Arches, Vaults, and Shells

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Arches, Vaults, and Shells

• 11/27/07

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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

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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.

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Catenary vs. Parabola
Parabola Catenary
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• 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

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Catenary vs. Parabola
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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)

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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.

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Corbeled and Voussoir Arches
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Keystone and Voussoirs
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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.

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Line of Thrust
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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.

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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.

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Voussoir Arch
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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.

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Fixing the Shape of the Arch
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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

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Types of Arches
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Pont du GardRoman Aqueduct, 100 AD?
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Arch of Septimius SeverusAncient Rome 203 AD
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Arch of TitusAncient Rome, 81 AD
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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
  load-bearing, allowing more openings for stained
  glass windows.

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Flying Buttresses
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Notre Dame
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Chartres Cathedral, France
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Flying buttresses at York Minster Cathedral
England 1220
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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 three-hinged
  arches

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Modern Arch Configurations
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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.

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ChaoTianMen Arch Bridge
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Santa Barbara the California Coast / Cold
Spring Arch Bridge
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Natchez Trace Double Arch BridgeFranklin,
Tennessee
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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.

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St. Louis Gateway Arch
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Making a Semicircular Arch

• Calculating arch segment angles

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Vaults and Shells
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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 well discuss it
  later.

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Load Distribution
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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)

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Cylindrical Vaults
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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.

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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

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Figure 14.4
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Barrel VaultArch of Sapor, Iraq
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Ruin of Basilica of Maxentius (aka Basilica of
Constantine)
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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

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Roman Groin Vault
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• One could also build a groin vault with different
  extents in the two dimensions

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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.

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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.

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Gothic Vault
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King's Arms HotelDorchester, England
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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.

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PPT Pantheon
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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.

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Pendentive
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Hagia Sophia
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Hagia Sophia
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Hagia Sophia
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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.

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Hagia Sophia
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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.

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Funicular Chain Model

• (Explain this to be an upside-down picture)

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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

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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.

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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.

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Catalan Method
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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.

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Tacoma Dome
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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.

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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 isnt 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.

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Kimball Art Museum
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Forces in dome shells

• Now lets 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.

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Stresses
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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.

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Deflection
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Hemispherical Shells
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Shallow or Deep?

• Which will have greater horizontal force at the
  base, a shallow (quarter-sphere) dome, or a
  deeper (hemisphere) dome?

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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.

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Houston Astro Dome
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300 foot tension ring