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Proceedings of OMAE2005 24th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2005) June 12-1

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Proceedings of OMAE2005 24th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2005) June 12-17, 2005, Halkidiki, Greece

OMAE2005-67502

INFLUENCE OF PRESSURE IN PIPELINE DESIGN – EFFECTIVE AXIAL FORCE Olav Fyrileiv and Leif Collberg Det Norske Veritas, Veritasveien 1, N-1322 Høvik, Norway

ABSTRACT This paper discusses use of the effective axial force concept in offshore pipeline design in general and in DNV codes in particular. The concept of effective axial force or effective tension has been known and used in the pipeline and riser industry for some decades. However, recently a discussion about this was initiated and doubt on how to treat the internal pressure raised. Hopefully this paper will contribute to explain the use of this concept and remove the doubts in the industry, if it exists at all. The concept of effective axial force allows calculation of the global behaviour without considering the effects of internal and/or external pressure in detail. In particular, global buckling, so-called Euler buckling, can be calculated as in air by applying the concept of effective axial force. The effective axial force is also used in the DNV-RP-F105 “Free spanning pipelines” to adjust the natural frequencies of free spans due to the change in geometrical stiffness caused by the axial force and pressure effects. A recent paper claimed, however, that the effect was the opposite of the one given in the DNV-RP-F105 and may cause confusion about what is the appropriate way of handling the pressure effects. It is generally accepted that global buckling of pipelines is governed by the effective axial force. However, in the DNV Pipeline Standard DNV-OS-F101, also the local buckling criterion is expressed by use of the effective axial force concept which easily could be misunderstood. Local buckling is, of course, governed by the local stresses, the true stresses, in the pipe steel wall. Thus, it seems unreasonable to include the effective axial force and not the true axial force as used in the former DNV Pipeline Standard DNV’96. The reason for this is explained in detail in this paper.

This paper gives an introduction to the concept of effective axial force. Further it explains how this concept is applied in modern offshore pipeline design. Finally the background for using the effective axial force in some of the DNV pipeline codes is given. Keywords: Pipelines, Effective axial force, Pressure effects, pipeline codes. INTRODUCTION The effective axial force is often considered as a virtual force in contrast to the so-called “true” axial force given by the integral of stresses over the steel cross-section. It is, however, a concept used to avoid integration of pressure effects over double-curves surfaces like a pipe deformed by bending. The main problem with the internal and external pressures is that the effect of these is often the opposite of what one instantaneously thinks is correct. Therefore these effects have been sources to misunderstandings and wrong designs. One example, as given by Palmer and Baldry (1974), is a straight pipeline restrained by rigid anchor blocks in each end. When this pipe is exposed to internal pressure, a tensile stress develops in the hoop direction. Due to Poisson’s effect, this hoop stress will tend to shorten the pipe. Since the shortening is prevented by the anchor blocks, the stress in the tensile direction also becomes positive. Despite of this the pipeline will buckle when the pressure reach a certain critical level as shown in the experiment conducted by the authors. The explanation of this contradiction is of course the effective axial force which becomes negative as the pressure builds up. The composite action of the fluid/gas pressure and the steel pipe axial force will cause buckling.

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Copyright © 2005 by ASME

As will be shown later, the effective axial force governs the structural response of the pipeline in an overall perspective, influencing on lateral buckling, upheaval buckling, anchor forces, end expansion and natural frequencies of free spans. For this reason it of outmost importance to understand its effects and be able to estimate it accurately in order to end up with a safe and reliable design. When it comes to local effects like local buckling, steel stresses and yielding, the true axial force governs. However, as seen later in this paper, the effective axial force may still be used to simplify the design criterion. Even though the effective axial force has been used in pipeline codes for several decades, see for example DNV’76 (1976), it is still misunderstood and misinterpreted when it comes to the effect of pressures. The effect of the internal pressure is treated by for example Palmer and Baldry (1974) and Sparks (1983, 1984), who seemed to introduce the term effective axial force. Quite recently Galgoul et al (2004) and Carr et al. (2003) claimed that the expression for the effective axial force and the way it is applied in some DNV codes like DNV-OS-F101 and DNV-RP-F105 is wrong. This is not the case as shown in the following. This paper will not come up with any new expressions or application for the effective axial force. The intention is solely to try to give the background for this concept and its application in the pipeline codes. By doing so, it is the hope that the design criteria become easier to understand and that it will not be misinterpreted in the future.

