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Projectile motion
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Dezi et. al (2010)
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Projectile motion
In this article, we'll explore the most famous bridge collapse in history, the Tacoma narrow bridge failure. First, we'll start with a brief history of bridge types and design to set the scene.
Introduction
Bridges are a part of human history. Engineers have revolutionized bridges, from simple wooden log bridges to ultra-slender suspension bridges that span kilometers. This development is the accumulated lessons of endless experiments and failures. Before we touch on a particular failure that changed bridge design forever, let’s rewind and observe the evolution of bridges:
1. As one of the earliest bridges, wooden bridges of all types have been constructed or formed naturally and used by humanity for centuries. From the improvised and sometimes natural wooded log crossing over rivers and streams to modern 30+ meters wide structurally designed wooden bridges, this is one of the oldest forms of bridges in human history.
2. Popular in ancient Rome from around 100 BC [1]. The Romans were famous for using stone and arch bridge structures to design aqueducts. These marvels are still around after 2000 years and inspire current viaduct designs. With the appearance of steel and reinforced concrete, modern arch bridges can span anywhere between 20 meters to 200 meters. Some arch bridges can achieve even longer spans by incorporating suspension cables or truss designs.
3. The first truss bridge was most likely constructed in the 13th century. But truss bridges didn’t rise to prominence until the 19th century when engineers and architects realized that trusses are easy to build, require fewer materials, and are efficient over long spans. It enabled some of the world's architectural marvels like the Eiffel Tower and Ikisuki Bridge. The Ikisuki Bridge in Japan is the world’s longest-spanning truss bridge, reaching over 400 meters in length.
4. The modern suspension bridge became popular in the early 1800s. This is due to increased economic demands for longer-span road connections, especially in the US. The longest-spanning suspension bridge is more than 2000 meters. This accolade belongs to the 1915 Çanakkale Bridge in Turkey.
Looking at the most modern suspension bridge types, are magnificent and elegant but also intimidating. Citizens or even engineers sometimes feel uneasy crossing that seemingly fragile thread and wonder why it has not been blown away by the wind yet. That is the case with the Tacoma Narrows bridge in 1940, a landmark structural collapse on a windy day. This has resulted in textbook lessons and stricter engineering tests for suspension bridges.
With that short history lesson complete, let’s explore the valuable engineering lessons from this iconic structural failure.
A famous bridge and a famous bridge collapse
In 1940, the Tacoma Narrows Bridge was counted as one of the superstructures of the time. It was the world’s third-longest suspension bridge, with a total length of 1810 meters.
During its short 4-month life span, the bridge had been swaying up and down noticeably even at average wind speeds. Despite several efforts to stabilize the bridge, it later collapsed, experiencing 70 km/h winds with a 40 cycles/minute gust frequency.
On the day of its collapse, onlookers were shocked to note a drastic change in its deflection motion from up and down wave-like movements to rigorous twisting motions - When the left side of the structure went down, the right side would rise and vice versa. You can observe the twisting and oscillating motions of the bridge below.
These motions exposed the structure to two failure mechanisms; aeroelastic fluttering and resonance. Eventually, the main support of the suspension bridge, the cables, could not handle the sway and failed individually. The central deck fell into the water with the reduced capacity of key structural cables, bringing about a dramatic and phenomenal end!
Through investigations, engineers learned that the bridge underwent overly aggressive ‘cost-saving’ efforts during design. Without these cost savings, key elements would have had more capacity and performed better under these extreme loads. If the designers put less emphasis on cost savings and aesthetics and more on dynamic analysis, the story could have been quite different!
Let’s take a moment to explore some of the key concepts at work here…
Resonance
The continuous wave-like motion of the bridge can be explained by a natural phenomenon called resonance. This is where an external force imposed at a particular frequency excites an object at its natural frequency. This excitation at the natural frequency causes an amplification of displacement. Every object’s natural frequency is determined by its shape, mass, and material properties.
Natural frequency is also when an object will remain vibrating if it’s excited at the same frequency by an external force.
This can happen when objects are subject to dynamic loads. Dynamic loads can be anything from gusts of wind to someone walking down a hallway. Unlike sustained static loads, dynamic loads cause vibrations in the subjected object or systems they’re imposed upon. In most instances, vibrations aren’t observable, but a great visual example of such vibrations occurs when a guitar string is flicked, and you can watch its resonance vibrations as its strings oscillate back and forth.
The charts below show that when the resonance reaches the natural frequency, the amplitude of vibration (how strong the object vibrates) is much higher than the normal amplitude.
In the case of a guitar string, the result of the vibration is a pleasing (or terrible) sound. In the case of the Tacoma bridge, the result was large-scale resonance, resulting in the violent swaying of the bridge deck and eventual structural collapse!
