ARCHES

THE USE OF ARCHES

Memorial arch Arch bridge Roof support structures Arched roof structures including domes Arched cables Wall penetration Arched buildings etc.

Garden scene, Suchou

The arch is part of the frame family, but distinguishes itself by providing a continuous one-member enclosure without having any abrupt kink points along the geometry. The internal forces flow smoothly along the arch and are not concentrated at points of sudden change of form, assuming that the external loads are distributed evenly; concentrated loads ideally should be located at kink points. The use of arches in architecture has a long history. In Europe, the semicircular arches of the Romans were adopted again as an essential part of architecture during the Romanesque period. Transformation from the round arch to the slender pointed arch happened in the Gothic period about nine hundred years ago. In contrast to high-pointed arches are the flat, segmental arches used for bridges during the Renaissance period in Italy. Today, arches have an important place in architecture in defining building spaces and as bridges in public spaces. The infinite possible applications of the arch principle can only be suggested by

HISTORICAL DEVELOPMENT

In the past, the arch together with the barrel arch and the arch-like vault were among the few structural systems that made it possible to span larger distances by using masonry with its low tensile capacity. Probably the first arches built were based on the corbelling principle, where horizontal masonry courses projected slightly beyond the previous course. These corbeled arches are false arches that do not develop lateral thrust, which is the basic characteristic of true arches.

Treasury of Atreus, c. 1325 BC, Mycenae

Ishtar Gate (reconstructed), Babylon, c. 575 BC

Palace of Ctesiphon, now Taq-i-Kisra, near Bagdad, Iraq, c. 400 AD

The Roman Aqueduct, Segovia, Spain, 50 AD

Coliseum, Rom, Italy, 80 AD

Constantine Basilica, Trier, Germany, 310 AD

Cathedral of Notre Dame de Paris, 1150 - 1220

Notre Dame de Paris,

1150 -1220

Thrusts in flying buttresses

(left) and structure of a groin

vault (above)

Amiens Cathedral, Amiens, France, 1269

Amiens Cathedral,

France, 1269

Notre Dame de Paris

St. Pierrre, Beauvais, 1247

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

Notre Dame de Paris. Schematic sections showing the flying buttresses

Bourges Cathedral,

France, 1214. Most

efficient flying

buttress system ever

constructed.

Sections through various French Gothic Cathedrals, showing progressive

development

Cathedral of Palma, Majorca - photoelastic Study by Robert Mark

St. Lorenz, Nuremberg, 1500,

St. Mary, Pirna, Germany, beginning 16th cent.

Construction

of a Gothic

cathedral

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

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

Dome of Santa Maria del Fiore,

Florence, is not hemispherical,

but is made up of 8 segments.

Centenary Hall, Breslau, Ger. (now Wroclaw, Pol.), Max Berg, 1913, Dyckerhoff & Widmann

Ponte Vecchio Bridge, Florence, 1367

Ponte Santas Trinita, Florence, 1569

Rialto Bridge, Venice, 1591

Anji Bridge located in Zhaoxian County of Hebei Province was built in the Sui Dynasty (581-618). Anji Bridge is a single span stone arch bridge in China, and it is also the oldest extant bridge of China. It is also known as the Zhaozhou Bridge with a history of about 1,400 years, it is reputed as The First Bridge Under Sky.

It may have been Robert Hook (1670), who was the first to relize from a scientific point of view that the catenary is the funicular response of the arch weight.

Christopher Wren introduced the concept of the catenary dome shape with the conical brick dome supporting the cupola of St. Pauls Cathedral, London (1970).

But Giovanni Poleni was the first to actually use a model of string and lead weights to obtain the thrust line of St. Peter in Rome (1743) and thus was able to make his recommendations for the number of tension rings required to prevcent bfurther cracking of the cupola.

Dome of St Peters Basilica, Rome, Michaelangelo, 1546

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

St Pauls Cathedral, London, 1710, Christopher Wren

Hookes hanging chain concept applied to the dome of Christopher Wrens

St Pauls Cathedral. The lantern on top of the dome distorts the chain

St Pauls Cathedral Dome

(3 domes inside each other)

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, CA)

Arched Bridge, the Summer Palace in Beijing, China, 1750

Antoni Gaudi (1852 1926) revived the idea of funicular curves of the loads in his search for the true nature of form. He derived arch shapes from suspended scale models so as to achieve purity of form and maximum efficiency of materials.

Gaudi also used parabolic arches as an approximation for catenary curves

Sagrada Familia Cathedral, Barcelona, 1982 - , Antoni Gaudi

The nearly 100-ft span cast iron bridge at Coalbrookdale over the Severn, UK, 1772, is often considered as a turning point from stone and brick as the dominant material for arches, to iron. The new material of iron and later steel made long spans and new building types possible.

St. Pancras Station, London, 1868, 240 ft (73 m)

Galerie des Machines (375 ft, 114 m), Paris, 1889, Dutert and Contamin

Grande Halle de la Villette, Paris, 1867

Viaduc de Garabit, Saint-Flour, Cantal, 1884, Gustave Eiffel

Eiffel Tower (300 m), Paris, 1889, Gustave Eiffel

Glass-vaulted Gallerias in Brussels, Milane, etc. end of 19th century

New Dresden Main Train Station, Dresden, 2006, Foster

Dresden Central Railway Sttation, 2006, Norman Foster Arch, Happold Eng.

