The Magic of Buoyancy and Hydrostatics –Buoyancy and Effective Forces

More than 2000 years after Archimedes formulated his famous law, there is today still some disagreement as to the physical understanding and actual acceptance and application of Archimedes’ Principle. This paper is prepared in the support of the full validity of Archimedes’ Principle, always calculating buoyancy as the weight of displaced fluid, without exceptions.

Available literature on buoyancy effect in an oil well reveals that there are different ways of understanding the physics of buoyancy and hydrostatics when assessing forces in a drill pipe.Published articles and books reflect confusion in the industry with two different schools, one being the buoyancy school based on Buoyancy' law related to the effect of displaced fluid, the second being the pressure-on-a-flat or piston method school.The latter claims that there need to be an exposed flat subject to pressure from below to have buoyancy.Many text books in physics support this view explaining buoyancy with higher pressure at the bottom than at top of an immersed object.In a free floating situation both methods will yield the same gross upthrust, however, the problem emerges if an element has no vertical flat exposed to pressure from below, like a cylindrical, vertical riser connected to the seabed.The flat-supporters claim there is no buoyancy while volume-supporters stick to the principle of Archimedes giving the riser an upthrust equal to the weight of the displaced fluid.Some authors go longer than just reflecting different understanding of the principle, some actually claim that Archimedes' Principle has shortcomings and does not always apply.As a matter-of-fact, there are views in which the critics express direct opposition to Archimedes' law!In 1980 Goins, after having performed some simple experiments, concluded that; "buoyancy is equal to the weight of the displaced fluid is not always true" (Goins, 1981).
In 2014 F.M.S. Lima et al. stated: "It would not be surprising that Archimedes, one of the greatest geniuses of the ancient world, had enunciated his original propositions with remarkable precision and insight, however there are some instances which he did not realize.One of these instances is shown here to be an exception to Archimedes Principle" (F.M.S. Lima et. Al, 2014).

Archimedes Principle
History says that Archimedes formulated his famous law more than 2000 years ago: "When a body is fully or partly immersed in a fluid, the buoyancy force is equal to the weight of the displaced fluid".This is simple and square cut.Easy to understand and easy to apply.Discussion could have ended with that statement.However, at some point of time an alternative definition of the magnitude of the buoyancy force emerged: "When a body is fully or partly immersed in a fluid, the buoyancy force is equal to the sum of external hydrostatic pressure forces acting on the object" arnfinn.nergaard@uis.noNote that the effective tension at a certain level can be derived by using equation (2) directly, thus not needing to go via the true tension of equation ( 3).Now, given case b), with the riser connected at the bottom, and with the same top tension, the bottom hydrostatic force of case a) seem to be replaced by a mechanical true compression force from the bottom.According to AP both cases have the full effect of buoyancy while the AP critics claim that only case a) has buoyancy.While the critics claim that the buoyancy in this case appears at the bottom flat only (line CB) the AP implies that the buoyancy develops over the full volume of the element, represented by triangle ABD, with full buoyancy seen only at the top, as line DA, identical for both cases.
Applying the above relations, and having the same top tension, the outcome is that cases b) and a) are equivalent mechanical systems implying that the force diagram shown represents both.
The following is concluded: A submerged riser pipe with a set top tension, with given properties and geometry is subject to the same true and effective tension independent of the pipe being connected to the bottom or not.The effective tension is calculated from equation (3) as above.Alternatively, it can be found at any level directly as weight of pipe element below the level plus weight of fluid in the element.Again, the true tension (or compression) in terms of pressure on flats is not part of the equation when dimensioning the riser for axial strength.When true tension is considered the hydrostatic element must be deducted to get to the dimensioning tension, T e , as demonstrated by equation 3.

Hydro-Mechanical Model for Effective Forces
C.P. Sparks derived the relation between true-and effective tension and internal and external pressures in the very simple equation: The hydro-mechanical model and understanding that supports the equation is presented in his publications (Sparks 1984(Sparks , 2005).The explanation is based on the acceptance of hydrostatic pressure 'invading' solid material and representing a force internally in the same manner as at the external flat.A simple illustration of this principle is presented in a force balance diagram in Fig 3.This shows that the Archimedian line not only coincides with effective tension, they are actually the very same line.This also shows that buoyancy is not depending on absolute pressure.Buoyancy is calculated on the basis of displaced fluid, however, pressure gradient may be used provided that internal hydrostatic pressures are included.
The principle shown in Fig 3, with hydrostatics invading a solid and having impact on vertical forces in the solid, is based on the 'assumption' that the hydrostatic pressure is isotropic, even inside solids.The fact that pressure in fluids is isotropic is fully and broadly accepted and understood.This is not the case for hydrostatic pressure/stress in solids.
The question is: How can it be that radial, horizontal pressure at the periphery of a vertical cylinder gives axial force inside?
In the following a demonstration of some effects supporting the fact that external peripheral pressure gives real axial forces are given:  This shows that axial deviatoric force resulting from lateral pressure at a given level is equal to the deviatoric force resulting from a mechanical tension force with a magnitude equal to the weight of the displaced fluid above the level.Letting σ 1 represent axial -and σ 2 , σ 3 lateral stress:

