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Queensborough Bridge

This bridge, one of a vast number in New York City, is very unusual. It has two unequal main spans which comprise cantilevers joined without a suspended span, making the bridge indeterminate. A central anchor span provides anchorage for two two inner cantilevers Do you think that joined cantilevers have any advantages over continuous beams? Over conventional cantilevers with suspended spans? Are there disadvantages? The Middlesbrough transporter bridge is another with no suspended central span.

Here is a footbridge based on two sets of I-beams. What has it to do with the Queensborough bridge? Note the join in the middle of the bridge. If the whole span is acting as a single beam, the shallowness of the I-beam compared with the span must place great shear stress on the bolts, though friction between the plates may relieve this. But if both halves are cantilevers, the fish-plates are merely connecting the halves. A third (unlikely) possibility is that one half is a cantilever and one is a beam. Look at the abutments. What suggests that cantilever action is possible?

Jacques Cartier Bridge

Like the Forth Bridges and the Quebec bridge, the Jacques Cartier Bridge in Montreal had to have enough clearance for the tall masts of big ocean going ships. The effects of such a construction can reach far inland, especially for railway bridges, because the gradient of the approaches have to be kept within strict limits. This adds to the cost of the structure. Here the Empress of Britain is about to pass under the bridge. These very old pictures have faded badly.

The appearance of cantilever bridges can suggest that the main span is a continuous truss. But look at this amended picture of the Quebec bridge (red = compression / blue = tension). The four members shown in yellow are not fundamental parts of the structure: the top and bottom chords are not continuous. If none of the four members were present, the suspended span could move longitudinally, so one of them is needed to prevent that happening. But the other three are for appearance only, and such members may even be allowed to slide at one end to allow for expansion, which can be significant over such a length. So strict functionality is not always adhered to in building a bridge: in fact some bridges have been provided with obviously superfluous adornment. Tower bridge is a well known example.

Having mentioned expansion, how much expansion and contraction is likely. At the Quebec bridge, the temperature could vary from -15C to +25C, a range of 40C. The total length of the bridge is about 850 metres. The coefficient of expansion of steel is about 11 parts per million per degree C. So the fractional change (which is what matters) for a 40C change is about 440 ppm. That doesn’t look to be very much. The actual change is 0.00044 x 850 = 0.37 m = 370 cm. That does look significant.

What would happen if no allowance were made for the changes in length? The Young’s modulus for steels is about 200 GN per square metre. We multiply this by the strain we already calculated, and we get a stress of 200 x 0.00044 = 0.088 GN/m² = 88MN/m². This is all very well, but what does it mean? It corresponds to a stress of about 8800 tonnes weight per square metre, to be added or subtracted from the calculated stresses, depending on whether the bridge is built in very hot weather or very cold weather, and which part of the bridge is being considered. To get some idea of what the stress means for the steel, we can compare the 88MN/m² with the ultimate tensile strength of steel, which depends on the type. A typical figure would be 1000MN/m², so we are approaching ten percent. In practice the effect would be worse, because the greatest stress would fall on that part of the load path that has the smallest cross-section. The cross-sections along the chords vary because the stresses vary. At the root of the lower chord, the compression stress is greatest, and the cross-section is also at its greatest. Out on or near the suspended span, the forces are much weaker, and so are the cross-sections. This is where temperature effects would strike first (and last, because one failure point would suffice).

Note the way in which the tension and compression are exchanged at the two suspension points. This exchange is found in no other type of bridge. Could this be in some way related to the fact that cantilever bridges can be longer than simple beam-truss bridges?

Now we repeat the list of long cantilever spans, with the three longest steel arches included. The two types appear to have a similar economic upper limit.

1800 580 640 0.356 Quebec

1710 680 350 0.204 Forth

1700 000 000 0.000 New River Gorge

1675 000 000 0.000 Bayonne

1670 000 000 0.000 Sydney Harbour

1644 Commodore Barry

1500 468 564 0.376 Howrah

You will have noticed that the lower chord of the Forth bridge appears to be curved, while that of the Quebec bridge is straight (though the first bridge used curves). The top chord of many truss bridges, such as the Queensborough bridge and the Jacques Cartier bridge, is curved, though the Forth bridge and the Quebec use straight chords. By "curved" here we mean composed of straight segments that are not collinear. So what is the "right" answer, if there is one? The solution in a given case is probably a compromise between "perfection" and economics.

The next picture shows an outline made from two parabolas, resulting in curved cantilever members and a lenticular truss. The suspended span is 3/8 of the total main span, which "looks" reasonable.

