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 vaul ...

in the shape of a parabola.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 parabola 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 polynomial ...

while a catenary is the

hyperbolic cosine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...

, , a sum of two exponential functions. 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 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 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.

Uses

In nature

A hen's egg can be fairly well described as two different paraboloids connected by part of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

.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 Tāq Kasrā ( ar, طاق كسرى, translit=ṭāq kisrā), also transcribed as ''Taq-i Kisra'' or ''Taq-e Kesra'' ( fa, طاق کسری, romanized: ''tâğe kasrâ'') or Ayvān-e Kesrā ( fa, ایوان خسرو, translit=Eivâne Xosrow, links=, ...

(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 Antoni Gaudí i Cornet (; ; 25 June 1852 – 10 June 1926) was a Catalan architect from Spain known as the greatest exponent of Catalan Modernism. Gaudí's works have a highly individualized, ''sui generis'' style. Most are located in Barcel ...

, 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 Sheffield Winter Garden is a large temperate glasshouse located in the city of Sheffield in South Yorkshire. It is one of the largest temperate glasshouses to be built in the UK during the last hundred years, and the largest urban glasshous ...

are often closer to true catenaries. * Palau Güell, 1886–88, Barcelona, 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 Wrocław Market Hall ( pl, Hala Targowa we Wrocławiu, german: Breslauer Markthalle) is a food hall in central Wrocław, Poland. Designed by Richard Plüddemann, it was built between 1906-08 as the ''Breslauer Markthalle Nr 1'', when the city was ...

, 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 The Sant Cugat Museum ( ca, Museu de Sant Cugat, es, Museo de Sant Cugat) was opened on 23 April 2003; its mission was to oversee the conservation and dissemination of the historic, artistic and cultural heritage of Sant Cugat del Vallès in Catal ...

, 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, 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 caree ...

Church of Saint Francis of Assisi The Church of Saint Francis of Assisi (''Igreja de Sao Francisco de Assis'', commonly known as the ''Igreja da Pampulha'') is a chapel in Pampulha region of Belo Horizonte, in the state of Minas Gerais, southeastern Brazil. It was designed by t ...

, 1943, Pampulha,

Belo Horizonte Belo Horizonte (, ; ) is the sixth-largest city in Brazil, with a population around 2.7 million and with a metropolitan area of 6 million people. It is the 13th-largest city in South America and the 18th-largest in the Americas. The metropol ...

, 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 anchor ...

, Mexico, Enrique de la Mora * Cosmic Rays Pavilion, 1951, Felix Candela with Jorge González Reyna, UNAM, 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 is a memorial park in the center of Hiroshima, Japan. It is dedicated to the legacy of Hiroshima as the first city in the world to suffer a Atomic bombings of Hiroshima and Nagasaki, nuclear attack at the end of World War II, and to the memorie ...

Kenzō Tange was a Japanese architect, and winner of the 1987 Pritzker Prize for architecture. He was one of the most significant architects of the 20th century, combining traditional Japanese styles with modernism, and designed major buildings on five cont ...

. * Dorton Arena 1957, Raleigh * St Mary's Star of the Sea Cathedral, 1958–62, Darwin, Australia, architect Ian Ferrier *

Toast Rack (building) The Toast Rack, or formerly known as the Hollings Building, is a Modernist building in Fallowfield, Manchester, England. The building was completed in 1960 as the Domestic Trades College, became part of Manchester Polytechnic then Manchester Metro ...

(originally Domestic Trades College, Manchester Polytechnic) 1960, Fallowfield, Manchester, United Kingdom, city architect,

Leonard Cecil Howitt Leonard Cecil Howitt (1896 – 1964) – often referred to as L. C. Howitt – served in both World Wars and was Manchester City Council's chief architect from 1946 until he retired in 1961. Life Leonard Howitt was born on 27 December 1896 in Is ...

* Theme Building, 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 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 St. Louis () is the second-largest city in Missouri, United States. It sits near the confluence of the Mississippi River, Mississippi and the Missouri Rivers. In 2020, the city proper had a population of 301,578, while the Greater St. Louis, ...

