parabolic arch

While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, {{math|cosh(x)}}, a sum of two exponential functions. One parabola is {{math|f(x) {{=}} x² + 3x − 1}}, and hyperbolic cosine is {{math|cosh(x) {{=}} {{sfrac|e^x + e^−x|2}}}}. The curves are unrelated.

Unlike a catenary arch, the parabolic arch employs the principle that when weight is uniformly applied above, the internal compression (see line of thrust) resulting from that weight will follow a parabolic curve. Of all arch types, the parabolic arch produces the most thrust at the base. Also, it can span the widest area. It is commonly used in bridge design, where long spans are needed.{{cite book|url=https://books.google.com/books?id=vgprkYFib4AC&pg=PA744|title=Developments in Mechanics of Structures & Materials|isbn=9789058096593|last1=Deeks|first1=Andrew J.|last2=Hao|first2=Hong|date=2004-11-15|publisher=Taylor & Francis }}{{Cite web |url=http://ebooks.narotama.ac.id/files/The%20Design%20of%20Prestressed%20Concrete%20Bridges/Chapter%2017%20The%20Design%20And%20Construction%20Of%20Arches.pdf |title=The Design of Prestressed Concrete Bridges/Chapter 17 The Design And Construction Of Arches |access-date=2016-04-24 |archive-date=2017-02-02 |archive-url=https://web.archive.org/web/20170202000349/http://ebooks.narotama.ac.id/files/The%20Design%20of%20Prestressed%20Concrete%20Bridges/Chapter%2017%20The%20Design%20And%20Construction%20Of%20Arches.pdf |url-status=dead }}

When an arch carries a uniformly distributed vertical load, the correct shape is a parabola. When an arch carries only its own weight, the best shape is a catenary.

File:Parabola graphed against a catenary upside down view.png|Parabola (red) graphed against a catenary (blue), view to simulate an arch.

File:Parabola graphed against a catenary upside-down, zoomed out.png|Parabola (red) graphed against a catenary (blue), view to simulate an arch. Zoomed out.

{{see also|Natural arch}}

A hen's egg can be fairly well described as two different paraboloids connected by part of an ellipsoid.{{cite journal | url=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/ps/52/1/10.3382/ps.0520127/2/poultrysci52-0127.pdf?Expires=1490584295&Signature=TnS-pUPw5394lZXTqG7sWihVya-imEJYlbt5YCmO6POV8FSTakjk5MOlJd-CorsOBeJURWw1nYqD2AiZU-WmFX4TapujkJHvZ~QXKaPqsT1YloE6tcwx~A4DNJvz0k6Vfum8wloctVH1smgvnZBFckKsZS3mLvJp6CZnY5m9VbSpC-Nj2wHoC4IIMCNIDGfuJ~Zbr~YIpoypF6hLsUKPLTOtajnR8unfinOCIcQMEHkur4YvLJOP-Ur~tcX~2JemMj7BJIKta5dGpCyqib9ZHEwNZpgaRKE8ALj3unNQCPe4vJSYNg64digA3B-31gXJjC2qafiRhGWSWnZxwem-ew__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q|title= Characterizing the Shape of a Hen's Egg|journal= Poultry Science|volume= 52|pages= 127–138| access-date=22 March 2017|doi= 10.3382/ps.0520127|year= 1973|last1= Rehkugler|first1= G. E.|doi-access= free}}Google Books: [https://archive.org/details/elementsgeometr00edwagoog/page/n286 https://books.google.co.uk/books?id=CIAAAAAAMAAJ&pg=PA264&lpg=PA264&dq="hen's+egg"+"parabolic+arch"&source=bl&ots=uOzbRBeDwm&sig=ohyUSTlHvU8sNUqTahR_BjU6gp4&hl=en&sa=X&redir_esc=y#v=onepage&q="hen's egg" "parabolic arch"&f=false], accessdate: March 22, 2017

File:Celler de Sant Cugat lateral.JPG ]]

