A parabolic arch is an

arch An arch is a vertical curved structure that spans an elevated space and may or may not support the weight above it, or in case of a horizontal arch like an arch dam, the hydrostatic pressure against it. Arches may be synonymous with vau ...

in the shape of a

parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...

.Article about parabolic arch by The Free Dictionary
Parabolic arch , Article about parabolic arch by The Free Dictionary
accessdate: March 2, 2017 In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.

Description

The mathematics

While a parabolic arch may resemble a catenary arch, a

is a

quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomi ...

while a

catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficia ...

is the hyperbolic cosine, , a sum of two

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...

s. One parabola is , and hyperbolic cosine is . The curves are unrelated.

The line of thrust

Unlike a catenary arch, the parabolic

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 A bridge is a structure built to span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, which is usually somethi ...

design, where long spans are needed.The Design of Prestressed Concrete Bridges/Chapter 17 The Design And Construction Of Arches
/ref>

Compared to catenary arches

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

. 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.

Uses

In nature

hen's egg Humans and human ancestors have scavenged and eaten animal eggs for millions of years. Humans in Southeast Asia had domesticated chickens and harvested their eggs for food by 1,500 BCE. The most widely consumed eggs are those of fowl, especia ...

can be fairly well described as two different

paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every pla ...

s connected by part of an ellipse.Google Books
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

Architectural examples

Interieur van de centrale hal met decoraties - Utrecht - 20349663 - RCE

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 Iraq,; ku, عێراق, translit=Êraq officially the Republic of Iraq, '; ku, کۆماری عێراق, translit=Komarî Êraq is a country in Western Asia. It is bordered by Turkey to Iraq–Turkey border, the north, Iran to Iran–Iraq ...

), the largest single-span vault of unreinforced

brickwork Brickwork is masonry produced by a bricklayer, using bricks and mortar. Typically, rows of bricks called ''courses'' are laid on top of one another to build up a structure such as a brick wall. Bricks may be differentiated from blocks by ...

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, 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 Oscar Ribeiro de Almeida Niemeyer Soares Filho (15 December 1907 – 5 December 2012), known as Oscar Niemeyer (), was a Brazilian architect considered to be one of the key figures in the development of modern architecture. Niemeyer was ...

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 Barcelona ( , , ) is a city on the coast of northeastern Spain. It is the capital and largest city of the autonomous community of Catalonia, as well as the second most populous municipality of Spain. With a population of 1.6 million within ...

, where Antonio Gaudi 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 Wilhelm E. Heinrich Küster (16 August 1870 in Hanover, 1 Juli 1956 in Görlitz) was a German architect and town officer (1909–1933 as ''Stadtbaurat'' of Görlitz). After his architectural studies (1890–1895) in high technical school in Hanov ...

, internal structure * Celler modernista, 1921, part of Sant Cugat Museum, Catalonia, Spain,

Cèsar Martinell i Brunet Cèsar Martinell i Brunet (Valls, 24 December 1888 - Barcelona, 19 November 1973) was a Catalan '' modernista'' architect. He was part of the small and selected group of architects that were connected to Antoni Gaudí, his most important teacher. ...

* Pinell de Brai Cooperative Winery, 1922,

Pinell de Brai El Pinell de Brai is a municipality in the ''comarca'' of la Terra Alta in Catalonia, Europe. This town is located in a small plain surrounded by mountains including Serra de Pàndols and Serra de Cavalls. This area suffered badly during the Ba ...

, Catalonia, Spain,

* Former main post office, 1919–24, Utrecht, main hall, designed by J. Crouwel Jr. * Priory Chapel at Saint Louis Abbey - July 2013

St. Engelbert,

Cologne Cologne ( ; german: Köln ; ksh, Kölle ) is the largest city of the German western state of North Rhine-Westphalia (NRW) and the fourth-most populous city of Germany with 1.1 million inhabitants in the city proper and 3.6 millio ...

, 1928−1932, by

Dominikus Böhm Dominikus Böhm (23 October 1880 – 6 August 1955) was a German architect specializing in churches. He built churches in Cologne, the Ruhr area, Swabia, and Hesse. Many of his buildings are examples of Brick Expressionism. Life and career B ...

