Beams and shells can be considered as three-dimensional solids, in which two or one of the dimensions are small compared to the remaining dimensions. This has given rise to beam and shell models in which the representation of the transverse displacement components is simplified. These models are formulated in terms of the translation and rotation of a reference curve or surface in the case of beams and shells, respectively. As it will appear, there are fundamental differences between the representation of translations and rotations. It is important to account for the special characteristics of rotations in the formulation and analysis of kinematically non-linear theories for beams and shells, and this chapter provides a concise presentation of the special properties of rotations needed for the development of general non-linear beam and shell theories. A detailed discussion of rotations and their various representations has been given by Argyris (1982) and by Géradin and Cardona (2001).

The translation of a point is described by a vector u, and the result of a sequence of translations Δu1, Δu2, … is simply the sum of the individual translation vectors. The result is independent of the order of the individual terms. Finite rotations cannot be described in this simple manner, as may be seen by considering two rotations of 90° about orthogonal axes, and comparing the result with that of the same rotations taken in the opposite order (Goldstein, 1980).