hollow tapered cylinder deflection

huangyhg

hollow tapered cylinder deflection?
i need a little help finding and or deriving some forumlas for a project we have been assigned.
we are to analyze and redesign a typical traffic light.  i have all the dimensions and loads that i need, but i need but am having a heck of a time deriving the formulas for deflection of a tapered hollow cylinder!
i need to know both torsional and bending deflections.   i would just draw this all up in pro/e, but i am not too familiar with pro/m to do the fea.
anyone have any clue how to derive these forumals or where i could find them?
well, without going into integrals (probably double integrals), knowing deflection is a function of moment of inertia, i would find the deflection if it was a uniform shape using both the smaller and the larger diameter at the extreme ends, then average them.
i am shooting from the hip.
or you can do a piecewise model from basic beam theory.
taking the average of the result from each end appeals to my lazy character, but the tip deflection is very sensitive to the conditions at the base of the cantilever... so an average is likely to be unreliable.
cheers
greg locock
greg,
10 bucks says it's not "unreliable".  this would be analyzed as a cantilever.  given length is constant, deflection as it relates to "i" is linear.  therefore, not shooting from the hip after further thinking about it, the average will work.
chip
10 bucks it is, payable to eng-tips.
you seem to think that the deflection of a uniformly tapered tube can be accurately represented as the average of the deflection of a beam whose section is uniform consists of the tapered tip of the beam, and another with the same section as the base of the beam.
without even bothering to look at a formula this is obviously ridiculous. a fully tapered beam does not have an infinite deflection, your theory suggests it would, since the deflection of the beam that is the thin end (d=0) would be infinite.
i may be wrong. it does happen.
cheers
greg locock
seems to me chipb is suggesting an approach where you use the average (i.e. at mid-length) diameter to compute inertia, and assume it's constant.  although my gut feel suggests this is slightly off the correct value, it's probably quite close.
i'm going to guess that you need to use the inertia at approx. 0.4 * l from the base - strictly a gut feeling.
no time to check it...
tg
dammit!!!  can i just make the $10 payable to papa john's in your honor (or some local micro-brewery of my choice)?
i neglected to take the l^3 factor into account, which, after you take the moment out is l^2, therefore it's not linear.  
i stand corrected!
best option is to model it in as greg has stated.
i just did a quick calc. with excel, using some sort of superpostion, and came to the following conclusion:
better use fea.
tg
chipb, i must confess even though i hadn't worked it out specifically, a long while back we looked at the resonant frequencies of conical chimneys, so i had a pretty good idea of how tapered cantilevers work. i also used to work on leaf springs.
  
cheers
greg locock
chipb, greglocock, trainguy - all of you are on the right track for a good numerical solution, it is just very tedious but could be written into a spreadsheet program.
you can break the tapered cylinder into short segments of different (constant) diameters, all stacked one on top of the next to simulate the complete structure. compute the deflection at the top of the bottom cylinder (cylinder with the largest diameter), here is the catch - compute the slope of the deflection curve at the top of this bottom cylinder. the next cylinder (with the second largest diameter)in not plumb but is "leaning" at the angle just computed. do the deflection same calcs for each cylinder in turn and do the trig to compute the "offset" of each cylinder because of the cumulatively increasing "lean". the sum of all cylinder deflections and all offsets can be a reasonable approximation.
it's been a long time since i have done this, just worked through a simplifed madeup example by hand and compared with a sofware solution - it works.
fea or a beam / frame analysis program is the fastest way to get an accurate solution, if you have access to one.
failing that, setting up a spreadsheet as suggested by slideruleera will do the same calculations as a full fea would provide - just takes a bit more work to set it up, but you could then use the spreadsheet every time you need to do such a check.
third option is to get a hold of one of the standard handbooks of engineering stress and strain formulae. e.g. my copy of "formulas for stress & strain" by roark and young (5th edition) has all the standard formulae for uniform beams (tables 1 to 12), and then has some tables of modifiers for tapered beams (tables 13a to 13d).
hope this helps.