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Liberty Bridge Analysis

This bridge is an example of a relatively new structure type: the ring cable bridge, invented by Jörg Schlaich, the former head of Schlaich Bergermann und Partner of Stuttgart, Germany.

Put simply, the bridge is a highly curved suspension bridge with the cables attached to only one side of deck.

The steel suspension cable is hung from the tops of two steel towers (each 100ft tall) and anchored in the ground at the ends of the bridge.

The 12ft wide concrete deck is 450ft long from end to end (measured along the curve) and is suspended 30ft in the air. As we will soon see, it is important to note that the deck is also anchored to the ground at the ends.

Below the deck, one sees a 4ft deep steel truss. The bottom chord of the truss is another cable, called the ring cable. The ring cable is also anchored to the ground at the ends.

4076014_orig

Structural Behavior

A good way to understand the structure is to first consider a typical cross section, then zoom out to consider the global behavior.

Forces in the Cross Section

Note that this is a section through the curved structure. As a result, I’ll refer to forces acting vertically (up and down) and radially (left and right in our section).

First of all, we can see that the weight (W) is balanced by a vertical force in the hanger cable (P.hc.vert). Because the hanger cable is inclined, the vertical force in the hanger cable creates a radial force as well (P.hc.rad).

Since W and P.hc.vert are separated, W would tend to rotate the deck about the base of the hanger cable. This tendency is counterbalanced by a radial force in the ring cable (P.rc.rad).
As a function of the geometry of the cross section, P.rc.rad is larger than P.hc.rad. This would tend to translate the deck radially inward. This tendency is countered by a radial force in the deck (P.deck.rad).

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W = weight
P.hc = force in hanger cable
P.hc.vert = vertical component of P.hc
P.hc.rad = radial component of P.hc
P.rc = force in ring cable
P.rc.rad = radial component of P.rc
P.deck = axial force in deck
P.deck.rad = radial component of P.deck

Global Behavior

Picture a very large bird sitting on a wire. The wire makes a “V” with the bird in the middle. The wire is in tension and the deviation (the bottom of the V) creates an upward force equal to the weight of the bird.

This is exactly what happens at the top of the hanger cable, where it meets the main suspension cable. Each hanger force (P.hc) plays the role of the bird and creates a slight deviation in the suspension cable. It is the sum of all these little deviations that gives us the familiar suspension cable shape.

We see the same effect in the ring cable. At each truss panel, the rotation of the deck is arrested by a horizontal force (P.rc.rad). This force is the result of deviating the ring cable into a suspension shape — but this time in the horizontal plane.

If we push our imagination a bit further, we can see that the also familiar arch shape is the suspension shape turned upside down. But this time instead of deviating tension, we are deviating compression. Recall that the tendency of the deck to translate radially was prevented by a radial force in the deck (P.deck.rad). This force is the result of deviating the deck into an arch shape — an arch in the horizontal plane.

Now we can see that the system doesn’t simply support a curved walkway. The curved walkway is an integral part of the system.

Suspension Cable Geometry

A normal suspension bridge features a straight deck and two suspension cables. Each cable is above an edge of the deck and exists solely in the vertical plane. The proper geometry is exceedingly simple to calculate, with the lesson taking up only a page or two in a standard textbook.

The three dimensional geometry of the suspension cable for the Liberty Bridge is completely different and is best found through experimentation. In the old days, the geometry of a complex tensile structure was found using various physical techniques. The practitioners were brilliant. See the works of Gaudi in the late 1800’s and Frei Otto in the 1960’s.

These days, as with so many other things, form finding is done with the help of a computer. Given the right software and a properly trained engineer, amazing things are possible at a fraction of the cost and much higher accuracies.