Arches, Vaults, and Shells
11/27/07
Arch
• A true arch is a funicular structure, similar to a
suspended cable
• While the cable was in pure tension (from its
own weight or a uniformly distributed load), the
arch is in pure compression
• From a practical perspective, arches are an
important way of allowing an opening to be
made in a wall
– admit people, light, or water, etc.
• Arches are also important aesthetically, and
have often been used as monuments
Catenary vs. Parabola
• Recall that for the cable the catenary can
be approximated with a parabola to
simplify the math.
• The same terminology of “catenary” vs
“parabola” can be applied to the arch,
depending on the loading conditions.
2
Parabola: a x
Catenary: a cosh((x-b)/a)
-20
-10
0
10
20
Catenary vs. Parabola
Parabola
Catenary
• Likewise, the forces at the ends of the
arch are found the same way as the forces
on the towers that support a suspension
cable
Catenary vs. Parabola
Corbeling
• So how do we arrange for a solid material, a
group of bricks for instance, to be entirely in
compression while having a gap beneath them?
• The early attempts used ordinary shapes for the
bricks over the opening, leading to Corbeling.
• You can make a gap in a masonry wall this way,
provided the angle is sufficiently steep (at least
45 degrees)
Voussoirs
• However, this primitive form of the arch creates
tension in the bricks.
• Why is this a problem?
– Recall that masonry is much weaker in tension than in
compression
• To obtain a true arch, we need bricks shaped
like wedges which are narrowest at the opening.
• Bricks or stones cut to this shape are called
voussoirs.
• Their wedge shape pushes outwards on each
neighbor, so that the load causes compression
throughout the arch.
Corbeled and Voussoir Arches
Keystone and Voussoirs
Line of Thrust
• Of course each voussoir pushes on the one next
to it.
• The top piece of the arch passes its weight to its
neighbors, the next in sequence pushes out with
the combined weight of the top piece and its
own, and so on.
• At each step the angle between the stones (and
so the angles of the applied forces) changes.
• A diagram showing the forces for all the pieces
at once is called the line of thrust.
Line of Thrust
Middle Third Rule
• Middle Third Rule
– The forces in a object, e.g. a bearing wall,
must remain in the middle third of the object
for it to remain in compression only
• The same rule applies to arches; the line
of thrust must remain in the middle third of
the arch for it to remain in compression
only.
Building an Arch
• How is an arch built?
• It would be awkward to stack stones on empty
air, so one provides a temporary scaffold that the
masonry can rest on.
• Then one adds the keystone, the piece at the
top of the arch that joins one side to another and
allows them to “fall against one another”.
• With the keystone in place, the arch is stable,
and the scaffolding can be removed.
Voussoir Arch
Making the Arch Shape Permanent
• The initial shape of the arch generally assumes uniform
loading
• If the load is applied non-uniformly (such as at one spot,
as some one stands there), the other stones can be
pushed out of position and the arch can collapse.
• To prevent this, material must be placed on top of the
arch
• This prevents the stones from pushing out, away from
the original shape.
• In this way the loads are channeled to the ground as
intended.
Fixing the Shape of the Arch
Non-funicular shapes
• Using this technique, one can make an
arch even using non-funicular shapes (i.e.
things other than a catenary or parabola).
• There are many possible arch shapes
• the most famous are
– the semi-circle favored by the Romans, and
– the pointed Gothic arch that came into vogue
in Medieval times
Types of Arches
Pont du Gard
Roman Aqueduct, 100 AD?
Arch of Septimius Severus
Ancient Rome 203 AD
Arch of Titus
Ancient Rome, 81 AD
Flying Buttress
• A closely related architectural element is the
“half-arch”.
• The half arch is not stable by itself, but its
inherent lean can counteract the outward push
of another element, such as a bearing wall.
• The half-arch is a device often seen in gothic
cathedrals, where it is called a “flying buttress”.
• This allowed walls to be thinner and less loadbearing, allowing more openings for stained
glass windows.
Flying Buttresses
Notre Dame
Chartres Cathedral, France
Flying buttresses at York Minster Cathedral
England 1220
Modern Arch Bridges
• Of course one can also build an arch without
using masonry. Indeed, this is how many
modern bridges are constructed.
