Skoltech researchers and their colleague from the University of Granada, Spain, have determined which ways to reinforce vaults and domes in architecture are more efficient. The team compared how well various traditional and unconventional patterns of stiffening ribs enable a structure to withstand both evenly distributed and asymmetric loads. Published in Thin-Walled Structures, the study relied on numerical analysis and physical experiments, and led its authors to propose an unprecedented rib pattern inspired by dragonfly wings, which surprisingly outperformed every other layout examined in the paper.

Stiffening ribs have been used in vaults and domes since ancient Roman times to enable thinner structures for both engineering and aesthetic reasons. This solution conserves material and allows for more intricate designs, bigger column-free floor spans, and larger windows — like those in Gothic cathedrals. The use of ribs to distribute the weight of the ceiling is not foreign to civil engineering, either. Some subway stations and industrial facilities offer a vivid example.

However, when it comes to selecting the geometric pattern for rib placement, it usually boils down to the old favorites, such as the barrel vaults with coffered ceilings — a long arch with a reinforcing square mesh of ribs on the inside — and cross vaults, familiar from early Roman architecture and Renaissance churches inspired by it. No complex analysis is usually attempted to identify potential for improvement.

“We decided to analyze several rib patterns to see which of them withstand vertical and asymmetric loads better,” the study’s lead author, Skoltech PhD student from the Mathematics and Mechanics program Anastasiia Moskaleva, said. “We conducted numerical simulations and experiments on the curved-surface polymer composite shells designed in last year’s study, fitting them with stiffening ribs positioned in five different ways, constraining the amount of material expended on ribs in each case to half the material used in the shell itself.”

Depicted above, the original shell was developed via an optimization technique called form-finding, in which the end shape is arrived at through a logical process inspired by processes in nature. It goes back to experiments like those done by Antoni Gaudí, who used to derive highly efficient forms by suspending models in the air to let them sag under their own weight. He then took the shape they assumed and inverted it. In effect, he let gravity do the work, so this approach is often called “form follows force.”

The five stiffening rib patterns initially investigated by the researchers included two time-honored designs — the coffered ceiling and the cross vault — along with two layouts obtained via topological optimization. The top pattern in the center column (see image below) was produced by optimizing the thickness of the shell at every point, effectively redistributing material to where it’s needed most. The bottom pattern was obtained by starting out with two shells on top of each other and only optimizing the bottom one as the seed structure for ribs. Finally, the fifth, biomimetic pattern comes up in turtle shells, dragonfly wings, and elsewhere, but not in its mathematically pure form known as the Voronoi diagram.

Both the physical experiment and the numeric simulation showed the topologically optimized designs to be superior to the conventional and biomimetic rib layouts for withstanding central load. But the tables turned when asymmetric load was applied, which roughly corresponds to snow piling up on one side of the roof or many people moving as a group from one spot to another. In that situation, the cross vault was king, followed by en bloc topological optimization. Importantly, the coffered ceiling and Voronoi pattern stood out as the two options whose performance suffered the least with the switch from symmetric to asymmetric load.

“This prompted us to combine the Voronoi pattern with the best optimized layout from the vertical load experiment in the hope of getting the best of both worlds,” Moskaleva commented. “We carefully examined the structure of the dragonfly wing, which does not actually follow the Voronoi pattern exactly, and found that the stiffening ribs in it can be thought of as forming two separate groups. There’s the more rigid type that counteracts twisting. And then there are thinner ribs, which ensure the overall structural integrity of the wing. And we thought we could reproduce that in vaults.”

To obtain the sixth hybrid pattern, the team first repeated topological optimization, but with a tighter constraint on material expenditure, allocating 70% of the rib material to these primary ribs. This was followed by an additional step where a parametric algorithm used up the remaining material, filling in the thinner secondary ribs according to the Voronoi pattern.

The idea worked so well that the new combined pattern outperformed each of the five initial layouts in both scenarios: for central and for asymmetric loads.

“This demonstrates that topological optimization could actually do a lot for structural design. And yet it is hardly ever used in civil engineering, only in mechanical engineering for things like automobile and aircraft parts,” Moskaleva said. “Sure, the optimized forms are quite intricate and therefore challenging in manufacture. But after the parts of a standard building, such a parking garage, have been optimized and can be reproduced on demand, it will pay off in the long run due to the material conserved. And then there’s the greater creative freedom for the architect.”