area. Other sectional forces like bending moments and shear forces are omitted for clarity as they will not enter the calculation of the effective axial force and the effect of the pressure.

N pe

=

N

pe

p eA e +

Figure 1 - Equivalent physical systems – external pressure As seen, the section with an axial force, N, and the external pressure, pe, (left figure) can be replaced by a section where the external pressure acts over a closed surface and gives the resulting force equal to the weight of the displaced water , the buoyancy of the pipe section (middle figure), and an axial force equal to N + peAe. Considering the effect of the external pressure in the way as shown in fig. 1 does not change the physics or add any forces to the pipe section. However, it significantly simplifies the calculation. The alternative would be to integrate the pressure over the double curved pipe surface. Note also that the varying pressure due to varying water depth over the pipe surface needs to be accounted for in order to get the effect of the displaced water, the buoyancy. A similar consideration, as for the external pressure, may be done for the internal pressure. However, as seen from fig. 2 when considering a section of a pipeline with internal pressure, the external forces acting on this section is the axial force, N, and the “end cap” force, piAi. Again other sectional forces like bending moment and shear forces are omitted for clarity. As the pressure acts in all direction in every point in the liquid, the internal pressure will always act on a closed surface. Further, the pressure at the cut away section ends will act as an external axial load in compression. From these considerations of the external and internal pressures acting on a pipeline section it becomes clear that the effect of these may be accounted for by the so-called effective axial force:

EFFECTIVE AXIAL FORCE CONCEPT The concept of effective axial force simplifies the calculation of how the internal and external pressures influence the behaviour of a pipeline. Therefore it is very important for the pipeline designer to fully understand this concept. Despite of this, the experience of the authors is that the effective axial force is quite often misunderstood and applied wrongly. The effective axial force is explained in detail in many papers, e.g. see Sparks (1983). However, a short description is included herein for the sake of completeness. The effect of the external pressure is most easily understood by considering the law of Archimedes: “The effect of the water pressure on a submerged body is an upward directed force equal in size to the weight of the water displaced by the body”. Archimedes law is based on the assumption that the pressure acts over a closed surface. Physically, Archimedes law can be proved by considering an arbitrary volume inside a larger liquid without any internal flow due to temperature /density differences. Since the effect of the pressure over the surface of this arbitrary volume is an upward force equal to the weight of this liquid, the arbitrary volume will be in equilibrium and will neither move up, down nor to any side. Of course the same conclusion is reached by mathematically integrating the external pressure over the surface of the volume. Now, consider a section of a pipeline exposed to external pressure as illustrated in fig. 1. The only sectional force included is the axial force, N, the so-called true wall force found by integrating the steel stress over the steel cross-section

S = N − p i Ai + p e Ae

(1)

In addition the integrated effects of the pressures over closed surfaces give: • Buoyancy (external pressure) • Weight of internal liquid (internal pressure)

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pi

From this it can clearly be stated that the effective axial force is a real force that can be measured and that have a physical interpretation.

pi

APPLICATION OF EFFECTIVE AXIAL FORCE Let us take a look at a few examples where the effective axial force concept may be used to simplify the calculation of the pipeline response. One example is already mentioned in the previous section; the external force to be applied to lock the pipeline axially or prevent any end expansion etc, see fig 3. Then any transfer of shear forces at the bend is neglected.

