# BRIDGES

## How modern arches work

Until the 18th century the story of the arch bridge is the story of the stone or masonry arch.

The first arch built not of masonry but of cast iron was at Ironbridge in Shropshire. It is like a masonry arch but with a framework of the new material. You may be able to spot the voussoirs made of cast iron ribs.

The modern arch, unlike a masonry arch, is usually made of a continuous material like steel or concrete. It can therefore resist bending.

So the modern arch is a special sort of beam - read more about beams.......

It is a special beam because of its arch shape - beams are usually straight.

However the structural designer can find a structurally efficient shape that keeps the internal forces due to bending (the bending moments) to a minimum.

For this reason the modern arch acts mainly in compression in a manner, similar to but different from, the masonry arch.

The development of the modern arch from the masonry arch also involves trusses - read more about trusses.....

So let's start with a masonry arch with a solid spandrel as in the diagram on the right.
The first arches were mostly semi-circular - though there were exceptions.

For an example see the masonry multi arch bridge at Bradford on Avon

Then designers, such as I K Brunel, began to make their arches flatter - as in the left diagram.

They made the spandrels lighter - firstly by making holes then by using vertical struts.

They also had the idea of introducing a pin to make the calculations possible since the structure is statically determinate.

The world famous Salginatobel Bridge designed by Robert Maillart is probably the best example.

However there is a limit to how flat you can make a 3 pinned arch. You can see this quite clearly in the diagram below and right because a flat arch becomes a beam and with a central pin is simply a mechanism not a structure.

Following on from the Ironbridge and using a truss to make the arch the length of the bridge could be considerably extended by building cantilever trusses from each support as shown in the diagram below.

Examples in France are the Viaur Railway Viaduct which is 3 pinned and the Garabit Viaduct which is two pinned.

Then came the idea to use the truss as a giant arch and to suspend the deck from it.

A world famous example is the Sydney Harbour Bridge.

But the arch could be so much more slender if it is a box girder - with 3 pins.

Or with 2 pins as at the Clyde Arc Bridge

It can even be made to open as the Gateshead Millennium Bridge.

Finally let's look briefly at the forces in a two pinned arch.

The lower right diagram shows a line diagram of an arch.

In the top diagram the arch has a load applied part way accross the span. The arch wants to spread out but is prevented from doing so by the pin supports. The horizontal reactions H are equal and opposite since there are no other horizontal forces shown. The vertical reactions are different - R1 and R2.

In the lower part of the diagram the arch has an imaginary cut at its apex. The internal forces at the cut are shown with dotted arrows.

The structure is statically indeterminate since we have 5 unknown forces (R1, H, F, M, R2) and we can only set up three equations of equilibrium by balancing the forces vertically, horizontally (actually we already have done this by making the horizontal reactions H equal) and by equating the turning effects of the forces (i.e. taking moments).

However a solution is possible when there is a hinge at the apex since M becomes zero. We can balance the forces vertically for the whole structure and take moments about one of the end pins. Then we can do the same thing for one half of the structure and find values for R1, R2, H and F. Most modern bridges, such as the Clyde Arc are two pinned structures analysed using computer techniques. You can read more about the finite element method in the book.