This looks very mathematical and simple, but there are some problems. The roadway does not correspond to any of the chords, which is very inconvenient. In the case of the Forth bridge, which is narrower at the top than the bottom, (no doubt in response to the collapse of the Tay bridge) the support of the railway track is not simple. It is much easier to make at least some of the chords align with the deck, so the suspended span is very rarely lenticular as shown here. Many cantilever bridges have the entire lower chord aligned with the deck, which has the added advantage that the entire span has a high clearance for shipping, if need be. The Forth bridge actually contains relatively thin truss bridges within the cantilever, resting on supports which carry their weight down to the bottom chords. But there is also a disadvantage to the horizontal-bottomed cantilever: in order to get the required depth of the trusses and the clearance for shipping, the towers have to be much taller than they would be with the sloping lower chords.

The diagram above hints at a relationship between a cantilever bridge and a combined suspension bridge and arch bridge. Is there anything in this? Consider other hints in this direction such as Brunel’s Saltash bridge (a lenticular truss that can be looked at as a self-anchored suspension bridge and a tied arch), and the Severn suspension bridge (with inclined suspenders, hinting at truss action). Remember that you can remove the suspended span of a cantilever bridge without anything falling down.

Consider a steel arch, which is capable of similar spans to steel cantilever bridges. In an arch, there is very little bending moment if it adheres to the funicular. But in a truss beam, increasing the span increases the bending moment, requiring a deeper truss. Eventually a point is reached where the truss is so deep that you may as well throw away the lower chord and simply build an arch, using the ground as the compression element. What the cantilever bridge achieves is to prevent the build up of bending moment by introducing the two hinges. The penalty (there is always a penalty) is the necessity of the balancing side-spans, though in fact these spans are useful in bridging the gap.

Since an arch is so much simpler than a cantilever, why not always build an arch? When the gap to be bridged is much greater than the 1800 feet of the Quebec bridge, some supports have to be placed in the water. If the water is deep, the clear span of an arch at the surface of the water will be less than that at the river bottom. Furthermore, if there is a great depth of mud, the arch will have to continue until good ground is reached, and even then, the abutments will have to be built to withstand the horizontal thrust. With a cantilever bridge, the foundations can be built by sinking vertical caissons, which is often fraught with difficulty. Providing a sloping support would be very much more difficult. Do you know of any arches that have their feet in the water? You cannot count multiple arches where the thrust is carried from span to span until it reaches the river bank.

If you still hanker after making a long beam, here is a re-arrangement of the parts of a cantilever bridge to make a beam bridge.

All we have done is to reverse each cantilever arm and change its shape to deal with the new forces, and deepen the central span, which is now rigidly a part of the beam. The average depth of the beam is far greater than that of the cantilever, requiring many very long struts which will need cross-bracing. So the whole thing is much heavier than the cantilever span. To achieve shipping clearance, the piers would have to be doubled in height. The alternative is to change the shape of the beam, as below.

Next comes a picture showing how the equivalent arch has to pass through water and mud to reach good ground, making the span much longer.

The maximum practicable beam-truss span is probably about 4/9 that of the maximal arch span. See also longest spans.

Perhaps it is worth trying a little calculation. Here is a diagram of one half of a cantilever bridge.

For simplicity, the two halves of the cantilever will be assumed identical, and the base of the tower will be assumed to be very narrow. We will assume that there is a single massive load which travels across the bridge. The weights we need to consider are as follows.

C = weight of the counterweight

L = weight of the load

S = weight of the suspended span.

The required distances are these –

CP = distance from counterweight to pivot = cantilever arm

LP = distance from load to pivot.

The half-weight of the suspended span can be considered to act at the end of the cantilever arm, because of the hinge at that point.

The equation for balance is as follows.

C x CP > L x LP + 0.5 x S x CP, so

C > (L x LP + 0.5 x S x CP) / CP, or

C > L x LP / CP + 0.5 * S

The moment to be balanced is clearly greatest when the load is at the end of the cantilever arm, and then

C > L + 0.5 * S

Because the suspended span is supported by hinged joints, the load is entirely supported by the one cantilever, but as soon as it moves on to the suspended span, its weight begins to be shared by both halves of the bridge. The moment to be balanced falls until the load reaches the centre of the bridge, at which point the load is shared equally by the two halves. The moment continues to fall until the load reaches the other end of the suspended span, at which point the moment is entirely resisted by the other cantilever.

The next diagram shows graphs of the moments (in green and purple) on the two cantilevers as the load moves across the bridge.

The next picture says a little more about the Forth railway bridge.

