response of a free span pipeline subjected to ocean currents
TRANSCRIPT
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CHAPTER 1
INTRODUCTION
Offshore pipelines are used to transport oil and gas between offshore platforms or
to transport oil and gas directly from offshore to land. Marine pipelines are used
for disposal of industrial and municipal wastewater into the sea, for cooling water
in nuclear power plants, and for the transportation of gas and crude oil from
offshore platforms. Marine cables, on the other hand, are increasingly used for
communication. Many offshore engineering activities involve the use of pipelines
in ocean along with offshore oil development, deep ocean mining, beach
replenishment harbour and entrance channel maintenance dredging etc. A pipeline
is a fixed asset with large capital costs. Once the pipeline is in place, though, the
operation, maintenance costs are relatively small, and the pipeline has an
operating life of 40 years or more.
The use of offshore pipelines is a more recent development of the latter
part of the twentieth century. The design of an offshore pipeline is
multidisciplinary and typically involves three fields of engineering such as
Structural mechanics, hydrodynamics, and soil mechanics. Planning of the route
demands a great deal of considerations of the life cycle of a pipeline must be
considered. During the life cycle from fabrication to abandoning the installed
pipeline after years of operation, the pipeline must provide safe transportation.
The pipelines are placed either to rest directly on the seabed or in trench or on
saddles depending on the bottom topography of the seabed along their routes. In
the water depth considered, beyond 300m, waves will not impose forces of any
significance to the pipeline. However, the ocean current causes a separated flow
around the free spanning pipeline. Hence pipelines cannot be designed and
constructed in a rational basis similar to pipelines on land due to the fact that sea
weather conditions may changes rapidly from smooth to rough seas. Therefore,
the submarine pipeline is in many ways more complex than land based structures.
2
The numerical simulation of unilaterally constrained structural systems is
receiving increased attention, mainly because direct solutions to the problem are
unattainable. During the operation and installation of offshore pipelines, high axial
forces and pressures are experienced, and their effects cannot be neglected. The
development of the technology of submarine pipelines has provided the possibility
of conducting projects under extreme conditions with respect to water depth and
environmental conditions. In these conditions, free spanning pipelines are often
unavoidable. Free spans may occur due to natural seabed irregularities present at
pipe installation or they may develop during operation due to erosion, scour, or
migrating sand waves.
Free spanning pipe section may subject to significant dynamic stresses due
to the presence of submarine currents and wave induced flows. Amplified
responses due to resonance between the vortex shedding frequency and natural
frequency of the free span may cause fatigue damage of the material and reduction
of the pipeline life. The sections of free spans may thus represent weak points of
the transport system, as they have a low reliability.
When a part of a subsea pipeline is suspended between two points on an
uneven seabed, it is always referred to as a free span pipeline. For a safe operation
of offshore gas or oil pipeline during and after installation, the free span lengths
should be maintained within the allowable lengths, which are determined during
the design stage. The determination of the critical length of spans under the
various environmental conditions along the pipeline thus becomes an important
element in pipeline design with a significant economical impact, especially in
deep water where the traditional maintenance and repair technology is inadequate.
In analysis and design of marine pipelines, free spanning analysis is one of
the scopes, besides determination of pipe size and wall thickness, on-bottom
stability and corrosion requirement. Free spanning analysis is performed to ensure
stability and fatigue life of the pipeline when exposed to wave and current forces.
The analysis and design of subsea pipeline is complex due to the facts that
a) Loading on the pipeline may be static, dynamic, transient, harmonic or
random.
3
b) Loading on the pipeline is location and time dependent.
c) The characteristics of sea floor vary along the pipeline corridor and sea
bottom is irregular.
d) The bottom contact points of the pipeline are not known a priori.
e) Material behaviour of the pipeline may be elastic or viscoelastic.
1.1 OCCURRENCE OF FREE SPANS
Free spans can be caused by:
Seabed unevenness.
Change of seabed topology (e.g. scouring, sand waves).
Artificial supports/rock beams etc.
Fig 1.1 Seabed unevenness 1.2 AIM AND SCOPE OF THIS THESIS
The aim of this thesis is to determine the fatigue damage of a free span pipeline
considering in-line and cross-flow force. This involves the following major steps.
Numerical modelling of the 3D frame work’s mass and stiffness properties
and obtaining the frequencies and corresponding mode shapes.
Deterministic regular wave fatigue analysis.
Deterministic irregular wave fatigue analysis using the rainflow cycle
counting method.
Parametric study is performed to determine the parameters, which
influence the dynamic response and fatigue of the free spanning pipeline.
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A MATLAB code is developed for calculating the fatigue damage of
pipeline for both regular and irregular wave conditions. This study provides the
understanding of the mechanisms that is involved in the free spanning analysis of
an offshore pipeline especially on the fatigue damage.
1.3 METHODOLOGY
1.3.1 Static analysis
To determine shape of the span after installation.
To ensure a stable equilibrium between the pipeline and the seabed.
To determine natural frequency of the free spanning pipeline.
1.3.2 Fatigue analysis
Calculate the stress ranges and verify that the magnitude of the maximum
stress is below yield stress of the steel pipe.
Calculate the number of stress cycles.
Determine the allowable number of stress cycles to from S-N curves.
Calculate the Damage by Palmgren-Miners rule.
1.3.3 Parametric study
Functional state
Damping ratio
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CHAPTER 2
LITERATURE REVIEW
2.1 GENERAL
During the life cycle from fabrication to abandoning the installed pipeline after
years of operation, the pipeline must provide safe transportation. In case of failure,
severe environmental pollution and great economic loss may occur. It is important
to consider the seabed conditions and the wave and current action on the pipeline
during route planning. The pipeline sections must comply the transportation
demand and at the same time have a bearing capacity and a proper protection to
resist the rather rough environment of the sea during installation and operation.
Structural configurations of the pipeline during construction will depend upon
the method and equipment used for installation. The pipes either short units or in long
floating strings are supplied by a ship to the laying vessel. The most important
characteristics of the laying vessel are the loading capacity, the motion characteristics,
pipe laying, and welding of the pipe segments. The different methods of laying a
pipeline are:
1. Stinger lay barge method
2. Reel barge method
3. Bottom pull method
4. Floating string method
During the operational state, the pipeline may subject to fatigue damage
due to cyclic loads. The maintenance is another important aspect of a pipeline life
cycle. Annual inspections are made by video whereas the repair work of the
seabed can be done by local rock dumping at the free span to prevent erosion in
the future.
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2.2 PREVIOUS WORKS ON SUBSEA PIPELINES
Three methods have been adopted in the past for handling the pipeline-seabed
random contact problem i.e. optimization, negative support reaction monitoring
and successive over-relaxation (SOR). Due to the statistical nature of the problem,
optimization methods have been extensively employed coupled with direct
solutions of the differential equation of flexure. Maier et al (1978) adopted
quadratic programming for the determination of the fundamental control variable
defining the relative position of the pipe to the rigid seabed geometry. Both elastic
and elasto-plastic material stress-strain relationships were incorporated into their
numerical model. Using a similar technique, Stavroulakis et al (1986) expanded
the optimal control theory to include the seabed-pipeline frictional effects during
the random contact process.
Chuang (1992) investigated the efficiency of the quadratic programming
method according to the flexibility and the stiffness formulation approach. In a
series of publications, Baniotopoulos et al (1985) investigated the static and
dynamic response of submarine cable configurations using the optimal control
theory. Mathematical programming has also been applied by Mahmoud et al
(1986) and Salamon et al (1989) to the study of other structural systems besides
submarine pipelines.
