Galloping Gertie: The Tacoma Narrows Bridge Collapse

Robert G. Fuller
University of Nebraska - Lincoln

Dean A. Zollman
Kansas State University

The Tacoma Narrows Bridge was not the first suspension bridge to collapse. In fact, a survey of the history of suspension bridges shows that several were destroyed by wind or other oscillating forces (Table 1).

However, the Tacoma Narrows Bridge was by far the longest and most expensive suspension bridge to collapse as a result of interaction with the wind. Perhaps this collapse seemed so striking because nearly 50 years had elapsed since the previous collapse of a bridge.

At the time of the Tacoma Narrows Bridge collapse in 1940, many theories were advanced to explain what had happened. What follows are excerpts from six different explanations. Each presents a slightly different view of the role of design and wind factors in the collapse.

Why Did The Tacoma Narrows Bridge Collapse? (Six Theories)

It is very improbable that resonance with alternating vortices plays an important role in the oscillations of suspension bridges. First, it was found that there is no sharp correlation between wind velocity and oscillation frequency such as is required in case of resonance with vortices whose frequency depends on the wind velocity. Secondly, there is no evidence for the formation of alternating vortices at a cross section similar to that used in the Tacoma Bridge, at least as long as the structure is not oscillating. It seems that it is more correct to say that the vortex formation and frequency is determined by the oscillation of the structure than that the oscillatory motion is induced by the vortex formation. A Report to the Honorable John M. Carmody, Administrator, Federal Works Agency, Washington, D.C. March 28, 1941.

The primary cause of the collapse lies in the general proportions of the bridge and the type of stiffening girders and floor. The ratio of the width of the bridge to the length of the main span was so much smaller and the vertical stiffness was so much less than those of previously constructed bridges that forces heretofore not considered became dominant. Board of Investigation, Tacoma Narrows Bridge, L.J. Sverdrup, Chairman, June 26, 1941.

Once any small undulation of the bridge is started, the resultant effect of a wind tends to cause a building up of vertical undulations. There is a tendency for the undulations to change to a twisting motion, until the torsional oscillations reach destructive proportions. Bridges and Their Builders, D. Steinman and S. Watson, Putnam's Sons, N.Y., 1941.

The experimental results described in a (1942) report indicated rather definitely that the motions were a result of vortex shedding. University of Washington Engineering Experiment Station Bulletin No. 116, 1952.

Summing up the whole bizarre accident, Galloping Gertie tore itself to pieces, because of two characteristics: 1. It was a long, narrow, shallow, and therefore very flexible structure standing in a wind ridden valley; 2. Its stiffening support was a solid girder, which, combined with a solid floor, produced a cross section peculiarly vulnerable to aerodynamic effects. Bridges and Men, J. Gies, Doubleday and Co., 1963.

Aerodynamic instability was responsible for the failure of the Tacoma Narrows Bridge in 1940. The magnitude of the oscillations depends on the structure shape, natural frequency, and damping. The oscillations are caused by the periodic shedding of vortices on the leeward side of the structure, a vortex being shed first from the upper section and then the lower section. Wind Forces on Buildings and Structures, E. Houghton and N. Carruthers, J. Wiley & Sons, N.Y. 1976.

Physical Principles

The general principle of the Tacoma Narrows Bridge collapse is straightforward: a resonance effect. However, the details of the physical mechanisms involved are neither trivial nor obvious.

Every system has a natural fundamental vibration frequency. If forces are exerted on that system at the right frequency and phase, sympathetic can be excited (Section 13.7). Oscillating forces at the right frequency and phase can cause sympathetic vibrations of catastrophic proportions. Thus, the amplitude of the bridge's oscillations increased until the steel and concrete could no longer stand the stress.

But how does the fluctuating force of just the right frequency arise as the wind blows across the bridge? The first idea that comes to mind is that the gusty wind arrived in pulses, thereby striking the bridge at just the appropriate frequency to cause the large oscillations. Closer examination of this explanation shows it cannot be correct. While all wind speeds fluctuate, these fluctuations tend to be random in phase and variable in frequency. Wind gusts are not the answer. Furthermore, the kinds of forces that must be exerted on the bridge are vertical forces -- transverse to the direction of the wind. The wind was blowing across the bridge (from side to side as shown in Figure 1), while the forces on the bridge were acting vertically. These oscillating vertical forces can be explained by a concept called vortex shedding. When a wind that exceeds a minimum speed blows around any object, vortices will be formed on the backside of that object (see Figure 2).

