On transfer matrix eigenanalysis of pin-jointed frameworks

Eigenanalysis of the state vector transfer matrix has previously been employed to obtain Saint-Venant decay rates and continuum beam properties of a repetitive pin-jointed framework. Decay eigenvalues occur as reciprocal pairs, the transfer matrix being symplectic, and three of the unity, transmission, eigenvalues pertain to the trivial rigid body displacements. By setting displacement or force components equal to zero at the remote right-hand end of the structure and, through use of a recurrence relationship, new displacement transfer matrices, S or C, are derived for the generic cell; these are one-half of the original size, well conditioned, and the redundant information is eliminated. The former, S, requires a large value of the recurrence index, i, to achieve accurate eigenvalues while the latter, C, retains trivial information pertaining to the rigid body displacements. An alternative force transfer matrix, M, derived from S, retains the maximum amount of relevant information and converges quickly. The method suppresses the redundant right-to-left decay eigenvectors, and calculation of the transmission vectors of tension, bending moment and shearing force is simplified by the need to calculate just one principal vector rather than four for the original eigenproblem. Finally, these transmission vectors are employed to determine the continuum beam properties of the framework.