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{{Short description|A catenary arch is an architectural pointed arch that follows an inverted catenary curve}} {{Short description|Architectural pointed arch that follows an inverted catenary curve}}


] catenary arch]] ] catenary arch]]
] (left) and a catenary arch, also a catenary curve (right). One points up, and one points down, but the curves are the same.]] ] (left) and a catenary arch, also a catenary curve (right). One points up, and one points down, but the curves are the same.]]


A '''catenary arch''' is a type of ] arch that follows an inverted ]. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of ]s and ]es in stone vaulted ] ]s and in ] ]s.<ref name="American Scientist">{{cite web|url=https://www.americanscientist.org/issues/pub/2011/3/the-perfect-dome|title=Letter to the Editors: The Perfect Dome|author=Handy, Richard L.|date=May 2011|work=American Scientist|access-date=2016-04-17|archive-url=https://web.archive.org/web/20160423193426/http://www.americanscientist.org/issues/pub/2011/3/the-perfect-dome|archive-date=2016-04-23|url-status=dead}}</ref> It is not a ]. A '''catenary arch''' is a type of ] arch that follows an inverted ]. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of ]s and ]es in stone vaulted ] ]s and in ] ]s.<ref name="American Scientist">{{cite web|url=https://www.americanscientist.org/issues/pub/2011/3/the-perfect-dome|title=Letter to the Editors: The Perfect Dome|author=Handy, Richard L.|date=May 2011|work=American Scientist|access-date=2016-04-17|archive-url=https://web.archive.org/web/20160423193426/http://www.americanscientist.org/issues/pub/2011/3/the-perfect-dome|archive-date=2016-04-23|url-status=dead}}</ref> It is not a ], although the non-] curves used in arch designs (parabola, catenary, and ]) look similar, and match at shallow profiles, so a catenary is often misclassified as a parabola{{sfn | Bradley | Gohnert | 2022 }} (per ], "the chain fits its parabola almost perfectly"{{sfn|Osserman|2010|p=220}}).


==In history== ==In history==
], holding a hanging ], which forms a catenary curve]] ], holding a hanging ], which forms a catenary curve]]


The 17th-century scientist ] wrote: "''Ut pendet continuum flexile, sic stabit contiguum rigidum inversum''", or, "As hangs a flexible cable so, inverted, stand the touching pieces of an arch." <ref name="The enigma of Robert Hooke">{{cite web|url=https://quantumfrontiers.com/2015/08/31/the-enigma-of-robert-hooke/|title=The enigma of Robert Hooke|author=|date=31 August 2015|work=Quantum Frontiers|publisher=Institute for Quantum Information and Matter, California Institute of Technology}}</ref> The 17th-century scientist ] wrote: "''Ut pendet continuum flexile, sic stabit contiguum rigidum inversum''", or, "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."<ref name="The enigma of Robert Hooke">{{cite web|url=https://quantumfrontiers.com/2015/08/31/the-enigma-of-robert-hooke/|title=The enigma of Robert Hooke|author=|date=31 August 2015|work=Quantum Frontiers|publisher=Institute for Quantum Information and Matter, California Institute of Technology}}</ref>


