A Beam Theory for Large Global Rotation, Moderate Local Rotation, and Small Strain

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

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

D. H. Hodges

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332

J. Appl. Mech 55(1), 179-184 (Mar 01, 1988) (6 pages) doi:10.1115/1.3173625 History: Received January 27, 1987; Revised July 06, 1987; Online July 21, 2009


Kinematical relations are derived to account for the finite cross-sectional warping occurring in a beam undergoing large deflections and rotations due to deformation. The total rotation at any point in the beam is represented as a large global rotation of the reference triad (a frame which moves nominally with the reference cross section material points), a small rotation that is constant over the cross section and is due to shear, and a local rotation whose magnitude may be small to moderate and which varies over a given cross section. Appropriate variational principles, equilibrium equations, boundary conditions, and constitutive laws are obtained. Two versions are offered: an intrinsic theory without reference to displacements, and an explicit theory with global rotation characterized by a Rodrigues vector. Most of the formulas herein have been published, but we reproduce them here in a new concise notation and a more general context. As an example, the theory is shown to predict behavior that agrees with published theoretical and experimental results for extension and torsion of a pretwisted strip. The example also helps to clarify the role of local rotation in the kinematics.

Copyright © 1988 by ASME
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