Firth of Forth Bridge (1708 ft), Scotland, 1890, Benjamin Baker, John Fowler

DOUBLE CANTILEVER STRUCTURES

Thonet's first bentwood rocking chair, upholstered, 1860, Thonet Brothers, Austria

The Chaise longue, c. 1928, Le Corbusier

Paimio chair, 1932, Alvar Aalto; bent laminated birch frame, solid birch, with painted bent plywood seat

Salignatobel Bridge, Switzerland, 1930, Robert Maillart

Arve Bridge, Vesay, Switzerland, 1935, Robert Maillart

Dorton (Raleigh) Arena, 1952,

North Carolina, Matthew Nowicki,

with Frederick Severud

Institute of Public Administration, Ahmedabad, India, 1963, Louis Kahn

Gateway arch (630 ft), St. Louis, 1963, Eero Saarinen 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

The Geometry of the Arch Arches may be composed of different types of curves. The most common ones are derived from conic sections. They are the circle, ellipse, parabola, and hyperbola but also the cycloid should be mentioned. Curves can be used as single-, double-, or multiple-curvature systems, in other words they can consist of various curvilinear segments. With respect to circular curves, arches can be one-centered (e.g. semi-circular, segmental), two-centered, and multi-centered. There are innumerable ways basic curves may be combined to yield various arch profiles. The most common arches are based on circular and parabolic geometry.

FROM THE HARMONY OF NATURE TO THAT OF ARCHITECTURE

National Stadium of Sports Affairs Council, Toyo Ito & Associates, Kaohsiung, Taiwan, 2009

National Stadium of Sports Affairs Council, Toyo Ito & Associates, Kaohsiung, Taiwan, 2009

CONTEMPORARY ARCHES

ARCH USE: BRIDGES, BUILDING ENCLOSURES, ROOF STRUCTURES, SUPPORT STRUCTURS, WALL ARCHES, COMPRESSION RINGS, SUSPENDED ARCHES, TREES, MONUMENTS, etc.

ARCHES

BRIDGES

Examples of arched bridges

Route 112 Bridge, Huntington, MA, using SAP

pedestrian bridge at Seattle's Museum of Flight, 2007, SRG Partnership

Based on an original drawing by Leonardo da Vinci, Oslo, 2001, Vebjrn Sand

Ponte della Constituzione , Venice, 2007 Santiago Calatrava

Bent wood bridge, Esslingen, Germany, 1986, R. Dietrich

Pedestrian bridge in Cologne, Germany

Barqueta Bridge, Seville, Spain, 1992, Santiago Calatrava

Bac de Roda Felipe II Bridge,

1987, Barcelona, S. Calatrava

La Devesa Footbridge, Ripoll, Spain, 1991, S. Calatrava, torsion

The 100-m span tied arch Japan Bridge in Paris (1993, Kisho Kurokawa) consists of the two

main inward leaning tubular steel arches, the walkway of triangular precast concrete panels

covered by a curved glass enclosure, and the support of the arched spatial cable-strut

network. The walkway and glass enclosure are suspended from the arches. The lateral arch

thrust is taken by the cable-strut network at the base. Torsion due to lateral loads is efficiently

resisted by the triangular cross-section of the bridge (i.e. torsion box).

Bridge over the Rhein-Herne-Chanel, BUGA 97 Gelsenkirchen, Germany, 1997, Stefan Polnyi

Brcke ber den Rhein-Herne-Kanal, BUGA 97 Gelsenkirchen

Asymmetrie bei Vollast wegen asymmetrischer Anordnung (Prof. Dr. Stefan Polnyi) Der Gehweg kreuzt den Kanal im Grundriss unter ca. 70/110 Winkeln. Die Bgen stehen genau senkrecht zum Kanal. Die Gesamtkonstruktion ist punktsymmetrisch zum Mittelpunkt des Gehweges. Beide Bgen sind identische Sttzlinienkonstruktionen fr Vollbelastung, jedoch ist der einzelne Bogen asymmetrisch. Wegen der Stellung des einzelnen Bogens zum Gehweg sind die identischen Bgen gegeneinander um 180 im Grundriss verdreht. Da die Bgen im Grundriss den Gehweg kreuzen (an jedem Ufer zwei Auflager an einer Brckenseite), mssen die Anlenkpunkte der Seile am Bogen so gewhlt werden, dass das Lichtraumprofil frei gehalten wird. Auerhalb der Anlenkpunkte sind die Bgen ideal gerade.