Well, why was that the case? Surely a bridge can withstand design wind loads? Normally, yes, but in Tacoma's case, the wind's vibration frequency was the same as the bridge's natural frequency. As a result, resonance was induced in the bridge, and the experienced vibrations were amplified and greatly exceeded the expected vibrations. In turn, key structural elements were subject to a greater magnitude of forces and deflections than they were designed for. They then failed, and the entire bridge collapsed.
Every structure has a natural frequency, and engineers ensure experienced vibrations under dynamic load are within allowable limits. And in modern design, great care is taken to ensure possible excitation frequencies do not converge on a structure's natural frequency. In the case of Tacoma, this didn’t happen!
Aeroelastic fluttering
The twisting motion of the bridge can be attributed to another phenomenon - aeroelastic fluttering. Wind airflow can be categorized into the turbulent flow (chaotic and unstable) and laminar flow (stable).
In the case of Tacoma, when hitting the bridge from the side, the wind flow was separated above and below the bridge, which created vortexes and turbulent flows. This chaotic airflow further accentuated the bridge’s deflection, and the different vortexes below and above the bridge caused the bridge's two sides to move up and down violently.
The substructure, cables, bridge deck, and towers were not designed for these intense movements, and the extreme forces and fatigue induced in all elements sent the entire bridge to the scrap heap!
The mechanism behind trusses in bridges
Why did other bridges with similar spans not encounter aeroelastic fluttering? Simply because the Tacoma bridge incorporated an ‘innovative’ idea that used plate girders to support the road deck instead of the originally proposed truss system.
Wind can easily pass through the truss structure for open trusses, remaining laminar and imposing minimal load. At the time, the importance of this aspect of truss supports on bridges had not yet been realized. The original application of trusses was only to support the bridge's side-by-side movement and vertical deflection, not to accommodate the wind. Furthermore, typical plate girders bridges were usually much shorter in length.
Plate girders, however, are an 8-foot solid web, not allowing airflow to pass through it and forcing wind flow to go around it. This generated turbulent flow and caused fluttering.
Ironically, the Tacoma Narrows Bridge’s original design included trusses below the deck. The reason for the switch to the plate girder solution was not only for aesthetic reasons but to reduce costs! The result? An unprecedented slender bridge with a depth-to-span ratio of 1:350 and width-to-span ratio of 1:72 with a construction cost of $6.4 million.
Comparatively, a suspension bridge with a similar span, the Bronx-Whitestone, cost $19.7 million to build and has a depth-to-span ratio of 1:209 and the width-to-span ratio of 1:31. One collapsed, and one didn’t…
Utilizing the plate girder on a longer, taller bridge meant that the Tacoma bridge engineers could reduce the complexity and cost of their design. But it exposed the bridge to a series of unknown risks. While the results were catastrophic, the. That is why the failure of the Tacoma bridge incident has opened up a whole new perspective for wind considerations in structural design.
How would modern engineers reconstruct the bridge?
As it turns out, modern engineers have already reconstructed the bridge twice! In 1950, the ‘new’ Tacoma Narrows Bridge was opened. In the 1990s, as the traffic increased, the government decided to build another next to the second Tacoma Narrows Bridge, completed in 2007.
With the lessons from the 1940s disaster front of mind, both bridges have incorporated deeper trusses to let the wind pass through under the deck! Fortunately, both new bridges still serve local transportation safely to this day.
Some forgiveness should be given to the designers of the Tacoma bridge. After all, it was mostly designed by hand! But if we were to design it with today’s design methodology, what processes and tools would we use?
Wind tunnel testing
Wind tunnel testing is a common experiment to simulate the actual wind situations (speed, flow, directions) and reactions of a structural model (vibrations, deflections, creep) in the laboratory during the design process.
Wind tunnel tests are performed in addition to computer simulations when there is uncertainty in design, such as applying new construction techniques, new materials, or building large-scale constructions. The test is usually applied for structures exposed to strong wind, such as skyscrapers, airplanes, and suspension bridges.
After the Tacoma bridge collapsed, the U.S. Government demanded that all bridges built with federal funds should have to conduct wind tunnel testing. Nowadays, U.S. engineers follow ASCE/SEI 49-12 Wind Tunnel Testing for Buildings and Other Structures to conduct the test. And all jurisdictions have similar codes that account for the local environment. You can see how Wind Tunnel testing was done in practice with the longest suspension bridge in the world, the 1915 Çanakkale Bridge, here.
When the U.S. government decided to build the new twin Tacoma bridge next to the 1950 bridge in the 90s, both the new and existing bridges underwent wind tunnel testing by RWDI. The purpose was to determine how the new bridge influenced the aeroelastic effects on the existing bridge. This consideration demonstrated how engineers have taken past lessons and applied them to subsequent designs.