• If we use metal or wood, then it becomes
possible to add one or more hinges to the arch.
• An arch without a hinge is called rigid
• There are also two-hinged arches and threehinged arches
Modern Arch Configurations
Hinged Arches
• Why would you hinge an arch?
• For much the same reason as using a roller
support in a bridge; it allows the arch to contract
and expand with temperature, or with deflections
due to various live loads.
• A two-hinged arch will reduce the bending at the
bottom of the arch (where the stress is greatest)
by allowing the ends to pivot – the rest of the
arch then bows up further if it expands, for
example.
• A three-hinged arch reduces bending throughout
the structure.
ChaoTianMen Arch Bridge
Santa Barbara & the California
Coast / Cold Spring Arch Bridge
Natchez Trace Double Arch Bridge
Franklin, Tennessee
St. Louis Gateway Arch
• Arched gateway to the historical American
West, on the bank of the Mississippi River.
• A 630 foot high graceful sweeping tapered
curve of stainless steel
• The tallest memorial in the US.
• Completed 1966.
• One of very few structures in St. Louis that
is built to withstand a serious earthquake.
St. Louis Gateway Arch
Making a Semicircular Arch
• Calculating arch segment angles
Vaults and Shells
The extension of elements
• Columns extended into 2D?
– the bearing wall
• Beams extended into 2D?
– the slab
• Arches extended into 3D?
– the vault
• As in the earlier cases, a vault is more stable than a
series of adjacent arches, because some of the load is
distributed at an angle to other parts of the vault.
• Like a true arch, a vault is designed to be in compression
(not tension), through shear resistance.
• A 3D arched structure that can also withstand tension is
called a shell, and we’ll discuss it later.
Load Distribution
Cylindrical Vaults
• We will begin with cylindrical vaults, which
curve only in one dimension.
• These can be
– barrel (extension of semicircular),
– Catenary, or
– pointed (Gothic)
Cylindrical Vaults
Instability
• Just as in cathedrals, building a 2D
structure like this (draw semicircle) means
that there will be some outward (lateral)
thrust at the base.
• In a small system the friction with the
group can be sufficient to provide this
• However, this can lead to a spread at the
top of the vault, which can be dangerous.
How to improve stability?
• Use a thicker wall at the base to provide support
• Add more material at the lower section of the
vault proper – this is called a haunch – closely
related is a solid buttress of the arch exterior.
• Add a flying buttress to the arch at the weak
point.
• Add a tie between the two sides which is in
tension
Figure 14.4
Barrel Vault
Arch of Sapor, Iraq
Ruin of Basilica of Maxentius (aka
Basilica of Constantine)
Intersection vaults
• In many cases builders wanted a
structure to having intersecting axes –
this was important for symbolism in
cathedrals, for example.
• This led to intersecting vaults, know
as groin vaults.
• The simplest groin vaults had the
same extent in both directions
Roman Groin Vault
One could also build a groin vault with different extents in
the two dimensions
Balancing the thrust forces
• It turns out that the case of equal lengths is
balanced even for the circular, Roman shape.
• However, the case of unequal lengths is not –
this leads to unbalanced thrust forces.
• To compensate, the builders of these
“Romanesque” vaults used massive buttressing
to support the exterior vault, resulting in a stable
but inefficient structure.
Stable Intersection
• A better solution was found in the middle
ages.
• To have a stable intersection of the vaults,
the two directions should rise into a point –
extension of the Gothic arch.
• Since the Gothic shape is much closer to
the ideal catenary than a semicircle is,
these vaults also needed less buttressing
than their Roman counterparts.
Gothic Vault
King's Arms Hotel
Dorchester, England
Dome Vault
• From these intersecting cylindrical vaults, it is a small
step to an actual dome vault.
• Here instead of translating the arch shape, we have
rotated it to get the final shape, resulting in a double
curved structure.
• In most cases the result is a dome that is a section of a
sphere (hemisphere, quarter-sphere, etc).
• As with any vault, the dome vault is in compression only,
and it requires lateral support to prevent spreading of its
base.