, 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 Brookfield Place (formerly BCE Place), is an office complex in downtown Toronto, Ontario, Canada, comprising the block bounded by Yonge Street, Wellington Street West, Bay Street, and Front Street. The complex contains of office space, and con ...

, 1992, Toronto, 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, Spain, Santiago Calatrava * L'Oceanogràfic aquarium, 2003, Valencia, Spain, Felix Candela. *

(catenary), 2003, Sheffield, 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 Buro Happold (previously ''BuroHappold Engineering'') is a British professional services firm that provides engineering consultancy, design, planning, project management, and consulting services for buildings, infrastructure, and the environment ...

*Fjordenhus, 2018,

Vejle Fjord Vejle Fjord is an East Jutland-type fjord in Denmark. It is approximately in length, and it stretches east from the town of Vejle at its head to the towns of Trelde Næs on the south side and Juelsminde on the north side at the mouth. The fjord' ...

, Denmark, Olafur Eliasson and Sebastian Behmann

Bridges

Bridges have used a variety of arches since ancient times, sometimes in very flat

segmental arch A segmental arch is a type of arch with a circular arc of less than 180 degrees. It is sometimes also called a scheme arch. The segmental arch is one of the strongest arches because it is able to resist thrust. To prevent failure, a segmental arc ...

ed 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 bridge A suspension bridge is a type of bridge in which the deck (bridge), deck is hung below suspension wire rope, cables on vertical suspenders. The first modern examples of this type of bridge were built in the early 1800s. Simple suspension bridg ...

s 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 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, Portugal, a railway bridge built in 1877. * Garabit viaduct, near

Ruynes-en-Margeride Ruynes-en-Margeride is a commune in the Cantal department in south-central France. Population See also *Communes of the Cantal department The following is a list of the 246 communes of the Cantal department of France. Intercommunaliti ...

, Cantal, France, designed by Gustave Eiffel, and built between 1882 and 1884. * Dell Bridge (footbridge),Alamy
para
accessdate: March 4, 2017 1894, Port Sunlight, Wirral, England. *

Puente Nuevo The ''Puente Nuevo'' (, "New Bridge") is the newest and largest of three bridges that span the chasm that carries the Guadalevín River and divides the city of Ronda, in southern Spain. Completed in 1793, the architect was José Martin de Aldehu ...

,Gimeno and Gutierrez. pag.122 1903, Murcia, 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 Jean-Camille Formigé (1845-1926) was a French architect during the French Third Republic.16th Street Bridge, 1905-10, Washington DC, the first parabolic arched bridge in the US. *

Victoria Falls Bridge The Victoria Falls Bridge crosses the Zambezi River just below the Victoria Falls and is built over the Second Gorge of the falls. As the river forms the border between Zimbabwe and Zambia, the bridge links the two countries and has border post ...

,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, Zimbabwe * 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 Springfield is a city in the Commonwealth of Massachusetts, United States, and the seat of Hampden County. Springfield sits on the eastern bank of the Connecticut River near its confluence with three rivers: the western Westfield River, the ...

* Tyne Bridge, 1928, Newcastle upon Tyne, UK. * Cape Creek Bridge, 1931, Lane County, Oregon, 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 Bayonne ( ) is a city in Hudson County, New Jersey, United States. Located in the Gateway Region, Bayonne is situated on a peninsula located between Newark Bay to the west, the Kill Van Kull to the south, and New York Bay to the east. As of ...

Othmar Ammann Othmar Hermann Ammann (March 26, 1879 – September 22, 1965) was a Swiss-American civil engineer whose bridge designs include the George Washington Bridge, Verrazzano-Narrows Bridge, and Bayonne Bridge. He also directed the planning and constru ...

and architect Cass Gilbert * 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 Sur ha ...

, 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 South Otago lies in the south east of the South Island of New Zealand. As the name suggests, it forms the southernmost part of the geographical region of Otago. The exact definition of the area designated as South Otago is imprecise, as the area is ...

, 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