File:Interieur van de centrale hal met decoraties - Utrecht - 20349663 - RCE.jpg

Self-supporting catenary arches appeared occasionally in ancient architecture, for examples in the main arch of the partially ruined Sassanian palace Taq Kasra (now in Iraq), the largest single-span vault of unreinforced brickwork in the world, and the beehive huts of southwestern Ireland. In the modern period, parabolic arches were first used extensively from the 1880s by the Catalan architect Antoni Gaudí,{{Cite journal|last=Chiuini|first=Michele|date=2015-05-13|title=The parabola of the parabolic arch|url=https://www.researchgate.net/publication/282801705|journal=IABSE Symposium Report|series=IABSE Conference, Nara 2015: Elegance in structures |volume=104|pages=372–373|doi=10.2749/222137815815775439|isbn=978-3-85748-138-3 }} deriving them from catenary arched shapes, constructed of brick or stone, and culminating in the catenary based design of the famous Sagrada Familia. Other Catalan architects then used them into the 1920s, and they appeared occasionally in German expressionist architecture of the 1920s-30s. From the 1940s they gained a new popularity in reinforced concrete, including in shell concrete forms often as hyperbolic parabloids, especially by Felix Candela in Mexico and Oscar Niemeyer in Brazil, but they could be found around the world, especially for churches, in the 1950s and 60s. Since the 1990s Spanish designer Santiago Calatrava has frequently used parabolas for his signature roof structures and bridges. Structures that are self-supporting arches like the Sheffield Winter Garden are often closer to true catenaries.

• Palau Güell, 1886–88, Barcelona, where Antoni Gaudí used parabolic arches in stone for the carriageway entrances, and in brick for the structure of the main hall.
• Casa Milà, 1906, where Gaudi used brick parabolic arches support the attic roof, used as a laundry space.
• Wrocław Market Hall, 1906-8, Richard Plüddemann and Heinrich Küster, internal structure
• Celler modernista, 1921, part of Sant Cugat Museum, Catalonia, Spain, Cèsar Martinell i Brunet
• Pinell de Brai Cooperative Winery, 1922, Pinell de Brai, Catalonia, Spain, Cèsar Martinell i Brunet
• Former main post office, 1919–24, Utrecht, main hall, designed by J. Crouwel Jr.
• File:Priory Chapel at Saint Louis Abbey - July 2013.jpgSt. Engelbert, Cologne, 1928−1932, by Dominikus Böhm
• Church of Saint Francis of Assisi, 1943, Pampulha, Belo Horizonte, Brazil, Oscar Neimeyer.
• Church of La Purísima, 1943, Monterrey, Mexico, Enrique de la Mora
• Cosmic Rays Pavilion, 1951, Felix Candela with Jorge González Reyna, UNAM, Mexico City
• Memorial Cenotaph, 1952, Hiroshima Peace Memorial Park,{{cite web |url = https://higashishinbun.blogspot.com/2008/04/lultimo-sogno-di-citt-ideale-tange.html|title = Art, culture and society from Far East |publisher= Modello Fantastico | access-date=8 Dec 2016}} Kenzō Tange.
• Church of St Mary and St Joseph, Poplar, 1954, London, United Kingdom, Adrian Gilbert Scott
• St Leonard's Church,Interior of St Leonard's church, St... (C) Julian P Guffogg :: Geograph Britain and Ireland: [https://www.geograph.org.uk/photo/4080009 Interior of St Leonard's church, St... (C) Julian P Guffogg :: Geograph Britain and Ireland], accessdate: March 2, 2017Organ, St Leonard's Parish Church (C) Julian P Guffogg :: Geograph Ireland: [http://www.geograph.ie/photo/3236009 para], accessdate: March 3, 2017 1955, St Leonards on Sea, United Kingdom, Adrian Gilbert Scott
• Dorton Arena 1957, Raleigh
• St Mary's Star of the Sea Cathedral, 1958–62, Darwin, Australia, architect Ian Ferrier
• Toast Rack (building) (originally Domestic Trades College, Manchester Polytechnic) 1960, Fallowfield, Manchester, United Kingdom, city architect, Leonard Cecil Howitt
• Theme Building, 1961, Los Angeles International Airport, Pereira & Luckman Architects, Paul Williams and Welton Becket
• Priory Chapel, Saint Louis Abbey, 1962, Creve Coeur, St. Louis, Missouri, United States, Hellmuth, Obata and Kassabaum (HOK), with Pier Luigi Nervi
• Gateway Arch, 1960-5, St Louis, a tall freestanding catenary arch designed by Eero Saarinen
• Allen Lambert Galleria, 1992, Toronto, Canada, Santiago Calatrava
• L'Umbracle (catenary shade house)Jean McConochie photos at pbase.com: [http://www.pbase.com/image/105581805 para], accessdate: March 3, 2017 2001, Ciutat de les Arts i les Ciències (City of Arts and Sciences), Valencia, Spain, Santiago Calatrava
• L'Oceanogràfic aquarium, 2003, Valencia, Spain, Felix Candela.
• Sheffield Winter Garden (catenary), 2003, Sheffield, UK, Pringle Richards Sharratt Architects and Buro Happold
• Fjordenhus, 2018, Vejle Fjord, Denmark, Olafur Eliasson and Sebastian Behmann{{Cite web|date=2018-06-04|title=Olafur Eliasson's first building is a castle-like office in a Danish fjord|url=https://www.dezeen.com/2018/06/04/olafur-eliasson-fjordenhus-office-vejle-fjord-denmark-architecture/|access-date=2020-08-12|website=Dezeen|language=en}}

