# Math in the Beauty and Realization of Architecture

## CONTENTS OF CURRICULUM UNIT 06.04.06

- Introduction to Unit
- Unit Objectives
- The History of Bridges
- Motivation for Unit
- Introduction Discussion with Students
- Bridge Language
- The Types of Bridges
- Interesting Famous Bridges
- How to Choose the Right Bridge for the Job
- Geometric Shapes in Bridge Trusses
- Icebreaker activity
- Bridge of the Day or Week and Daily Bonus Question
- Understanding the Strength in Columns
- Building Stability - Columns
- Building Stability - Beams
- Finding the Forces in a Truss
- Final Project: Designing a Community Friendship/Peace Bridge
- Final Project Rubric
- Bibliography

### Unit Guide

## The Math in the Design and Building of Bridges

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## Interesting Famous Bridges

A handout of a world map is given out to each student. The following bridges will be located and labeled on this map.

### London Bridge

London Bridge has an almost 2000 year rich history, beginning with a Roman built bridge in 43 AD. In 1176, the first stone bridge stood for 600 years. Its construction and upkeep was paid by rents and tolls. In 1824 a five arch bridge was built and replaced in 1972. The old bridge sank ¼ inch each year and so by 1968 this became alarming to the authorities. The old bridge was dismantled and bought by McCullough (chain-saw manufacturing) who moved it 10,000 miles to Lake Havasu, Arizona. How did he do it? Each of the over 10,000 pieces were carefully removed and coded. The bridge was reconstructed according to the original blueprints. Originally it took 800 men and 7 ½ years to build this bridge but only 400 men and 2 years to rebuild it, although they did not have to sink foundations. It is in the Guinness World Records as the world's largest antique. There is an excellent web site: (http://www.oldlondonbridge.com/timeline.shtm) which shows an amazing time line, in full color pictures, of London Bridge's eventful history. Another site shows the reconstructed bridge: www.londonbridgeresort.com/history/london_bridge.htm

### Math

Find the total depth the London Bridge sank over the 1824 to 1968 at ¼ inch per year.

1968 - 1824 = 144 years / .25 = 38 inches

Calculate how many feet this is.

38 / 12 = 3.17 ft.

### Brooklyn Bridge

The famous Brooklyn Bridge in New York City, over the East River, has been celebrated in stories, poems and paintings. It may be the most photographed bridge. The design of its towers and graceful pattern of the stays create a bridge of beauty. It is close enough to New Haven to visit and walk over. An engineering wonder, it was the longest suspension bridge in the world at its 1883 opening. It had to have a large clearance of 135 feet above the river. The idea came to its designer, John A. Roebling, when he was crossing that river on a ferry and it became stuck in the ice. Ferries would stay tied up for weeks because of the ice in the river. All new techniques were used. It was the first bridge built of steel. It cost double the original estimate, about $1 billion in today's dollars, and took 14 years to build. It was supposed to take only 5 years to build.

### Math

How many extra years did people have to wait to use it? 14 - 5 = 9

Discuss if structures are commonly completed past their due date.

The huge opening ceremonies were elaborate with the US President, Chester A. Arthur, attending and 14 tons of fireworks exploding in the sky. For a virtual tour go to: (www.newyork.com/attraction/panorama/tour4a/htm)

Each cable is made up of 3,515 miles of steel wire, each galvanized with zinc to protect them from the adverse effects of weather.

Each wire was the thickness of a pencil. Each strand is made up of 286 wires. One strand was at the center, surrounded by 6 more in a hexagonal shape. Surrounding those were 12 more.

### Math

Calculate
**
**
the amount of wires in one cable.

286 x (1+6+12)

286 x 19 = 5,434 wires

### Math

Each cable has a thickness of 15 ¾ inches. Students will cut a 2 inch strip of paper the circumference length of a 15.75 diameter circle.

C = pd = 3.14 x 15.75 = 49.5 in. rounded to the nearest tenth.

