buckling of rod

Please make a sketch your explanation is not totally clear.

I am trying to get the vuckling formula for this rod:

http://img42.imageshack.us/i/knekklol.jpg/

It is supposed to be pi*pi*e*I/L/L. I want to be able to calculate it with differential equationsb ut I get the answer wrong. At the end when I use the initial conditions I get one answer giving me pi*pi*e*I/L/L but another 4*pi*pi*e*I/L/L. So I was wondering if anyone knew any places on the web where the euler load for this rod is caluculated with differential equations. I can't find any on google.

What you have is two cantilevered columns of length L/2. If the length is L, the Euler buckling load is π²EI/4L². But in your case, the length is L/2 so you must substitute L/2 for L. This gives you the correct expression.

If you need the differential equations for a cantilevered column, see Chapter 2 of Theory of Elastic Stability by Timoshenko and Gere. You can download the entire book for free.

Hi,

any buckling problem will have higher order solutions.

So the lowest is likely to be near reality.

Be cautious if using any of the computer codes, these do not find correct solutions as in Euler buckling some severe simplifications are made that cannot be solved to a correct result by the differential equation solvers. (Non-constant length may be the worst).

I had months of wasted time until realising this (in the calculation of thin-bladed flexure hinges).

Be also cautious in matching to reality: any non-symmetric load application will largely change the resulting load at a given deformation.

RHABE

Thanks for all the answers, what I had doen wrong was that I had usen a wrong moment-variation in the particular sollution of the differential equation.

I have another question that I hope don't will take much of your time.

Here is an exact sollution for the bucling of a beam which is pinned in one end and free in the other. And it has rotational springs in both ends not translational.

http://img177.imageshack.us/i/knekkhjelp.jpg/

Now here is the tricky part, if we look at global equilibrium there can be only one horisontal reactional force and we have no horisontal forces on the problem. This would suggest that y'''=0 at the bottom(no shear force here). But if y'''=0 then y'=0 because we get a sine and a cosine function from the homogenous sollution, and my particular sollution is a constant. So how are the boundary conditions here?

Is it ok if we ask these kind of theoretical questions on this site or is it supposed to be work-related help here most for people who work?

Hi,

this "exact" solution is not at all exact!

Euler buckling assumes infinite elastic deformation perpendicular to undeformed direction but does not take into account the finite length of the beam.

Your question with rotational springs at the ends: as one end is translational free there is no horizontal force- as you state.

But I doubt about your statement of y''' = 0 as there is a bending moment from the lengthwise force and this is changing with the length-coordinate.

So there is a curvature (y'') and a changing curvature (y''').

Much better you try any math program and add to the lengthwise force a small crosswise force and/or moment. Then you will see the very first stages of buckling being extremely dependent on the crosswise force or moment which may result from a small eccentricity in the buckling-force with respect to the beams neutral axis.

RHABE

Is it ok if we ask these kind of theoretical questions on this site or is it supposed to be work-related help here most for people who work?

The questions are not necessarily restricted to work-related topics but CR4 is not a site where students can expect answers to homework assignments. If that is what you are doing, we can't help you.

The above sketch shows you how you can find if your solution ir right or wrong. Sorry but I was busy and did not have the time to react earlier.

The horizontal force which appears since the initial beam form is straight, does not affect the buckling limit since the bending moment it gives is proportional to the axial position of the section considered and NOT with the distance between the section and the undiformed beam line which enters the differential equation. If you have such other problems where you need a help the best is to write what you already did to show that you worked on your own and do not ask to get all done by others and then ask your question.

With this sketch you see that the 2 cases have same component from the stability point of view and thus should have same critical force.

I hope it will help to understand better how to analise more complex buckling problems.