19. Bridge towers

When architects design and construction engineers build towers, they make them vertical. By “vertical” we mean straight up and down or, more formally, in line with the direction of gravity. A tall, thin structure is most stable if built vertically, as then the centre of mass is directly above the centre of the base area.

If the Earth were flat, then vertical towers would all be parallel, no matter where they were built. On the other hand, if the Earth is curved like a sphere, then “vertical” really means pointing towards the centre of the Earth, in a radial direction. In this case, towers built in different places, although all locally vertical, would not be parallel.

The Humber Bridge spans the Humber estuary near Kingston upon Hull in northern England. The Humber estuary is very broad, and the bridge spans a total of 2.22 kilometres from one bank to the other. It’s a single-span suspension bridge, a type of bridge consisting of two tall towers, with cables strung in hanging arcs between the towers, and also from the top of each tower to anchor points on shore. (It’s the same structural design as the more famous Golden Gate Bridge in San Francisco.) The cables extend in both directions from the top of each tower to balance the tension on either side, so that they don’t pull the towers over. The road deck of the bridge is suspended below the main cables by thinner cables that hang vertically from the main cables. The weight of the road deck is thus supported by the main cables, which distribute the load back to the towers. The towers support the entire weight of the bridge, so must be strong and, most importantly, exactly vertical.

The Humber Bridge

The Humber Bridge from the southern bank of the Humber. (Public domain image from Wikimedia Commons.)

The towers of the Humber Bridge rest on pylons in the estuary bed. The towers are 1410 metres apart, and 155.5 metres high. If the Earth were flat, the towers would be parallel. But they’re not. The cross-sectional centre lines at the tops of the two towers are 36 millimetres further apart than at the bases. Using similar triangles, we can calculate the radius of the Earth from these dimensions:

Radius = 155.5×1410÷0.036 = 6,090,000 metres

This gives the radius of the Earth as 6100 kilometres, close to the true value of 6370 km.

Size of the Earth from the Humber Bridge

Diagram illustrating use of similar triangles to determine the radius of the Earth from the Humber Bridge data. (Not to scale!)

If this were the whole story, it would pretty much be case closed at this point. However, despite a lot of searching, I couldn’t find any reference to the distances between the towers of the Humber Bridge actually being measured at the top and the bottom. It seems that the figure of 36 mm was probably calculated, assuming the curvature of the Earth, which makes this a circular argument (pun intended).

Interestingly, I did find a paper about measuring the deflection of the north tower of the Humber Bridge caused by wind loading and other dynamic stresses in the structure. The paper is primarily concerned with measuring the motion of the road deck, but they also mounted a kinematic GPS sensor at the top of the northern tower[1].

GPS sensor on Humber Bridge north tower

Kinematic GPS sensor mounted on the top of the north tower of the Humber Bridge. (Reproduced from [1].)

The authors carried out a series of measurements, and show the results for a 15 minute period on 7 March, 1996.

Deflections of Humber Bridge north tower

North-south deflection of the north tower of the Humber Bridge over a 15 minute period. The vertical axis is metres relative to a standard grid reference, so the full vertical range of the graph is 30 mm. (Reproduced from [1].)

From the graph, we can see that the tower wobbles a bit, with deflections of up to about ±10 mm from the mean position. The authors report that the kinematic GPS sensors are capable of measuring deflections as small as a millimetre or two. So from this result we can say that the typical amount of flexing in the Humber Bridge towers is smaller than the supposed 36 mm difference that we should be trying to measure. So, in principle, we could measure the fact that the towers are not parallel, even despite motion of the structure in environmental conditions.

A similar result is seen with the Severn Bridge, a suspension bridge over the Severn River between England and Wales. It has a central span of 988 metres, with towers 136 metres tall. A paper reports measurements made of the flexion of both towers, showing typical deflections at the top are less than 10 mm[2].

Deflections of Severn Bridge towers

Plot of deflection of the top of the suspension towers along the axis of the Severn Bridge. T1 and T2 (upper two lines) are measurements made by two independent sensors at the top of the west tower; T3 and T4 (lower lines) are measurements made by sensors on the east tower. Deflection is in units of metres, so the scale of the maximum deflections is about 10 mm. (Reproduced from [2].)

Okay, so we could in principle measure the mean positions of the tops of suspension bridge towers with enough precision to establish that the towers are further apart at the top than the base. A laser ranging system could do this with ease. Unfortunately, in all my searching I couldn’t find any citations for anyone actually doing this. (If anyone lives near the Humber Bridge and has laser ranging equipment, climbing gear, a certain disregard for authority, and a deathwish, please let me know.)

