~~ HSE

Health & Safety Executlve

OTI 93 613

STRUCTURAL ANALYSIS OF PIPELINE SPANS

Prepared by J P Kenny & Partners Ltd for the Health and Safety Executive

Offshore Technology lnformation

Health and Safety Executive

STRUCTURAL ANALÍSIS OF PIPLEINE SPANS

OTI 93 613

Development of Guidelines for Assessment of Submarine Pipeline Spans

Background Document Two

Report prepared by

J P Kenny & Partners Ltd Thames Plaza, 5 Pinetrees

Chertsey Lane, Staines Middlesex TW18 3DT

HSE BOOKS

Health and Safety Executive - Offshore Technology lnformation

© Crown copyright 1993 Applicationsfor reproduction should be made to HMSO

First puhlished 1993 1SBNO 7176 0639 2

This report is published by the Health and Safety Executive as part of a series of reports of work which has been supported by funds fonnerly provided by the Department of Energy and lately by the Executive. Neither the Executive, the Department nor the contractors concemed assume any liability for the reports nor do they necessarily reflect the views or policy of the Executive or the Department.

Publications in the Offshore Technology lnformation (OTI) series are intended to provide background information and data arising from offshore research projects funded by the Department, or the Executive, and major companies.

Results, including detailed evaluation and, where relevant, recominendations stemming from their research projects are published in the OTH series of reports.

FOREWORD

A repon in the Offshore Technology series (OTH 86 231) "The Development of Guidelines for the Assessment of Submarine Pipeline Spans - Overall Summary Repon", was published in 1986 by the Depanment of Energy, who at that time were responsible for authorising the construction and operation of pipelines io UK waters.

It descrihed the maio fiodiogs of a major study on the problems associated with pipeline spans. The work was carried out by J P Kenny and Panners Ltd and comprised a wide­rangiog programme of theoretical and experimental work, designed to provide a basis for the development of guidelines for the assessment of pipeline spans.

There have heen a numher of requests for access to the detailed information on which that repon was based and the theoretical development and experimental data are obviously valuable reference sources. Acconlingly the Offshore Safety Division of HSE ha ve agreed to publish the back ground repons prepared at the time the project was completed. These comprise:

l. "Evaluation ofVonex Sheddiog Frequency aod Dynarnic Span Response" OTI 93 614

2. "Structural Analysis ofPipeline Spans" OTI 93 613

3. "Vibration of Pipeline Spans" OTI 92 555 •

The first two of these deal with the principal aspects of the spans project aod the third describes io detail the experimental work carried out on the full scale pipeline span test rig built in the Severo Estuary.

• The last repon describes work done by HR Wallingford Ltd, which at the time of the study was called Hydraulics Research Ltd ·

lll

CONTENTS

Page

SUMMARY ix

1 . INTRODUCTION 1

2. SPAN FORMATION, BOUNDARY CONDITIONS ANO 3 LOADINGS

2.1 lntroduction 3 2.2 Pipeline installation methods 3 2.3 Development of spans 5 2.4 Types of span 7 2.5 Span end conditions 14 2.6 Loading conditions 14 2.7 Types of loading 14 2.7.1 Submerged weight 17 2.7.2 Effective mass 17 2.7.3 Interna! and externa! pressure loading 18 2.7.4 Thermalloading 21 2.7.5 Residuallay tension 21 2.7.6 Non-linear sag tension 23 2.7.7 Soil/pipeline interaction 23 2.7.8 Seawater/pipeline interaction 27 2.7.9 Seismic loading 31 2.7.10 Trawl board pullover or hooking 31 2.8 Span analysis load summary 32

3. STA TIC ANALYSIS 35 3.1 1 ntroduction 35 3.2 Equilibrium of supported and free pipes 35 3.2.1 Equilibrium equations 35 3.2.2 Boundary conditions 37 3.3 Non-dimensional parameters 39 3.3.1 Vertical equilibrium equation 39 3.3.2 Effective axial force, J3 0 40 3.3.3 Horizontal equilibrium equation 41 3.3.4 Boundary conditions 41 3.3.5 Summary of non-dimensionalised parameters 41 3.4 Method of solution of equilibrium equations 42 3.5 Static response and strength 42 3.5.1 Static response 43 3.5.2 Yielding 43 3.5.3 Buckling 43 3.5.4 Serviceability 50

4. DYNAMIC ANALYSIS 55 4.1 lntroduction 55 4.2 Methods of dynamic analysis 55 4.2.1 Dynamic equilibrium equations 55 4.2.2 Energy method 57

V

4.2.3 4.2.4 4.3 4.4 4.5

4.5.1 4.5.2 4.5.3 4.5.4 4.5.5

Rayleigh method Limitations of the Rayleigh method Methods of estimation of natural frequency Dynamic span response to vortex shedding Effects of pipe, span and seabed characteristics on natural frequency Effects of axial force, T Effects of soil conditions Effects of seabed geometry Static and dynamic non-linear effects Multiple spans

5. PARAMETRIC ANALYSIS OF SPANS 5.1 lntroduction 5.2 Ranges of parameters 5.2.1 Linear axial force parameter, ~' 5.2.2 Seabed slope parameter, 1.. 5.2.3 Soil stiffness parameter, y 5.2.4 Characteristic displacement parameter, 1; 5.2.5 Pipe/soil friction parameter,l) 5. 3 Sensitivity studies 5.4 Non-dimensional design curves 5.5 Span analysis methods

6. CONCLUSIONS

APPENDIX A • Notatlon and deflnltlons

APPENDIX B · References

LIST OF TABLES

Table 2.1 Typical ranges of E,, v, and k. for various types of soil 2.2 Typical ranges of axial pipe/soil friction coefficients for

various types of soil 3.1 Summary of non-dimensional parameters 4.1 Frequency-static displacement relationship for difieren!

end conditions 4.2 Comparisons between exact and approximate values of

natural frequency 4.3 Dependence on bi-linear soil stiffness ratio on the pipe

end support conditions 5.1 Critica! span lengths 5.2 Variation of slope parameter 5.3 Limits of soil stiffness parameters 5.4 Recommended analysis factors

vi

Page

57 58 58 59 60

60 60 66 70 76

79 79 79 79 80 83 84 84 84 89 94

97

99

107

Page

24 25

42 58

59

63

81 81 83 89

LIST OF FIGURES

Page

Figure 2.1 Pipe installation techniques 4 2.2 J-lay pipe laying method 6 2.3 Typical formation of spans due to scouring action 8 2.4 Typical mechanisms of pipe burial due to sand movement 9 2.5 Typical examples of single span geometry 10 2.6 ldealised span geometries 11 2.7 Typical examples of multiple spans 12 2.8 Pipe span al plalform base 13 2.9 Typical span end conditions 15 2.10 ldealised end condition models 16 2.11 Pressure effects on a pipe 19 2.12 Lateral out-of-balance loads on curved pipe due lo 20

interna! pressure 2.13 Development of axial force in pipe spans 22 2.14 Pipe/soil friction behaviour 26 2.15 Soil pressure distribution around a buried pipe 29 2.16 Typical ranges of suitability for various wave theories 30 2.17 Functional plus environmentalload summary 33 3.1 Equilibrium of pipe element 36 3.2 Notation and convention for span geometry and supports 38 3.3 Typical static behaviour of pipe spans 44 3.4 Local buckling of axially compressed pipes 45 3.5 Collapse of pipes subject to bending 46 3.6 Collapse of pipes subject to externa! hydrostatic pressure 47 3.7 Collapse of pipes subject to combined bending and externa! 48

pressure 3.8 Theoretical and experimental information for pipes subject 48

to combined bending and externa! pressure 3.9 Relationship between propagation pressure and D/1 49 3.1 o Local and propagation buckling test results for pipes subject 50

lo combined bending and externa! pressure 3.11 Buckling strains of steel pipes subject to bending 52 3.12 Strain-flattened relationship for tubes subject lo bending 53 3.13 Externa! pressure- curvatura relationship 54 4.1 Behaviour of pipe span under dynamic loading 56 4.2 Basic span configurations for unburied and buried pipes 61 4.3 Variation of natural frequency of pipeline spans with axial 62

force, assuming rigid soil supports 4.4 Variation of natural frequency with soil stiffness and axial force 64 4.5 Pipe span with bi-linear soil stiffness 65 4.6 Graph showing effect of bi-linear soil stiffness on natural 67

frequency of span 4.7 Typical distribution of frictional force 68 4.8 Effect of soil friction on natural frequency 69 4.9 Static non-linear effects 71 4.10 Variation of non-linear tension with characteristic 72

displacement for different friction coefficients 4.11 Dynamic non-linear effects 73 4.12 Variation of natural frequency with amplitude of vibration 74 4.13 Effect of vibration amplitude on side-span length 76 4.14 Multiple spans 78 5.1 Approximate dependence of critica! span length on 82

pipe diameter 5.2 Variation of frequency and moments with friction parameter 85 5.3 Variation of non-linear tension with friction parameter 86 5.4 Variation of natural frequency and bending moments 87

with slopes

vii 1

viii

5.5 5.6 5.7 5.8 5.9 5.10 5.11

Variation of non-linear tension with slopes Non-linear axial force design curves for y= 107

Non-linear axial force design curves for y= 1 O' Non-linear axial force design curves for y= 1 O" Non-linear axial force design curves for y= 1 CJ' Bending moment ratio (M/Mc) design curves Natural frequency ratio (file) design ctlrves

NOTATION AND DEFINITIONS REFERENCES

Paga

88 90 91 92 93 95 95

110 115

SUMMARY

The formation of spans either during installation or by subsequent scouring or movement, may have a critical influence on the safety and integrity of a submarine pipeline. The early detection, assessment, and where necessary, correction of spans is importan! to the safety and life expectancy of pipeline systems. There is therefore a need for guidance which will enable reliable assessments to he made of the structural response of a pipeline to particular span conditions which may be random in nature. This report presents a method of analysis which allows the safe evaluation of the static and dynamic characteristics of pipeline spans.

The various methods of pipelaying are considered such as laybarge, reeling, J-I ay and tow­out. The development of spans during laying or within the service life of a pipeline is discussed. The almost infinite number of span geometries resulting from these conditions are classified into a relatively small number of idealised formations in arder to carry out the analytical procedures.

The span end conditions are discussed with regard to axial soil stiffness, lateral soil stiffness and rotational soil stiffness. The end conditions may change during the lifetime of a span according to the various loading and environmental conditions occurring in its installation and operational phases.

The main loading conditions during the lifetime of a pipeline are identified as instaiiation, water-filled, hydrotest and operational. The types of loading are classified as functional, which are the loads due to the existence of the pipe (eg weight of pipe, coating, contents, etc), and environmentalloads (eg waves, currents, trawl-board hooking, etc). The various combinations of fnnctional and environmentalloadings are addressed and it is concluded that the most importan! forces applied to the pipe are !hose in the vertical plane.

The pipeline's response to these loads is seen to vary considerably depending on the interactions with the environment, configuration and geometry of the particular span. This complexity means that pipeline behaviour may be itself much more complex than is predicted by sorne of the simple analytical methods in use, particular! y those based on engineering bearn theory. It was therefore necessary to develop a more rigorous static analysis procedure. This set out the exact differential equilibrium equations and boundary conditions describing the behaviour of supported and free-spanning pipes and their various interactions. The static response and strength of pipes is discussed with special reference to yielding and buck:ling hehaviour. ·

A similarly rigorous approach was adopted for the development of a dynamic analysis of pipeline spans resting on elastic supports including the effects of friction and non-linearities such as sag tension are difficult. In this case it was desirable to introduce simplifications which would enable acceptable approximations to be made. In this way the effects of pipe, span and seabed on the natural frequency are considered in detail and then assessed to identify those areas where further more accurate information may be needed for reliable estimations of natural frequency.

The results of the work previously described were then applied to the development of a parametric span analysis method. The variables which influence the hehaviour of a pipeline span are expressed in non-dimensional format in order to reduce the total number of independent variables and provide generalised solutions to pipe spanning problems. The structural quantities of interest are axial force, bending moments and natural frequency.

Because of the complexity of the pararnetric method of span analysis described, a two stage approach is suggested. A conservative analysis assuming that the span is simply supported is carried out. Spans which are unacceptable by these criteria are considered lo be critica! and require the full pararnetric analysis methc:id to he applied.

IX

1. INTRODUCTION

Pipeline spans on the seabed required assessment to determine whether or not remedia! action is required to avoid damage to the pipeline. lt is in the interests of both the operators and the regulatory body concemed, to ha ve the ability to · predict the state of a span as accurately as possible;. The Department of Energy commissioned J P Kenny to carry out a research project in order to provide the technical background for the development of guideline. This comprised theoretical studies, laboratory model testing of spans, and full scale testing of a 20 inch diameter pipeline span.

This report addresses the detailed structural analysis of submarine pipeline spans, and presents the technical background toa parametric method of structural analysis. A separate background report deals with the hydrodynamic and span response aspects of pipeline span behaviour.

The parametric method of structural analysis provides a means of evaluating the state of stress and natural frequency, for the purpose of span assessment. The analysis is a complex procedure but it is only necessary to apply it to critical spans. A simple and conservative method of span assessment has been developed which assumes that the span is simply supported and checks the following failure modes:

• yield • bar buckling • flow induced vibrations

Spans which are found unacceptable by this simple analysis are considered to be critical and are therefore subject to the detailed non-dimensional, parametric method of span analysis which has been developed and is described in this report.

This report is arranged as follows:

• Summary of the work. • Section l is the Introduction. • Section 2 describes the mechanics of span formation and the loading conditions to

which a span is subjected during its design lifetime. • Section 3 establishes the static response of a span, and its strength against yielding and

buckling. • Section 4 pro vides methods for determination of the natural frequency of a span which

govems its dynamic response.

• Section 5 presents a method for structural analysis of pipeline spans by means of five controlling parameters.

• Section 6 sets out the conclusions from this part of the study.

2. SPAN FORMATION, BOUNDARV CONDITIONS ANO LOADINGS

2.1 INTRODUCTION

The formation of spans in submarine pipelines, either due to seabed irregularities during installation or due to subsequent scouring and movement, may have a critica! influence on the safety and integrity of the pipeline. Usually potential spans can be identified prior to installation by proper gathering of data and marine surveys along the proposed pipeline route [19, 20]. Available methods of analysis [!] may then be used to investigate the static and dynamic characteristics of these pipe spans, to ensure that the pipe can be maintained at an acceptable safe state. If cases where the required safety cannot be ensured then remedia! actions in the forro of rerouting, span correction, use of gravity mattresses, etc., are used to make certain that design criteria regarding stress levels and potential for vortex shedding induced response are not exceeded

However, spans may be created subsequent to installation due to scour action or pipe movement. In such cases their configuration is usually random, depending on the seabed topography and composition, wave and curren! action and pipe properties [19,26], The early detection, assessment and correction of a span is important to assure the continued safe use of the pipeline. In view of the random nature and the unpredictability of the extent of spans, they can constitute a potential hazard to the pipeline, in that increasing span lengths are accompanied by increasing pipe displacements, possibility of vortex shedding induced oscillations and eventual damage. In addition to the importance of detecting such span early, there is a need for guidance in assessing their effect on the structural response of the pipeline, to enable engineers to make decisions regarding the need for intervention [23]. Such guidance should preferably be based on simple, yet sufficiently accurate, methods to allow the safe evaluation of the static and dyn~c characteristics of these spans.

It is the intention of this Chapter to provide such information in an easily accessible and explicit form. The following sections provide background information on the mechanics of span formation and response, with special emphasis on the assessment of the various types of loading to which pipe spans may be subjected.

