Analogies Between Coupled-Mode Gate Vibration and Coupled-Mode Flutter: The Need for Dynamic Design

Analogies Between Coupled-Mode Gate Vibration and Coupled-Mode Flutter: The Need for Dynamic Design

DOI: 10.4018/978-1-5225-3079-4.ch013
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In this chapter the similarities between the Tacoma Narrows Bridge failure in 1940 and the Folsom Dam gate failure in 1995 are examined. In both cases, static design guidelines were followed in the design of the structure under the assumption that large, massive structures would not be susceptible to dynamic excitation. Fundamentals of two-dimensional coupled mode flutter are presented. The frequency mode coalescence that occurs in two-dimensional flutter is noted. It is seen to have some resemblance to the mode-coupling in the coupled-mode instability of Tainter gate. The need for development of dynamic design guidelines for Tainter gates is argued to be parallel to the need for dynamic design guidelines for suspension bridges in the wake of the Tacoma Narrows failure.
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In the previous chapter, we described how fluid-structure interaction caused gates at the Folsom Dam and Wachi Dam to fail. Forty-five years before the Folsom gate failure, the first Tacoma Narrows Bridge collapsed due to the effects of moderate winds, well below the design wind speed. What do these disasters have in common? We suggest that, in both cases, the structures were designed for static stability without taking into account the dynamic forces exerted by the fluid on the structure. Table 1 shows at least 10 major suspension bridge failures in the 19th century due to the wind. It is unlikely that the bridge builders who were professionally active in the mid-1930s and 1940s had any first-hand knowledge of these earlier failures. In fact, the deflection theory of bridge design that was popular at the time probably lulled the younger generation of engineers into a false sense of confidence that increasingly elegant bridges could be constructed with less material and longer spans. However, paraphrasing George Santayana, those who do not learn from the mistakes of history are bound to repeat them.

When the Tacoma Narrows Bridge was opened in 1940 with a span of nearly 860 m, it was the third longest suspension bridge in the world at that time. Carrying only two lanes of traffic, the bridge was very slender. The bridge deck was supported by 8-ft high, solid girders, giving it an H-shaped cross section.

On 7 November 1940, four months after the opening of the bridge, with a moderately strong wind of about 42 mph (18.8 m/s) blowing, the bridge underwent large amplitude heaving (vertical) vibration. After more than an hour of heaving vibrations, the vibration mode abruptly changed from a heaving mode to an anti-symmetric, full-span torsional mode with a torsional node at the mid-point of the center span. Initially, the torsional mode frequency was 14 cycles/min (0.233 Hz), decreasing later to about 12 cycles/min (0.20 Hz). In this torsional mode, the bridge failed in a little more than one hour. The conclusion of the forensic report by Ammann et al. (1941) was that a center cable tie became loose in the heaving mode, allowing the cables on opposite sides of the center span to become asymmetric. The strong, steady wind blowing across the bridge increased the torsional motion that lead to complete destruction of the bridge.

The Tacoma Narrows Bridge had already shown a propensity for vertical (heaving) wind-induced vibrations while under construction. These vibrations continued, even with relatively low speed winds, over the four months the bridge was open to traffic. The vibrations were not predictable; sometimes they were present and other times not, under apparently similar conditions. Methods of attenuation were explored and a number were implemented, including adding diagonal ties at the center span, with mixed success. The most common mode of vibration was a half-wavelength heaving mode shape between the towers, with a maximum mid-span amplitude of about 2 ft (0.61 m). With increasing wind speed, the vibration mode often, but not always, changed to one with a shorter spatial wavelength with an increase in frequency and a corresponding increase in the number of nodes, up to nine nodes. A maximum peak-to- peak amplitude of about 5 ft (1.52 m) was observed in a mode with two nodes between the towers at a frequency of vibration of about 12 cycles/minute.

Table 1.
Failures of suspension bridges in the 1800s, followed by a gap of more than 50 years between the failure of Keefer’s Niagara Bridge and the Tacoma Narrows Bridge
BridgeDesignerSpan (ft)Span (m)Failure Date
Dryburgh Abbey (Scotland)John and William Smith26079.251818
Union (England)Samuel Brown.449136.851821
Nassau (Germany)Lossen and Wolf24574.681834
Brighton Chair Pier (England)Samuel Brown.25577.721836
Montrose (Scotland)Samuel Brown432131.671838
Menai Strait (Wales)Thomas Telford580176.781839
Roche-Beruard (France)P. LeBlanc641195.381852
Wheeling (USA)Charles Ellet1010307.851854
Lewiston-Queenston (USA)Edward Serrell1041317.301864
Niagara-Clifton (USA)Samuel Keeler1260384.051889
Tacoma Narrows (USA)Leon Moisseiff2800853.441940

Adapted from Ge &Tanaka (2013).

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