# Math, Engineers, Bridges, and Hand Waving

After seeing the alarming video of the Minnesota I-35 bridge collapse in class today, I’m reminded from my recent reporting on the flattening of the Golden Gate Bridge 25 years ago on how flexible numbers can be. Searching for a simple detail revealed how much hand waving goes on that I hope is reserved for the press and not actual analysis.

In 1987, hundreds of thousands of people crammed onto the iconic Golden Gate Bridge. Image from wikimedia commons, taken by Cary Bass.

To quickly summarize: hundreds of thousands of people crossed the Golden Gate Bridge on foot to celebrate its 50th anniversary. The weight, more than two times of cars in bumper to bumper traffic, was enough to make the middle sag 7 feet, flattening the suspension bridge’s slight curve.

The number I was looking for was pretty simple: just the amount of weight the bridge was designed to support. The original design load of 4,000 pounds per foot of bridge is pretty well documented, but renovations during the mid-80s removed a dense concrete layer and replaced it with lighter and stronger steel frameworks. Those renovations should have boosted the design capacity of weight, and I just wanted to know what it was. But no one I talked to (including officials and engineers that oversee the bridge) could tell me what that new number was. More surprisingly, three of the four engineers I talked to for the story gave me starkly different ideas about the bridge’s strength and how to think of it that day, none of which particularly agreed with the chief engineer’s numbers as reported by the Merc at the time. (The other engineer I talked to recused himself from analysis because he wasn’t familiar with the Golden Gate’s particulars).

Off the bat, all the experts agreed that it would be practically impossible for the Golden Gate to break and drop people to their deaths from simply cramming more and more weight onto it. But they disagreed at the point where

the bridge would start getting damaged– at what load metal might start to permanently bend (like a bent paper clip), rivets might break, joints get overstressed, all which would require maintenance.

According to the Merc’s story in that era, the day after the bridge flattened, the bridge’s chief engineer said that the bridge was no where close to being damaged. He said that the bridge was designed to hold 5,700 pounds per foot of bridge, while the crowd weighed about 5,400 pounds, using rather generous estimates. But even then there’s a buffer where the bridge can gain even more weight without any deleterious effects. Some engineers call it the factor of safety. “There’s no way we could have gotten enough people on the bridge to cause any problem,” the chief engineer said at the time.

But now, one engineer insisted that the bridge was close to being damaged. Partly because the bridge flattened, which should never happen, he said, and also because he estimated that the weight on the bridge was substantially higher than the standards set by AASHTO (the wordy American Association for State Highway and Transportation Officials), which he didn’t think the Golden Gate far exceeded. He said that the load already significantly intruded upon the factor of safety, and was close to completely exhausting it. But he also said that because the bridge went back to its original shape, that was a sign that it wasn’t permanently damaged.

Another engineer, who analyzed the bridge in the years after the bridgewalk, remembered that the load on the bridge was about the same as the design load. But his concept of factor of safety was a little bit different. Dipping into the factor of safety meant that the bridge would start getting damaged, but not completely fall apart. In a way, this idea is similar to the previous idea that the bridge was being close to damage, but the difference of such a simple, standard idea, was a little odd.

The current chief engineer gave me a far different answer, saying that the bridge is and was capable of being completely filled with 36 ton, 28 foot long trucks, which would be far heavier than the 1987 crowd of people. Rough calculations would put that capacity at more than 15,000 pounds per foot of bridge, more than 2.5 times the 1987 estimate. On top of that there’s an additional safety factor which would put that capacity through the roof, before the bridge would begin to get damaged. In short, the engineer said, there was no practical way to put enough weight on the bridge to get it to crack.

Now all of these experts are pretty well established– they’re either professionals who design or analyze bridges for a living, or professors at prestigious universities. What are we to do if they can’t agree on what seems to be a pretty straightforward concept?