Oberbaumbruecke, Berlin, 1995, Santiago Calatrava

Proposal train station, Florence, Italy, 2007, Arata Isozaki, Mutsuro Sasaki

Bus Stop, Aachen, 1998, Peter Eisenman

ARCHES as

ROOFSUPPORT

STRUCTURES

The Metro station at Blaak, Rotterdam, 1993, Harry Reijnders of Movares; the arch

spans 62.5 m, dome diameter is 35 m

Ice hokey stadium, Munich, 1985, Kurt Ackermann

Lanxess Arena, Cologne, 1998, Peter Bhm Architekten

Olympic Stadium OAKA, Athens, Greece, 2004, Santiago Calatrava

Complex canopy

ARCHES AS

ROOF

STRUCTURES

Arched structure, computer model

Media and Study Centre, D. Hosiassohn (Sketch program)

St. Dominque, 2001, Gifu Design: ,Takenaka

Deutsche-Med, Rostock, 2004, Helmut Jahn, Werner Sobek

Student Housing IIT, Chicago, 2003, Helmut Jahn

Office building of the European Investment Bank, 2009, Luxembourg, Ingenhoven Architects

Office building of the European Investment Bank, 2009, Luxembourg, Ingenhoven Architects

Museum of Contemporary Art, Helsinki, Finland, 1998, Steven Holl, Arup + Nordenson

Ningbo Air Terminal

Sportscenter Dalian, China

Beijing Capital International Airport - Terminal 2, 1999

Inchon Airport, Seoul , 2002, Terry Farrel

Beijing International Airport Terminal 3, 2008, Norman Foster, Arup

EXPO-Dach Hannover, Arch.: Herzog und Partner, Ing.: Julius Natterer, 2000

Autobahnraststtte, Arch. & Ing.: Heinz Isler, Deitingen 1968

Bodegas Protos, Peafiel, Valladolid, Spain, 2008, Richard Rogers, Arup

Ferrari Restaurant, Maranello, Italy. 2008, Marco Visconti

Olympic Stadium Montreal, Canada, 1975, Roger Taillibert

Bordeaux Law Courts, 1998, Richard Rogers, Arup

Harajuku Protestant Church, Kita-Aoyama, Tokyo, Ciel Rouge Creation, Kaneko Fumiko & Henri Gueydan, Tokyo, 2005

Sustainable towers in Malaysia, 2008, Studio Nicoletti

Allianz Stadium Railway Station Froettmanning, Munich, 2006

Airport Terminal Newark

Lehrter Bahnhof, Berlin, 2006, von Gerkan, Marg and Partners

Peek & Cloppenburg, Cologne, Germany, 2005, Renzo Piano

National Museum of the Marine Corps and Heritage Center, Quantico, Virginia USA, 2006, Fentress Bradburn Architects, Weidlinger

San Giovanni Rotondo, Foggia, Italy, 2004, Renzo Piano

Pilgrimage church Padre Mio, Renzo Piano The Dome is supported by 21 prestressed stone arches. In plan view, the structure appears spiral shaped converging into a dome structure consisting of 11 arches along the outer ring and 10 arches along the inner ring The shape of the arches corresponds to the pressure line (asymmetrical) -> the arch is only under compression

Konstruktionaufnahme von Druck:jeweils 5 Steine sind zu einem Segment verschraubt (Montage) Segmente werden ber Zapfenverbindung in Position gehaltenLehrgerst fr Montage fr Lastfall Erdbeben werden nach Montage 2 Stahlseile durch den Bogen gefhrt und gespannt druckbeanspruchte Steinbgen Vouten am Sto von Segmenten haben gestalterische Funktion Fundamente zur Aufnahme der Horizontalkrfte ausgebildet

Center Paul Klee, Bern, Switzerland, 2007, Renzo Piano Building Workshop , Arup

Barajas Airport, Madrid, Spain, 2004, Richard Rogers,

Anthony Hunt Associates (main structure), Arup (main

faade)

Floating Pavilion, Groningen, Netherland, 1997, Fumihico Maki

Milwaukee Art Museum, Santiago Caloatrava

Lisbone Orient Station, Lisbone, Portugal, 1998, Santiago Calatrava

Lige Guillemins TGV Station, Lige, Belgium, 2008, Santiago Calatrava

Documentation Center Nazi Party Rally Grounds, Nuremberg, 2001, Guenther Domenig

ZhongGuanCun West Office, Beijing, 2006, Kohn Pederson Fox Assoc

Central Chinese Television (CCTV) Tower, Beijing, 2008, Koolhaas and Ole Scheeren/OMA

Iglesia de la Medalla Milagrosa, Navarte, Mexico City, 1955, Felix Candela

Satolas Airport TGV Train Station, Lyons, France, 1995, Santiago Calatrava

BCE Place, Toronto, 1992, Santiago Calatrava

City of Arts and Sciences, Valencia, Spain, 1996, Santiago Calatrava

Les Halles, Paris, 1979, Claude Vasconi and Georges Pencreac'h

Vaillant Arena , Davos, Switzerland, 1981

United Airlines Terminal at

OHare Airport, Chicago, 1987, H. Jahn

Minute Maid Field, Houston, 2000, HOK Sport

Atrium, Germanisches Museum, Nuremberg, Germany

Jaegerpassage, Leipzig, Germany

The Kimmel Center for Performing Arts, Philadelphia, 2003, Vinoly

Neue Messe Leipzig, 1996, Gerkan, Marg und Partner

National Grand Theater, Beijing, 2005, Paul Andreu

Olympic Stadium, Montreal, 1976, Roger Taillibert

Montreal Biodome (The Montreal Olympic Velodrome ), 1978, Roger Taillibert

Bangkok International Airport, 2006, Murphy/Jahn, Werner Sobek

The new International Terminal at San Francisco International Airport, 2001, SOM

Stuttgart Airport, Terminal 1, Germany, 1991, von Gerkan, Schlaich

Beijing Capital International Airport - Terminal 3, 2008, N. Foster

Kansai International Airport Passenger Terminal Building, 1994, Renzo Piano, Ove Arup (Peter Rice)

