The history of this formula, despite of its simplicity, even on the conceptual standpoint, is however unexpectedly jagged and sometimes controversial.

The subject, indeed, even though apparently consolidated by, I would say, a secular use, is really affected by some confusion due to the various interpretations that over the years have superimposed each other and stratified (on this subject see ref. [4]).

A denomination issue is, first, faced by whom have studied engineering in Italian Universities. In Italy, the designation “Mariotte’s formula” is assigned to the equation for computing the primary hoop stress in cylindrical shells, whereas, in English technical literature, the designation “Barlow’s formula” is assigned to the same equation. Outside Italy, the only known “Mariotte’s formula” is the one for compressed gases. Based on what I know, the same occurs even in France.

Going back to the origins, that is to textbook “Traité du mouvement des eaux” by Mariotte (1700), ref. [1] (>>), at page 348 we find the following paragraph: « II. Discours, De la force des Tuyaux de conduite, et de l’épaisseur qu’ils doivent avoir suivant leur matière et la hauteur des réservoirs », whose reading does not provide any formula, but only the following rule: « I. Règle : Si la hauteur du réservoir est double, il y aura deux fois autant de poids d’eau, & par conséquence il faudra deux fois autant d’épaisseur de métal dans le tuyau afin qu’il y ait deux fois autant de parties à séparer. Si le diamètre du tuyau est 2 fois plus large, il faudra 2 fois plus d’épaisseur : car les mêmes parties du fer blanc ne feront pas plus chargées, & elles sont seulement doubles. ». This rule exhibits in nuce some of the elements that will be found in the formula, but not all of them, since, for example, the principle (equilibrium) is not declared, and the material strength is missing.

where: r is the internal radius, c is the material cohesive strength, x is the thickness and p is the internal pressure.

The current “Barlow’s formula” has been obtained by Goodman who, in his textbook “Mechanics applied to Engineering – 1914”, ref. [5], upon application of the so called Barlow’s theory to thick-walled piping, gets the formula pr_o=f_it where we find the outer radius,r_o, and the hoop stress, f_i, on the internal edge.

Goodman obtained this equation by imposing that the hoop stress on the internal edge of a thick-walled piping be equal to the hoop stress of a thin-walled piping, where really that computed is the stress averaged over the thickness, not a local stress. For the thin-walled piping, even Goodman, based on equilibrium considerations, obtained the same equation shown at the beginning of this article, with the inside diameter or radius, that implies that the computed stress is averaged over the thickness. A consequence of the Goodman’s equivalence is that if the hoop stress on the inside edge is shown, then in the formula the outer diameter shall appear, since the theory used to derive it is based on the principle that the hoop stresses have values in inverse proportion to the square of the radius where it is computed (the inside stress stands to the outside stress alike the square of the outer radius stands to the square of the inner radius). Should Goodman have imposed the equivalence with the thin-walled piping considering the average stress of the thick-walled piping, he would have obtained the equation with the inside diameter for the thick-walled cylinder, too. All these considerations are developed in detail in the technical paper (About Barlow’s and Mariotte’s Formulas) attached to this article (downloadable by whom is interested).

In technical literature (see attached paper) three “Barlow’s formulas” are considered: with the inside diameter, with the mean diameter and with the outside diameter (which is the most frequent one). The Barlow’s formula is usually considered valid for thin-walled piping only (D/t > 20). This statement anyhow is strictly related on the diameter adopted and how the stress so obtained is classified and used, as a matter of fact:

Even though the same hoop stress formula is used for either cylindrical pressure vessels or piping, the designation “Barlow’s formula” is almost exclusively adopted for piping only in both technical literature and standards / codes; practically, no pressure vessel design handbook and related standard / codes makes use of this designation.

The Barlow’s formula is used also to evaluate the piping limit pressure (bursting pressure).

Two limit states are to be considered for piping: the elastic service limit state (gross deformation or plastic collapse) and the collapse (bursting) limit state. The elastic service limit state is governed by the yielding strength; it is important in presence of threading connections (for example, the so called OCTG = Oil Country Tubular Goods) where the gross deformation may cause leakage. The collapse limit state is governed by a strength value that can be close to the ultimate tensile strength in case of materials having low strain hardening exponent, whereas has a value between yielding and ultimate tensile for other materials.

For steels having high strain hardening exponent, , as the austenitic stainless steels, it is necessary to use more suitable equations that appropriately consider this exponent (Zhu-Leis, Klever, Faupel, etc.). In the last equation, ε_ys is the yielding strain. For carbon steels, then we obtain: ε_ys = 0.2%, n ≤ 0.1, Y/T ≥ 0.745.

The determination of the collapse (bursting) pressure is, however, a so sensitive and complex subject that cannot be run out with the few hints here above, especially for materials with high strain hardening exponent.

Conclusions

References

Attachment - Technical paper – About Barlow's and Mariotte's Formula >>

pdf of this article