Three-dimensional fiber reinforcement shapes obtainable from flat, bidirectional fabrics without wrinkling or cutting. Part 2: a single n-sided pyramid, cone, or round box

Abstract

Abstract Criteria are derived for the adoption, without wrinkling or having to be cut, of the shapes of regular and truncated n-sided pyramids, cones, and round boxes by bidirectional fabrics. The flat surfaces of pyramids result in distinct deformation zones that can be compared with experiment and allow simple geometrical visualization and explanations. Only trellis shear, the major mode of fabric deformation, was considered. As with four-sided pyramids, the criterion for a fabric to cover a non-truncated n-sided pyramid is a match in angles between the fabric and the pyramid. Needing to be matched are the sum of angles of the n faces and the sum of angles of the four quadrants of the fabric that are determined by the two fibers crossing at the apex. The fabric is least deformed everywhere when the fabric is given the same deformation over the entire surface of the pyramid. The covering of a cone is analogous with n→∞. For truncated pyramids and prisms, the fabric deformation is never uniform. With deformation increasing away from the top surface, short truncated pyramids can be covered without wrinkling even when a taller pyramid having sides of similar steepness cannot. The rules for covering a truncated pyramid or prism and the first appearance on the side of specific degrees of fabric deformation are derived.