Some General Considerations on the Natural Mode Technique

Abstract

The application of the matrix displacement method to large displacements and strains, as formulated in refs. 1, 2, 3, is based on the idea of natural modes. This method was significantly generalised to triangular and tetrahedronal curvilinear elements in TNs 17 and 18 (ref. 4), where the important new concept of the local sub-element was first presented. A further obvious extension of the approach should cover curved beams in space, and, what is much more important, shells. However, the realisation of a theory which includes such elements is not straightforward and requires some subtle considerations on the natural modes of a local sub-beam. In order to give a clear presentation of the underlying physical assumptions, we first discuss in this note the natural mode technique on a generalised basis and apply it then to a curved beam in space. The idea of the local sub-element emerges thereby in all its significance. The examples investigated here cover an arbitrary curved beam in space with initial curvature and twist and the special case of a circular arch. To limit the length of the note we have to exclude the effects of shear deformability in bending, and warping in torsion. Moreover, to simplify the formulae we assume that bend ing is defined about the principal axes. The beams in question may be classified as slender and are part of the ASKA library (CUBA for curved beam in space and CIRCA for circular aroh). Part II of the note will include the effect of large displacements and show how the ideas of ref. 4 may be adapted to curved beams and similar structural elements.