A note written by ] in 1788 reads, "I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientific work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the ] is in perfect equilibrium".<ref>{{cite book|last1=Jefferson|first1=Thomas|title=Memoir, Correspondence, and Miscellanies, from the Papers of Thomas Jefferson, Volume 2|date=1830|publisher=Gray and Bowen|location=Boston|page=416|url=https://books.google.com/books?id=Qjw-AAAAYAAJ&pg=PA416|language=en}}</ref> A note written by ] in 1788 reads, "I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientific work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the ] is in perfect equilibrium".<ref>{{cite book|last1=Jefferson|first1=Thomas|title=Memoir, Correspondence, and Miscellanies, from the Papers of Thomas Jefferson, Volume 2|date=1830|publisher=Gray and Bowen|location=Boston|page=416|url=https://books.google.com/books?id=Qjw-AAAAYAAJ&pg=PA416|language=en}}</ref>
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Architecturally, a catenary arch has the ability to withstand the weight of the material from which it is constructed, without collapsing.<ref name="St. Louis Gateway Arch">{{cite web|url=http://www.enchantedlearning.com/history/us/monuments/stlouisarch/|title=St. Louis Gateway Arch|author=|date=|work=enchantedlearning.com|accessdate=27 April 2016}}</ref><ref name="strath.ac.uk">{{cite web|url=http://personal.strath.ac.uk/j.wood/Biomimetics/Engineering%20Areas/structures/PROP%201_files/PROP%201.htm|title=Building an arch that can stand up by itself|author=|date=|work=strath.ac.uk|accessdate=27 April 2016}}</ref> For an arch of uniform density and thickness, supporting only its own weight, the catenary is the ideal curve.<ref name="The inverted catenary arch">{{cite web|url=http://zonedome.com/?p=167|title=The inverted catenary arch|author=|date=|work=zonedome.com|accessdate=27 April 2016}}</ref> Architecturally, a catenary arch has the ability to withstand the weight of the material from which it is constructed, without collapsing.<ref name="St. Louis Gateway Arch">{{cite web|url=http://www.enchantedlearning.com/history/us/monuments/stlouisarch/|title=St. Louis Gateway Arch|author=|date=|work=enchantedlearning.com|accessdate=27 April 2016}}</ref><ref name="strath.ac.uk">{{cite web|url=http://personal.strath.ac.uk/j.wood/Biomimetics/Engineering%20Areas/structures/PROP%201_files/PROP%201.htm|title=Building an arch that can stand up by itself|author=|date=|work=strath.ac.uk|accessdate=27 April 2016}}</ref> For an arch of uniform density and thickness, supporting only its own weight, the catenary is the ideal curve.<ref name="The inverted catenary arch">{{cite web|url=http://zonedome.com/?p=167|title=The inverted catenary arch|author=|date=|work=zonedome.com|accessdate=27 April 2016}}</ref>


Catenary arches are strong because they redirect the vertical force of gravity into compression forces pressing along the arch's curve. In a uniformly loaded catenary arch, the ] runs through its center.<ref name="Build an arch that can stand up by itself">{{cite web|url=http://www.exploratorium.edu/files/cmp/exnet/exhibits/group1/catenary_arch/media/catenary_g.pdf|title=Build an arch that can stand up by itself|author=|date=|website=|accessdate=}}</ref><ref name="Getting the arch back into architecture">{{cite web|url=http://publications.lib.chalmers.se/records/fulltext/203465/203465.pdf|title=Getting the arch back into architecture|author=Karl Robin Nilsson|date=|website=|accessdate=}}</ref> Catenary arches are strong because they redirect the vertical force of gravity into compression forces pressing along the arch's curve. In a uniformly loaded catenary arch, the ] runs through its center.<ref name="Build an arch that can stand up by itself">{{cite web|url=http://www.exploratorium.edu/files/cmp/exnet/exhibits/group1/catenary_arch/media/catenary_g.pdf|title=Build an arch that can stand up by itself|author=|date=|website=|accessdate=|archive-date=2016-04-25|archive-url=https://web.archive.org/web/20160425032122/http://www.exploratorium.edu/files/cmp/exnet/exhibits/group1/catenary_arch/media/catenary_g.pdf|url-status=dead}}</ref><ref name="Getting the arch back into architecture">{{cite web|url=http://publications.lib.chalmers.se/records/fulltext/203465/203465.pdf|title=Getting the arch back into architecture|author=Karl Robin Nilsson|date=|website=|accessdate=}}</ref>


