A strain-amplification phenomenon sometimes associated with pressure surges in liquid petroleum pipelines may cause a latent defect in a pipeline to fail at an unexpectedly low pressure. Pipeline operators need to account for this phenomenon in their integrity management plans for those pipelines or pipeline segments that are likely to be exposed to significant pressure surges.
Most liquid pipelines and some gas pipelines are subject to pressure surges during their operation. Pressure surges can occur during start-up and shutdown, and under transient conditions such as loss of power to the pumps, inadvertent rapid valve closure, and vapor pocket collapse. Any event that rapidly disrupts the steady-state velocity of fluid flow in a pipeline can cause a pressure surge.
Pressure surges, slug flow, and vapor pocket collapse can damage supports, anchors, bends, or the pipe itself. A pressure surge may cause a pipeline to rupture simply because the pressure increase associated with the surge reaches the failure pressure level of a defect in the pipeline.
Occasionally, a rupture induced by a surge occurs at a total pressure (surge pressure plus operating pressure) that is less than the pressure of a recent hydrostatic test, less than the allowable pressure of 110% of the maximum operating pressure (MOP), or even less than a recent high operating pressure of a liquid pipeline. In such cases it appears that the defect that caused the rupture failed at a dynamic pressure less than the static pressure at which it might have survived.
Either one or both of two phenomenon are thought to contribute to this occurrence. One possible effect of a surge on a defect arises from the extremely high strain rate associated with the incremental pressure increase created by the surge. The other may arise if the sonic wave velocity of the pressure surge is greater than a characteristic velocity of the pipeline.
In the latter case, the dynamic hoop strain is amplified beyond the static hoop strain calculated for the static pressure (equivalent to the surge plus operating pressure). This latter phenomena is discussed in this article.
Prior to the discussion of strain amplification effects, it is useful to note that high strain rates can cause a defect in a carbon steel material that would cause it to fail in a brittle manner at a lower-than-expected level of applied stress. Since a surge produces an extremely high strain rate, it is possible that the strain caused by a pressure surge could cause a sharp defect to propagate in a brittle manner, while the same defect might not fail at all if subjected to the same stress level under quasistatic loading.
If a defect in a pipeline ruptures in a brittle manner in response to a surge, it may therefore not be possible to assess whether the failure occurred because of the strain-rate-induced brittleness, the strain amplification effects discussed here, or both. Whether or not a surge-induced rupture appears to have occurred in a brittle manner, however, one should consider that an unexpectedly low failure stress could be associated with strain amplification effects.
Dynamic hoop strain
Dynamic hoop strain-stress amplification in a pipeline is due to the passage of a pressure wave traveling in the internal fluid medium. The dynamic hoop strain-stress amplification is not the pressure magnification effect that occurs when a pressure wave meets an obstruction in the pipeline, such as a valve, reducer-expander, elbow, bend, or blanked-off end. The dynamic hoop strain-stress amplification effect is a dynamic phenomenon that occurs when the pressure wave’s velocity in the fluid exceeds the velocity of a characteristic wave of the pipe itself.
A study analyzed motion of a tube under internal traveling pressure loads utilizing a linear elastic steady-state model and dynamic shell equations accounting for rotary inertia and shear deformations.1 Four critical velocities were determined. When the pressure wave traveled at any of these velocities, the solution became unbounded. In order of increasing magnitude these four velocities are the flexural wave velocity (vc0); the modified shear wave velocity (vc1); the dilatational wave velocity in a bar (vc2); and the plane stress dilatational wave velocity (vc3).
The first critical velocity, vc0, corresponds to the group velocity of the structural waves and the phase velocity of the traveling pressure wave. The traveling pressure wave produces various dynamic strains in the tube that in some cases are greater in magnitude than the strains that a static analysis would predict. The dynamic amplification factor is defined as the ratio of the maximum dynamic strain to the equivalent static strain (in this case the hoop strain) calculated from the static formula using the measured peak pressure of the traveling wave (Equation 1, see Equation box).2
This factor goes from about 1 below vc0 to unbounded at vc0 and drops to about 2 between vc0 and vc1. However, the unbounded characteristic occurs because no damping was included in the model.