5/1/2015 328

Shenyang Airport

Exchange House, London, 1990, SOM

Berlin Stock Exchange, Berlin, Germany, 1999, Nick Grimshaw

Tekla Xsteel

Cathedral of Christ the Light, Oakland, CA, 2008, SOM

The Response of Roof Arches to Loading

FUNICULAR COMPRESSION SYSTEMS

BRACED ARCHES

COMPOSITE SYSTEMS AND FORM-RESISTANT STRUCTURES

ARCHES WITH PRESTRESSED TENSILE WEBS

Construction of a Circular Arch with SAP 1) Just draw a single line between the arch supports. 2) Select the arch member (only one member may be selected at a time) to be meshed. 3) From the Edit menu select Mesh Curved Frame/Cable command to access the Curve Parameters form. From the Type of Circular Curve drop-down list select a predefined curve such as Circular Arc Planar Point & Radius 4) Enter the Radius and a value for the Number of Divisions in this edit box to specify the meshing of the generated curved frame/cable element. 5) Click the Insert button and SAP2000 will calculate the coordinates of the curve automatically based on the Number of Divisions specified. The curve will be displayed in plan in the display area on the right-hand side of the Curve Parameters form. 6) Click OK button and the arch will appear on the screen but not in the xz-plane, it must be rotated to its proper location: select all (i.e. arc), then Edit, then Replicate, then Radial, then Rotate About XLine, then check Coordinates of Point on YZ Plane, then check Angle of rotation using increments of 450, and check Delete Original Objects, then OK.

Graphic statics

Funicular Compression Structures By arranging material along the opposite profile of a sagging cable, it is possible to make a spanning structure that works in pure compression.

It is common to use arches with pin supports and an internal hinge connection. This configuration is called a three-hinged arch. They were particularly popular in the 19th and early 20th centuries because they are statically determinate. Unbalanced Loading Unlike cables, which can reconfigure to a different stable profile when the load pattern changes, arches cannot. A loading that does not correspond to the arch's funicular profile will be called a non-funicular loading or unbalanced loading. There are two possible responses to unbalanced loading:

The structure develops internal shear and moment to compensate for the difference between the funicular profile of the load and its own profile. The structure becomes unstable and collapses.

Since the first option is preferred, it is generally necessary to design arch structures for shear and moment. Typically, the profile is based on the funicular profile corresponding to dead load acting alone, and the arch rib is designed to resist shear and bending moments resulting from unbalanced live loads. Arch structures sometimes vary from the dead-load funicular profile for architectural or functional reasons.

a) Bgen werden vor allem auf DRUCK belastet.

b) Die umgekehrte Seillinie heit STTZLINIE. Wenn ein Bogen die Form der Sttzlinie hat, wird er nur auf Druck belastet.

Vorteil: verhltnismig schlanke Konstruktionen, weil nur druckbelastet.

Problem: bei wechselnden Lasten (Verkehrslast, Wind) entstehen Momente im Bogen

auch Bgen mssen stabilisiert werden! c) Bgen lsen HORIZONTALKRFTE in den Auflagern aus (Druck

nach auen). die Gre der Horizontalkrfte ist vom Stich des Bogens

abhngig: groer Stich - kleine Horizontalkrfte

kleiner Stich - groe Horizontalkrfte

wenn die Horizontalkrfte nicht aufgenommen werden knnen, kann ein gebogenes Tragwerk nicht als Bogen - also druckbeansprucht - wirken. (es ist dann ein gekrmmter Biegetrger, der nur momentenbeansprucht ist)

FUNICULAR LINE

The Response of Roof Arches to Loading Parabolic arches and circular arches behave differently under loading unless there are shallow and have a height-to-san ratio of h/L 1/8, in which case the circular arch can be treated as a parabolic one for preliminary design purposes. Under uniform, gravity load action on the horizontal projection, the pressure line coincides with the centroidal axis, or the parabolic arch is the funicular shape for the given loading. Hence there is no bending and no shear along the arch; the forces are resisted in purely axial manner. The maximum axial force Nmax appears at the reaction. The dead load, however, acts as a uniform load along the arch, for which the funicular response is a catenary. Because of the complex mathematical nature of the form, it is often approximated by a second-degree parabola, especially when the parabola is shallow. Hence, dead load action may be approximated as a horizontal uniform load and considered similar to snow loading The critical moments for a parabolic arch occur under asymmetrical uniform live loading across one-half of the arch. The maximum and minimum moments for a three-hinge arch are located at quarter spans and are equal to, M = wLL

2/64 The same equations can also be used for preliminary design purposes for two-hinged and fixed arches.

