Math Behind The Tallest Buildings.

I was checking out the tallest buildings in the world which got me to thinking about the math behind them.  There has to be some, otherwise they wouldn't having buildings as tall as the Burj Khalifa in Dubai, towering 2,717 feet above the ground.  The building itself has 163 floors made up of a hotel, offices, and homes.

Traditionally, the ratio for height to width is in the 8:1 or 9:1 range but with taller buildings it could be 15:1 such as 432 Park Place.  To put this in perspective for children, you might want to stand a ruler on the table so its smallest width is on the table with the longer edge going up to show a 12:1 ratio.

The problem with building that have a ratio higher than 9:1 is the cost increases due to needing thicker walls and additional technology to reduce the sway of buildings.

The biggest enemy of tall buildings is wind.  When a new skyscraper is designed, a model is created and put into a wind tunnel to see how it will react to the occasional 100 mph winds.

According to research, they've discovered they do not want smooth buildings because the a design with vertices, rounded edges, and layers such as in the picture break the wind so it is not as forceful.  Furthermore, they incorporate dampers which act as counterweights to shift and stabilize buildings.  This is done by building in a  300 to 800 ton piece of steel or concrete on a floor near the top.

Engineers use two different types of dampers in today's buildings.  The first is a tuned mass damper that works much like a swinging pendulum in a clock, and has been used since 1958, while the other is a slosh damper  or slosh tank that uses water to absorbs vibrations.    Tuned mass dampers often use small motors to adjust the building in the opposite direction to the wind and some use magnets to do the same thing.  These dampers can add up to $5 million to the cost of the project.

In addition, over the years, engineers have made concrete both lighter and stronger by changing the additives to include fly ash, slag, and micro-silica thus improving it.  Furthermore, they are playing with the way concrete and steel are put together to improve its strength without adding extra weight.  There are buildings being built right now that are expected to be higher than the current 2,717 feet because people want to be known as the one with the highest building.

The current title holder used a buttressed core design which is sort of a three wing spear due to its stability, usable space, and less loss of building to structural necessities.  The company that uses this particular design feels that buildings could be build so they are two to three times the current record.  However, engineers and designers are working on other technologies that might allow additional heights while using the current technologies.  Architects have come up with other designs but the technology has not caught up yet with the designs.

Unfortunately, elevators are one of the current limiting factors on tall buildings because there is a limit to the length of the steel cable that moves elevators up and down.  The current limitation for rope is about 500 meters or just over 1600 feet.  An elevator can weigh up to 60,000 pounds which puts quite a strain on the steel cable and uses over 130,000 kilowatts of energy each year.

There is a company in Finland working on creating a new carbon based elevator rope but its still in development as far as I can tell.  Once they've got it up and working, it will remove the current limit and allow elevators to travel longer distances.

Due to travel, I won't be here tomorrow, other than having an in transit picture.  As you can see tall buildings do require quite a bit of math.  Let me know what you think, I'd love to hear.