Selection of Principal Stress
Selection of Principal Stress
(OP)
Hi,
I am currently working on a cylinder with external pressure applied on it. It is a thick cylinder and I am using Lame's equations to solve for the maximum pressure at yield point.
Now the three stresses are the tangential/hoop, radial and the axial. When I combine these stresses using Von Mises criterion to solve for maximum pressure, which one should be selected as sigma_1, sigma_2 and sigma_3?
I chose the tangential stress as sigma_1 in attached solution, but am not completely sure if it is right.
Any help would be appreciated.
Thanks,
Mike
I am currently working on a cylinder with external pressure applied on it. It is a thick cylinder and I am using Lame's equations to solve for the maximum pressure at yield point.
Now the three stresses are the tangential/hoop, radial and the axial. When I combine these stresses using Von Mises criterion to solve for maximum pressure, which one should be selected as sigma_1, sigma_2 and sigma_3?
I chose the tangential stress as sigma_1 in attached solution, but am not completely sure if it is right.
Any help would be appreciated.
Thanks,
Mike





RE: Selection of Principal Stress
Let von Mises stress==sM,
sigma_1==s1, etc.
then
2*(sM^2)=(s1-s3)^2+(s2-s3)^2+(s3-s1)^2
Then compare sM to the yield stress, sY. If sM<sY, then the "Maximum Energy of Distortion Theory of Failure" predicts no failure. Whether this is the correct Theory of Failure to use, that depends on your application. Some design codes call for one Theory of Failure to be used (for instance, max. shear); some design codes use the typical industry practice to determine Theory of Failure.
If you must select one stress over the other to be s1, s2 and s3, then use the largest stress, (hoop or tangential as you call it) as s1, s2 is the longitudinal stress (along the axis of the cylinder), and s3 is radial stress. This of course assumes the shear stresses are zero; otherwise the calculation of "sM" is more complicated.
RE: Selection of Principal Stress
So, in this case would you still select the tangential stress as sigma_1?
RE: Selection of Principal Stress
RE: Selection of Principal Stress
RE: Selection of Principal Stress
RE: Selection of Principal Stress
btrueblood & others should be correct, however it is normally assumed that S1 = hoop, S2 = axial and S3 = radial. Cases in which radial becomes larger than hoop can only happen for bending-dominated problems, but not in your case of a cylindrical pressure vessel.
Regards
RE: Selection of Principal Stress
Here would be my back of the envelope, so I could judge orders of magnitude. We know the 'thin shell' approximations are very easy to compute--for an internal pressure on a thin cylinder, the hoop and axial stresses are approx. pr/t and 0.5*pr/t, respectively. What is the radial stress? In this case, it goes from "p" at the inside of the cylinder, to zero at the outside--the most it could be is "p", while those other two stresses, hoop and axial, are much larger. So from the thin cylinder approximation, you can guess that hoop stress>axial stress>radial stress--standard practice is to set the principal stresses sigma_1>sigma_2>sigma_3. I realize you are doing the thick cylinder analytical solution, nevertheless, I think the thin cyl. approximations are very good for estimating relative magnitudes of the stresses, which should give you pretty good idea what the relative magnitudes of the stresses computed the thick cylinder equations. Would you agree that this is a good way to ballpark the estimate of the stresses?
BTW--in the thin cylinder approximations, they write "p" but what they really mean is "delta-p"--that is, the difference between outside and inside pressures.
RE: Selection of Principal Stress
btrueblood is right. I get the same answer even if I select sigma_1 = axial and sigma_2 = hoop, in both cases i.e using Von Mises failure theory and Max Shear Stress theory as I did using sigma_1 = hoop and sigma_2 = axial.
What is not clear to me is the theoretical reason behind this.
prost:
From what I got from the books is like you said, that generally the highest stress is selected as sigma_1 which is hoop here, but then why select the highest as sigma_1 if I get the same answer by switching them?
RE: Selection of Principal Stress
Regards,
Cory
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RE: Selection of Principal Stress
RE: Selection of Principal Stress
RE: Selection of Principal Stress