tapered compression member

the compressive stress is just the force divided by the area, and this should be kept below the yield strength of the material.

However slender columns (more than 5 diameters tall I think) fail by buckling. The Euler column analysis assumes constant section modulus. You could probably repeat it with a non-constant section, but then you might have to make a different assumption about the shape of the buckling member and re-derive the stability criterion, which makes my head hurt. If you assumed the Euler sinusoidal shape and then calculated the relative moment at each position along the length you could then reduce the section to keep the ratio of actual to allowable stress constant. Intuitively I think it will work. For a circle (for example) this could be done by either thinning the walls or by reducing the diameter. I would recommend the reduction in diameter to avoid local crippling (intra-element) buckling.

Of course it could collapse and kill millions of people too.

Maya, I am going to make some suppositions here. There probably is a standard formula for what you ask, but I could not find it in my stuff. But there is a standard formula for compressive strength in a column with both ends rounded. This is like the ends being pinned, except the column can move in any direction rather than just at right angles to the pin. A safer formula anyway. It is P=Pi^2 times E times I divided by length^2. P is total ultimate load. I is least moment of inertia. E is modulus of elasticity of the material. ^ means to the power (^2= 2 squared). [this page is unfriendly to mathematical symbols] You are going to solve for I given the E of your choice of material. OK, now starting with that, suppose you calculate the size (cross section) of the column at its full length, say 10 meters (or whatever) for the full required load (plus safety factor). Now, calculate the same necessary cross section as though the column was only 9 meters long (it would come out a little smaller). Then calculate it at 8 meters long, then 7, then 6, etc. Now you can use these sizes to design a column that has the 10 meter cross section at the center 2 meters, then it can have the cross section of the 9 meter calculation for the next meter out from the center in both directions, then the 8 meter cross section for the next meter from the center at both sides of center, and etc for the whole thing. You will get a stepped column. Then just connect the outermost points of those steps to create a smooth taper, and I think you will have your most weight efficient column design. But I am sure there is a much better way to do this because this is not a new problem. --jer

See "Den Hartog - Advanced Strength of Materials" (also Strength of Materials). Both books are very reasonably priced. They are not too advanced (See books by Timoshenko and Gere which are the bibles for many engineers) and quite readable for text books.

Den Hartog tackles the problem from an energy solution by Rayleigh's Method. Basically, the varying inertia of the cross section will affect the buckling shape. Based on the formula for P_crit (maximum load) you can iterate to find a shape that suits.

This is assuming that your slenderness is above 30 i.e. length of section divided by radius of gyration not greater than 30. Otherwise, your column is rather stubby and will be governed more by crushing at the ends (Hertzian stresses on the pins).

Best to go and buy the Den Hartog book. I promise that you won't regret it. It is now used by all the people in our office.

See the following link too which I found after a quick look on the web:

http://www.springerlink.com/content/p76jk64kt1465u67/

You can find some indications in "Roarke".

The solution to your problem can also be "estimated" as follows:

Let us consider the column with the smallest diameter ans assume that the wall thickness is constant. The critical load - provided that the column will fulfil the slenderness conditions for the Euler formula will be Pcr= E J/(∏*H)^2.

If the diameter increases then the J value grows up with its 3rd power i.e. very rapidly. This will be equivalent to a shorter column with the smallest diameter and thus will accept a higher load. When the estimation comes under the limit of the Euler formula then only the elastic limit is the limit with the relationship of Tetmeyer-Jasinsky.

WHAT are you talking about?

Please be more detailed in the definition of your "what"

Thanks

Since I have the feeling that you did not understood WHAT I meant here is an explanation which will bring you the requested clarity.

If you have any other question it will be a pleasure to answer.

Nick Name, I appreciate your sending that page, but the resolution of what is shown is not high enough to be able to read it. But I would like to study it, so could you re-scan it and send it directly to my email address. I can always stand to learn something. --jer

Did I make myself clear enough or is it needer a more detailed explanation to your question?

I do not like to let people in the dark and since it is always difficult to explain I can understand your problem. I am never the less ready to answer any question able to eliminate the misunderstanding.

Nick Name,

Thank you for the information, if you could email me a higher resolution picture that would appreciated. Thank you very much,

10maya

Your answer is somewhat confusing and will not lead to a "safe" answer.

Roark's Formulas for Stress and Strain 6th Edition doesn't seem to cover the subject. It does talk about eccentrically loaded columns using the secant formula and other imperfection formulae. Perhaps there is now a 7th edition?

Better look in Den Hartog.