2.2 PIPELINE INSTALLATION METHODS

There are various methods of submarine pipeline installation that have been or are being developed, including lay-barge, reel vessel, J-lay and tow method. The appropriateness of each method depends on the circumstances of the pipeline, particular! y the water depth and pipe size. The most common installation method used at the present time, especially in the Nortb Sea, is laying by a specialised lay-barge.

In the lay-barge method [19], the pipe is jointed on the vessel using conventional welding techniques, tested and field-joint coated. Once welded, the pipe is fed into the sea over a curved supporting stinger, by moving the vessel forwards on its anchors. The pipe takes up an S-bend configuration, as shown in Figure 2.1 (i), which can be controlled to preven! overstressing and buckling. The stinger radius controls the overbend curvature and the sagbend curvature is controlled by the tension applied to the pipe by tensioners on the barge. The required tension depends on the water depth, the weight of the pipe and the allowable radius of curvature of the sagbend. The pipeline is general! y in a state of residual !ay tension as it reaches the sagbend. The maximum depth to which pipelines can be layed by lay barges is govemed by their tension capacity, ranging, for example, from 50m to 300m [2I].

3

LAY BARGE

STINGER

PIPE

' ...... .

i l PIPE LAYED BY LAY BARGE

HOLD-BACK VESSEL TOW VESSEL

ii} OFF -BOTTOM TOW

Figure 2.1 Pipe installatlon techniques

4

One of the oldest methods of pipelay is by reeling [ 1 9]. The pipeline is manufactured on shore and reeled round a large drum on a purpose built vessel. The pipe rnay undergo plastic deformation during this process, but is straightened using a special ramp prior to installation. From then on the method of installation is similar to the lay barge, but the stinger is generally steeper effectively eliminating the overbend. Therefore, residual lay tension levels are relatively lower than those arising from lay barge methods. Because of restrictions in curren! Codes [1] regarding plastic deforrnation ofpipelines, the maximum diarneter pipe that can be reeled is 0.4m, with D/t < 25.

For short pipelines or bundles, towing methods, described in References [19,22]. rnay be preferable to the conventionallay barge methods. Surface, below-surface, off-bottom and on-bottom tow methods ha ve been developed and used successfully in the recent past. These methods enable the pipe to be fabricated onshore and then pulled out by vessels, as for example shown in Figure 2.1 (ii) for off-bottom tow. Residual lay tension levels arising from such methods are usually small or even zero.

With the discovery of new oil and gas fields in very deep water new methods of pipe lay are necessary. One of the methods presently being developed is the J-lay method. A detailed description of the method is presented in Reference [30]. As the narne suggested, during installation the pipelines will take up a J-configuration as opposed to the usual S-shape, as shown in Figure 2.2. This is achieved by lowering the pipe almost vertically into the water, thus totally eliminating the overbend. This reduces the total stress in the pipe, allowing greater depths to be reached [32]. It also has the additional effect of reducing residual lay tension levels in comparison to lay barge methods. Dynamic positioning may be used to keep the J-lay barge on course, thus alleviating the problems of requiring anchoring in deep water.

In general, as can be seen from the descriptions above there can be considerable variations in the levels of residual la y tension depending on the method employed for pipe laying.

In addition, in places of uneven seabed, pipelaying rnay be followed by the development of spans, that may have a critica! effect on the pipeline response and integrity during its lifetime. ·

2.3 DEVELOPMENT OF SPANS Spans can develop during laying, because of seabed irregularities, or during the service life of the pipeline, due to scouring and, in sorne cases, due to horizontal movements. The mechanics of development and behaviour of spans are discussed below. During pipelaying, discussed in the previous section, the pipeline takes up a configuration on the seabed which is dependen! on the seabed profile, the type of soil, the residual tension, the pipe flexura! stiffuess and its submerged weight. ·

U neven areas in the seabed constitute serious problems for the structural safety of pipelines as they enhance the forrnation of free spans. Route selection, therefore, plays an importan! part in design, as discussed in Reference [23]. However, it is usually impossible to selecta totally obstruction free route. In most situations intervention work may be needed in order to reduce the likelihood of spans occurring during installation. This. for exarnple, can take the forro of clearing the routing corridor prior to installation.

The effect or residual tension on span creation is closely linked to the pipe weight. A large residual lay tension tends to generate more spans, and to increase span length, whereas a heavy pipe will norrnally rest on the seabed, thus minimising the number and length of spans. However, if the pipe is. beavy. greater tension is required during installation in order to prevent overstress.

5

6

J- SHAPED

SAG BEND

PIPE

Figure 2.2 J·lay pipelaylng method

;,.·

J-LAY VESSEL

.:·. ,·,.

The amount of residual tension in the pipe in contact with the seabed al so depends on the soil friction. The ends of the installed pipeline and the pipe at the sides of spans are free to move. If this movement is nüt resisted by soil friction, it will have the effect of reducing the value of the tensile force. In general, at sorne point along the pipeline sufficient soil friction develops to preven! further movement, usual! y in the first kilometre from the pipe ends. This is normally termed as the 'anchor point'. The full residual lay tension remains effective in the rest of the pipe. A relatively stiff pipe will tend to develop more and longer spans than a less stiff pipe on the same irregular seabed. The value of the residual !ensile force also depends on the method of installation. With sorne la y methods residual tension in the installed pipe may be small or even zero, as for example with mid-depth tow.

Subsea surveys immediately after installation sometimes show the presence of these span features [24]. However, subsequent surveys carried out at regular intervals, may show changes in the configuration of these primary spans. Sorne of them may alter both in length and depth, while other fill-in complete! y, and in cases, new ones appear for the first time. This occurence is usually dueto scour effects.

Scour is caused by the flow of steady and wave-induced currents around the pipelines. In this event turbulence is set up due to the obstruction to the flow path by the pipe. As a result, the seabed may be eroded away from both the upstream and downstream sirle of the pipe, as discussed in Reference [28]. This is especially evident with granular materials such as sand.

The scour can be caused by long term or short term effects, with the span configurations constantly changing. As scour boles develop around the pipe, gaps form under it, creating small spans. At such gaps fluid jets from the upstream si de may be forced under the pipe, thus leading to "tunnel" erosion. Further erosion under the span is initially very rapid. As a result, substantially long free spans can be formed by scouring, as shown in the illustration in Figure 2.3 obtained from experimental results reported in Reference [26]. These effects have also been shown in tests perforrned on small scale models [25]. In addition, these tests have also shown that continuous current and wave action may also result in eventual burial of the pipe, as illustrated in Figure 2.4, [26].

Full scale tests, reported in Reference [26]. show that most scouring takes place shortly after installation. This is because of the immediate imposition and interference of the pipe on the flow re gime near. the seabed. It is therefore importan! to assess the extent of span creation from scour effects and to examine the seabed Configuration as soon after installation as possible. Timely detection of spans, and remédial action, where necessary, are important to ensure pipeline integrity and continuous safe operation.

2.4 TYPES OF SPAN

It follows from the discussion on span development that the irregular nature of the seabed topography and the randomness of scouring, can give rise toa theoretically infinite number of possible span geometries. Severa! typical examples of single spans are shown in Figure 2.5. indicating the influence of seabed slopes. Single spans can be idealised in analysis as shown in Figure 2.6.

Pipeline spans are not limited to the single types addressed abo ve. Figure 2. 7 shows sorne possible multiple span configurations, in which adjacent spans are located in sufficiently close proximity for interaction between them to be possible. In such situations each span cannot, therefore, be considered individual! y, but rather the entire system must be analysed as a unit. In addition to these simple cases; more complex forros may occur in practice, including, for example, spans created at the base of platform risers, as shown in Figure 2.8. However, the range of span types is usually much more limited in practica! situations. A survey carried out on the Frigg pipelines l and 2 (D = 0.8m) and reported in Reference [7]., assessed the frequency of type occurrence and dimensions of the spans present along the pipelines. The most cornmon type observed was the single pipe span lying on the top of the seabed, and which on average makes up 75% of the total number of spans. In addition, the slope of the seabed at the sides of the spans were found to be relatively small. However, spans in different regions may have different characteristics and the boundary conditions at either side of the span may show wider variation:

7

.• .. -" ' ~ ,- .... -.

/

····' •'

_;~:,;.;·_ ....... - _,.,..

··t'•F_ .. i'·-r 'Y.; .. .:

il

li)

Figure C2.3 Typlcal formation of spans due to scouring actlon

8

1 )

ji)

.. "

;;o -: .·:

.. · .. -./ ·<_.":- ~_:... . ~-.¡e'~-~-~-_, .. _ ... ~-~~.¿,·

- ......-- ~-- .. -.-,_ ·, j

:·. . ... ·· ·<

-- ~-·; -.

~.1·

Figure 2.4 Typlcal mechanisms of pipe burlal due to sand movement

9

L

i) PIPELINE CROSSES SEABED DEPRESSION

L

ii) PIPELINE CROSSES SEABED WITH CHANGE IN SLOPE

iii) PIPELINE CROSSES SEABED DEPRESSION WITH SLOPING ENDS

Figure ·2.5

Typical examples of single span geoinetry

10

i)

i i)

i);~' ,, :t

iv)

vi)

._,,_ ~·-'-

ix)

Figure 2.6 ldealised span geometries

11

i) PIPELINE CROSSES SEABED ROCK OUTCROP

ii) PIPELINE TOUCHES DOWN AT CENTRE OF SEABED DEPRESSION

¡¡¡) PIPELINE CROSSES HIGHLY UNEVEN SEABEO REGION

Figure 2.7 Typlcal examples of multiple spans

12

RISER

\4----STRUCTURE

SEALS TUBE

PIPE

l&~~S.~'~:t~1-f~('ii]:¡¿~~~2;.}~~~~\.~H?i!~ih·:~:"i;á~~<~~~~:~~:1~1t~11li~~}~iál~~~~~~~~~r*~t~~;;_ ~1;~-.(·:'?-,\-·_;¡.\~-:i:%{~~;¿_02:~~-: SOIL

Figure 2.8 Pipe span at platform base

13

2.5 SPAN END CONDITIONS

Two of the simplest types of span end condition likely to be encountered in practice are shown in Figure 2.9. Figure 2.9 (i) shows the simple case of an unburied pipeline in contact with and lying on top of the seabed. Figure 2.9(ii) shows the altemative case of a pipeline which is completely buried in the seabed soil. This situation could arise as a result of scouring if the pipe has originally been purposefully buried during installation, or could altematively be due to natural deposition of material on top of an initially unburied pipeline, for example, as a result of sediment transportation due to wave and current action. In analysis these end conditions can be idealised by employing end springs representing the stiffness and frictional resistance ofthe soil, as shown in Figure 2.10.

The conditions at the sides of spans may change duting its lifetime, according to the various loading and environmental conditions, to which the pipeline may be subjected in the installation and operational phases.

2.6 LOADlNG CONDITIONS

During the lifetime of a submarine pipeline it is subject to four loading conditions, narnely:

• Installation:

This represents the period during which the pipeline is installed, but prior to any flooding taking place. The main loads experienced by the pipe during this period will. be wave and steady current loading and self weight. The combination of pressure and bending loads, as a result of the laying method and the formation of spans, may have a signification effect on pipeline behaviour at this stage.

• Water filled condition:

This condition occurs before and after hydrotesting of a pipeline. It may also occur if the line is to be te111porarily abandoned at any stage, in which case it may be filled with (inhibited) seawater to provide increased on-bottom stability. The increase in submerged weight may critically affect pipeline configuration and stress condition.

• Hydrotest:

Hydrotesting involves subjecting the pipeline to an interna! pressure, which exceeds the design operating pressure by a factor chosen according to design Codes [1, 2]. The pipe is usua11y water filled for this operation. As a result of the increased submerged weight and very high interna] pressures the pipe may, at this stage, experience its most severe loading.

• Operation:

Following installation and testing the pipeline enters its operational phase. During operation, the pipe may be subjected to high interna! pressure and temperature. Normal operating conditions are expected to apply for most of the design life ·of the pipeline.

2.7 TYPES OF LOADING

The loadings to which a pipeline is subjected are divided into two groups:

• Function loado;:

14

These are loads that arise a< a result of the existence of the pipeline and the conditions of its use, without considering the influence of the various environmental effects. Functionalloads are, therefore, those dueto weight, including weight of pipe, coating and contents, due to pressure and thermal effects, as a result of its use, and any prestressing due to laying.

SECTION A-A

A

ii) BURIED PIPE ......

