Discrete membrane O surface theory and graphic statics for a development of architectural surface design method

Reference No. 2022a034
Type/Category Grant for Young Researchers and Students-Short-term Joint Research
Title of Research Project Discrete membrane O surface theory and graphic statics for a development of architectural surface design method
Principal Investigator Yoshiki Jikumaru(Kyushu University, Institute Mathematics for Industry / Postdoc)
Research Period May 12, 2022. - May 12, 2022.
May 26, 2022. - May 26, 2022.
June 09, 2022. - June 09, 2022.
June 23, 2022. - June 23, 2022.
July 07, 2022. - July 07, 2022.
Keyword(s) of Research Fields Architectural surface design, equilibrium shape, constructibility, shell membrane theory, discrete differential geometry, integrable geometry, graphic statics
Objectives and Expected Results In architectural surface design and structural engineering, it is desirable to derive the shape of shell and membrane structures covering large spaces from mechanical and geometric properties.
To utilize the stiffness of a shell structure efficiently, the ideal shape does not bend under the loading and in equilibrium in-plane membrane stresses only.
However, conventional shape determination methods based on geometric properties have dealt only with very narrow classes of surfaces.
On the other hand, C. Rogers and W. K. Schief showed that when a constant load acts in the normal direction on a smooth surface, the nonlinear system of equations obtained by pairing the Gauss-Codazzi equation with the membrane equilibrium equation, which does not occur in-plane shear stresses along the curvature line coordinate directions, forms an integrable system called a "membrane O surface".
Moreover, a discretization theory that preserves the integrability has also been proposed in [2].
Membrane O-surfaces are a class of surfaces that includes linear Weingarten surfaces and other surfaces that are said to have favorable mechanical properties but have not been utilized in architectural surface design.
In the Short-term Cooperative Research Program "Differential Geometric Approaches to Shell Membrane Theory and Their Application to Architectural Surface Design (Principal Investigator: Kentaro Hayakawa)," we have conducted the first study on its application to architectural surface design.
In this joint project, the effectiveness of introducing the perspective of "Graphic statics" which visualizes the correspondence between the equilibrium shape and the diagram of the force acting on the structure, was confirmed.
However, the implementation was limited to the equilibrium state of the so-called "pure shear" proposed in [2] and has not been implemented utilizing discrete membrane O-surfaces.
In this joint research project, the goal is to implement a novel geometric shape generation method on the CAD software "Rhinoceros" which possesses the desirable mechanical properties and constructibility for a given load, utilizing the theory of discrete membrane O-surfaces proposed and the method of graphic statics.
Two designers and developers from Taiyo Kogyo Corporation, a world leader of practical applications of the membrane structures, will participate in this joint research project and will discuss issues in future practical applications of this research positively, aiming for continued collaboration through the formulation of structural mechanical and geometric problems.

References.
[1] C. Rogers and W. K. Schief, On the equilibrium of shell membranes under normal loading.
Hidden integrability. Proc. R. Soc. Lond. (2003) 59, 2449–2462.
[2] W. K. Schief, Integrable structure in discrete shell membrane theory, Proc. R. Soc. A (2014) 470: 20130757.
Organizing Committee Members (Workshop)
Participants (Short-term Joint Usage)
Yohei Yokosuka(Kagoshima University / Associate Professor)
Kazuki Hayashi(Kyoto University / Assistant Professor)
Kentaro Hayakawa(Kyoto University / PhD student)
Takanori Yagi(Taiyo Europe GmbH / General Manager of Global Engineering)
Shono Suzuki(MakMax Taiyo Kogyo Corporation / Design job)
Yusuke Sakai(Sony computer science laboratory / Associate researcher)
Keisuke Mizutani(University of Tokyo / PhD student)
Adviser Kenji Kajiwara (IMI, Kyushu University / Professor)