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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Copyright © 2005 by ASME
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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Copyright © 2005 by ASME
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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Copyright © 2005 by ASME

 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
5
Copyright © 2005 by ASME
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)<1) in
connection with global analyses.
• Compared to the original formulation, the utilisation is
unaltered for bending dominant cases.
• The limit state is now applicable also in case of axial
force and pressure only. The format and axial force
utilisation is consistent with traditional stress-based
von Mises criteria.
• Note that the pressure terms are not factorised.
Extreme pressure cases are handled in the (system)
burst check.
By considering equilibrium of lateral forces and the
bending moment acting on the beam section shown in fig. 6, the
dynamic equation of motion for the beam section exposed to an
axial tensile force, P, and a distributed lateral force, q, reads:
EI
f 0 = C1
f 0 = C1
(28)

P 

⋅ 1 −

 Pcr 
EI
mL4
(29)

P 

⋅ 1 + C 2⋅
PE 

What happens when the beam or pipe is put in water and
subjected to external and internal pressures in addition to the
axial force?
Looking at the governing differential equation it is obvious
that most of the terms remain the same; the effects of the
bending and axial force do not change. Note that the axial
force, P, is strictly speaking the so-called true axial force, N,
given in DNV-OS-F101 and DNV-RP-F105. This force is the
integrated axial stresses over the steel cross-section and
contains terms from residual lay tension, temperature expansion
effects, Poisson effects from hoop stresses and any axial
expansion/sliding effects.
For this case with the pipe in water and exposed to internal
and external pressures, the vibrating mass will also involve the
mass of the pipe content and the hydrodynamic added mass (the
effect of moving the pipe through water and creating pressure
effects around the circumference).
R
M+dM
P
EI
Q+dQ
P
EI
mL4
where L is the span length and Pcr is the critical buckling axial
force where the span buckles. Note that P and Pcr are positive in
tension, and that Pcr and that the constant C1 depends on the
boundary conditions of the span, for more specific guidance see
DNV-RP-F105 or Fyrileiv and Mørk (2002).
The Pcr term may be replaced by the Euler buckling force
for a pinned-pinned span, PE = ¶2EI/L2 (positive in
compression) and a second boundary condition constant, C2:
q
M
(27)
where v is the lateral displacement, x the axial co-ordinate, t is
time and EI and m the bending stiffness and dynamic mass per
unit length, respectively.
The classical solution to this differential equation without
any lateral force (free vibrations) gives the following
fundamental vibration frequency:
DNV-RP-F105 FREE SPAN RESPONSE
The most important parameter when it comes to structural
response of free spans is the natural frequency. This parameter
enters the reduced velocity and thereby governs whether vortex
induced vibrations will occur or not, given the pipe outer
diameter and the flow velocity normal to the span. By changing
the natural frequency, e.g. by introducing additional span
supports or reducing the span length, the VIV response may be
mitigated.
From textbooks like Clough and Penzien (1975), it is
obvious how the axial force influence on the dynamics and
frequencies of a free spanning pipeline. However, there have
been a lot of mistakes when it comes to the effect of the internal
pressure in various papers and reports. The latest example is the
paper by Galgoul et al. (2004) where it is claimed that the
internal pressure will increase the frequency.
Since the discussion is about how the pressure effects
influence on the frequencies or more precisely if the internal
pressure tends to increase or decrease the frequencies, it is of
outmost importance to get the signs right, considering the
governing equations of dynamics for the beam.
v
∂ 4v
∂ 2v
∂ 2v
− P 2 + m 2 = q ( x, t )
4
∂x
∂x
∂t
Q
∂ 4v
∂ 2v
∂ 2v
−
N
+
m
= q ( x, t )
eff
∂x 4
∂x 2
∂t 2
(30)
As mentioned, the external and internal pressure will have
an indirect effect on this equation through the hoop stress and
the Poisson effect on the true axial force.
However, as shown by Galgoul et al. (2004), the effect of
the internal pressure can be seen as a lateral force, q. Consider
the infinitesimal length of the pipe as shown in fig. 6 which is
deformed with a radius R = ∂2v/∂x2. Due to the deformation,
the pipe fibres at the compression side become shorter than the
x
dx
Figure 6 – Forces acting on an infinitesimal length of a
beam.
6
Copyright © 2005 by ASME
ones at the tensile side. The relative shortening becomes (Rr·sinφ)/R where φ is defined in fig. 7.
of the axial force, P, from the beginning. Then the cumbersome
integration of double-curves surfaces would have been avoided.
Let us consider eq. (27) without the dynamic term (inertia
term). Having no external distributed load but including
external and internal pressures will lead to an equation similar
to eq. (32) but without the inertia term:
R=1/d2v/dx2
EI
∂ 4v
∂ 2v
−
(
N
−
p
A
+
p
A
)
=0
i i
e e
∂x 4
∂x 2
(35)
Solving this equation gives the following critical load,
buckling load, for a pinned-pinned boundary condition:
q
S cr = N cr − p i Ai + p e Ae =
φ
pi
Figure 7 – Deformed pipe cross section with internal
pressure and with a bending radius R.
The net effect of the internal pressure will be similar to a
distributed load, q. The integrated effect becomes:
2π
q = pi
∫
0
R − ri sin φ
∂ 2 v (31)
ri sin φdφ = − p i Ai / R = − p i Ai 2
R
∂x
where ri denotes the internal pipe radius. The governing
equation of motion now becomes:
EI
∂ 4v
∂ 2v
∂ 2v
−
(
N
−
p
A
)
+
m
=0
i i
eff
∂x 4
∂x 2
∂t 2
(32)
and the fundamental frequency:
f 0 = C1
EI
m e L4

