ARCH Structures including sap2000 Prof. Wolfgang Schueller
For SAP2000 problem solutions refer to “Wolfgang Schueller: Building Support Structures – examples model files”: https://wiki.csiamerica.com/display/sap2000/Wolfgang+Schueller%3A+Building+S upport+Structures+If you do not have the SAP2000 program get it from CSI. Students should request technical support from their professors, who can contact CSI if necessary, to obtain the latest limited capacity (100 nodes) student version demo for SAP2000; CSI does not provide technical support directly to students. The reader may also be interested in the Eval uation version of SAP2000; there is no capacity limitation, but one cannot print or export/import from it and it cannot be read in the commercial version. (http://www.csiamerica.com/support/downloads) See also, (1) The Design of Building Structures (Vol.1, Vol. 2), rev. ed., PDF eBook by Wolfgang Schueller, 2016, published originally by Prentice Hall, 1996, (2) Building Support Structures, Analysis and Design with SAP2000 Software, 2nd ed., eBook by Wolfgang Schueller, 2015. The SAP2000V15 Examples and Problems SDB files are available on the Computers & Structures, Inc. (CSI) website: http://www.csiamerica.com/go/schueller
THE USE OF ARCHES •
Memorial arch
•
Arch bridge
•
Roof support structures
•
Arched roof structures including domes
•
Arched cables
•
Wall penetration
•
Arched buildings
•
etc.
Arch geometries
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
Arches as enclosures
Arches
Examples of arch structres
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
Interior, Coliseum, Rom, Italy, 80 AD
Arch of Constantine, Forum, Rome, AD315
Constantine Basilica, Trier, Germany, 310 AD
Palatine Chapel, Aachen Cathedral, Aachen, Germany, c.800 AD, Odo of Metz Arch
Basilica of the Holy Apostles, Cologne, Germany, 11th – 13th century
Cathedral of Notre Dame de Paris, 1150 - 1220
Notre Dame de Paris, 1150 -1220
Notre Dame de Paris. Schematic sections showing the “flying buttresses”
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
17 m
13 m
Thrusts in flying buttresses (left) and structure of a groin vault (above)
Construction of a Gothic cathedral
Amiens Cathedral, Amiens, France, 1269
Amiens Cathedral, France, 1269
Amiens Cathedral, Amiens, France, 1269
St. Pierrre, Beauvais, 1247
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.
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.
Arched Bridge, the Summer Palace in Beijing, China, 1750
Charles Bridge, Prague, Czech Republic, c. 1400
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. Paul’s 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 Santa Maria del Fiore, Florence, , Italy, 1436, Filippo Brunelleschi Arch
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 St Peter’s Basilica, Rome, Michaelangelo, 1546
“Hanging chain” analysis of Dome of St Peter’s, by Giovani Poleni, 1742
St Paul’s Cathedral, London, 1710, Christopher Wren Arch
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”
St Paul’s 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) Mission San Carlos Borromeo, Carmel, CA, 1793, Manuel Estevan Ruiz Arch
Centenary Hall, Breslau, Ger. (now Wroclaw, Pol.), Max Berg, 1913, Dyckerhoff & Widmann
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, William Henry Barlow Designer
