The following figures are a geographic map of the site of the Tacoma Narrows Bridge as well as pertinent technical specifications and component drawings.
Figure B1: a geographic map of the site of the Tacoma Narrows Bridge
Figure B2: Technical specifications of the Tacoma Narrows Bridge and component drawings
WHY THE BRIDGE COLLAPSED, COMPETING THEORIES
Initial suggestions as to the cause of the TNB collapse came from the
FWA commission. Without drawing any definitive conclusions, the commission
explored three possible sources of dynamic action; aerodynamic instability
(negative damping) producing self-induced vibrations in the structure; eddy
formations which might be periodic in nature; and the random effects of
turbulence, that is, the random fluctuations in velocity and direction of the
wind. Each source was considered separately in seeking the causes of the
vertical and torsional oscillations. The commission appeared to have identified
the leading possible contributors to the destructive oscillation, since all
competing theories which followed to date fit into one of the above categories.
The standard textbook explanation for the collapse attributes the cause of the failure to the phenomenon of resonance. Like a mass hanging from a spring, a suspension bridge's deck hanging from its cables oscillates at a natural frequency, or more than likely being multi-modal, has several natural frequencies. In order for a resonant phenomenon to exist, the driving force would have to be periodic, that is, varying regularly with respect to time. The mathematical model that most simply illustrates this type of behavior is represented by the following differential equation:
This model, known as a single-degree-of-freedom oscillator, characterizes the motion of a system (the TNB in this case) based on an input force that varies with time explicitly. With this model, resonance, or maximum response amplitude, occurs when the external forcing frequency, omega-subscript f, approaches the square root of k/m, representing the system natural frequency. When resonance occurs, a small input force can produce large deflections in a system. Several proposed solutions to the TNB problem build their theoretical foundation on this concept.
According to several accounts [Billah and Scanlan, 1991], the
turbulent wind blowing over the bridge deck produced a fluctuating force 'in
tune' with one of the structure's natural frequencies, steadily increasing the
amplitude of its oscillations until the structure was ripped apart. The specific
characteristics of the 42 mph wind that blew over the bridge on November 7 were
not recorded in the same detail as the collapse, and much speculation has
accompanied the nature of the flow [Peterson, 1990]. Allowing for periodic wind
gusts, the turbulent conditions that were created could produce a flow pattern
whose time-varying pressure matched a natural frequency of the structure. This
explanation of the disaster appears to fundamentally flawed, however. Resonance
is a very precise phenomenon, requiring the frequency of the driving force to be
near one of the system natural frequencies in order to create large
oscillations. A steady wind still has enough variability in its motion that many
find it difficult to accept the idea of the wind alone having the periodicity
needed to set up resonance in the structure. Applied mathematician P. Joseph
McKenna of the University of Connecticut in Storrs notes that the elegance of
this explanation is too simplistic:
|This explanation has enormous appeal in the mathematical and scientific community. It is plausible, remarkably easy to understand, and makes a nice example in a differential equations class....It is hard to imagine that such precise, steady conditions existed during the powerful storm that hit the bridge.|
Von Karman was convinced that the oscillations
that contributed to the failure were due to the shedding of turbulent vortices
in a periodic manner. This vortex shedding has the potential to produce the
necessary periodicity to establish a resonant condition. Through experimental
observations, von Karman and others had shown that bluff (blunt) bodies like
bridge decks do in fact shed periodic vortices in their wake . Figure 3
illustrates this type of flow pattern around a spherical body.
Figure 3. Natural vortex shedding off a smooth, non-streamlined body suggested as the source of periodicity responsible for the collapse of the Tacoma Narrows Bridge [Petroski, 1991].