By considering full soil-pipe contact, Bianchi et al (1988) used a node-by-
node negative seabed reaction elimination method for identifying the free pipeline
span development along the complete subsea route. With the aid of a successive
over relaxation algorithm, Kalliontzis et al (1996) examined the static installation
bending stresses developed in a pair of highly pressurized submarine pipelines
along a strait crossing. Direct pipe bending stress solutions can be obtained if the
structure-soil contact points are assumed to be known beforehand. Following this
method, Pranesh et al (1995) developed an analytic stress theory, aimed at
determining the optimal subsea pipeline routing.
Park et al. (1997) analyzed static and dynamic free spans of pipelines, and
proposed an allowable length of free span. The variation of allowable lengths is
examined for specialized boundary conditions, where free spanning is modeled as
a beam with transversal and rotational springs at each end. Kapuria et al. (1998)
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used the beam equation to determine the allowable free span or a pipe loaded both
tensile as well as compressive axial forces. An allowable spanning length is
prescribed to ensure enough strength and to prevent resonance happening. In
pipeline assessment, fatigue life shall be evaluated if VIV occurs.
Choi et al. (2001) derived a closed form solutions of the beam-column
equation for the various possible boundary conditions by considering tension and
compressive force. The natural frequency is calculated by energy balance method.
Some calculations are to present the sensitivity of the axial forces on the allowable
free spanning lengths. For free spanning pipeline, thermal expansion leads to
compression and reduces the natural frequencies and make the pipeline vibrate at
frequencies much lower than that without axial force.
Kim J. Mork et al (2001) proposed a rational design criteria and guidance
on fatigue design methods for free spans subjected combined wave and current
loading. Soreide et al (2002) investigated the dynamic properties of in the vertical
and horizontal direction of the free span due to the sag effect of a long free span.
Model tests on long free spans have revealed the importance of the interaction
between in-line and cross-flow VIV responses.
Furnes et al (2002) formulated a time domain model to examine
dynamical features of free spanning pipelines subjected to current forces.
Coupling between the cross-flow and the in-line motions was carried out by
considering the time varying part of the axial tension caused by current induced
deflections.
Abbas et al (2007) investigated the effects of sea bed formation along with
axial force on Natural Frequency of offshore pipelines. Based on this assessment a
new simple formula is proposed and compared the results with the allowable free
span length of Qeshem Island pipeline calculated based on DNV (1998) and ABS
(2001) guidelines.
J. Zhou et al (2008) has been derived a discrete equation of free spanning
submarine pipeline. The spatially varying earthquake ground motion, the internal
pressure, and the thermal load were imposed on the FE model. The nonlinear
material constitutional relationships of the pipe and the soil as well as the large
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displacement effect were considered. The effect of internal pressure, thermal
loading, and the interaction between pressure and temperature on the multi-
support input response of the submarine pipelines was studied.
XU Jishang et al (2010) carried out both static and dynamic analysis
methods to study the maximum allowable free span length (MAFSL) in the deep
water regime of the pipeline off the west coast of Hainan Island. The static
analysis was carried out by considering the maximum bending moment to
calculate the MAFSL. MAFSL Estimated by Cross-Flow Induced VIV is based on
partial safety factor design criteria.
2.3 CYCLE COUNTING
According to ASTM E 1049 – 85 (2011), Standard Practices for Cycle Counting
in Fatigue Analysis, cycle counting is used to summarize often-lengthy irregular
load versus time histories by determining the number of times cycles of various
sizes occur around varying mean stress levels. Cycle counts can be made for time
histories of force, stress, strain, torque, acceleration, deflection, or other loading
parameters of interest..
Schütz (1993), considers the rainflow counting procedure to be the
optimum counting procedure, because it counts the stress ranges and the
associated mean stresses correctly. The author cautions however that one aspect of
the rainflow counting procedure requires consideration i.e., it will always give the
load variation between the lowest trough and the highest peak as the largest range
counted cycle. However, consider the lowest trough occurs very early in the load
sequence and the highest peak at the end.
ASTM E 1049 – 85 (Reapproved 2011), defines basic fatigue loading
parameters, the range as the absolute value between successive valley and peak
loads, peak as the point at which the load – time history changes from a positive to
a negative sign with a valley being the opposite, reference load the steady state
condition on which loads are superimposed, reversal the point at which the load –
time history changes sign and mean crossings as the number of times the load –
time history crosses the mean load level during a given time history, as shown in
Fig. 2.1.
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Fig 2.1:Basic Fatigue Loading Parameters
2.4 CUMULATIVE DAMAGE
According to Heuler & Klätschke (2005), cumulative damage has been studied
for decades and it is well known that data and models that characterise the fatigue
behaviour of materials and structures under baseline constant amplitude loading
may not be appropriate or sufficient to adequately assess their fatigue performance
under irregular variable amplitude loading.
Further it is generally agreed that the structural load variations should be
characterised in the time domain since in most cases the range of a load, stress or
strain cycle plus its respective max or mean value can be considered as fatigue
relevant. The sequence or mixture of load cycles of different ranges and means
must not be neglected.
Issler (2009), indicates that a variable amplitude load – time history must
be broken down into a constant amplitude load - time history using an accepted
cycle counting method such as rainflow cycle counting. This results in a load
history consisting of various stress amplitudes across different mean levels.
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CHAPTER 3
NUMERICAL MODELLING OF SUBSEA PIPELINE
3.1 GENERAL
The modelling of a subsea pipeline is multidisciplinary and typically involves
three fields of engineering
a) Structural mechanics
b) Hydrodynamics
c) Soil mechanics
In this analysis, it is assumed that the pipeline is located at intermediate
water depth. Fig 3.1 and 3.2 explains the design condition of free spanning
pipeline.
Fig 3.1: Design condition of free spanning pipeline
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Fig 3.2: Cross section of pipeline
3.2 STRUCTURAL MODELLING
The numerical data taken here is that of a dual submarine pipeline crossing along
the Revithoussa -Aghia Triada strait in Greece carrying liquefied natural gas. The
horizontal distance between upstream and downstream points of the submarine
pipeline is subdivided into a number of segments known as finite elements. For
modeling, the following engineering assumptions were made.
1. The pipeline is a two-nodal 3-D Bernoulli-Euler linear elastic beam.
2. The pipeline cross section remains circular even after bending
3. The seabed is idealized by the Winkler foundation model
4. The sea floor characteristics are considered to be same along the length
of the pipeline under consideration
5. The flexural rigidity of the pipe, EI, is constant
6. The other external forces on the pipelines, such as pull due to anchors,
are neglected.
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The equation governing deformations of the linearly elastic slender pipe is
given by
EId ydx − T
d ydx + k y − w = 0 (3.1)
Subject to the contact or suspension condition of
(y − z)q = 0 (3.2)
Where, y is deflection of the pipe, y1 is pipe elevation relative to an
arbitrary reference datum, z is seabed elevation, q is seabed reaction per unit
length, T is tension in pipeline, E is young's modulus of pipe material I is moment
of inertia pipe section. Fig 3.3 shows a beam segment of a space frame having six
degrees of freedom per node, translation along all-axis (ux, uy, uz) and rotation
about all axis (Өx, Өy, Өz,) are considered .The element has a constant moment of
inertia I, modulus of elasticity E, density ρ and length L.
The element stiffness and mass matrices of the elements in the Matlab
Model are described in the appendix A. The elements are based upon the theory of
3-D, straight Bernoulli-Euler beams.
