Eigenanalysis and continuum modelling of a curved repetitive beam-like structure

A curved repetitive pin-jointed structure is analysed using a state variable transfer matrix technique. Within a global Cartesian coordinate system, the transfer matrix G of each cell is different, but within a polar coordinate system they are identical, implying circumferential symmetry. Eigenanalysis provides the rates of decay of self-equilibrated end loading, as anticipated by Saint-Venant's principle, two real unity eigenvalues associated with rigid body rotation and pure bending, and repeated conjugate complex unity eigenvalues, associated with the rigid body displacements, and tension and shear. Interpretation of the eigen- and principal vectors, and also combined vectors from different eigenspaces, allows one to determine the equivalent continuum beam properties, e.g. second moment of area, location of the neutral axis, cross-sectional area, and shear coefficient. The transfer matrix approach is validated by comparison with what may be regarded as exact finite element predictions, and also compared with a (believed novel) thick curved beam strain energy analysis employing the derived equivalent continuum properties and the use of Castigliano's theorems. Agreement is found to be excellent.