Buoyancy and von Mises Stress
Substituting these into the von Mises stress equation: This gives the von Mises stress: This means; the isolated effect of external fluid hydrostatic pressure on a vertical cylinder gives a von Mises equivalent or effective stress equal to the hydrostatic pressure.Over the cross section of the cylinder this stress corresponds to a force T = ρghA = Weight of the displaced fluid above the level considered.

Effect of Lateral Pressure -the Bridgman Experiment
Bridgman's paradox: In 1912, Percy Bridgman performed an experiment in which a cylindrical bar of glass was exposed to lateral pressure in a high-pressure test cell as shown in Fig 5 (Bridgman, 1912).The chamber was pressurized until the bar parted.Bridgman demonstrated how sufficiently high pressure in the chamber made the bar part as if in axial tension.This test specimen was clearly exposed only to radial pressure and hence, the surprising effect became known as Bridgman's paradox., 2013).The fracture seems to happen at different (random) locations between the glands of the test cell.Another important observation is that fracture surfaces are perfectly perpendicular to the longitudinal axis of the bar with a flat and shiny appearance.As a part of the project (Fossli, 2014) test specimens were also subject to bending fracture and to mechanical tensioning fracture in a tension test machine.In the bending experiment, the fracture surface had distinct tension and compression zones, the tension zone being perpendicular to the longitudinal axis of the rod.
In the tension machine experiment, the fracture surface was found to be perfectly perpendicular to the longitudinal axis of the rod as in the hydrostatic tests.Finally, the fracture surfaces were examined in a SEM (scanning electron microscope) with the following result; quote "The tension surface is similar to the fracture surfaces in both the Bridgman experiment and the tension side of the bending experiment".

Conclusion
This paper deals with the effect of radial pressure on axial forces in a cylindrical element.Effective tension theory says that external radial pressure on a cylinder gives an effective axial tension force equal to the pressure multiplied by the cylinder cross sectional area.This paper shows that effective tension in a vertical submerged cylinder corresponds with Archimedian tension.This is supported by demonstrating that lateral pressure at a given level gives axial tension at this level as if mechanically tensioned by a force equal to the weight of the displaced fluid above this level.Furthermore, Bridgman experiments at University of Stavanger have verified that the effective tension from lateral pressure tension the test specimen until fracture at a pressure equal to the tensile strength of the material.Finally, it is shown that applying effective tension for stress analysis is supported by classic analysis in the form of deviatoric stress theory and the von Mises criterion.Consequently, this implies that buoyancy develops over the volume of a submerged body and not from hydrostatic pressure on the bottom of the object.
This paper supports the full validity of Archimedes' principle, always calculating buoyancy as the weight of displaced fluid, without exceptions.

Figure 1 .
Figure 1.Immersion with and without flat Fig 1 have zero effective tension at the bottom.This assures that there is no buckling-or tension force.The true compression force at the bottom is a hydrostatic isotropic component not contributing to principal forces and hence not a part of the equation when dimensioning the riser for axial strength.Now, consider a closed end pipe with fluid inside only, Fig. 2. The mechanical systems of Cases a) and b) are likewise found to be equivalent, however, as might be expected, with opposite effect as compared to the external fluid case.

Figure 2 .
Figure 2. Fluid-filled pipe with and without 'bottom'

Figure 3 .
Figure 3. Force balance with internal hydrostatic forces and Deviatoric ForcesFor demonstrating the effect of external pressure on deviatoric forces in a vertical pipe, a demo example is shown in Fig 4.Imagine a solid cylinder with a starting stress (compression) from own weight equal to σ.In case a) the cylinder is exposed to external fluid only, in case b) to mechanical top tension T only.

Figure 4 .
Figure 4. Submerged vs dry riser Consider case a) of Fig 4, a pipe exposed to lateral pressure only.