Fig 3.3: Three-dimensional beam element
Fig 3.2 shows the definition sketch of a unilaterally supported submarine
pipeline with its finite element discretisation. The pipeline is divided into
segments of length L, known as finite elements by S nodes. Along the one-
dimensional grid, the length L of each segment is assumed constant. Two benefits
result from such an assumption. Firstly, arithmetic performance tends to be
13
particularly efficient when using regular grids and secondly, compact program
coding is facilitated.
Fig 3.4: Sketch of a unilaterally supported submarine pipeline and its
finite element discretisation grid
If rigid sea floor geometry is assumed, then the seabed reaction q is
calculated after the pipeline deformation has been determined. A more general
approach accounting for variable sub-soil characteristics, is to model the seabed
reaction as a series of springs using the standard force-displacement relationship
q = k ∗ y (3.3)
Where, ks and y respectively denotes spring stiffness or sub grade modulus
and soil displacement or settlement.The sub grade modulus k of the linear spring,
the magnitude of which represents the resistance offered to vertical pipeline
movements, is usually linearly related to the shear modulus G of the sub-soil.
Hence
k = α ∗ G (3.4)
Where, α is a coefficient ranging from 1.0 to 3.0. It is customary to assume
a value of 3.0 for vertical springs and 2.0 for horizontal springs.
The seabed is idealized by Winkler foundation medium given by linear
vertical springs as shown in Fig 3.3.
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Fig 3.5: Pipeline model on elastic foundation
3.3 HYDRODYNAMIC MODELLING
In this session, the flow condition for a cylinder near a wall is discussed. This is
an important aspect of determining the hydrodynamic forces because the seabed
proximity has a large effect on the hydrodynamic forces that affect the pipeline
free span. The conventional model for determining hydrodynamic forces on
cylindrical structures in the offshore industry is the Morison Model. The
hydrodynamic forces affecting the pipeline have components in two directions the
in-line force and the cross-flow force. Fig 3.6 shows the forces affecting the
pipeline. The hydrodynamic force coefficients are determined according to DNV-
RP-F105.
Fig 3.6: Hydrodynamic forces acting on the pipeline
15
Although the real sea is random, the wave environment can be described by two
methods. In the deterministic method, the sea is described as composed of regular,
individual waves. In the spectral method, the sea is described as a function of sea
surface elevation due to regular waves combining to form an irregular sea.
3.3.1 Simulation of Regular Wave
In this analysis, regular waves are simulated by discretization of a continuous
wave spectrum as explained in the section 3.3.2. Fig 3.7 explains the wave
characteristics,
Fig 3.7: Wave Characteristics
)sin(. tkxa (3.5)
Fluid velocity component in the z-direction,
cosh( ( )). sin( )
cosh( )zagk k d zV t kx
kd
(3.6)
Fluid velocity component in the y-direction,
sinh( ( )). cos( )
cosh( )yagk k d zV t kx
kd
(3.7)
16
Fluid acceleration component in the z-direction,
sinh( ( )). cos( )
cosh( )zk d za agk t kx
kd
(3.8)
Fluid acceleration component in the y-direction,
sinh( ( )). sin( )
cosh( )yk d za agk t kx
kd
(3.9)
Where, a is the wave amplitude, T is the time period of wave, k = L/2 ,
is the wave number, L is the wave length, d is the water depth, z is the point at
which water particle kinematics is to be determined with SWL as origin, is the
amplitude of the wave, is the frequency of wave.
3.3.2 Simulation of Random Sea State
A uni-directional random wave train may be simulated as a sum of component
regular wave trains, all propagating in the same specified direction but with
different amplitudes, frequencies, and phases. In the present study, simulation was
done by deterministic spectral model or the random wave phase spectrum method.
In this, waves are assumed stationary, homogenous and erdodic in the statistical
sense.
Fig 3.8: Discretization of a continuous wave spectrum
17
The short-term, stationary, irregular sea states may be described by a wave
spectrum. The JONSWAP spectrum was developed by Hasselman, et al. (1973)
was to relate wave amplitude as a function of wave frequency and with further
analysis the spectral energy density S (f) as a function of the wave frequency as
given by Equation 3.10, for the Joint North Sea Wave Project. The formula is to
be derived from the modified Pierson-Moskowitz spectrum formula.
S(f) =αg
(2π) f exp −1.25(ff ) (3.10)
Where, α=.0081,g=9.81,,f = frequency and fo =peak frequency (Hz).
Wave profile or surface elevation is represented by
ˆ( , ) .cos( 2 )1
Mx t a k x f ti i i ii
(3.11)
dkgktf iii tanh).ˆ.2( 2 (3.12)
Here, M being a sufficiently large number, ai denotes the amplitude of the
component wave in the ith frequency,
if is the ith representative frequency, which
is evenly distributed in the range of ),( 1
ii ff , i is phase angle. In this equation,
wave number of the ith component, ki, can be determined from the dispersion
relationship after knowing the corresponding representative frequency and water
depth d.
The wave amplitude ai is determined from a given function of the
frequency spectrum
)( ifS by,
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iii ffSa )ˆ(2
(3.13)
2/)ˆˆ(ˆ1 iii fff (3.14)
)ˆˆ( 1 iii fff (3.15)
Fluid acceleration component in the z-direction is given by,
u(t) =
H2 (2π푓) sin(k x− 2π푓t + i )
cosh (k y)sinh(k h)
(3.16)
Fluid acceleration component in the y-direction is given by,
v(t) = −H2 (2π푓) cos(k x− 2π푓t + i )
sinh (k y)sinh(k h) (3.17)
Fluid velocity component in the z-direction,
u(t) = H (π푓) cos(k x− 2π푓t + i )cosh (k y)sinh(k h) (3.18)
Fluid velocity component in the y-direction,
v(t) = H (π푓) sin(k x− 2π푓t + i )sinh (k y)sinh(k h) (3.19)
Where, ai is the wave amplitude, ki = L/2 , is the wave number, L is the
wave length, h is the water depth, y is the point at which water particle kinematics
is to be determined with SWL as origin, is the amplitude of the wave and
if is
the ith representative frequency.
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3.3.3 Random Wave Validation
After generating the random waves, validation of the simulated random wave
profile is needed to assure the precision and efficiency of the numerical
simulation. This is conducted by comparing the simulated or predicted frequency
spectrum with the target frequency spectrum presented in equation (3.10) after
considering that all the typical random wave characteristics, such as significant
wave height and period can be determined by the frequency spectrum.
After knowing the water surface elevation ( , )x t , there are two methods,
which can be applied to determine the simulated (predicted) spectrum: auto-
correlation method and FFT method. In this study, FFT is used.
3.3.4 Hydrodynamic Coefficients
This section describes the determination of force coefficients independent of time,
which are the estimated values that are typically implemented in the Morison
Model. The drag coefficient CD and inertia coefficient CM to be used in
Morison’s equation are functions of Keulegan Carpenter number, the current flow
ratio, α, the gap ratio, (e/D).In order to get all contributions from the
hydrodynamic load, three different force coefficients need to be determined
a) Drag force coefficient
b) Inertia force coefficient
c) Lift force coefficient
In reality, the hydrodynamic force coefficients vary with time and depend
on multiple parameters, including the Reynolds number, the Keulegan-Carpenter
number, the current-flow velocity ratio, seabed proximity, and the pipe roughness.
Table 3.1 shows typical values for the surface roughness
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3.3.4.1 Drag Force Coefficient
The drag force coefficient CD is determined according to DNV-RP-F105.The drag
force coefficient is determined as
CDA
CDproxi
CDk
oD DCC ... (3.20)
Where, DoC is the basic drag coefficient.