Figure 2.9 Typical span end conditions

15

SOIL FRICTION

SPRING1

-%~·~~~~~~~~­Illl __ SOIL STIFFNESS rrr1 SPRING

il UNBURIED PIPE

-~~~~ t

~~~~;_, .. >----SOIL "STIFFNESS SPRING

ii l BURIED PIPE

Figure 2.1 O ldeallsed end condltlon models ·

16

ROTATIONALL y FREE SUPPORT

PIPE BURIED AT SUPPORT

ROTATIONAL RESTRAINED SUPPORT

• Environmentalloads:

These are loads that arise as a result of environmental phenomena, such as wind, waves. current, etc. They are usually random in nature, and are often evaluated on this basis of probabilistic methods. Environmental design data values are general! y specified in terms of a 'return period', which is the average time interval between successive events of the design value being equalled or exceeded. The retum period specifi.ed for a given loading condition is normally related to the expected duration of the loading condition itself. For normal operating conditions, the retum period would generally be a multiple of the design life of the pipeline, whereas for temporary pha.;;es, a multiple of the expected duration of the phase would be more appropriate.

For any given loading condition, it is therefore necessary to consider the combination of the environmental design loads appropriate to that condition, together with the functionalloads. Since it is generally difficult to predict which combinations of functional and environmentalloads ·wm produce the worst overallloading situation, it is usually necessary to analyse all the various possible combinations and select the worst. In arder to facilitate this analysis it is appropriate to first consider and assess the various types of environmental and functionalloads separately.

There are ten types of loading which need to be taken into account during the assessment of a submarine pipeline span. Functionalloadings result from:

• submerged weight of pipe • effective mass ofpipe • extemal and intemal pressure

• thermal strains • residuallay tension

• sag tension • soil/pipe interaction for unburied and buried pipeline.

Environmentalloadings result from the following:

• seawater/pipe interaction, hydrodynamic forces due to current and wave action • loadings dueto seismic activity

• trawl board pull over or hooking.

The pipeline span loadings are based on evaluations of the environment, construction methods, operational parameters and test requirements. The evaluation of these loadings is discussed in detail in the following text.

2. 7.1 Submerged weight

The submerged weight of the pipe can be calculated in a straightfoiVIard manner based on known or specified pipe data. For example, for a concrete coated pipe the submerged weight, q, may be obtained from:

1t 2 2 2 2 2 2 q = ¡ g [p,(D - D, )+p,(Do- D )+p,D,- p.D"] Eqn 2.1

The submerged weight is a uniformly distributed static load which, especially for a heavy pipe; can have a critical influence in detennining on bottom stability and the behaviour of pipe spans.

2. 7. 2 Effective mass

In dynamic analysis, the effective mass of the pipe is the importan! parameter, as opposed to the weight which is needed for static analysis. This, in addition to the mass of pipe, m.. coating, lllc, and contents, ID¡, includes an added mass, lila, and it is given by:

m., = m. :t m., + . rn +rn. Eqn 2.2

The first three components can be calculated simply by knowing the pipe dimensions, thickness and density of coating, and density of contents. The added mass, rn., corresponds to the mass of water which vibrates with the pipeline, and is a function of the following:

17

• vibration frequency and amplitude

• Keulegan-Carpenter number

• Reynolds number

• proximity of boundaries.

However, when the clearance between the pipe and the seabed is larger than three diameters the added mass is usual! y taken as being equal to the mass of water of the same volume as the pipe and coating [l]. The validity of this assumption is discussed in reference [27]. An increase in effective mass has a detrimental effect on dynamic behaviour as it reduces the natural frequency of the span.

2. 7. 3 Interna! and externa! pressure loading

A subsea pipeline is normally subjected to internal and externa! pressures during its operating life as shown in Figure 2.11(i). The internal pressures, P¡, is a result of pressurised contents passing through the pipe during operation and hydrotest. The external pressure, Po. is dueto hydrostatic pressure as a result of the depth, H, of the pipe below the water surface, and is given by:

Po = p.gH Eqn 2.3

As a result of pressure loading, hoop and axial stresses are induced in the pipe.

The average hoop stress, ae, is induced by the pressure acting radially on the pipe wall, as shown in Figure 2.11(ii), and is defined by:

Eqn 2.4

The hoop stress is tensile when P¡ > Po·

If the associated axial strain in the pipe, due to the Poisson effect, illustrated in Figure 2.11 (iii), is completely restrained, for example as a result of pipe/soil friction, an axial stress will be induced, given by:

_ VP;D; O'xv - 2t Eqn 2.5

The externa! pressure does not induce a Poisson stress because the extemal pressure Poisson strains are present before the line is restrained by the pipe/soil friction. A conservative value of the Poisson induced axial force, T, is given by:

Eqn 2.6

In cases where the pipe has an overall curvature, K, the effect of pressure loading will be to induce lateral pressure loads, qp. per unit length, in the direction of the radius of curvature, as shown in Figure 2.12. This may be calculated as follows.

lt is assumed that an interna! pressure P; is acting on a strip element along the pipe, of area (R + r;sin9)8tf>r;OB where R is the radius of curvature of the bent pipe and r; is its interna! radius.

Its componen! in the vertical y direction is P;sin9(R +r;sin9)8$r;cSe so that the total vertical out-of-balance force, Q,, due to pressure acting on a curved pipe is:

f27t Q, = 0 P;sin9(R + r;sin9)8q,r;d9

= 1tr;P;84>

1tD: P;O$ 4 =

The lateral pressure load, q,, may then be obtained from:

18

n_ - _Q,_. ~ - R8<!>

2 rtD¡P;

4R

i)

iil SECTION AA.

¡¡ i)

' '

O¡

L-----LONGITUOINAL CONTRACTION DUETO POISSON'S EFFECT

Figure 2.11 Pressure effects on a pipe

19

1 K=­

R

R s,¡.

R.Sr¡,

Figure 2.12

R R+ r¡ Sine

y

SECTION AA

Lateral out-of-balance loads on curvad pipe due to Interna! pressure

As (!IR) is the curvature of the pipe, K, then

20

q, = 1tD~P¡K

4

An externa! pressure has a similar effect but acting in the opposite direction, so that:

Eqn 2.7

2. 7. 4 Thermal loading Inlet temperature of hydrocarbon products may be significantly elevated above ambient seawater temperature, so that when these are fed into a pipeline, significan! thermal expansion of the pipe material may take place. If the pipeline is restrained axial! y and laterally, due for exarnple to soil friction or mechanical restraint, then thermal expansion effects are suppressed, thus inducing compressive forces, Te, in the pipe. These may be calculated from:

Te=-aAEt.e Eqn 2.8

In parts of the pipeline in continuous contact with the seabed, thermal effects may induce buckling, a phenomenon which is well known in railway tracks [34]. On the other hand, the temperature increases in pipeline spans will usually result in a combination of compressive forces and increased displacements. However, as the distance away from the wellhead or platform increases, the temperature of the pipe contents decreases, due to heat loss, until it becomes equal to the ambient seawater temperature. In these regions, of course pipe spans will not be affected by temperature effects.

It is, therefore, evident that when considering thermal effects on pipe behaviour, it is necessary to consider the distance of the span from the wellhead or platfOrm relative to the heat affected region.

2. 7. 5 Residual lay tension

The residual terision present in the pipeline is dependen! on the method used for the installation of the pipe as discussed in Section 2.2. This is a parameter that is difficult to quantify in view of the many effects that can influence its value. The magnitude of the nominallay tension applied by the !ay barge during installation is generally under tight control and known accurately. The residual value at the sea bed is a factor of this tension and the weight of the pipe at the bottom of the S oc J curves.

An estimate of the residual tension, T,.,, can be obtained approximately by subtracting the weight of the pipe from the laybarge tension, T ""' [32] that is:

Eqn 2.9

where q and H. are the weight per unit length of the pipe in air and the height of the tensioners above the water surface respectively.

This tension will remain effective provided that the pipe remains axially restrained. However, many effects will influence its value during the lifetime of the pipe. For this reason, it may be unrealistic to assume that the residuallay tension indefinitely retains the value immediately after laying. For example, random barge motions at the time of installation will provide initial deviations from the nominal tension, and further deviations will occur as the pipe adapts itself to the seabed configuration and interacts with the environmental conditions. During hydrotesting, movement and permanent deformations of the pipe may occur, resulting in stress relieving. In addition, creep and current [72], thennally or pressure produced movements throughout the life of the pipeline can also effect the value of residual la y teosion. These effects are alllargely unquantifiable and make reliable estimates of residual tension very difficult. For this reason, it is advisable to ensure that appropriate conservative assumptions are made during the assessment of a span condition.

21

EQUILIBRIUM POSITION

AXIAL fORCE• \AY

i) SPAN CAUSED BY IRREGULARITIES IN SEABED TDPOGRAPHY

INITIAL

_L_EQUILIBRIUM POSITION

FORCE = T LAY

- -

AXIAL FORCE = T LAY + T NL

ii l SPAN CAUSED BY SCOURING

1· l ·1

L

iii l DEFLECTED SHAPE OF A TYPICAL SPAN

Figure 2.13 Development of axial force in pipe spans

22 ,,

2. 7. 6 Non-linear sag tension Non-linear sag tension, Tnh is induced in the span as the result of additional deflections of the span being axially restrained at the end supports. Non-linear sag tension is induced when seabed scour causes either existing spans to increase in size or new spans to develop.

The axial force in a span is initially equal to the residual lay tension, Tlay, as shown in Figure 2.13(i). If scouring under the pipeline occurs the span sag is increased. The additional deflection of the span is axially restrained by the friction at the side of the span. The effect of this is the development of non-linear sag tension in the pipe span. Tbe non­linear sag tension, Tnh is additional to the residual lay tension, T1ay. as shown in Figure 2.13(ii). Sag tension is truly non-linear, as it is related to x and y displacements by second arder differentials.

Tbe sag tension can be calculated by considering the additional deflection of the pipe from its initiallay position, as shown in Figures 2.13(ii) and (iii).

The non-linear strain, En~o in the pipe is given by:

1 <!Y 2 e,1=;:(dx)

The associated change in 1ength, OL, of a pipe with span L, which is restrained axial! y, may be obtained from:

oL = d~ (;¡;)' dx

The non-linear sag tension, Tnh can, therefore, be calculated from:

oL T,1 = EAL

EA JL [9Y]' dx 2L O dx

Eqn 2.10

For a pipe with clamped ends, for which the deflected profi1e may be approximated by:

1 ( ltX) y =;:y, 1- cos2L Eqn 2.11

Equation 2. 9 gives

Eqn 2.12

When the pipe ends are not fully restrained and sorne axial movement against soil friction is al1owed to take place, the magnitude of T,1 will be reduced in comparison to that given by Equation 2.12. Sorne usefu1 information on this is given in References [6,15].

2. 7. 7 Soillpipeline interaction The evaluation of the interactive effects between the pipeline and the seabed, when complete contact exists, is important in establishing the potential destabilising effect of hydrodynamic forces and also the expansion due to the pressure and temperature of the pipe contents. In cases of pipeline spans, as discussed in Section 3.5, soil stiffness and pipe/soil friction are important in determining the response of the pipe to various imposed loads.

The currently availab1e data on the behaviour of soils is empirically based and highly dependent on test conditions. In addition the composition of the soil is a prime factor in determining the magnitude of friction coefficients and soil stiffnesses, as is the embedment of the pipe. In general, results from the tests show a considerable scatter band even for nominally the same soil. Indeed the present stilte of information is such that the mechanism of soil behaviour cannot be defined with sufficient precision to permit accurate analysis. For this reason conservative approaches are used in design, by employing soil pararneters such that the associated displacements and forces are calcu1ated as upper bounds. In general, if the choice of soil parameters is crucial for design it is advisable to obtain soil data from in-situ tests.

Relevant information regarding soil stiffnesses and friction coefficients, relevant to offshore pipelines, is given below.

23

Soil Stiffness:

The stiffness of soil, k., in resisting the normal forces applied by a pipe in contact with the seabed, may be obtained on the basis of simple settlement theory. Thus, according to Reference [ 10], and by defining stiffnesses as the ratio of the normal force per unit length to the corresponding soil settlement, k. can be expressed as:

k.=~ 2(1-v,)

Eqn 2.13

where E, is the modu1us of elasticity and v, is the Poisson's ratio of the soil.

Typical ranges of va1ties of E, and v, for various types of soi1 are given in References [9, 12]. These together with the corresponding values ofk. are Iisted in Table 2.1.

The soil stiffnesses in Table 2.1 are more applicable to pipes resting on the seabed or. partially embedded in soil. For buried pipes soil stiffness increases with depth of burial. However. as no definitive methods exist for estimating this depth dependence of les. it may be conservative to assume that the values in Table 2.1 are applicable irrespective of depth. In addition, for buried pipes k, becomes effective for both downward and upward movements, as suggested by the springs model in Figure 2.1 O (ii).

Tabla 2.1

Typlcal ranges of E., v8 and ks for various types of soil

Soil Type E, (MPa) v, k.. (MPa)

V. Soft Clay 2- 15 0.4 - 0.5 l- lO

(saturated)

Soft Clay 5- 50 3- 33

Medium Clay 15- 50 9- 33

Hard Clay 50- lOO 30- 67

Sandy Clay 25 - 250 0.2 - 0.3 l3 - 140

Sand (Loose) lO- 24 = 0.3 5- l3

Sand (Dense) 48- 81 0.2 - 0.4 25- 48

Silt 2- 20 0.3 - 0.35 l - ll

Roe k 103- lo' 0.2 - 0.3 550 - 52000

24

Pipe/Soil Friction

Soil friction has the effect of restraining the pipe against horizontal movement. In pipes it may have· both axial and lateral components, with the coefficient of friction varying accordingly. From the point of view of pipe spans it is the axial component that is of particular interest.

N; with soil stiffnesses available information on pipe/soil friction coefficients is mainly empirical and highly dependent on the test conditions, the soil composition, pipe size, the degree of pipe embedment and pipe roughness. Test results show a considerable scatter band of friction coefficients even for nominally the same soil and pipe. In practice it is advisable to employ conservative approaches such that the adopted friction coefficients result in upper bound estimates of displacements.

A typical range of axial pipe/soil friction coefficient for various types of soil, obtained from a number of reported experimental studies [35, 36, 37, 38], is given in Table 2.2.

SAND

SILT

CLAY

ROCK

Table 2.2 Typical ranges of axial pipe/soil frlction

coefficients for varlous types of soil

Soil Type Longitudinal Coefficient Friction (~)

0.3 - 1.2

0.3 0.9

0.2 0.6

0.6

When a pipe resting on soil is pulled horizontally, as shown in Figure 2.14 (ii), movement will be resisted by the development of friction. Thus, at very small displacements friction will develop very rapidly until a limited value, F 1, is reached. Further horizontal movement, in general, will not affect F,, significantly. A typical experimentally observed response for loose sand, reported in Reference [10], is shown in Figure 2.14 (i). This behaviour can be modelled as shown in Figures 2.14 (ii) and (iii). The initial response of an idealised pipe can be considered to be elastic and described by:

F = G,.u

where G, = E,/[2(l+v,)] is the soil shear modulus. The limiting response is given by:

F = F,

For a pipe resting on the seabed:

Fl = ~q Eqn 2.14

25

~

0·8

0·7

o 6

0·5

"'- 0·4

0·3

o 2

e o 1

s u

; 1 r

I/

1

! ' i 1 ' 1 i 1 1

i 1 : T 1 T í

d ':'~~n~----rv~t=f----¡;;: ;::¡;;:1 =<>"'1-~[3::¡::-:~ :,----,: 1

zg; O 2·5 50 7·5 10·0 125 O SHEAR DISPLACEMENT (rr,m)

i) TYPICAL RESULTS FROM A DIRECT FRICTION TEST ON A LOOSE SANO