These vortices generate alternating high and low pressure regions on
the lee side of the body, which resonates in consequence. Vortices produced in
this manner are termed Strouhal vortices and the rate at which they are shed off
the body in question is governed by the following relation:
For the case of the TNB, the characteristic dimension, that is, the dimension directly associated with the vortex formation, was 8 feet, the depth of the side plate girders and the main obstruction to smooth flow over the bridge deck. Noting that the wind speed was 42 mph on the morning of the collapse, this relation dictates a vortex shedding frequency of about 1 Hz (cycle/sec). According to several sources [Petroski, 1991], the wake region reinforced structural oscillations that grew until the bridge deck could no longer hold itself together. How this would occur is fairly simple. Natural vortex shedding can create a phenomenon known as 'lock-on'. When the frequency of vortices being generated around the body closely matches one of the resonant frequencies of the structure, the driving force feeds off of the structure's motion and vortex-induced vibration can occur that can build to destructive amplitudes. This explanation goes a long way in describing the events that took place in the destruction of the TNB. One of the questionable aspects of this solution, however, is that as the amplitude of structural motion increases, the local fluid boundary conditions are modified in such a way as to generate compensating, self-limiting forces [Scanlan and Billah, 1991]. Figure 4 depicts the various modes that existed for the TNB structure. In all modes except one, the production of self-limiting forces is evidenced by an increase in amplitude up to a specific wind velocity for each mode, followed by a decline in amplitude prior to the manifestation of the next mode. Because of these structural characteristics, the motion of the TNB was restricted to fairly benign amplitudes during its lifetime. The slipping of the cable band on November 7, however, created an unbalanced loading condition that, married with the unstable torsional mode shown in Figure 4, allowed the twisting motions of the bridge to increase steadily to failure. The 'lock-on' theory proposed by von Karman does not appear to account for the fact that observations made at the scene of the accident show that the oscillation frequency of the torsional mode was only around 0.2 Hz, substantially different than the Strouhal frequency of 1 Hz. Thus, it does not seem likely that the power behind the destruction of the TNB can be wholly attributed to the natural vortex shedding of the structure. Even the Federal Works Administration report of the investigation concluded that "It is very improbable that resonance with alternating vortices plays an important role in the oscillations of suspension bridges" [Ross, 1984]. With portions of the natural vortex shedding theory in doubt, a more recent theory was published that disputes the idea that the TNB failure was a case of simple forced resonance.
Robert H. Scanlan, a professor of Civil Engineering at Johns Hopkins
University believes that the forces at work behind the collapse of the TNB were
highly interactive ones. In a paper published in the American Journal of Physics
in 1991, Messrs. Scanlan and K. Yusuf Billah attribute the behavior of the
bridge to a phenomenon known as self-excitation. This concept differs from the
above theory of vortex-induced vibration in the fact that the driving force for
oscillation is not purely a function of time, as described by Equation (1.1),
but is rather a function of bridge angle during torsional oscillation and the
rate of change of that angle. For torsional motion, the behavior is described
mathematically by the relationship:
According to this theory, the motions of the TNB built to destructive amplitudes based on an intimate interaction of the wind and the structure; the wind supplying the power needed for movement, and the movement supplying the power-tapping mechanism [Billah and Scanlan, 1991].
Figure 4. Modal response of Tacoma Narrows Bridge [Billah and Scanlan, 1991].
The distinctions made between the self-excitation theory and natural
vortex shedding theory are founded in the composition of the wake region of the
structure. Experimental testing has shown that bluff bodies in oscillatory
motion shed vortices at both the oscillation and the Strouhal frequencies.
According to Scanlan and Billah, under high amplitudes of oscillation the
periodicity of the Strouhal vortices is interrupted and the vortices resulting
from the periodic motion of the body predominate. With the TNB, as represented
by the body shape in Figure 5, it can be seen that when the shape changes angle
of attack in a fluid stream, it will shed new vorticity in its wake that cannot
be described by natural vortex shedding. The motion that results from such
interaction is a form of separated-flow flutter which tends to excite the
torsional degree of freedom, the unstable mode for the TNB. In contrast with
airfoil-type flutter, in which the high wind speeds will create aerodynamic
forces that can reach magnitudes comparable to the structural inertial
resistance and stiffness, bridge flutter can occur at much lower wind speeds.
Because of the sheer weight of bridge structures, the aerodynamic forces that
develop have little effect on the response modes or their frequencies. These
wind generated forces, however, can influence the overall damping of the
structure, reversing the sign of the middle term in brackets in Equation (1.3),
producing a response whose solution increases without bound. For the case of the
TNB the unstable torsional mode shown in Figure 4 was pushed to destructive
amplitude as a result of the interactive, self-excitation phenomenon.
Figure 5. Self-excitation flow patterns around the Tacoma Narrows Bridge deck. Note
that the vortex formations result from interaction with the bridge1s motion [Petroski, 1991].
The presence of contrasting theories about a structural failure that happened not only several generations ago, but that had the benefit of extensive documentation, underscores the importance that has been assigned to the TNB. In fact, the activity that has occurred in the engineering profession as a result of this specific accident has produced several important advancements in the design of similar structures.
Pinpointing the true cause(s) of the TNB collapse is more than just an academic debate. The need for practicing engineers to have a complete understanding of nature's interaction with their designs has led to new problem solving methods. Though the sensational photographs and film made the TNB an "irresistible pedagogical example," its destruction has brought many advances to the engineering community [Civil Engineering, Dec. 1990]. Now, designers look not only at static loads but also review the implications of aerodynamic effects of their structures. Few bridges, buildings or other exposed structures are currently constructed without testing a model in a wind tunnel. In fact, if a bridge is built with federal grant money, preliminary design must include at least a two-dimensional wind tunnel analysis of the structure, with a three-dimensional model that includes the surrounding terrain being preferred.