Table 3.1: Surface roughness
Pipe surface Steel, painted Steel, un-coated
(not rusted)
Marine growth
k [metres] 10-6 10-5 1/200 → 1/20
For small gap ratios, the seabed will have influence on the flow around the
pipe. The drag force generally increases with decreasing gap ratios. The correction
factor for the seabed proximity is taken as
CDproxi =
0.9 +. 5
1 + 5 eD
, eD < 0.8
1, else (3.21)
Where, e is the gap between the pipe and the seabed in m, De
is gap
ratio.The current-flow velocity ratio is determined as
α = mc
c
UUU
(3.22)
Where, Uc is the Velocity of the current in m/s, Um is the Maximum
velocity of the wave in m/s.
21
3.3.4.2 Inertia Force Coefficient
The inertia force coefficient is determined according to DNV-RP-F105.The
inertia force coefficient is determined as
CMproxi
CMkM
oM CC .. (3.23)
Where, MoC is basic inertia force coefficient, CM
k is correction factor for
the pipe roughness, CMproxi is correction factor for the seabed proximity.
The basic inertia force coefficient is determined as function of the current-
flow velocity ratio α and KC.
MoC = f(α) +
5 2 − f(α)KC + 5 (3.24)
f(α) = 1.6− 2α, α ≤ 0.5 0.6 α > 0.5 (3.25)
For increasing pipe roughness, the inertia force will decrease. For increasing
pipe roughness, the inertia force will decrease. The correction factor for the pipe
roughness is taken as
CMk = 2-
CDk (3.26)
For decreasing gap ratios, inertia force will increase. The correction factor for
the seabed proximity is taken as
CMproxi =
⎩⎪⎨
⎪⎧0.84 +
0.8
1 + 5 eD
, eD < 0.8
1 eD > 0.8
(3.27)
22
3.3.4.3 Lift Force Coefficient
DNV-RP-F105 does not specify values for the lift force coefficient. In this project,
the maximum absolute value of CL when the gap is small (e / D < 1) and the
cross-flow force arises from wall proximity.
Fig 3.9: Lift force coefficient for a near-wall cylinder
3.3.4.4 Keulegan-Carpenter
The Keulegan-Carpenter number is defined as
KC =U + U
D T (3.28)
Where, T is the wave period, Um is the in-line flow velocity for the wave in m/s,
Uc is the maximum in-line flow velocity for current in m/s.
3.3.5 Estimation of In-Line Force
For a cylindrical structure with infinite stiffness, the Morison Model for
determining the in-line force reads,
f = g . U|U| + g . U (3.29)
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Where, 푓 is the in-line force per unit length of the pipe as function
of time in mN , g is the drag force term, 2m
kg , g is the inertia force term,mkg , U is
the in-line flow velocity for wave and currentsm .
The force terms for drag, inertia and added hydrodynamic mass are
defined as
g = C14 ρ D (3.30)
g = C14 ρ D (3.31)
Where, CD and CI respectively denote the drag force, inertia force
coefficients.
3.3.6 Estimation of Cross-Flow Force
When the cylinder approaches the wall, the change in pressure along the pipe
perimeter causes a resulting cross-flow force that is directed upwards. The
Morison Model for the cross flow force reads
f = g . (U) (3.32)
Where, 푓 is the cross-flow force per unit length of the pipemN
, g is the lift force term, mkg .
The lift force term is determined as
g = C14 ρ D (3.33)
Where, C is the lift force coefficient.
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3.3.7 Current
The current that affects the pipeline is determined by assuming that the velocity
profile is polynomial and can be formulated as
u(z) =87 u .
zh (3.34)
Where, u is current velocity affecting the pipe,(m/s), z is vertical
coordinate with origin at the water surface, h is water depth, uc is Basic current
parameter.
The velocity profile, which uses a 1/7th-power profile, is generally in good
agreement with a logarithmic formulation of the velocity profile. However, it is
however that the logarithmic velocity profile tends to be more accurate because
the seabed roughness is included as an additional parameter in this formulation.
The basic current parameter varies according to the return period of the design
wave. The current that affects the pipe is taken at an elevation 1 m above the
seabed, i.e. z = −h +1m. Table 3.2 shows the steady current that affects the pipe.
Table 3.2: Basic current parameter
Current
Extreme current
associated
with 1-year design
wave
Extreme current
associated
with 10-year
design wave
Extreme current
associated
with 100-year
design wave
Basic current
parameter
uc (m/s)
0.25 0.45 0.60
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CHAPTER 4
ANALYSIS OF FREE SPAN PIPELINE 4.1 GENERAL
Considering the overall free span analysis procedural aspects, the first step is to
undertake a static analysis to determine the sag deflections and tension
distribution in the pipeline. After identifying the unknown contact points and
pipeline deformation, a fatigue analysis is performed to determine the damage
ratio due to wave and current forces.
4.2 STATIC ANALYSIS
In general, four methods have been used for the static analysis of the suspended
pipelines. They are listed as follows, the stiffened catenary method, the finite
beam segment method, the finite difference method, and the finite element
method. In the present analysis, finite element method is adopted for
determination of stresses in laid pipelines for static loads.
A 3D beam finite element model of the pipe is applied. The analysis will
start from a stress free (horizontal straight lined) configuration without any
seafloor contact. A sequence of loads will be applied in such a way that the final
condition will represent the real pipeline as accurately as possible. Linear bottom
springs take care of the pipe/seafloor interaction. A large number of load
increments may be needed in order to maintain a stable solution during all
intermediate conditions. Iterative analysis is continued till a stable condition is
attained. Fig 4.1 shows the line diagram of stable condition attained after static
analysis.
26
Fig 4.1: Line diagram of stable condition attained after static analysis
WVKsKgK (4.1)
Where, W is the load vector and, K , gK , sK are, respectively, the
stiffness matrix, the geometric matrix and soil stiffness matrix with typical
coefficients.
In this Project, static analysis is done by modifying the MATLAB code
developed by Subin et al (2011), in which the pipe element is considered as 2
node linearly elastic beam subject to vertical loads within a vertical plane having
three degrees of freedom per node. But for the fatigue analysis 3 degrees of
freedom per node is not sufficient. Hence the same program is modified to a 2
noded linearly elastic beam having six 6 degrees of freedom and is validated with
the unknown contact points and pipeline deformation in literature. Thereby the
formulation of stiffness matrix, geometric matrix, and soil stiffness is validated.
Under free vibration, the natural frequencies and the mode shapes of a
multiple degree of freedom system are the solutions of the eigenvalue problem
0][][ 2 MK (4.2)
Where, is the angular natural frequency is the mass matrix and is the
mode shape of the structure for the corresponding natural frequency. The element
stiffness, the geometric and mass matrices of the pipe elements is described in the
appendix A.
27
4.3 FATIGUE ANALYSIS
Offshore structures, which are subjected to cyclic hydrodynamic loading, suffer a
reduction in strength, which may eventually cause failure, through a process
called fatigue. Fatigue is the systematic degeneration over time, which can
eventually lead to fracture and failure of components exposed to varying or cyclic
loads, which never reaches a level sufficient to cause failure in a single
application. The reliability of systems relies heavily on accurate fatigue life
prediction of related components. Fatigue damage in free spanning pipeline is
predominantly a result of wave loads. Fatigue life prediction is a complicated
process requiring the correct methodology to determine accurate and reliable
predictions. The Palmgren Miner damage accumulation hypothesis is widely used
in determining the fatigue life of components exposed to variable loading
conditions.
