# Cross-sections of Beams

The cross section of a beam has a significant effect on how easily the beam will deform. For example, everyday experience tells us that a flat metal ruler will flex much more easily than a piece of metal tubing with walls of the same thickness.

Because it is difficult to use the "shape" of a cross-section directly as a variable in an equation modeling beam behavior, we instead compute two quantities that describe properties of the cross section. These quantities are the centroid and the moment of inertia , and we will incorporate these two quantities into our model.

## Centroids

If you think of a cross section as a two-dimensional figure cut out of stiff cardboard, the centroid is the point at which is will balance on the point of a pin. In many physical settings where the most important properties are mass and position , the actual cardboard cross-section will have the same effects as a point mass located at the centroid. For example, if you are constructing a mobile, it will balance just as well if you attach a cutout of the Botticelli's Birth of Venus to one of the arms at her centroid, or clamp a fishing sinker of equivalent weight there.

Equations modeling physical phenomenon can often be greatly simplified in situations where a cross-section can be treated as a point mass.

## Moment of Inertia

In physical systems where objects are moving and accelerating, both the mass of the objects and how the mass is distributed is important. The moment of inertia measures how hard it is to rotate an object around an axis. Again drawing on every day experience, it is much easier to twirl an umbrella around when it is folded than when it is open. The further a mass is from an axis of rotation, the more it will resist rotating.

Similarly, a force applied far from the axis of rotation is much more effective in causing rotation than a force applied close to the axis. Again drawing on common experience, imagine trying to open a heavy door. If you push on the door close to the hinge, then the door will barely budge. It is much easier to move the door if you push far away from the hinge. That is why door knobs are placed far away from the door hinges.

To see how the centroid and moment of inertia of a cross section might effect the way a beam bends, imagine the beam built up out of many thin cross-sectional slices. Bending the beam amounts to slightly tilting and sliding the slices. Part of how resistant a beam is to bending comes from how much each slice resists tilting.

These concepts will be explored in more detail in the sections on Centers of Mass and Centroids and Moments of Inertia.