Simple Equation For Calculating Skyscraper Sway in an Earthquake

Skyscrapers swaying during earthquake in Japan from woahdude

A major and disasterous earthquake has just hit Indonesia. It is the job of engineers to calculate and incorporate – as far as possible- safety margins into the structures of buildings, dams, power plants and even pipe lines. 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 vomit inducing (barfomatic?).

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.

Earthquake Engineering Maths

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.

Calculus Equation here.

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.

How far does the tallest buidling in the world SWAY in an earthquake?

Earthquake Engineering Maths

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.

Death by Caffeine: The Maths Part 1 METRIC

Death by Caffeine: The Math  Part 1 USA UNITS here.

The lethal dose for caffeine is about 150 milligrams per kg of body weight. While the average person’s caffeine consumption is around 200 milligrams a day, the Mayo Clinic advises against exceeding 500 to 600 milligrams per day.

This work titled Timing is Everything is by street artist ABOVE in London.

You may want this much energy but be warned, energy drinks can kill.

It would take a ridiculously huge number of ANGRY WOMBAT energy drinks to kill a 14 year old weighing, say, 48 kg. Nevertheless, some kids are more sensitive to caffeine than others and then much lower levels of consumption of energy drinks can be fatal. 

See Death by Caffeine: The Maths Part 2 METRIC

Here is the work (see below) TIMING IS EVERYTHING by street artist ABOVE in London showing the timing.

 

Death by Caffeine: The Math Part 1 USA UNITS

Death by Caffeine: The Maths  Part 1 METRIC here

The lethal dose for caffeine is about 68 milligrams per lb of body weight. Average consumption of caffeine is about 200 milligrams a day. The Mayo Clinic advises against exceeding 500 to 600 milligrams per day.

 This work titled Timing is Everything is by street artist ABOVE in London.

You may want this much energy but be warned, energy drinks can kill.

It would take a ridiculously huge number of ANGRY WOMBAT energy drinks to kill a 14 year old weighing, say, 110 lb. Nevertheless, some kids are more sensitive to caffeine than others and then much lower levels of consumption of energy drinks can be fatal. 

See Death by Caffeine: The Math Part 2 USA UNITS

Here is the work (see below) TIMING IS EVERYTHING by street artist ABOVE in London showing the timing.

 

QWERTY Math for MAC

QWERTY Math uses KEYBOARD SHORTCUTS

There are 10 math questions

(pdf version and pdf ans below).

The boxed sections are keyboard math symbol shortcuts.

When the students push these keys the Qs finally make sense.

eg. option v = square root sign

BUT:

1. They must have a computer with a KEYBOARD (not a tablet)

2. The computer must be a MAC*

*Mathspig tried to find the QWERTY Math shortcuts on her old PC and it went crazy!!!! 

QWerty Math for MAC pdf

Qwerty Math for MAC ANS

WINTER OLYMPICS: How ski Jumpers Use Math to Increase their Jump Length

While air resistance has little impact on aerial skiers it is a significant factor used by ski jumpers to increase their jump distance.

The significant maths for ski jumpers is therefore X-section area.

Here is the jump at Pyeong Chang, 2018. Just imagine going down that at top speed!!!

              A ski jumper is set to jump in Pyeongchang.

                   Casey Larson USA Pyeong chang 2018

Ski jumpers increase their speed going down the ramp by reducing their X-section area:

           Lindsey Van, USA, practicing in a wind tunnel

Once they leave the ramp, ski jumpers try to increase their X-section area like Ski Divers to slow their vertical fall. But they have to land safely so they keep their skis at a minimum  angle.

Abby Hughes, USA, practicing in a wind tunnel.

          Abby Hughes, USA, in the air

Here are the X-section areas for Abby Hughes*:

Here is the formula for Air Resistance of Drag:

D = ½CApv²

Where C is the drag coefficient or constant, which depends on the shape and spin of an object. It is found by testing the object in a wind tunnel.

A is the X-section Area,

p is the density of the air and

v the velocity of the object.

More here.

As Abby Hughes has tripled here X-section area in the air she will have tripled the vertical drag during her jump. This will slow here decent.

*Mathspig calculated the X-section area by the old fashioned method of counting squares and rounding off the final count. Mathspig sized the two pics of Abby Huges so that her head was the same size in both pictures.

Winter Olympics: Why do figure skaters spin so fast?

Here is the sensational pairs team from the 2018 Winter Olympics.

But why do ice skaters spin so fast?

Here is the math!

REASON  1:

Well, mathspiggies, the girl in this video is right. Angular momentum remains constant unless external forces are applied.

L = mvr

L = angular momentum

v = linear velocity

r = separation of object

Louisa Barama, USA

Let’s have a look at this equation:

The fastest spin on ice skates was achieved by Natalia Kanounnikova (Russia) with a maximum rotational velocity of 308 RPM (rotations per minute) at Rockefeller Centre Ice Rink, New York, USA on 27 March 2006. See Guinness Book of Records.

 Record spin :  v_r = 308 RPM

Other spins include:

Mao Asada, Japan, triple Axel

Triple Axel spin v_r = 220 – 280 RPM

……………………………………………………………………………………………..

 Maximum Triple Axel spin v_r = 402 RPM

Skaters can spin faster during a triple axel jump because there is no friction from the ice slowing their spin.

To complete a quad axel, it’s estimated that the skater would have to rotate in the air at:

540 rpm.

…………………………………………………………………………………

Camel spin v_r = 90 RPM

More info here.

Kim Yuna, South Korea

…………………………………………….

REASON 2:

How can a figure skater move from

a camel spin into a very fast standing spin?

Now, mathspiggies, you must separate Linear Velocity (v₁ ) from Angular Velocity (v_r ). Linear Velocity is measured in m/sec ie. it is the speed of, say, a skaters foot around the circle. Angular Velocity is measured in either RPM (Revolutions Per Minute) or degrees or Radians per minute. Ie. It is the rate of spin. We can’t judge how many m/sec a skaters foot is moving in a circle. We can only see how fast they spin. In other words, we see their Angular Velocity. When a skaters foot is in the Camel position that foot travels in a very big circle.

But when that same foot is in a Triple Axel postion it moves in a very, very small circle.

Patrick Chan, Canada, Camel Spin

Patrick Chan, Canada, Triple Axel

By halving the radius, firstly, a skater’s Linear Velocity doubles due to the conservation of angular momentum.

Then, secondly, by halving the radius the circumference of the circle moved by , say, the skaters foot is halved.

Overall, by doubling the velocity around the circle and halving the circumference a skater increases their rotational velocity by a factor of 4.

Look at the numbers:

Camel spin v_r = 90 RPM

…………………………………………………………………………………

Triple Axel spin v_r = 4 x 90 RPM = 360 RPM

That’s about right.