Explicit exact analysis of infinite periodic structures under general loading

Abstract

This paper deals with the explicit analysis of infinite periodic structures under arbitrary loadings. In the context or structural stiffness optimization, with its inherent problem of multiple reanalysis, the purpose is to obtain expressions for the stress resultants anywhere in the infinite structure as an explicit function of the stiffnesses of the elements. Following the method of the representative cell the analysis of an infinite structures is reduced to the analysis of single module under transformed loading and boundary conditions by using the discrete Fourier transform. This produces the equilibrium, strain-displacements and constitutive equations in terms of complex-valued displacements, generalized strains and generalized stresses transforms. Next an existing formula is used to write the stress resultants transforms explicitly in terms of the stiffnesses. Finally one computes the stress resultants wherever needed in the real structure by means of the inverse Fourier transform. The exact formula for the stress resultants is usually impractical due to the large number of terms involved in the analytical expressions. What makes the approach practical herein is the very reduced size of the repeating module that is to be analysed, which renders the analytical formula more tractable in many cases. The technique is illustrated with the explicit analysis of an infinite truss with 1D translational symmetry and of an infinite grid of orthogonal beams on elastic supports with 2D translational symmetry.