* Church of Saint Francis of Assisi, 1943, Pampulha, Belo Horizonte, Brazil, Oscar Neimeyer. * Church of La Purísima, 1943,

Monterrey Monterrey ( , ) is the capital and largest city of the northeastern state of Nuevo León, Mexico, and the third largest city in Mexico behind Guadalajara and Mexico City. Located at the foothills of the Sierra Madre Oriental, the city is ancho ...

, Mexico, Enrique de la Mora * Cosmic Rays Pavilion, 1951, Felix Candela with

Jorge González Reyna Jorge is a Spanish and Portuguese given name. It is derived from the Greek name Γεώργιος (''Georgios'') via Latin ''Georgius''; the former is derived from (''georgos''), meaning "farmer" or "earth-worker". The Latin form ''Georgius' ...

UNAM The National Autonomous University of Mexico ( es, Universidad Nacional Autónoma de México, UNAM) is a public research university in Mexico. It is consistently ranked as one of the best universities in Latin America, where it's also the bigges ...

, Mexico City * St Leonard's Church,Interior of St Leonard's church, St... (C) Julian P Guffogg :: Geograph Britain and Ireland
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
para
accessdate: March 3, 2017 1953-61, St Leonards on Sea, United Kingdom. * Memorial Cenotaph, 1952, Hiroshima Peace Memorial Park, Kenzō Tange. *

Dorton Arena J. S. Dorton Arena is a 7,610-seat multi-purpose arena located in Raleigh, North Carolina, on the grounds of the North Carolina State Fair. It opened in 1952. Architect Maciej Nowicki of the North Carolina State University Department of Architect ...

1957, Raleigh * St Mary's Star of the Sea Cathedral, 1958–62,

Darwin Darwin may refer to: Common meanings * Charles Darwin (1809–1882), English naturalist and writer, best known as the originator of the theory of biological evolution by natural selection * Darwin, Northern Territory, a territorial capital city i ...

, Australia, architect Ian Ferrier * Toast Rack (building) (originally Domestic Trades College, Manchester Polytechnic) 1960,

Fallowfield Fallowfield is a suburb of Manchester, England, with a population at the 2011 census of 15,211. Historically in Lancashire, it lies south of Manchester city centre and is bisected east–west by Wilmslow Road and north–south by W ...

, Manchester, United Kingdom, city architect, Leonard Cecil Howitt *

Theme Building The Theme Building is a structure at Los Angeles International Airport (LAX), considered an architectural example of the Space Age design style. Influenced by " Populuxe" architecture, it is an example of the Mid-century modern design movement ...

, 1961,

Los Angeles International Airport Los Angeles International Airport , commonly referred to as LAX (with each letter pronounced individually), is the primary international airport serving Los Angeles, California and its surrounding metropolitan area. LAX is located in the W ...

, Pereira & Luckman Architects,

Paul Williams Paul Williams may refer to: Authors * Paul O. Williams (1935–2009), American science-fiction author and poet * Paul L. Williams (author) (born 1944), FBI consultant, journalist * Paul Williams (journalist) (1948–2013), American founder of mu ...

and

Welton Becket Welton David Becket (August 8, 1902 – January 16, 1969) was an American modern architect who designed many buildings in Los Angeles, California. Biography Becket was born in Seattle, Washington and graduated from the University of Washingt ...

* 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 Eero Saarinen (, ; August 20, 1910 – September 1, 1961) was a Finnish-American architect and industrial designer noted for his wide-ranging array of designs for buildings and monuments. Saarinen is best known for designing the General Motor ...

* Allen Lambert Galleria, 1992,

Toronto Toronto ( ; or ) is the capital city of the Provinces and territories of Canada, Canadian province of Ontario. With a recorded population of 2,794,356 in 2021, it is the List of the largest municipalities in Canada by population, most pop ...

, Canada, Santiago Calatrava * L'Umbracle (catenary shade house)Jean McConochie photos at pbase.com
para
accessdate: March 3, 2017 2001, '' Ciutat de les Arts i les Ciències'' (City of Arts and Sciences),

Valencia Valencia ( va, València) is the capital of the autonomous community of Valencia and the third-most populated municipality in Spain, with 791,413 inhabitants. It is also the capital of the province of the same name. The wider urban area al ...

Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = '' Plus ultra'' ( Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , ...

, Santiago Calatrava * L'Oceanogràfic aquarium, 2003,

, Spain, Felix Candela. * Sheffield Winter Garden (catenary), 2003,

Sheffield Sheffield is a city status in the United Kingdom, city in South Yorkshire, England, whose name derives from the River Sheaf which runs through it. The city serves as the administrative centre of the City of Sheffield. It is Historic counties o ...

, UK,

Pringle Richards Sharratt Pringle Richards Sharratt is an architectural firm that was formed in 1996 by John Pringle, Penny Richards and Ian Sharratt. Based in London, the practice has worked on public buildings, art galleries, museums, libraries, archives, university a ...

Architects and Buro Happold *Fjordenhus, 2018, Vejle Fjord, Denmark, Olafur Eliasson and Sebastian Behmann

Bridges

Bridges 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 A through arch bridge, also known as a through-type arch bridge, is a bridge that is made from materials such as steel or reinforced concrete, in which the base of an arch structure is below the deck but the top rises above it. It can either be lo ...

) first appeared in the 1870s, and have been used occasionally ever since; examples include: * Maria Pia Bridge,

Gustave Eiffel Alexandre Gustave Eiffel (born Bonickhausen dit Eiffel; ; ; 15 December 1832 – 27 December 1923) was a French civil engineer. A graduate of École Centrale des Arts et Manufactures, he made his name with various bridges for the French railway ...

and

Théophile Seyrig François Gustave Théophile Seyrig (19 February 1843 to 5 July 1923) was a German engineer, best known for designing bridges. On 6 October 1868, he founded ''Eiffel and Company'' with Gustave Eiffel. Seyrig contributed to the Eiffel and Company ...

Porto Porto or Oporto () is the second-largest city in Portugal, the capital of the Porto District, and one of the Iberian Peninsula's major urban areas. Porto city proper, which is the entire municipality of Porto, is small compared to its metropo ...

Portugal Portugal, officially the Portuguese Republic, In recognized minority languages of Portugal: :* mwl, República Pertuesa is a country located on the Iberian Peninsula, in Southwestern Europe, and whose territory also includes the Macaronesian ...

, a railway bridge built in 1877. * Garabit viaduct, near Ruynes-en-Margeride,

Cantal Cantal (; oc, Cantal or ) is a department in the Auvergne-Rhône-Alpes region of France, with its prefecture in Aurillac. Its other principal towns are Saint-Flour (the episcopal see) and Mauriac; its residents are known as Cantalians ( ...

France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan ar ...

, designed by

, and built between 1882 and 1884. * Dell Bridge (footbridge),Alamy
para
accessdate: March 4, 2017 1894,

Port Sunlight Port Sunlight is a model village and suburb in the Metropolitan Borough of Wirral, Merseyside. It is located between Lower Bebington and New Ferry, on the Wirral Peninsula. Port Sunlight was built by Lever Brothers to accommodate workers in its ...

, Wirral, England. * Puente Nuevo,Gimeno and Gutierrez. pag.122 1903,

Murcia Murcia (, , ) is a city in south-eastern Spain, the Capital (political), capital and most populous city of the autonomous community of the Region of Murcia, and the List of municipalities of Spain, seventh largest city in the country. It has a ...

, Spain, civil engineer José María Ortiz *

Viaduc d'Austerlitz Viaduc d'Austerlitz ( en, Austerlitz Viaduct) is a single-deck, steel arch, rail bridge that crosses the Seine in Paris. Its usage is solely dedicated to Line 5 of the Paris Métro. It links Gare d'Austerlitz on the Rive Gauche to Quai de la Ra ...

, 1903-4, Paris, engineers Louis Biette and Fulgence Bienvenüe, architect Jean-Camille Formigé * 16th Street Bridge, 1905-10,

Washington DC ) , image_skyline = , image_caption = Clockwise from top left: the Washington Monument and Lincoln Memorial on the National Mall, United States Capitol, Logan Circle, Jefferson Memorial, White House, Adams Morgan, ...

