Beam Buckling 1

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.

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The String Equation

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.

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Derivation of the Euler-Lagrange Equation

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.

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Further Application — The Catenary Problem

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?

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Beam Bending Final

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.

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