Uniform load on projection -> pressure line (funicular line) is a parabola -> centroidal line of arch is a parabola -> arch is a parabola hence, the arch does not deviate from the pressure line -> there is only compression ! The arch does not carry any moments Single load

-> pressure line deviates from the geometry of the arch -> hence the arch must carry moments in addition to compression -> moments at any location are equal to axial force, N, times the distance of the pressure line from the centroidal axis of the arch, e: M = Ne

Uniform load on half the span -> pressure line has a belly -> hence the parabolic arch deviates from the pressure line -> hence the arch must carry moments in addition to compression

Arch is slender -> pressure line are outside the member section -> the moments increase with the increase of (e), the distance between centroidal line and pressure line -> Because of the larger moments material must be able to resist tension: e.g. reinforced concrete, steel, wood

Thick arch -> the pressure line falls within the arch section -> moments are small tensile stresses are small -> masonry can be used Arch action primarily in compression

w/2

/2

MmaxMmin = Ne e

N

Av

H H

h

Bv

L

wL

Typically, an arch's profile is based on the funicular profile corresponding to dead load acting alone, and the arch rib is designed to resist shear and bending moments resulting from unbalanced live loads. The arch must be designed to resist both extremes.

Gaudi Only compressive construction (brick) construction follows the pressure line.

Colonia Guell crypt, Barcelona, 1915, Antoni Gaudi

Funicular shape under uniform loads

Berlin Stock Exchange, Berlin (Grimshaw, London) - The main support structure is a 2-hinge arch

- The floor framing hangs on the arches in a uniform manner (assuming uniform floor loads) -

Neue Messe Leipzig Arch has a deep cross section, therefore the pressure line falls within the section causing primarily compression in the trussed member fixed arch

STATICALLY DETERMINACY Dreigelenkbogen statisch BESTIMMTES System ohne Zwngungen alle Sttzlinien gehen durch die 3 Gelenke (Momenten-Nullpunkte) gnstig fr den Transport (1/2 Bogen kann vorgefertigt und transportiert werden) Zweigelenkbogen 1-fach statisch UNBESTIMMT geringer Unterschied zum 3-Gelenkbogen, Kfteverlauf sehr hnlich, da sowieso kaum Momente auftreten Eingespannter Bogen 3-fach statisch UNBESTIMMT Einspannung (= Momente) in den Auflagern Ausfhrung: - bei sehr groen Krften - wenn Aussteifung quer nicht mglich

BCE Place, Toronto, 1992, Santiago Calatrava -> Two-hinge arch -> Steep arch with small thrust forces

w/2

/2

MmaxMmin = Ne e

N

Av

H H

h

Bv

L

wL

For the preliminary design of three-hinged circular roof arches considering only gravity loading, one may use the following approximations: Shallow arches (h/L 1/8): treat circular arches as parabolic arches, use M = wLL

2/64 Intermediate arches (1/8 < h/L 1/3) for the rise-to-span ratios between steep and shallow roof arches, circular arches may be considered as parabolic arches for first-approximation purposes, although the effect of dead load causing bending must be considered. Mmax - wDh

2/8 - wLL2/64

Ignored are safely the difference in location between the two moments. Steep circular arches (h/L > 1/3) use

Mmax = - wh2/8

Design of: parabolic arch: Mmax = wLL2/64 = 0.5(40)2/64 = 12.5 k-ft

0.75(12.5 + 14.55) = 20.29 k-ft (COMB4)

S 1.15Mb/Fb = 1.15(20.29)12/24 = 11.67 in.3

try W8 x 15, Sx = 11.8 in.3

semicircular arch: -Mmax = wh2/8 = 1.0(20)2/8 = 50 k-ft

0.75(-50 - 8) = -43.5 k-ft (COMB3) S 1.10Mb/Fb = 1.10(50)12/24 = 27.5 in.

3 try W8 x 35, Sx = 31.2 in.

3

Parabolic arch:

Mmax wwh2/5.5 = 0.2(20)2/5.5 = 14.55 k-ft

Semicircular arch:

Mmax wwh2/4.5 = 0.2(20)2/4.5 = 17.78 k-ft

Mmin -wwh2/10 = -0.2(20)2/10 = -8 k-ft

wind loading

Parabolic arch: Mmax= -PL/16 = -2(40)/16 = -5 k-ft

Semicircular arch: Mmax -PL/10= -2(40)/10 = -8 k-ft

Single load at crown

Intermediate arches, h/L = 10/40 = Parabolic arch Design based on left side:

+Mmax = wLL2/64 =

0.5(40)2/64 = 12.5 k-ft S 1.25Mb/Fb = 1.25(12.5)12/24 = 7.81 in.3 try W8 x 13, Sx = 9.91 in.

3

Circular arch:

Design based on right side: Mmax - wDh

2/8 - wLL2/64 = - 0.5(10)2/8 12.5 = -6.25 12.5 = -18.75 k-ft

S 1.20Mb/Fb = 1.20(18.75)12/24 = 11.25 in.3

try W8 x 15, Sx = 11.8 in.

3

PARABOLIC ARCH

CIRCULAR ARCH

Parabolic arch: Mmax= - PL/16 = -2(40)/16 = -5 k-ft Semicircular arch: Mmax 1.2(- PL/16) = 1.2(-5) = -6 k-ft

Single load at crown

Treat the shallow arches as parabolic arches for preliminary design purposes because of the rise-to-span ratio h/L = 5/40 = 1/8

Mmax = wL L2/64 = 0.5(40)2/64 = 12.5 k-ft

S 1.35Mb/Fb = 1.35(12.5)12/24 = 8.44 in.3

try W8 x 13, Sx = 9.91 in.3

SAP requires a W8 x 15

PARABOLIC ARCH

CIRCULAR ARCH

Single crown load: Mmax= PL/16 = 2(40)/16 = 5 k-ft

PARABOLIC ARCH

CIRCULAR ARCH

Bent beam -> The connection at the top is hinged and cannot resist any horizontal forces -> the column carries only vertical reaction forces -> the right foundation only carries vertical reaction forces -> the structure is a bent beam

The arch is a bent beam which cannot resist lateral thrust forces -> the reaction forces are only vertical -> the bent member acts as a beam carrying only moments The walls cannot provide resistance to horizontal loads -> the arch is simply a bent beam.

