Shell and membrane theory from the viewpoint of differential geometry and its application to architectural surface design

Reference No.	20200029
Type/Category	Grant for Young Researchers-Short-term Joint Research
Title of Research Project	Shell and membrane theory from the viewpoint of differential geometry and its application to architectural surface design
Principal Investigator	Kentaro Hayakawa（Dept. of Architecture and Architectural Engineering, Kyoto University・Ph. D Student）
Research Period	February 18,2021. ～ February 18,2021. February 27,2021. ～ February 27,2021. March 6,2021. ～ March 6,2021. March 12,2021. ～ March 12,2021. March 20,2021. ～ March 20,2021.
Keyword(s) of Research Fields	architectural structure, equilibrium shape, constructability, shell theory, membrane theory, finite element, linear Weingarten surface, integrable geometry, discrete differential geometry, geometric variational problem
Abstract for Research Report	In architectural design and structural design, the surface shape of shell structures and membrane structures covering large spaces should be derived from mechanical and geometric perspectives in addition to aesthetics. In order to efficiently utilize the stiffness of the shell structure, it is desirable that the curved surface of the structure achieves an equilibrium state with only in-plane membrane stresses. However, existing methods of determining surface shape by geometric properties are limited to classical surface classes such as inverted hanging surfaces. In the design of the membrane structure, the uniform stress distribution is desirable and the minimal surface is often used. On the other hand, the membrane structure is constructed by connecting planer membrane sheets. In this respect, the developable surface may be desirable. However, the only surface which is both minimal and developable is a plane, and it is a major issue to achieve both mechanical performance and constructability. Mathematically, it has been shown by Rogers and Schief that the surface equilibrating with in-plane stresses under constant loading in the normal direction is a special case of the "O-surfaces" presented by Schief and Konopelchenko obtained by coupling the Gauss-Codazzi equations for surfaces with the equilibrium equations of shell and membrane surfaces. However, there are no applications in architecture that utilize the mechanical properties of these surfaces, and it is necessary to show their usefulness. In this study, by understanding the shell and membrane theories from both mathematical and architectural viewpoints. we construct a fundamental theory that relates the mechanical and geometric properties of curved surfaces and investigate its applicability to architectural and structural design. In the analysis of shell and membrane structures, surfaces are generally discretized into triangular and quadrilateral meshes. Therefore, in order to implement the constructed theory in the framework of finite elements, a method based on the theory of discrete differential geometry is considered. A new method may be developed for determining the shape of surface that satisfies the equilibrium condition or achieves both mechanical performance and constructability, and the class of surfaces used in architectural design may be expanded.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Yoshiki Jikumaru（Institute of Mathematics for Industry, Kyushu University・Post-doctoral Fellow） Yohei Yokosuka（Graduate School of Science and Engineering, Kagoshima University・Associate Professor） Takashi Kagaya（Institute of Mathematics for Industry, Kyushu University・Assistant Professor） Kazuki Hayashi（Dept. of Architecture and Architectural Engineering, Kyoto University・Ph. D Student） Yusuke Sakai（Dept. of Architecture and Architectural Engineering, Kyoto University・Ph. D Student）
Adviser	Kenji Kajiwara (IMI, Kyushu University / Professor)