~

SOIL

u HHHH+ t?it\'$¡;,}'\;;;;Niééi''\;!c¡¡¡~'f ~/~r~F ~tgtgt~· ~t~. ~ T'';'/¡¡;P,Jo/.'~Zg:"jifi!gt(~i,~

1 1 L------~...l/

1

1

ii) PIPE/SOIL FRICTION RESPONSE

F

LIMITING FRICTION --::::;;;;;;------

REAL PIPE

LONGITUDU'JAL u l u DISPLACEMENT

iii) THEORETICAL REPRESENTATION OF PIPE/SO/l FRICTION BEHAVIOUR

26

Figure 2.14 Plpe/soll frlctlon behavlour

However, development of friction in a real pipe/soil interaction will follow the so lid curve shown in Figure 2.14 (iii). This can be modelled by the relationship:

u F = F,.-

u+c

where e is a constant to be defined as u ~O

and therefore:

or using Equation 2.13:

:: 1 u'= O F, --= Gs e

e = F, G,

F, e=:-~--,­

k,(!- v,)

Assuming that for soils v,= 0.5, then

2F, e =

k. So that the pipelsoil friction can be expressed as:

F = Fl. 2F, u+-

k.

u Eqn 2.15

This relationship can be used in pipeline analysis. It is, of course, an approximate friction modelling but it has the advantage of allowing friction to develop gradually, as observed in experiments, rather than assuming that limiting friction develops immediately after the application of load.

The developed equations may be applied to both pipes resting on the seabed or partially embedded, or to completely buried pipes. However, in ·the later situation the limiting friction, F1, will be affected by soil pressure.This has been accounted for in a theoretical derivation based on ideal soil conditions, developed in Reference [40]. The assumed soil pressure distribution is shown in Figure 2.15, and the resulting limiting friction is given by:

1t Fl = Jl [-

2 Eqn 2.16

where Ps is the submerged soil density and Hs is the depth of the pipe below the soil surface; K., is the coefficient of lateral soil stress at rest, give by:

K, = l - sin<I>F

where <I>F is the angle of interna! friction [ l 0].

2. 7. 8 Seawater/pipeline interaction

Eqn 2.17

Subsea pipelines exposed to wave and curren! action experience hydrodynamic forces, that is lift, drag and inertia forces. These influences may also give rise to vortex shedding, which is discussed in more detail in Reference [5].

The process of evaluating hydrodynamic forces is by its nature extraordinarily complex and one which relies heavily on empiricism. The conventional steps followed are:

• Definition of the nature of the environment with regard to wave and currents, by referring to sorne statistical extreme state.

• Definition of the resultan! steady and oscillatory components of curren! at the seabed, using appropriate boundary !ayer and wave theories.

• Evaluations of imposed loadings using hydrodynamic force coefficients of drag, lift and inertia, or, in the event of vortex shedding, fluctuating force coefficients.

27

Each step has inherent uncertainties and, consequendy, conservative estimates are usually made to ensure that upper bound, and therefore safe, values are used in design analyses.

With regards to the prediction of extreme wave conditions, the practice is to extrapolate from measured or observed data using an appropriate statistical model. The accuracy of this approach depends heavily on the quantity and quality of base data. An important factor in such an analysis is lhe selection of lhe retum period; this is usually taken lo be 50 or 100 years, or al least three times the design life of the system, thus ensuring an inherent degree of safety. For the analysis of a pipeline system in its "as installed" condition, prior to commissioning, 1 or 5 year extremes are normal! y used.

A wave may then by represented by the maximum or the significan! force, obtained using statistical methods. However, in pipeline stability analysis it may be unreasonable to expect that the maximum wave should be taken as the basis for hydrodynantic loading calculations. This is because, in general, the wave traverses only a short section of a pipeline, possible at sorne angle, and with a finite Iength of crest. Consequently, a factored maximum or significan! wave is normally appiied. Such approaches have been used extensively in practica! design situations and ha ve been found to be generally safe .. However, hydrodynamic loadings arise not only from the action of waves, but more general! y from the combined effects of waves and currents.

The definition of the steady and oscillatory components of current involves additional complications because of the disturbances of the flow due to ·wave action and bottom roughness. In practice sea bed roughness cannot be readily quantified and simple boundary !ayer models are often used, based on the appropriate wave and boundary !ayer theories. The choice of wave theory depends on the relationships of the wavelength, water depth and, sometimes, wave height.

Existing literature sometimes concentrates on the definition of ranges of applicability of particular wave theories for prediction of w~ve induced particle motions. A summary of suitable ranges, suggested in Reference [42], is shown in Figure 2.16. Recognised theories include:

e Airy's small arnplitude wave theory

e Stoke's finite arnplitude wave theory (various orders)

• non-linear shallow wave theories, such as conoidal and solitary wave theories

• . strearn function theory.

These are described in detail in Reference [42]. However, such wave theories may be of limited use for conditions at the seabed, which merit careful consideration before the appropriate theory is seiected. This panicular area is inadequately addressed in curren! literature and in available Codes. ·

Other parameters which influence conditions at the seabed are boundary effects, directional spreading of wave induced motions and the effects of wave-current interactions. All of these are highly complex phenomena, the influences of which have not yet been deterrnined to a degree useful for general pipeline design purposes. For this reason the evaluation of hydrodynarnic forces introduces additional uncertainty into the analysis, which is usually compensated by making upper bound estimates to ensure safety. In this context hydrodynantic effects are evaluated as quasi-static forces in the form of lift, drag and inertia forces, dueto the combined effects of waves and currents. The analysis is normally carried out using Morison's equations, which take the following form for forces per unit length of member.

• Drag force, which acts in Iine with the direction of current flow, and is a function of the flow velocity U, so that:

• 1

FD = 2 p.D.CoUIUI Eqn 2.18

• Inertia force, which aiso acts in the direction of current flow and is a -function of the flow acceleration, aw, and therefore only dependent on wave action, given by:

1 2 F1 = 1t ¡ p w D 0 C 1 a w Eqn 2.19

28

NOTES:

P5 = SOIL OENSITY

H5 = OEPTH OF PPE BELDW SOIL SURFACE

k0

= COEFFICJENT OF LATERAL SOIL STRESS AT REST [ 40]

Figure 2.15 Soil pressure dlstrlbutlon around a buried pipe

29

30

0·05-r------.-------------.----,

0·02

0·01

0·005

STOKES' 2nd ORDER 1

0·002

0001

0·0005

00002

o 0001

LINEAR THEDRY

1

1

1

1

IDEEP

IWATER WAVES

INTERMEDIATE DEPTH WAVES

o·ooo05_¡:=:::=;~1f::;===:;==::;:::::=;:==::=;;=j::;==:j o oo1 o oo2 o·oo5 o o1 o o2

H

gT.;

o·os o1

Hw= WAVE HEIGHT (m)

T,. = WAVE PERIOD ( s)

H = WATER DEPTH (m)

g = GRAVITY

Lw = WAVE LENGTH (m)

H0 = HEIGHT OF BREAK!NG WAVE (m)

REFERENCE ( 42]

Figure 2.16

( mfs2)

Typical ranges of suitability for varlous wave theorles

• Lift force, which acts normally to the direction of curren! flow and is a function of the flow velocity U and the proximity to the seabed, obtained from:

Eqn 2.20

U is the design velocity derived by combining the effects of waves, U., and steady state currents, Us, in the form

U = Uw + U, Eqn 2.21

However, theory shows that the wave induced velocity, Uw, and acceleration, aw. are ninety degrees out of phase, so that in a wave cycle with velocity amplitude, U., and period, T,

Eqn 2.22

Eqn 2.23

where Sw is the wave phase angle. lt is normally necessary to use an iterative procedure to determine the phase angle 9w which will produce the worst combination of drag and inertia forces.

A typical method for the selection of force coefficients Co, e[ andeL is given in the DnV Rules [1]. These are dependen! on the Reynolds's and Keulegan-Carpenter numbers and the proximity of the pipe to the seabed. The approach given by Dn V [ 1] should, however, be treated with sorne caution since the values given are not fully consistent with a 'steady state' type loading condition. In general terms:

• Drag coefficient C0 , is dependent on the marine growth and surface roughness of the pipe as well as Reynolds number. Thc seabed proximity effect becomes negligible when the clearance between the pipe and the seabed is larger than one diameter.

• Inertia coefficient, CI is dependent piimarily on seabed proximity. When the clearance is larger than three diameters the value can be taken as approximately two.

• Lift coefficient, CL, is extremely dependent on bed proximity and becomes very small when the pipe clearance is more than one diameter. · ·

Additional relevan! information is presented in References [42, 43]. The lift force is induced by asymmetries in flow caused by the pipe proximity to the seabed and acts away from the seabed, thus opposing self-weight. As the clearance. between the pipe and the seabed increases the tlow becomes more symmetrical and the lift force, therefore, decreases significantly. For these reasons it is generally conservative to ignore lift force in span analysis.

2. 7. 9 Seismic loading In general seismic effects do not need to be considered during span assessment, since they are normally small. However, there is available information on the effects of seismic activity on pipelines, reported in Reference [33], which should be consulted for designs in areas prone to significant earthquakes.

2. 7.1 O Trawl board pullover or hooking Trawl board pullover or hooking needs to be considered in areas of fishing activity. Trawl board loads can be calculated using the information described in Reference [73]. The probability of a trawl board impact may be shown to be sufficiently low for trawl board loadings to be disregarded during span assessment.

31

2.8 SPAN ANALYSIS LOAD SUMMARY

In assessment of pipeline spans it is necessary to consider combinations of the functional and environmentalloads described in Section 2.7. These loadings act on the pipeline span as summarised in Figure 2.17, and are listed below:

• combined axial force, Ta

• externa! pressure, P,

• intemal pressure, P¡

• submerged weight, q

• lateral pressure load, q¡,

• hydrodynantic force, FH

• soil friction force, F

• soil reaction force, ksY

The combined axial force, T, is made up of three linear components and one non-linear component, namely:

• residuallay tension, Ttay

• axial force due to thermal effects, T 9

• axial force dueto Poisson effect associated with pressure loading, Tv

• non-linear sag tension, Tnt

so that:

T, = T1,, +Te+ T, + T,, Eqn 2.24 The pressure loads, P, and P, induce the hoop stress, cre, the Poisson axial force, T,, and the lateral pressure load, qp. The non-linear sag tension is normally only present in spans that develop as the result of scour.

The vertical loading of the span. q, consists of the submerged weight, q and the lateral pressure load, qp. as discussed in Section 2.7.3, so that :

q,=q+q¡, Eqn 2.25

An additional force acting in the vertical plane is the lift force, FL, discussed in Section 2.7.8. However, it has been argued in the sarne Section that FL acts upwards, opposing self­weight and that it becomes very small when the clearance between the pipe and the seabed is greater than one diameter. For these reasons, it is conservative to ignore lift forces in the analysis.

The horizontalloading of the span, qh, consists of the maximum combined drag and inertia forces, FH, and the lateral pressure load, q,, so that:

Eqn 2.26

In the assessment of pipeline spans, it is important to estímate the maximum value of FH. As the maximum values of F0 and F1 do not occur at the same value of wave phase angle, 9., as discussed in Section 2.7.8, it is necessary to use iterative procedures to determine the maximum value of FH.

In addition to the loads acting on a pipe span element, the pipe/soil frictional and reaction forces at positions of contact between the pipeline and the seabed, as shown in Figure 2.17(i). are considered by the structural analysis.

1t is apparent that a free span is subjected to bi-axial loading. A rigourous solution to the static problem should be based on analysis that takes into account the bi-axial nature of loading. This is achieved by making use of the symmetry of the pipe cross-section. Solutions cOrresponding to the loads in the vertical and horizontal planes are obtained independently, and then combined according to the principie of superposition.

In the following sections oniY forces in the vertical plane are considered. This is the plane containing the inertia forces that determine the natural frequency response of the pipe span in its worst response mode. These methods may. then be u sed to evaluate the static response in the horizontal plane. The appropriately superposed results yields the complete static solution.

32

i) TYPICAL PIPE SPAN

cp q.P

iil LOADS ON PIPE SPAN ELEMENT

q. + q.p

iiil LOADS ON SECTION AA

Figure 2.17 Functlonal plus envlronmental load summary

33

3. STA TIC ANAL YSIS

3.1 INTRODUCTION

During installation and operation and the different intermediate stages, a pipeline may be subjected to various loads which are discussed in detail in Section 2. The pi peJines response to these loads will vary considerably depending on the degree of interaction with its environment, its conditions of use and its configuration and geometry. For these reasons, pipeline behaviour may be considerably more complex than that predicted by the various simplified analytical methods [6, 13, 14, 15, 44, 45]. Many of these methods are based on different versions of simple engineering beam theory. While, as a first approximation, this may provide useful results, the validity becomes highly questionable when effects such as soil flexibility, pipe/soil friction and sag tension are considered.

For this reason, it is necessary to investigate the influence of such parameters by using more accurate results, these methods can provide reliable information, against which the simplified approaches can be gauged.

The exact differential equilibrium equations and boundary conditions which describe the behaviour of supported and free spanning pipes, and their interaction with flexible foundations, including non-linear sagging and hogging effects, are given in detail in this Section. These equations are subsequently non-dimensionalised, in order to reduce the number of independent variables and facilitate parametric investigations, and hence determine the importan! pararneters governing the pipeline hehaviour.

Following a brief description of the method of solution the static response and strength of pipes is discussed with special reference to yielding and buckling behaviour, serviceability conditions are also considered.

3.2 EQUILIBRIUM OF SUPPORTED ANO FREE PIPES

3. 2.1 Equilibrium equations The equi1ibrium of supported pipe elements [ 17] can be discussed with reference to Figure 3.1. In deriving the equations it is assumed that the pipe rotations and the initial inclination of the pipe element to the horizontal,"'" are small.

In addition to the forces discussed in Section 2.7, there is a shearing force V anda bending moment M acting on the sides of the element. The pipe/soi1 interaction is represented by the frictional force, F, and the normal reaction force, N. The positive directions and notation for forces and displacements are shown in Figure 3.1.

Thus, summing up forces in the x direction gives:

Consideration of equilibrium in the y direction results in:

::-N +q +q,- [F- d!'] [~ + 'l'•] +T,~ =O

Taking moments about point b gives:

dM-V=O dx

Eqn 3.1

Eqn 3.2

Eqn 3.3

35

36

--- J 'its ~ 7--------1

Sx

i) FORCES

. .¡, ,------ __ \

5 ., x{u)

---

dM _\M+ dXSx

~+dTA.., A dx oX

_} __ _ ----,-y

~y y

dx , a

ii) DISPLACEMENTS ANO ROTATIONS

Figure 3,1 Equllibrlum of pipe element

2 dy d y " -+=oX dx dx

Combining Equation 3.2 with Equations 3.1 and 3.3 to eliminate V and by neglecting second order terms the following equilibrium equations are provided:

d2M ciT dx' - N+ q + q, + T, dx' = O Eqn 3.4

and

Eqn 3.5

Tbe bending moment, M, and the combined axial force, T., can be expressed in terms of the vertical deflection, y, and the axial displacement, u, as follows:

M=-EI~ Eqn 3.6

Eqn 3.7

where Ae is the temperature increment from ambient temperature and a is the linear coefficient of thermal expansion.

Tbe normal reaction force, N, appearing in Equation 3.4 can be expressed, for a pipe resting on soil, in the form:

N = k,y Eqn 3.8

where y is the pipe displacement be1ow the seabed surface. For a pipe buried in a bilinear soil