- the tie rod resist horizontal forces

-> the arch can transfer the lateral thrust forces and acts only in compression -> the forces in the columns carry only the vertical reaction forces The horizontal arch forces are resisted by the tie rod

Between 2 arches, the H-forces cancel each other - However, at the end there is no counteracting H-force available

-The reaction force of the arch will be outside the brick

- However, there acts also the superimposed brick loads from above

- With this superimposed load the reaction load stays within the brick wall -> it is a true arch

- the brick column at the end is too thin so that no horizontal forces can be resisted

- therefore it cannot be a arch from a statical point of view

- because only vertical forces can be resisted, a lintel must be provided on top of the arch (hidden) to act as a beam to transfer the forces

Cargolifterhalle, Berlin Brand

Largest free-span hall on Earth (2000) width: 210 m length: 360 m height: 107 m Arches consist of 4 trusses with a fabric membrane spanning between

a b

c d

8'

40'

First, the geometry input for modeling the arches must be determined. The radius, R, for the shallow arch (Fig. 7.7A) according to Eq. (7.7), is R = (4h2 + L2)/8h = (4(8)2 + 402)/8(8) = 29 ft The location of the span L as related to the center of the circle is defined by the radial angle o according to Eq. (7.8). sin o= (L/2)/R =20/29 = 0.69, o = 43.600 Now three grid spaces with the following grid spacing along radial angles are selected, o/n = 43.600/3=14.530 The circular arch length, l, according to Eq. (7.9), is l = R(o/900) = (29)43.60/90 = 44.14 ft The arch length, l, for the semicircular arch, is l = R = (20) = 62.83 ft

To model the geometry of the arches in SAP the following values are selected: Global grid system: grid spacing in X direction: 4 ft using 24 spaces grid spacing in Y direction: 4 ft using 24 spaces grid spacing in Z direction: 4 ft using 15 spaces Cylindrical grid system: CSYS1 for case A: grid spacing along Radius: 29 ft using 1 space radial angles along Theta: 14.53 deg using 6 spaces spacing of curves along Z direction: 29 ft using 1 space The circular sector must be rotated 90 43.60 deg = 46.40 deg counterclockwise about the Z axis. CSYS2 for Case C: grid spacing along Radius: 20 ft using 1 space radial angles along Theta: 15 deg using 12 spaces spacing of curves along Z direction: 20 ft using 1 space Duplicate full arches and delete portions to obtain the one-half arches.

Mmax wL2/162 = 0.8(240)2/162 = 284 k-ft (SAP 310 k-ft)

Where the triangular load: w (0.020 + 0.018)240/36 = 0.8 k/ft

S 1.10Mb/Fb = 1.10(284)12/24 = 156.2 in3 Try W24 x 76, S = 176 in3

798 Beijing Art Factory, Beijing, 1956

COMPOSITE SYSTEMS AND FORM-RESISTANT STRUCTURES An example of an asymmetrical arch system is shown in the next slide. Here the supports are at different levels and a long-span arch and a short arch support each other, in other words the crown hinge is located off-center. The relatively shallow asymmetrical arch system constitutes a nearly funicular response in compression under uniform load action since the circular geometry approaches the parabolic one; notice that the location of the hinge is of no importance. Hence, live loading for each arch separately must be considered in order to cause bending, while the dead load is carried in nearly pure compression action; the long arch on the right side clearly carries the largest moments. Superimposing the pressure lines of the two loading cases results in a composite funicular polygon that looks like the shape of two inclined bowstring trusses, hence suggesting a good design solution. For long-span arches the use of triangular space trusses may be advantageous. Under asymmetrical loading on the long arch, the long arch acts in compression and the bottom chord in tension to resist the large positive bending moment. However, the bottom chord of the short arch acts in compression and the top chord in tension under the negative bending moment. But should the bottom member be straight, then it resists directly the compression force due to the live load in funicular fashion leaving no axial force or moment in the arch. Under asymmetrical loading on the short arch, the bottom chord of the long truss will resist the compression force directly, hence causing no moment or axial force in the arch if it would be a compression member. But since it is a tension member, there must be enough tension due to the weight of the long-span in the member to suppress the compression force!

Plan view

Pressure lines in elevation

It is common to vary the depth of the rib member according to the pattern of the moment for unbalanced loading.

Lehrter Bahnhof, Berlin, 2002, Gerkan, Marg & Partner, Mero

20' 17.32'

2.68'

10'

10'

30 deg

60 deg30 deg

17.32'

17.32'

5.86'

27.32'10'

4.29'

7.32'

a.

b.

2.68'C.

