At any capped end or valve, you'll develop longitudinal stresses unless the cap or valve is independently supported.
When you put hoop stresses in a cylinder, the cylinder gets shorter from Poisson's effect. If the pipe is restrained, this will give longitudinal stresses about 1/3 of the hoop stress.
At an unblocked ell or tee, you'll develop longitudinal stress in a pipe due fluid flowing.
As I recall from strength of materials , remember the "law of the hot dog",ie , the hot dog always cracks in the longitudinal direction because the circumferential stress is equal to twice the longitudinal stress.
for circumeferncial stress, a free body diagram of a cylinder of length L, diameter D, wall thickness t and internal pressure P has the following stress relationship:
P*D*L=2*S,c*t*L, or S,c= PD/(2t)
For the same cylinder, the longitudinal stress is found by a similar free body diagram:
P*pi*D^2/4= pi*D*t*S,l, or S,l=P*D/(4t)
so the circumferntial stress is twice the longitudinal stress
The longitudinal stress in a pressure vessel is simply the reaction of the bulkhead to pressure. In the absence of such end caps, your situation for an infinitely long pipeline, I can see your point.
For pipeline computations, I have often simply dropped the longitudinal stress stated in Thick Walled Pressure Vessels, then applied Von Mises-Hencky. Of course this is in the absence of themal fluxuations which may cause stresses due to restraining. Also, JStephen correctly points out the implication of Poisson's Ratio.
Wow, "Law of the Hot Dog"! I never quite thought of the phenonema as such, probably won't forget about it now! "Law of the Hot Dog", sounds more like Mr Smarty-Pants in hockey getting his just beets.
Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada
From a Stress Intensity (Tresca) perspective, the longitudinal stress exists, but is irrelevant. Stress intensity is defined as "the difference between the algebraically largest principal stress and the algebraicaly smallest principal stress at a given point." [VIII-2 4-112(a)] So if you take your principal stresses from highest to lowest as P1, P2, and P3 then stress intensity is P1-P3.
If you agree that the circ stress (P1) is the highest tensile stress, and longitudinal is half that (P2) (and tensile), and that the radial stress is close to zero (except for thick wall cylinders) then the stress intensity = circ stress - radial stress. The longitudinal stress plays no role, though it does exist.
The von Mises criteria offers a similar approach to Tresca, and if you play with some numbers, you'll find that the longitudinal stress has little impact on the von Mises stress also.
Longitudinal stress does not exist "no matter how long the pipe". It depends on the pipe conditions. For example, water pipe is typically laid in 20' sections with push-on gasketed joints between sections. In that case, there should be very little longitudinal stress. The ends are restrained by thrust blocking or by friction against the soil, not by stress in the pipe. Also, as far as I know, in most cases, the allowable stress will be based on the maximum hoop stress or longitudinal stress as established by some code, not by theories of failure.
Longitudinal stress exists and is relevant when considering loads other than pressure. e.g. combined stresses between pipe supports/vessel saddles; combined stresses on a column due to wind/seismic; nozzle loads; etc. For example,if you had a very large longitudinal moment on a nozzle, longitudinal stress could govern when adding the combined effect of pressure plus external loads.
Longitudinal stress is also important when using the new high temperature creep resistant ferritic materials. These new alloys have strong parent matieral , but the weld heat affected zone of a butt weld has only 60% of the strength of the parent material. So, even if the butt weld is exposed to only 50% of the circumferential stress, it can become overstressed if one adds longitudinal gravity loads, bendign momnets, shear stresses, and other pipeline stresses.
