Karpov, E.G.
(2002)
Invariant functional solutions for the statics of regular lattices.
*University of Southampton, School of Engineering Sciences, Doctoral Thesis*.

## Abstract

This research is concerned with the static behaviour of regular lattice structures. A group of effective analytical approaches are presented within a single framework of the invariant functional description. Within this formulation, the regular beam- or plate-like lattice structure considered formally as a segment of a boundless cyclic or infinite domain, and the realistic static response is obtained in terms of the solutions for the latter. The static solutions are obtained in the form vectors of displacements or stresses, as functions of the discrete spatial parameters - numbering of typical structural joints. The governing equation is written as a single operator form with the physical stiffness operator fully representing the lattice statics; that can be easily adapted to the specifics of a particular problem by considering only a representative substructure stiffness matrix. Boundary conditions do not affect the equation form, and are taken into account at a subsequent stage of the analysis. The translational or cyclic symmetry of the boundless domains are utilised to write the stiffness operator as a simple convolution sum for all regular lattices to admit effective solution techniques by discrete transform approaches. The technique of virtual load and substructure is proposed to emulate the cyclic or translational symmetry and to assure the equivalence of responses of the symmetric and original lattices. The boundless lattice response is compelled to satisfy the original boundary conditions by including additional compensating loads into the final solution forms. The latter are written compactly, as convolution sums over the boundless lattice Green’s function and the actual external and virtual loads. Found on a symmetric structural domain, this Green’s function describes its basic response behaviour, and features invariance with respect to the values of spatial parameters and the original boundary conditions. Straightforward techniques, based on discrete Fourier and z-transforms, are proposed for the invariant Green’s function evaluation on the cyclic and infinite domains, respectively. An alternative solution strategy is presented for the analysis of end-loaded beam lattices, where the governing stiffness operator equation is reduced to a transfer matrix. Its general solution is decomposed over a number of characteristic modes to represent the intrinsic structural behaviour, regardless of the lattice size and boundary conditions. A method of constructing the transfer matrix for a wide class of structures, and relation between the characteristic modes and infinite beam Green’s function are demonstrated. As further applications, an approach to modelling failures and disordering in regular lattices is presented, and problems with non-rectangular boundary shapes and initial (pre-load) member stress in random geometrically imperfect lattices are considered. For the latter the spatial invariance of the statistics is shown to arise on the cyclic domains, and parameters of the corresponding probability distributions are obtained analytically. The results obtained using the present approach require less computer efforts with a factor at least equal to the number of typical repeating units in a lattice under analysis – if compared with the direct matrix approach. For the probabilistic analysis and for multiple solving of large lattice problems, this factor is higher. Since the approach is exact in the analytical sense, the numerical accuracy of the results can be affected only by the precision of chosen finite element models for the representative substructures. Numerous examples demonstrate practical applications.

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