This principle has been employed architecturally to create arched structures that follow exactly, and in a visibly apparent way, the form of an inverted catenary. A significant early example of this is the arch of ]. The catenary, spun 180 degrees, forms the structure of simple domed building such as the ]s of the ], Ireland. This principle has been employed architecturally to create arched structures that follow exactly, and in a visibly apparent way, the form of an inverted catenary. A significant early example of this is the arch of ]. The catenary, spun 180 degrees, forms the structure of simple domed building such as the ]s of the ], Ireland.
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The principle of the catenary is also the underlying factor in the much more complex architectural systems of the Medieval and Renaissance architecture. Buildings that have heavy roofs that are arched in shape and deliver a strong outward thrust must comply with the form of the catenary curve in order not to collapse. This does not imply that the arches themselves are catenary in form, but that the total system of walls or buttresses that support the roof or dome contain a catenary curve, which delivers the downward thrust. The principle of the catenary is also the underlying factor in the much more complex architectural systems of the Medieval and Renaissance architecture. Buildings that have heavy roofs that are arched in shape and deliver a strong outward thrust must comply with the form of the catenary curve in order not to collapse. This does not imply that the arches themselves are catenary in form, but that the total system of walls or buttresses that support the roof or dome contain a catenary curve, which delivers the downward thrust.


In the 15th century ] designed the pointed, octagonal, Gothic dome on ] in a manner that utilised the principle of the catenary arch. In the 17th century, ] designed the dome of ] based directly on a catenary curve. The vaulted roof and buttresses of ], Cambridge, have been discovered to comply with the formula of the catenary arch. In the 15th century ] designed the pointed, octagonal, Gothic dome on ] in a manner that utilised the principle of the catenary arch.{{fact|date=April 2022}} In the 17th century, ] designed the dome of ] based directly on a catenary curve. The vaulted roof and buttresses of ], Cambridge, have been discovered to comply with the formula of the catenary arch.{{fact|date=April 2022}}


{{clear}} {{clear}}


==Examples== ==Examples==
The classification of non-circumferential curves in actual arches is hard. González et al. provide an example of (well-studied) ], where researchers do not agree on classification of the arches or claim the prominence of parabolic arches, while the accurate measurements show that just two of the 23 arches designed by ] are actually parabolic.{{sfn | González | Samper | Herrera | 2018 | pp=174, 184}}

⚫ === Ancient arches ===
⚫ ]]]
The curves close in shape to catenary were used in arches of ] (]) and ] (]).{{sfn|Hourihane|2012|p=130|loc=Catenary}}

⚫ The unfinished ] has a catenary shape.<ref name="Saqqara ostracon">{{cite web|url=http://www.egyptorigins.org/catenarycurve.htm|title=An Ancient Egyptian Catenary Construction Curve|author=|date=1926|website=|accessdate=}}</ref>


===Cathedrals and churches=== ===Cathedrals and churches===
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===Natural arches=== ===Natural arches===
], ]]] ], ]]]


] in the U.S. state of ] has a natural catenary shape, possibly produced by ] in high-stress areas.<ref name="Arctic"/> ] and ], also in Utah, have a catenary shape as well.<ref name="The Dimensions of Kolob Arch">{{cite web|url=http://www.naturalarches.org/archinfo/kolob.htm|title=The Dimensions of Kolob Arch|author=Jay H. Wilbur|date=|website=|accessdate=}}</ref><ref name="Landscape Arch">{{cite web|url=https://www.geocaching.com/geocache/GC20EX7_landscape-arch?guid=24fcdaef-373c-47ab-92d0-3d3b42440f9f |title=Landscape Arch|author=Cincinnati Cache Collectors|date=|website=|accessdate=}}</ref> ] in the U.S. state of ] has a natural catenary shape, possibly produced by ] in high-stress areas.<ref name="Arctic"/> ] and ], also in Utah, have a catenary shape as well.<ref name="The Dimensions of Kolob Arch">{{cite web|url=http://www.naturalarches.org/archinfo/kolob.htm|title=The Dimensions of Kolob Arch|author=Jay H. Wilbur|date=|website=|accessdate=}}</ref><ref name="Landscape Arch">{{cite web|url=https://www.geocaching.com/geocache/GC20EX7_landscape-arch?guid=24fcdaef-373c-47ab-92d0-3d3b42440f9f |title=Landscape Arch|author=Cincinnati Cache Collectors|date=|website=|accessdate=}}</ref>
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===Human-made arches=== ===Human-made arches===