wheny<O Eqn 3.9

when y> O

However, as the behaviour of unburied pipes is more important from the point of view of safety, the definition of N in Equation 3.8 will be assumed in the rest of this document.

Combining Equations 3.4, 3.6, 3.8 and 2.7 gives a vertical equilibrium equation in the form:

Eqn 3.10

Thus the equi1ibrium equations for a supported pipe e1ement are 3.5 and 3. 10. Tbe same equations can be used for a free e1ement by neg1ecting the pipe/soi1 interaction terms F and

k,y.

3. 2. 2 Boundary Conditions At positions of discontinuity in the seabed and span slopes it is necessary to ensure that the continuity conditions for the pipe are satisfied [17]. Thus, with reference to support A in Figure 3.2(ii) and taking downwards disp1acements positive, pipe continuity is satisfied when:

~ 1 d_y 1 dx L + IJI A + dx R + '1'·

or

Eqn 3.11

where AIJI is the re1ative ang1e between the seabed and the span, given by:

AIJIA = IJIA- '1'· Eqn 3.12

37

A

--L - B

i) SE ABE O j SPAN GEOMETRY

ii) SUPPORT A

lii) SUPPOR T B

Figure 3.2 Notatlon and conventlon for span geometry and supports

38

Similar equations are obtained when considering support B in Figure 3.2(iii). Suffices L and R indicate slopes evaluated at the left and right of the supports respective! y.

3o3 NON-DIMENSIONAL PARAMETERS

The equilibrium equations and boundary conditions for a pipe element on flexible foundations have been derived in Section 3.2. It can be seen from these equations that there are several variables that effect the behaviour of a span such as pipe properties and dimensions, soil properties, seabed configuration and imposed loads. In order to simplify the analysis, by limiting the number of independent variables, it is advantageous to non­dimensionalise these equations. This procedure has the effect of reducing the number of pipe parameters that need to be considered in analysis and has the additional advantage of providing a general solution applicable to severa! pipeline and seabed configurations.

3 o 3 o 1 Vertical equilibrium equation The vertical equilibrium equation 3.1 O can be written in a conveniently modified form as:

<!'Y ~ - -El dx'- T, dx2 + k,y- q Eqn3.13

The effective tension, or axial force, combines the combined axial force, Ta, and the pressure axial force, Tp as given below:

Eqn 3.14

where:

Tp = ¡ (PoD2 - P,D,2

) Eqn 3.15

The pressure axial force, T,, represents the effect of pressure acting on a curved pipe.

Non-dimensionalisation of Equation 3.13 may then proceed by assuming a set of non­dimensional coordinates (x*,y*), given by:

x* = -"- and y* = ':L. Lo Lo

Eqn 3.16

where Le is sorne characteristic pipe length. Thus, Substituting these parameters into Equation 3.13 and after sorne rearrangement the equation becomes:

~ T,L2

d'y* + k,L' -. K Eqn 3_17 dx*4 - EI dx*2 El y = EL

Assuming that the characteristic length, Le. is defined by:

Eqn 3.18

and letting ~ and y be the non-dimensional axial force and soil stiffness parameters, given respective! y by:

TL' El Eqn3.19

ksL4

"( = - Eqn 3.20 El

then Equation 3.17 can be expressed in a complete! y non-dimensional formas:

<i'r. d'y* -* dx*' - ¡3, dx*' + 'YY = 1 Eqn 3.21

Note that the axial force parametefs, ~a. ~e. ~P• ~ 1 and ~81 associated with each of the different axial forces T., T" T,, T¡ and T,¡ may be obtained from Equation 3.19.

Thus, the number of parameters has been reduced from five in Equation 3.13 to two in Equation 3.21.

39

3. 3. 2 Effective axial force, 13.

The effective axial force, T" defined by Equation 3.14, is given below:

Te = Ta + Tp

Substituting for T., Equation 2.24, gives:

Te = T1ay + Te + Tv + Tn1 + Tp The effective axial force, Te. contains two types ofterm, namely:

• Those that are linear in terms of displacement given by TI, where: du V1t 2

T1 = T,,, + EA [dx. - at.8] + z (P,D, )+ T,

• Those that are non-linear in tenns of displacements, given by T ni. where:

Tnl = EA ¡<!Y¡' 2 dx.

Eqn 3.22

Eqn 3.23

Non-dimensionalisation of Equation 3.22 is of no special significance, and it is sufficient to define a linear axial force parameter, ~ .. using Equation 3.19. The non-linear axial force parameter, J3,¡, can be defined in a similar way. Thus when Equation 3.23 is non­dimensionalised it takes the form:

But for pipes with D/t > 10 a good approximation to (A/1) is

A 8 ¡=O'

so that:

J3,, = 4 [:;]' [~]' Replacing L., by the physically more convenient displacement of a clarnped pipe, y,, given by:

yields:

~-.h.. yc = 384EI - 384

2 [YE]' [<!Y"]' J3,, = 768 D dx*

Eqn 3.24

The term (y JO) can he considered a displacement parameter, characteristic for a particular pipe. Hence, by defining a characteristic displacement pararneter, ¡;,as

the non-linear tension can be expressed as:

y,. D

[~.]' J3,¡ = 7682/;

2 UA

The useful outcome of thiS non-dimensionalising process is the derivation of the characteristic displacement pararneter, 1;.

40

Eqn 3.25

Eqn 3.26

3.3.3 Horizontal equilibrium equation For the convenience of non-dimensional analysis and without loss of significance, the horizontal equilibrium Equation 3.5 ~an be simplified into the form:

Eqn 3.27

by assuming that the pipe slope is small, so that the weight term can be ignored, and, that the friction is at its limiting condition, F1, given by:

F1 = llk,y

where 11 is the coefficient of pipe/soil friction.

Eqn 3.28

Using Equations 3.16 and 3.28 and multiplying by (L 3/EI), Equation 3.27 can be non­dimensionalised in the form:

_<!__ [T,L'] = dx* El

or, using Equation 3.25,

L, * .y L

.!ID,_ = 38411 º- y~ y* dx* · . L

1 Defining a friction parameter, 11. such that•

then

Eqn 3.29

Eqn3.30

Thus non-demineralisation of the horizontal equilibrium equation has led to the derivation of friction parameter, 11·

3.3.4 Boundary conditions The non-<limensionalised pararneter pertaining to seabed and span slopes can be derived by considering the boundary condition Equation 3.11, which by using Equation 3.16 may be ·expressed as:

º-Y':.I º-Y':.I - ~,. - -ª-dx* R - dx* L - dljiA L. - dljiA qL'

Defining a non-dimensional seabed slope parameter, A, as:

then

El ).. =. dljl qL'

º-Y':.I º-Y':.I dx* R .- dx* L = )..A

A siiJlilar expression can be obtained for the support B in Figure 3.2(iii).

3. 3·. 5 Summary of non-dimensionalised parameters

Eqn 3.31

Eqn 3.32

The non-dimensional parameters derived in Section 3.3 should provide a complete description of the static response of a subsea pipe. Tbey have the effect of generalising the · analysis andas such they will be used extensively in subsequent Sections of this document. A summary of these parameters is given in Table 3.1.

41

Table 3.1 Summary of non-dimensional parameters

Non-Dimensional Parameter Symbol

y

Tf

Relationship

TL' El

.)\;

D

D ~­L

Description

Axial force parameter

Soil stiffness parameter

Characteristic displacement

parameter

Soil friction parameter

Seabed slope parameter

3.4 METHOD OF SOLUTION OF EQUILIBRIUM EQUATIONS

The differential equilibrium equations. that have been derived in Section 3.2 and non­dimentionalised in Section 3.3, were solved numerically using a computer programme [17]. Because of the non-linear nature of the equations an iterative method was employed, in which the displacements obtained from one step were used to pro vide new estimates of the effective tension and the pipe/seabed interaction forces, which in tum were used as input into the next step. This procedure was repeated until acceptable convergence was obtained.

The derived results were then used to compute bending moments and axial forces, including sag tension, that described the static behaviour, and estimates of the natural frequency, discussed in Section 5.

3.5 STATIC RESPONSE ANO STRENGTH

An important consideration in the assessment of pipe spans, is the evaluation of their structural static response when subjected to the various loading conditions. which the pipeline encounters during its design life. The pipeline and especially the pipe spans are expected to withstand these condition without damage. and within the permissible limitations, specified in Design Codes, typically [ 1]. From the point of view of static strength, these limitation are in the forro of permissible stresses to avoid excessive yielding and buckling, and limitations on deformations for serviceability reasons. These are discussed in the following sub-sections, together with a description of static behaviour.

It is usual in design to employ usage factors when carrying out stress calculations, to account for variabilities in the pipe parameters and the conditions of use. Such factors ha ve not been considered in this document. but typical values are given in References [1.2].

42

3. 5. 1 Static response

A pipe spanning between supports can be idealised as shown in Figure 3.3(i). with the support frictional, F, and normal reaction, N, forces replaced by l~near springs.

· A submarine pipeline span subject to an effective axial force, T,, behaves in an ordinary Euler coturno. Thus when the pipe is assumed to be straight and the effect of self-weight, q. is neglected, the pipe will suffer ""bar buckling'' at the Euler critica! load, TE· When q is included in the analysis it will have an effect similar to that of an initially deflected coturno. with the pipe following path II in Figure 3.3(ii).

The presence of frictional force, F, has an axial restraining effect, which resists axial displacements, and as a consequence, limits lateral displacement, Yo. As a result of this there is a build-up of sag tension in the span, which depends on the degree of axial restraint, and thus an apparent stiffening of the pipe. Under such circumstances the pipe will follow equilibrium path 1 in Figure 3.3(ii), with the combined axial force, T,. increasing rapidly with Y,. in contras! with an axial! y unrestrained pipe.

This type of behaviour, including the effect of friction, is taken into account explicitly by the method of analysis developed in Section 3.2.

3.5.2 Yielding

In addition to the behaviour described in Section 3.5.1, a subsea pipe subjected to the loads discussed in Section 2.7 will develop axial, bending, hoop and shear stresses. If these stresses are large, failure may occur dueto excessive yielding, provided that local buckling does not occur first. For this reason Codes [1.2] specify permissible limits to stress levels to ensure that this form of failure is avoided. Usually these limitations are expressed in tenns of an "equivalent stress", <J'E, calculated according to the von Mises criterion as:

1

a E = [ a;y + a~ - OxCJy + 3-t:!y 1 2 Eqn 3.33

where crll, O"y, and 't~y are the maximum longitudinal stress (incorporating axial and bending . effects). the hoop stress and the shear stress respective! y, acting at the sarne position in the pipe. The limitations placed on crE are usually in the form:

Eqn 3.34

where ur is a usage or design factor to account for variabilities in the pipe parameters and Oy, the yield stress of the pipe material. Por typical values of uf see, for example, Reference [ l].

The pipe stress appearing in Equation 3.33 may be calculated as foltows:

• Maximum longitudinal stress T, MD a, = A + 21

Eqn 3.35

• Hoop stress

Eqn 3.36

• Shear stress. This is usually small and is neglected in analysis. However, if shear forces and torsional moments are significant, then it may be necessary to include 'txy·

3.5.3 Buckling

Considerable research has been devoted in recent years to obtaining theoretical and experimental information on the buckling behaviour of circular cylindrical members such as pipes. Sorne of the results of this research have been incorporated in Codes [1,2]. while other results are too recent for such modification. Most of the relative information is summarised in Reference [46]. The buckling behaviour under different loading conditions is discussed in the following:

43

44

i) AXIALLY COMPRESSED PIPE WITH END RESTRAINTS

z o ¡;;

"' ... "' !i o <.>

o

~ ¡;; z ... 1-

1-+---------- Yo

ii) TYPICAL AXIAL FORCE- OISPLACEMENT RELATIONSHIP

1- AXIALLY RESTRAINED PIPE

R- AXIALLY FREE PIPE

FOR SIMPLE SUPPORTS

Figure 3.3 Typlcal statlc behavlour of pipe spans

Axial Compression

When pipes are subjected to large axial compressive loads, they are likely to develop large lateral deformations in a mode known as "bar buckling", discussed in Section 3.5.1, or localised bulging deformations of the pipe wall. E ven though this latter mode is not very likely to occur in pipe spans. it may occur in certain instances during installation. or in situations where an axially compressed pipe is restrained from developing lateral deformations. Available experimental information is summarised in Figure 3.4. lt can be seen that for pipes with D/t < 50 buckling in a local mode is unlikely to be a problem. However, the theoretical curve, recommended in Reference [53], provides a safe lower bound to the test results. In the range D/t < 100 this curve can be described accurately by the following linear equation:

cr~ = 1 - 0.0024 (~) Cly t

Eqn 3.37

where O'xc is the local buckling stress for axially compressed pipes.

1·2

~ 1·0 u

tl' (/) 0·8 (/) w tr: ... (/)

"' 06

z ::; " '-' "

04 a> ..J

" ,02 g 00

o

Pure Bending

• ' y ' . . . .......... :--- . .. . .· ... •

.~

.............. .~----·- . . ' .......... . ----...... -...... ~----..... _____ .

·-·- . ·-·-·-·---. lli: e TEST RESULTS [s 3l

API [54]

REFERENCE [53]

--- Dnv [o]

50 100 150 200

DIAMETER TO THICKNESS RATIO, 011

Figure 3.4 Local buckllng of axlally compressed pip,es

250

When a pipe is subjected to bending, as in a span, collapse can occur either due to maximum moment reached because of the combined effects of ovalisation and plasticity, or due to local buckling, depending on the geometrical and material properties of the pipeline. Considerable information is available regarding the strength of tubular elements subject to bending and Figure 3.5 shows a collation of test results. lt can be seen that the curve recommended for design in Reference [53] provides a clase lower bound to experimental results. This curve, which includes the practica] range forpipelines, may be approximated to:

M., D - = 1 - 0.0024 -Mo t

Eqn 3.38

where M., is the recommended collapse moment and Mo is the full plastic moment capacity of the section, given by:

M 0 = (D - t)2t crv Eqn 3.39

The right hand side ofEquation 3.38 is identical to that of Equation 3.37, which reflects the clase sintilarity in buckling behaviour between pipes subjected to bending and axial compression. It can also be seen from Figure 3.5 that pipes with D/t < 25 can develop over 95% of their full plastic moment capacity.

45

Extemal Pressure

o :E ....

u 1·2 :E

• •• • • .....

• • ••

• • .... ,.. 1 •• ¡-.... • •• • ...... Z~ 1 o ¡-· ... -'-....;:"'.~t:,·;;·~:·~=-·!=-~ .. ,;~~~~:-~----..__j .... _::r-4-- .,. • 1

~ OB ...... o ~-........ -• z ~ .... -~

' • • • • • • • .. • • " •

~ O ·6 KEY : ..,.--~ .... ..,. "' -.-TEST RESULTS [53,59] ............ .., !; o 4 ~

5 :::> 0·2

- REFERENCE (53)

·--DnV [1]

00+-----~-------r------~----~-------r------~ o 50 100 150

OIAMETER TO THICKNESS RATIO, 0/t

Figure 3.5

Collapse of pipes subject lo bending

The effect of extemal pressure loading on perfect pipes is first to cause a uniform radial contraction, and then al a critical pressure, dependent on the material and geometric characteristics, to instigare a bifurcation type buckling. Pipes with initial out-of-roundness imperfections will start ovalising immediately with the application of pressure and will eventually collapse in a snap-through manner. Considerable research effort, both experimental and theoretical, has been expended on the effects of pressure loading. As a result, the relationship between pipe characleristics and pressure is adequately understood and documented [53]. Available results are summarised in Figure 3.6.

The theoretical externa! pressure, P "' required to buckle a perfectly round elastic tube is:

2E [ t ]' Poc = (l-v2) D-t Eqn 3.40

However, in the presence of small initial flatlening imperfections analysis by Timoshenko [3] has shown that the collapse pressure, Pe, may be obtained from:

2 3 D Pe- [P, + (1 + l e, t) P"] P, + PcrP, = O Eqn 3.41

where Py is the pressure required for development of full plasticity in the hoop direction, given by:

t P, = 2crv(D-t)

ande, is the out of roundness of the pipe, defmed as:

l>D eo==2--

D

Eqn 3.42

Eqn 3.43

The value of e, recommended in DnV [1] is 0.02 for pipes with D < 500mm. Using this value, Equation 3.41 appears to provide clase predictions of experimental behaviour in Figure 3.6.

46

Note that IP6 [2) defines out-of-roundness as equal to eJ2, and recommends a value of 0.0 l for out-of-roundness. Thus care must be taken over the definition of out-of-roundness being u sed.

1·0-r----r----r----.---r::-----.. .·· f/ .·. ·. :\ .. . :.:. ... . oet---+---+--~~~~JL---1 . ,. v···. ). ·ti·(·.·: . . ·:.:.~. . . . . . 06+-------+-------+-~~·~~·~----~~----~

