Parabolic arch

Key takeaways

Parabolic arch belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Parabolic arch to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Parabolic arch from memory before moving on to harder problems.

Reference excerpt

A parabolic arch is an arch in the shape of a parabola. 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 while a catenary is the hyperbolic cosine, cosh(x), a sum of two exponential functions. One parabola is f(x) = x2 + 3x − 1, and hyperbolic cosine is cosh(x) = ⁠ex + e−x/2⁠. The curves are unrelated.

The line of thrust 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.

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.

Uses

In nature

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

Architectural examples

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, 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. St. 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, Kenzō Tange. Church of St Mary and St Joseph, Poplar, 1954, London, United Kingdom, Adrian Gilbert Scott St Leonard's Church, 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 Allen Lambert Galleria, 1992, Toronto, Canada, Santiago Calatrava L'Umbracle (catenary shade house) 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

Bridges

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Parabolic arch

Start with the simplest possible case. Write down what Parabolic arch claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Parabolic arch before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Parabolic arch ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Parabolic arch

In research: Parabolic arch appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Parabolic arch in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Parabolic arch is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arches and vaults, Architectural history, Parabolas, so understanding it makes those chapters shorter.
In everyday life: Look for Parabolic arch outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Parabolic arch in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Parabolic arch means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Parabolic arch out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Parabolic arch in simple terms?

A parabolic arch is an arch in the shape of a parabola. 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.

Why does Parabolic arch matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Parabolic arch?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Parabolic arch.