A new three-dimensional shell theory in general (non-lines-of-curvature) coordinates for analysis of curved panels weakened by through/part-through holes

Abstract

Abstract A heretofore unavailable differential geometric formulation for representing the geometry of a doubly connected and doubly-curved three-dimensional shell-type deformable body is presented. This results in a three-dimensional shell theory in general non-lines-of-curvature coordinates, endowed with non-vanishing geodesic curvature and twist. Such a theory is useful in modeling exactly the geometry of, e.g., a perforated or semi-perforated shell. Using the Weingarten–Gauss relationship and Serret-Frenet formulas, the curvature, of a coordinate space curve lying on the reference (bottom) surface, is related to the normal curvature of the reference surface in that direction and also the geodesic curvature, while its torsion is related to the twist of the reference surface. The metrics of the reference surface are related to the geodesic curvature, normal curvature and twist of the reference surface, using the concept of the Darboux vector and the associated moving tri-hedron. Using the Weingarten–Gauss relationship, quantities of interest at a parallel surface are then derived in terms of the reference (generating) surface geometry and the thickness coordinate. One important consequence of this is that unlike its counterpart for the generating (reference) surface, the first fundamental form coupling quantity of a parallel surface does not vanish. The included angle between the projections of the orthogonal coordinate curves onto a parallel surface is then easily derived. Finally, the kinematic or strain–displacement relations are obtained in terms of displacement components, normal curvature, geodesic curvature and twist of the reference surface. Once these quantities are defined, an expression for the total potential energy can easily be obtained, which forms the basis for a variational principle, used in conjunction with the finite element discretization. A C 0 -type triangular composite degenerated shell element, based on the assumptions of a locally non-twisting approximation, transverse inextensibility and layer (piece)-wise constant shear-angle theory (LCST), is employed to analyze an edge-loaded deep cylindrical panel weakened by an internal (or embedded) part-through elliptical hole. The same triangular element formulation is employed, based on the assumptions of locally non-twisting approximation, transverse inextensibility and constant shear angle through thickness, for analysis of a perforated cylindrical panel under normal external pressure.