Something I did find concerned the Verrazzano-Narrows Bridge in New York City. It has a slightly smaller central span than the Humber Bridge, with 1298 metres between its two towers, but the towers are taller, at 211 metres. The tops of the towers are reported as being 41.3 mm further apart than the bases, due to the curvature of the Earth. There are also several citations backing up the statement that “the curvature of the Earth’s surface had to be taken into account when designing the bridge” (my emphasis).[3]

Verrazzano-Narrows Bridge

Verrazzano-Narrows Bridge, linking Staten Island (background) and Brooklyn (foreground) in New York City. (Public domain image from Wikimedia Commons.)

So, this prompts the question: Do structural engineers really take into account the curvature of the Earth when designing and building large structures? The answer is—of course—yes, otherwise the large structures they build would be flawed.

There is a basic correction listed in The Engineering Handbook (published by CRC) to account for the curvature of the Earth. Section 162.5 says:

The curved shape of the Earth… makes actual level rod readings too large by the following approximate relationship: C = 0.0239 D2 where C is the error in the rod reading in feet and D is the sighting distance in thousands of feet.[4]

To convert to metric we need to multiply the constant by the number of feet in a metre (because of the squared factor), giving the correction in metres = 0.0784×(distance in km)2. What this means is that over a distance of 1 kilometre, the Earth’s surface curves downwards from a perfectly straight line by 78.4 millimetres. This correction is well known among civil and structural engineers, and is applied in surveying, railway line construction, bridge construction, and other areas. It means that for engineering purposes you can’t treat the Earth as both flat and level over distances of around a kilometre or more, because it isn’t. If you treat it as flat, then a kilometre away your level will be off by 78.4 mm. If you make a surface level (as measured by a level or inclinometer at each point) over a kilometre, then the surface won’t be flat; it will be curved parallel to the curvature of the Earth, and 78.4 mm lower than flat at the far end.

An example of this can be found at the Volkswagen Group test track facility near Ehra-Lessien, Germany. This track has a circuit of 96 km of private road, including a precision level-graded straight 9 km long. Over the 9 km length, the curvature of the Earth drops away from flat by 0.0784×92 = 6.35 metres. This means that if you stand at one end of the straight and someone else stands at the other end, you won’t be able to see each other because of the bulge of the Earth’s curvature in between. The effect can be seen in this video[5].

One set of structures where this difference was absolutely crucial is the Laser Interferometer Gravitational-Wave Observatory (LIGO) constructed at two sites in Hanford, Washington, and Livingston, Louisiana, in the USA.

LIGO site at Hanford

The LIGO site at Hanford, Washington. Each of the two arms of the structure are 4 km long. (Public domain image from Wikimedia Commons.)

LIGO uses lasers to detect tiny changes in length caused by gravitational waves from cosmic sources passing through the Earth. The lasers travel in sealed tubes 4 km long, which are under high vacuum. Because light travels in a straight line in a vacuum, the tubes must be absolutely straight for the machine to work. The tubes are level in the middle, but over the 2 km on either side, the curvature of the Earth falls away from a straight line by 0.0784×22 = 0.314 metres. So either end of the straight tube is 314 mm higher than the centre of the tube. To build LIGO, they laid a concrete foundation, but they couldn’t make it level over the distance; they had to make it straight. This required special construction techniques, because under normal circumstances (such as Volkswagen’s track at Ehra-Lessien) you want to build things level, not straight.[6]

So, the towers of large suspensions bridges almost certainly are not parallel, due to the curvature of the Earth, although it seems nobody has ever bothered to measure this. But it’s certainly true that structural engineers do take into account the curvature of the Earth for large building projects. They have to, because if they didn’t there would be significant errors and their constructions wouldn’t work as planned. If the Earth were flat they wouldn’t need to do this and wouldn’t bother.

UPDATE 2019-07-10: NASA’s Jet Propulsion Laboratory has announced a new technique which they can use to detect millimetre-sized shifts in the position of structures such as bridges, using aperture synthesis radar measurements from satellites. So maybe soon we can have more and better measurements of the positions of bridge towers![7]

References:

[1] Ashkenazi, V., Roberts, G. W. “Experimental monitoring of the Humber bridge using GPS”. Proceedings of the Institution of Civil Engineers – Civil Engineering, 120, p. 177-182, 1997. https://doi.org/10.1680/icien.1997.29810

[2] Roberts, G. W., Brown, C. J., Tang, X., Meng, X., Ogundipe, O. “A Tale of Five Bridges; the use of GNSS for Monitoring the Deflections of Bridges”. Journal of Applied Geodesy, 8, p. 241-264, 2014. https://doi.org/10.1515/jag-2014-0013

[3] Wikipedia: “Verrazzano-Narrows Bridge”, https://en.wikipedia.org/wiki/Verrazzano-Narrows_Bridge, accessed 2019-06-30. In turn, this page cites the following sources for the statement that the curvature of the Earth had to be taken into account during construction:

[3a] Rastorfer, D. Six Bridges: The Legacy of Othmar H. Ammann. Yale University Press, 2000, p. 138. ISBN 978-0-300-08047-6.