The shortcomings of the 'deflection theory' in properly compensating for the loading conditions forced engineers to advance methods that would account mathematically for stresses in all components of a structure, a process that was heretofore an extremely time consuming if not impossible task to accomplish by hand. With the advent of electronic computers after World War II, a numerical solution technique known as the finite element method was able to be routinely applied to bridge designs. This method allows a structure to be mathematically or graphically reduced to a large number of small, interconnected elements. When the overall deflections of the structure are too complex to solve for directly, the finite element method can solve for the deflections of each small piece of the structure, and then sum them up to produce the overall deflection and state of stress. With the advancement of graphics capabilities and processing speed, this testing can now done on desktop computers in any design office [Schlager, 1994].
Additionally, more complex analytical models accounting for the
non-linear behavior of structures like the TNB are currently being proposed
[Peterson, 1990]. McKenna is currently developing a mathematical model that will
hopefully reproduce the behavior of the TNB. When designing suspension bridges
in the past, engineers assumed that the cable stays would remain in tension
under the bridge's weight, acting like rigid rods. This assumption allowed the
designer to use relatively simple, linear differential equations to model the
bridge's behavior. When a structure like the TNB starts to oscillate, the cable
stays alternately lossen and tighten, producing a non-linear effect and changing
the nature of the forces acting on the bridge. According to McKenna ,
non-linear modeling of bridge behavior will provide less predictable solutions:
|Linear theory says that if you stay away from resonance, then in order to create a large motion, you need a large push. Non-linear theory says that for a wide range of initial conditions, a given push can produce either small or large oscillations.|
In combination with other modeling techniques, accurate non-linear models will let engineers observe the response of a structure to a multitude of environmental conditions, such as those that existed during the final hours of the TNB.
Finally, the push towards wind tunnel testing bridge deck section models has led to an abundance of data on flutter response characteristics of various deck shapes [Scanlan and Jones, 1990]. These data assist in guiding a bridge designer's understanding of the general behavior of a shape under various flow conditions. In some cases, the necessity for wind tunnel testing at the initial design stages may be avoided if a sufficiently aerodynamically-similar bridge deck is used.
The theories presented in this paper represent only two of several suggestions about the behavior of the TNB on November 7, 1940. Natural vortex shedding was selected to illustrate the viewpoint of the aeronautical engineering profession in the years following the collapse. The concept of self-excitation, while not entirely new, was presented to illustrate the effect of additional years of testing and analysis on the advancement of scientific methodology.
Bridge design paradigm case studies performed by Sibly and Walker  demonstrate the need for engineers to acknowledge the design history of the structures they create. By studying the temporal cycle of suspension bridge design, there was a period, in the early examples of the structural form, in which aerodynamic force analysis was of secondary importance. Over time, as designers extended the limits of this form, aerodynamic factors became of prime importance and, unheeded, led to catastrophic failure. The collapse of the TNB happened, not because the designer neglected to provide for sufficient strength as dictated by accepted practice at the time, but rather by the introduction of a new type of behavior that was not completely understood. Thus, the trend toward 'streamlining' in the 1930s took suspension bridge design away from the excessively stiff structures of the late 19th century and back to the ribbon-like decks and aerodynamic problems of a hundred years earlier.
Architects and engineers today, recognizing the importance of including a complete analysis of aerodynamic interactions with the structures they design, are able to use advanced modeling tools to assist them in their calculations. Some of these advancements grew out of the events of November 7, 1940 at Tacoma Narrows. Recognizing that scientists and engineers still argue the actual cause of the collapse shows the continued relevance of the Tacoma Narrows Bridge failure on the advancement of the 'scientific method'. This debate further underlies the fact that natural events are complex phenomena that cannot necessary be explained with simplistic equations. Hopefully, this evaluative review has offered engineers some guidance in recognizing potential lapses in their analyses of structures.
In 1950, the state of Washington opened a new 18 million dollar bridge
on the site of the first Tacoma Narrows Bridge. Tested in wind tunnels at the
University of Washington, the four-lane, 60 ft. wide deck and 25 ft. deep
stiffening trusses form a box design that resists torsional forces.
Self-excitation is controlled by hydraulic dampers at the towers and at midspan.
Using the same piers as the original bridge, the new structure was evidence that
the lessons learned about the collapse of 'Galloping Gertie' were being
rigorously applied to new designs.
Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks
all 7 pages in an Adobe pdf file: Billah-Scanlan.pdf (345KB)