Galerie des Machines (375 ft, 114 m), Paris, 1889, Ferdinand Dutert Arch, Victor Contamin Struct. Eng
Frames: 375-ft (114 m) span steel arches for the Galerie des Machines (1889), Ferdinand Dutert Arch, Victor Contamin Struct. Eng
Grande Halle de la Villette, Paris, 1867
Atrium, Glasgow School of Art, Glasgow, UK, 1909, Charles Rennie Mackintosh Arch
The Stock Exchange Amsterdam, 1903, Hendrik Petrus Berlage Arch
Viaduc de Garabit, Saint-Flour, Cantal, 1884, Gustave Eiffel
Firth of Forth Bridge (1708 ft), Scotland, 1890, Benjamin Baker, John Fowler
DOUBLE CANTILEVER STRUCTURES
Eiffel Tower (300 m), Paris, 1889, Gustave Eiffel
Glass-vaulted Gallerias in Brussels, Milane, etc. end of 19th century
Sagrada Familia Cathedral, Barcelona, 1882 - , Antoni Gaudi
Casa Mila, Barcelona, Spain, 1910, Antoni Gaudi Arch
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
Casa Batlló, Barcelona, Spain, 1906, Antoni Gaudi Arch
Colegio Teresiano, Barcelona, Spain, 1889, Antoni Gaudí Arch
St Thérèse, Metz, France, 1954, RogerHenri Expert Arch
St Martin Donges, Colmar, Haut-Rhin, France, 1957, Jean Dorian Arch
Grundtvig’s Church, Copenhagen, Denmark, 1930, Peder Vilhelm Jensen-Klint Arch
Church of the Holy Cross, GelsenkirchenÜckendorf, Germany, 1929, Josef Franke Arch
Hohenzollernplatz Church, Berlin, Germany, 1933, Fritz Hoeger Arch
St. Paul’s Episcopal Church, Seattle, WA, 1962, Robert Theirault Arch, Jim Harriott Struct Eng
New Dresden Main Train Station, Dresden, 2006, Norman Foster Arch, Happold Struct. Eng.
Dresden Central Railway Sttation, 2006, Norman Foster Arch, Happold Eng.
St.-Josefskirche, Zabrze (Hindenburg OS), Polen, 1931, Dominikus Böhm Arch
Palazzo della Civiltà Italiana, Rom, Italy, 1953, G. Guerrini, E.B. La Padula, M. Romano Arch
Institute of Public Administration, Ahmedabad, India, 1963, Louis Kahn Arch
Parlaments Buildingde, Sher-eBanglanagar, Dhaka, Bangladesch,1974, Louis I. Kahn Arch
Zitadelle, Sher-eBanglanagar, Dhaka, Bangladesch, 1974, Louis I. Kahn Arch
Ayub Hospital in Dhaka, Bangladesch, Louis I. Kahn Arch
Gravity Load distribution
Museum of Roman Art, Mérida, Spain 1985, Jose Rafael Moneo Arch
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
Parabolic Slides, Math and Computer Science College, TU Munich
Flamingo sculpture, Chicago Federal Center Stabile, 1974, Alexander Calder Artist (Chicago Federal Center by Mies van der Rohe)
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 onecentered (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
CONTEMPORARY ARCHES ARCH USE:
BRIDGES, BUILDING ENCLOSURES, ROOF STRUCTURES, SUPPORT STRUCTURS, WALL ARCHES, COMPRESSION RINGS, SUSPENDED ARCHES, TREES, MONUMENTS, etc.
ARCHES
ARCH BRIDGES
Traditional bridge, Chhina
Salignatobel Bridge, Schiers, Grisons, Switzerland, 1930, Robert Maillart Designer
Arve Bridge, Vesay, Switzerland, 1935, Robert Maillart
Route 112 Bridge, Huntington, MA, using SAP
Blennerhassett Island Bridge, Route 50, Belpre Township, WV, 2008, E.L Robinson, consult eng
Oberbaumbruecke, Berlin, 1995, Santiago Calatrava Designer
Borneo-Sporenburg Python Bridge , Amsterdam, 2001, Adriaan Geuze Arch
Pedestrian bridge at Seattle's Museum of Flight, 2007, SRG Partnership
Ponte della Constituzione , Venice, 2007 Santiago Calatrava
Japan Bridge, Paris, France, 1993, Kisho Kurokawa Arch, Hugh Dutton /SOM Struct. Eng
