1. Engineering Structures 101 Structural Engineering From the Beginning: Beams, Arches, Domes Professor Martin Fahey School of Civil & Resource Engineering, Room A1.08 (e-mail: fahey@civil.uwa.edu.au) www.civil.uwa.edu.au/for/students/undergraduate/online_teaching_resources/semester_two/610101 (The pictures contained in this presentation were either downloaded from the Internet, or scanned in from books. The sources are too numerous to list).

2. Newgrange, Ireland, 3200 BC 80 m diameter burial mound, Boyne Valley (where I grew up!), 40 km from Dublin, built by pre-Celtic neolithic people (Tuatha de Dannan?). The lack of knowledge of the people who built makes it a very eerie place to visit.

3. Newgrange, Ireland, 3200 BC Exterior view of entrance, and interior of burial chamber. Note stone lintel. At sunrise on summer solstice (21 June) sun shines through window above entrance, down the long passage, and strikes an altar at the centre of the chamber.

4. Stonehenge, Salisbury Plain, England. Between 3000 BC and 1500 BC. Purpose?

5. Stonehenge: Stone beams supported by stone columns

6. Mesopotamia: (“Land between two rivers” - the Euphrates and the Tigris) Start of “modern” civilisations? about 7000 BC. Ice age just finishing in Europe. Very fertile then - now desert (Iran/Iraq) Modern humans appeared about 160,000 years ago in Africa Did not flourish until extinction of the Neanderthals about 35,000 years ago

7. Ziggurat (temple) at Ur, 2125 BC Mesopotamia (Sumerians, 3500 to 1900 BC)

8. Pyramids of Khafre & Khufu at Giza, Egypt (Old Kingdom: 2686- 2181 BC)

9. Great Pyramid of Khufu, Giza, Egypt (Old Kingdom: 2686-2181BC). Angle 51°52’ 146 m high, 2.3 million stone blocks, each 2.5 tonnes. Base is almost perfect square, 229 m sides. Aligned perfectly with cardinal points (N,S,E,W)

10. Climbers on the Great Pyramid at Giza (note sizes of blocks) Originally, smooth surface - faced with limestone - now weathered away

11. Bent Pyramid at Dahshur, Egypt, 2680-2565 B.C Angle changes from 54 to 43 degrees (attributed to foundation problems?). If it had been completed to original plan, it would have been the biggest pyramid in Egypt.

12. Temple of Horus, Edfu, Egypt (3 stages between 237 BC and 57 BC)

13. Beams: Tension and Compression Top half of beam in compression: Rock: strong in compression Bottom half of beam is in tension: Rock: weak in tension Maximum tensile stress mid-span Value varies in proportion to L 2 Therefore, beams must be short if poor tensile strength Egyptian & Greek columns close together - column spacing < 2 x beam depth - very cluttered space

14. « Galileo's Discorsi, his Dialogues Concerning Two New Sciences, were published in Leyden in 1638. The second new science is concerned with the mechanics of motion; the first gives the first mathematical account of a problem in structurai engineering. Galileo wishes to compute the breaking strength of a beam, knowing the strength of the material itself as measured in the tension test shown in the illustration. The drawing does not encourage belief that Galileo ever made such a test (although Galileo himself never saw the illustration - he was blind by the time the book was printed). The hook at B would have pulled out of the stone long before the column as a whole fractured. In the same way, it is thought that Galileo did not in fact drop balls of different weights from the Leaning Tower of Pisa. It is not known that Galileo ever designed crucial experiments of this sort, in order to prove or disprove a theory. What he did was to make crucial observations, from which ensued brilliant advances in every subject he touched ». Jacques Heyman « The Science of Structural Engineering » Imperial College Press This is the famous illustration for Galileo's basic problem - the breaking strength of a beam. Again, the drawing is not really representational, although there is a wealth of circumstantial detail. In this case the hook C may well have been able to carry the load, but the masonry at AB looks insufficient to resist the turning moment at the wall. It is interesting to note that Galileo actually got the statics completely wrong – he did not understand that the stresses on the cross section had to give zero net horizontal force. He thought that the stress distribution went from a maximum at the top, to zero at the bottom. He would have failed 1st year statics!