The cantilever system is symmetrical, but the loads are not. The left side bears half the weight of the suspended span, plus the weight of the railway train that has been added to the picture. This throws more weight on the piers marked with red arrows, and less on the others. In the event that the left purple arrow represents the maximal sustainable force, the bridge will lift off the right hand piers.

In the Forth bridge, this is prevented by hanging a heavy weight from the inshore end of the cantilever. This weight can be less than the sum of the weights of the train plus half the suspended span (ignoring the need for a safety factor). Why can it be less? When the train is at the inshore end, that cantilever is supported by its masonry pier. In the central span there are no piers. Therefore the central tower is made much wider than the two outer ones, so that it cannot be overturned by any realistic load. It does have one advantage over the outer towers; both its arms hold hold a suspended span, so the only unbalance is from railway trains.

If we look at the main members of these great cantilevers, we can see that the compression members, struts, are massive tubes, while the tension members, or ties, are light trussed girders. But now look at the towers. All the main diagonal members are massive tubes. Are they all struts? When the train is in the position shown, the strut action occurs in the two tubes that slope from bottom left to top right. But when the train is at the right hand side of the picture, the other two tubes become the struts.

In many trussed structures there is no clear division between some of the members as ties and struts, because of moving loads, and so they have to be built to act as struts, which is the worst case. The upper and lower chords are of course always in compression and tension respectively.

Contrast with the Forth Bridge

A very common type of bridge, over or under roads, is the concrete cantilever bridge. It’s simple, clean appearance is a great contrast to that of the trussed cantilever. These bridges are also built over rivers. Bridstow bridge near Ross-on-Wye is a good example.

Cantilever Foot-bridges

Here is a cantilever bridge of a type which is found in several places in Gloucestershire and Wiltshire. This one crosses the Barnwood bypass, east of Gloucester. There is another one about a kilometre to the east. The narrow roads leading to these exhibit cracks typical of slumping on embankments. The local ground is largely clay. Some pictures at right show this. The last picture shows cracking due to drying out of the soil in a nearby field.

The M11 motorway is crossed by several bridges which look similar, though they have much longer span.

Here is a slender cantilever footbridge made possible by the use of steel wires inside the concrete. See also pre-stressed bridges.

The diagram above represents the response of a simple cantilever bridge to subsidence. In this case the joints allow stress-free movement, so nothing is distorted.

Jacking might still be provided. In a very slender foot-bridge, the slightest error could be noticeable, and so some adjustment may be needed.

After a bridge has been completed, jacks may be concreted over, or they may be left as usable adjusters. The Eiffel tower is a good example of the jacking requirement. The stresses, and therefore the strains, at the base, changed markedly during construction. Jacking enabled the builders to compensate as the work progressed. This subject is developed further in Indeterminacy.

This is what happens if the load produces a moment that is greater than the cantilever can supply: fine in a see-saw, catastrophic in a bridge. The answer is to tie down the ends. They can in fact be pulled down with sufficient pre-stress that there will always be compression between the bridge and the ground. The outer ends of the Forth rail bridge are pulled down by counterweights, a trick that cannot be used elsewhere in a cantilever bridge. That is why the central tower of the Forth is so wide: even with the heaviest train in the centre of a span, the central cantilever pair is stable.

Why are the towers of big cantilevers so high? Consider the weight of the bridge and it load, pulling down at some point a long way from the tower. This creates a moment which will pull the bridge down unless it is resisted. An opposite moment is created by the tower pulling the top chord and pushing the bottom chord. Since any moment is the product of distance times force, increasing the height of the tower decreases the forces. But increasing height will eventually add more weight than is gained by the reduction in force. In practice, the ratio of length to height does not vary all that much. Look at some pictures of cantilevers and try to work out the ratio in each case.

Note that once the load has moved on to the suspended span, its moment at the tower does not increase even if its distance increases. Why is that? In that case, why don’t bridges have much shorter cantilevers and much longer suspended spans?

Cantilevers can be used during construction of structures which, when finished, do not include them. This enables the channel below to be free for navigation. The penalty is the introduction of temporary forces that are not present in the final structure. This was dramatically demonstrated by collapses of several box girder bridges during construction in the 1960s.

The two halves of the Sydney harbour bridge were held back by cables until the time came for them to be connected. Cantilever construction has been used in trusses, Bailey bridges, box girders, truss arches and cable-stayed bridges. The diagram below shows an example of cantilever construction at two different stages.

Vertical Cantilevers

Vertical structures may be regarded as columns, but when there is a wind, they have to act as cantilevers. Generally, the movement is very small, but some very tall buildings can sway perceptibly. Some even include active compensation in the form of masses which are made to move in response to the accelerations detected by sensors. All the objects in these pictures moved in response to the wind, but in only one case was the movement visible.