Based on an original drawing by Leonardo da Vinci, Oslo, 2001, Vebjørn Sand
Bent wood bridge, Esslingen, Germany, 1986, R. Dietrich
Pedestrian bridge in Cologne, Germany
Crown Prince Bridge, Berlin, 1996, Santiago Calatrava Arch
La Devesa Footbridge, Ripoll, Spain, 1991, S. Calatrava, torsionbeam
Bac de Roda Felipe II Bridge, La Sagrera, Catalonia, Spain, 1987, Santiago Calatrava Designer
Barqueta Bridge, Seville, Spain, 1992, Santiago Calatrava Designer
La Devesa Footbridge, Ripoll, Spain, 1991, S. Calatrava, torsion beam
James Joyce Bridge on Liffey River, Dublin, 2003, Santiago Calatrava Designer
Nordsternpark Bridge, BUGA, Gelsenkirchen, Germany, 1996, Feldmeier + Wrede Arch, Stefan Polónyi Struct. Eng
Infinity Loop Bridge, Zuhai, China, Infinity Loop Bridge, Gordon Affleck & Adrian Boot Arch, Buero Happold Struct. Eng
Proposal train station, Florence, Italy, 2007, Arata Isozaki, Mutsuro Sasaki
ARCHES as PART OF THE BUILDING STRUCTURE
Garden scene, Suchou
Documentation Center Nazi Party Rally Grounds, Nuremberg, 2001, Guenther Domenig
Stadelhofen Railway Station, Zurich, 1984, Santiago Calatrava
Jubilee Church, Rome, Italy, 2003, Richard Meier Arch
Ludwig-Erhard-Haus, Berlin, Germany, 1999, Nick Grimshaw Struct Eng
Exchange House, London, UK, 1990, SOM - Srinivasa “Hal” Iyengar Struct. Eng
New Entrance Hall, Van Gogh Museum, Amsterdam, 2015, Kisho Kurokawa + Hans van Heeswijk Arch, Arcadis Struct. Eng
Office building of the European Investment Bank, 2009, Luxembourg, Ingenhoven Architects, Werner Sobek Struct Eng
Bordeaux Law Courts, 1998, Richard Rogers Arch, Arup Struct E
Arch Building, Band University, Gold Coast, Queensland, Australia, 1989, Arata Isozaki Arch
La Grande Arche, Paris, 1989, Johan Otto von Sprechelsen/ Peter Rice for the canopy
Central Chinese Television (CCTV) Tower, Beijing, 2008, Koolhaas and Ole Scheeren/OMA Arch, Ove Arup Struct Eng
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
ARCHES as ROOF/BUILDING Support STRUCTURES
Tramway stop Marktplatz-Bohl, St. Gallen, 1999, Santiago Calatrava Arch
Metro Blaak station, Rotterdam, 1993, Harry Reijnders + Marja Haring Arch , László Vákár Struct. Eng
Ice hokey stadium, Munich, 1983, Kurt Ackermann Arch, Sclaich Bergermann Struct. Eng
Lanxess Arena, Cologne, 1998, Peter Böhm Arch
Olympic Stadium “OAKA”, Athens, Greece, 2004, Santiago Calatrava
Olympic Stadium “OAKA”, Athens, Greece, 2004, Santiago Calatrava
The Olympic Velodrome, Athens, Greece, 2004, Santiago Calatrava
Jumbo Maintenance Hangar, Deutsche Lufthansa, Hamburg Airport, von Gerkan, Marg & Partner Arch
ARCHES AS ROOF STRUCTURES
Arched structure, computer model
Dorton (Raleigh) Arena, 1952, North Carolina, Matthew Nowicki Arch, Frederick Severud Struct. Eng
Nürnberg Passage
Hamburg Airport, Terminal 2, 1993, Gerkan Marg Arch, Schlaich Bergemann Struct Eng
SOSC Shanghai Oriental Sport Center (Natatorium), 2011, Gerkan Marg Arch, Schlaich Bergemann Struct Eng
10' 10'
10'
10'
10'
a
b
c
d
e
f
Fig. 8.15 Multiple-Bay, Single-Story Frames and M-Diagrams for Uniform Gravity Loads (SAP2000)
USAFA Cadet Chapel, Colorado Springs, 1962, Walter Netsch/SOM Arch
Berliner Bogen, Hamburg, Germany, 2002, Bothe Richter Teherani Arch, Binnewies Struct. Eng
Messehaus 9 and Galleria , Frankfurt, Germany, 1983, Oswald Mathias Ungers Arch, Stefan Polonyi Struct. Eng.
