Beams Arches and Domes

7/30/2019 Beams Arches and Domes

1/93

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.


2/93

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.


3/93

Stonehenge, Salisbury Plain, England. Between 3000 BC

and 1500 BC. Purpose?


4/93

Stonehenge:

Stone beams supported by stone columns


5/93

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


6/93

Ziggurat (temple) at Ur, 2125 BC

Mesopotamia

(Sumerians, 3500 to 1900 BC)


7/93

Pyramids of Khafre &

Khufu at Giza, Egypt

(Old Kingdom: 2686-

2181 BC)


8/93

Great Pyramid of Khufu, Giza, Egypt (Old Kingdom: 2686-2181BC). Angle 5152

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)


9/93

Climbers on the Great Pyramid at Giza (note sizes of blocks)

Originally, smooth surface - faced with limestone - now

weathered away


10/93

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.


11/93

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


12/93

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 L2

Therefore, beams must be short if poor tensile

strength

Egyptian & Greek columns close together

- column spacing < 2 x beam depth

- very cluttered space


13/93

Galileo'sDiscorsi, his DialoguesConcerning 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 columnas 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 turningmoment 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!


14/93

Temple of Horus, Edfu, Egypt


15/93

Temple of Horus, Edfu, Egypt

Hypostyle Hall

(hall of many columns)


16/93

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


17/93

The Parthenon stands atop the Acropolis, in Athens, Greece


18/93

Parthenon


19/93

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 Ioniccapital

Corinthiancapital


20/93

Parthenon: Doric order; stone architrave, frieze and cornice


21/93

Wooden beams

Wooden planks

Compacted clay Tiles

Roof structure of Greek Temple - very short spans

Stone columns

Stone architrave


22/93

Arches:

Achieving large spans while

avoiding tension


23/93

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 merelyresponded 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.


24/93

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.


25/93

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


26/93

Packhorse Bridge, Scotland, 1717

Arch is inherently stable


27/93

Arch is inherently stable


28/93

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


29/93

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.


30/93

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 agradient of 0.4 per cent (0.4m per 100 m length).


31/93

Possible falsework (or centering) scheme used for the Pont du Gard


32/93

Pont du Gard: The three levels were built in dressed stone without mortar. The

projecting blocks supported the scaffolding during construction.


33/93

Elements of a Roman Arch Bridge


34/93

Aqueduct, Segovia, Spain. Built by Romans, 1st century AD.

39 m high


35/93

Segovia, Spain


36/93

Pons Fabricus (Ponte Fabrico), Rome, Tiber. Built in 62 B.C. by

L.Fabricius. Oldest surviving bridge in Rome. Still used by

pedestrians


37/93

Pons Fabricius (Ponte Fabricio), Rome, Tiber. Built in 62 B.C. by

L.Fabricius. Oldest surviving bridge in Rome. Still used by

pedestrians

I i ti P F b i i


38/93

Inscription on Pons Fabricius

The building inscription, found on four places, reads L(ucius)

FABRICIUS C(ai) F(ilius) CUR(ator) VIAR(um) FACIUNDUMCOERAVIT 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 floodof 23. Perhaps the small arch on top of the pier is meant.


39/93

Pont St Martin, Aosta, Italy. 25

BC. Longest span Roman Arch

bridge (32 m).


40/93

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.


41/93

Colleseum, Rome,

70-80 AD, Emperor

Vespasian

187 m long, 155 m wide, 49 m high


42/93

Arch of Titus, Rome, AD 81.

Triumphal Arch, celebrating victory in war


43/93

Arc de Triomphe, Paris

Commissioned in 1806 by

Napoleon I, shortly after his

victory at Austerlitz, it

was not finished until 1836

L G d A h ( h G


44/93

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


45/93

Culverts and underpasses: soil provides support (pressure from all

sides - circular shape efficient).

R A h d G hi A h


46/93

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


47/93

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


48/93

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)


49/93


50/93

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


51/93


52/93

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


53/93

Notre Dame de Paris.

1150 -1220.


54/93

Notre Dame de Paris


55/93

Notre Dame de Paris:

North Rose Window.

Suspended in perfectequilibrium on a web of

stone, the immense north

rose window remains

intact after 700 years, itsintricately 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


56/93

Notre Dame de Paris. Schematic sections showing the flying butresses


57/93

Decorative features on tops of

columns (statues, pinnacles, as inNotre Dame, below) have

stabilising function


58/93

Construction

of a Gothic

cathedral


59/93

Bourges Cathedral,

France, 1214. Most

efficient flyingbuttress system ever

constructed.


60/93

Sections through various French Gothic Cathedrals, showing progressive

development


61/93

Amiens Cathedral,

France, 1220.


62/93

Thrusts in flying buttresses

(left) and structure of a groin

vault (above)


63/93

Dome: 3-dimensional equivalent of an arch.

Pantheon, Rome, 118-128 AD. Temple to all the gods


64/93

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


65/93

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


66/93

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


67/93


68/93

Pantheon: Interior.

Biggest clear span until 19th

century


69/93

Pantheon: Interior.

Light provided by circular

hole (occulus) in the top


70/93

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


71/93

Hagia, Sophia, Istanbul,

537 AD. Interior,

showing support

system for central dome


72/93

Hagia, Sophia, Istanbul, 537 AD. Schematic showing support

system for central dome


73/93


74/93

Comparison of sizes of various domes

Underground water storage cistern, built by Justinian, Istanbul


75/93

Underground water storage cistern, built by Justinian, Istanbul


76/93


77/93

Santa Maria del Fiore,

Florence, Italy.

Begun in 1296.

Segmented domeadded by Brunelleschi

in 1436.

42 m span, 91 m high.

Built without

centering


78/93

Santa Maria del Fiore,

Florence, Italy.

Begun in 1296. Dome

added by Brunelleschiin 1436.

42 m span, 91 m high.

Built withoutcentering

Shape is arch a quinto

acuto


79/93

Dome of Santa Maria del Fiore, Florence, is not hemispherical,

but is made up of 8 segments.


80/93

St Peters Basilica, Rome, Michaelangelo, 1546


81/93

Dome of St Peters Basilica, Rome, Michaelangelo, 1546

Interior of St Peters


82/93

Basilica, Rome, showing

dome resting on four arches

supported by four great

pillars


83/93

Hanging chain analysis of Dome of St Peters, by Giovani Poleni, 1742


84/93

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.


85/93

Interior of Carmel

Mission. Built in 1793it 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 SantaLucia Mountains.

(Carmel, California)


86/93

St Pauls Cathedral, London (1675-1708). Christopher Wren


87/93

St Pauls Cathedral Dome


88/93

St Pauls Cathedral Dome

(3 domes inside each other)


89/93

Hookes hanging chain concept applied to the dome of

Christopher Wrens St Pauls Cathedral. The lantern on top of

the dome distorts the chain

Segrada Familia (Holy Family)Cathedral, Barcelona

hi GA


90/93

Architect: ANTONI GAUDI

1852-1926

Started 1882still not finished

Many examples of Gaudi work

in Barcelona


91/93


92/93


93/93

Documents

Beams Arches and Domes