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What is a shell?
Published in Sigrid Adriaenssens, Philippe Block, Diederik Veenendaal, Chris Williams, Shell Structures for Architecture, 2014
The Gaussian curvature is the product of the two principal curvatures on a surface and they are of opposite sign on a cooling tower. Gauss's theorem (Theorema Egregium) tells us that the Gaussian curvature of a surface can be calculated by only measuring lengths on a surface, and therefore inextensional deformation does not change the Gaussian curvature. A developable surface is a surface with zero Gaussian curvature which can be laid out flat. Examples include cylinders and cones. Gauss's Theorem is derived in Appendix B.
Map projections in light of surface theory
Published in Martin Vermeer, Antti Rasila, Map of the World, 2019
Stated again differently, the curvature of a surface may be characterized by one number K, which may be calculated from only the metric intrinsic to the surface — by well-known operations like differentiation, multiplication, addition — without knowing anything about the properties of the space surrounding the surface or the way in which the surface is embedded in it1 . This conceptually important result observed by Gauss is known by the name theorema egregium, “remarkable theorem,” Bhatia (2014).
Adaptive multimodal surface patterning on a cylindrical shell panel using piezoelectric actuators
Published in Mechanics of Advanced Materials and Structures, 2022
Surfaces having an inherent curvature are better suitable as substrates for pattern formation, considering their ability to form bistable/multi-stable configurations [14]. Among different curved surfaces, a cylindrical shell structure can exhibit either a localized or periodic pattern in the form of dimples if the loading conditions are appropriately tailored. An explanation for this nature of the distribution of pattern, that is, localized or periodic, for the structure undergoing deformation, may be sought from an understanding of the Gaussian curvature and is linked to the distribution of stresses in the sheet via Gauss’s Theorema Egregium; a consequence of the theorem is that surfaces with positive Gauss curvature (e.g., a spherical cap) exhibits localized patterns, whereas surfaces with negative Gauss curvature (e.g., a saddle surface) deforms nonlocally along characteristic lines that extend through the entire system when subjected to local transverse loads [15]. A cylindrical shell surface is having zero Gaussian curvature that may exhibit either local or global buckling response owing to its positive principle curvature in the circumferential direction and zero principle curvature in the axial direction. This inherent versatility for the generation of either a local or global pattern makes the cylindrical surface, an ideal structure for the creation of adaptive patterns. The response of different primary structures, when subjected to localized transverse load, is shown in Figure 2. A detailed discussion on the response of shells of different principle curvatures under lateral poking may be found in Vaziri et al. [15].