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How arches work

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

Modern arch bridges work differently - read more.....

Stone arch bridges seem to have only three chapters - but in fact there are four.

  • The first chapter contains the voussoirs which are the wedge shaped pieces of stone or brick that form the line of the curved arch known as the arch rib or ring - red on the diagram.

  • The second chapter is the roughly triangular area between the outer curve of the arch rib and including the bridge deck known as the spandrel - light blue on the diagram.


  • The third chapter contains the foundations - green on the diagram.


  • The fourth chapter may be unseen and hidden. It is the supporting structure, often called 'centering' that holds all of the components in position as the arch is built.


  • The picture below is reproduced by kind permission of Andreas Kessler, Igis, Bannwaldweg 30, CH-7206 Igis, Switzerland.
    It shows the centering scaffolding built by only 6 men in 1929 to support the arch of the Salginatobel Bridge as the concrete was being cast.


    The arch derives its strength from its shape so that is why it will stay in place for a very long time as long as the foundations don't move.
    The secret of the stability of stone or masonry arches, such as at Bradford on Avon shown lower right, lies not in the mortar but how the stones are packed together to allow the internal forces to flow.

    Dry stone walls are so-called because they have no mortar.

    To build them a shallow narrow trench is dug and a base of small stones laid. The wall is then built up in horizontal layers, each narrowing slightly towards the centre of the wall.

    The centre is filled with small stones or rubble and part way up, stones are laid across the width of the wall to tie it together. The topping is generally a row of slanting or vertical stones.

    Each stone in the wall bears on the stones below and hence compresses them.

    This pressing down force gets bigger as you go down the wall and follows what is called a 'line of thrust' which is the line along which the internal forces flow.
    At any point in the wall the thrust line is due to the weights of all of the stones above and is vertical and downwards and acts at the centre of gravity - the point which represents the weight of all of the stones at a given level in the wall.

    When you push the wall sideways the thrust line becomes the combined effect of your push and the weight of the wall.
    The thrust line is then inclined at a very small angle to the vertical because normally your push will be small compared to the weight of the wall.
    In order to push the wall over completely and as a whole, you would have to disturb the thrust line so much that it no longer passes through the base of the wall. In other words you would have to push the wall past its tipping point.

    Each voussoir of an arch interacts with its neighbours - just like the dry stonewall - but the thrust line is more complex.
    Each stone is pushed down by the stone above and can only push back if it can push down on the stone below.
    That is the reason why arches only work when they are complete because each voussoir reacts off its lower neighbour.

    The very bottom stone has to react against a foundation - that is why the ground must remain firm.
    The difference between the arch and a dry stonewall is that in an arch the direction of the line of thrust changes as the force from the stone above combines with the weight of the voussoir plus the weight of any infill rubble or stones from the spandrel.
    These three forces combine to bear down on the stone below and the stone below pushes back.

    So how do we know where the thrust line goes in an arch?

    Robert Hooke found the solution in 1675.
    He realised that if you hang a chain between two points then you get an upside down version of an arch.

    He wrote, 'As hangs the flexible chain, so - but inverted - will stand the rigid arch'.

    Just as the line of thrust for our dry stonewall had to lie within the base of the wall for it to remain stable so the thrust line for the arch must fit within the line of the voussoirs of the arch.

    In an arch each voussoir stone has four forces acting on it but we will simplify them down to three - two actions and one reaction see the diagram below.


    The first is the internal action force (which I have called P in the right hand diagram) from the neighbouring voussoir stone above pressing down on it.

    The second is the two forces of the self-weight of the voussoir and the weight bearing down from the mass of masonry in the bridge combined into one external action force (W).

    The two action forces must be resisted by a third force (R) that is the internal force reaction from the voussoir stone below if equilibrium is to be maintained.



    Of course, in turn, R will bear down on the voussoir immediately below it.

    In that way the internal forces P and R define the thrust line in that part of the arch.

    We can draw arrows of the three forces (the two actions and one reaction) to make a closed loop - a triangle of forces as in the left hand diagram.



    In the next diagram (on the right) we can see in the upper diagram that all of the voussoirs in an arch bridge have these three forces acting on them.

    I have drawn Robert Hooke's hanging chain underneath.

    There are many possible thrust lines within the arch.
    Fortunately we don't need to know exactly where the real one is as long as we know it lies within the voussoirs from the key stone at the top to the springing stones at the base.

    What happens when the thrust line is at the very edge of the voussoirs?
    In the diagram below this is happening in 5 places.

    When the thrust line force is at the very edge of the stone then the voussoirs are at their tipping point. In other words they are about to rotate about their edges which are in contact.




    If the thrust line goes outside of the line of the voussoirs, as in the diagram below left, then the voussoirs will rotate about the edge - rather like a door hinge.
    Once an arch bridge is built and in place then we know that the thrust line is within the voussoirs - that is our starting point.

    The key to the continued success of the bridge is then in the foundations.
    If the foundations cannot cope and move slightly outwards then the thrust line will also move.
    If it strays to the edge of the stones then the joint at that point will open up and a crack will appear.

    If only one hinge forms in the arch then it may be a bit unsightly, but the arch will not collapse.
    Indeed if three hinges form the bridge will not collapse either.

    For the whole arch to collapse the arch must have four or more hinges because then it becomes a mechanism - an assembly of moving parts as shown in the diagram.

    Normally the foundation movements and consequent cracks and hinges in the masonry do not happen quickly so there is time for those people in charge of the bridge to read the signs and make repairs.

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