Japan

027

Akashi-Kaikyo Bridge, Kobe, Japan

34° 36′ 59″ N, 135° 1′ 13″ E

The World’s Longest Suspension Bridge

There are suspension bridges all over the world, from the Clifton Suspension Bridge in Bristol, UK, to the Golden Gate Bridge in San Francisco, California. But only one can be the longest, and that honor goes to the Akashi-Kaikyō Bridge in Japan, which links the city of Kobe on the main Japanese island of Honshu with Awaji Island across the Inland Sea. The bridge was built to cross a dangerous waterway that had previously been plied by ferries and was the site of tragic sinkings in stormy weather.

The Akashi-Kaikyō Bridge has a main span of 1991 meters (by contrast, the main span of the Clifton Suspension Bridge is 214 meters, and that of the Golden Gate Bridge is 1,280 meters). The original plan was for a main span of 1,990 meters, but in 1995, after the main pillars were built but before the bridge was installed, a large earthquake hit the Kobe area and moved the pillars apart by an extra meter. Overall, the bridge is 3,911 meters long.

To help prevent swaying (especially in an earthquake), the bridge has pendulums inside the towers that act to dampen any movement (more on damping via pendulums in Chapter 34).

Avoiding corrosion is an important concern because of the salty air. The main cables consist of multiple cables bunched together, and dry air is pumped into the interior spaces to prevent any buildup of moisture. Painting the steel structure is also vital, and is achieved using a robot that can paint 500 square meters per day. Underwater portions of the bridge are protected from corrosion using sacrificial anodes (see pages 316–317).

As with other modern bridges (see, for example, Chapter 14), the Akashi-Kaikyō Bridge has extensive monitoring systems, with multiple anemometers measuring wind speed at the top of the towers and at the deck level. The exact position of the bridge is monitored using GPS, and accelerometers measure the movement of the towers and the anchors.

Visitors can, of course, drive over the bridge after paying the toll. But to get a real sense of its scale and construction, it’s best to visit the Bridge Science Museum on the Kobe side of the bridge. There’s a small park from which the bridge can be observed, and a tunnel that takes visitors underneath the bridge and out across the water to a glass-floored observation area.

It’s also worth seeing the bridge at night, when it’s illuminated in a pattern of ever-changing colors.

Practical Information

The Bridge Science Museum is situated on the Kobe side of the Akashi-Kaikyō Bridge, a short walk from the JR Maiko station. There is some information in English, but you might consider arranging for an interpreter to understand all the displays. The Japanese National Tourist Office has information about visiting the bridge at http://www.jnto.go.jp/eng/location/sit/hyogo/2821.html.

The Shape of Suspension Bridges

Before the deck of a suspension bridge is added, the main cables are draped across the pylons and hang down under their own weight to form a catenary (see page 408). But when the deck is added, the shape changes to a parabola. It only takes a few simplifying assumptions and a little bit of mathematics to understand why.

Start by considering the point where the curve of the suspension bridge cable is at its lowest, and use that point as the origin for a set of (x, y) coordinates (Figure 27-1). Here, (0, 0) is the lowest point; the x-axis runs parallel with the bridge’s deck, and the y-axis runs vertically.

Figure 27-1. Mathematics of a suspension bridge

To simplify things, imagine that the cable has no weight (in fact, the cable is much lighter than the deck, so the deck’s weight will be the factor that most influences the cable’s shape). Also imagine that the deck is entirely uniform and that the weight of the deck acts on the cable uniformly (this isn’t entirely true, because there isn’t an infinite number of vertical rods linking the main cable to the deck, but it’s close enough).

Now take a point P somewhere up the cable to the right of the origin at a distance x along the x-axis, and consider the portion of cable from the origin to P. To determine the shape of the cable, we need to determine the angle of this portion of cable (which we’ll assume is short enough that we can pretend that it’s straight).

There are three forces acting on the portion of cable (Figure 27-2). At the origin, there’s a purely horizontal force coming from tension in the cable, which we’ll call T. The weight of the section below the cable, W, pulls down on the cable uniformly. Finally, there’s tension up the cable at P, which we’ll call F.

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Figure 27-2. Forces on a small section of cable

Since the cable is not moving, these three forces must be in equilibrium, and therefore F must equal W/T. T is a constant force (i.e., it doesn’t matter what point we choose for P) and W depends on the weight of the deck. Since we assumed that the deck has a uniform shape, we can specify a linear density D for the deck, and get a weight of Dx for the section between the origin and point P. So the force F is Dx/T.

Now imagine that the curve of the cable is defined by some unknown function f such that y = f(x) is the cable’s equation. At a distance x, the cable is at a height y (refer back to Figure 27-1). The slope of the cable at x is the derivative dy/dx, which is df(x)/dy. Since we know the slope from the forces acting on the cable at distance x, we know the value of dy/dx: it’s Dx/T (Equation 27-1).

Equation 27-1. Cable shape differential equation

It’s just a matter of integrating both sides of this equation to recover f(x) (with an unknown constant c). Although the actual tension, T, in the cable is unknown, it’s easy to see that the shape is a parabola (Equation 27-2).

Equation 27-2. Shape of a suspension bridge’s cable