Variational asymptotic analysis for plates of variable thickness

Abstract

It is a common assumption in plate theory that the effect of varying thickness on plate elastic constants can be simply accounted for by incorporating the thickness distribution in the formulae obtained for a plate of uniform thickness. For plates whose thickness varies linearly in the global directions, the variation described by a set of taper parameters, a plate theory has been developed using the variational-asymptotic method. In a first of its kind development, both the 8×8 stiffness matrix and the 3D stress and strain recovery are shown to contain these taper parameters explicitly, as opposed to their merely being present by virtue of the varying plate thickness. Furthermore, terms such as bending-shear and twist-shear coupling are shown to be present, which are typically disregarded in published works on tapered plates. It is later clearly demonstrated using the full 3D finite-element solutions from a test case that an analysis of this kind is needed to accurately determine the complete stress distribution. In conclusion, merely changing the thickness distributions in results for uniform plates may work for some problems but it is clearly not sufficient in general, nor can it guarantee good agreement with all aspects of plate analysis. This is because these kinds of approximations are not in tune with the mechanics of such a problem, as they disregard the tilt of the outward-directed normals in the boundary conditions for top and bottom surfaces. Furthermore, as always, a theory developed using the variational-asymptotic method does not make any assumptions regarding the deformation and is generally well suited for capturing such effects without fundamental changes in the procedure.