As the wind increases in speed, the vortices form on alternate sides of the backside of the object, break loose, and flow downstream. At the time a vortex near the object breaks loose from the backside of the object, a transverse force is exerted on the object. The frequency of these fluctuating eddies is about 20% of the ratio of the velocity of the wind to the width of the object. These lateral forces can be as much as twice as large as the drag forces. Thus, vortex shedding allows us to understand the origin of the fluctuating vertical forces on the Tacoma Narrows Bridge even though the wind was blowing across it in a transverse, horizontal direction.

Model of the Tacoma Narrows Bridge

Let us create a model of the bridge in which the bridge is suspended by two springs having equal force constants, as shown in a cross-sectional view in Figure 3. We can then write the equations of motion using Newton's second law for translation (Eq 1), and for rotation (Eq 2)

(Eq 1)

(Eq 2)

We then make a small-angle approximation (Eq 3),

(Eq 3)

and assume simple harmonic motion forms for the solutions to the simultaneous equations. Thus, we can write the solutions as

and

We can write in a standard way the two normal mode solutions for this cross section of the bridge. The vertical motion, in which the amplitudes of oscillation of the two sides are equal in magnitude and direction, has a frequency ,

(for A1 = A2) (Eq 4)

The torsional motion, in which the amplitudes of the two sides are equal in magnitude but opposite in direction, has a frequency , where is given by

(for A1 = A2) (Eq 5)

The latter frequency describes the twisting motion that ultimately caused the bridge to fall down. The exact values for these oscillation frequencies depend on the characteristics of the bridge. On the basis of the physical properties of the first Tacoma Narrows Bridge, we find the values appropriate for this analysis are: mass per unit length = kg/m, width of the bridge = 12 m, bridge radius of gyration of the bridge = 4.8 m, effective spring constant = N/m. These numerical values result in a vertical normal-mode frequency of 8 cycles per minute and a torsional motion of 10 cycles per minute. The approximate equality of these two frequencies played an important role in the fate of the Tacoma Narrows Bridge.

In a real system, the normal-mode frequency will not be a single frequency but rather a distribution of frequencies. The energy per unit time that is accepted by a mode of oscillation is given by the following equation:

(Eq 6)

where is the width of the resonance response curve at half maximum. The maximum increases with the tendency of the system to resist oscillations. For the original Tacoma Narrows Bridge, the tendencies of the original bridge to resist vertical and torsional motions were different. Hence, the functions for vertical and torsional motion have different values for, the normal-mode frequency and for. Using the constants given above, the values for and , for vertical and torsimal motion can be computed. Graphs of and versus using typical values for are shown in Figure 4. The area of overlap of the two curves indicates the tendency of the vertical motion to pump energy into the torisonal motion.

As can be seen in Table 2, the ratio of torsional to vertical frequencies for other long bridges is significantly larger than the ratio for the first Tacoma Narrows Bridge.

Even before the Tacoma Narrows Bridge was opened, the vortex shedding forces were pumping energy into the vertical motion of the bridge. Vertical oscillations were noticed early, and many people avoided using the bridge. However, the torsional oscillations did not occur until the day of the collapse. On that day a mechanical failure allowed the torsional oscillations to begin. Because this motion was closely coupled to the vertical motion of the bridge, it quickly led to its destruction. The photographs (Figure 5) show the collapse of the Tacoma Narrows Bridge and provide a vivid demonstration of mechanical resonance.

Public Perceptions

It is interesting to recall some public perceptions involving the bridge collapse. Locals had already noticed for some time that, when driving across this technological marvel, they would experience an undeniable bounce. Some tried to deal with this humorously. For example, the bridge was often referred to as "Galloping Gertie."