The cables of the Verrazano-Narrows Bridge are 3 ft. thick (diameter). Find the circumference. 3.14 x 3 ft = 9.42 ft.

Convert these feet to inches. 9.42 x 12 = 113 inches

Compare the circumferences of the cables of the two bridges.

D = 15.75in. C = 49.5 in

D = 3 ft = 36 in. C = 113 in. Thus roughly double.

Next students will find the cross sectional area.

A = pr 2 = (15.75/2) x 3.14 = 24.7 in2

### Math

Workers traveled over 10,400 times from each end (anchorage) to loop the wire.

How long did it take to just stretch the wires?

10,400 x 10 = 140,000/60 = over 2,333 hrs. 2,333/24 = 97 days

Roebling used pneumatic caissons modeled after the ones Eads used on the St. Louis suspension bridge. The ones for the Brooklyn Bridge were approximately 170 by 100 feet. The walls of the air chamber at the bottom were 9 feet thick on top. The extremely thick roof in one of the caissons the chamber was 22 feet thick to support the masonry of the tower foundations. There were sharp iron triangles on the bottom of the caisson to cut deeper as more river bottom was removed to get to the bedrock.

### Activity

Take an empty glass. Turn it over in a pan of water to experience an air lock.

Roebling died from complications in a ferry accident, so his son took over. His son became paralyzed by the bends from the caissons and could not leave his home for years. His wife, Emily, had to quickly learn the business enough to become the on site director.

### Activity

Rewrite these parts of a suspension bridge in the order of which they are built:

1. Suspenders

2. Towers

3. Foundations

4. Deck

5. Cables

A fascinating hands-on web site demonstrates step by step the building of this bridge.

Another time line enables you to see the progression of the different types of traffic at six different times in its history. Originally only cable cars and horse and buggies crossed with a center elevated lane for pedestrians. Trucks have been banned. Discuss why. (http://www.brooklynexpedition.org/structures/buildings/bridge/bl_construct_ b.html)

### Lake Pontchartrain Bridge

The longest bridge in the world is near New Orleans, LA, and crosses Lake Pontchartrain.

It is 24 miles long. The piles were driven 70 ft. down into the sandy bottom. It has a maximum vertical height of 50 ft. and a minimum of 15 ft.

### Math

What is the range of the clearances? 50 - 15 = 35.

Why is there a difference in clearance? It costs less to make a bridge lower but boats also need to pass under at some point.

### Math

The bridge length, 24 miles, is 1/1000th of the earth's circumference.

How long is the earth's circumference? 1000 x 24 = 24,000 miles.

### The Pearl Bridge

The longest suspension bridge in the world, the Akashi Kaiko Bridge, also called the Pearl Bridge, was built in Japan in 1998 and cost $4.3 billion. It crosses 110m deep water with a very swift tidal current, heavy winds and lies only 150m from a potential 8.5 earthquake. The towers alone are over 282m. tall. The cables over the towers have a diameter of 1.12m. Each hanger cable/rope has a diameter of 7mm.

### Math

Find the circumference of each and cut a 2 inch strip of paper the length of those circumferences and tape them into loops to feel the size of these cables.

C = 3.14 x 1.12 = 3.5m.

Compare these to the cable circumference on the Brooklyn Bridge.

The Pearl Bridge has a total length of 12,828 ft. and a span of 6,527 ft. (1991 m). This is almost a quarter mile (366 m.) longer than the second longest suspension bridge, the StoreBaelt in Denmark built in 1998, 580 meters longer than the Humbers Bridge in England, 692m longer than the longest US suspension bridge, the Verrazano Bridge in New York, and 710 m longer than the Golden Gate Bridge in San Francisco, CA.

### Math

How many meters long is the Golden Gate Bridge? 1991 - 710 = 1281m

This is an amazing site showing a very clear satellite map of the world and points you can click on to see a bridge photo taken there.

http://geoimages.berkeley.edu/wwp904/index.html