To help us understand why buildings move, we first have to learn some basic physics concepts. The way that buildings move is much the same as other household objects. For example, take a ruler and clamp it down just like a diving board. Then push down on it with your finger. You will notice that the more you push down on the ruler, the more it will bend and the distance the tip moves downward continues to increase. A building behaves in the same way. Its like a vertical diving board.

When an earthquake or strong wind pushes on the side of the building, the building is moves to the side. The more that this force pushes or pulls on the building, the more it will move or displace, just like the diving board. The wind or earthquake, as they are pushing on the building, exert a force, labeled F. The amount the building moves due to this force is termed its displacement, labeled with an x. By applying a steady force that does not increase or decrease, you are placing a static load on the building. So our first observation is that a building will displace more as more force is applied. How much the building moves when a force pushes on it depends on what the building is made of. Its geometry and material will dictate its stiffness, we represent with the variable k. And this name makes sense. If something is stiff, it is hard to move. Thus, a stone building would be stiffer than a building of wood, because you can imagine it takes a much larger force to move a stone building than a wood building. The stone building would have a bigger stiffness. Engineers can use the relationship between the force applied to a building and how much it moves to help in design. The relationship between force and displacement and stiffness is summarized by an equation called Hooke’s Law.

Hooke’s law is essential, because knowing a building’s geometry and material, the engineer can determine its stiffness and predict how much the building will move under the worst possible forces or loads imaginable – an important step in design.

However, we are not only interested in these static displacements. Instead, we are more concerned about the structures dancing – so called dynamic motion. These dynamic motions are ones that change, often quickly, with time. If you were to take the same ruler and give it a quick push and then let the diving board bounce up and down or oscillate, you can see that this simple ruler has the ability to dance. This is the same thing you will observe when you jump off a diving board and see (and hear) it bouncing up and down in your wake. In the same way, the rapid impact of an earthquake or a sudden gust of wind can send even the world’s largest building into crazed motions, causing the building to sway to and fro. These dynamic motions are those responsible for the horror stories of windows falling, as buildings do sometimes dance so much that they even loose parts of themselves.

This concept of stiffness of a building is quite helpful in describing the building’s behavior. It is a simple way of predicting how much a building will move when impacted by a given load. We just learned that stiffness, through Hooke’s Law, has a simple way of predicting the static behavior. However, stiffness is also an important parameter in predicting the dynamic response of our building.

When a building dances in such dynamic fashion, it performs its dance at a certain pace. Just as you and I have a natural pace at which we walk or run, some of us faster and others slower, buildings have such a pace. This pace is called the natural frequency. Everything in the world has such a frequency or pace that it naturally moves to. Taking that same ruler we used before and setting into motion, you can time, with a very fast stop watch, how long it takes the ruler to go up and down in one oscillation or cycle. By doing this, you are determining its pace or natural frequency. Amazingly, buildings do the same thing. If you had a well-trained eye, you can watch even the Sears Tower move back and forth. If you had a stopwatch you could see that it takes about 7 seconds for it to go through one cycle of motion.

Now the stiffness plays an important role in determining the natural frequency or pace of the building. In this next equation, the frequency, f, can be determined once you know this stiffness and the mass, m, of the building.

Since everything in the world has stiffness, though some things like steel and stone are stiffer than others (wood, paper), and everything has some weight or mass, you can see that this equation will allow you to determine the frequency of just about anything. Everything has some natural frequency and if you were strong enough, you could punch anything from a building to a bridge to a football stadium and make it dance at this pace or frequency.

So it is natural for things to move. Rulers, diving boards, slinkies and even buildings can move if they are pushed hard enough. The question is, what pushes huge skyscrapers enough to make them dance? And what does it take for these motions to become so crazy that they cause windows to pop out of the building?

There has to be a force involved because It Takes Two to Tango!