Researchers from Skoltech and the University of Granada have found a way to speed up the architectural design of vaults and domes with wavy patterns, while also conserving construction materials. By extending the applicability of a technique known as the force density method to this new class of curvy objects, called corrugated surfaces, the team offers architects a way to unleash creativity and — by the same token — keep the budget in check. The study came out in the journal Engineering Structures.

“In architecture, sparing resources and maxing out aesthetics are often seen as two mutually exclusive possibilities. On this view, you economize by opting for more primitive shapes and you express your creativity by choosing more structurally involved and expensive solutions, which are not necessarily justified in terms of overall structural integrity. We show that visual appeal and economy need not be in conflict. A structure can be both interesting to look at, stable, and easy to manufacture,” said the lead author of the study, Anastasiia Moskaleva of Skoltech Materials, who holds a PhD in mathematics and mechanics from Skoltech.

When four walls or four columns are topped by a curved surface — that is, a vault or a dome — the shape itself makes the structure stronger than a flat rectangular slab of concrete. That surface can be further strengthened by making it thicker or by fitting it with stiffening ribs — additional bars of material that thicken the surface at strategic points. Previously, the Skoltech-University of Granada team optimized rib configurations to strengthen the curved shells it designed via the force density method.

In their new study, the researchers adapt that same method to the design of wavy, or corrugated surfaces, whose strength comes not from stiffening ribs or extra material but from the curved geometry of the shell itself.

“In this paper, we study how the geometric templates that we call q-patterns can reinforce shells serving as vaults and domes in architecture,” Moskaleva said. “We propose a new approach for making the shells more stable. Namely, the shell is formed with a preset load distribution pattern, which fuses ribs, waves, or folds into the structure. The added curvature strengthens the surface making it less prone to buckling and deformation when loaded.”

The researchers computed the stability for domes with five different fold geometries (see image), each for the case of a four-wall contour or a set of four columns serving as the underlying support. Prior research had pinpointed loss of stability as the single most common mode of failure for such structures.

Calculations run by the team highlighted the corrugated dome shapes that were the most stable cover for four walls and for four columns, with different geometries coming out best depending on the type of support used. One of the five investigated patterns proved inferior in terms of stability regardless of underlying supports.

“Our findings can expand the applicability of folded surfaces in architecture. We give more freedom to the architect and spare computational resources and construction materials, compared with adding material across the entire surface,” Moskaleva said. “Furthermore, the simplification applies not just to the design of the structures, but also to their manufacture. It is easier and cheaper to make an intrinsically curved surface from metal, concrete, or even plastic in the case of miniature structures, rather than fitting a basic template structure with stiffening ribs, which require mechanical fixtures, welding, or other additional manipulations. Manufacturing the curved shell in one go, for example by molding or casting, cuts the costs and speeds up construction.”

The study relied on a modified version of the force density method, tailored to the design of corrugated surfaces using q-patterns, which reflect the force distribution within the structure. The approach used is particularly well-suited to structures made of isotropic materials — those whose properties remain the same when tested in different directions. One of such materials, steel was presumed in the numerical modeling and final-element analysis run by the team in this latest study.

That said, the method is not limited to steel. It would work with concrete, plastic, and other isotropic materials. Plastic shells are an option for miniature architecture: gazebos, canopies, pavilions. Curved surfaces made of steel are useful for industrial structures, for example for fuel storage or other liquids. The researchers believe that their method could even be adapted to composite materials, including fiber-reinforced plastics, but that would require a more detailed numerical model to account for these materials’ anisotropy.