Geometry of Bridge Construction

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Geometry of Bridge Construction




The four kinds of bridges and some combinations




A. The beam or truss bridge is, in effect, a pair of girders supporting a deck spanning the gap between two piers. Such a beam has to withstand both compression in its upper parts and tension in its lower parts. Where it passes over supports, other forces come into play. A beam may be a hollow box girder or an open frame or truss.



B. An arch bridge can be designed so that no part of it has to withstand tension. Concrete is well suited to arched bridge design. When reinforced concrete is used, a more elegant and sometimes less costly arch can be designed and most concrete arch bridges are reinforced.



C. A suspension bridge consists, basically, of a deck suspended from cables slung between high towers. The cables of high tensile steel wire can support an immense weight. The towers are in compression and the deck, often consisting of a long slender truss (used as a hollow beam), is supported at frequent intervals along its length.



D. A cantilever bridge is generally carried by two beams, each supported at one end. Unlike a simple beam supported at both ends, the cantilever must resist tension in its upper half and compression in its lower.



A fifth type arrived on the scene in 1952 the first modern cable-stayed bridges were built in Germany and Sweden. There are also many other composite forms of bridges. The bridle-chord bridge is a combination of a long beam (usually a trussed girder) partially supported by steel wires from a tower at one end, or from towers at each end. Most cantilever bridges are designed so that a gap remains between two cantilevered arms that reach out from their abutments: the gap is bridged by a simple beam.

Some Examples



An early history of bridge building




The Romans' legacy to bridge building was the heavy masonry arch bridge, hundreds of which were built throughout Europe. In this, large stone blocks were wedged against each other to form an arch. The central stone at the top of the arch was known as the keystone. The finest surviving example of such a bridge is the Pons Fabricius in Rome. Completed in 62 BC, the bridge (now called the Ponte Quattro Capi) has two fine semicircular arches each spanning 78 feet. A small "relief" arch in the central sponging of the two main arches releases excess water in times of flood.

So prolific and efficient was Roman building that it was hundreds of years before Europeans took to bridge building anew. Then, in the 12th century Catholic priests and professionals took over the building of bridges because the Church recognized the advantages of good road communications in a developing society. In France a group of interested priests formed a new order, the Freres du Pont, to design and build lasting bridges. Most famous of this order's works was the Pont d' Avignon, built in 1177 over the Rhone River. It had 21 arches in all, the longest spanned 115 feet. Similarly in England it was Peter de Colechurch who designed and built the first stone bridge over the Thames, the famous London Bridge. Another was done by the architect-priest Giovanni Giocondo (c. 1433-1515) who used the segmental arch in Paris' first masonry bridge, built in 1507.

Until the late 17th century bridges continued to be designed and built largely by priests or architects with a flair for engineering. But such complex and essential work could not rest in the hands of gifted amateurs forever. In 1716 French army engineers took the lead on the rest of the world in bridge building.

References

Encyclopedia Britannica Chicago: 1959 edition, Vol. IV pp 123-137
Random House Encyclopedia New York: 1977 edition pp 1756-1759
Brown, David Bridges New York: Macmillan Reed International Books; 1993
Eves, Howard An Introduction ot the History of Mathematics: New York: Holt; 1964





Simple Truss (or Beam) bridges




Arch Bridges
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Suspension Bridges


Cantilever Bridges




Various Bridges


The seven Bridges of Konigsberg



The Konigsberg Problem and the beginning of Network theory.



In 1736 Euler resolved a question as to whether it was possible to take a walk in the town of Konigsberg in such a way that every bridge in the town would be crossed once and only once and the walker return to his starting point. The town was located close to the mouth of the Pregel River, had seven bridges, and included an island. Euler reduced the problem to that of tracing the associated graph in such a way that each line of the graph is traced once and only once, and the tracing point ends up at its starting point.



In considering the general problem, the following definitions are useful. A node is a point of a graph from which lines radiate. A branch is a line of a graph connecting two consecutive nodes. The order of a node is the number of branches radiating from it. A node is said to be even or odd according as its order is even or odd. A route consists of a number of branches that can be traced consecutively without traversing any branch twice. A graph that can be traced in one route is said to be unicursal; otherwise it is said to be multicursal. About these concepts Euler succeeded in establishing the following propositions:



1. In any graph the number of odd nodes is even.
2. A graph with no odd nodes can be traversed unicursally in a route that terminates at its starting point.
3. A graph with exactly two odd nodes con be traversed unicursally by starting at one of the odd nodes and then terminating at the other.
4. A graph with more than two odd nodes is multicursol.


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