theories has its limit, especially when long-term dynamic responses as well as strains and stresses in three-dimensional level are needed for design purposes. In this case, more refined curved beam theories should be used. In this paper, a geometrically exact finite-strain curved and twisted beam theory with large displacements/rota- tions is re-examined and extended using orthonormal frames and the rigid cross-section assumption. The Computational Mechanics 30 (2003) 428–443 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0421-8 428 Received: 17 June 2002 / Accepted: 21 January 2003 R. K. Kapania (&), J. Li Aerospace and Ocean Engineering Department, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA 24016-0203 e-mail: rkapania@vt.edu The work was partly sponsored by a grant (CDAAH04-95-1-0175) from the Army Research Office with Dr. Gary Anderson as the grant monitor. We would also like to thank Prof. Raymond Plaut of Dept. of Civil and Environmental Engineering at Virginia Polytechnic Institute and State University for his technical help.