How crowds can make bridges wobble and sway

In a new study published in the journal Nature Communications, a group of Georgia State mathematicians led by professor Igor Belykh along with colleagues at the Universities of Bristol, Cambridge, and Leicester in Great Britain show how a bridge -- even one as highly engineered as the Golden Gate or the Brooklyn Bridge -- can become suddenly unstable.

If enough people are crossing by foot, each walking at their own natural speed, they transfer so much energy into the bridge that it may start to oscillate. Then, as each individual walker adjusts their steps to try not to fall, they destabilize the bridge even more.

Bridge wobbles can lead to mass panic -- in 1987, on the 50^th anniversary of San Francisco's Golden Gate Bridge, 300,000 celebrants tried to cross and the bridge groaned and swayed, causing people to panic, vomit, and start throwing bicycles and even baby strollers into the ocean to lighten the bridge's load.

The new work topples the long-accepted explanation for shaky, shimmying bridges -- which for the last 20 years was thought to be due to mass synchronization of footsteps.

"Our work shows that very tiny vibrations from each person walking can get amplified significantly," said Belykh.

At that point, people notice the wobble and adjust their footsteps to keep their balance. That only makes things worse, and the bridge can become unstable. The new work relies on data from 30 different bridges and utilizes complex mathematical formulas that predict the precise tipping point for a particular bridge -- the exact number of people that will start to make it move. The study also shows that bridges in general may be more vulnerable than once thought.

Bridges have natural frequencies of vibration, due to phenomena like air currents and traffic. Predicting their point of instability is not easy.

The new insights will help engineers and bridge designers "build better, safer bridges," said Belykh. "The geography of these crowd-induced instability events is truly worldwide."

In 2003, for example, New York City had a blackout and a crowd of pedestrians walked home across the Brooklyn Bridge in the dark. This caused the famed bridge to vibrate and rock so much that some felt seasick and could not keep their balance if they stood still. And in 2000, London's newly unveiled Millennium Bridge was nicknamed wobbly bridge because of its swaying motion as crowds crossed on opening day, causing authorities to immediately close the bridge and not re-open it for two years.

It was the Millennium Bridge that inspired the long accepted explanation of bridge shakiness -- that humans were walking in lockstep, synchronized like swinging pendulums. This is called phase-locking. Looking at videos of the walkers, their heads and torsos did seem to move in unison like a wave. It was thought this massive left-to-right movement tipped a bridge back and forth. In fact, the Albert Bridge, which was built across London's Thames River, was nicknamed "The Trembling Lady" and boasts a sign instructing nearby soldiers who are marching across to break step.

"This explanation was so popular," said Belykh, "it has been part of the scientific zeitgeist."

One reason it was popular is the idea that coherent behavior can emerge in complex systems as they oscillate -- whether it is neurons or fireflies or human footsteps. And yet the theory was instantly questioned by Nobel Prize winning physicist Brian Josephson, only four days after the Millennium Bridge incident. He suggested what the current work proves: it was people trying to keep their balance on the swaying bridge that intensified the wobble.

The new paper builds on insights first published by Belykh and colleagues in 2017, which found that when 164 people walked on Millennium Bridge, it could remain steady -- but adding one more person tipped the balance. The models take into account both side-to-side (lateral) and forward motion.

"Think of passengers walking on a boat rocking side-to-side in a stormy sea," said Belykh. "They will adapt their motion both laterally and in a forward direction in response to the shaking of the boat. In particular, they will slow down their forward motion."

As they do this, they impact bridge stability through a mechanism called "negative damping," which essentially means that pedestrian movements excite the bridge and increase its oscillations. Belykh compared this to a rusty playground swing that was hard to move, but if enough parents gave it a shove, it would start swinging on its own.

"Bridge designers should be aware there could always be dangerous instances of negative damping," said Belykh, if the crowd is sufficiently large. "Our formula provides useful estimates, given the expected number of pedestrians using a bridge."

Engineers should be able to plug in details about a bridge and potential users and build accordingly. The mathematical formulas rely on the concept that humans function a bit like inverted pendulums (with two pendulum legs) and thus can be simulated as 'crash test dummies' on videos and in diagrams of bridges.

Looking ahead, the researchers plan to study the effect of human-to-human interactions and movement in dense crowds. They also plan to harvest a bridge's inherent energy, since all bridges oscillate as they are used. Working with Heriot-Watt University in Edinburgh, and researchers at Georgia Institute of Technology, Belykh plans to harvest this unused energy to power small sensors that can monitor the structural integrity of a bridge.

The calculations and simulations for the study were performed in part by Georgia State Ph.D. student Kevin Daley and former Ph.D. student Russell Jeter, now at a robotics company Motus Nova. The work was funded by National Science Foundation Grants, including a grant on bridge stability and a grant on energy harvesting for bridge sensors.