Currently, there are no codes, equations, or technical papers that mention the Bourdon effect. We have hunted in vain for such a thing for years without success. Therefore we have a phenomenon that "everybody" knows how to use, how and when to apply, etc. but that nobody will put their name to. The reason, we believe, is because the traditional Bourdon implementation that everybody uses does not match up with reality.
Bourdon has little to do with the pressure translation of ovalized cross-sections into circular cross-sections - that is the pressure stiffening effect that most codes use to modify bend SIFs and flexibility factors. Bourdon is an attempt to take into account the strain that the piping undergoes when subjected to pressure. This strain is due to two components - the axial strain due to the pressure end cap effect (roughly PD/4tE) and then the Poisson effect (axial shrinkage due to radial and hoop expansion under pressure).
Virtually all pipe stress programs implement the Bourdon effect in the manner first developed for the MEC-21 pipe stress program, circa 1960. This method applies pressure elongation as a uniform strain to the entire piping system, in a way similar to thermal expansion (some codes, such as BS 7159, actually instruct the user to convert pressure strain to "equivalent temperature"). The actual pressure strain calculation is done as:
e = P(Ri * Ri) / (Ro * Ro - Ri * Ri) / E - V P(Ri) / (Ro - Ri) / E , or,
slightly more exact,but virtually identical for "thin" wall pipe:
e = [P(Ri * Ri) / (Ro * Ro - Ri * Ri) / E] (1 - 2 V)
Where:
e = uniform pressure strain
P = pressure
Ri = internal radius
Ro = outer radius
E = modulus of elasticity
V = Poisson's ratio
In the Bourdon method, this strain is then applied throughout the piping system, in the same manner as a thermal strain would be. The upshot of this is
(think piping systems loaded with thermal strain):
1) On an unrestrained system (i.e., a cantilever), this leads to no stress, non-zero displacements, and no anchor loads.
2) On a restrained system (straight pipe anchored at both open ends), this leads to compressive stresses and compressive forces on the restraints and zero displacements.
3) On a restrained system (straight pipe with intermediate anchors),this leads to compressive stresses, zero anchor loads, and zero displacements.
In real life, the situation would be:
1) On an unrestrained system (i.e., a cantilever), there would be tensile stress equal due to the end cap effect, non-zero displacements,and an anchor load (pressure thrust load).
2) On a restrained system (straight pipe anchored at both open ends),there would be tensile stresses equal to the Poisson's effect (due to hoop stress), tensile loads on the restraints, and zero displacements.
3) On a restrained system (straight pipe with intermediate anchors),there would be tensile stresses equal to the Poisson's effect (due to hoop stress), zero loads on the intermediate restraints, and zero displacements.
(Note that real life piping systems are much more complicated than any
of these three scenarios.)
For all load cases containing pressure (whether Bourdon is activated or not), CAESAR II (and probably most other pipe stress programs) then adds the constant value P(Ri * Ri) / (Ro * Ro) to the stress due to other loads (since this is required by most piping codes). So looking at the implications of different scenarios:
1) On an unrestrained system (i.e., a cantilever), with no Bourdon activated, this leads to a stress of P(Ri * Ri) / (Ro * Ro), no displacements, and no anchor loads. Technically this is correct for stress, incorrect for displacements, and incorrect for anchor loads.
2) On an unrestrained system (i.e., a cantilever), with Bourdon activated, this leads to a stress of P(Ri * Ri) / (Ro * Ro),displacements equal to
length * P(Ri * Ri) (1 - 2V) / (Ro * Ro - Ri * Ri) / E, and no anchor loads. Technically this is correct for stress, correct for displacements, and incorrect for anchor loads.
3) On a restrained system (straight pipe anchored at both ends), with no
Bourdon activated, this leads to a stress of P(Ri * Ri) / (Ro * Ro), no displacements, and no anchor loads. Technically this is incorrect (but conservative, as intended by most codes) for stress
(the stress should probably actually be tension equal to only the
Poisson term: -V P(Ri) /(Ro - Ri), correct for displacements, and incorrect for anchor loads.
4) On a restrained system (straight pipe anchored at both ends), with Bourdon activated, this leads to a stress equal to the end cap tension, less the Bourdon compression, or
just the Poisson effect: V P(Ri) / (Ro - Ri), no displacements, and compressive anchor loads. So this would be correct for the stress, correct for displacements, and incorrect for anchor loads.
5) On a restrained system (straight pipe with intermediate anchors),
with no Bourdon activated, this leads to a stress of P(Ri * Ri) / (Ro * Ro), no displacements, and no anchor loads. Technically this is incorrect (but conservative, as intended by most codes) for stress (the stress should probably actually be tension equal to only the
Poisson term: -V P(Ri) / (Ro - Ri), correct for displacements, and correct for
anchor loads.
6) On a restrained system (straight pipe anchored at both ends), with Bourdon activated, this leads to a stress equal to the end cap tension,less the Bourdon compression, or
just the Poisson effect: V P(Ri) / (Ro - Ri), no displacements, and compressive anchor loads. So this would be correct for the stress, correct for displacements, and incorrect for anchor loads.
In our opinion, the correct answer is to model pressure elongation as two distinct effects:
(1) a primary (force driven) load equal to the pressure end cap thrust load, modeled at every elbow, valve seat, or other thrust surface; and
(2) a secondary (displacement driven) uniform strain equal to the Poisson's effect of the hoop stress.
This sort of model would make each of the above layouts (as well as all in between) work out correctly. The problem would be that this would buck a forty-year old trend, and probably would not be easily implemented by most pipe stress software available today, without modification. A secondary by-product is that this sort of analysis would not provide the sort of conservatism that is currently allocated to longitudinal pressure stress by most codes.
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