# Question regarding bending of beams (solid mechanics)

I'm still in High-school, and we've just finished a chapter on solid-mechanics.

A scenario we were supplied with by our teacher, involves a cuboidal beam supported at two ends, like this:

Now if a load/weight $W$ were placed at the center of the beam, and the center of the beam sags by $d$, then the relation between $W$ and $d$ is (or so he says):

$$d = \frac {Wl^2}{4Ybt^3}$$

Where $Y$ is the Young's modulus of the beam, and $l,b,t$ are the length, breadth and thickness of the beam respectively.

Furthermore, had a cylinder of radius $r$ and height/length $l$ been used, the formula would've turned out to be:

$$d = \frac {Wl^2}{12Yπr^4}$$

We asked him (our teacher) how we ought to go about deriving the formula. He said he didn't really know it himself, and that he picked up the formula from some random workbook.

I can't for the world figure out how this was derived (can't seem to approach it intuitively either).

I Googled "Bending of Beams" and that threw up a Wikipedia article on the "Euler-Bernoulli Beam theory"... which is currently, beyond my level of understanding.

So I wanted to know:

1) Are the formulas provided correct? If so, how are they derived?

2) If they aren't correct, then what would they look like once corrected?

The formulas are correct and are special cases of the center deflection of a center-loaded simply supported beam, which is $$\delta=\frac{Wl^2}{48YI}$$ in your nomenclature (but replacing $$d$$ with $$\delta$$). Here, $$I$$ is the second moment of area or the area moment of inertia, which is a measure of how far the regions of a beam's cross section lie from the centroid. Intuitively, the more spread out the cross section, the harder it is to bend the beam.

This equation is in turn a special case of the solution to the Euler-Bernoulli beam deflection equation, which is $$\frac{d^2}{dx^2}\left(YI\frac{d^2\delta(x)}{dx^2}\right)=q(x)$$ where again I'm using $$\delta$$ to avoid confusion with the differential operator and where $$q(x)$$ is a distributed force on the beam. (In your case, this distributed force arrives in a single lump at the center of the beam.)

The derivation of this general equation and its solution is part of the content of a first- or second-year university course in physics or engineering. It involves a free-body diagram of a small portion of the beam and a force balance to express the curvature of that small portion in terms of the force distribution at either end and along its length. The force balance requires an integral across the cross section (to accommodate the compressive forces in the top half and the tensile forces in the bottom half), which is where $$I$$ comes from. It also incorporates Hooke's Law, $$\sigma=Y\epsilon$$, to translate deformations into forces and vice versa. In addition, it relies on the assumption that the curvature $$\rho\approx(d^2\delta(x)/dx^2)^{-1}$$ and a host of other assumptions.

If you've been studying differential equations and how to solve them with boundary conditions, and you can construct a free-body diagram, you may be able to anticipate this content.

The mid-point deflection of this beam is actually several layers deep into beam theory and is not the best starting point to analyze this problem. The starting point is really to analyze a diving board, or a "simple cantilever". Your beam can then be analyzed as two diving boards back-to-back (and not necessarily in the most obvious way...think about it.)

And the key to analyzing the diving board is not to focus on the endpoint deflection, but rather the curvature at any point, which is a simple function of the deflecting leverage. If you understand how the diving board curves in a general way, you have the fundamentals of beam theory.

As for exactly how much it curves, you don't want to start by analyzing a cuboidal or spherical cross-section. You want to start with the simplest possible beam, which is the ideal I-beam. If you don't know why this is the simplest beam to analyze, then that's where you should start off.

When I first took on this problem, I made the simplifying assumption that the beam (the I-beam, that is) bends into an arc of a circle. It's wrong, but it's still a pretty good starting point if you really only know about Young's Modulus.