General translation of cellular to shellular polyhedral structures using reciprocity

Shell-based cellular (shellular) funicular structures (SFSs) are single-layer 2-manifold efficient structures with anticlastic curvature, designed in the context of graphic statics. This research proposes a comprehensive methodology for designing these efficient structures in the context of graphic statics. Due to the significant challenges in the process of designing these structures, and the ease of using 3D graphic statics in designing cellular funicular structures, this article proposes a general technique to translate any cellular funicular structure (CFS) to a shellular version (SFS). To address this transition, this study presents an integrated methodology coupled with a computational algorithm. This technique proposes a new tetrahedralization method using the reciprocal relationship between the force and the form diagrams, generalizing the translation technique. As a result, the research explores a spectrum of shellular funicular structures, under pure compression or tension states. Diverse design techniques are introduced, enabling the creation and manipulation of these structures through their three-dimensional spatial connectivity graphs, termed "labyrinths". A comparison between the structural performance of cellular and shellular funicular structures with similar volume density is performed displaying that for the same boundary condition, a shellular specimen can tolerate forces three times more than a cellular structure. To emphasize the practical utility of this design methodology, the study delves into its application at micro and meso scales. Specifically, it showcases the utilization of the shellular technique in the design of the midsole structure of a sneaker. This innovative approach draws inspiration from the pressure patterns exerted by the soles of the feet, emphasizing the adaptability and versatility of the proposed design technique. The results display that shellular funicular structures, with their lightweight and efficient nature, demonstrate superior structural capacity compared to their cellular counterparts and are applicable across micro, meso, and macro scales.