N piAi

Figure 2 - Equivalent physical systems – internal pressure EFFECTIVE AXIAL FORCE – A REAL FORCE? Some of the confusion among pipeline designers regarding the effective axial force may be due to the use of the term “true force” for the steel wall force, N, and the perception of effective axial force as a fictitious, non-physical force. This is also claimed by Galgoul et al. (2004) stating that the term piAi is a lateral force and not an axial one. The question to be answered is really; “Is only forces causing stresses in the pipe wall real forces?” Let us consider the same example as Sparks (1983), a reinforced concrete beam. Since concrete only resist insignificant tensile stresses, the load-bearing capacity of concrete elements may be improved by using pre-stressed (tensile) reinforcement. Without any external loads, the concrete will be in compression and the reinforcement in tension. But what is the real axial force in the beam? Is it in compression due to concrete stresses? It is obvious that an integration over the total cross-section, including both concrete and reinforcement stresses gives an axial force equal to zero. This composite axial force has clear similarities to the effective axial force concept. Thus the answer from this is that the effective axial force is a real force. But how could it be interpreted or measured? If strain gauges are mounted to the pipeline, the steel strain and thus, steel stresses and thereby the steel wall force or “true” axial force may be deduced. Let us, however, consider a pipeline terminated with a blind flange which could be the case during the pressure test of the pipeline system. If the end is free to move, no external loads act on it (assuming no external pressure, i.e. above sea water). When the pipeline is pressurised what will the axial force become? From simple equilibrium calculations it is obvious that the “true”, steel wall force will be equal to N = piAi (in tension) and the effective axial force will be Seff = N - piAi = 0. This means that the effective axial force is the force one may measure at the blind flange. Further it is the effective axial force that must be counteracted in case axial expansion of pipeline ends are to be avoided or the pipeline is to be locked axially with intermittent rock berms, e.g. to section a hot pipeline and ensure controlled lateral buckling. All types of pipeline anchoring and end terminations must consider the effective axial force.

S

pi

Figure 3 - Force to prevent any axial expansion at bends etc. Fig. 4 shows the pipeline during installation using a typical S-lay barge with the pipeline bent over the stinger and in the sag bend near the seabed. Flt

S

Figure 4 – Pipeline during conventional S-lay installation. By simple considerations using the effective axial force concept, the axial force in the pipeline after installation can be estimated. Equilibrium of horizontal forces requires that the true axial force, N, in the pipeline after installed, i.e. at the seabed, is equal to the barge tension, Flt, in addition to the integrated effect of the external pressure. In addition the horizontal components of the roller contact forces at the stinger must be accounted for, but these are omitted here for simplicity. As mentioned earlier the integration of the varying external pressure over the double-curves pipe is complicated. However, by closing the integration surface, it can easily be shown that the effective axial force at bottom must be: S = N + p e Ae = Flt

(2)

Accounting for the horizontal force components of the rollers and any other relaxation due to axial sliding etc this

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leads to the so-called residual lay tension, H. This force is to be considered as an effective axial force. Now, using H as the effective axial force after installation, the true axial force becomes: N = H − p e Ae

At the time when the pipeline is laid down, it does not rely on any force transfer between the soil and pipe, hence, the effective axial force is equal to the bottom tension force, H (effective force). The equation can then be re-written as:

(3)

ε l ,1 =

As the pipeline is operated, the true axial force gets into compression due to the thermal expansion (-Asα∆ΤΕ) and into tension due to the hoop stress and Possion’s effect (νσhAs) if not allowed to slide axially (fully restrained). The true force becomes: N = H − p e Ae + νAs

p i Di − As α∆TE 2t

=

1 S − p e, 2 Ae + p i , 2 Ai E As

(5)

p i , 2 Di − p e, 2 D pi , 2 + p e, 2 − ν − 2t 2

which is the same expression as given in DNV-OS-F101. Note that in DNV-OS-F101, pi is replaced by ∆pi. This does not mean the differential pressure between the outer and the internal pressure, but the change in internal pressure from installation, accounting for a potential water-filled installation with a hydrostatic pressure inside the pipe.