File:ML Bixby Creek Bridge.JPG parabolic arch|alt=]]

File:Garabit.jpgBridges have used a variety of arches since ancient times, sometimes in very flat segmental arched forms but rarely in the form of a parabola. A simple hanging rope bridge describes a catenary, but if they were in the form of a suspension bridges they usually describe a parabola in shape, with the roadway hanging from the inverted arch. Modern suspension bridges were built from the early 19th century, beginning with chains and progressing to more and more elegant steel rope examples, and are still in use today. Parabolic arches that support the roadway from below (or in the form of a through arch) first appeared in the 1870s, and have been used occasionally ever since; examples include:

• Maria Pia Bridge, Gustave Eiffel and Théophile Seyrig, Porto, Portugal, a railway bridge built in 1877.{{Cite web |title=Ponte Maria Pia Bridge |url=https://www.asce.org/project/ponte-maria-pia-bridge/ |website=Invention and Technologu |access-date=2020-08-09 |archive-date=2021-03-05 |archive-url=https://web.archive.org/web/20210305131345/https://www.asce.org/Project/Ponte-Maria-Pia-Bridge/ |url-status=dead }}
• Garabit viaduct, near Ruynes-en-Margeride, Cantal, France, designed by Gustave Eiffel, and built between 1882 and 1884.{{Cite web |last=Weber |first=Jutta |date=May 2009 |title=The Engineer's Aesthetics – Interrelations between Structural Engineering, Architecture and Art |url=http://www.bma.arch.unige.it/internos/PDF/CONSTRUCTION_HISTORY_2009/VOL3/Weber-Jutta_layouted.pdf |website=Proceedings of the Third International Congress on Construction History |access-date=2020-08-09 |archive-date=2017-08-08 |archive-url=https://web.archive.org/web/20170808203018/http://www.bma.arch.unige.it/internos/PDF/CONSTRUCTION_HISTORY_2009/VOL3/Weber-Jutta_layouted.pdf |url-status=dead }}
• Dell Bridge (footbridge),Alamy: [http://www.alamy.com/stock-photo-the-iconic-port-sunlight-dell-bridge-and-on-the-other-side-the-lyceum-89416210.html para] {{Webarchive|url=https://web.archive.org/web/20210603004313/http://www.alamy.com/stock-photo-the-iconic-port-sunlight-dell-bridge-and-on-the-other-side-the-lyceum-89416210.html |date=2021-06-03 }}, accessdate: March 4, 2017 1894, Port Sunlight, Wirral, England.
• Puente Nuevo,Gimeno and Gutierrez. pag.122 1903, Murcia, Spain, civil engineer José María Ortiz
• Viaduc d'Austerlitz, 1903-4, Paris, engineers Louis Biette and Fulgence Bienvenüe, architect Jean-Camille Formigé
• 16th Street Bridge,{{cite news|last=Bisbort|first=Alan|title=The Draw of Bridges|newspaper=The Washington Post|date=April 10, 1992|pages=A8–A9}} 1905-10, Washington DC, the first parabolic arched bridge in the US.
• Victoria Falls Bridge,The Victoria Falls Bridge: [http://www.tothevictoriafalls.com/vfpages/devel/bridge.html To the Victoria Falls - The Victoria Falls Bridge], accessdate: March 2, 2017Livingstone News: [http://www.livingstonetourism.com/blog/a-brief-history-of-the-victoria-falls-bridge/ central], accessdate: March 2, 2017Best Bridge Africa Victoria Falls Bridge: [http://bestbridge.net/Africa_en/victoria-falls-bridge.html Best Bridge Africa Victoria Falls Bridge], accessdate: March 2, 2017 1904-5, Victoria Falls, Zimbabwe
• Memorial Bridge,Arch Bridges on Waymarking.com: [http://www.waymarking.com/waymarks/WMBBZB_Memorial_Bridge_Springfield_West_Springfield_MA Memorial Bridge - Springfield/West Springfield, MA - Arch Bridges on Waymarking.com], accessdate: March 4, 2017 1920, Springfield, Massachusetts
• Tyne Bridge,{{cite web |title=Tyne Bridge |website=BBC Inside Out |date=24 September 2014 |url=https://www.bbc.co.uk/insideout/northeast/series3/tyne_bridge_northeasthistory.shtml |access-date=3 March 2017}} 1928, Newcastle upon Tyne, UK.
• Cape Creek Bridge,{{cite book |title= Style & Vernacular: A Guide to the Architecture of Lane County, Oregon |publisher= Western Imprints, The Press of the Oregon Historical Society |year= 1983 |page= 151 |isbn= 978-0-87595-085-3}} 1931, Lane County, Oregon, United States, engineer Conde McCullough
• Bayonne Bridge,The New York Times: [https://www.nytimes.com/2014/03/23/nyregion/going-up-a-bridge-makes-way-for-bigger-ships.html?_r=0 parab], accessdate: March 3, 2017 1931, Bayonne, New Jersey, Othmar Ammann and architect Cass Gilbert
• Bixby Creek Bridge,Bixby Creek Bridge on Highway One from the Pat Hathaway Photo Collection: [http://www.caviews.com/bixby.htm Bixby Creek Bridge on Highway One from the Pat Hathaway Photo Collection], accessdate: March 6, 2017 1931-2, Big Sur, California, highway engineer C. H. Purcell and engineer F. W. Panhorst
• Balclutha Road Bridge,Heritage New Zealand: [http://www.heritage.org.nz/the-list/details/5180 www.heritage.org.nz/the-list/details/5180], accessdate: March 3, 2017 1933-35, Balclutha, South Otago, New Zealand
• Juscelino Kubitschek Bridge, 2002, Brasilia, Brazil, Alexandre Chan and structural engineer Mário Vila Verde

