‘Effective tension’ is one of the key concepts in pipeline and marine riser engineering since it tells the user how to account for the pressure in the fluid inside and outside the pipe. Although known and understood for at least 60 years, engineers still sometimes overlook it and get into trouble. Prof Andrew Palmer and Agustony Sabtian discuss the outcome of a recent National University of Singapore experiment aimed at removing any remaining uncertainty.
It has been known for a very long time, at least since the early work of Lubinski (1950), that buckling calculations about the longitudinal force in a pipeline or a riser need to take into account the longitudinal force in the fluid contents as well as the longitudinal force in the wall of the pipe. That is why a longitudinally-constrained pipe can buckle under internal pressure alone, as can readily be demonstrated by experiment (Palmer & Baldry, 1974). In that instance, the longitudinal force in the pipe wall is tensile because of the Poisson effect, but there is a larger longitudinal compressive force in the contents, and so the resultant longitudinal force is compressive and the pipe can buckle if the resultant longitudinal force is large enough. It turns out that the condition under which buckling can occur is a simple one: the constrained pipe buckles when the resultant compressive force, counting both the pipe wall and the contents, is equal to the calculated buckling force that corresponds to the pipe dimensions, material and end conditions.
The reality of this effect is understood and recognised, and it routinely forms part of calculations (see, for example, Sparks, 2007) and of codes (see for example, Det Norske Veritas, 2010, section 4 G 306). Neglect of it has occasionally led to accidents. However, arguments about the point still surface occasionally. They are often based on an assertion that the contained fluid cannot exert a lateral force on a pipe, and that in consequence the pipe cannot buckle, or alternatively that a pipe cannot buckle if the pipe wall is in tension.
The objective of the experiments described here is to demonstrate the effect by a simple theoretical analysis, followed by a straightforward experimental demonstration, designed to leave no room for argument about the details.
It is helpful first to consider an initial straight column rather than a pipe. That example has the advantage of not being controversial: nobody argues that an Euler column cannot buckle because there is no lateral force on it. The classical structural analysis shows that the column buckles under a load that can readily be calculated, and many experiments confirm that..
Figure 1(a) shows a weightless column. Initially the column was straight and stress-free, and lay along the chain-dotted line. Under load, the column has deflected away from the original line. It is loaded by equal and opposite compressive forces that act along that line; there is no lateral load. AB is an element of the column between two parallel lines drawn perpendicular to the chain-dotted line. Figure 1(b) is an expanded view of the element AB and the lines that define it, and shows the forces and moments that act on it. At the upper end A, there is a downward force equal to the compressive force in the column, as can be confirmed by considering the equilibrium of the column section above A. At the lower end B, there is an upward force equal to the compressive force in the column. Because the column is deflected, the downward force at A is not in line with the upward force at B. That offset creates a couple: if element AB is to be in equilibrium that couple has to be balanced by bending moments at A and B. It is because those moments create curvature that the column can buckle, even though there is no lateral load.
The analysis of Euler buckling looks for the condition under which an initially-straight elastic column can remain in equilibrium even though it has deflected sideways. The lowest critical load at which that can happen is close to the load at which real slender columns buckle. At the cost of increased mathematical complexity, the analysis can be made more sophisticated by taking account of large deflections, but the practical conclusion is essentially the same.
Consider next the pipe shown in Figure 1(c), internally pressurised by a weightless fluid and deflected away from its initial position in which the tube axis lies along the chain-dotted line. Again define an element of the pipe and its contents AB between two parallel intersecting planes perpendicular to the line. That element is redrawn in Figure 1(d). The downwardpointing heavy arrow at the top represents the force exerted by the pipe and the fluid above A across the plane A and onto the element AB. The upward-pointing heavy arrow at the bottom represents the force exerted by the pipe and the fluid below B across the plane B and onto the element AB. Those forces are not in line, and again the offset creates a couple.
Figure 1(e) shows the fluid (not the tube and fluid together) between the parallel planes A and B. The heavy arrows represent the forces exerted on the fluid element between A and B, by the fluid above the element at A and below the element at B. Again those forces are not in line, and together they create a couple. There is no shear across the fluid boundaries at A and B. The element of fluid is in equilibrium. There must therefore be a balancing couple, represented by the circle with arrowheads, which is the resultant couple that corresponds to the forces exerted by the wall of the tube on the element of fluid. An equal and opposite couple is exerted by the fluid element onto the wall of the tube between A and B. Exactly as in the case of the Euler column in Figure 1(a), it is that couple that can create buckling, and that needs to be accounted for in calculations.
These issues are explored at greater length in numerous publications. They take account of additional factors such as the weight of the fluid contained within the tube, but again without altering the underlying conclusion. Lubinkski’s original work (1950) did not use the term effective tension, and that came in later (Lubinski, 1977). In retrospect, it was perhaps not the ideal choice of words, but it is nowadays so widely used that it would be counter-productive to try to change it.
Figure 2(a) is a uniform straight tube. Two possible loading systems are illustrated in Figure 2(b) and 2(c).
Loading 1 in Figure 2(b) loads the tube as a pin-ended column, as in the classical Euler buckling analysis; the internal pressure is atmospheric. The axial load is transmitted to the pipe wall by end plugs. The tube wall is under longitudinal compression, and the longitudinal stress is compressive and equal to the axial load divided by the cross-section area of the pipe wall. The circumferential hoop stress is zero.
