The facts Arup estimates that up to 100,000 people crossed the Millennium Bridge on its opening day, with a maximum of 2,000 people on the crossing at any one time, a density of around 1.5/m 2.
Short term lateral oscillations at between 0.5Hz and 1.0Hz were observed, with maximum accelerations of 250milli-g.
Groups of people were seen walking in unison, a phenomenon Arup later dubbed 'lock-in'.
Arup later uncovered a number of cases of lateral sway on very different crossings when pedestrians were present.
Arup concluded the problem was 'synchronous lateral excitation' caused by the input forces from pedestrians.
Dr Richard Bassett, reader in geotechnical engineering at University College, London
During the last 12 months we have constructed a small scale model of the bridge to explore its dynamic responses. The bridge is not really a suspension bridge as conventionally envisaged as the connections between the cables and the deck are not flexible hinges but stiff 'wing arms' to enable the key horizontal curvature to be applied to the cables.
The usual deck unit consists of two near circular axial tubes welded to two pairs of 'wing arms' one at the centre of the unit and one at one end. To connect all these units together while keeping the vertical alignment, yet allowing vertical articulation, there is a 'quasi pin' connection at each end of the two tube members. This connection is absolutely key as it replaces the flexibility of the hanger on a conventional suspension bridge.
Arup argued that the stiffness of the bridge comes from the very high tension in the main cables. Our test results tended to confirm this argument, as we had considerable difficulty in stimulating any noticeable horizontal oscillation in either the northern or central spans. The southern span, about which Arup said little, is a different matter.
It has a complete mix of deck units due to its division into two approach walkways and therefore appears to have very variable stiffness along its length.
This is unusual for a suspension bridge and our model work suggests that the south span is very sensitive and can quite easily be triggered into a vertical oscillation mode, coupled with a smaller torsional mode.
The torsional mode due to geometry appears to develop a horizontal component. In our tests the dynamics of the south span stimulated the centre span quite noticeably. In my opinion this is because of the use of very odd single wing arm units at the piers, which act as very flexible dynamic transmitters between adjacent spans, and I note the proposed damping system has very strong ties to the piers to overcome this problem.
Robert Benaim, chairman of structural engineer Robert Benaim Associates
Engineers must innovate. However, innovation carries risks and sometimes things go wrong.
Most failures are private, and the lessons learnt by only the few.
Arup has, to its credit, disseminated its analysis so that all may learn. However, my instincts tell me that the explanation is not complete.
My understanding of suspension bridges is that the stiffening girder needs bending and torsional characteristics that preclude the cables vibrating out of phase, Tacoma Rapids style. As the Millennium Bridge has no stiffening girder, what prevents this out of phase vibration?
Furthermore, I question whether the traditional calculations for the stability of bridge decks are applicable when the cables only provide torsional and bending stiffness themselves.
On opening day pedestrians reported a rolling motion, with horizontal movement on the centre line, and additional vertical movement at the parapets. Such movement is not compatible with pure horizontal vibration of the cables as postulated by Arup;
it is compatible with out of phase vibration of the two sets of cables. There may be some other explanation, but it has not been offered.
Arup has asserted that lateral vibration may be triggered by a critical mass of pedestrians, and that once started it will be reinforced by people adapting their walking patterns.
This is a form of instability to which I believe the Millennium Bridge would be particularly susceptible. Due to the lateral inclination of the cables the deck will move horizontally under any vertical load or impulse that is not applied on the bridge axis.
It is unnecessary to seek out tiny horizontal impulses from normal walking; the much bigger vertical impulses will excite the bridge laterally, with the change in walking patterns magnifying the effect.
Lacking restraint from a torsionally stiff deck, the cables may then vibrate out of phase with combined vertical and horizontal motion.