## Cool Formula For Calculating Skyscraper Sway

March 21, 2011

The world is in turmoil mathspigs with earthquakes, tsunamis and nuclear power plant meltdown worries. It is the job of engineers to calculate and incorporate – as far as possible- safety margins into the structures of buildings, dams and power plants. Observers have noted that the skyscrapers in Fukushima wobbled during the recent 8.9 magnitude earthquake in Japan.

This is intentional, as rigid structures can snap in strong winds or during earthquakes.

But the maths used to calculate SKYSCRAPER SWAY is straightforward.

The Earthquake Engineering website offers a simple explanation.

Short, rigid buildings are damaged in earthquakes because they shake very fast. 10 story buildings have a period of oscillation of about 1 second the same as the earthquake pulse. This is VERY dangerous.

Tall, flexible buildings can withstand an earthquake because they can sway. They are like a very large, slow moving tuning fork. If they are TOO RIGID they snap. If they are too flexible the people on the 100th floor would be throw all over the place.

The 59-story steel-construction Citicorp Centre, NY (pictured) has an oscillation time of 6.7 seconds. Details Google Books.

The 102-story brick clad Empire State Empire Building sways about 8cm ( 3 inches) whereas the 110-story steel -mesh World Trades Centre Towers, NY, before they collapsed swayed over 1 m ( 3 ft 5 inches).

One more thing. You want buildings to have springy foundations so they don’t snap at the base and fall over.

# Earthquake Engineering

The idea is not to strengthen the building, but to reduce the earthquake generated seismic forces acting upon it. This can be done in 3 ways.

## 1. Base Isolation. Rubber pads or Rollers. Are used so the base does not feel the full shake or jump off foundations.

Details Base Isolation Specialists

## 2. Shock absorbers or dampers are added to the structure to dissipate the seismic shock.

Details Damper Supplier

## 3. Active Tuned Mass Dampers use a computer controlled counter moving weight to actively move against the building sway.

The 508m (1,667-foot) Taipei 101 Tower would sway back and forth up to 60cm (2 feet) each way within five seconds. This according to Wired magazine is highly barfogenic or spewdacious, perhaps, in Aussieland.

The Taipei 101 engineers included a 662 tonne (730-ton) counter giant pendulum to act as a counter weight.Some buildings use a big block of concrete.

It is pushed in the opposite direction to the building sway to dampen the oscillation.

# Take 1:

Wired magazine includes the equation for Skyscraper Sway acceleration (See definition of terms @ Wired link):

But I’m going to use a student friendly equation from Wind Engineering for Large Structures.

Mathspigs, you can just look at this equation and see how to change it to make a building EARTHQUAKE SAFE. Keep in mind that k, the stiffness constant actually decreases for taller buildings.

Imagine you are designing a building to withstand the 8.9 magnitude earthquake. You have already added base isolation. Now you have three options to work with: building mass (m), damping constant (c) and stiffness constant (k). Remember the earthquake force is constant. If you change just the stiffness of the building (k) what happens to the distance of sway(x)?

Engineers have to come up with the optimum design for the strongest structure with least acceleration (but enough building mass for strength), greatest damping and least sway at the lowest cost.

# Take 2:

Structural Engineer Ron Klemencic explained on the Discover News that a simple rule of thumb for calculating skyscraper sway was to simply divide the buildings height in by 500 because the building codes demand the building fit a 1:500 sway ratio.

The tallest building in the world at 2,716 feet (828m), the Burj Khalifa, Dubai, would sway back and forth about 5.5 feet or 1.7 m.

Ahhhhhhhhh!  But you would have to drag Mathspig onto the 168th floor screaming.

But mathspigs you can work out the sway on the top ten tall buildings in the world.

## The shop is 50 Smoots on your left

June 4, 2009

According to the New Scientist  magazine ‘AS A prank in 1958, Matthew Dirckx tells us, a group of students from Massachusetts Institute of Technology used the body of Oliver Smoot, the shortest new recruit to their fraternity, to measure the length of a bridge across the Charles river connecting Boston to the MIT campus. The bridge was found to be 364.4 smoots long, plus or minus an ear.’ Fifty years later a plaque was placed in the new bridge officially recognising the Smoot.

Meanwhile in the most recent Dr Who episode “Planet of the Dead’ one scientist named a very big amount of weird energy after himself. Instead of Joules or calories he called his energy unit the Malcolm. Good idea.

Mathspigs we definitely need more units of length such as the Werner, Chinock, Wu and Clutz. Claim your own height as a unit of length. Name it and then convert the heights of the following monuments into your unit:

Empire Ste Building: 381m

Statue of Liberty: 93 m

Great Pyramid of Giza: 147m

Eiffel Tower: 300m

Gustave Eiffel was an engineer who not only designed the Eiffel Tower (Built 1889) but was also involved in the design and construction of the Statue of Liberty.

Twin Towers New York: 420 m

Mathspig travelled to the top of the Twin Towers in 1999 and found the view terrifying. But a little known fact about the Twin Towers is that they were built using half a dozen  Australian designed cranes called the Favco Climbing Crane.

Burj Dubai: 818 m