This section describes how to determine the fatigue damage of the pipeline
free span according to the design criteria given in [DNV-RP-C203 2005]. The
procedure of a fatigue damage check for the pipeline free span is:
1. Calculate the stress ranges and verify that the magnitude of the
maximum stress is below the yield stress of the steel pipe.
2. Calculate the number of stress cycles.
3. Determine the allowable number of stress cycles to failure from S-N
curves.
4. Calculate the damage by Palmgren-Miners rule.
5. Verify that the damage criterion is satisfied.
4.3.1 Nominal Stress
The nominal stress component of pipes is a linear combination of the axial and
bending stresses given by
σ(t) = σ (t) + σ (θ, t) (4.3)
28
Axial stress is given by,
σ (t) =T(t)
(A − A ) (4.4)
To ensure that the maximum stress range does not exceed the yield stress
of the steel pipe, the Mises criterion for yielding is applied. The Von Mises
equivalent stress safety criterion is adopted for examining pipe material non-
yielding conditions. Based on the proposals of Det Norske Veritas and BS 8010,
the von Mises equivalent stress formula is applied for testing the overall pipeline
stability. Hence,
σ ≤ ησ (4.5)
σ = ( (σ ± |σ |) + σ − (σ ± |σ |)σ (4.6)
Where, σ is the equivalent stress, σ is the yield stress of steel, σ is the
hoop stress of steel and η the usage factor, a coefficient less than or equal to unity
and σ the bending stress given by
σ (θ, t) =f R (M (t))
I (4.7)
Where, f is a stress reduction factor, which accounts for residual strains
and according to Ref.7, may be taken as 0.85. M is the bending moment about the
local y and z axes.
An increase of the internal fluid pressure relative to the ambient salt-water
hydrostatic pressure yields a hoop or circumferential stress, which for thin walls is
defined as,
훔퐡 =ퟐ∆퐏퐀퐢
(퐀퐨 − 퐀퐢) (4.8)
Where, A0 and Ai, are the outer and inner pipe steel section areas, ∆P the
mean pressure difference assumed to be constant. For zero longitudinal strains the
tensile stress is given by,
29
σ = μσ (4.9)
Where, μ is the Poisson ratio for steel. Consequently, the tensile force is
obtained from
T = σ (A − A ) = 2∆PA (4.10)
Second moment of area the composite section shown in Fig 4.2 may be
determined from
I =βπ4 (R − R ) (4.11)
Where,Ro and Ri are the outer and inner radius of steel part of the pipe
respectively.
Fig 4.2:Composite pipeline cross section parameters
The parameter β is given by,
β = 1 + [sin(φ)R
R ] +R tπR t
[(π2 − φ)(1 + 2sin φ)) −
32 sin (2φ)] (4.12)
30
Where,
φ = 2R t /π(2R t + R t ) (4.13)
R =12 (R + R ) (4.14)
t =tf (4.15)
Here, te is effective concrete thickness, tc thickness of concrete portion of
the pipe, fe is the young’s modulus ratio of steel to concrete, Rc radius of the pipe
up to neutral axis and ts is the thickness of steel portion of the pipe.
Table 4.1: Allowable range of bending stress
Functional state η Lim|휎 |
N/mm2
Empty .72 321.64
Water filled 0.72 322.56
Pressurized .96 330.87
4.3.2 Deterministic Regular Wave Fatigue Analysis
This method may be considered as a simplified version of the spectral method.
The main simplification involves how wave-induced load effects are
characterized. In wave-by-wave method, a discrete set of regular waves are
selected to represent the typical sea spectrum. In this analysis, regular waves are
selected at equal frequency increments by discretization of a continuous wave
spectrum. Each wave will be the same frequency difference away from its
neighbours, but each wave will have a different height corresponding to the
energy within its frequency increment.
31
In order for a structure to suffer from fatigue damage, it must undergo a
displacement process. Anytime a structure is displaced in a cyclic manner, the
possibility of the displacement of the structure being dynamically amplified exists.
This analysis involved an evaluation of the range of frequencies over which the
hydrodynamic loading might dynamically amplify the response of the free-
spanning submarine pipeline structure. Hence, in order to account the dynamic
behavior of pipeline under the effect of the environmental wave forces Dynamic
Amplification Factors (DAFs) has been determined, which will be applied to the
static analysis.
Airy wave theory was employed to describe surface-wave induced water
particle kinematics. This theory allowed water particle velocity and acceleration to
be determined at the elevation of the free-spanning pipeline. The structure is then
analyzed to determine the stress S for each of the load cases and, hence, the
fatigue life using Miner’s Rule.
4.3.2.1 Number of wave cycles in each bin
In this analysis, the waves are divided into five characteristic sea states. Let the
list of sea state spectra and associated durations be denoted Si and Di respectively,
for i= 1, 2 … k where k is the number of sea states. The probability of occurrence
of sea state q is given by
p(Sq) = Dq / ∑Di. (4.16)
A general definition of the nth order moment of an energy density
spectrum is given as
m = f S(f)df (4.17)
Where n = 0,1,2…… This further used to derive the significant wave
height (Hrms) and average zero cross periods (To) as below. A mean period is
defined as To,1 ,
32
Where,
T , =mm (4.18)
The root mean square value is calculated from spectrum and is given by
the formula,
H = 2 2m (4.19)
On the assumption of a narrow-band spectrum, the probability density of
wave height, H, having a period, T is
p(ξ, τ) =ξ√2π
exp [−12ξ (1 + τ )] (4.20)
Where, 휉 is normalized wave height, τ is a non dimensional variable
defined by Longuent-Higgins (1962).
ξ =H
H (4.21)
τ =T − T ,
ν T , (4.22)
Where, ν is defined in terms of the moments of the spectrum by
ν =m m − m
m (4.23)
The total number of occurrences Oij for each bin is given by
O = p(ξ, τ) D
T (4.24)
Where, Dtotal is the total duration.
33
4.3.3 Deterministic Irregular Wave Fatigue Analysis Using
The Rainflow Cycle Counting Method
Dynamic analysis is used when inertia forces are comparatively important and can
be done either in the frequency domain or in the time domain. In the frequency
domain, transient effects are neglected and steady-state solutions are obtained.
The method assumes linear system. In the time domain, on the other hand, the
nonlinear drag is taken into consideration by the time integration of the design
wave. Thus, the transient effects as well as nonlinearities are considered.
Using finite element, subsea pipeline analysis may typically be formulated
within the framework of structural dynamics as follows:
[M]{x} + [C]{x} + [K]{x} = {F(t)} (4.25)
Where, {f(t)} is force vector,[M] is mass,[C] is damping,[K] is stiffness
matrices,{x} is acceleration,{x}, {x} is displacement vectors for the whole
structure.
The solution of the dynamic equations of a linear system may be found by
the numerical integration of the dynamic equations.
4.3.3.1 Stress cycles
Counting methods have initially been developed for the study of fatigue damage
generated in aeronautical structures. Since different results have been obtained
from different methods, errors could be taken in the calculations for some of them.
Level crossing counting, peak counting, simple range counting and rainflow
counting are the methods which are using stress or deformation ranges. One of the
preferred methods is the rainflow counting method.
Since each stress cycle causes a certain amount of damage to the structure,
it is necessary to determine the number of stress cycles. For harmonic loading, this
is relatively simple, but for irregular stress cycles, the counting of stress cycles is
usually ambiguous. Cycle counting is used to summarize irregular load-versus-
time histories by providing the number of times cycles of various sizes occur. The
definition of a cycle varies with the method of cycle counting. Various methods to
obtain cycle counts are level-crossing counting, peak counting, simple-range
34
counting, range-pair counting and rainflow counting. Cycle counts can be made
for time histories of forces, stress, strain, torque, acceleration, deflection, or other
loading parameters of interest.