Messehaus 9 and Galleria , Frankfurt, Germany, 1983, Oswald Mathias Ungers Arch, Stefan Polonyi Struct. Eng.
Agora , Athens, Greece, 2004, Santiago Calatrava Arch
Lisbon Oriente Bus Station , Lisbon, 1998, Santiago Calatrava Arch
Santa Maria Assunta, Riola Vergato, Italy, 1978, Alvar Aalto Arch
Media and Study Centre, D. Hosiassohn (Sketch program)
Motorcar Verona Showroom, Ponte Forio, Italy, Andrea Casellato Arch
National Stadium of Sports Affairs Council, Kaohsiung, Taiwan, 2009, Toyo Ito Arch
Gare d’Orleans, Paris, 2007, Jean-Marie Duthill , Etienne Tricaud AREP
Burj-Al-Arab Hotel, Dubai, United Arab Emirates, 1998, Tom Wright Arch (W.S. Atkins), W.S.Atkins Struct Eng
St. Dominique, 2001, Gifu Design,Takenaka
Focus Media Center (Deutsche-Med-Platz) Rostock, 2004, Helmut Jahn Arch, Werner Sobek Struct. Eng
Student Housing IIT, Chicago, 2003, Helmut Jahn Arch
Museum of Contemporary Art, Helsinki, Finland, 1998, Steven Holl Arch, Arup + Nordenson Struct. Eng
Santa Caterina Market, Mollet del Valles, Barcelona, 2001, Enric Miralles and Benedetta Tagliabue Arch
Ningbo Air Terminal
Beijing Capital International Airport - Terminal 2, 1999
Inchon Airport, Seoul , 2002, Terry Farrel Arch
Beijing International Airport Terminal 3, 2008, Norman Foster Arch, Arup Struct Eng
New York World Trade Center Transportation Hub, 2016, Santiago Calatrava Arch, Thornton Tomasetti Struct Eng
EXPO-Dach Hannover, 2000, Thomas Herzog Arch, Julius Natterer Struct Eng
Autobahnraststätte, Deitingen, Switzerland,1968, Heinz Isler Arch. & Eng.
Bodegas Protos, Peñafiel, Valladolid, Spain, 2008, Richard Rogers Arch, Arup Struct E.
Ferrari Restaurant, Maranello, Italy. 2008, Marco Visconti Arch
Olympic Stadium Montreal, Canada, 1976, Roger Taillibert Arch
Biodome (Olympic Velodrome), Montreal, Canada, 1976, Roger Taillibert Arch
Parc des Princes, Paris, France, 1972, Roger Taillibert Arch
Harajuku Protestant Church, Kita-Aoyama, Tokyo, 2005, Ciel Rouge Creation, Kaneko Fumiko & Henri Gueydan
Sustainable towers, Kuala Lumpur, Malaysia, 2013, Studio Nicoletti
Subway Station Munich-Fröttmaning, Munich, 2005, Peter Bohn Arch
Airport Terminal Newark
Berlin Hauptbahnhof, Berlin, 2006, von Gerkan, Marg Arch, Schlaich Bergermann Struct Eng
Peek & Cloppenburg, Cologne, Germany, 2005, Renzo Piano
National Museum of the Marine Corps and Heritage Center, Quantico, Virginia USA, 2006, Fentress Bradburn Architects, Weidlinger Struct Eng
San Giovanni Rotondo, Foggia, Italy, 2004, Renzo Piano Arch, Peter Rice/Ove Arup concept designer
Center Paul Klee, Bern, Switzerland, 2005, Renzo Piano Arch , Arup Struct Eng
Waterloo Station, London, UK, 1993, Nicholas Thomas Grimshaw Arch, Anthony Hunt Struct Eng
Mmax
Mmin
7.70 k
5.86'
Pressure lines in elevation 4.29'
Plan view
10'
27.32'
(2) Fig. 9.10 EXAMPLE 9.2: Axial Forces (COMB1), Moments (COMB1)