·lV = . r 04+-------+-~~·~~·~------+-----~~----~

:Y 02+---~--~---r--~--~

OL o 02 04 06 OB 1·0

THEORETICAL COLLAPSE PRESSURE Pe /Py

KEY: • TEST RESULTS

EQUATION 3• >1)

Figure 3.6 Collapse of pipes subject to externa! hydrostatic pressure

Combined Externa/ Pressure, Bending and Axial Compression

For pipes subjected to combined bending and pressure loads comparisons between reported theoretical numerical results and recommendations made by Dn V [ 1] are shown in Figure 3.7 for pipes with D/t < 30. A close correlation appears to exist.

The relation~hip recommended by Dn V accounts for combinations of axial compression, bending and pressure loads and may be written as

Eqn 3.44

where cr, is the maximum longitudinal stress, defined by Equation 3.35, and P is the appl~ed extemal pressure. Index n is defined by Dn V as

n = l + 300 [i] (:J Eqn 3.45

Equation 3.44 is valid only when both <Jx and P produce compressive effects.

47

1·0

o." ' ~ 1 w 1 "' 1 :::>

"' 1 "' 1 ... "' 1 .. 1 ..J

05 1 ..

z ~;

1 "' 1 ...

1 1-- NUMERICAL ANALYSIS [ 60,61) X 1

"' 1 --- DnV [1] 1

1 D/t' SO 1

1 \

O· O 1·0 0·0 05

BENOING MOMENT, MIMe

Figure 3.7 Collapse of pipes subject to combined bending and externa! pressure

Propagation Buckling

lt is strictly true, that provided that the pipe dimensions are such that it will not suffer local buck:ling due to imposed loads and that it is free of defects, there is no need to consider the incidence of propagation b~ckling. This is because the pressure to cause a buckle to "ruÍl" is always lower than that to cause local buckling. However, in combined loading situations, for example involving externa! pressure and bending, a local buck:le caused by bending may be converted to a propagating buckle because of the action of externa! pressure. Thus, with reference to Figure 3.8 at very low externa! pressures local buckling caused by bending will remain localised. As the externa! pressure is increased there will be a transition zone above which a local buckle will be converted to a propagating buckle. Its most critical feature is that once initiated it will propagate under a considerably lower pressure than that required to initiate it, P, referred toas the propagation pressure, P p·

... §

"' "' "' g:

~ ffi I­X

"'

KEY: e LOCAL BENDING 8tJCKLING J - O PROI'AGATION BUCKLING TEST RESULTS

- NUMERICAL ANAL.YSIS (e.o] ' --- EQUATION (3 36)

..Q. • 50 1

e

[so]

L, TRANSITION ZONE ==~iji'x::l=

1 le 1 1 1 1

0·0+---------r--------T--------,-----_¿~~----' 0·0 0·5 1·0

BENOING t.IOMENT, M 1 M0

Figure .3.8 Theoretical and experimental lnformation for pipes subject to cOmblned bendlng

and externa! pressure ·

48

The propagation pressure is a characteristic pressure for a pipe and because deformations during buckle propagation are largely plastic, in addition to being dependen! on Dlt, it is al so a function of the material properties. Empirical relationships ha ve been suggested in references [1, 60, 62]. These are compared with available experimental results in Figure 3.9. It can be seen that the expression developed by Kyriakides and Babcock [62], which for steels with low strain hardening takes the form

t 2.25 P, = 10.7 crv (0)

provides a safe lower bound to available experimental results.

Eqn 3.46

The initiation pressure, P1, is the lowest externa! pressure at which a buckle will initiate and propagate when the pipe is subjected to combined externa! pressure and bending loading. As a consequence of experimental investigations, the results of which are summarised in Figure 3.10, an empirical relationship was suggested in reference [51], which can be expressed as

t 2.064 P, = 0.02E (0) Eqn 3.47

This is shown in Figure 3.10 to provide a lower bound fit to the available test results for which propagation buckling was the mode of failure.

0020

t> 0·010 ' o." 0008

~ o 006

"' "' 0·004 w g: z Q

~ 0002 C>

~ o g:

\\\ ';. \\ \ \ \

~' '-V \ t ·. ~

\~ ' \ ~\.

\. 0·001

10 20 30 40 60 BO 100

OIAMETER TO THICKNESS RATIO, 0/ t

KEY· • EXPERIMENTAL RESULTS (60,62.]

REFERENCE (60)

- EOUATION 3 46)- REFERENCE (62)

Do V [o)

Figure 3.9 Relatlonshlp between propagation pressure and D/t

49

·~ o o o o o

~ o e co ..

' o <o o .. "' Ir

" U> U>

"' "' .. ...J

"' z Ir 02 "' .... X • "' •

00 20 40 60 80

OIAMETER TO THICKNESS RATIO, 0/t

KEY. • LOCAL BUCKLING } TEST RESULTS [so] o PROPAGATION BUCKLING

EQUATION (3-47) -REFERENCE [so]

Figure 3.10 Local and propagation buckling test resulta for pipes subject to combined bendlng

and externa! pressure

The complete understanding of propagation buckling is very important in pipeline design. Because its consequences are so significant from a financial viewpoint, care must be taken to ensure its elimination. This is influenced greatly by the level of externa! pressure to which the pipeline is subjected, P, in relation to the initiation, PI, and propagation, P, pressures. Buckling may be initiated in pipeline spans due to large bending deformations and their interaction with extemal pressure.

Thus if P < Pp. even, though the pipe may still develop a local buckle due to bending loading, it is certain that it will not propagate and buckle arrestors are not needed in this range.

However, when Pp < P < P1 propagation buckling is possible, depending on the magnitude of initial flattening imperfections and possible damage. Extensive experimental information on the effects of local pipeline damage on the initiation of propagating buckles is given in reference [63]. Provision ofbuckle arrestors may be necessary in this range.

Propagation buckling is very likely when P, < P, < P, even in the absence of significan! fabrication imperfections and damage. Protection of the pipeline from extensive damage by the provision of buckle arrestors is imperative. Information on the design and performance of buckle arrestors is given in references [64,65].

Of course the altemative is always available to the engineer in designing his system so that P < Pp. otherwise provision of buckle arrestors may be necessary.

3.5.4 Serviceability

In certain situations where significant deformations of the pipe cross-section may occur as a result of loading or excessive plastic action during installation or operation, it is usual to specify limits of acceptable deformation to ensure continued and satisfactory serviceability of the pipeline. For example, limitation of ovalisation and flattening of pipes is important to the unhindered operation of pigging. The important deformation states corresponding to various loading conditions are discussed in the following.

50

Pipes Subject to Pure Bending

In certain situations, mainly associated with pipelaying, where significan! cold-bending may be necessary, and pipeline spans where large bending deformations may occur, the pipe is required to be able to sustain a specified curvature, or bending strain, without suffering collapse. This problem has been examined extensively, both theoretically and experimentally [59, 70, 71]. Experimental results reported in reference [49] are shown in Figure 3.11, illustrating the dependence of the bending buckling strain, e,, on the pipe diarneter to thickness ratio, D/t. A lower bound fit to the experimental results appears to be provided by an equation of the form:

t 2 e, = 15 (D) Eqn 3.48

The corresponding curvature may then be obtained from the well known relationship between curvature, K, and strain, E, in the form

2 K= -e n· Eqn 3.49

In pure bending situations the severity of the ovalisation or flattening of the cross-section is of particular importance, especial! y with regards to pigging operations. For this reason considerable research work [55, 56, 57, 58] has been concemed with establishing a relationship between applied curvature, or bending strain, e, and the associated ovalisation of the cross-section, e". Particular! y simple relationships can be developed using the method pioneered in Reference [55]. Based on this the following expression was developed in reference [58],

e = l [!2. e] ' o 8 t

Eqn 3.50

which is claimed to be valid, as a first approximation, for the bending of pipes in the plastic range. Comparisons with available experimental results in Figure 3.12, for pipes with D/t < 35, sbow that this is indeed the case initially, but considerable disparities occur in the advanced plastic range. In any case Equation 3.50 appears to forro an upper bound to the considerable scatter of experimental results, and as such may be useful in design. However, considerably more research may be needed in this area.

51

0·05

()-04 • • 0·03 •

• 002 • • •• .&> • "' ~ ... .. • a: 0·01 .... •• Ul

" O·OOB ' • ji!; •• _,¡

" 0·006 u ::>

"' " z 0·004 i5 z "' "'

• 0002

' 2 Eb•15l¡¡l

o 001 +-----,r--...---.,.--,~r--...,.--.-...1 20 40 60 BO 100 120 160200

OIAMETER TO THICKNESS RATIO, D/1

KEY • TEST RESULTS (59, 70 J - EQUATION ( 3 · 48)

Figure 3.11 Buckllng stralns of steel pipes subject to bendlng

52

0·06~--------.r--------------------~

004 .. , • o

"' "'

•• • z 002 • •• z • "' • • 1- • 1-<! -' ... 000

0·0 01 0·2 03 04 ( .Q E )2 1

KEY • TEST RESULTS [59,72] - 0/t < 35

EQUATION (3 50)

Figure 3.12 Straln-llattened relationship lor tubes subject to bendlng

Effect of Combined External Pressure and Bending

Tbe presence of externa! pressure has a considerable effect on the capacity of a pipe to sustain a specified curvature without collapse. A vailable experimental and theoretical information indicates that this effect is very strong, with collapse curvature decreasing significantly even at low extemal pressures, in comparison to its value under pure bending, K •. This is corroborated by the results from Reference [61], which have been plotted in Figure 3.13. These and other results reported in Reference [61] suggest a linear interaction between curvature and extemal pressure, of the form

where K., using Equations 3.48 and 3.49, can be expressed as

K = 30 (_!_) 2

• D D

Eqn 3.51

Eqn 3.52

A similar influence of extemal pressure on ovalisation is expected to exist but there is insufficient information to quantify this in any way.

53

54

1·0

u e: C>8 ..

...J

"' z ffi 0·4 .... )(

"' o 2

e TEST RESULTS} - THEORY RE~RENCE(61)

--- EOUATION 1 3 SI) O lt • 34 · 7 • ALUMINIUM ALLOY

06 0·8 1·0 1·2

CURVATURE, K/K0

Figure 3.13 Externa! pressure - curvature relationship

4. DYNAMIC ANAL YSIS

4.1 INTRODUCTION

The exact dynamic analysis of pipeline spans resting on elastic supports, including the effects of friction and non-linearities, such as sag tension, is extremely difficult. For this reason it is desirable to introduce simplifications to the problem that will enable approximate solutions to be obtained. At the same time it is necessary to ensure that these approximations are small and within acceptable bounds. For these reasons this Section initially describes the exact dynamic equilibrium equations and then considers available approximate methods, assessing at each step the levels of approximation introduced.

The pararneter that is of greatest importance in dynamic analysis is the natural frequency, as this determines the response of the system to time-dependent excitation forces. For example, the response of free pipeline spans to vortex shedding is dictated by the closeness of the vortex shedding frequency to the natural frequency.

In order to understand the physical nature of the problem, the effects of pipe, span and seabed characteristics on the natural frequency are considered in detail, using available published information, both for single and multiple spans. Assessment of these results can then help to identify those areas where further, more accurate, information may be needed for reliable estimations of natural frequency.

4. 2 METHODS OF DYNAMlC ANAL YSlS

4, 2.1 Dynamic equilibrium equations The dynamic equilibrium of a free vibrating pipe can be obtained by considering the response of a pipe element, in a way similar to that used for the static analysis in Section 3.2. Thus, with reference to Figure 3.1, the difference between dynamic and static response is that in the former the submerged weight is replaced by an inertia force. This is given by the product of the effective mass, m.,, incorporating added mass effects as described in Section 2.7.2, and the transverse acceleration of the beam element, Yt(t), as indicated in Figure 4.1 (i).

For a pipe element, vibrating freely, and subjected to an effective axial force, T" the dynamic equilibrium equation becomes:

El <!'y.,, - T <f.lí dx e dx.Z = ITle)'t Eqn 4.1

which is similar to Equation 3.13, but with the weight replaced by the inertia term. A dot (.) above y indicates derivatives with respect to t.

When a pipe vibrates transversely in one of its natural modes, its deflection, Yt. varies harmonically with time [47], such that:

y, = y [A cos (mt) + B sin (Olt)] Eqn 4.2

where ro is the angular velocity of the vibration. Substitution of Equation 4.2 into Equation 4.1 results in:

El ~ - T' ~ = mew'y Eqn 4.3

Solution of this equation should yield the exact dynamic displacemeni function, y, and the angular velocity, Ol, and thus the natural frequency, f, from

f = .fQ_ 21t

Eqn 4.4

·55

me X ' . .p .. PIPE SPAN

·iiiE L

X

' 1¿' DYNAMIC DEFLECTED SHAPE

X

' 1(( ACCELERATION

Yt

INERTIA FORCE

il OYNAMIC RESPONSE PARAMETERS

z 14 RESONANCE Q 1-

1 u ... ..J

1

UNDAMPED RESPONSE ... ... (IDEAL PIPE) Q

~ w DAMPED RESPONSE ¡¡:

lE 1 (REAL PIPE)

::>

1 lE FREQUENCIES x .. fu= VORTEX :E SHEDDING

FREQUENCY RATIO fn =NATURAL

fulfn

¡¡ l DEFLECTION FREQUENCY RELATIONSHIP FOR FORCED PIPE VIBRATIONS

Figure ·4.1 Behaviour of pipe span under dynamic 'loadlng

56

Methods of solution of Equation 4.3 when T, ; O and with simple boundary conditions are described in Reference [47]. However, wheil boundary conditions are more general and T, T- O, exact solution may become tedious and cumbersome. Simpler, approximate methods, that produce sufficiently accurate results, may be better suited for use in such situations. Such methods are discussed in the following Sections.

4.2.2 Energy method The dynamic analysis of vibrating pipes can be carried out using conservation of energy principies. In this, assuming that the total energy of the system remains constan!, the maximum potential energy, (PE), must be equal to the maximum kinetic energy, (KE), so that:

Eqn 4.5

For vibrating pipe with no axial forces, and assuming that the maximum potential energy is due mainly to bending action:

(PE)~, ; ~ f El [~] 2 dxlmax Eqn 4.6

The corresponding maximum kinetic energy is given by:

(KE)m,. ; ~ f m. (y,)'dxlmax Eqn 4.7

Substituting y,, using Equation 4.2, in Equations 4.6 and 4.7, evaluating maximum values of (PE) and (KE) with respect to time, and finally substituting the resulting expressions in Equation 4.5, yields: ·

Eqn4.8

Thus, the angular velocity, ro, and therefore natural frequency of the pipe may be obtained using Equation 4.8, provided that the dynamic displacement function, y, is known. A convenient approximation introduced by Rayleigh [11] simplifies solution of Equation 4.8 further.

4. 2. 3 Rayleigh method The basis of the Rayleigh approximation when applied to vibrating pipes, is that the dynamic deflection shape, y, can be assumed to be identical to the deflection shape, y, due to the static application of loads.

This assumption makes it possible to express the maximum potential energy of the system in terms of the externa} work of the statically applied submerged weight, q, (which is now equal to the bending strain energy) so that:

~Ed [!*]' ctx; ~qLctx Thus Equation 4.8 can now be re-written in the simpler forro:

q J ydx ; m.ro' J y2dx

so that ro can be obtained from

ro' - _<¡_ fydx - m. !y'dx Eqn4.9

Thus by knowing the static deflection shape of the pipe it then becomes possible to calculate ro using using Equation 4.9. A further simplification can be introduced by expressing the deflected shape in the forro:

y ; y,Y(x) Eqn4.10

57

where Yo is the maximum static deflection and Y(x) is the deflection function of the pipe. Use of this expression reduces Equation 4.9 to the form:

2 _g_ JY(x)dx ro = m.,y. JY'(x)dx Eqn 4.11

which indicates that the angular velocity and, therefore, the frequency of vibrations is inversely proportional to the square roo! of the maximum static deflection of the pipe.

4. 2. 4 Limitations of the Rayleigh method The choice of a definite shape for the dynamic deflected curve of the pipe, as a result of the Rayleigh approximation, [ 11], is equivalent to constraining its degrees of freedom. This in tum has the effect of increasing the stiffness and therefore the frequency of vibration of the pipe.

Therefore use of Equation 4.11 should always result in overestimation of the frequency in comparison to exact methods. However, severa] examples examined in Reference [47] indicate that this error is very small; for practical pipeline span geometries it is less than 1.5%.