( N − p i Ai ) 

⋅ 1 + C 2⋅
PE


(33)
In the same way the effect of the external pressure can be
incorporated. It is seen that by using the definition of the
effective axial force:
S = N − p i Ai + p e Ae
(34)
both the governing differential equation and the expression for
the frequency are simplified.
Of course the same expressions would have been deducted
if the concept of effective axial forces has been applied in stead
7
π 2 EI
(36)
L2
This shows that there is a clear relationship between how
the internal (and external) pressure affects both global buckling
and the natural frequencies of a span.
It is also quite obvious that the natural frequency will
decrease as the internal pressure increases. Finally, as the
effective axial force approaches the critical buckling load, the
frequency approaches zero, 0. However, this occurs for the
theoretical case with pinned-pinned support conditions. In
reality, a pipeline span will experience a gradual deflection or
sagging that will give raise to non-linear effects not accounted
for in the expressions above. By this reason, the natural
frequency of free spans should be estimated by use of other
tools, e.g. non-linear FE analysis, when the effective axial force
reaches a certain value and non-linear effects become
significant, as recommended in the DNV-RP-F105.
Another comment to be given to eq. (36) is that this
expression gives the critical buckling force for a pinned-pinned
span. For more realistic boundary conditions, DNV-RP-F105
gives some advice for a single span supported on soil (flexible
span shoulders). In this case, a C2 coefficient equal to 0.25 is
used in combination with an effective length of the span longer
than the appearing span length. However, the intention of the
expressions for the structural response given in the DNV-RPF105 is to support calculation of VIV and fatigue and not to
give exact guidance on span buckling.
For most real spans, the span deflection will gradually
increase as the effective axial force tends towards the critical
buckling value as correctly stated by Palmer and Kaye (1991)
and Galgoul et al (2004). This deflection will release some of
the axial force and may also cause the span to vanish as the
span height is limited. Anyway, as the effective axial force
increases in compression and approaches the theoretical
buckling limit, the pipeline response becomes complicated and
highly non-linear. Therefore, expressions based on linear beam
theory can not be applied and must be replaced by for example
non-linear FE analysis.
Due to this, the range of applicability for the expressions
based on beam theory in DNV RP-F105 is limited to short and
moderate span lengths L/D < 140, moderate deflections δ/D <
2.5 and moderate geometrical stiffness effects C2Seff/PE < 0.5This is explained in more detail in the paper by Fyrileiv and
Mørk, 2002.
Copyright © 2005 by ASME
CONCLUSIVE SUMMARY
The following comments can be used to summarise the
discussion within this paper:
• Effective axial force is a very simple and efficient concept
to account for pressure effects.
• The expression for the effective axial force in the DNV
code is correct although some simplifications have been
made.
• The local buckling is influenced by the true steel wall force
and the hoop stress.
• The DNV-OS-F101 local buckling criterion is based on the
effective axial force. In this way the criterion is simplified
compared to the alternative using the true axial force.
• The natural frequency will drop as the internal pressure
increases.
• The pressure effect on the frequency of a span is accounted
for by the effective axial force concept.
DNV-OS-F101, DNV Offshore Standard “Submarine Pipeline
Systems”, Det Norske Veritas, January 2003.
DNV-RP-F105, DNV Recommended Practice, ”Free Spanning
Pipelines”, Det Norske Veritas, March 2002.
Fyrileiv, O. and Mørk, K., “Structural Response of Pipeline
Free Spans based on Beam Theory”, Proceedings of
OMAE2002, Oslo, Norway, 2002.
Galgoul, N.S., de Barros, J.C.P. and Ferreira, R.P., “The
Interaction of Free Span and Lateral Buckling Problems”,
Proceedings of IPC 2004, Calgary, Canada, 2004.
Galgoul, N.S., Massa, A.L.L. and Claro, C.A., “A Discussion
on How Internal Pressure is Treated in Offshore Pipeline
Design”, Proceedings of IPC 2004, Calgary, Canada,
2004.
Hobbs, R.E.., “In-Service Buckling of Heated Pipelines”,
Journal of Transportation Engineering, vol. 110, No 2,
March 1984.
Palmer, A.C. and Baldry, J.A.S., “In-Service Buckling of
Heated Pipelines”, Journal of Petroleum Technology, pp.
1283-1284, November 1974.
Palmer, A.C. and Kaye, D., “Rational Assessment Criteria for
Pipeline Spans”, OPT, Offshore Pipeline Technology
conference, 1991.
Sparks, C.P., “The Influence of Tension, Pressure and Weight
on Pipe and Riser Deformations and Stresses”,
Proceedings of OMAE, 1983.
Sparks, C.P., “The Influence of Tension, Pressure and Weight
on Pipe and Riser Deformations and Stresses”, ASME
transaction vol 106, pp. 46-54, 1984.
REFERENCES
Carr, M., Bruton, D. and Leslie, D.., “Lateral Buckling and
Pipeline Walking, a Challenge for Hot Pipelines”,
OPT2003, Offshore Pipeline Technology conference,
Amsterdam, The Netherlands, 2003.
Bruschi, R., Spinazze, M., Vitali, L. And Verley, R., “Lateral
Snaking of Hot Pressurized Pipelines – Mitigation for
Troll Oil Pipeline”, Proceedings of OMAE1996, 1996.
DNV’76, “Rules for the Design, Construction and Inspection of
Submarine Pipelines and Pipeline Risers”, Det Norske
Veritas January 1976.
DNV’96, “Rules for Submarine Pipelines systems”, Det Norske
Veritas, December 1996.
8
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