Heathrow Terminal 5, London, UK, 2008, Richard Rogers Arch, Ove Arup Struct Eng
Kansai International Airport Terminal, Osaka, 1994, Renzo Piano Arch, Ove Arup Struct Eng
Lyon-Satolas Airport Railway Station, Lyon, France, 1994, Santiago Calatrava Arch + Struct Eng
Barajas Airport, Madrid, Spain, 2005, Richard Rogers Arch, Anthony Hunt Associates (main structure), Arup (main façade)
Floating Pavilion, Groningen, Netherland, 1997, Fumihico Maki
Milwaukee Art Museum, 2001, Santiago Calatrava Arch, Graef Anhalt Schloemer Struct Eng
Lisbone Orient Station, Lisbone, Portugal, 1998, Santiago Calatrava Designer
Liège Guillemins TGV Station, Liège, Belgium, 2008, Santiago Calatrava
Zhongguancun Financia Center, Beijing, 2006, Kohn Pederson Fox Arch
Iglesia de la Medalla Milagrosa, Navarte, Mexico City, 1955, Felix Candela
BCE Place Galleria, Toronto, 1992, Santiago Calatrava Designer
City of Arts and Sciences, Valencia, Spain, 1996, Santiago Calatrava Arch
Les Halles, Paris, 1979, Claude Vasconi and Georges Pencreac'h
Vaillant Arena , Davos, Switzerland, 1979, Krähenbühl Arch, W. Bieler Struct Eng
United Airlines Terminal at O’Hare Airport, Chicago, 1987, H. Jahn Arch, Lev Zetlin Struct E
Minute Maid Field, Houston, 2000, HOK Sport Arch, Walter P. Moore Struct E
Miller Park , Milwaukee, WI, 2001, Dan Meis Arch/NBBJ + HKS, Arup/Flad Struct Eng
Atrium, Germanisches Museum, Nuremberg, Germany, 1993, me di um Arch
Jaeger-Passage, Leipzig, Germany
The Kimmel Center for Performing Arts, Philadelphia, 2003, Rafael Vinoly Arch, Ove Arup + Dewhurst Macfarlane Struct Eng
Neue Messe Leipzig, 1996, Gerkan, Marg + Ian Ritchie Arch, Stefan Polónyi Struct Eng
National Grand Theater, Beijing, 2007, Paul Andreu
Bangkok International Airport, 2006, Murphy/Jahn Arch, Werner Sobek Struct Eng
International Terminal at San Francisco International Airport, 2001, SOM Arch + Struct E
Stuttgart Airport, Terminal 1, Germany, 1991, von Gerkan Marg Arch
a.
b.
c.
(2) Fig. 10.1a Example of Space Frame Structures: tree construction in SAP2000
Shenyang Taoxian International Airport, 2002, KOHLSTRUNG INTERNATIONAL ARCHITECTURE & URBAN DESIGN CONSULTING CO
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
Tekla Xsteel
Complex canopy
Cathedral of Christ the Light, Oakland, CA, 2008, SOM Arch + Struct Eng
Cathédrale Notre-Dame de Créteil Extension, Paris, France, 2015, Architecture Studio
The Response of Roof Arches to Loading • SIMPLE LOADING CONDITIONS • FUNICULAR COMPRESSION SYSTEMS • BRACED ARCHES • COMPOSITE SYSTEMS AND FORM-RESISTANT STRUCTURES
• ARCHES WITH PRESTRESSED TENSILE WEBS
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 = wLL2/64
The same equations can also be used for preliminary design purposes for twohinged and fixed arches.
Pressure line responses to various load actions: funicular shapes
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.
