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 hyperboloid
structures, nature's organic shapes,[7] and the fluidity
of water into his architecture. While designing
buildings,
he observed the forces of gravity and related catenary
principles.[8] (Gaudí designed many of his structures
upside down by hanging various weights on
interconnected strings or chains, using gravity to
calculate catenaries for a natural curved arch or
vault.[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.
MIT
Dr. John Ochsendorf
Catenaries
This is the basis for understanding both
suspension bridges and masonry buildings
and structures.
Interesting Side-Note
One of Ochsendorf’s groups endeavors.
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.
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.
• 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 chain
decreases because
the chain is supporting
less weight.
Gateway Arch
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
•
•
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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.
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.
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.
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
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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
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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.