November 2, 2013

Napkin Math: Will the Leaning Tower of Pisa Ever Fall?

by Dan Stone

N 45° 26' 23", E 12° 18' 53"

There are many symbols of Italy—gondolas, Sophia Loren, gelato—but none is more iconic than the Leaning Tower of Pisa. I’ve always thought it strange that an engineering screw-up based on poor planning and shoddy construction has become one of the Seven Wonders of the World.

Our question in Italy: Will it lean forever, or will it eventually fall? Over a couple Italian beers, we plotted the physics.

The tower was built back in 1173 with marble, so the assumptions of modern engineering don’t really apply. Not long after it was built, the ground under one side started to sink under the weight of the stones. Workers tried to compensate by making the columns on the other side slightly bigger to even out the lean. But that didn’t work.

The structure continued to decline into the 20th century, when engineers realized they had a problem. The tower was leaning more, about 1/20th of an inch every year, which would increase as the tower’s center of gravity shifted. Eventually, it would simply fall. That started a few different restoration efforts. One was to inject cement into the ground under the tower to stabilize it. That didn’t work so well, it just made the ground heavier, which made it subside more. Then engineers in the 1990s figured they needed to remove soil underneath the non-leaning side, inject more cement, and stabilize the structure with steel cables. Working delicately for about four years, they reduced the lean by 17 inches.

Now back to the physics: In terms of angles, the tower now leans at 3.99 degrees. Considering the weight and height of the tower, physicists have said the maximum angle would be 5.44 degrees before the tower falls (at its worst, the tower once leaned briefly at 5.5 degrees. Everyone is still baffled that it didn’t collapse).

With the current restoration, engineers now think the tower can stand for another 200 years. Advanced technology by then could extend its life even further. The payoff is obvious if you consider the billions of Euros it brings to Italy in tourist revenue and swag. One thing less clear: How many more kitschy photos will be taken of people holding it up?

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