Use the Von Mises-Hencky Equation thus giving you a tri-axial state of stress on the wall element. Note that you need to model longitudinal stress as a function of applied load IN ADDITION to the hoop and tangential stress.
I would start at the base of the taper, do the calculation and then move down the cone, say 25% total overall length. Redo the calculation and continue until reaching the apex of the cone. Graph the results and see if you can mathematically model this triaxial stress as a function of length displaced down the cone from its base.
Okay, the model itself is basically Von Mises which gives you twice the gradient of stress squared equalling the cross product of stress vector with itself. You need to resolve the stress vector as hoop (x), radial
and longitudinal (z) directions, for example. These quantities are directly related to Thick Wall Pressure Vessel. Unlike TWPV theory where longitudinal stress is the result of end cap reaction to internal pressure, you would use APPLIED LOAD as the input variable and solve for stress accordingly. Note that the other two terms follow in similar fashion.
I've done a similar problem, finding the equation boils down to the following: Stress = sqrt(3) P [R^2 / (R^2 - 1)] when R = OD/ID for the tubing. P = applied longitudinal load/area of taper base, OD = outer diameter, ID = inner diameter at the slice of cone in question. My piece was large, NPS 12 pipe thick walled that was purposely machined with a taper through the ID. I put strain gauges to the OD and started to apply the longitudinal load with a hydraulic cylinder. Ultimately my measurement was something like 2.25% error, well within scientific measure for acceptance. I used a tremendous amount of points to pull it into Excel and graph the results.
Note that the model does not address shear, so these are principle stresses experienced by the wall of your tapered piece. If you have shear, you must add three times the shear stress squared as a second term to Von Mises.
At any rate, hope this helps somewhat. It is a lot of mathematics to grind through, but algebraically it is quite simple. Pay attention to the (D^2 - d^2) term in the denominator of the hoop and longitudinal stress. Multiplying the radial by this, i.e. P [(D^2-d^2)/(D^2 - d^2)] would collect terms while adding the numerator to the equation, P(D^2 - d^2).
This forum just doesn't lend itself well to mathematical modelling of typical engineering problems. Hope it is not too confusing, good luck with it.
Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada
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