Thursday, July 31, 2008

Long-span bridge: new ideas

Suspension bridges.

I've been thinking a bit about long-span suspension bridges, and have come up with a few minor innovations that may be worth checking out. I don't claim any of these as new ideas, but one or two may be.

For long span bridges, suspension bridges seem to be the only way to go at present. The longest current span is 1991 m for the Akashi-Kaikyo bridge. Cable-stayed bridges can't handle spans anywhere near as large as the largest suspension bridges.

It's well known that a major problem with suspension bridges is lack of stiffness. The catenary cable and hangers can take the load but provide bugger-all stiffness, so the truss under the deck has to be extremely stiff, which means big and heavy. The truss under the world's longest span bridge is extremely deep, heavy and the thick truss members add a lot of wind load to the structure.

My first thought was to see if the suspension structure of catenary and cables could be modified to increase the stiffness of the bridge, so that the deck and truss could be made lighter and less prone to wind loading.

Innovation 1.
Converting the ends of the span next to the pylons to a cable-stayed design is a no-brainer. Advantages are extra stiffness and shorter pylons. Disadvantages are longer cables there carrying heavier loads. See Figure 1. But for a really long span bridge as shown the extra stiffness is limited to the end regions.

Fig. 1. Innovation 1. Using cable-stays at the ends of the bridge.

Innovation 2.
Sloped hangers. Triangulating the hangers, as one would triangulate a truss, provides extra stiffness that resists vertical deflection. See Figure 2. The crossing arrangement of cables show at left provides greater stiffness than the simple triangulated cables at right, but at a price. The price is that the cables carry a heavier load. Even with a prestress offset, some of the cables are going to be carrying at least dead load + 1.5 * live load, instead of the normal dead load + live load. If the live load is large relative to the dead load then this could cost. This problem doesn't occur with the triangulation of cables shown at right.
Fig. 2. Innovation 2. Use of crossed or triangulated hangars for extra stiffness.

Coupling Innovations 1 and 2 allows the end cables to be sloped towards the tops of the pylons, which slightly reduces pylon height.

Innovation 3.
So much for vertical stiffness, but what about the stiffness needed in the horizontal deflection to resist wind loads. For quite a while I couldn't see an easy solution to that, but I finally figured it out. Use four catenaries instead of the usual two. The four catenaries combine to produce lateral stiffness that resists wind loading. From the side (Figure 3) the bridge looks just the same as a standard two-catenary bridge, but viewed from the end the pylons above the bridge are W-shaped, with one catenary strung off each end of the W and two off the centre. The four catenaries and bridge deck are also shown in a view from above in Figure 3. I don't know how much of the side loading due to wind forces can be carried by the catenaries in this way, but certainly a lot more than in a conventional two catenary bridge.
Fig. 3. Innovation 3. Use of four catenary cables for resisting wind loads.

Summary so far. Innovations 1, 2 and 3 together would greatly reduce the need for stiffness of the truss under the bridge deck. In a very real sense together they minimise the dead load of the bridge deck and truss, and almost certainly reduce the total dead load and wind load of the whole structure. (Innovation 5 will reduce the wind load even more).

Innovation 4. (Incompatible with Innovation 3)
However, let's turn the whole process on its head and suppose that the governing factor is live load rather than dead and wind load, and suppose that torsional strength of the truss is adequate. Then a considerable cost saving can be had by reducing the two catenaries of the standard design to one (left side of Figure 4). Cable-staged bridges are already made by using one row of hangers down the centre instead of one on each side, the same principle can be applied to suspension bridges. The result would be aesthetically beautiful. The total wind load on the combined catenary and hangers would be reduced by 30%.
Fig. 4. Innovation 4. Use of one catenary cable instead of two.

A key to whether Innovation 4 can be applied is the torsional strength of the truss. If this is critical then use an intermediate design with one catenary and two rows of hangers (right side of Figure 4). In that case the advantage over the traditional two catenary design is not clear, the cost of the single catenary is less but ir had to be further above the bridge deck so the pylons and hangers have to be taller, and the torsional stability isn't as good when there's heavy traffic on one side of the bridge and no traffic on the other.

Innovation 5.
If it had just been for the four innovations above I probably wouldn't have bothered writing this. But innovation 5 is almost as radical as innovation 3.

The bridge with the world's longest span has an exceptionally deep truss under the deck. The strength of a beam is proportional to the cube of the depth, so even a big result from Innovations 1, 2 and 3 together may not reduce the depth of the truss much. This truss is taking a lot of wind loading, so minimising the wind loading needs to be a major priority. The two most familiar methods of strengthening a bridge deck are using a truss and using a tube (or box section). The tube has the advantage of torsional rigidity and the truss has the advantage of flow-through to reduce wind loading. But even the wind loading on the truss is too large. The only obviouis alternative of using an I-beam with holes in the web has the disadvantages of both and the advantages of neither.
Fig. 5. Innovation 5. Design of the bridge deck based on wing design for a biplane.

Innovation 5 is to design the bridge deck as you would the wing of a biplane, with stiffened lightweight panels top and bottom and very slender wires and struts in between (Figure 5). The wind load on the top and bottom panels will be very small because they're horizontal. Applying the method to suspension bridge design may not cut the total weight by much but should cut the wind load considerably. It should also have aesthetic appeal. Design the struts to have least drag when the wind is perpendicular to the bridge, as that currently gives the greatest bending moment.

Innovation 6.
This again is so obvious it barely needs stating. A tension member such as the catenary will be just as strong no matter what the shape, so when it's running nearly horizontal over the centre of the bridge it is better to be squashed vertically and extended horizontally to minimise the wind load (Figure 6).
Fig. 6. Innovation 6. Flatten the catenary cable near mid-span to minimise wind load.

Innovation 7.
Avoid rectangular pylons. The wind load is much worse than on circular or elliptical plyons, or pylon shapes with long-radius corners.

Innovation 8.
This innovation is a pair of ideas that I developed some years ago in a study on optimising long compression members. Neither is ideal, but either may be better than what is used now. One problem with compression members now used in pylon and truss construction is that they're very heavy. The aim of the earlier study was to develop compression members that were almost as light-weight as tension members by using geometrical considerations to overcome buckling. For a single member the result was as shown (not to scale) in Figure 7. Such an element could be made by extrusion, eg of glass fibre in epoxy, carbon fibre in epoxy, or high strength aluminium. Or it could be made by assembly, eg. by gluing many little stainless steel boxes together. It would be ideal for a long span bridge except for one thing - the large radius and solid construction means that it has an exceptionally high wind drag.
Fig. 8. Innovation 7. Ideal lightweight compression element, possibly suitable for pylons.

The alternative is to use a conventional truss of tubular elements, optimised for minimum weight, which could be used for pylons in place of current heavy concrete and box steel designs.

One of the primary problems in suspension bridges is lack of stiffness. Innovations 1, 2 and 3 would work together to use four catenaries and hangers to provide extra stiffness, allowing the truss design to be replaced by a low-wind-load deck design from Innovation 5.

A completely different alternative is to use a single catenary from Innovation 4 in place of the usual two. Because of the loss of torsional stiffness this could be coupled perhaps with a deck in which vehicles travel inside an aerodynamically optimised tube.

Innovations 6 and 7, like 5, are there to help minimise wind loads.

Innovation 8 is presented as a possible alternative to standard pylon construction, and may not be any better than the standard design. The principle there is minimum weight.