Effect of Earth's Curvature on Suspension Bridge Dimensions

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Introduction

Figure 1: Schematic of the Verrazano Narrows Bridge.

Figure 1: Schematic of the Verrazano Narrows Bridge.

I have received a number of questions recently on how the curvature of the Earth affects building construction. In general, the effects of the Earth's curvature are ignorable because most man-made construction is on too small of a scale to notice the effects of the Earth's curvature. One well documented exception is the Verrazano-Narrows bridge, whose design took into account that the bridge towers are 1 5/8 inch farther apart at the top than at the bottom. In this post, I will show how to compute this value.

The calculations here are straightforward. My intent is to show that there are some structures that must take the Earth's curvature into account. There are two other examples that I know of: Stanford Linear Accelerator (Source), and Fermilab's neutrino communication experiments (Source). For more examples, see this comment.

Analysis

Figure 2 shows how to compute the 1 5/8 inch of additional separation based on the drawing in Figure 1.

Figure 2: Calculations for the Additional Separation Between the Bridge Towers.

Figure 2: Calculations for the Additional Separation Between the Bridge Towers.

 Conclusion

The Earth's curvature will only have significant effects on massive structures that are sensitive to small errors. With the Verrazano-Narrows bridge, we are talking about a structure with a size on the order of a 1000 feet and the effect of the Earth's curvature is ~1 inch.

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6 Responses to Effect of Earth's Curvature on Suspension Bridge Dimensions

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  2. Ray Brown says:

    Why must this adjustment be adhered to. What effect would it have leaving out. I would imagine that it has to do with the "pull" of the suspension cables.

     
    • mathscinotes says:

      I am not a bridge designer, but I have read from many sources (example) that the VN bridge design had to compensate for the curvature of the Earth. If I had to speculate, I would guess that the compensation has to do with tolerancing (e.g. making metal members with holes line up for fastening). For example, I personally have seen bridge decks that could not be fastened together until nighttime when temperatures cooled and the deck members shrunk enough for holes to line up. That being the case, both temperature and curvature must be taken into account.

      One of the most interesting tales of compensating for the Earth's curvature involves an early H-bomb test. A 9000' long Krause-Ogle box (link) was built that extended from the bomb site to some test instruments. The box was filled with bags of hydrogen gas. The idea was to provide a portal for measuring the first radiation associated with the explosion before the instruments were destroyed by the blast. Early tests showed that the box did not provide proper alignment between the bomb and the instruments. It was quickly discovered that the box design had not corrected for the Earth's curvature. Once the design was corrected for the Earth's curvature, the box and instruments worked properly.

      Temperature is a more common problem than the Earth's curvature. I often have to deal with temperature expansion and tolerancing. I used to work in the automotive metrology field (e.g. measuring transmission parts). All measurements were made to 0.0001" and temperature really matters (see this web page for an example). For an interesting historical situation where temperature compensation was important, see this example from the Trinity atomic bomb test. The plutonium core did not fit until all the temperatures were equalized.

      mark

       
      • Sparky says:

        You know when you are installing fence posts and how you use that leveler tool and line the bubble up in the middle to set the post in exactly vertical? So when you do this on a suspension bridge of great length, even all your pillars are set vertical, they will not be exactly parallel. So if you don't account for this when calculating the dimensions of the suspension cables, I suppose you could end up with some problems. Your suspension cables might end up being slightly too short.

         
        • mathscinotes says:

          You got it. Plumb and level exist for a point on the surface. When you have more than one fence post, technically, each is pointing at a slightly different angle relative to the center of the Earth. Of course, the difference is insignificant for relatively small objects. However, there are large objects where it must be taken into account — think suspension bridges, atom smashers, tunnels.

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  3. Pingback: The Verrazano-Narrows Bridge and The Curvature of The Earth – FlatEarth.ws

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