Ah

Av

Bh

Bv

5.86'

27.32'10'

4.29'

10.1

0 k

7.70 k

Mmax

Mmin

Waterloo Terminal, London, 1993, Nicholas Grimshaw

+ Anthony Hunt

BRACED ARCHES When arches are braced or prestressed by tensile elements, they are stabilized against buckling, and deformations due to various loading conditions and the corresponding moments are minimized, which in turn results in reduction of the arch cross-section. The stabilization of the arch through bracing can be done in various ways as suggested in Fig. 7.15 and 7.16. Typical examples of braced arches with non-prestressed web members are shown in Fig. 7.15. The most basic braced arch is the tied arch (b). Arches may be supported by a single or multiple compression struts or flying columns (c, d)). Slender arches may also be braced against buckling with radial ties at center span (e) as known from the principle of the bicycle wheel, where the thin wire spokes of the bicycle wheel are prestressed with sufficient force so that they do not carry compression and buckle due to external loads; the uniform radial tension produces compression in the outer circular rim (ring) of the wheel and tension in the inner ring. However, in the given case, the diagonal members are not prestressed. Here, the three members at center-span are struts.

Beams obove or below the arches carry the moments due to the varying live loads

Arches may also be supported by a dense network of overlapping diagonal tensile members (f); notice, this case represents a pure tensile network. When loaded on one side the diagonals under the load fold while the diagonal members on the non-loaded side are placed under tension. In general, however, depending on the arch proportions (Fig. 7.16) the tensile webbing may have to be prestressed to remain in tension under any loading condition and to increase the load carrying capacity and stiffness of the arch. Initially, forces are applied to certain members during the construction stage so that the structure counteracts stresses created by external loads. The design of the unbraced arched portal frame in (a), is controlled by full uniform gravity loading; here the lateral thrust at the frame knees is resisted completely in bending. However, when the relatively shallow portion of the arch is braced by a horizontal tie rod (b), the lateral displacement under full uniform gravity loading is very much reduced, that is bending decreases substantially although axial forces will increase. For the tied arch cases without or with flying column supports for cases (b, c, d)), the design of the critical arch members is controlled by gravity loading or the combination of half gravity loading together with wind whereas the design of the web members is controlled by gravity loading. It is apparent, as the layout of the arch webbing gets denser the arch moments will decrease further as the structure approaches an axial system. If a vertical load large enough is applied to the intersection of web members in case (e) to prestress the radial rod web members, then the entire web members form a radial tensile network. For further discussion refer to Problem 7.4.

Heavy construction -> moving loads hardly have an influence on the change of pressure lines

Several typical examples of braced arches with non-prestressed web members are shown in Fig. 9.12. The most basic braced arch is the tied arch (b). Arches may be supported by a single or multiple compression struts or flying columns (c, d)). Slender arches may also be braced against buckling with radial ties at center span (e) as known from the principle of the bicycle wheel, where the thin wire spokes of the bicycle wheel are prestressed with sufficient force so that they do not carry compression and buckle due to external loads; the uniform radial tension produces compression in the outer circular rim (ring) of the wheel and tension in the inner ring. However, in the given case, the diagonal members are not prestressed. Here, the three members at center-span are struts. Arches may also be supported by a dense network of overlapping diagonal tensile members (f); notice, this case represents a pure tensile network. When loaded on one side the diagonals under the load fold while the diagonal members on the non-loaded side are placed under tension. SAP takes into account the redistribution of forces by treating the cable network in case (f), for example, as tension-only members by performing a nonlinear static analysis. In general, however, depending on the arch proportions the tensile webbing may have to be prestressed to act more efficiently under any loading condition and to increase the load carrying capacity and stiffness of the arch. The cable-braced, latticed, tied-arch in Fig. 9.12g approaches the behavior of a truss; the cable network substantially reduces bending moments in the arch and tie beam where the bottom loads prestress the arch. For fast approximation purposes use the beam analogy (see also Fig. 6.3g).

a d

b e

c f

L = 40'

10'

6'

12'

10'

g

ARCHES WITH PRESTRESSED TENSILE WEBS The spirit of the delicate roof structure of the Lille Euro Station, Lille, France as shown in the following conceptual drawing (1994, Jean-Marie Duthilleul/ Peter Rice), reflects a new generation of structures aiming for lightness and immateriality. This new technology features construction with its own aesthetics reflecting a play between artistic, architectural, mathematical, and engineering worlds. The two asymmetrical transverse slender tubular steel arches (set at about 12 m or 40 ft on center) with diameters of around one-hundredth of their span, are of different radii; the larger arch has a span of 26 m and the smaller one 18.5 m. The arches are braced against buckling similar to the spokes of a wheel by deceitfully disorganized ties and rods; this graceful and light structure, in harmony with the intimate space, was not supposed to look right but to reflect a spirit of ambiguity. The roof does not sit directly on the arches, but on a series of slender tubes that are resting on the arches which, in turn, carry the longitudinal cable trusses that support the undulating metal roof. The support structure allowed the gently curved roof almost to float or to free it from its support, emphasizing the quality of light.

ad e

b c

500 50 0

500

500 50 0

50 0

20'

10'

