# Design of a Sustainable and Strong structure Bridges and Domes Triangles are the strongest building component for providing support We use these components.

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Design of a Sustainable and Strong structure

Bridges and Domes Triangles are the strongest building component for providing support We use these components for bridges (trusses) and roof supports We expect stresses We are constrained by size & material Compression and Tension

Hemispheric domes Figure 8(a) A hemispherical dome is formed by rotating a circular arch. (b) A point dome is formed by rotationg a pointed arch. In physical structures, domes, unlike arches, have both longitudinal stresses from top to bottom, and circumferenctial (or hoop) stresses around the domes..

United States Capitol, DC

Goal of this Project Build a roofing structure using ‘Pure’ compression and tension Use Hooke’s law Create a parametric model of this structure Make a multimedia presentation of the overall project to the class

"As hangs the flexible line, so but inverted will stand the rigid arch.“ Robert Hooke - 1675 This may or may not be Robert Hooke. Robert Newton instigated the removal of Hooke's portrait in the Royal Society.

Antoni Gaudí 1852-1926 The Architect of Barcelona He integrated the parabolic arch and hyperboloidparabolic archhyperboloid structuresstructures, nature's organic shapes,[7] and the fluidity[7] of water into his architecture. While designing buildings, he observed the forces of gravity and related catenarygravitycatenary principles.[8] (Gaudí designed many of his structures[8] upside down by hanging various weights on interconnected strings or chains, using gravity to calculate catenaries for a natural curved arch or vault.[7])[7] From online encyclopedia - Wikipedia

Gaudi's weights and string stuctural models

Antoni Gaudi Gaudi's work is admired by architects around the World as being one of the most unique and distinctive styles. His work has greatly influenced the face of Barcelona Architecture and you will see Gaudi's work all over the city.Antoni Gaudi was born in Reus in 1852 and received his Architectural degree in 1878. From the very beginning his designs were different from those of his contemporaries.

A hallway in Casa Mila. These arches were modeled after catenaries. Throughout his architecture, catenaries are a prominent design, proving Gaudí's fascination with mathematics.

La Pedrera vaulted attic

Arch and spiral staircase Gaudi

Then MIT Got Involved A group led by Ochsendorf studies buildings and sustainable structures based on the history of still standing designs.

Dr. John Ochsendorf MIT

Catenaries This is the basis for understanding both suspension bridges and masonry buildings and structures.

Interesting Side-Note This is a 500 year old Inca bridge Made completely of grass strung Into cord. When the Spanish came To South and Central America they Realized that there was no equivalent To this type of structure in Europe. One of Ochsendorf’s groups endeavors.

King’s College Chapel an image of the magnificent fan-vaulting on the ceiling of King's College Chapel (completed in 1515), which spans 42 feet and hovers 84 feet above the pavement, yet its constituent blocks are only four inches thick. It's been standing for 500 years."

Hyperbolic Functions Hyperbolic functions are useful because they occur in the solutions of some important linear differential equations, notably that defining the shape of a hanging cable, the catenary.differential equationscatenary

In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). The chain is steepest near the points of suspension because this part of the chain has the most weight pulling down on it. Toward the bottom, the slope of the chainmathematicsshapechaincablegravitationalslope decreases because the chain is supporting less weight.

Gateway Arch

The stainless steel-plated arch is in the shape of an inverted, weighted catenary curve. It spans 630 feet at ground level from outer edge to outer edge and is 630 feet high, making it the tallest man-made monument in the US. Designed by Eero Saarinen

Hyperbolic Function In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc., in analogy to the derived trigonometric functions. The inverse functions are the inverse hyperbolic sine "arsinh" (also called "arсsinh" or "asinh") and so on.mathematics trigonometric Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola. Hyperbolic functions are also useful because they occur in the solutions of some important linear differential equations, notably that defining the shape of a hanging cable, the catenary, and Laplace's equation (in Cartesian coordinates), which is important in many areas of physics including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.hyperbola differential equationscatenaryLaplace's equationCartesian coordinatesphysicselectromagnetic theoryheat transferfluid dynamicsspecial relativity The hyperbolic functions take real values for real argument called a hyperbolic angle. In complex analysis, they are simply rational functions of exponentials, and so are meromorphic. hyperbolic anglerational functions exponentialsmeromorphic

Cosh & Sinh

Activity Build an arch to cover a building Start with an inverted hyperbola –We will take a piece of material cut in a polygon of your choice – about 14” across –Design a framework about 11” across to match the number of corners in your material –Dip the material in our glue and allow to drain –Clip the material to the frame with clothespins and let dry

Following are examples of where you might want to begin

One way might work

Another possibility

Cut Once – Use front and back

When it’s OK to say “Oh Crap”

Perhaps a wireframe

evaluation What did you learn from this experience How strong is your roof? How can we test it? How would you do this different if you did it again?

Related Vocabulary Domes Geodesic domes US Capitol Catenary Hyperbolic Function Parametric model Hooke’s Law Gaudi

Geodesic dome: "Windstar" dome A 20th-century invention changed dome engineering forever. In the 1950s, a radical new design -- the geodesic dome -- changed the way engineers looked at domes for the first time in 2,000 years. Invented by American engineer and architect Buckminster Fuller, the geodesic dome is a partial sphere shape structured from a series of triangles, rather than a series of arches.

Expo 57 This geodesic dome was built for the United States pavilion at Expo ’67, the world’s fair in Montréal, Canada. Geodesic domes have no internal supports, making them ideal structures for holding large groups of people. They are made up of standardized, interlocking shapes that can be assembled and taken apart quickly.

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