How does the critical loading force for a long column vary with the length of the column?
As we get into this question, let’s start out by considering just what a critical loading force is. Say that we have a column standing on the ground. And then imagine that we start to load the column with weights. The column is designed to be able to support some amount of weight. But as we keep increasing the weight pressing down on it, eventually we reach a point where there’s so much stress along the length of the column that the column is in danger of buckling.
When we’re at this point, when the column is still intact but is right at the edge of bending and starting to break, then we say that the load on the column is the critical load it can handle. And the force that that load exerts is the critical loading force. The critical loading force is sometimes abbreviated 𝑃 sub cr.
And there’s a mathematical relationship giving us this critical load in terms of parameters of the column. Here’s what that equation looks like. And let’s walk through it variable by variable, so we can understand just what the critical load depends on.
We see that in the numerator of the right-hand side of this equation, we have 𝜋 squared times capital 𝐸. That capital 𝐸 is Young’s modulus. This modulus gives a measure of how elastic the deformation of our column is. That is, when the column is compressed or otherwise changed its shape, how likely is it to return to its original shape?
That term, capital 𝐸, is multiplied by 𝐼, which is the moment of inertia of the column. And then, all that numerator is divided by the denominator of 𝐾𝐿 all squared. Here, 𝐾 is an effective length factor that has to do with the shape that our column takes when it begins to buckle. And capital 𝐿, as we might guess, is the length of the column. It’s this variable that holds the key to our answer, just how the critical loading force varies with the length of the column.
If we pull out this variable from our equation for critical load, we can say that the critical load is proportional to one over 𝐿 squared.
And then writing this out in words, we can say that the critical loading force is proportional to the inverse of the column length squared. That’s how the critical loading force varies with the length of the column that’s being loaded.