In experimental trials on aluminum and stainless steel tubes, amplification factors of the hoop strain as high as 3.9 were observed when the wave velocity was near the first critical velocity.3 Factors of about 1.5 were observed in the steady-state region between the flexural wave velocity (vc0) and the modified shear wave velocity (vc1).
Other experiments showed amplification factors ranging from 1.5 to 1.65 in the steady-state region between vc0 and vc1.4
Pipeline ramifications
A simple model has been developed for the motion of the tube under an internal traveling pressure wave whose analysis neglected rotary inertia and shear deformation.5 Equation 2 shows the closed-form solution for the first critical velocity.
Equations 3-5 show the remaining three critical velocities for a thin steel tube.
Table 1 shows the first critical velocities determined from these equations for several different pipe diameters, with two wall thicknesses for each diameter. The velocities in Table 1 are probably higher than actual velocities since they do not include rotary inertia and shear deformation or account for the added mass of the material surrounding the pipeline if it is buried.
For comparison, the bulk sonic wave velocities for four typical liquids were calculated from the bulk modulus-mass density velocity equation for typical pipeline fluids that are relatively incompressible (Equation 6).
Table 2 shows the results. The sonic wave velocity in the pipe is less than the bulk sonic wave velocity due to the resilience of the pipe wall and the manner in which the pipe is anchored. Equation 7 provides the wave velocity.
For most representative pipe sizes, the pipe sonic wave velocity vp, is higher than the first critical velocity, vc0. For example, for D/h = 100 with Ψ = 1, the sonic wave velocity in the pipe, vp, for water is approximately 71% of the bulk wave velocity, vw, and rises to approximately 84% for gasoline. As the D/h decreases, the sonic wave’s velocities as a percent of the bulk wave velocity increase.
This relationship means that the pipe sonic wave velocities, vp, for these common pipeline fluids would still exceed the first critical velocities, vc0, for typical pipeline diameters and wall thicknesses even while they are much less than the second critical velocity. Consequently, for the steady-state region between vc0 and vc1 and if the hoop strain remains in the linear-elastic region, Equation 8 can determine the maximum hoop stress produced by the passage of the pressure wave.
Equation 9 provides a simple formula for the dynamic amplification factor in the region vc0
The above equation is based on a simple shell model. Its validity was tested by comparing the amplification factor for a 30-in. OD, 0.5-in. WT pipeline operating at 1,000 psig undergoing a 300-psig pressure surge with the results of a 3D finite-element analysis (FEA). The model looked at a methane-filled pipe where the surge was in the subcritical region (v
Analytical example
Consider a 30-in. OD, 0.5-in. WT pipeline operating at 1,000 psig that undergoes a 300-psig pressure surge on top of the operating pressure. Equation 9, for the dynamic amplification factor for a water or gasoline-filled pipe, gives Φ = 2, resulting in a dynamic hoop stress of 48,000 psi; 18,000 psi more than the hoop stress of 30,000 psi at the normal operating pressure. If the problem were treated as a static case, the hoop stress from the surge would be 39,000 psi, or only 9,000 psi above the operating hoop stress.
Suppose the operator, which had just completed an in-line inspection for metal loss, were planning to expose and remediate only those anomalies with calculated failure pressures below 1,600 psig (a hoop stress level of 48,000 psi), and based on anomaly dimensions provided by the inspection. The safety factor for the worst-case anomaly not exposed would be 1.6 for a steady-state operating pressure of 1,000 psig. Ostensibly, even if a surge of 300 psig occurred, the safety factor in the absence of strain amplification would be 1.3. When the possibility of a strain amplification factor of 2 is taken into account, however, the safety factor is reduced to 1.0. In other words, there is no safety factor.
Under circumstances where a strain-amplifying surge could actually raise the hoop stress to 48,000 psi, the prudent pipeline operator might elect to remediate anomalies with calculated failure pressures not lower than 1,900 psig.
Strain amplification may explain why some pipelines fail at a dynamic surge pressure that is less than a previous hydrostatic test pressure current, MOP, or previous maximum operating pressure, all of which were greater than the surge pressure.
