I. Concluding Remarks

We have taken a detailed journey through the process of creating beam theories and cross-sectional models for those theories. The asymptotic framework has added a degree of rigor not possible within the usual paradigms of beam analysis. In particular, we have created two sets of equations from a common framework, that of the three-dimensional geometrically nonlinear theory of anisotropic elasticity. One addresses the geometrically exact behavior of one-dimensional continua, and the other addresses the connection of that model to the three-dimensional world through a two-dimensional analysis over the crosssectional plane. The former is in an elegant, canonical form that, despite its geometrical exactness, can be written in a compact matrix notation in a few lines. The latter is sufficiently powerful that it can boil down topologies as complex as realistic helicopter rotor blade cross-sections to an equivalent beam model by means of a two-dimensional finite element model over the cross-sectional plane. The cross-sectional analysis also provides stress recovery over the cross-section. The power of the method has been demonstrated herein in a wide variety of examples.

The method presented herein was made possible by two separate developments of the 1970s and 1980s. First was the variational-asymptotic method of Berdichevsky (1976, 1981), who was evidently the first to plainly state that the three-dimensional problem could be split into separate one-dimensional and twodimensional analyses by virtue of small parameters. Second was the work of Danielson and Hodges (1987) who showed that decomposition of the rotation tensor into global and...