Generally, a wind tunnel test for a long-span bridge includes:
- A support system (bridge rig) that simulates the bridge's real stiffness. This is mainly achieved using spring supports in the vertical direction to mimic the sway behavior and the horizontal direction to mimic the torsional behavior of the bridges. The supports will be connected on both sides of the bridge’s suspension model (it could be a part of the bridge or a full-scale model).
- Measurement of forces and torques acting on the model, mainly at the support system.
- Subject the bridge to various wind speeds, from ordinary wind in the area to powerful cyclones.
- Mimicking other effects on the bridges, such as vehicles, pedestrians, and earthquakes, to ensure the aerodynamic instability is accounted for. Engineers achieve this by subjecting the model to artificial motions and forces before subjecting them to the wind. Then they will observe if the motion will grow or decay in certain wind simulations.
- Applying the test by simulating the terrains. In the case of the Tacoma bridge, the testing involves two models to ensure that the current bridge won’t negatively affect the newly built bridge.
Finite Element Analysis (FEA)
Before the historical Tacoma failure, engineers still used the ‘deflection theory’ to design suspension bridges. The theory only accounts for deflection under gravity and static wind loads, leading to the insufficient conclusion that the longer the span, the less stiffness is required for the deck. However, just because the bridge failed does not mean the theory was completely rejected. The deflection theory is still largely used in suspension bridge design regarding static load analysis and cable stress analysis. It especially helps Finite Element Analysis (FEA) develop as one of the most common tools for engineers to calculate deflections, deformations, stress, etc.
Nowadays, FEA theory has been applied widely as a calculation mechanism behind engineering software to analyze structures with much higher accuracy than hand calculations. Almost every structure around the world in recent years, including long-span bridges, utilize FEA software for design.
Computational Fluid Dynamics (CFD)
Nowadays, in bridge design, engineering simulation plays a crucial part in the testing process. If an experimental test cannot be carried out, CFD software can simulate wind load conditions with high accuracy on a computer.
Combining the CFD software to simulate the complex wind load and the FEA software to investigate the structural behaviors, the designs of the Tacoma bridge today would be much more thorough and result in a far more robust structure.
Vibration analysis
Not only have bridge designs evolved, but so have the way ways in which they are maintained. Vibration analysis is a method to detect bridge damage by monitoring vibration levels and investigating vibration patterns. Vibrations are collected using various sensor types installed throughout the structure. Sensors measure vibrations as a result of winds or traffic.
This technique is commonly used in railway bridge maintenance because they experience stronger vibrations at a higher speed [2]. The sensor will present the vibrations under the amplitude-frequency graph, which can detect abnormal vibrating patterns. For instance, a crack developed on a roller bearing outer race will lead to periodic collisions with bearing rollers.
This allows maintenance teams to identify small local issues before they proliferate into large macro issues that impact asset performance.
How has the Tacoma bridge impacted structures today?
The Tacoma Narrows Bridge failure shocked engineers of the day. While costly, it did provide many lessons and sparked many studies that advanced structural design. Countless structures benefited from these lessons, so here are just a few close calls!
The Whitestone Bridge:
This suspension bridge finished one year before Tacoma Bridge in the US and also used solid plate girders. After Tacoma collapsed, the Whitestone Bridge was immediately strengthened by adding trusses and openings below road decks to let the wind pass through. This decreased the oscillations and extent of deflections experienced by the structure, and the bridge is still in use even today.
The Mackinac Bridge:
The Tacoma collapse revealed how wind-sensitive long-span bridges could be. As a result, it kick-started the growth of bridge aerodynamics-aeroelastic study, an area of research that influenced the design of long-span bridges for years to come [3]. In the next three years after the failure, the designer of the 8,038-metre-long Mackinac Bridge, David Steinman, published a theoretical analysis of suspension-bridge stability problems. The paper recommended that future bridge designs include deep stiffening trusses to support the bridge deck or an open-grid roadway to reduce wind resistance. The Mackinac Bridge, finished in 1957, achieved both features in its design, and it is still the longest suspension bridge between anchorages in the Western Hemisphere until now.
Every suspension bridge since then is either designed as aerodynamically streamlined, stiffened against the torsional motion, or both. This opened up new areas of study, new design methods, and testing regimes that have enabled modern bridges to reach farther and be safer.
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📚 References
- [1] Babic, M. (2013). Ancient Roman bridges and their social significance. Acta Antiqua, 53, 61-72. 10.1556/AAnt.53.2013.1.4.
- [2] Svedholm, C. (2017). Efficient Modelling Techniques for Vibration Analyses of Railway Bridges [Working Paper]. KTH School of ABE. https://www.diva-portal.org/smash/get/diva2:1073647/FULLTEXT01.pdf
- [3] Shuyang, C., & Jixin, C. (2017). Toward Better Understanding of Turbulence Effects on Bridge Aerodynamics. Frontiers in Built Environment, 3. https://doi.org/10.3389/fbuil.2017.00072