It may, therefore, be concluded that use of Rayleigh's method should result in sufficiently accurately predictions of the frequency of vibrations of pipes, certainly within the error bounds described above.

4.3 METHODS OF ESTIMATION OF NATURAL FREQUENCY

Further simplifications can be made in the method of estimating natural frequency, by extending the Rayleigh approximation, described in Section 4.2.3. That is, in general, it can be assumed that any reasonable approximation of the deflected shape of the pipe should lead toa close estimate of the true frequency of the vibrations [47]. This implies that the ratio of the integrals in Equation 4.11 should be approximately the same irrespective of, say, the flexible boundary conditions of the pipe. Thus from Equation 4.11 and with reference to Equation 4.4:

f'v.m ~ = constant

q Eqn4.12

The validity of the assumptions can be investigated by examining the ratio (f2yorn.Jq) for a pipe with different degrees of flexural restraint at its ends, as shown in Table 4.1, assuming that q = m.,g.

Table 4.1 Frequency · static displacement relationship

for different end conditions

Support Condition

Simple- Simple (SS)

Simple - Clamped (SC)

Clamped - Clamped (CC)

f'y,mJq

1.268

1.285

1.303

% Difference From CC

2.7%

1.4%

As in most practica! situations of pipeline spans the end support conditions will be within the range of those considered in Table 4.1, it can be concluded that the simplification of Equation 4.12 is valid, within the accuracy ofthe present analysis.

58

The implication of relationship 4.12 is that if the frequency, f,, and maximum static deflection, y" of a clamped-clamped axial! y free pipe are adopted as the standard values, with respect to which results from pipes with other support conditions (f, Yo) are compared, then:

Eqn4.13

Estimates of the error involved in calculating f using Equation 4.13, are listed in Table 4.2 for (SS) and (SC) end supports.

Table 4.2 Comparisons between exact and approximate values of natural frequency

f/fc

Support Condition Exact Ref Equation % Difference [47 4.13

SS 0.441 0.447 1.4

se 0.689 0.694 0.7

Sirnilarly, small errors are found to occur even when support flexibility is considered, as shown in Figure 4.5(iii), using solutions described in Reference [47].

Thus the approximations introduced in the dynamic analysis may in general result in small overestimations of natural frequency. However, in the context of vortex shedding calculations lower bound estimates are required. For this reason, it is recommended that a design factor is applied to the natural frequency, calculated according to Equation 4.13, to account for the following errors:

• Rayleigh approximation; 1.5% error. • Approximation involved in Equation 4.13, 1.5% error. • Small errors due to support flexibility.

The frequency design factor that would appear to be appropriate is 0.95.

Thus, with its validity established, Equation 4.13 can be used very conveniently to obtain the natural frequency of a pipe span with any end support conditions, provided that its maximum static displacement is known. Such a simplification is of a great value in structural response analysis, because it means that both the static and dynamic response of pipe spans can be evaluated, simply by carrying out a static analysis.

4.4 DYNAMIC SPAN RESPONSE TO VORTEX SHEDDING

The analysis of free vibrating pipeline spans was considered in previous Sections, as were methods for evaluating their natural frequencies. However, if the pipe is subjected to time­dependent, periodic, disturbing forces, its response is referred to as forced vibrations. In subsea pipelines, depending on the pipe and span characteristics, and the current and wave conditions, fluid tlow around the circular pipe may result in vortices occurring in the wake, at regular intervals. This is referred toas vortex shedding and it induces cyclic forces on the pipe. Methods for calculating the vortex shedding frequency are described in detail in the companion background document, Reference [5].

The effect ofthe vortex shedding frequency, f, on the pipe response can be examined with reference to Figure 4.l(ii). When f, is very small in comparison to the natural frequency, f,, then the effect of vortex shedding is small and pipe deflections are not very different from those due to static action. However, as the frequency ratio, fvlfn, approaches unity the amplitude of the resulting forced vibrations increases very rapidly, reaching a maximum when fv =fu. referred toas resonance. These very large detlections may cause pipe failure

59

due to yielding, buckling, concrete spalling, or combinations of these. In addition fatigue damage may occur due to fluctuations in the stress levels in the pipe [48, 49]. For tbese reasons it is necessary to allow a safety margin between the vortex shedding and natural frequencies to avoid such effects. Recommendations to achieve this are presented in Background Document One, Reference [5].

The assessment of vortex shedding and methods of evaluating the associated forced dynamic response of pipeline spans and their consequences are also discussed in detail in Background Document One, Reference [5].

4.5 EFFECTS OF PIPE, SPAN ANO SEABED CHARACTERISTICS ON NATURAL FREQUENCY

The natural frequency of pipe spans, so crucial in detennining their response to forced periodic loading, may be influenced by the pipe and span characteristics and the seabed conditions. The effects of these parameters on the natural frequency of single and multiple spans are reviewed in the following sub-sections, using available published information. Most of the available literature related to single spans [6, 13, 14, 15, 18, 45], with very little published on multiple spans [16]. lt is shown in these References that the most importan! parameters that affect the natural frequencies are:

• axial force • soil conditions and degree of pipe embedment in soil • seabed geometry • static and dynamic non-linear effects • multiple spans - separation length.

4. 5. 1 Effect of axial force, T Most of the available information on the effects of axial force on natural frequency, corresponds to single pipe spans with either surface resting supports, as shown in Figure 4.2(i), or buried supports, as shown in Figure 4.2(ii), in rigid soil. These effects are illustrated in Figure 4.3 for various end conditions, using results reported in References [6, 13, 14, 18, 44, 45]. In all cases the pipes were assumed to be umestrained axially whilst subject to an axial force.

As expected, a reasonable lower bound to the different curves is provided by the response of a pipe with pinned ends. Increasing the bending restraint at the end supports has the effect of increasing natural frequency. The large variation in values shows that simplification of boundary conditions may not be justifiable, either producing results that may be very conservative when simple supports are adopted, or producing results that may be unconservative if stiffer end conditions are assumed.

The effect of axial tension, as shown in Figure 4.3, is to increase natural frequency. Therefore, in cases where the axial tension cannot be estimated accurately, it may be conservative to calculate the natural frequency assuming f3::; O. However, the same cannot be said for axial compression which is shown to have a detrimental effect on natural frequency. Conservative estimates of natural frequency can, therefore, be obtained only if the maximum axial compression is adopted in calculations.

This dependenc~ of natural frequency on axial forces will be modified if the effects of axial restraint are incorporated. These effects are examined in Section 4.5.4.

4. 5. 2 Effects of soil conditions In most practica! situations for a subsea pipeline, support is provided by flexible soils. There are very few available studies which consider the effects of flexible soil on pipe span response and in many design procedures it appears to be completely neglected. The two main parameters of interest are soil stiffness and soil/pipe friction. Their effect on natural frequency was examined in References [13, 14, 15] and the results are examined in the following. ·

60

.~I~;&;·!.o-.:;,¡;~~~~J.:.i):.W~~::;~

{':,;- ', :;

,.;,:~=""'=~,...,-......J,'-~ ~~l'.:~:~:~::~~~->~·-:· ... , .. ,._;_:~~:-~?~ /

L

i) TYPICAL PIPE, SPANNING BETWEEN HORIZONTAL RIGIO SUPPORTS

ii) BURIED PIPE WITH LINEAR SOIL SPRINGS

Figure 4.2 Baslc span configurations for unburied and buried pipes

61

CLAMPED

i fe

........ - .......... ......... ...... ~

-~-----.-...-----· ······· ---------1·0

-· -· ..... -· -· -· -· -· -· -·- . ··•·····••

~--... ~

~.":!:::·

0·5 ---.._.--..,===--=~=1 _______ _!INNED_ ------ OA

03

o 5 10

COMPRESSON TENSION

KEY: REFERENCE SUPPORT CONDITION

6 SURFACE RESTING SPAN ON.RIGIO SOIL

---- 13 SURFACE RESTING SPAN ON R IGID SOIL

----- 14 BURlE O SPAN IN RIGID SOIL

---- 18 SPAN WITH PINNED ENDS

............. 44 SURFACE RESTING SPAN ON RIGID SOIL

• 45 SPAN WITH CLAMPED ENDS

AXIALL V UNRESTRAINED PIPES

Figure 4.3 Variation of natural frequency of pipeline spans wlth axial force, assumlng rigid

soil supports

62

Buried Linear Soil Stiffness

The span configuration employed in the analysis of the effects of soil stiffness on natural frequency is shown in Figure 4.2(ii). This is based on a pipe with buried ends, linear soil springs and no axial restraint. The dependence of natuial frequency on soil stiffness is shown in Figure 4.4. It can be seen that as the soil becomes more flexible, and the soil stiffness parameter, y decreases, the natural frequency decreases. If, in addition to this, the effect of axial forces is considered, there can be large variations in the value of the natural frequency. Thus, a combination of soft soil and large axial compression may result in very low frequencies.

It is evident from Figure 4.4 that soil stiffness has an important effect on the natural frequency and should be considered qrrefully in analysis. There can be as much as 250% difference between the values off corresponding lo soft and hard soils.

Partially B~ried or Unburied Bi~inear Soil StiffneSs.

In earlier Sections the span was considered as either resting on the surface of the seabed, or with its ends completely buried. The behaviour ·af a span with end conditions somewhere between unburied and total! y buried, is considered in this Section.

An approximate solution to this situation has been developed in Reference [15]. In this, the pipe span was modellOO, as shown in Figure 4.5 resting on bilinear soil springs, with the spring stiffness in the upwiird directión denoted by ~ and that in the downwards direction by

~; their ratio is denoted by r = kJk;. Values of r corresponding to different end support conditions are listed in Tablé 4.3.

Table 4.3 Dependence on bi·linear soll stiffness ratio on the pipe end support conditions

Pipe End Support Conditions k~ • k,

Resting on flexible soil o k, o

Resting on rigid soil o ~ o

Buried in linear soil k, k,

Buried in bilinear soil k; • k,

Buried in rigid soil

The case of pipe ends buried in two different types of soil, for example sand resting on clay, is modelled by assunting a pipe is buried in bilinear sóil.

In the analysis reported in Reference [15] the bilinear soil stiffness effects were modelled approxiÍnately. lt was assumed that the springs in Figure 4.5(i) can be replaced by equivalen! translational, k,,, , and rotational, k22, springs, as shown in Figure 4.5(iii). The stiffnesses of these springs, k,, and k22, were expressed as functions of k; or k:, depending on whether the average deflection of the pipe over the support was downwards or upwards. In addition, the pipe was assumed to be completely unrestrained axially.

63

u -.... 11: w 1-..., :lE <( 11: <( Q.

,_ u z w ::;) o ..., 11: "-

64

1·5

1·4

1·3

1·2

,., 1·0

09

O·B

0·7

0·6

0·5

04

0·3

0·2

0·1

CLAMPED BEAM (3 = 40

----¡3 = 20

¡3 =o

/3 = -4

/3=""8

/J = -,2

11 = -,6

/3 =-20

10° 101 102 103 Kf 105 106 107 108 109 1010 1011

SOIL STIFFNESS PARAMETER, 1f

NOTES

AXIALLY UNRESTRAINED BURIED PIPE SPAN . WITH LINEAR SOIL STIFFNESS

,REFERENCE (14)

Figure ·4.4

Varlatlon of natural frequency wlth soll sllffness and axial force

}~'"' COMPRES StO N

il SPAN RESTING ON BILINEAR SOIL SPRINGS

l~o---DOWNWARD STIFFNESS, k:

BILINEAR SOIL STIFFNESS

UPWARD STIFFNESS, k~ RATIO,

iil BILINEAR SOIL STIFFNESS RELATIONSHIP

I iiil MODELLED END CONDITIONS

Figure 4.5 Pipe span with bl·llnear soll stlffness

65

Even though these are severe limitations on the method, the results obtained from the analysis should still be useful in providing an approximate assessment of the effects of bilinear soil stiffness on natural frequency. The results are shown in Figure 4.6, and they correspond to a soil with "( = 106 These results were obtained by considering a number of typical pipe geometries and span lengths, assuming in all cases that [3 =O. A small scatter was exhibited about the curve shown in Figure 4.6, but always less than ±1 %. lt can be seen that as the soil stiffness ratio, r, increases, the effect of bilinearity on the natural frequency rapidly becomes very small; for r > 0.5 it practically disappears, with the frequency becoming equal to that of a pipe totally buried in linear soil.

What is also interesting, is that the natural frequency of a pipe resting on the surface of the supports, provides a reasonable lower bound to the frequency of a totally buried pipe; for the example in Figure 4.6 the difference is only 14%. This result suggests that in cases where the degree of pipe embedment and the resistance to pipe deflections due to the soil head cannot be quantified accurately, it may be conservative, but not unreasonably so, to base natural frequency calculations on a surface resting pipe support model.

Effect of Soil Friction

In general, the axial movement of a pipe resting on or buried in soil is restrained by frictional forces. These forces are dependen! on the normal reaction force, N, between the pipe and the soil, and the coefficient ofpipe/soil friction, ¡.t, as discussed in Section 2.7.8.

For pipes resting on the surface or buried in the soil at the supports, the development of friction is as illustrated in Figure 4.7. Immediately adjacent to the span, where the support reactions are large, frictional forces of large magnitude develop, decreasing rapidly away from the span, until they reach a limiting value dependent on the submerged weight of the pipe. Further away, as axial movement decreases and becomes zero, no frictional forces develop. The position where this occurs is referred to as the anchor point.

The effect of pipe/soil friction on the natural frequency has been examined in reference [15}. In arder to simplify the analysis, an approximate linear modelling of the development of frictional forces .was adopted in Reference [15]. This modelling may be a reasonable approximation for' pipes completely buried at the supports, but may be unrepresentative of surface resting pipes, as can be seen in Figure 4.7(i). In Reference [15] the frictional

resistance was modelled as a linear friction spring with stiffness, kr = ..J 2~qEA. The results of this approximate analysis have been plotted in Figure 4.8. In this the values of ¡..t > 1 were included to model the effect of soil pressure on the development of frictional forces in búried pipes, discussed in Section 2. 7 .8.

Thus, in Reference [15] instead ofusing Equation 2.22 to define limiting friction for buried pipes, Equation 2.20 was used but with ¡.t modified to account for the difference. The comparisons of Figure 4.8 show that, depending on the pipe and span geometry, there can be a strong influence of J..l on frequency. One important reason why this may be so, is the development of very large sag tension forces due to the increasing axial restraint provided by increased friction. However, these results may, to a certain extent, be misleading, by exaggerating the effects of friction, for the following reasons:

• The geometries for which large variations occur in f with increasing m, correspond to long slender pipes, which may be unrealistic.

e The deflected shape was fixed in all cases to correspond to a pipe with rotationally restrained ends.

• With small ¡.t, implying a surface resting pipe, the assumption of full rotational restraint may b~ too conservative.

Thus, the influence of friction on buried pipes can be exarnined consistently only if considered together with the effects of bilinearity in the soil stiffness, for more practica! pipe and span geometries.

4. 5. 3 Effects of seabed geometry The seabed is often uneven with several changes in slope. The effects of the seabed slope adjacent to spans on natural frequency ha ve been investigated in Reference (6], together with the effects of axial force. The pipe was considered to be restingon rigid sloping supports as shown in Figure 3.2(i).

66

1·00

....) --....) ID

o ~ Ir >- 0·90 u z UJ :::> fil Ir "-