15. Temple of Horus, Edfu, Egypt

16. Temple of Horus, Edfu, Egypt Hypostyle Hall (hall of many columns)

17. Parthenon, Athens, Greece, 447 BC. Deep stone beams, over closely-spaced columns

18. The Parthenon stands atop the Acropolis, in Athens, Greece

19. Parthenon

20. Three types of columns (three “orders”) used in Greek buildings: Doric, Ionic and Corinthian The top (“capital”) of each column type is different - in fact, whole style & proportions of each are different Doric capital Ionic capital Corinthian capital

21. Parthenon: Doric order; stone architrave, frieze and cornice

22. Wooden beams Wooden planks Compacted clayTiles Roof structure of Greek Temple - very short spans Stone columns Stone architrave

23. Arches: Achieving large spans while avoiding tension

24. A simple masonry arch is made from identical wedge-shaped voussoirs - it is built on falsework, since it cannot stand until the last stone, the keystone, is in place. Once complete, the falsework (the ‘centering’) may be removed, and the arch at once starts to thrust at the river banks. Inevitably the abutments will give way slightly, and the arch will spread. Figure (b), greatly exaggerated, shows how the arch accommodates itself to the increased span. The arch has cracked between voussoirs - there is no strength in these joints, and three hinges have formed. There is no suggestion that the arch is on the point of collapse - the three-hinge arch is a well-known and perfectly stable structure. On the contrary, the arch has merely responded in a sensible way to an attack from a hostile environment (gravity). In practice, the hinges may betray themselves by cracking of the mortar between the voussoirs, but larger open cracks may often be seen.

25. An arch supports vertical forces by generating compression between the “voissoirs” of the arch. The arch abutment must be capable of supporting the resulting horizontal thrust.

26. An arch with three hinges can be stable - in fact many arches are built this way deliberately Four hinges are required in an arch for collapse. Picture shows “snap-through” failure

27. Packhorse Bridge, Scotland, 1717 Arch is inherently stable

28. Arch is inherently stable

29. A stone beam with small span-to-depth ratio (such as those in the Parthenon) may act as a three-pin arch if it cracks at the centre, and may not necessarily collapse

30. Pont du Gard, Nimes, southern France. Aqueduct. Built by Romans, -15 BC to 14 AD. The Romans perfected the use of the arch, and used it widely.

31. This aqueduct, over the river Gard, is 275 metres long and 49 m high. Part of an aqueduct nearly 50 km long that supplied Nimes with water. On its first level it carries a road and at the top of the third level, a water conduit, which is 1.8 m high and 1.2 m wide and has a gradient of 0.4 per cent (0.4m per 100 m length).

32. Possible falsework (or “centering”) scheme used for the Pont du Gard

33. Pont du Gard: The three levels were built in dressed stone without mortar. The projecting blocks supported the scaffolding during construction.

34. Elements of a Roman Arch Bridge

35. Aqueduct, Segovia, Spain. Built by Romans, 1st century AD. 39 m high

36. Segovia, Spain

37. Pons Fabricus (Ponte Fabrico), Rome, Tiber. Built in 62 B.C. by L.Fabricius. Oldest surviving bridge in Rome. Still used by pedestrians

38. Pons Fabricius (Ponte Fabricio), Rome, Tiber. Built in 62 B.C. by L.Fabricius. Oldest surviving bridge in Rome. Still used by pedestrians

39. Inscription on Pons Fabricius The building inscription, found on four places, reads “L(ucius) FABRICIUS C(ai) F(ilius) CUR(ator) VIAR(um) FACIUNDUM COERAVIT EIDEMQUE PROBAVEIT”, meaning that Lucius Fabricius as curator of the roads ordered the construction of the bridge. In smaller letters is added that Marcus Lollius and Quintus Aemilius Lepidus, the consuls of 21 BCE, improved the bridge. This may refer to adjustment made necessary after the great flood of 23. Perhaps the small arch on top of the pier is meant.

40. Pont St Martin, Aosta, Italy. 25 BC. Longest span Roman Arch bridge (32 m).

41. Anji, (or Great Stone) Bridge, Jiao River, China, 610 AD, Li Chun. Still in use. Described by Ming Dynasty poet as “new moon rising above the clouds, a long rainbow drinking from a mountain stream”.

42. Colleseum, Rome, 70-80 AD, Emperor Vespasian 187 m long, 155 m wide, 49 m high

43. Arch of Titus, Rome, AD 81. Triumphal Arch, celebrating victory in war

44. Arc de Triomphe, Paris Commissioned in 1806 by Napoleon I, shortly after his victory at Austerlitz, it was not finished until 1836

45. La Grande Arche (the Great Arch), La Defense, Paris, is not actually an arch. One of the great projects initiated by Francois Mitterand, President of France, in the 1980s

46. Culverts and underpasses: soil provides support (pressure from all sides - circular shape efficient).

47. Roman Arch: semi-circular (“Romanesque” architecture) B 4/5B B Gothic Arch: Pointed. Example shown is “a quinto acuto” - two circular segments with radius = 4/5 of the base Roman Arch compared to Gothic Arch

