The cause of tacoma bridge collapse

The Tacoma Bridge was built between 1938 and 1940. It was 5000 feet long, 39 feet wide, with two lanes of traffic, and 2800 feet long. The bridge was built in the state of Washington. According to data on the history of suspension bridges, the findings show that many were demolished due to the wind factor. The Tacoma was not an exception only to the fact that it was the most costly and longest bridge ever to have collapsed. The bridge collapsed more than 50 years since the last one.
According to Scheer, Joachim, and Linda (2011), the main cause for the crash of 1940 was disproportionate versatility. The incident was related to a slippage of the cable bands on the northern tower. They separated into two parts of unequal lengths. As a result, the girders twisted. During the collapse of the bridge, its main suspension cables twisted side to side and then they broke loose 100 feet down. The cables glided from their spots and the saddles went atop of each mast. The north cable broke over 350 wires (Scheer et al., 2011). Also, other cables were sternly stressed and the main ones were totally destroyed. The materials that left after the collapse consisted only of scrap metals. The structure’s fundamental weakness was the aspect of flexibility. There was an extreme vertical torsion, and the side spans were too long when compared to the diameter of the centre span. More so, the deck was lighter and too shallow. At eight feet, it was approximated at 1/350 ratio when compared to the centre span (Scheer et al., 2011).
One of the factors that contributed to the failure of the bridge structure – torsion – is also known as vortex shedding. Principally, this case of vortex shedding can be explained as follows: as the wind struck the side of the galloping deck, a small amount of wavering occurred due to elasticity of steel and its subsequent changes under high stress. As a result, a vortex formed led to a whirling wind forces which eventually twisted the deck. Normally, it would naturally return to its earlier position. However, as it reverted, the speed did not match the lifting force, thus resulting in a lock-on event. When the bridge motion changed, the structure absorbed more energy from the wind. The motion controlled the wind vortex produced by the force – also known as self-excited movement. Vortex and torsional motion sequentially transpired at low wind speeds. Now, the wind was way beyond its natural ability (Melchers & Beck, 2017). The Galloping Gertie induced more motion which intensified to a point where the bridge structure could not hold any further.
To further elaborate the details of the collapsed bridge, the paper discusses the following sub-topics:
1. Suspender cables. The tragedy broke many suspender cables and a significant number of them was completely destroyed.
2. Towers. The East and West towers, which included bracing struts, were twisted due to the stress beyond its elastic limit. As a result, the metal holding the tower was buckled and permanently distorted (Fuller et al. 2008).
3. Deck-floor system. Steel and concrete at the centre of the span were already destroyed and they were laying on the bottom of the narrows. The rest of the concrete on the side spans was beyond repair. Sections of the floor system were bent and overstressed.
4. Side spans and piers. Once the centre section was destroyed, plummeting of the side spans followed. The plate girders and floor beams were severely buckled and beyond repair. However, both East and West piers sustained minimal damage. After the centre span had collapsed, it resulted in fractional sheering of the rivets which directly connected the turrets to the top of the piers (Sachs, 2013).
5. Anchorage. Surprisingly, the anchorage that linked the main cables remained undamaged. In order to build a replacement for the bridge, the architects would have to remove part of the concrete in order to spin new cables. More so, the plate girder and deck functioned as aero-foils during the failure, thus creating lift-to-drag effect.

Figure 2. Cause map (Think Reliability, 2008)
After the incident, bridge engineers documented that light and narrow bridge structures were theoretically and functionally sound. Failure of the bridge revealed the drawbacks of deflection theory. As a result, aerodynamic stability has reinforced the approach (Ausoni, Farhat, Escaler, Egusquiza, & Avellan, 2007). For self-anchored bridges, a fabrication chamber needs to be subjected to live loads, which are articulated after an improved initial state. A three-span suspension bridge should be steadily derivative of the tower effects. The theory also applies the unstrained length technique. This approach keeps all components stable during non-linear iteration. Deflection theory assumes that all the bridges are likely to remain within the limits of proportionality. Also, the cables are assumed to be elastic. Although deflection theory is still an integral part of bridge engineering, it serves as a benchmark for better and complex analytical techniques such as finite element programs. Such programs help engineers in calculation problems that evaluate stress in a suspension cable (Sachs, 2013).
The actual measurement of the main-span was 2,800 feet while the width was 39 feet. During the construction of the bridge, there were times when up and down waves were experienced at unusual amplitude (Arioli & Gazzola 2013). The physics behind the explanation on the failure of the Tacoma Narrows Bridge revolves around fluid flow as well as resonant vibrations. Tacoma Narrows Bridge’s extreme flexibility was one of the leading factors to its failure. The flexibility nature of the bridge was observed in both vertical and torsion aspects of suspension. The high flexibility of the bridge was due to the shallow reinforcing girders that were used as well as the narrow roadways with respect to the length of the span (Plaut, 2008). There were theories used to explain the failure of the Tacoma Narrows Bridge.
One of these theories is the random turbulence theory. According to this theory, the collapse of the bridge is attributed to the wind pressure that caused natural frequencies of the bridge (Dalen, 2016). Vortex shedding theory is also used in explaining the collapse of the bridge. According to this theory, vortex shedding triggered bridge oscillations. Low mechanical dumping allowed the structure to vibrate for surprisingly long periods. The H-shape of the cross-section area and its slender design contributed to its collapse.

