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CivilComp Proceedings
ISSN 17593433 CCP: 102
PROCEEDINGS OF THE FOURTEENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by:
Paper 219
Preserving the Scalar Product of Vectors and the Nonlinear Finite CurvedBeam Element Y.L. Pi and M.A. Bradford
Centre for Infrastructure Engineering and Safety
Y.L. Pi, M.A. Bradford, "Preserving the Scalar Product of Vectors and the Nonlinear Finite CurvedBeam Element", in , (Editors), "Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 219, 2013. doi:10.4203/ccp.102.219
Keywords: curvedbeam element, scalar product of vectors, geometric nonlinearity, rotation matrix.
Summary
To develop a useful finite curvedbeam element for geometric nonlinear large deformation
analysis, accurate finite strains is essential. Using vector analysis, the strain
tensor at a material point of the beam element can be expressed in terms of the position
vectors of the point in the undeformed and deformed configurations. The calculation
of the finite strain tensor consists of scalar products of vectors. It is known that for
the calculation, the position vectors have to be transformed to the same vector space
using rotation matrices. To obtain an accurate strain tensor, the scalar products of the
position vectors and their derivatives have to be preserved during the rotation. Group
theory shows that a special orthogonal rotation matrix that satisfies the orthogonal and
unimodular conditions will preserve the scalar products of vectors and so the rigid
body movement can be excluded from the finite strain tensor. This paper provides a
derivation of a special orthogonal matrix that is expressed by the three dimensional
deformations and that can preserve scalar products of vectors, and it is used to develop
a finite curvedbeam element that can predict the nonlinear large deformation
behaviour accurately.
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