• Arch bridge
• Catenary arch
• Catenoid
• Dome
• Gothic arch
• Gothic architecture
• Lancet arch
• Lancet window
• Mathematics and architecture
• Musgum mud huts
• Nubian vault
• Overhead line
• Simple suspension bridge
• Steel catenary riser
• Stressed ribbon bridge
• Suspension bridge
• Truss arch bridge
• Vault (architecture)
• Voussoir

{{reflist}}

{{Commons category-multi|Parabolic arches|Arches}}

• [http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page5.htm On why suspension bridges are parabolic]
• [http://www.worldarchitecturemap.org/tags/name/parabolic-arch Many parabolic arches]
• [http://mathforum.org/library/drmath/view/65729.html One the difference between a parabola and a catenary]
• [http://arch-re-review.blogspot.com/2010/09/catenary-and-parabola.html On parabolic versus catenary arches]
• [http://www.cs.rutgers.edu/~mcgrew/dimacs/slides/Amadeo_Huylebrouck.pdf On a variety of curves, parabolas, catenaries, hyperbolas, and ellipses]
• [https://www.youtube.com/watch?v=-GVKzvv7XMc YouTube video]
• [https://www.youtube.com/watch?v=0HRA_y_hfUQ Another Youtube video]

• {{Citation

| last = Gimeno Díaz de Atauri

| first = Jorge

|author2=Gutiérrez Andrés, Juan

| title = El Puente de la Pólvora y otros puentes

| url =https://openlibrary.org/b/OL23850640M/El_Puente_de_la_Pólvora_y_otros_puentes

| year = 2001

| publisher = Colegio de Ingenieros de Caminos, Canales y Puentes. Demarcación Murcia

| place = Murcia

| isbn = 978-84-607-3209-9

}}

{{clear}}

Category:Parabolas

Category:Arches and vaults

Category:Architectural history