Response of typical gable frame roof enclosures to gravity loading
Simple Arch Loading
Response of the arch to single-load action
Parabolic arch under uniform load action
Circular arch under uniform load action
Graphic statics
Colonia Guell crypt, Barcelona, 1915, Antoni Gaudi:
compressive construction (brick) construction follows the pressure line
Funicular shape under uniform loads
Arches under loading: exercises
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 dropdown 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 xzplane, 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.
Minor masonry arches: • arch action in compression
Thick arches:
• the pressure line falls within the arch section moments are small tensile stresses are small masonry can be used • arch action primarily in compression
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
Radial forces and stresses
STATICALLY DETERMINACY 3-hinged arch statically determinate all pressure lines pass through the 3 hinges where the moments are zero. convinient for transportation (prefab ½ arches) 2-hinged arch once indeterminate: moments occur at the eliminated hinge. force flow is rather close to 3-hinged arches since the moments are rather small. Fixed arch thrice indeterminate moments occur at the fixed supports.
Indeterminate Arches
BCE Place, Toronto, 1992, Santiago Calatrava • 2-hinged arch • steep arch with small thrust forces
wL w
M min = Ne
Mmax h
e
N H
H Av θ/2
Bv
θ/2
θ
L
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 = wLL2/64 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 ≈ - wDh2/8 - 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
PARABOLIC ARCH
Intermediate arches, h/L = 10/40 = ¼
Parabolic arch CIRCULAR 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
Circular arch:
try W8 x 13, Sx = 9.91 in.3
Design based on right side:
Mmax ≈ - wDh2/8 - 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:
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
PARABOLIC ARCH
CIRCULAR ARCH
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
Uniform loading on inclined member
Folded/bent beam structures: loading and moments
8'
a
b
c
d
90 40'
Arches and bent beams
Case d: 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 20'
8'
a
b
c
d
40'
Arches EXAMPLE: 9.1
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 θo according to Eq. (7.8).
sin θo= ±(L/2)/R =20/29 = 0.69,
θo = 43.600
Now three grid spaces with the following grid spacing along radial angles are selected,
θo/n = 43.600/3=14.530 The circular arch length, l, according to Eq. (7.9), is
l = πR(θo/900) = π(29)43.60/90 = 44.14 ft The arch length, l, for the semicircular arch, is
l = πR = π(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
Ribbed Domes
Circular dome using Etabs
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!
Berlin Hauptbahnhof, Berlin, 2006, von Gerkan, Marg Arch, Schlaich Bergermann Struct Eng
Space Truss Arch: axial force flow
Pressure lines in elevation
Plan view
Asymmetrical arch
2.68'
C. 10'
30 deg 17.32'
60 deg
Bh Bv 10' 30 deg
a.
Ah
20'
Av
17.32'
2.68'
7.32' 5.86' 17.32'
4.29'
b.
10'
27.32'
EXAMPLE: 9.2
10
.10
k
Mmax
Mmin
7.70 k 5.86'
4.29'
10'
27.32'
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. 9.12 and 9.14.
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.
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 9.1.
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 .
10'
a
d 6'
12'
b
e c
f
10'
L = 40'
g
Problem 9.1: Braced arches
Beams above or below the arches carry the moments due to the varying live loads
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.
TGV Lille-Europe Station, Lille, France, 1994, JeanMarie Duthilleul/ Peter Rice
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 prestressing 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).
20'
10'
a
500 0
0
50
50
50 0
b
50 0
c
0
50
Braced Arches Problem 9.3
d
e
Introducing to the semicircular arch a horizontal tie rod (Problem 9.3) at midheight, 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 , 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. 9.14d, 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
Munich Airport Center, Munich, Germany, 1997, Helmut Jahn Arch
Kempinski Hotel, Munich, Germany, 1997,
H. Jahn/Schlaich
Stone Pavilion of the Future, EXPO 92, Seville, Spain, Peter Rice Eng. (Ove Arup)
Stone Pavilion of the Future, EXPO 92, Seville, Spain, Peter Rice Eng. (Ove Arup)
student project
