Pointed  Arches

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This page could be read in conjunction with the pages about arches, funicular, and arches in religious buildings.

Arches are generally of two types, rounded at the crown, and pointed at the crown.  The same could be said of domes.   Modern arch bridges are generally round, though some concrete ones are segmented.  Pointed arch bridges are mainly medieval.  The shapes of domes also vary considerably, and in fact much more extremely, if we count a spire as an extreme case of a pointed dome.  Many domes have a lantern or other structure at the top, which obscures the shape.  Inside a spire we may see a clue to the construction of all these structures - it may contain many wooden struts, because a straight cone does not follow a funicular.

Pointed arches are found in the architecture of many cultures, particularly in buildings with a religious purpose.  The shells of Sydney opera house are modern versions of the same idea.

Before looking into these arches, we can look at these two pictures, showing necklaces, one with, and one without, a pendant.  Both follow funicular curves.  We see that the discontinuity in the slope of the chain is associated with a point load.  Inverting the chain, we see that a pointed arch should correspond to a large load at the top.  In fact, the reverse is usually true - precisely because of the shape, there is less material above a pointed arch than above a round one which follows roughly the same line, apart from the point.

Some bridges do have segmental arches, because the load is applied to them at a few discrete points by columns or walls.  The segments are often straight, as the main mass is in the thick deck which stiffens the whole structure.  The force down each wall is large compared with the weight of the arch, and of course the thrust in the arch is even bigger, because of its slope.  Thus the force in a segment is far greater than its weight, and it is more or less like a strut in a truss.

The diagrams below show the crown of an arch, and then a separation to show the forces which each half exerts on the other.

ArchPicsA2.gif (7444 bytes)

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The next picture shows the same thing for a pointed arch, with two extra details showing the crown of a pinned arch, with two positions of the pin.

ArchPicsA4.gif (20126 bytes)

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Diagram B above shows that the horizontal forces at the crown cannot go far before they reach the outer edge of the arch.  Then what?  Forces cannot just disappear.  In C this has been emphasised by adding a pin at the top.  Diagram D is similar, but with the pin at the bottom.  In C, the pin pushes the half arch horizontally.  But almost immediately the line of that force hits the edge of the stone.  How can it be deflected downwards?  There is hardly any weight above this point that can add a downward force.  The answer is, the arch has to provide the force itself.  The only way an object can create a force is by deforming.  The point of the arch is highly stressed, and it might even break.  The pin position D is much kinder.  Removing the pin allows the faces of the arch halves to meet.  If they are perfectly flat and vertical, they will in principal touch at all points.  In view of the results already seen, how do you think the pressure will vary, if at all, over the interface?

So the pointed arch cannot behave as a pure arch, that is, one in which the forces are entirely parallel to the material at every place.

In the next diagram, the pointed arch is a part of a wall or a bridge, which the the normal habitat of this type of arch.  If we now think about the horizontal force at the crown, it is entirely believable that the thrust goes outside the voussoirs into the masonry.  This is probably true at some points in any arch that is part of a masonry structure.

ArchPicsA5.gif (22465 bytes)

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Perhaps we can look a little more into pointed structures like gothic arches and Sydney opera house.  Here is a catenary, the funicular for a shell of uniform density and thickness, curved only in one direction.

FuniButtAA.gif (2164 bytes)

Let us now remove the top part and butt the remainder together to make a pointed shell.

FuniButtAA2.gif (2072 bytes) FuniButtAA3.gif (1843 bytes)

That was easy.  We took two pieces of a funicular, and we joined them to make a new one.  Easy - but wrong.  As you can read in the page about funiculars, a funicular is not a thing, and it is not a mathematical curve.  The curve that we call a funicular depends on the forces which are applied to a member.  Here we have removed the weight of the top part of the original shell, and what remains is not a funicular.  A funicular cannot change direction without an external applied force.  The forces at the apex are horizontal, as we saw earlier in this page, and not along the shell.  What can we do?  Here is a hint.

FuniButtAA4.gif (1957 bytes)

We have taken the part that we removed and put it back on top, upside down so that it can rest neatly.  Since the two halves are now supporting the same weight as they did originally, the forces in them must be the same.  So they must be truly funicular.  This is hardly a solution that is visually acceptable, so we must replace the top part by a compact weight, running along the ridge of the shell.

FuniButtAA5.gif (1787 bytes)

The question now is this - would it be better to remove that extra weight, and use the material to stiffen the shells against bending.  They would need at least some stiffening in any case, to prevent buckling, and to resist wind loads.

One lesson from this is that departure from the funicular must be paid for, in resistance to bending moment, which must be paid for, literally, as this will cost money. The ultimate departure from the funicular is probably the beam.

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Bolton.jpg (46148 bytes)BoltonAbbey.jpg (47296 bytes)GlosOldArch.jpg (62872 bytes)The robust nature of masonry arches is well illustrated by the longevity of the many ruins in Britain, many dating from the dissolution of the monasteries, which was begun in 1538 by Henry VIII.  The pictures here show Bolton Abbey in Yorkshire, and a ruin in Gloucester.

BristolBombedA.jpg (161946 bytes)And here is the result of mid-20th century bombing.

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