[3b] Caro, R.A. The Power Broker: Robert Moses and the Fall of New York. Knopf, 1974, p. 752. ISBN 978-0-394-48076-3.

[3c] Adler, H. “The History of the Verrazano-Narrows Bridge, 50 Years After Its Construction”. Smithsonian Magazine, Smithsonian Institution, November 2014.

[3d] “Verrazano-Narrows Bridge”. MTA Bridges & Tunnels. https://new.mta.info/bridges-and-tunnels/about/verrazzano-narrows-bridge, accessed 2019-06-30.

[4] Dorf, R. C. (editor). The Engineering Handbook, Second Edition, CRC Press, 2018, ISBN 978-0-849-31586-2.

[5] “Bugatti Veyron Top Speed Test”. Top Gear, BBC, 2008. https://youtu.be/LO0PgyPWE3o?t=200, accessed 2019-06-30.

[6] “Facts about LIGO”, LIGO Caltech web site. https://www.ligo.caltech.edu/page/facts, accessed 2019-06-30.

[7] “New Method Can Spot Failing Infrastructure from Space”, NASA JPL web site. https://www.jpl.nasa.gov/news/news.php?feature=7447, accessed 2019-07-10.

18. Polar motion

The Earth rotates around an axis, an imaginary straight line that all points not on the line move around in circles. The axis passes through the Earth’s North Pole and the South Pole. So the positions of the two Poles are defined by the position of the rotation axis.

Earth rotation and poles

The Earth’s North and South Poles are defined as the points where the axis of rotation passes through the surface of the planet. (Earth photo is a public domain image from NASA.)

Interestingly, the Earth’s rotation axis is not fixed – it moves slightly. This means that the Earth’s poles move.

The positions of the Earth’s poles can be determined by looking at the motions of the stars. As we’ve already seen, if you observe the positions of stars throughout a night, you will see that they rotate in the sky about a central point. The point on the Earth’s surface directly underneath the centre of rotation of the stars is one of the poles of the Earth.

Star trails in the northern hemisphere

Star trails above Little Hawk Lake in Canada. The northern hemisphere stars rotate around the North Celestial Pole (the point directly above the Earth’s North Pole). The bright spot in the centre is Polaris, the pole star. The circles are somewhat distorted in the upper corners of the photo because of the wide angle lens used. (Creative Commons Attribution 2.0 image by Dave Doe.)

Through the 19th century, astronomers were improving the precision of astronomical observations to the point where the movement of the Earth’s rotational poles needed to be accounted for in the positions of celestial objects. The motion of the poles was also beginning to affect navigation, because as the poles move, so does the grid system of latitude and longitude that ships rely on to reach their destinations and avoid navigational hazards. In 1899 the International Geodetic Association established a branch known as the International Latitude Service.

The fledgling International Latitude Service established a network of six observatories, all located close to latitude 39° 08’ north, spread around the world. The initial observatories were located in Gaithersburg, Maryland, USA; Cincinatti, Ohio, USA; Ukiah, California, USA; Mizusawa, Japan; Charjui, Turkestan; and Carloforte, Italy. The station in Charjui closed due to economic problems caused by war, but a new station opened in Kitab, Uzbekistan after World War I. Each observatory engaged in a program of observing the positions of 144 selected reference stars, and the data from each station were cross referenced to provide accurate measurements of the location of the North Pole.

International Latitude Service station in Ukiah

International Latitude Service station in Ukiah, California. (Public domain image from Wikimedia Commons.)

In 1962, the International Time Bureau founded the International Polar Motion Service, which incorporated the International Latitude Service observations and additional astronomical observations to provide a reference of higher accuracy, suitable for both navigation and defining time relative to Earth’s rotation. Finally in 1987, the the International Astronomical Union and the International Union of Geodesy and Geophysics established the International Earth Rotation Service (IERS), which took over from the International Polar Motion Service. The IERS is the current authority responsible for timekeeping and Earth-based coordinate systems, including the definitions of time units, the introduction of leap seconds to keep clocks in synch with the Earth’s rotation, and definitions of latitude and longitude, as well as measurements of the motion of the Earth’s poles, which are necessary for accurate use of navigation systems such as GPS and Galileo.