Introducing to the semicircular arch a horizontal tie rod (Fig. 7.16b) at mid-height, reduces lateral displacement of the arches due to uniform gravity action substantially, so that the combination of gravity load and wind load controls now the design rather than primarily uniform gravity loading for an arch without a tie. Also the moments due to the gravity and wind load combination are reduced since the tie remains in tension as it transfers part of the wind load in compression to the other side of the arch. In contrast, when the arch is braced with a trussed network (Fig. 7.16c) then the arch is stiffened laterally very much, so that the uniform gravity loading case controls the design with the corresponding smaller moments. Similar behavior occurs for the arch placed on the diagonal (Fig. 7.16d, e). As a pure arch its design is controlled by bending with very small axial forces as based on gravity loading, in other words it behaves as a flexural system. However, when prestressed tensile webbing is introduced the moments in the arch are substantially reduced and the axial forces increased, now the arch approaches more the behavior of an axial-flexural structure system requiring much smaller member sizes; also here the controlling load case is gravity plus prestressing although the design of some members is based on dead load and prestressing. For further discussion refer to Problem

PRESTRESSING TENSILE WEBS To model tensile webs of arches, the web members may have to be prestressed by applying external prestress forces, or temperature forces. With respect to external prestress forces, run the structure as if it were, say a trussed arch, and determine the compression forces in the web members, which it naturally cannot support. Then, as a new loading case, apply an external force, which causes enough tension in the compression member so that never compression can occur. With respect to temperature forces, run the structure without prestressing it, then determine the maximum compression force in the cable members which should not exist, then apply a negative thermal force (i.e. temperature decrease causes shortening) to all those members thereby pre-stressing them, so that they all will be in tension. To perform the thermal analysis in SAP, select the frame element, then click Assign, then Frame/Cable Loads, and then Temperature; in the Frame Temperature Loading dialog box select first Load Case, then Type (i.e. temperature for uniform constant temperature difference).

Munich Airport Center, Munich, Germany, 1997, Helmut Jahn Arch

Kempinski Hotel, Munich, Germany, 1997, H. Jahn/Schlaich

Medieval masonry arch passes assessment with LUSAS masonry arch structure linear and nonlinear analysis load capacity of structure proved

Sheffield City Councils Design and Property (Structures Division) used LUSAS Bridge to undertake an assessment (strength) check of High Bridge, a 13th century quadripartite arch bridge on behalf of Lincolnshire County Council, the bridges owner. High Bridge, which carries Lincolns main pedestrian thoroughfare, had been structurally assessed twice before with conflicting outcomes. The LUSAS analysis provided an independent third assessment and proved the structure was safe for the imposed loading. The first of the earlier analyses failed the structure and concluded that the bridge was inadequate for pedestrian loading due to permissible tension in the masonry being exceeded. Next, Lincolnshire County Council carried out an in-house assessment using a serviceability limit state check under nominal loads with a line of thrust method. Unlike the first analysis, this took the formation of a hinge as the failure criterion, implying an acceptance of tension up to the formation of the first hinge. This assessment deemed the structure to have passed. Lincolnshires approach was felt to be a more rational basis for determining failure than the onset of tension since many masonry arches regularly experience tension and remain perfectly viable structures. Because of this, Sheffield used the plane stress concrete model in LUSAS to mimic the behaviour of the masonry. An ultimate limit state, rather than serviceability state, analysis was applied, because the issue was considered to be primarily one of structural safety. The technique used was to trace crack formation as nominal loads, i.e. without partial load factors, were applied incrementally with manual amendment of the model between each analysis. Repeated re-appraisal of the structures stiffness between runs was necessary since the model allows cracks to develop and propagate as the load increases and the structure degrades. As the model developed, the load factor achieved increased. This procedure continued with the aim of reaching an acceptable value. To reduce processing time, a series of linear analyses were done prior to a full nonlinear analysis. The initial linear analyses determined that out of balance effects from applying pedestrian loading to quarters of the plan area were minimal. The worst load case was shown to be pedestrian loading of 5kN/m2 over the entire plan of the structure. This allowed a simpler 3D quarter model to be employed thereafter, giving faster results. Additional linear analyses found that support conditions were critical to the mode of failure. With the supports rigid in respect of vertical settlement, as initially modelled, failure in the structures legs occurred very early in the loading regime. Truly rigid supports were felt to be unrealistic, so springs were introduced, resolving the premature leg failure. A final refinement was to introduce spring lateral restraints. These replaced the earlier rigid ones (which modelled lateral soil pressures) since it was felt that complete rigidity was unrealistic and furthermore had caused problems for the legs.

The cumulative effect of all the modelling changes was to raise the structures load factor to 3.43, an acceptable figure, and proving the safety of High Bridge for pedestrian loading. Achieving this outcome depended upon certain assumptions, as well as lateral support derived from the surrounding soil and adjacent buildings. This point was made clear to the Client in case future construction or demolition work nearby affected this beneficial soil pressure. In this analysis the following assumptions are pertinent: A condition factor of unity was assumed. Lateral soils pressure was earth pressure at rest. The only live load was pedestrian at 5 kN/m2. A linear stress/strain model was assumed, but a parabolic profile is more likely for masonry. The compressive strength of masonry was taken as 15 N/mm2. The formation of tension and hinges were accepted. Failure was deemed to occur when the analysis failed to converge at the nth iteration. The load at the n-1th iteration, divided by the nominal load, provided a load factor value. It is implicit in this approach that cracking is permitted, as is the formation of one or more hinges, but not a mechanism.

Inflatable arches

China Central Television (CCTV) Headquarters, Beijing, 2008, Rem Koolhaas and OMA, Arup