The ] in the American city of ] (]) is an inverted catenary arch.<ref name="Modern Steel Construction">{{cite web|url=http://msc.aisc.org/globalassets/modern-steel/archives/1961-1995/1963v04.pdf|title=Modern Steel Construction|author=|date=|website=|accessdate=}}</ref> The ] in the American city of ] (]) is a catenary arch<ref name="Modern Steel Construction">{{cite web|url=http://msc.aisc.org/globalassets/modern-steel/archives/1961-1995/1963v04.pdf|title=Modern Steel Construction|author=|date=|website=|accessdate=}}</ref> Due to more strength needed at the bottom, the bottom of this arch is wider than the top, so the actual shape is technically a "]".<ref name="How the Gateway arch got its Shape">{{cite web|url=http://www.msri.org/people/staff/osserman/papers/NNJ_v12n2_Osserman_pp167-189.pdf|title=How the Gateway arch got its Shape|author=Robert Osserman |date=|website=|accessdate=}}</ref>

Due to ], the top being thinner than the bottom, its actual shape is technically a "]".<ref name="How the Gateway arch got its Shape">{{cite web|url=http://www.msri.org/people/staff/osserman/papers/NNJ_v12n2_Osserman_pp167-189.pdf|title=How the Gateway arch got its Shape|author=Robert Osserman |date=|website=|accessdate=}}</ref>

⚫ ===Ancient Egyptian===
⚫ ]]]
⚫ The unfinished ] has a catenary shape.<ref name="Saqqara ostracon">{{cite web|url=http://www.egyptorigins.org/catenarycurve.htm|title=An Ancient Egyptian Catenary Construction Curve|author=|date=1926|website=|accessdate=}}</ref>


===High-rises=== ===High-rises===
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===Igloos=== ===Igloos===
{{further|Igloo#Engineering}} {{further|Igloo#Construction}}


] are designed with a catenary arch cross-section.<ref name="What house-builders can learn from igloos">{{cite web|url=http://news.bbc.co.uk/2/hi/uk_news/magazine/7326031.stm|title=What house-builders can learn from igloos |author=Dan Cruickshank|website=|accessdate=1 May 2016}}</ref><ref name="Arctic"/> This shape offers an optimal balance between height and diameter, avoiding the risk of collapsing under the weight of compacted snow.<ref name="Arctic">{{cite journal|last1=Handy|first1=Richard L.|title=The Igloo and the Natural Bridge as Ultimate Structures|journal=Arctic|date=Dec 1973|volume=26|issue=4|pages=276–281|url=http://pubs.aina.ucalgary.ca/arctic/Arctic26-4-276.pdf|doi=10.14430/arctic2926}}</ref> ] are designed with a catenary arch cross-section.<ref name="What house-builders can learn from igloos">{{cite news|url=http://news.bbc.co.uk/2/hi/uk_news/magazine/7326031.stm|title=What house-builders can learn from igloos |author=Dan Cruickshank|website=|date=2 April 2008 |accessdate=1 May 2016}}</ref><ref name="Arctic"/> This shape offers an optimal balance between height and diameter, avoiding the risk of collapsing under the weight of compacted snow.<ref name="Arctic">{{cite journal|last1=Handy|first1=Richard L.|title=The Igloo and the Natural Bridge as Ultimate Structures|journal=Arctic|date=Dec 1973|volume=26|issue=4|pages=276–281|url=http://pubs.aina.ucalgary.ca/arctic/Arctic26-4-276.pdf|doi=10.14430/arctic2926|access-date=2017-01-15|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304201726/http://pubs.aina.ucalgary.ca/arctic/Arctic26-4-276.pdf|url-status=dead}}</ref>


===Other architecture=== ===Other architecture===
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===Monuments=== ===Monuments===
]]] ]]]
In ], the ] has the shape of a catenary arch.<ref name="Taq Kasra">{{cite web|url=http://people.bath.ac.uk/abscjkw/LectureNotes/what-is-a-shell.pdf|title=Taq Kasra|author=Chris J K Williams|date=|website=|accessdate=}}</ref> In ], the ] has the shape of a catenary arch.<ref name="Taq Kasra">{{cite web|url=http://people.bath.ac.uk/abscjkw/LectureNotes/what-is-a-shell.pdf|title=Taq Kasra|author=Chris J K Williams|date=|website=|accessdate=|archive-date=2017-01-04|archive-url=https://web.archive.org/web/20170104105137/http://people.bath.ac.uk/abscjkw/LectureNotes/what-is-a-shell.pdf|url-status=dead}}</ref>