The motion of Earth’s poles can be broken down into three components:

1. An annual elliptical wobble. Over the period of a year, the Earth’s poles move around in an ellipse, with the long axis of the ellipse about 6 metres in length. In March, the North Pole is about 6 metres from where it is in September (though see below). This motion is generally agreed by scientists to be caused by the annual shift in air pressure between winter and summer over the northern and southern hemispheres. In particular there is an imbalance between the Northern Atlantic ocean and Asia, with higher air pressure over the ocean in the northern winter, but higher air pressure over the Asian continent in summer. This change in the mass distribution of the atmosphere is enough to cause the observed wobble.

Annual wobble of North Pole

Annual elliptical wobble of the Earth’s North Pole. Deviation is given in milliarcseconds of axial tilt; 100 milliarcseconds corresponds to a bit over 3 metres at ground level. (Figure adapted from [1].)

2. Superimposed on the annual elliptical wobble is another, circular, wobble, with a period of around 433 days. This is called the Chandler wobble, named after its discoverer, American astronomer Seth Carlo Chandler, who found it in 1891. The Chandler wobble occurs because the Earth is not a perfect sphere. The Earth is slightly elliptical, with the radius at the equator about 20 kilometres larger than the polar radius. When elliptical objects spin, they experience a slight wobble in the rotation known as free nutation. This is the sort of wobble seen in a spinning rugby ball or American football in flight (where the effect is exaggerated by the ball’s exaggerated elliptical shape). This wobble would die away over time, but is driven by changes in the mass distribution of cold and warm water in the oceans and high and low pressure systems in the atmosphere. The Chandler wobble has a diameter of about 9 metres at the poles.

The combined effect of the annual wobble and the Chandler wobble is that the North and South Poles move in a spiralling pattern, sometimes circling with a diameter up to 15 metres, then reducing down to about 3 metres, before increasing again. This beat pattern occurs over a period of about 7 years.

Annual _ Chandler wobble of North Pole

Graph showing the movement of the North Pole over a period of 4500 days (12.3 years), with time on the vertical axis and the spiralling motion mapped in the x and y axes. The motion tickmarks are 0.1 arcsecond in rotation angle of the axis apart, corresponding to about 3 metres of motion along the ground at the Pole. (Public domain image from Wikimedia Commons.)

3. The third and final motion of the Earth’s poles is a systematic drift, of about 200 millimetres per year. Since 1900, the central point of the spiral wobbles of the North Pole has drifted by about 20 metres. This drift is caused by changes in the mass distribution of Earth due to shifts in its structure: movement of molten rock in the mantle, isostatic rebound of crust following the last glacial period, and more recently the melting of the Greenland ice sheet. The melting of the Greenland ice sheet in the last few decades has shifted the direction of polar drift dramatically; one of the serious indications of secondary changes to the Earth caused by human-induced climate change. Changes in Earth’s mass distribution alter its rotational moment of inertia, and the rotational axis adjusts to conserve angular momentum.

Motion of North Pole since 1900

Plot of motion of the North Pole since 1900. The actual position of the Pole from 2008 to 2014 is shown with blue crosses, showing the annual and Chandler wobbles. The mean position (i.e. the centre of the wobbles) is shown for 1900 to 2014 as the green line. The pole has mostly drifted towards the 80° west meridian, but has changed direction dramatically since 2000. (Figure reproduced from [2].)

Each of the three components of Earth’s polar motion are: (a) observable with 19th century technology, (b) accurately measurable using current technology, and (c) understandable and quantitatively explainable using the fact that the Earth is a rotating spheroid and our knowledge of its structure.

If the Earth were flat, it would not be possible to reconcile the changes in position of the North and South Poles with the known shifts in mass distribution of the Earth. The Chandler wobble would not even have any reason to exist at close to its observed period unless the Earth was an almost spherical ellipsoid.

References:

[1] Höpfner, J. “Polar motion at seasonal frequencies”. Journal of Geodynamics, 22, p. 51-61, 1996. https://doi.org/10.1016/0264-3707(96)00012-9

[2] Dick, W., Thaller, D. IERS Annual Report 2013. International Earth Rotation Service, 2014. https://www.iers.org/IERS/EN/Publications/AnnualReports/AnnualReport2013.html