Understanding the Deflection of Beams

Thanks to Brilliant for sponsoring this video. Beams are used to carry loads in a whole range of different structures, and it’s important for engineers to be able to predict how much a beam will deflect when loads are applied to it. Allow a beam to deflect too much and it could cause damage to other parts of the structure,

feel unsafe to the user, or fail to meet its intended function, even if stresses in the beam are low and there’s no risk of failure. Construction codes typically define what the maximum allowable deflection is, and so it’s up to the engineer to figure out how much the beam will deflect. There are quite a few different approaches that can be used to do this.

I'll cover five of the more common methods in this video. All of these methods make the same important assumption, which is that deflections are small and stresses remain in the elastic region. For this reason, the deformed shape of the beam is called the elastic curve. We can define x as the distance along the axis of the undeformed beam and y as the vertical

deflection of the beam. The objective is to figure out how y varies along the length of the beam. We will assume that there’s no displacement along the x axis. We’re also interested in the angle of rotation of the beam, Theta. Let’s take a closer look at the angle. We can define the slope of the beam as dy divided by dx.

Since we’re limiting ourselves to scenarios where deflections are small, we can apply the small angle approximation, and so we can assume that the angle of rotation is equal to the slope of the beam. In other words, the angle of rotation is equal to the first derivative of y with respect to x. We can start thinking about how beams deflect under load by reviewing the moment-curvature