1 ε l = [σ l −ν (σ h + σ r )] + α∆T E

(7)

(8)

σ h = ( p i Di − p e D) / 2t

(9)

σ r ≈ −( p i + p e ) / 2

(10)

+ α∆T2

1 H − p e,1 Ae + p i ,1 Ai E As p i ,1 Di − p e,1 D pi ,1 + p e,1 + α∆T1 = − ν − 2t 2 1 E

(14)

S − p e, 2 Ae + pi , 2 Ai As

p i , 2 Di − p e, 2 D p i , 2 + p e, 2 − ν − 2t 2

+ α∆T2

Further, the external pressure is the same, i.e. pe,1=pe,2 and the differential temperature from reference (i.e. ∆T2-∆T1) can be denoted as ∆T and the equation can be simplified as:

where εl is the axial (longitudinal) strain , σl, σh and σr are the axial, hoop and radial stresses, ν is Poisson’s ratio, α is the thermal expansion coefficient and ∆T is the temperature difference. Further, the following relationships apply:

σ l = N / As

(13)

Since the pipe is axially restrained; εl,1=εl,2 and S can be solved from (setting Eq. (12) equal to Eq.(13)):

DNV-OS-F101 EFFECTIVE AXIAL FORCE An alternative proof of the effective axial force for a fully restrained pipeline will be given in the following. This is based on two fundamental expressions; Effective axial force and Hooke’s law:

H + p i ,1 Ai

p i ,1 Di pi ,1 = − ν − As 2 2t S + p i , 2 Ai pi , 2 Di pi , 2 + α∆TE − ν − As 2 2t

(15)

Introducing ∆pi as the internal pressure difference from laying (i.e. pi,2-pi,1) the equation reads:

where As is the steel cross sectional area of the pipe, D and Di are the external and internal diameters and t is the nominal wall thickness. Let the as-installed pipeline condition be denoted with an index 1. The only un-known in this condition is the longitudinal strain which is given by Hooke’s law as: (11) 1 ε l ,1 = σ l ,1 −ν (σ h,1 + σ r ,1 ) + α∆T1 E

[

1 H − p e,1 Ae + p i ,1 Ai E As

ε l ,2 =

(6)

(12)

where H is the (effective) bottom tension. During operation, denoted index 2, the longitudinal strain will be, similar to the above:

S = H − p i Ai + νAs

S = N − p i Ai + p e Ae

N pi ,1 Di − p e,1 D p i ,1 + p e,1 + α∆T1 − − ν 2t 2 As

p i ,1 Di − p e,1 D p i ,1 + p e,1 + α∆T1 − ν − 2t 2

(4)

From the definition of the effective axial force, the following is deducted: p i Di − As α∆TE 2t ≈ H − p i Ai [1 − 2ν ] − As α∆TE

1 E

0=

S − H + ∆p i Ai ∆p ∆p D − ν i i − i + α∆TE As 2 2t

(16)

As Di − 1 − Asα∆TE 4 t

(17)

S = H − ∆p i Ai + 2∆piν

]

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D + Di Di − 1 π t 2 t − A α∆TE S = H − ∆p i Ai − 2ν s 4

•

(18)

The external pressure will not affect the effective axial force in the operational condition.

Both Palmer and Baldry (1974) and Hobbs (1984) are correct and according to the DNV formulae when it comes to internal pressure.