, the first parabolic arched bridge in the US. * Victoria Falls Bridge,The Victoria Falls Bridge
To the Victoria Falls - The Victoria Falls Bridge
accessdate: March 2, 2017Livingstone News
central
accessdate: March 2, 2017Best Bridge Africa Victoria Falls Bridge

accessdate: March 2, 2017 1904-5,

Victoria Falls Victoria Falls (Lozi language, Lozi: ''Mosi-oa-Tunya'', "The Smoke That Thunders"; Tonga language (Zambia and Zimbabwe), Tonga: ''Shungu Namutitima'', "Boiling Water") is a waterfall on the Zambezi River in southern Africa, which provides hab ...

Zimbabwe Zimbabwe (), officially the Republic of Zimbabwe, is a landlocked country located in Southeast Africa, between the Zambezi and Limpopo Rivers, bordered by South Africa to the south, Botswana to the south-west, Zambia to the north, and Mozam ...

* Memorial Bridge,Arch Bridges on Waymarking.com
Memorial Bridge - Springfield/West Springfield, MA - Arch Bridges on Waymarking.com
accessdate: March 4, 2017 1920, Springfield, Massachusetts * Tyne Bridge, 1928,

Newcastle upon Tyne Newcastle upon Tyne ( RP: , ), or simply Newcastle, is a city and metropolitan borough in Tyne and Wear, England. The city is located on the River Tyne's northern bank and forms the largest part of the Tyneside built-up area. Newcastle is a ...

, UK. * Cape Creek Bridge, 1931,

Lane County, Oregon Lane County is one of the 36 counties in the U.S. state of Oregon. As of the 2020 census, the population was 382,971, making it the fourth-most populous county in Oregon. The county seat is Eugene. It is named in honor of Joseph Lane, Orego ...

, United States, engineer

Conde McCullough Conde Balcom McCullough (May 30, 1887 – May 6, 1946) was an American civil engineer who is primarily known for designing many of Oregon's coastal bridges on U.S. Route 101. The native of South Dakota worked for the Oregon Department of Tr ...

* Bayonne Bridge,The New York Times
parab
accessdate: March 3, 2017 1931, Bayonne, New Jersey, Othmar Ammann and

architect An architect is a person who plans, designs and oversees the construction of buildings. To practice architecture means to provide services in connection with the design of buildings and the space within the site surrounding the buildings that h ...

Cass Gilbert Cass Gilbert (November 24, 1859 – May 17, 1934) was an American architect. An early proponent of skyscrapers, his works include the Woolworth Building, the United States Supreme Court building, the state capitols of Minnesota, Arkansas and W ...

* Bixby Creek Bridge,Bixby Creek Bridge on Highway One from the Pat Hathaway Photo Collection
Bixby Creek Bridge on Highway One from the Pat Hathaway Photo Collection
accessdate: March 6, 2017 1931-2,

Big Sur Big Sur () is a rugged and mountainous section of the Central Coast of California between Carmel and San Simeon, where the Santa Lucia Mountains rise abruptly from the Pacific Ocean. It is frequently praised for its dramatic scenery. Big S ...

, California, highway engineer C. H. Purcell and engineer F. W. Panhorst *

Balclutha Road Bridge The Balclutha Road Bridge, which spans the Clutha River in Balclutha, South Otago, New Zealand, is one of the best-known road bridges in New Zealand's South Island. The bridge is often simply known as "The Clutha Bridge", despite the fact that ...

,Heritage New Zealand
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 Alexandre Chan (born ) is a Brazilian architect best known for designing the President Juscelino Kubitschek Bridge in Brasília, Brazil with structural engineer Mario Vila Verde, and the Bridge of Knowledge in Rio de Janeiro. He has been describ ...

and structural engineer Mário Vila Verde

References

External links

On why suspension bridges are parabolic

Many parabolic arches

On a variety of curves, parabolas, catenaries, hyperbolas, and ellipses

YouTube video

Another Youtube video

Bibliography

* {{clear Parabolas Arches and vaults Architectural history Catholic architecture