===Airports=== ===Airports===
The roof of ] is a suspended catenary curve.<ref name="Dulles Airport">, {{cite web|url=http://architecture.about.com/od/findphotos/ig/Eero-Saarinen/Dulles-Airport-.htm|title=Dulles Airport|author=Jackie Craven|date=|website=|accessdate=}}</ref> The roof of ] is a suspended catenary curve.<ref name="Dulles Airport">, {{cite web|url=http://architecture.about.com/od/findphotos/ig/Eero-Saarinen/Dulles-Airport-.htm|title=Dulles Airport|author=Jackie Craven|date=|website=|accessdate=|archive-date=2016-05-05|archive-url=https://web.archive.org/web/20160505204914/http://architecture.about.com/od/findphotos/ig/Eero-Saarinen/Dulles-Airport-.htm|url-status=dead}}</ref>


A catenary steel cable system supports the roof of ].<ref name="Shell Structures">{{cite web|url=http://www.coloradomagazineonline.com/Travel/DIA_2010/DIA_2010.htm|title=Denver International Airport|author=|date=|website=|accessdate=}}</ref> A catenary steel cable system supports the roof of ].<ref name="Shell Structures">{{cite web|url=http://www.coloradomagazineonline.com/Travel/DIA_2010/DIA_2010.htm|title=Denver International Airport|author=|date=|website=|accessdate=}}</ref>


===Train stations=== ===Train stations===
]’s ] has a roof in the form of a catenary arch.<ref name="Pennsylvania Station">{{cite web|url=https://www.nytimes.com/2016/01/14/nyregion/penn-stations-5th-redesign-fails-to-charm-some-critics.html?_r=0|title=Penn Station's 5th Redesign Fails to Charm Some Critics|author=David W. Dunlap|date=1926|website=|accessdate=}}</ref> ]’s ] has a roof in the form of a catenary arch.<ref name="Pennsylvania Station">{{cite news|url=https://www.nytimes.com/2016/01/14/nyregion/penn-stations-5th-redesign-fails-to-charm-some-critics.html?_r=0|title=Penn Station's 5th Redesign Fails to Charm Some Critics|author=David W. Dunlap|newspaper=The New York Times |date=1926|website=|accessdate=}}</ref>


===Banks=== ===Banks===
On the ], the building has been remodeled, but still visible is the catenary arch suspending the original building.<ref name="100 Years of the Ninth District Fed - Federal Reserve Bank of Minneapolis">{{cite web|url=https://www.minneapolisfed.org/about/locations/minneapolis/100-years-of-the-ninth-district-fed|title=100 Years of the Ninth District Fed - Federal Reserve Bank of Minneapolis|author=|date=|work=minneapolisfed.org|accessdate=27 April 2016}}</ref> On the ], the building has been remodeled, but still visible is the catenary arch suspending the original building.<ref name="100 Years of the Ninth District Fed - Federal Reserve Bank of Minneapolis">{{cite web|url=https://www.minneapolisfed.org/about/locations/minneapolis/100-years-of-the-ninth-district-fed|title=100 Years of the Ninth District Fed - Federal Reserve Bank of Minneapolis|author=|date=|work=minneapolisfed.org|accessdate=27 April 2016|archive-date=21 November 2019|archive-url=https://web.archive.org/web/20191121200015/https://www.minneapolisfed.org/about/locations/minneapolis/100-years-of-the-ninth-district-fed|url-status=dead}}</ref>


