Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor

[+] Author and Article Information
D. A. Danielson

Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943

D. H. Hodges

Rotorcraft Dynamics Division, Aeroflightdynamics Directorate, US Army Aviation Research & Technology Activity (AVSCOM), Ames Research Center, Moffett Field, CA 94035

J. Appl. Mech 54(2), 258-262 (Jun 01, 1987) (5 pages) doi:10.1115/1.3173004 History: Received May 03, 1986; Revised September 02, 1986; Online July 21, 2009


A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

Copyright © 1987 by ASME
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