48. “Hanging chain” (catenary) shape (Pure tension - no bending) Inverted “hanging chain” shape (pure compression - no bending). Arch in this shape would have no bending in any part. Gothic “a quinto acuto” arch An “inverted catenary (chain) is the ideal shape for an arch. Gothic arch “a quinto acuto” is very close to ideal shape - therefore can be very thin and still be stable

49. For stability, a circular Roman arch supporting only its own weight must be thick enough to contain an equivalent “inverted catenary” arch Therefore, Romanesque architecture typically very massive (“heavy”)

50. Romanesque: Church of Sainte-Foy, Conques, France, 1050-1120

52. La Madeleine, Vezelay, France: interior, nave, 1120-1132. Typical Romanesque church

53. Cathedral of Notre Dame de Paris. 1150 -1220. Example of Gothic Architecture

54. Notre Dame de Paris. 1150 -1220.

55. Notre Dame de Paris

56. Notre Dame de Paris: North Rose Window. Suspended in perfect equilibrium on a web of stone, the immense north rose window remains intact after 700 years, its intricately interlocking blocks so exact they ring when struck. Though individual blocks may be removed for repairs without collapsing the whole, only minor buckling has occurred 13 m 17 m

57. Notre Dame de Paris. Schematic sections showing the “flying butresses”

58. Decorative features on tops of columns (statues, pinnacles, as in Notre Dame, below) have stabilising function

59. Construction of a Gothic cathedral

60. Bourges Cathedral, France, 1214. Most efficient flying buttress system ever constructed.

61. Sections through various French Gothic Cathedrals, showing progressive development

62. Amiens Cathedral, France, 1220.

63. Thrusts in flying buttresses (left) and structure of a groin vault (above)

64. Dome: 3-dimensional equivalent of an arch. Pantheon, Rome, 118-128 AD. Temple to “all the gods”

65. Pantheon, Rome, 118-128 AD. Construction of the dome (concrete).

66. Interior of dome of Pantheon is semi- circular (hemispherical)

67. Outward thrust of the dome taken by 8 m thick composite heavy wall

69. Pantheon: Interior. Biggest clear span until 19th century

70. Pantheon: Interior. Light provided by circular hole (“occulus”) in the top

71. Hagia Sophia, Istanbul, 537 AD. Largest church for 9 centuries.

72. Hagia, Sophia, Istanbul, 537 AD. Interior, showing support system for central dome

73. Hagia, Sophia, Istanbul, 537 AD. Schematic showing support system for central dome

74. Hagia, Sophia, Istanbul, 537 AD.

75. Comparison of sizes of various domes

76. Underground water storage “cistern”, built by Justinian, Istanbul

78. Santa Maria del Fiore, Florence, Italy. Begun in 1296. “Segmented dome” added by Brunelleschi in 1436. 42 m span, 91 m high. Built without “centering”

79. Santa Maria del Fiore, Florence, Italy. Begun in 1296. Dome added by Brunelleschi in 1436. 42 m span, 91 m high. Built without “centering” Shape is arch “a quinto acuto”

80. Dome of Santa Maria del Fiore, Florence, is not hemispherical, but is made up of 8 segments.

81. St Peter’s Basilica, Rome, Michaelangelo, 1546

82. Dome of St Peter’s Basilica, Rome, Michaelangelo, 1546

83. Interior of St Peter’s Basilica, Rome, showing dome resting on four arches supported by four great pillars

84. “Hanging chain” analysis of Dome of St Peter’s, by Giovani Poleni, 1742

85. Gateway Arch, St Louis, USA. This free-standing arch is 630 ft. high and the world's tallest. Built of triangular section of double-walled stainless steel, the space between the skins being filled with concrete after each section was placed. Looks like perfect “inverted catenary” shape.

86. Interior of Carmel Mission. Built in 1793 it is an interesting design in that the walls curve inward towards the top, and the roof consists of a series of inverted catenary arches built of native sandstone quarried from the nearby Santa Lucia Mountains. (Carmel, California)

87. St Paul’s Cathedral, London (1675-1708). Christopher Wren

88. St Paul’s Cathedral Dome

89. (3 domes inside each other)

90. Hooke’s “hanging chain” concept applied to the dome of Christopher Wren’s St Paul’s Cathedral. The “lantern” on top of the dome distorts the “chain”

91. Segrada Familia (Holy Family) Cathedral, Barcelona Architect: ANTONI GAUDI 1852-1926 Started 1882 – still not finished Many examples of Gaudi work in Barcelona