===Mud huts=== ===Mud huts===
]'s ] have a catenary cross-section.<ref name="musgum earth architecture">{{cite web|url=http://www.designboom.com/architecture/musgum-earth-architecture/|title=musgum earth architecture|author=|date=|website=|accessdate=2 May 2016}}</ref><ref name="Architecture Addiction, The Official Blog of">{{cite web|url=http://blog.architectureaddiction.com/index.php/a/2010/02/21/|title=Architecture Addiction, The Official Blog of|author=Katy Purviance|date=|website=|accessdate=2 May 2016}}</ref><ref name="Masonry Design">{{cite web|url=http://masonrydesign.blogspot.com/2010/05/arch-designs-corbels-and-catenaries.html|title=Masonry Design|author=|date=11 May 2010|website=|accessdate=23 December 2016}}</ref> ]'s ] have a catenary cross-section.<ref name="musgum earth architecture">{{cite web|url=http://www.designboom.com/architecture/musgum-earth-architecture/|title=musgum earth architecture|author=|date=18 February 2010|website=|accessdate=2 May 2016}}</ref><ref name="Architecture Addiction, The Official Blog of">{{cite web|url=http://blog.architectureaddiction.com/index.php/a/2010/02/21/|title=Architecture Addiction, The Official Blog of|author=Katy Purviance|date=|website=|accessdate=2 May 2016|archive-date=4 January 2017|archive-url=https://web.archive.org/web/20170104105535/http://blog.architectureaddiction.com/index.php/a/2010/02/21/|url-status=dead}}</ref><ref name="Masonry Design">{{cite web|url=http://masonrydesign.blogspot.com/2010/05/arch-designs-corbels-and-catenaries.html|title=Masonry Design|author=|date=11 May 2010|website=|accessdate=23 December 2016}}</ref>


==See also== ==See also==
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==References== ==References==
{{Reflist|30em}} {{Reflist|30em}}

== Sources ==
* {{cite book | last1=Bradley | first1=R.A. | last2=Gohnert | first2=M. | title=Current Perspectives and New Directions in Mechanics, Modelling and Design of Structural Systems | chapter=Parametric study of the catenary dome under gravity load | publisher=CRC Press | publication-place=London | date=2022-09-02 | isbn=978-1-003-34844-3 | doi=10.1201/9781003348443-130}}
* {{cite journal | last1=González | first1=Genaro | last2=Samper | first2=Albert | last3=Herrera | first3=Blas | title=Classification by Type of the Arches in Gaudí's Palau Güell | journal=Nexus Network Journal | volume=20 | issue=1 | date=2018 | issn=1590-5896 | doi=10.1007/s00004-017-0355-7 | pages=173–186}}
* {{cite book|last=Hourihane|first=C.|title=The Grove Encyclopedia of Medieval Art and Architecture|publisher=Oxford University Press|volume=1|year=2012|isbn=978-0-19-539536-5|chapter-url=https://books.google.com/books?id=FtlMAgAAQBAJ&pg=PA131|access-date=2024-12-24|chapter=Arch|pages=129-134}}
* {{cite journal|last=Osserman |first=Robert |title=Mathematics of the Gateway Arch |url=https://www.ams.org/notices/201002/rtx100200220p.pdf |date=February 2010 |journal=] |issn=0002-9920 |volume=57 |issue=2 |pages=220–229 }}


==External links== ==External links==

Latest revision as of 05:45, 24 December 2024

Architectural pointed arch that follows an inverted catenary curve
A mudbrick catenary arch
A catenary curve (left) and a catenary arch, also a catenary curve (right). One points up, and one points down, but the curves are the same.

A catenary arch is a type of architectural arch that follows an inverted catenary curve. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of vaults and buttresses in stone vaulted Gothic cathedrals and in Renaissance domes. It is not a parabolic arch, although the non-circumferential curves used in arch designs (parabola, catenary, and weighted catenary) look similar, and match at shallow profiles, so a catenary is often misclassified as a parabola (per Galileo, "the chain fits its parabola almost perfectly").

In history

Painting of Robert Hooke seated in a study, holding a small chain suspended between his hands by the ends
Robert Hooke, holding a hanging chain, which forms a catenary curve

The 17th-century scientist Robert Hooke wrote: "Ut pendet continuum flexile, sic stabit contiguum rigidum inversum", or, "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."

A note written by Thomas Jefferson in 1788 reads, "I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientific work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium".