In this project, the stress cycle counts are determined by using rainflow-
counting method as described in Ref 9. The Rainflow cycle counting method
extracts the composition of a variable amplitude stress history. Rules for the
rainflow counting method described in ASTM E–1049 Standard Practices for
Cycle Counting in Fatigue Analysis are given as follows:
Let X denotes range under consideration; Y denotes previous range
adjacent to X; and S, starting point in the history.
1. Read next peak or valley. If out of data, go to Step 6.
2. If there are less than three points, go to Step 1. Form ranges X and Y using
the three most recent peaks and valleys that have not been discarded.
3. Compare the absolute values of ranges X and Y.
a. If X<Y, go to Step 1.
b. If X_Y, go to Step 4.
4. If range Y contains the starting point S, go to step 5; otherwise, count
range Y as one cycle; discard the peak and valley of Y; and go to Step 2.
5. Count range Y as one-half cycle; discard the first point (peak or valley) in
range Y; move the starting point to the second point in range Y; go to Step
2.
6. Count each range that has not been previously counted as one-half cycle.
35
(a) (b)
(c) (d)
(e) (f)
Fig 4.3: Practical definition of rainflow cycle counting
36
4.3.4 S-N Curves
S-N curves determine the fatigue resistance of a local part of the structure as a
function of the amount of cyclic loading. The S-N curves are based upon the
following relationship
(4.26)
Where, N is the maximum allowable number of cycles at the ith stress
range , is the ith stress range in Mpa, is the intersect parameter of the S-N
curve with the log N axis, is the Negative slope parameter of the S-N curve
The S-N curves depend upon the detail category. This considers the type
of constructional detail, welding type, the site conditions during the welding as
and the loading type of the structure. The S-N curves for a steel structure in
seawater with cathodic protection and in different detail categories are shown in
Fig 4.4. The parameters for the S-N curve in category D are given in Table 4.2.
Fig 4.4: S-N curves in seawater with cathodic protection
37
Table 4.2: Parameters for S-N curve for category D, DNV-RP-C203 2005
Number of cycles Stress range m Log10 a
N ≤ 106 cycles ∆σ ≥ 83.4MPa 3.0 11.764
N ≥ 106 cycles ∆σ ≤ 83.4MPa 5.0 15.606
4.3.5 Fatigue Damage
The damage caused by cyclic loading is determined by Palmgren-Miners
accumulation rule. This way of determining damage assumes that the order of
stress cycles does not have influence on the damage of the material. The stress
range distribution is replaced by a histogram with a chosen number of blocks with
constant stress ranges. The fatigue damage is then determined by
D = nN ≤ α (4.27)
Here, Ni denotes the fatigue life under constant amplitude loading with
amplitude, ni is the number of load cycles at this amplitude and is the allowable
damage ratio. Failure due to fatigue damage is assumed to occur when fat D =1.
The allowable damage ratio according to DNV is shown in Table 4.3.
Table 4.3: Allowable damage ratio for fatigue [DNV-OS-F101 2007, p50]
Safety class Low Medium High
α 1/3 1/5 1/10
The damage ratio can be calculated according to DNV-RP-C203 2005, p10, by
combining the S-N curves and Palmgren-Miners rule which provides,
D =1a n (∆σ ) ≤ α (4.28)
38
CHAPTER 5
RESULTS AND DISCUSSION
5.1 GENERAL
For identifying unknown contact points and pipeline deformation, a Matlab
program developed by Subin et al (2011) for two dimensional pipeline element
was taken and modified to three dimensional pipeline element. A Matlab program
is developed for analysis of subsea pipeline considering wave-by-wave method
and rainflow counting method. A parametric study is performed to understand the
parameters governing the fatigue for the free span pipeline.
5.2 STRUCTURAL AND FUNCTIONAL DATA
The formulated finite element model was subsequently applied to free span
analysis of a dual submarine pipeline crossing along the Revithoussa-Aghia
Triada strait in Greece.
Table 5.1 Pipeline data
E σy μ Ri Ro Rc Length
of pipe
Depth
(D)
2x108kN/m2 448N/mm2 0.3 29.23cm 30.50cm 36.50cm 102.5m 80m
The functional data for the pipeline is determined for three functional
states: empty, water-filled, and operational state. Table 5.2 shows the details of
functional states.
39
Table 5.2 Functional Data
Functional state η )/( 2mmNP We ( kN/m)
Empty .72 -0.20 0.476
Operational .72 5.00 1.130
Water filled .96 0.00 3.109
It is assumed that the pipeline is located at intermediate condition there for
the depth/wavelength ratio (d/L) is taken as .40. The wave data considered for the
fatigue analysis is shown in Table 5.3.
Table 5.3 Wave data for fatigue analysis
Hs
(m)
Tz
(sec)
Duration
[hours/year]
Sea state 4 7.0 11.6 47
Sea state 3 5.0 9.8 362
Sea state 2 3.0 7.6 2668
Sea state 1 1.0 4.4 5707
5.3 PARAMETRIC STUDY FOR WAVE BY WAVE METHOD
This parametric study is performed to identify the governing parameters of the
dynamic response and fatigue for the pipeline. The parametric study is divided
into the following parts
a) Functional state
b) Damping
40
5.3.1 Functional State
This analysis is made to find the most critical functional state for the pipeline free
span considering fatigue damage. The functional states that are considered are: (1)
Empty condition, (2) Water-filled condition, (3) Operational condition.
Static analysis is validated by comparing the results obtained from
literature with a MATLAB program incorporating three dimensional analysis of
pipeline. The comparison of results is given in Table 5.4. and Table 5.5.
Table 5.4 Gap between Seabed and pipeline under operational condition
Operational condition
Free span length= 60m
Spanning length Le/3 Le/2 Le3/2
Gap ( Ref:) 0.28593m 0.68669m 0.30634m
Gap ( 3D frame) 0.28593m 0.68669m 0.30634m
Table 5.5 Gap between Seabed and pipeline under empty and water filled
Empty condition Water filled condition
Free span length=75m Free span length=50m
Spanning length Le/3 Le/2 Le3/2 Le/3 Le/2 Le3/2
Gap 0.67472m 0.53452m 0.39941m 0.53683 0.63559 0.45045
Fig 5.1 shows the final profile obtained after static analysis for each
functional state.
41
Fig 5.1: Final pipe profile obtained after pipe analysis
42
Once the static equilibrium position obtained by static analysis, eigenvalue
analysis can be perform to determine the natural frequency of pipeline. This
further used to determine Dynamic amplification factor due to hydrodynamic
forces. First six natural frequency of pipeline after empty pipe analysis is listed in
the Table 5.6.
Table 5.6: The four lowest eigen frequencies for each functional state.
Functional
state f1 (Hz) f 2 (Hz) f 3 (Hz) f 4 (Hz) f 5 (Hz) f 6 (Hz)
empty 0.021138 0.058497 0.11486 0.19 0.28391 0.39659
Operational 0.019556 0.053899 0.07556 0.17461 0.26081 0.36423
Waterfilled 0.01296 0.03587 0.07044 0.11653 0.17413 0.24324
Fig 5.2 shows the first two mode shapes of pipeline.
Fig. 5.2 :First two mode shapes of pipeline under operational condition
For validation of mass matrix, a fixed beam is modeled as shown in Fig
5.3 in the same program. Problem is taken from Structural dynamics theory and
computation by Mario Paz.