D D − 1 − 3 t t − A α∆TE = H − ∆p i Ai 1 − 2ν s 2 D − 2 t

DNV-OS-F101 LOCAL BUCKLING The local buckling criterion in DNV’96 for internal over pressure is based on a pure von Mises calculation, considering the two dimensional stress stage in the pipe wall. In general terms this becomes: 3 σh ⋅ 4 SMYS 2

M c = M p ⋅ f M = SMYS ⋅ D 2 ⋅ t ⋅ 1 −

To this point, the only simplification is that the radial stress is assumed constant across the wall thickness. However, the above equation can be approximated with: S ≈ H − ∆pi Ai [1 − 2ν ] − Asα∆TE

1 σ h N π − ⋅ 2 π ⋅ SMYS ⋅ D ⋅ t 2 SMYS Cos 2 3 σh 1 − 4 SMYS

(19)

The order of error introduced (of the pressure contribution) by this simplification is: D D − 1 − 3 t t 1 − 2 ⋅ν ⋅ 2 D − 2 t e= 1 − 2 ⋅ν

(20)

Concentrating on the bracket in the nominator of the Cosine term, this has been split up into functional and environmental loads in DNV’96 as:

γ F ⋅ N F + γ E ⋅ N E 1 γ F ⋅σ h − ⋅ 2 SMYS π ⋅ SMYS ⋅ D ⋅ t

and it is plotted in fig 5. The error of the simplification is less than 1% for D/t larger than 15!. Note that the error is valid for the internal pressure term only and is really negligible. It is also worth to mention that in most HTHP cases, it is the thermal expansion term that dominates the effective axial force.

(22)

This can be re-written by including the definition of the effective force, defined as: S = N − p i ⋅ Ai + p e ⋅ Ae

(23)

Split into functional and environmental parts this becomes S F = N F − p i ⋅ Ai + p e ⋅ Ae

New

(24)

SE = NE Including these definitions into the inner bracket of the cosine term, it becomes:

1.1

1.05 Relative error

(21)

γ F ⋅ (S F + p i ⋅ Ai − p e ⋅ Ae ) + γ E ⋅ S E 1 γ F ⋅ σ h − ⋅ π ⋅ SMYS ⋅ D ⋅ t 2 SMYS

1

(25)

Assuming thin walled pipe this can be re-written as ≈

0.95

γ F ⋅ S F + γ E ⋅ S E γ F ⋅ ( p i − p e ) ⋅ Ae 1 γ F ⋅ σ h + − ⋅ = π ⋅ SMYS ⋅ D ⋅ t π ⋅ SMYS ⋅ D ⋅ t 2 SMYS = ..... +

0.9 0

10

20

30

40

50

60

70

(26)

γ F ⋅ ( p i − p e ) ⋅ Ae ⋅ 2 ⋅ D 1 γ F ⋅ σ h − ⋅ = 2 SMYS π ⋅ SMYS ⋅ D ⋅ t ⋅ 2 ⋅ D

D2 ⋅2 γ ⋅ ( pi − pe ) ⋅ D 1 γ ⋅σ h 4 = ... + F ⋅ − ⋅ F = π ⋅ SMYS ⋅ D ⋅ D 2 SMYS t⋅2 1 1 γ ⋅σ h = ... + γ F ⋅ σ h ⋅ − ⋅ F = 2 ⋅ SMYS 2 SMYS γ ⋅ S + γ E ⋅ SE = F F π ⋅ SMYS ⋅ D ⋅ t

π⋅

D/t

Figure 5 - Estimate of error due to simplification of axially restrained pipe stress formula The conclusions from this deduction are: • The given formula, Eq. 5.4 in DNV-OS-F101 is correct if the pipe can be considered as a thin walled pipe- the error due to this simplification can be seen in Figure 5.

Hence, the bracket in the nominator can be simplified by using the effective force. This local buckling formulation has

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been further developed by Taylor expansion, including strain hardening and other minor adjustments to fit with FE analyses in DNV-OS-F101 based on the same principles as explained above. Hence, the local buckling formula has been derived at by use of the “true” pipe wall force. However, the result turns out to be simpler if converted to effective force. The following comments apply: • The new format is “simpler” to use and understand. It is well suited for performing design checks (g(x)