Figure 1. (a) The flutter derivative for a Tacoma-like H-section (calculated using Morgenthal's VXFLOW). (b) A zoom at low reduced velocity (Morgenthal, 2002)
A tunnel test formed a big part of the report. Upon calculation of static wind forces, the bridge was not strong enough to withstand them. The importance of such dynamical studies had not yet been absorbed by the engineering community (Melchers & Beck, 2017).
Another explanation of the bridge’s collapse is the theory of minimum energy. A bridge normally twists fully or it may divide into half spans which allows for reverse rotations. However, nature prefers half -span options since it involves minimal energy. The Tacoma buckled during the torsional mode. Precisely, the 600-feet centre span broke loose. There is a possibility that the span experienced fatigue failure due to stress reversals. The other argument elaborates on the increased instability due to the angular displacement. Principally, the two explanations are interrelated. For instance, accumulated fatigue is likely to lower the yield and increase ultimate stress limits (Voigt, 2012).
The ultimate failure of the Tacoma Narrows Bridge was related to the phenomenon of the aerodynamically induced state of self-destruction. The aeroelastic phenomenon involved in the failure is presented as interactive in the sense that it accommodated wind forces that were powerfully linked to the structural motion (Voigt, 2012).
As it is aforementioned, strong wind was the major contributing factor of the bridge failure. The 35-mile-per-hour winds prompted crosswise vibration style on the bridge, with an amplitude of 1.5 feet (Dalen, 2016). Over time, the swiftness of the wind amplified to 42 miles per hour. The supporting cable positioned at the mid span of the structure broke, thus ensuing in an unhinged loading situation. Due to the unbalanced position, it resulted in a 0.2 Hz torsional pulsation with a breadth of up to 28 feet (Arioli & Gazzola, 2013).
The structure was eventually split into parts in the result of the torsional effect. The halves of the bridge began vibrating out of phase with one other; one 1/2 rotated clockwise while the other half rotated counter-clockwise (Ge & Tanaka, 2000). Due to the different motions and rotation of the two halves, the bridge spanned in alternate polarities. The law of minimum energy provides an explicit explanation on the bridge spanning in alternate polarities. According to the law of minimum energy, a suspension bridge has the possibility of twisting as a whole or dividing into sections that rotate in opposite directions (Ausoni et al. 2007).
Conclusion
In any structural or engineering design, a human error is always a factor to consider. Moreover, it is important to understand the dynamics, local conditions and materials when accessing the failure of any structural component. For instance, the failure of the Tacoma Narrows Bridge happened due to the combination of structural and aerodynamic stability problems. The designer of the bridge did not account for the aerodynamic forces within the local surrounding that would have affected the stability of the structure. The actual failure mode requires the use of complex mathematical analysis to calculate all the associated degrees of freedom and set of load imposed.References
Arioli, G., & Gazzola, F. (2013). Old and new explanations of the Tacoma Narrows Bridge collapse. In Atti XXI Congresso AIMETA, Torino, 10.
Ausoni, P., Farhat, M., Escaler, X., Egusquiza, E., & Avellan, F. (2007). Cavitation influence on von Kármán vortex shedding and induced hydrofoil vibrations. Journal of fluids engineering, 129(8), 966-973.
Dalen, M. A. (2016). Aerodynamic stability of suspension bridges (Master's thesis, Norwegian University of Science and Technology).
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Sachs, P. (2013). Wind forces in engineering. Elsevier.
Scheer, Joachim, and Linda Wilharm. (2011). Failed Bridges: Case Studies, Causes and Consequences. Foreword by Christian Menn. Berlin: Ernst, Wilhelm & Sohn. Internet resource. Retrieved from: https://www.wsdot.wa.gov/TNBhistory/Machine/machine3.htm
Think Reliability. (2008). Cause map. Retrieved from https://www.thinkreliability.com/wp-content/uploads/2017/05/CM-TacomaNarrows.pdf
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