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.

 

Death by Caffeine: The Maths Part 2 METRIC

Death by Caffeine: The Math  Part 2 USA UNITS HERE.

Caffeine content sources Caffeine Content Data Base

On April 12 2017 16-year-old Davis Cripe collapsed at school in South Carolina and died later in hospital.  In the span of two hours, Davis drank a cafe latte from McDonald’s and a large Mountain Dew, then “chugged” a 16-ounce energy drink when he got back to art class.

Here, assuming Davis drank large sized drinks, is the lethal caffeine consumption.

NOTE: Davis lived in the USA where standard drink volumes are slightly different to Australia & UK. The USA volumes are used here converted to litre and ml.

The official cause of death was “caffeine-induced cardiac event causing a probable arrhythmia,” the coroner concluded. Source: Washington Post

Caffeine can kill.

WARNING: “Mixing caffeine with alcohol is a dangerous practice because it may lead to higher levels of alcohol consumption as the person often believes and feels they are more alert,” said Dr Robert Glatter, ER doctor at Lenox Hill Hospital, NYC . “The risk of alcohol poisoning increases as people consume more alcohol because they feel the caffeine will keep them awake and alert.” Source: USA Today

Death by Caffeine: The Math Part 2 USA UNITS

Death by Caffeine: The Math  Part 2 METRIC HERE.

Caffeine content sources Caffeine Content Data Base

On April 12 2017 16-year-old Davis Cripe collapsed at school in South Carolina and died later in hospital.  In the span of two hours, Davis drank a cafe latte from McDonald’s and a large Mountain Dew, then “chugged” a 16-ounce energy drink when he got back to art class.

Here, assuming Davis drank the large sized drinks, is the lethal caffeine consumption. 

The official cause of death was “caffeine-induced cardiac event causing a probable arrhythmia,” the coroner concluded. Source: Washington Post

Caffeine can kill.

WARNING: “Mixing caffeine with alcohol is a dangerous practice because it may lead to higher levels of alcohol consumption as the person often believes and feels they are more alert,” said Dr Robert Glatter, ER doctor at Lenox Hill Hospital, NYC . “The risk of alcohol poisoning increases as people consume more alcohol because they feel the caffeine will keep them awake and alert.” Source: USA Today

I Can’t Sleep: Caffeine Math Part 3 USA UNITS

I Can’t Sleep Math: Caffeine Maths METRIC HERE.

Sensitivity to caffeine varies for individuals, but in healthy adults the half life for caffeine is approx 6hrs meaning your body eliminates half the caffeine you have drunk in 6 hrs. Ref: Caffeine Pharmacology

Caffeine interferes with sleep. One study found that consuming caffeine 6 hours before bedtime reduced total sleep time by 1 hour.

Caffeine content sources Caffeine Content Data Base

Street art by JEKS, South Carolina

The Half life equation is an exponential function.

The Mayo Clinic advises up to 400mg Caffeine a day for adults, which would be;

10 black teas

7 instant coffees

4 cappuccinos

2 dark roast brewed

less than one cup of Black Insomnia Coffee.

Approx half this amount of caffeine for 14 yos.