• Furthermore, a dome which is part of a sphere (not a
funicular shape) will tend to buckle outwards above the
base (the haunch, as we saw above) – so this needed to
be braced.
PPT Pantheon
Building an Arch
• In Byzantine architecture it became
popular to use parts of domes, for
example the Pendentive.
• This is created by taking a hemisphere
and slicing out sections to give it square
sides.
• Then, we add a smaller half-dome to each
wall.
• We repeat the process with the top to
generate the Pendentive.
Pendentive
Hagia Sophia
Hagia Sophia
Hagia Sophia
Lateral Support
• To maintain the proper shape of a
shell under non-funicular loading
conditions, it is usually necessary to
stiffen the shell around its perimeter
(ends and edges).
• As with an arch, the barrel shell also
needs lateral support at its base.
Hagia Sophia
Vault Shapes
• Vaults can also be built in other shapes.
• For example, in some cases, there is a desired shape for
the walls of the building, but the form of the roof needs to
be added.
• The best structure will of course be the funicular
structure, the one for which the loads are naturally all
compression or tension.
• One could try to construct a model of the roof extending
over a building.
• But it turns out to be easier to hang the roof from the
same perimeter shape and take a picture of it.
Funicular Chain Model
• (Explain this to be an upside-down picture)
Non-funicular shapes
• Using this technique, one can make an
arch even using non-funicular shapes (i.e.
things other than a catenary or parabola).
• There are many possible arch shapes
• the most famous are
– the semi-circle favored by the Romans, and
– the pointed Gothic arch that came into vogue
in Medieval times
Roofs
• Since arches and suspension cables are
merely mirror images of one another, if
you can find the best arrangement for the
hanging chains, you can turn it upside
down to find the best form for the roof.
Vault Construction
• Usually a vault is constructed much the same way as an
arch is.
• A temporary scaffold is put in place to support the
material until the vault is complete.
• However, some vaults are created without a scaffold,
using a method called Catalonian vaulting.
• Here one constructs one series of tiles around the
perimeter at a time.
• The tiles are held together by mortar, and once the first
set is dry, the second can be added, and so on.
Catalan Method
Lamella Vaults
• Vaults may also be constructed out of
connected pieces of wood or metal.
• Short pieces are assembled in a diagonal
(basket-weave) pattern that can then be
used to support roof panels.
• Vaults of this type are called Lamella
vaults.
Tacoma Dome
Shells
• A shell looks much like a vault but can
also withstand tension.
• Obviously this means the shell must be
made of a material that can take tension,
such as metal, wood, or reinforced
concrete.
Types of Shells
• A barrel-shaped shell is a type of developable
shell.
• A short barrel shell has a diameter greater than
its depth, and isn’t really that different than an
arch.
• A long barrel shell (with a depth greater than its
diameter) acts much the same as a simply
supported beam
• it has compression on one side and tension on
the other side.
Kimball Art Museum
Forces in dome shells
• Now let’s go on to dome shells, technically
known as synclastic shells.
• In addition to the forces that run down
from the top of a dome to the supports
(like an arch, and so called arch lines or
meridians), we also have forces that act in
horizontal circles.
• These are then called hoop forces.
Stresses
Compression and Tension
• Under uniform loading, the arch lines are
always in compression (just as a vault).
• For a hemispherical dome the hoop lines
are in compression near the top of the
dome, but near the bottom they are
actually in tension.
• This hoop tension is what prevents a shell
from buckling at the haunch the way a
vault does.
Deflection
Hemispherical Shells
Shallow or Deep?
• Which will have greater horizontal force at
the base, a shallow (quarter-sphere)
dome, or a deeper (hemisphere) dome?
Tension Ring
• As with other funicular structures, the more
vertical the structure is at the base, the less the
horizontal force.
• Thus deeper domes have less horizontal thrust,
and the hoop tension is sufficient to prevent the
dome from spreading out at the base.
• A shallow dome, with greater horizontal thrust,
needs additional help in the form of a tension
ring at the base – a thickening of the shell to
strengthen it.
Houston Astro Dome
300 foot tension ring