An editorial in the Tacoma Times dated August 25, 1940, (soon after the bridge was opened) made the following comment regarding the tolls being charged:

Before the bridge collapse, bridges had been considered secure, so much so that a local insurance agent who had arranged a second $800,000 auxiliary policy on the bridge had never bothered to pay the premium. Instead, he pocketed the money and was sent to jail following the disaster.

The collapse of the Tacoma Narrows Bridge was a watershed in the design of suspension bridges. The debates as to who was responsible and whether anything could have been done to prevent the collapse continue.

Suggested Readings

Aerodynamic Stability of Suspension Bridges, University of Washington, Engineering Experiment Station, Bulletin No. 116, Parts I, II, III, IV, and V, University of Washington Press, Seattle, 1949, 1950, 1952, and 1954.

O'Connor, C. Design of Bridge Superstructures, New York, John Wiley & Sons, 1971.

The Failure of the Tacoma Narrows Bridge. A Reprint of Original Reports, School of Engineering, Texas Engineering Experiment Station, College Station, Texas, Bulletin No. 78, 1944.

Fuller, Robert G., Dean Zollman, and Thomas C. Campbell, The Puzzle of the Tacoma Narrows Bridge Collapse, New York, John Wiley & Sons, 1982.

Houghton, E.L. and N.B. Houghton. Wind Forces on Buildings and Structures: An Introduction, New York, Halsted Press, 1976.

Scigliano, Eric. "Galloping Gertie," Pacific Northwest, January 1989.

Simiu, E. and R.H. Scanlan, Wind Effects on Structures: An Introduction to Wind Engineering, New York, John Wiley & Sons, 1978.

Steinman, D.B. "Suspension Bridges: The Aerodynamic Problem and its Solution," American Scientist, July 1954, pp. 397 - 438.

Wind Effects on Bridges and Other Flexible Structures, Notes on Applied Science, No. 1, National Physics Laboratory, London, 1955.

Essay Questions

1. The Tacoma Narrows Bridge was a two-lane bridge. How would the bridge have behaved had it been a four-lane bridge?

2. Based on current knowledge, what would have been your advice for a quick fix for the bridge that could possibly have averted the disaster?

Essay Problems

1. Derive the frequency relations of Equations (Eq 4) and (Eq 5) by solving Equations (Eq 1), (Eq 2), and (Eq 3). Assume that the solutions are simple harmonic vibrations.

2. Calculate the effect of stiffening the bridge suspension by increasing the effective spring constant of the Tacoma Narrows Bridge by 50 percent.

3. How much change in the total mass of the Tacoma Narrows Bridge is necessary to bring the vibrational frequency down to that of the Golden Gate Bridge? Assume uniform mass distribution.

Figure Captions

Figure 1
Direction of wind blowing across the bridge.

Figure 2
Vortex shedding.

Figure 3
Cross-sectional view of the bridge.

Figure 4
Typical values for .

Figure 5
(a) High winds set up vibrations in the bridge, causing it to oscillate at a frequency near to one of the natural frequencies of the bridge structure.

(b) Once established, this resonance condition led to the bridge's collapse. (UPI/Bettmann Newsphotos)

TABLE 1 Collapsed Suspension Bridges
Bridge	Designer	Span Length (ft)	Failure Date
Dryburgh (Scotland)	J. and W. Smith	260	1818
Union (England)	Sir Samuel Brown	449	1821
Nassau (Germany)	L. and W.	245	1834
Brighton (England)	Sir Samuel Brown	255	1836
Montrose (Scotland)	Sir Samuel Brown	432	1838
Menai (Wales)	T. Telford	580	1839
Roche (France)	LeBlanc	641	1852
Wheeling (USA)	C. Ellet	1010	1854
Niagara (USA)	E. Serrell	1041	1864
Niagara (USA)	S. Keefer	1260	1889
Tacoma Narrows	L. Moisseff	2800	1940

TABLE 2 Ratio of Torsional to Vertical Frequencies in Suspension Bridges
Bridge	Length (m)	fv(min-1)	ft(min-1)	Ratio ft/fv
Verrazano	4260	6.2	11.9	1.9
Golden Gate	4200	5.6	11.0	1.9
Severn	3240	7.7	30.6	3.9
Tacoma Narrows (1st)	2800	8.0	10.0	1.25

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