1. Arches, Vaults, and Shells
11/27/07

2. 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

4. 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.

5. Catenary vs. Parabola
Parabola Catenary

6. 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

7. Catenary vs. Parabola

8. 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)

10. 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.

11. Corbeled and Voussoir Arches

12. Keystone and Voussoirs

13. 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.

14. Line of Thrust

15. 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.

16. 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.

17. Voussoir Arch

18. 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.

19. Fixing the Shape of the Arch

20. 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

21. Types of Arches

22. Pont du Gard Roman Aqueduct, 100 AD?

23. Arch of Septimius Severus Ancient Rome 203 AD

24. Arch of Titus Ancient Rome, 81 AD

25. 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.

26. Flying Buttresses

27. Notre Dame

28. Chartres Cathedral, France

29. Flying buttresses at York Minster Cathedral England 1220

30. 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

31. Modern Arch Configurations

32. 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.

33. ChaoTianMen Arch Bridge

34. Santa Barbara & the California Coast / Cold Spring Arch Bridge

35. Natchez Trace Double Arch Bridge Franklin, Tennessee

36. 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.

37. St. Louis Gateway Arch

38. Making a Semicircular Arch
Calculating arch segment angles

39. Vaults and Shells

40. 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.

41. Load Distribution

42. 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)

43. Cylindrical Vaults

44. 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.

45. 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

46. Figure 14.4

47. Barrel Vault Arch of Sapor, Iraq

48. Ruin of Basilica of Maxentius (aka Basilica of Constantine)

49. 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

50. Roman Groin Vault

51. One could also build a groin vault with different extents in the two dimensions

52. 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.

53. 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.

54. Gothic Vault

55. King's Arms Hotel Dorchester, England

57. 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.

58. PPT Pantheon

59. 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.

60. Pendentive

61. Hagia Sophia

62. Hagia Sophia

63. Hagia Sophia

65. 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.

66. Hagia Sophia

67. 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.

68. Funicular Chain Model
(Explain this to be an upside-down picture)

69. 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

70. 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.

71. 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.

72. Catalan Method

73. 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.

74. Tacoma Dome

75. 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.

76. 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.

77. Kimball Art Museum

78. 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.

79. Stresses

80. 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.

81. Deflection

82. Hemispherical Shells

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

84. 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.

85. Houston Astro Dome

86. 300 foot tension ring