Last month we laid out the three degrees of freedom (DOF) in 2D: horizontal translation, vertical translation, and rotation. We also established that a structure’s supports need to restrain against all three DOF. Read that article now if you missed it, and then let’s look at a simple bridge to see how its supports work.

(Since bridges are actually 3D, this model is just a starting point. Even so, it will account for vertical forces like vehicle weights and snow accumulation, and horizontal forces like accelerating/braking and seismic activity. What’s missing are lateral forces like wind, which would be addressed later in design.)

Most beam bridges use two of the three most common supports. These bridges have a roller on one end of the span and a pin on the other end. (They’re easy to see in older bridges – look for them next time you get a chance!) That adds up to three restraints – two vertical, one horizontal. Three DOF, three restraints.

Hold on though. We’ve clearly restrained against horizontal and vertical translation, but where is the restraint against rotation? To answer, let’s do a thought experiment. Imagine trying to rotate the bridge around the left support. To rotate clockwise, the right support would have to move down, and to rotate counterclockwise, the right support would have to move up.

But the right support can’t move down or up because it’s fixed against vertical translation. Same goes for rotation about the right support. Thus, two vertical restraints prevent rotation.

What would happen if we used two pin supports instead of a pin and a roller? We’d still restrain against all the DOF. But we’d have a big problem with thermal expansion. All materials grow and shrink when the temperature gets very hot or very cold. It’s typically a tiny number – steel for example grows 0.0012% longer for every degree Celsius of temperature increase. But over a 100-meter bridge span and a 30-degree temperature swing that adds up to 3.6 centimeters, which is certainly a noticeable change. Without the roller the bridge has nowhere to expand, and it buckles and breaks.

The choice of supports is crucial to the design of any structure.