Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures

Stephen, N.G. (2006) Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 462, (2072), 2245-2270. (doi:10.1098/rspa.2006.1669).


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Transfer matrices are used widely for the dynamic analysis of engineering structures, increasingly so for static analysis, and are particularly useful in the treatment of repetitive structures for which, in general, the behaviour of a complete structure can be determined through the analysis of a single repeating cell, together with boundary conditions if the structure is not of infinite extent. For elastostatic analyses, non-unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of self-equilibrated loading, as anticipated by Saint-Venant's principle. Multiple unity eigenvalues pertain to the transmission of load, e.g. tension, or bending moment, and equivalent (homogenized) continuum properties, such as cross-sectional area, second moment of area and Poisson's ratio, can be determined from the associated eigen- and principal vectors. Various disparate results, the majority new, others drawn from diverse sources, are presented. These include calculation of principal vectors using the Moore–Penrose inverse, bi- and symplectic orthogonality and relationship with the reciprocal theorem, restrictions on complex unity eigenvalues, effect of cell left-to-right symmetry on both the stiffness and transfer matrices, eigenvalue veering in the absence of translational symmetry and limitations on possible Jordan canonical forms. It is shown that only a repeating unity eigenvalue can lead to a non-trivial Jordan block form, so degenerate decay modes cannot exist. The present elastostatic analysis complements Langley's (Langley 1996 Proc. R. Soc. A452, 1631–1648) transfer matrix analysis of wave motion energetics.

Item Type: Article
Digital Object Identifier (DOI): doi:10.1098/rspa.2006.1669
ISSNs: 1364-5021 (print)
Related URLs:
Keywords: transfer, symplectic, matrix, elastostatic, pseudo-inverse, jordan canonical form
Subjects: T Technology > TJ Mechanical engineering and machinery
Q Science > QC Physics
Divisions : University Structure - Pre August 2011 > School of Engineering Sciences
ePrint ID: 23809
Accepted Date and Publication Date:
July 2006Published
Date Deposited: 17 Mar 2006
Last Modified: 31 Mar 2016 11:44

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