buckling load of a fishing rod

huangyhg

buckling load of a fishing rod
does anyone know the formula for the buckling load of a fishing rod, or, rather, of a cantilever column carrying a weight via a rope over a pulley at its top?
consider a cantilever column like a flag pole, of total height "l" .  a smooth rope is independently anchored very close to the base of the pole, and runs up beside the pole through several frictionless eyelets to a frictionless pulley attached to the top of the pole.  the rope turns 180 degrees over the pulley and runs down a small way to where it has a hanging weight "w" attached to it.  the eyelets are rigidly attached to the pole, and are spaced uniformly l/n apart:  they prevent the rope from moving laterally relative to the pole, but can impose no vertical force on it.
ignore all eccentricity effects, such as those from the eyelets' offset from the pole's centreline, and the finite size of the pulley.  if the hanging weight was simply attached to the top of the pole then the pole would buckle at a w value of
p^2*e*i/(2l)^2   (where p = pi)
which is the standard euler result.  however the presence of the pulley, the rope and the eyelets has two confounding effects.  the first effect is that the looping of the rope over the pulley means that the axial compression in the pole is 2w, not just w.  the second effect is that the rope tension acts through the eyelets to impose a stabilising set of forces on the pole once any deflection begins.
i have done some algebraic manipulations for the case where n is infinite, which would apply if the pole was hollow and the rope ran up inside it.  these convinced me that for this special case the two effects cancel each other out exactly, so the buckling weight remains as given in the above formula.  however i cannot crack the case where n is finite, since i have to allow somehow for potential buckling modes between the eyelets and cannot see how to assess the degree of anti-buckling constraint that these eyelets provide.  i cannot even get my mind around the other special case, that with n=1.
any thoughts out there in the ether?
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what a great question!
i'm not a structural engineer, but i've tackled a few of these types of problems before.  i'll give you a few general comments that may help you to look at the problem from a somewhat different perspective.
my first comment is that you cannot - in reality - assume that the eccentricity, e, is zero.  buckling only occurs when eccentricity is present in either the load or the pole.  go back and look at how euler solved the problem.
eccentricity can/will be different for each eyelet, the pulley, and the connection point of the rope to the ground.  two kinds of eccentricity can be present: the 1)smallest and 2) largest separation between the rope and pole at the pulley, eyelets, and rope anchor point.  in my "vision" of the problem, only one "e" exists at the pulley and rope anchor point.  my gut says that if the rope's anchor point has the least eccentricity (i.e. is is "at" the pole), then the point with greatest eccentricity will control the failure mode.
and i haven't even considered the "e" at the weight "w"!
i don't think that you will find a simple numerical solution to this problem.  you're probably looking at a series of calculations/computer simulations to find a "family" of solutions.  or one "hairy" set of integration problems!
hi denial
interesting thread my suggestion to try and solve this would be to consider the pulley at the top of the rod to provide an equal offset of the rope to both sides of the rod in doing this then you trully have 2w acting on the top
of the rod and with no offset to consider, as the bending moment on the rod would be cancelled out and i would also ignore any contribution of stabalisation from the eyelets. next i would do a calculation based on the rods second moment of area and assuming the rod is tapered i would use the mean value for the rod. finally i would now revert to practical methods by placing the load i calculated onto the cable and observing the effect on the rod. whilst the method i suggest may not be accurate it should at least be somewhere near, which at best is all that you will achieve with the buckling formula as they only estimate the buckling loads also.
regards desertfox
thanks for your comments, chaps.  i had a sneaking suspicion that nobody would be able to point me towards a theoretical perspective on the problem.
however, part of me is still a bit surprised.  the "n=1" case i referred to in my original post is really just a cantilevered jib crane with its arm close to vertical (which is how i encountered it).
what you all seem to be missing is that instability is a matter of energy: for the critical load the energy made available by the displacement of the load is equal to the elastic energy stored in the bar for the same displacement.
now it's true that the load on top of your pole is 2w, but when is starts to buckle its travel will be half that of a w load fixed on top, and consequently the energy available is the same!
i'm not surprised of your result for n=∞ , as the change in length of the pole at buckling (the only effect that can change the balance of energy by displacing the load) should be an infinitesimal of the second order.
the conclusion is that the buckling of a jib crane, as you suspected, is exactly the same as if the load was sitting on top of it.
prex
seems to me to be a problem best solved by test and experiment....in other words, pack up your rods and bait and go fishin.....
i think trussdoc should get the "first annual piscatorial award".  anyone that can come up with another great reason (excuse) to go fishing deserves the award, especially if it is engineering related!  worms anyone?