Structural properties

Architecturally, a catenary arch has the ability to withstand the weight of the material from which it is constructed, without collapsing. For an arch of uniform density and thickness, supporting only its own weight, the catenary is the ideal curve.

Catenary arches are strong because they redirect the vertical force of gravity into compression forces pressing along the arch's curve. In a uniformly loaded catenary arch, the line of thrust runs through its center.

This principle has been employed architecturally to create arched structures that follow exactly, and in a visibly apparent way, the form of an inverted catenary. A significant early example of this is the arch of Taq Kasra. The catenary, spun 180 degrees, forms the structure of simple domed building such as the beehive homes of the Dingle Peninsula, Ireland.

The principle of the catenary is also the underlying factor in the much more complex architectural systems of the Medieval and Renaissance architecture. Buildings that have heavy roofs that are arched in shape and deliver a strong outward thrust must comply with the form of the catenary curve in order not to collapse. This does not imply that the arches themselves are catenary in form, but that the total system of walls or buttresses that support the roof or dome contain a catenary curve, which delivers the downward thrust.

In the 15th century Brunelleschi designed the pointed, octagonal, Gothic dome on Florence Cathedral in a manner that utilised the principle of the catenary arch. In the 17th century, Christopher Wren designed the dome of St Paul's Cathedral based directly on a catenary curve. The vaulted roof and buttresses of Kings College Chapel, Cambridge, have been discovered to comply with the formula of the catenary arch.

Examples

The classification of non-circumferential curves in actual arches is hard. González et al. provide an example of (well-studied) Palau Güell, where researchers do not agree on classification of the arches or claim the prominence of parabolic arches, while the accurate measurements show that just two of the 23 arches designed by Gaudi are actually parabolic.

Ancient arches

Saqqara ostracon

The curves close in shape to catenary were used in arches of Ramesseum (13th century BC) and Arch of Ctesiphon (6th century AD).

The unfinished Saqqara ostracon has a catenary shape.

Cathedrals and churches

St Paul's Cathedral's dome

Natural arches

Landscape Arch, Utah

Rainbow Natural Bridge in the U.S. state of Utah has a natural catenary shape, possibly produced by weathering in high-stress areas. Kolob Arch and Landscape Arch, also in Utah, have a catenary shape as well.

Human-made arches

The Gateway Arch in the American city of St. Louis (Missouri) is a catenary arch Due to more strength needed at the bottom, the bottom of this arch is wider than the top, so the actual shape is technically a "weighted catenary".

High-rises

Marquette Plaza in Minneapolis

Marquette Plaza in Minneapolis used catenary arches.

Kilns

Kilns are often designed with catenary arch cross-section.

Igloos

Further information: Igloo § Construction

Igloos are designed with a catenary arch cross-section. This shape offers an optimal balance between height and diameter, avoiding the risk of collapsing under the weight of compacted snow.

Other architecture

The inside of Budapest’s Keleti Railway Station forms a catenary arch.

The Nubian ton is a burial vault, of Nubia, For greatest stability, the structure’s cross-section follows a catenary arch.

The beehive homes (clocháns) of Ireland’s Skellig Michael have a cross-section that follows the style of a catenary arch.

Homes

A beehive home (a clochan) on Dingle Peninsula, Kerry, Ireland
Rice House

The Rice House has catenary arches.

Hotels

The Icehotel in Sweden employs catenary arches.

Bridges

A catenary bridge has the form of a catenary arch.

One famous example is the An-Lan Bridge, in China.

Monuments

Taq Kasra

In Iraq, the Taq Kasra has the shape of a catenary arch.

Airports

The roof of Washington Dulles International Airport is a suspended catenary curve.

A catenary steel cable system supports the roof of Denver International Airport.

Train stations

New York City’s Pennsylvania Station has a roof in the form of a catenary arch.

Banks

On the Federal Reserve Bank of Minneapolis, the building has been remodeled, but still visible is the catenary arch suspending the original building.

Mud huts

Cameroon's musgum mud huts have a catenary cross-section.

See also

References

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Sources

External links

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