a bridge

Bridges

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Bridges

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## How do Suspension or Hanging Bridges work |

There are basically two types of hanging bridges - suspension bridges and cable stayed bridges. On the left is perhaps the most recognisable suspension bridge in the world - the Golden Gate Bridge in San Francisco USA. On the right is the more modern cable stayed bridge with a definite 'wow' factor. They are exciting because they can be light and daring with long spans. Click on Notable Bridges above to see some more examples. Hanging bridges are absorbing because they can be refined works of art. Click on the header Bridges as art above for more on this. They are interesting because at one level the flow of forces in hanging bridges is easy to understand yet when we dig down to a deeper level we find that they are statically indeterminate and are a blend of many time dependent complex processes. Wobbly bridges, like the swing bridge in the picture in New Zealand on the right, may be acceptable in remote places or as tourist attractions but they are of little use as major highway bridges. The breakthrough for hanging bridges came when in 1801 Judge James Finley built one with a level stiff bridge deck in the USA. The ideas spread quickly in the USA and UK. The chapters of the book of a hanging bridge are the suspension system, towers, stiffening girder, bridge deck and the foundations including bearings and anchors. Click on the diagram to the left to enlarge it and find further explanation. The principle of the suspension bridge is that of a simple clothes line. We stretch a rope between two anchor points and prop the line up in one or more places then hang objects from it. Ropes, chains or cables are suspended between two towers with vertical (or occasionally inclined) hanger cables attached at intervals stretching down to a stiff road deck. The deck distributes the forces from the traffic on the bridge to the hangers and onto the cables. Cable stayed bridges hang in a different way. There are no curved cables slung between the towers but rather straight cables from the tower to the deck as at the Millau Viaduct in France. There are basically two types the fan and the harp but there are others. The fan system has cables radiating out from a single point on the tower. The harp has parallel stays from tower to deck. The chapters, paragraphs of the Millau Bridge are shown on the right. Suspension cables may be spun on site. They generally contain thousands of wires. The job of the towers is to create the height from which the bridge deck may be hung - they are the props on the clothes line. They come in many shapes and sizes. There is one important difference we should note between suspension bridges and cable stayed bridges. In a suspension bridge the main cable tension is concentrated in the anchorages at each end. By contrast the cable tension in a cable stayed bridge is distributed over many anchorages along the deck. The more stays the more supports there are to the deck and the lighter and less costly the bridge. Suspension bridge main cables may be anchored at their ends to earth through anchorage blocks or to piers or to the bridge deck. In some bridges the back stay cables react off piers as at the Bay bridges in San Francisco. Consequently foundations may have to resist large uplift forces. Anchorages are a major issue for suspension bridges, far less so for cable stayed bridges. Cable stays are usually anchored to towers to make erection easier. Consequently the forces each side can become unequal and so bend the towers as a vertical cantilever. The towers therefore need to be flexible enough to resist. The horizontal component of the internal tension in the inclined stays of cable stayed bridges creates a considerable compression in the bridge deck. Many modern bridges decks are box girders of steel or concrete to give torsional strength and streamlining in wind. Hanging bridges are not easy to build so the erection process plays a big part in design thinking. That is why many modern bridges are cable stayed - they are easier to erect than suspension bridges - but they still represent a significant challenge. If you have ever erected a tent with rope stays you will know the problem. Just as you tighten one stay another loosens. The tensions in the stays are interdependent. If you change one then you change another. However the situation is worse than that. If you get the tensions wrong your bridge could collapse. So when bridge builders erect the deck girder of a cable stayed bridge they have to be very careful to get the tensions in the stays right. You can read more in the book - for details click on Book at the top of this page. Now let's consider how a cable suspended between two towers like a clothes line behaves. It will work quite differently from a cable being pulled in axial tension only because forces are applied laterally along its length. A rope can only resist a lateral load by changing its shape. Let's imagine what happens to the rope (a) in the diagram when a large truck on the bridge moves onto the bridge. The truck is a point load applied directly to the rope shown in (b). The wiggly lines represent a load distributed horizontally. This