When the earth trembled near Izmit, Turkey, in August 1999, more than 17,000 people died and 77,000 homes and businesses were destroyed. Some buildings were reduced to rubble; others collapsed into pancakelike stacks. But some of the destruction followed a strange pattern. In Adapazari, about 25 miles from Izmit, dozens of buildings pitched over at crazy angles but were not severely damaged. It was as if a giant hand had gently tipped them over.
Engineers had a clue from earlier earthquakes about what might cause buildings to tilt: Under certain conditions, soil can liquefy, turning into a gooey jelly that can no longer support the weight above. It’s a simple matter of balance, says Chris Rorres, a mathematician at the University of Pennsylvania. The structures in Adapazari weren’t designed to float, so they did exactly what a badly designed ship would do in roiling seas—they capsized.
When he began studying flotation in 1995, Rorres simply wanted to fill in the gaps in a classic book of ancient Greek science, On Floating Bodies. Written by Archimedes, a mathematician who lived in the third century B.C., the book set forth two laws of buoyancy that have become the foundation of modern shipbuilding. The first law states that any floating object will displace a volume of water whose mass equals the object’s mass. An iceberg that is 90 percent as dense as seawater, for instance, displaces 90 percent of its volume, so 90 percent of it lies below the surface. The second law determines how objects of different shapes and densities orient themselves as they float. Archimedes knew, for instance, that a shallow, bowl-shaped paraboloid will float in an upright position no matter what its density. But a deeper one will float in different positions, depending on its density. Some will float straight upright; some will list gently to the side; others will tilt over violently, like a capsizing ship.
When exactly will an object capsize and in what position will it settle? Archimedes couldn’t fully answer this question, so Rorres set out to do it for him. Using a notebook computer, he drew the equilibrium surface of a floating paraboloid—a graph showing the angle at which it would tilt, depending on its density (compared with the liquid around it) and its aspect ratio (height divided by base). Deep, bowl-shaped objects, he found, are so unstable that very small changes can cause sudden, catastrophic shifts.
Going from bowl-shaped floats to predicting what happens to buildings in liquefied soils seems like a stretch, but in Rorres’s virtual world it requires only two minor modifications: turning the paraboloid upside down (to make it domelike) and giving it a density typical of buildings, which are mostly hollow. In his computer model, such structures topple over as soon as the aspect ratio reaches 1.4. That’s not so different from what reconnaissance teams observed at Adapazari. “A strong correlation was shown between aspect ratio and the degree of tilting,” says Joseph Wartman, a geotechnical engineer at Drexel University. “Buildings with an aspect ratio greater than 1.8 overturned—it was almost a universal rule. Buildings with an aspect ratio of 0.8 only had a little differential settlement. That suggests there’s a threshold being crossed, at which all the buildings are falling over. And you see that in the computer models. When you cross the threshold, you get catastrophic change.”
Some who study soil liquefaction are skeptical. Ross Boulanger, an engineer at the University of California at Davis, argues that even after soil liquefies, it is still a two-phase material: a dense slurry of sand particles in highly pressurized water. So Rorres’s model, which treats the soil as a classic fluid, may be just as misleading as one that treats the soil as a solid. Not only that, the ground may liquefy in patches, creating “sand boils,” or it may liquefy 10 to 20 feet underground. All of this makes predicting how real buildings will float a lot more difficult.
Should buildings in earthquake zones be designed to float? Perhaps not. “It could be dangerous,” says Les Youd, an engineer at Brigham Young University. “If too much weight was placed on one side of the building, the building could tip over.” Yet many buildings, such as San Francisco’s recycling facility at Pier 96, have so-called floating foundations. This bargelike foundation extends from the building’s bottom and employs Archimedes’ principles of buoyancy but not Rorres’s newer findings. The alternative is to keep soil from liquefying under new buildings by adding cement to it or to support the building on pilings. But many older buildings remain vulnerable. Just how far they’re likely to tilt when the earth becomes more like the sea is a question that Rorres—with a little help from Archimedes—may have an answer to in coming years.