I suggested several ways of resolving the factor of 2 discrepancy in the previous post, but one thing I did not suggest in that section was what I used to start the post: Allow for the possibility of the beam being simultaneously compressed and bent.
The intuitive way I understand beam buckling:
Put force on a beam. It compresses (linearly). It takes a certain amount of energy to compress it linearly. At some point, it might have so much energy stored in compression that it may be able to relieve some by returning to a longer length, but bent. At that point, the beam has buckled.
Apparently, a developing theme of this blog is that I don’t like using Newtonian methods for continuum mechanics problems.
Let’s now derive the string equation. Wikipedia, say, does it with vector diagrams (this source uses the same derivation, too). We’ll do it with a method that’s closer to what we’ve been using for other problems.
Here’s the model:
We divide the string into a bunch of springs.
We can also use those tools to develop the Euler-Lagrange equation itself. In my humble opinion, this is a more straight forward derivation than is usually shown.
If L = T – V, with T a particle’s kinetic energy and V its potential energy, L is called the Lagrangian and the integral of the Lagrangian over a path in time is called the action. The principle of least action says that the path a particle actually takes is a minimum in action.
For a simple example, the Lagrangian of a particle in a uniform gravitational field is
Jumping to the answer, the Euler-Lagrange equation is
Plugging in, we eventually get
Which of course is obvious, from Newton’s second law.
Now let’s derive the equation on our own.
The curve of fastest descent: Drop a point object on a curved slope — what slope gives the minimum time for it to travel between two points?
This problem is different from the others:
So, we derived the Euler-Bernoulli Beam equation, and solved some beam bending problems, but in the process we developed some general tools, so we really should try to use them for more things, if possible.
We’ll start with the catenary problem. Catenary comes from the latin word for chain. A chain is hanging from two fixed points. What shape does it take?
This is wrapping up, I think. There’s not much more to do before trying to move on to more practical calculations. I suppose I could compare numerical large angle results with the small angle analytic results; that’s the main other thing I can think of. Or maybe I could look at other extensions, like buckling.
Anyway, this post has one more worked example, plus the derivation of the Euler-Bernoulli beam equation from the method I have been using.
Another worked example.