Modeling of planar nonshallow prestressed beams towards asymptotic solutions W. Lacarbonara a, * , A. Paolone a , H. Yabuno b a Dipartimento di Ingegneria Strutturale e Geotecnica, Universit a di Roma La Sapienza, via Eudossiana 18, 00184 Roma, Italy b Institute of Engineering Mechanics and Systems, University of Tsukuba, Tsukuba-City 305-8573, Japan Received 20 November 2003 Abstract Employing the geometrically exact approach, the governing equations of nonlinear planar motions around non- shallow prestressed equilibrium states of slender beams are derived. Internal kinematic constraints and approximations are introduced considering unshearable extensible and inextensible beams. The obtained approximate models, incor- porating quadratic and cubic nonlinearities, are amenable to a perturbation treatment in view of asymptotic solutions. The different perturbation schemes for the two mechanical beam models are discussed. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Cosserat rod; Prestressed arch; Post-buckling; Initial curvature; Direct perturbation approach 1. Introduction Resonant large-amplitude planar or spatial motions of slender beam structures are often encountered in engineering. These motions are characterized by finite displacements and rotations whereas the actual strains usually remain small. The most comprehensive mathematical theory today available to describe overall motions of rods in space is the special Cosserat theory of rods also known as the director theory (Villaggio, 1977; Antman, 1985). The beam is conceived as a one-dimensional continuum with a local rigid structure. Because the beam sections are assumed locally rigid neglecting in-plane and out-of-plane deformations, the sections cannot undergo distortion and warping deformations; therefore, the theory is mainly restricted to beams with closed cross sections. Consequently, the motions of a rod in space can be described by three vector-valued functions, the position vector of a material point of the axis and two orthonormal vectors defining the configurations in space of the individual section. In the literature, there have been various implementations of the special Cosserat theory of rods. In computational mechanics, different finite element formulations have been presented to address nonlinear Mechanics Research Communications 31 (2004) 301–310 www.elsevier.com/locate/mechrescom MECHANICS RESEARCH COMMUNICATIONS * Corresponding author. Fax: +39-06-488-4852. E-mail address: walter.lacarbonara@uniroma1.it (W. Lacarbonara). 0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.11.004