Loading 2 in Figure 2(c) loads the tube by applying internal pressure. The pressure is contained by frictionless end plugs at both ends, but now the plugs are slightly different, so that they are free to move axially within the tube. The plugs have to be constrained so that the internal pressure does not push them out of the tube, but the plugs are free to rotate in the plane of the diagram, so as to provide the same pin-ended end conditions as in loading 1. The longitudinal stress is zero. The circumferential hoop stress is tensile. The force on each plug is the internal pressure multiplied by the internal cross-sectional area of the tube.
In summary, then, the longitudinal force is carried wholly by the pipe wall in loading 1 but wholly by the pipe contents in loading 2. The accepted theory of effective longitudinal force tells us that the resultant longitudinal force at which the pipe will buckle will be the same in loading 2 as in loading 1.
If, on the other hand, the longitudinal force in the contents does not need to be taken into account, as objectors to the effective force concept occasionally argue, then the pipe will not buckle at all in loading 2, and the internal pressure can be increased up to the point at which the pipe bursts. This allows a straightforward experimental test of the theory.
The tube was Swagelok 316L stainless steel, outside diameter 9.53mm, wall thickness 1.25mm. The elastic modulus measured in a tension test was 188.5GPa. Significant departure from linearity began when the stress reached 175MPa, the 0.5% yield stress was 280MPa, and the ultimate tensile stress 580 MPa.
Sabtian (2012) describes the experiment in detail. Figure 3 is a sketch showing how the tube was mounted in the Tritech 100kN testing machine, for the test under internal pressure, loading 2. The same set-up was used for loading 1, but with the pump, valve and pressure gauge removed. Two different tubes were used for loading 1 and loading 2: they had the same specification and identical dimensions. Figure 4 is a photograph of the end bearings in loading 1. The tube was mounted so that it could laterally deflect freely in any direction, and the central deflection was measured by two perpendicular pairs of Sakkae 13FLP100A potentiometers, so that absolute deflection could be calculated as the vector sum of the two deflections. Figure 5 is a detailed drawing of the top end plug in loading 2.
The scheme described in Figure 2(c) is an ideal experiment with frictionless end plugs. A practical implementation has to contend with some friction in the end plugs.
Figure 6 plots the longitudinal force against the lateral movement at the mid-height of the tube, for loading 1 (longitudinal compression applied externally) and loading 2 (internal pressure). The longitudinal force in loading 1 is the force measured by the load cell. The longitudinal force in loading 2 is the force measured by the load cell, but because of the effect of friction on the end plugs that force is a little smaller than the measured internal pressure multiplied by the internal cross-sectional area. At the end of the test, the load cell force is 571N and the pressure multiplied by the internal area is 660N, and therefore the tube wall carries a longitudinal tension of 89N. It is to be expected that the tube will be put into tension, because in the ideal situation with frictionless plugs (Figure 2(c)) the pipe wall would be in pure circumferential tension, and the tube would therefore shorten longitudinally because of the Poisson effect. Partial frictional restraint therefore puts the tube into longitudinal tension.
It can be seen that the loads at which the deflections become large agree well. It is not to be expected that the deflections will coincide perfectly. Although the two tubes in the two tests were nominally identical, each of them has an initial out-of-straightness, and the out-of-straightness will not be exactly the same: this point is quantified below.
The same graph includes the calculated buckling load. The Euler buckling load of an elastic pin-ended column with length L, elastic modulus E and second moment of area I is π2EI/L2. In this instance E is 188.5GPa (1.885×105N/mm2), I is 285.0mm4 and L is 920mm. The corresponding buckling load is 626N.
A real column can never be perfectly straight, and so it begins to deflect at loads less than the buckling load. Southwell (1936) put forward an elegant method for analysing buckling tests on columns. His method makes it possible to determine the elastic critical load and the initial out-of-straightness without continuing the test to the point at which the deflections become large and the column begins to deform inelastically. The relationship between the load and the lateral deflection becomes (Britvec, 1973)
where P is the compressive force in the column Pcr is the Euler critical load Δ0 is the initial out-of-straightness when the compressive force P is 0. u is the additional lateral deflection that occurs when the compressive force is increased from 0 to P. Equation (1) rearranges into
Southwell plots u/P against u. The slope of the resulting line is 1/Pcr, and the intercept on the u/P axis is Δ0/Pcr.
Figure 7 shows the Southwell plots for loadings 1 and 2. In the loading 1 case, the buckling load derived from the plot is 652N, and the initial out-of-straightness is 0.9mm. In the loading 2 case, the buckling load derived from the plot is 630N, and the initial out-of-straightness is 1.9mm.
The linearity of the Southwell plot depends on the dominance of the first Fourier component of the out-of-straightness. That component becomes dominant as the load approaches the Euler critical load: that is discussed in detail by Britvec (1973). The linearity will not be retained once plastic deformation begins.
The results allow a three-way comparison between two independent experiments and theory, as follows:
buckling load (N)
experiment, loading 1 652
experiment, loading 2 630
A further comparison can be made with Figure 6.
The agreement is good. It is consistent with the accepted effective tension theory of how to treat longitudinal force in internally pressurized pipes, and it vindicates that theory. It gives no support to those who have argued that internal pressure cannot create buckling. According to the counterargument, there ought to be no buckling under loading 2: that is incontrovertibly not the case. Hopefully this will resolve the argument once and for all. OE