Snorgy,
Well, the formula the OP has used is not based on a natural sag bend. That's a permanently bent pipe, as that Sa value kind'a says its a yield stress with no safety factor, and the stress formula is based on maximum remaining allowed axial stress after subtracting the axial component of pressure stress.
sopha,
Since you mention "natural sag" I think you're trying to calculate natural sag, which for a beam with pinned ends the maximum would be 5WL^3/(384EI), where W is a distributed load on the pipe, not an internal pressure (which would cause no sag). Problem is that you've used an end moment bending equation M/EI to get radius of curvature which for end moments is not 5WL^4/384EI.
So is this,
1) a natural sag bend problem
2) a permanent bend radius problem
3) an elastic bend radius problem
The formula you are using appears to be derived from a total axial stress equation with radial stress subtracted, to get the allowable bending stress. The problem with the equation is that it only shows bending stress minus pressue stress and pressure stress isn't = PD/4/t. That pressure (radial) stress is PD/2t. But since we want axial stress and axial stress due to radial stress is Poisson's ratio * radial stress, it can be reduced. Since Poisson's ratio (u) is 0.3 for steel, and 1/2 is 0.5, it seems that 1/2 the radial stress is being subtracted, which is conservative, so it could be used, if you want a conservative result.
Now we have Sa - u * PD/2/t in there and if u = 0.5 then,
Sa - 1/2 * PD/2/t
If we go back to
Bend Radius, Rb_in = E_psi * R_in / Sb_psi
and substitute the remaining allowable stress for Sb
E_psi * R_in / (Sa - 1/2 * PD/2/t)
E_psi * R_in / (Sa - PD/4/t)
So, that should be conservative, if we chose Sa properly.
Sa here is Allowable Axial Stress, which for some codes should be limited to 0.9 * SMYS, but see your design code to be sure if axial stress or combined stress, or both need to be checked and what all the allowable stresses are for whichever stress you are considering must be.
We are saying the bending stress can be maximum when equal to Allowable axial stress - pressure stress in the axial direction.
If we are checking this stress to find bend radius with 0.9 SMYS as the allowable, we are assuming that the pipe is under pressure and elastically bent, not bent permanently. Radial stress and axial stress must be checked according to the code, either individually or both combined. This also assumes no temperature change.
If you calculate the bend radius using the allowable equal to SMYS (I assume 52 ksi), its a permanent bend, but when the bending force is removed, the bend stress goes to zero, so if checking stress for the pipe allowable operating pressure, only radial stress PD/2t remains; there is no axial stress from bending, only axial stress from u * PD/2t, again assuming thermal stress is zero.
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"Pumping accounts for 20% of the world’s energy used by electric motors and 25-50% of the total electrical energy usage in certain industrial facilities."-DOE statistic (Note: Make that
99% for pipeline companies)
