On a geometrically exact curved/twisted beam theory under rigid cross-section assumption |
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Authors: | R K Kapania J Li |
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Affiliation: | (1) Aerospace and Ocean Engineering Department, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA 24016-0203 e-mail: rkapania@vt.edu, US |
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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 curvature 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 resultant and couple are defined in the classical sense and the reduced
strains are obtained from the three-dimensional 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 considered for the engineering strain and stress conjugate pair at the three-dimensional
beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the
initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also
summarized in a manner that is easy to understand.
Received: 17 June 2002 / Accepted: 21 January 2003
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. |
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Keywords: | Curved beam theory Geometrically exact Rigid cross-section Finite strain Finite rotation Curved beam element Orthonormal frame |
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