43
Fig 5.3:Fixed beam modelled for validating Mass matrix formulation
Table 5.7 Validation of Natural frequency
Literature Natural Frequency(Hz) 7.23 20.09 39.86 75.4 124.71 200.93
Program Natural Frequency(Hz) 7.2307 20.089 39.856 75.403 124.71 200.93
In wave-by-wave method, regular waves used for static deterministic wave
load calculations. In order to determine the wave static load, the spectrum is
discretized into number of bins and the particle velocities from each bin are
determined for different sea states. Number of wave cycles in each bin is
determined as explained in section 4.3.2.1. Fig 5.4 to 5.7 shows the number of
waves in each bin for different sea states.
Fig 5.4 :Number of waves in sea state 1
44
Fig 5.5 :Number of waves in sea state 2
Fig 5.6 :Number of waves in sea state 3
Fig 5.7 :Number of waves in sea state 4
45
Fatigue analysis was carried out by discretizing the spectrum into a number of
waves. For showing the results, from each sea state, only those waves having
maximum number of cycles are summarized in Table 5.8 to 5.10.
Table 5.8 shows the values of force coefficient considered in the operational state
Table 5.8 KC number and force coefficients for operational condition
Sea state KC CD Cm CL
Sea state 1 20 1.3 1.81 0.9
Sea state 2 9 1.3 1.74 1.9
Sea state 3 2.22 1 2.19 2
Sea state 4 1.5 1 2.24 2
Damage caused by the stress ranges is determined at the mid span. Table 5.9
shows the dynamic amplification factor obtained for most probable wave height
under operational condition. Table 5.10 shows the contribution of most probable
wave height from each sea state.
Table 5.9 Dynamic amplification factors for most probable wave height
under operational condition
Sea
state
Hm
(m) DAF
T
(sec) f1 (Hz) f 2 (Hz) f 3 (Hz) f 4 (Hz) f 5 (Hz) f 6 (Hz)
1 7.185
1 1 1 1.5032 1.1769 1.0835 9.861
2 4.944
1 1 1 1.8374 1.2577 1.1175 8.4394
3 3.23861
1 1 1 3.6276 1.4888 1.2028 6.8573
4 .77
1 1 1 2.5231 2.5631 1.4588 4.8632
The number of cycles of the most probable wave in sea states 1 to 4 are 1300,
16724, 348902, and 471094 respectively.
46
Table 5.10 Summary of fatigue analysis for most probable wave height under
operational condition
Fz is the horizontal wave force. Fy wave force in Z direction. Table 5.10 shows
the damage for each functional state respectively.
Table 5.11 Damage at the mid span of pipeline under functional state
ζ = 0.05
Functional
state
Sea states Cumulative
Damage
ratio 1 2 3 4
Empty 0.15 0.013 0.0093 0.0023 0.1746
Operational 0.0024 0.094 0.2 0.27 0.5664
Water
filled 0.0019 0.054 0.084 0.18 0.3116
Sea
state
Hm
(m) Fz
(N/m)
Fy
(N/m)
Nominal stress
( Mpa) Damage
ratio T
(sec)
Maximum
in vertical
Minimum
invertical
Maximum
in lateral
Minimum
in lateral
1 7.185
112.066 85.291 60.4711 3.1849 74.6652 34.0092 0.00039 9.861
2 4.944
55.2118 72.3661 49.3575 13.2687 44.2205 16.4058 0.0013 8.4394
3 3.23861
31.1801 91.7763 55.0888 16.5354 37.6673 22.9570 0.045 6.8573
4 .77
30.4318 91.2942 59.7822 -24.1656 43.2734 -2.6569 0.083 4.8632
47
Table 5.9 Table 5.10 shows the damage caused by 3- 4 sea state is
significantly lower than the damage caused by the other two sea states in empty
condition. The number of cycles is higher in 3-4 sea state and stress ranges are
lower than stress ranges caused by the other two sea states, which shows that the
magnitude of stress ranges in this case significant on damage on pipeline. In the
case of water-filled condition, number of cycles is significant on damage on
pipeline.
Validation of nominal stress calculation is done by considering a frame as
shown in Fig 5.9. Members 2 and 3 are subjected to external distributed load of 6
kN/m.
Fig 5.8 :Frame considered for validation of nominal stress
Results obtained from the MATLAB program and literature is given in the
Table 5.10.
Table 5.12 Validation of nominal stress calculation
Element
no:
Length
(m)
Area
(m2)
EI
(kNm2)
Nominal
stress
Nominal stress
(Ref )
1 3 10000 1 27.8904 27.89
2 3.1623 10000 1 -39.7608 -39.76
3 3.1623 10000 1 17.6888 17.69
4 3 10000 1 -39.7608 -39.76
48
5.3.2 Effect of Damping
The dynamic response is compared for three different damping ratios:
ζ = 0.05
ζ = 0.10
ζ = 0.15
Table 5.13 Damage at the mid span of pipeline for different damping ratio
Functional
state
Sea states Cumulative
Damage
ratio 1 2 3 4
ζ = 0.10 Operational 0.0024 0.094 0.2 0.27 0.5664
ζ = 0.15 Operational 0.0024 0.094 0.2 0.27 0.5664
Table 5.12 shows the change in material damping ratios does not affect the
calculated stress of the model.
5.4 IRREGULAR WAVE FATIGUE ANALYSIS USING
RAINFLOW CYCLE COUNTING METHOD
In practice, a spanning analysis of a pipeline free-span is conducted by assuming
that the hydrodynamic forces are induced by regular waves. Waves are irregular in
nature and hence this study is carried out in order to investigate the implications of
this simplification of random waves to regular waves. Only operational condition
is considered for the analysis. The critical fatigue stresses are determined by the
normal stresses in the mid-section of the pipeline, where the magnitude of the
normal stresses is largest. Since the response history is irregular, the Rain Flow
Counting Method is used for determining stress ranges and cycles.
The irregular sea states 1-5 are modelled for 3.0 hours duration and the number of
cycles has been scaled by the ratio between the actual and modeled duration of the
irregular sea states.
49
Fig 5.9 :Simulation of wave profile for sea state 1
Fig 5.10 :Simulation of velocity profile for sea state 1
Fig 5.11 :Simulation of acceleration profile for sea state 1
50
Fig 5.12 : Inline force history for sea state 1
Fig 5.13 : cross flow history for sea state 1
Fig 5.14 : Inline response for sea state 1
51
Fig 5.15 : Cross flow response for sea state 1
For validation of response calculation, a fixed beam as shown in Fig 5.3 is
modeled and analysed in SAP software.
A sin wave of amplitude 500 and time period 20 is applied in the Y direction of
the beam.Time history analysis done in SAP2000 and MATLAB.Results are
showed in the Table:9.4
Table 5.14 Validation of response
Fig 5.16:Comparison of response obtained from SAP and MATLAB
Max response at mid span (SAP) Max response at mid span(MATLAB)
0.003386m 0.0032m
52
Fig 5.17 : Nominal stress history in lateral direction for sea state 1
Fig 5.18 : Nominal stress history in vertical direction for sea state 1
Table 5.15 Damage at the mid span of pipeline under functional state using
rainflow counting
ζ = 0.05
Functional
state
Sea states Cumulative
Damage
ratio 1 2 3 4
Empty 0.0897 0.0003 0.0027 0.0011 0.0938
Operational 0.0012 0.094 0.1386 0.1767 0.4105
Water
filled 0.0001 0.0336 0.0579 0.18 0.1258
53
For validation of stress cycle counting, a problem is shown in Fig 5.16, taken from
John Wægter etal. Stress cycles obtained from literature is given in Table
5.14.Table 5.15 shows the stress cycles obtained from MATLAB program.
