Differential geometry and structural action of vaults and shells

Doctoral thesis, 2024

Masonry vaults and bridges, concrete shells, and steel and timber grid shells are all examples of shell structures. Their beauty, spatial qualities, structural efficiency and manufacturing are closely tied to their geometry. Geometry is central since it does not only describe the shape of the shell but also the building blocks, which together form a pattern or grid dictating its construction and structural behaviour.

History has shown how architects and builders successfully utilized curved shapes to combine architectural qualities with structural efficiency through the use of simple building blocks of local materials. Differential geometry offers a mathematical framework for the investigation of such architectural qualities in both historic and new structures. The focus of this thesis is to investigate how differential geometry can contribute to the design and production of shell structures in the digital age.

During the thesis work, different directions for the investigation of differential geometry in architecture and engineering have been explored. Out of these explorations, two main directions have evolved, each of them with a more limited and specific research question:

How can differential geometry be used to design structurally efficient shells and shells with surface patterns for simple production?

How can differential geometry be applied in a design process supporting the quick production of a grid shell?

These questions have been treated and investigated in studies reported in the appended Paper A-F. Paper A, D and E cover different aspects of how to design shells where simple building blocks can be used. Paper A and D show different possibilities for designing new brick shells and masonry bridges, while Paper E investigates the architectural application of shells whose boundaries subtend a constant solid angle. Paper B and C describe and discuss the design and construction process of two different timber grid shell structures built of straight planar laths. Paper F takes its point of departure from the fact that shells are generally statically indeterminate structures, allowing several force and moment distributions that can fulfil equilibrium. Our study offers a framework for designing shells using the force method, where the redundant forces and moments in the shell are represented as two surfaces.