A maximum strain amplification factor of two is probably the value to be used in most cases for a liquid-filled pipeline experiencing an internal traveling pressure wave. Very seldom is the velocity of the pressure wave close to the first critical velocity, unless the pipe wall is abnormally thick or the diameter is relatively small with a thicker wall than normal. In this case, as has been reported, amplification factors can approach a value as high as 4. In general for large values of pressure-wave velocity but small compared to the shear wave velocity, the amplification factor appears to tend towards a limiting value of 2.
As seen in this article, the amplification factor increases the hoop strain produced by the pressure wave over the static hoop strain produced by the operating pressure prior to the passage of the pressure wave. While one cannot rule out effects of lower dynamic toughness resulting from the dynamic loading, it is evident that the phenomenon of strain amplification should be considered a contributing factor in cases where surge-induced pipeline rupture occurs at an unexpectedly low total pressure level.
The example provided is an extreme case, and for the vast majority of pipelines, the impact of the surge-strain-amplification phenomenon is likely to be negligible. Clearly, however, a pipeline operator should at least consider the exposure of pipeline segments to the surge-strain-amplification phenomenon as a part of its integrity management plan.✦
References
1. Tang, S., “Dynamic Response of a Tube Under Moving Pressure,” Proceedings of the American Society of Civil Engineers, Engineering Mechanics Division, Vol. 5, October 1965, pp. 97-122.
2. Chao, T. W., and Shepherd, J. E., “Fracture Response of Externally-Flawed Cylindrical Shells to Internal Gaseous Detonation Loading,” ASME Pressure Vessels and Piping Conference, Vol. 462-2, Vancouver, August 2002, pp. 85-98.
3. Beltman, W., and Shepherd, J.E., “Linear Elastic Response of Tubes to Internal Detonation Loading,” Journal of Sound and Vibration, Vol. 252, No. 4, 2002, pp. 617-55.
4. Chao, T. W., and Shepherd, J.E., “Detonation Loading of Tubes in the Modified Shear Wave Regime,” Proceedings of the 24th International Symposium on Shock Waves, Paper 1642, Beijing, July 11-16, 2004.
5. Simkins, T., Resonance of Flexural Waves in Gun Tubes, Technical Report ARCCB-TR-87008, US Army Armament Research, Development and Engineering Center, Watervliet, NY, July 1987.
6. Beltman, W.M., and Shepherd, J.E., Structural Response of Shells to Detonation and Shock Loading, GALCIT Report FM 98-3, Apr. 1, 1998.
The authors
Raymond E. Mesloh has been a senior engineer for 10 years at Kiefner & Associates Inc. (KAI), Worthington, Ohio. Previously, he worked at the Battelle Memorial Institute, Columbus, Ohio, for 33 years as a program manager, section manage, projects leader, and researcher. He holds a MS in engineering mechanics (1958) and a BS in electrical engineering (1956), both from Virginia Polytechnic Institute, Blacksburg, Va. Mesloh is a registered professional engineer in the state of Ohio. He is a member of the US Naval Institute and a past member of the ASME and the Society for Experimental Mechanics.
Robert B. Francini joined Kiefner & Associates Inc. as a senior pipeline specialist in January 2005. Prior to coming to KAI, Francini worked as a principle research scientist for Battelle Memorial Institute in Columbus, Ohio, and Richland, Wash., for 13 years. During that time Francini had been a principal investigator in numerous projects related to pipeline integrity, mechanical damage, and composite repairs. Francini has a BS in physics from Miami University in Oxford, Ohio (1980), and an MS in engineering mechanics from Ohio State University in Columbus, Ohio (1985).
John F. Kiefner is the principal advisor of Kiefner & Associates Inc., a firm that he founded in 1990. Over the past 38 years, Kiefner has specialized in developing and using analytical methods for assessing pipeline integrity and in conducting research on pipeline material behavior and pipeline defects and repair methods. Kiefner received his BS and MS degrees in civil engineering from Purdue University in West Lafayette, Ind., and his PhD from the University of Illinois in Urbana-Champaign, Ill. He is a registered professional engineer in Ohio and a member of ASME and NACE.
