Keywords: curved-beam element, scalar product of vectors, geometric nonlinearity, rotation matrix.

To develop a useful finite curved-beam 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 curved-beam element that can predict the nonlinear large deformation behaviour accurately.

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