might be the the dead weight of the bridge or the weight of a line of traffic on the bridge (which we call the 'live load' because it can move). The rope will change its shape and the internal tension in the rope will increase. As the ends of the rope are held in place then the horizontal external reaction forces will also increase. If the sag of the rope gets bigger then the value of the horizontal reaction at the ends gets smaller. This is because the ends of the rope will be more vertical and will tend to pull down on the end more. This will increase the required vertical reaction. For most hanging bridges the size of the first wiggly load, the dead load, is much bigger then the second wiggly load the live load. The dead load is also permanent. So what happens is that the bridge settles down under the dead load and then changes as the live loads move across are smaller. The effect of the stiff bridge deck is to both increase the self weight of the bridge and its stiffness. We can calculate the force in the cable approximately as follows. The next diagram on the right shows the right hand half of the cable. In effect I have cut the cable in half to reveal the internal force H - the tension in the cable.At the other end is the external reaction force T.This is the combination of the horizontal and vertical external reaction forces shown previously. Again the wiggly line is the dead weight of the bridge which I have called w kN/m (i.e. w kiloNewtons per metre).Because we are assuming it is uniformly distributed along the span the total weight on the half span is w times the length of the half span L/2 which is wL/2 and is shown as the vertical downward arrow.It effectively acts halfway along the half span i.e. at the quarter point. Now let's look at the turning effect of these forces about the right hand end of the bridge. Recall that the turning effect or moment is a force times a distance. The force H acts horizontally and the distance between it and the right hand end of the bridge is d.Therefore the moment of H about the right hand end is H times d and it acts clockwise.The weight of wL/2 acts vertically at the quarter point of the span.The distance from where it acts to the right hand end is (L/4).Therefore the turning effect or moment of the weight is the total weight of (wL/2) times the distance (L/4).This is (wL/2)*(L/4) i.e. (wL and is anticlockwise.^{2}/8)Now the turning effect of the force H and the weight (wL/2) must balance.This means that H*d = (wL.^{2}/8)From this we conclude that H = (w* L.^{2}) / (8*d)How do we interpret this formula? It is just a mathematical way of saying three things. Firstly it tells us that the tension in the cable increases directly with the dead load. Secondly it tells us that the increase is even bigger as the span increases - indeed it increases with the square of the span. So if the span is doubled the tension is quadrupled. Finally the formula tells us that the tension is inversely proportional to the dip - so the larger the dip the smaller the tension. We noted that earlier by physical reasoning - now we have it through more formal mathematical reasoning. The ratio of the span to the sag of the cable at centre span (L/d) is interesting. It is sometimes used as a measure of the form of the bridge and is implicit in our formula for the tension since H = (w*L/8)*(L/d).In the next diagram on the right I have cut through a suspension bridge vertically to expose the internal forces. A suspension bridge is statically indeterminate because there are more than three unknown internal forces on each of the two separated parts. We know the load but we have six unknown forces. Two of them are external reactions - the reaction at the base of the tower and the anchor force needed to hold the cable down. Four of the unknowns are internal forces in the cable and the deck. In 1888 an Austrian Josef Melan, put forward a more advanced 'deflection theory' for suspension bridges. He argued that the deflection of the girder was significant. The mathematics of differential calculus developed by Newton and Leibnitz in the mid 17th century was used to produce so-called 'differential equations' to model the forces flowing through a suspension bridge. Unfortunately the theory rapidly became too complicated for anyone but specialists to follow. Various approximate solutions were developed and important mathematical solutions and techniques produced particularly by Stephen Timoshenko. Today we have powerful computer techniques based on finite element analysis. Now let's look at the internal forces in cable stayed bridges I have made cuts in three places to expose the internal forces. The forces shown are indicative since the actual values will depend on the detailed geometry and loading. The section cut from the centre of the bridge again shows the upward tensile forces from the cables, the downward forces from the dead and live load and the internal forces in the beam bridge deck. The section cut at the top of the mast shows the internal tensile forces in the cables pulling down and the upward internal compression force in the pylon or mast together with bending and shear. You can read more about hanging bridges click on Book at the top of the page for more details. |

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