Table 5.16 Cycle count in literature
Literature Full cycle Half cycles
2-3-3a, 4-5-5a, 6-
7-7a, 9-10-12b
and11-12-12a
1-8, 8-13 and 13-
14
Fig 5.19:Rain flow counting problem in literature and matlab
54
Table 5.17 Cycle count obtained from program
Amplitude Mean Cycle count
0.5000 2.5000 1.0000
1.5000 2.5000 1.0000
1.0000 4.0000 1.0000
3.5000 2.5000 0.5000
0.5000 -2.5000 1.0000
1.5000 -2.5000 1.0000
5.5000 0.5000 0.5000
6.0000 1.0000 0.5000
55
CONCLUSIONS
The major findings of the study on response of a free span pipeline subjected to
ocean currents are summarized as follows:
The functional state of an offshore pipeline has been studies under three
conditions, viz. empty condition, operational condition and water-filled
condition.
Among these functional states, the operational condition of the pipeline
has been found to be the most critical state when considering fatigue
damage ratio.
The cumulative damage ratio for operational, empty and water-filled
conditions were determined and were found to lie in Low, Normal and
high Safety classes respectively, according to DNV-OS-F101 2000. Hence
the free span has to be reduced for fatigue damage reduction.
Temporary measures such as providing sandbags as intermediate supports
and permanent measures such as piled supports for the pipelines could be
suggested as suitable remedial measures.
The dynamic behavior of the free span pipeline was found to be only
slightly affected by the effect of fluid and material damping.
56
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57
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15 Recommended Practice Det Norske Veritas: “Rules for Submarine Pipeline
Systems”, Reapproved 2011.
16 Recommended Practice ASTM E1049-85: “Standard Practices for Cycle
Counting in Fatigue Analysis”, Reapproved 2011
17 S.K Chakrabarti. “Hydrodynamics of Offshore Structures”,2003
18 Schütz W, “The significance of service load data for fatigue life analysis”,
Fatigue Design ESIS 16, Mechanical Engineering Publications Limited, pp 1-
17, 1993.
19 Sonsino C M, “Principles of Variable Amplitude Fatigue Design and Testing”.
Journal of ASTM International, Vol. 1, No. 10, Paper ID JAI19018, pp 7–8,
2004.
20 Tao Xu, “Wave-induced fatigue of multi-span pipelines”, Marine Structures
12, pp 83-106, 1999.
21 XU Jishang, “Calculation of Maximum Allowable Free Span Length and
Safety Assessment of the DF1-1 Submarine Pipeline”, Ocean University of
China, Science Press and Springer-Verlag Berlin Heidelberg, pp 1-10, 2010.
58
APPENDICES
A ELEMENT MATRICES IN MATLAB MODEL
B RAYLEIGH DAMPING
59
A ELEMENT MATRICES IN MATLAB MODEL
Element Stiffness Matrix
The stiffness matrix for a three-dimensional uniform beam segment is readily
written by the superposition of the axial stiffness matrix, the torsional stiffness
matrix, and the flexural stiffness matrix
in which Iy, Iz are respectively cross-sectional moments of inertia with respect to
the principal axis labeled as y and z in Fig 3.2 and L,A and J are respectively the
length, cross-sectional area, and torsional constant of the beam element.
60
Element Mass Matrix
22
22
22
22
4L 0 0 0 22L- 0 3L- 0 0 0 13L- 0 0 4L 0 22L 0 0 0 3L- 0 13L 0 0
0 0 140Ia 0 0 0 0 0 70Ia 0 0 0 0 22L 0 156 0 0 0 13L- 0 54 0 0
22L- 0 0 0 156 0 13L 0 0 0 54 0 0 0 0 0 0 140 0 0 0 0 0 70 3L- 0 0 0 13L 0 4L 0 0 0 22L 0
0 3L- 0 13L- 0 0 0 4L 0 22L- 0 0 0 0 70Ia 0 0 0 0 0 140Ia 0 0 0 0 13L 0 54 0 0 0 22L- 0 156 0 0
13L- 0 0 0 54 0 22L 0 0 0 156 0 0 0 0 0 0 70 0 0 0 0 0 140
420mL
e
where, m = distributed mass/unit length in kg/m
Aps mmmm
sm = mass including steel and concrete/unit length
pm = mass of oil or gas/unit length.
Am = hydrodynamic added mass.
2
4* DCm AA
ρw
AC =added mass coefficient.
ρw =density of sea water.
Ia=Io/A,
Io=polar mass moment of inertia,
61
Soil Stiffness Matrix
LL4k 0 0 0 L 22k- 0 L3k- 0 0 0 L 13k- 0
0 LL4k 0 L22k 0 0 0 L3k- 0 L13k 0 0 0 0 140ka 0 0 0 0 0 70ka 0 0 0 0 L22k 0 156k 0 0 0 L13k- 0 54k 0 0
L22k- 0 0 0 156k 0 L13k 0 0 0 54k 0 0 0 0 0 0 140ka 0 0 0 0 0 70ka
LL3k- 0 0 0 L13k 0 L 4k 0 0 0 L22k 0
0 L3k- 0 L13k- 0 0 0 L 4k 0 L 22k- 0 0 0 0 70ka 0 0 0 0 0 140ka 0 0 0
0 L13k 0 54k 0 0 0 L 22k- 0 156k 0 0 L13k- 0 0 0 54k 0 L22k 0 0 0 156k 0
0 0 0 0 0 ka*70 0 0 0 0 0 140ka
420
tt2
tt
tt2
tt
tttt
tttt
tt2
tt
2tt
2tt
tttt
tttt
LK s
ka is the linear spring stiffness in axial direction 2MN
kt is the linear spring stiffness in transversal direction 2MN
62
4.4 Element Geometric Matrix
LKg 30
22
22
22
22
4L 0 0 0 3L- 0 L- 0 0 0 3L 00 4L 0 3L 0 0 0 L- 0 3L- 0 00 0 0 0 0 0 0 0 0 0 0 00 3L 0 36 0 0 0 3L 0 36- 0 03L- 0 0 0 36 0 3L- 0 0 0 36- 0
0 0 0 0 0 0 0 0 0 0 0 0L- 0 0 0 3L- 0 4L 0 0 0 3L 0
0 L- 0 3L 0 0 0 4L 0 3L- 0 00 0 0 0 0 0 0 0 0 0 0 00 3L- 0 36- 0 0 0 3L- 0 36 0 0
3L 0 0 0 36- 0 3L 0 0 0 36 00 0 0 0 0 0 0 0 0 0 0 0
T=Tensile force in N
4.5 Load Matrix
L6q61-
L6q61
q q q q
L6q61
L6q61
q q q q
2LW
y
z
w
z
y
x
y
z
w
z
y
x
qx, qy, qz are respectively uniform distributed load corresponding to x,y and z axis
as labeled in Fig 4.3
63
B RAYLEIGH DAMPING
The Effect of Viscous Damping
The global damping is implemented in the numerical models by Rayleigh’s
damping model which assumes that the damping matrix can be written as a linear
combination of the mass and stiffness matrix as
C = a0M+ a1K
where,
C is the damping matrix
M is the mass matrix
K is the stiffness matrix
a0 , a1 are the Rayleigh coefficients
Rayleigh coefficients can be calibrated perfectly for two eigenmodes by
Two different damping ratios, 1 = 0.05 and 2 = 0.05 are used to show the effect
of the damping ratio on the dynamic magnification factor.