o 80 l---r--.--,.--.--,---,----¡--..,----,,-.,..---0 o 1 o 2 0·3 04 0·5 0·6 0-7 0·8 0·9 1·0

BILINEAR S~L STIFFNESS RATIO, r (=k; 1 k: )

_ FREOUENCY ~ SPAN WITH f BL - BILINEAR SOIL STIFFNESS

_ FREQUENCY CF SPAN WITH -LINEAR SOIL STIFFNESS (BURIED)

l!

(J

= UPWARD STIFFNESS

DOWNWARD STIFFNESS

= ;o• =O

AXIALLY UNRESTRAINED PIPE

REFERENCE (15]

Figure 4.6 Graph showlng effect of bl-linear soil stlffness on natural frequency of span

67

FcxJ ---- .... ·'

FtxJ VARIATION OF FRICTIONAL FORCE

i) FRICTIONAL FORCE FOR SPAN RESTING ON SOIL

BURIED SPAN

• • + ,rf11 Fcxl VARIATION OF FRICTIONAL FORCE

ii) FR!CTIONAL FORCE- FOR BURIED SPAN

Figure 4.7 Typlcal dlstrlbutlon of frlctlonal force

68

.. " -" ... -o ¡::: ... 0:

>-u z "" ::> o '" 0: "-

L/0 RATIO

I·O +---+---+---+-....,.-----+-50

0·8

100 100

200 400

0·7+1---r---1---t-.--~--~-------, NOTES

fF = FREQUENCY WITH SOIL FRICTION

fe•= FREQUENCY OF AXIALLY RESTRAINEO BURIEO PIPE SUPPORTS

{$ = 3 PIPE SUPPORTS BURIED IN RIGIO SOIL ANO RESTRAINEO ROTATIONALLY

osl----+----+----+-===~===í==~----~ 2 4 6 8 10

COEFFICIENT OF FRICTION fL

Figure 4.8 Effect of soil friction on natural frequency

69

The reported results from Reference [6] show that seabed slopes of less than zo have an appreciable effect on natural frequencies, sometimes up to 15% in comparison to horizontally supported pipes, for zero axial forces.

However, these large variations may be dueto the approximations made when obtaining the solution to the problem, particular! y with regards to the estimation of sag tension and friction effects. In the calculations it was assumed that the displacement shape and amplitude of the freespan remained constant irrespective of the variations in axial forces and the seabed slopes. In addition, the effect of friction at the edges of the sirle spans, illustrated in Figure 4.7(i), was completely ignored; this may in cases be very important depending on the reaction forces at these positions, which are usually large. For these reasons, in general small variations in seabed slope are expected to have a smaller effect that that suggested in Reference [6]. The effects of seabed slope on pipeline span are exarnined further in Section Six.

4.5.4 Static and dynamic non-linear effects When the ends of a pipe span are restrained against axial movement, for example due to the. development of pipe/soil friction, then application of loads will induce sag tension, the magnitude of which increases parabolically with the amplitude of the deflected shape. The effect of an increase in tension is to stiffen the pipe and increase the natural frequency as discussed in Section 4.5.1.

This dependence of sag tension and natural frequency on the pipe span deflection is of a non-linear nature, and it depends on the degree of axial restraint. Sag tension can develop as a result of static effects, due, for exarnple, to the response of a pipe span to submerged weight, or as a result of dynarnic effects, for example caused by vortex shedding oscillations. In the latter case both stiffening and destiffening effects may occur, depending on whether the movement is downwards or upwards during a vibration cycle. Aspects of this behaviour have been examined in Reference [15], and are discussed in the following.

Effect of Static Sag Tension

The effect of static sag tension on static deflections and thus on natural frequency, has been examined in Reference [15], by considering completely clamped pipe spans, as shown in Figure 4.9(i). lt was found that when the non-linear effects were included, the static deflection, y,, could be obtained from:

Eqn 4.14

where l3 is the axial force parameter, defined by equation (3.19), and ~ is the characteristic displacement parameter, defined by Equation 3.25. It was found that, for this particular example, variations of 13 in Equation 4.14 did not have a strong effect on Yo•· This is illustrated for different pipe geometries and span lengths in Figure 4.9(ii). Because of the assumed boundary conditions very large sag tensions are induced in the pipe spans, thus overshadowing the effect of the applied axial force. This is a very unrealistic example. In practica} situations the axial and rotational restraints will have finite values depending on the development of axial friction and the soil stiffness of the supports. In such situations the effects on non-linear and linear tensions will be expected to be comparable.

This is illustrated in Figure 4.10, which shows the effect of characteristic displacement parameter, 1;. on non-linear axial force parameter, 13ru, for different values of the coefficient of friction, ~- The results in Figure 4.10 were obtained in Reference [15], assuming pipes with ends buried in rigid soil supports and subjected to axial tension corresponding to 13 ; 3. lt can be seen that the degree of friction development has a very strong effect on 13.,. For soils with ~ ; 0.5 and pipes with ~ < 2; it appears that 13., is of the same order as 13. However, the practica! significance of this graph is limited because of the assumptions made regarding rotational restraint and soil stiffness.

70

il AXIALLY RESTRAINED PIPE

-5 -4 -3 -z -, COMPRES S ION

05

o

:~;;r~~~i.t:ii.tr-!~~;~;.:(j'ffj ;'t, RIGID SUPPORTS

~~+------

2 3 4

TENSION

iil VARIATION OF STATIC NON-LINEAR DEFLECTION WITH AXIAL FORCE PARAMETER ,¡:¡.

Figure 4.9 Static non·linear effects

L=50m

71

72

...J z ~

a: "' .... "'

20 ::E <1 a: Cf z o ¡¡; z "' .... a: <1

"' z :::; ' 10 z

o z

~ •<>o

~ = 5·0

~ =0·5

CHARACTERISTIC DISPLACEMENT PARAMETER, ~ (= Yc/Dl

NOTES'

PIPE BURlE O IN RIGID SOIL

fl = 3

Figure 4.10. Varlatlon of non-linear tenslon wlth characterlstlc dlsPiacement for dlfferent

friction coefficlents

il VIBRATION OF PIPE WITH yd < Y05

· ....

iil VIBRATION OF PIPE yd > Y05

m

RESTORING FORCE

·.· ...

FIXED ENDS

FIXED END

CASE li

1 DYNAMIC DE LECTION RATIO yd 1 y05

iiil VARIATION OF RESTORING FORCE WITH AMPLITUDE OF VIBRATION

. Figure 4.11 Dynamlc non-linear effects

73

• -' "" .. -·o ¡::

"' "' ,_ u z w ::> o w

"' "-

74

1·5

1·0

D = 1·0 m

:'-----O= 05m L= lOOm 0·5

'--,,----O= O· 25m

o+-------~------+-------r-------,-------0 0·5 10 1·5 20

OYNAMIC TO STATIC OISPLACEMENT AMPLITUDE RATIO,

NOTES'

® YELO OCCURREO AT (J y = 350 M Po

,;e = 3

Y os

AXIALLY RESTRAINED PIPE BURIEO IN RIGIO SUPPORTS

Figure 4.12 Varlatlon of natural frequency wlth amplitude- of vibratlon

Dynamic Non-Linear Effects

Pipe spans subjected to static loads will deflect to a static equilibrium position, for example shown by the salid curve in Figure 4.ll(i). However, when subjected to a periodic disturbing force, for example to vortex shedding loading, they will vibrate about this equilibrium position, as shown in Figure 4.ll(i). Thus the dynamic response will be superimposed on the static deflected shape of the pipe, as indicated by the broken lines in Figure 4.11 (i). Downwards dynamic displacements will result in increases in sag tension and thus increases in stiffness, indicated by curve I in Figure 4.ll(iii). Upwards dynamic displacements will result in reductions in sag tension and stiffness, indicated by curve 11. However, when these upward displacements are large, with yiyos>l, so that shape reversa! occurs, as shown in Figure 4.11 (ii), sag tension and pipe stiffness will start increasing again, as shown by curve II1 in Figure 4.11 (iii). Increases and decreases in sag tension will be accompanied by restoring forces being generated in the pipe, the effect of which is always to restare the pipe to its original static equilibrium position. Their variation with the amplitude of vibration, y,, is shown in Figure 4.11 (iii).

The variations in sag tension and pipe stiffness, which are functions of the vibration amplitude, y,, will have an effect on the natural frequency study of the system. This is shown in Figure 4.12, obtained from results reported in Reference [15). In general, the frequency, fsd, decreases in comparison to the value, which occurs at yd =O, fs. It reaches a minimum value at yd =Y os. and increases again for Yd > Y os· This behaviour is closely associated with the variation in pipe stiffness, indicated by Figure 4.ll(iii). Reductions in frequency as muchas 20% can occur because of this non-linear destiffening associated with dynamic displacements. When the amplitude of the vibrations is significant, compared to the static deflection, y,., yielding of the pipe, and possibly collapse, may occur during its dynamic response. Positions where yielding of the pipe occurs, assuming that allowable stress is 350 MPa, are shown in Figure 4.12. For this reason, it is advisable that large amplitude vibrations are prevented, by ensuring that a safe separation margin exists between the vortex shedding frequency, and the pipe natural frequency as discussed in Reference [5]. For ydiYos < 0._2 the effect of dynamic displacements on the natural frequency is insignificant.

An additional effect of vibrations on span behaviour is that illustrated in Figure 4.13, associated with situations where surface resting pipes form small side-spans. During vibrations the deflection, y,, and length, L, of the side-spans change, increasing with downward movement and decreasing with upward movement of the central span. Because of the interaction between the si de and central span response, when the amplitude of motion is significan! the time-varying support length may have an effect on the natural frequency. This behaviour was examined in Reference [15), assuming a surface resting pipe, on rigid supports, and zero axial force. It was also assumed that the amplitude of vibrations in the central span was limited to a value, such that the amplitude of vibrations in the side span, y,, was equal to the static deflection, y,, to relevan! problems associated with contact with the seabed. Within these limitations and for the exarnple considered in Reference [15], the effect of time varying support length on natural frequency was reported to give a 6.5% underestimation of the value. However, the practica! value of these results is very limited, due to the very idealised support conditions considered in the analysis.

In general, it has been shown that non-linear actiOn associated with vibration of pipe spans can have a significan! effect on natural frequency. However, it is usual practice to preven! vibrations of pipe spans, because of the potential of catastrophic collapse associated with fatigue or buckling, when significantly large vibration amplitudes occur. This is achieved by ensuring a safe separation margin between the vortex shedding frequency and the natural frequency, recommendations about which are made in Reference [5]. In such cases, therefore, it may be concluded that non-linear dynamic effects have insignificant influence on natural frequency.

75

y,

.Ó.Lo

~r)i'fiCf.•CC>Mj' ,,

;;&if.S;t,i,;i'lw~fi!\j;\?X1'if:~·;::::i;1't"'W~6· Lo L Lo

Figure 4.13 Effect of vibration amplltude on slde·span length

4. 5. 5 Multiple · spans There may be occasions where spans form adjacent to each other along the pipeline route, especially when the seabed is very uneven, or dueto scouring, as discussed in Section 2.3. If such spans are sufficiently clase together, with relatively short intermediare support lengths, they may interact, responding as a system. When this occurs, the whole multiple span system must be considered as a unit in calculating the natural frequency. Tbis interaction of adjacent spans occurs through the coupling influence of the intermediate support length. It is therefore important to know whether the spans are uncoupled, thus behaving independently, or if they are coupled in which case they must be analysed together using more accurate methods of analysis, for example [17].

Buried Supports

A study was carried out in Reference [ 16], concerning the above phenomenon. In order to simplify the analysis, the following assumptions were made.

a. the sea bed was horizontal b. the soil supports behaved as linear elastic springs c. the pipeline remained in contact with the supports d non-linear effects were ignored e. both spans existed simul taneously [, the frrst fundamental frequency mude was considered only g, frictional effects were ignored.

Assumption (e) was made because the order in which the spans are formed can have an effect on the results. For instance, if a single span occurs and is followed by a second adjacent span, the sag of the first span would be reduced during the development of the second span.

The multiple span geometry which was considered in the analysis is shown in Figure 4.14(i) with Lo and L, being the adjacent span lengths, and L; the intermediate support length. The pipe spans were assumed to have soil spring supports of stiffuess k,.

The natural frequencies of a number of multiple span cases were calculated in the study, including the effects from variations of span lengths, L1 and L2, soil stiffuess, k,, and axial force, T,

76

The spans were considered to be uncoupled when the frequency of the fundamental mode did not change with the support length L;. It was found that a good approximation to the critica! uncoupling support length, L¡, is given by:

21t L;o = [::-2::---c(c:-k-, -_EI-n--',f'é!2"-) -:-~-----=-]­

+ _! ~ El'

The spans were considered as uncoupled when:

L;o < L¡

Eqn4.15

Eqn 4.16

The above equations can be non-dimensionalised using the parameters derived in Sections Four and Five, by expressing L;o in terms of L¡. The resulting equations can be simplified by noticing that in most cases k.. >> f'. Thus, combination of Equations 4.15 and 4.16 gives the condition for which span decoupling is ensured as:

2y, i + J3, > 41t2 Eqn 4.17

where y; and ¡3; are respectively the soil stiffness and axial force parameters corresponding to the intermediate support. Provided expression 4.17 is satisfied interaction between adjacent spans will be avoided.

It was also found. in Reference [ 16], that the effect of interaction between adjacent spans is to reduce the natural frequency, relative to that of the individual spans. As this is more critica!, being unconservative from the point of view of vortex shedding, it is importan! to analyse spans which are coupled, using accurate methods such as those described in Reference [17].

When spans are uncoupled they can be treated as independent single spans, and can be analysed using the methods described in Sections 3 and 4.

Surface Resting $upports

The approximatioo to the critical decoupling support length, L;c, given by Equation 4.15, was obtained on the basis of the assumption that vibrations of the central support length did not extend into the adjacent spans. In the case of pipe resting on the surface of the supports, it was shown in Reference [ 13] that side-spans form of length approximately equal to L/2, as illustrated in Figure 4.14(ii). Thus assuming in this case that span uncoupling occurs when vibrations of the intermediate support lenglh do not extend into these side-spans, equation 4.16 takes the form:

1 Lto + :;: (L, + L,) < L, Eqn4.18

which after non-dimensionalisation and rearrangement becomes:

1 41t2

2Y:;:+ ¡3, > [ _ (L 1 + L2)]'

l 2Li

Eqn 4.19

Thus, provided that expression (4.19) is satisfied, uncoupling between adjacent spans can be ensured. As in most practica! situations it is difficult to establish whetber intermediate supports are completely buried, partially buried, or surface resting, it is recoinmended that Equation 4.19 should be used in all cases to provide conservative estimates of span uncoupling.

If span coupling is a problem, due to its adverse effects on frequency more accurate methods of analysis should be used, as for exarnple those in Reference [17]. However, the results from pararnetric studies presente<! in Reference [16] indicate that conservative estimates of natural frequency may be obtained by assuming simple supports and analysing the spans individually.

77

L, L¡

i) BURIED SUPPORTS

L, L¡

ii) PIPE RESTING ON SURFACE OF SUPPORTS

78

Figure 4.1~

Multlple sp~ns

5. PARAMETRIC ANALYSIS OF SPANS

5.1 INTRODUCTION

The variables which influence the behaviour of a pipeline span are identified in Sections 3 and 4. They have been expressed into a non-dimensional forma! in an effort to reduce the total number of independent variables and provide generalised solutions to pipe spanning problems. These non-dimensional parameters include the axial force parameter, ~. soil stiffness pararneter, y, seabed slope parameter, /., characteristic displacement parameter, ~. and soil friction parameter, 11· It is necessary to limit the values of these non-dimensional parameters to a practica! range, in arder to ensure realistic results. This is achieved by examining values of the variables as used in practice, under normal conditions.

The methods of analysis established in Sections 3 and 4 have been employed to carry out parametric investigations of span behaviour, by varying the above mentioned parameters within chosen limits.

The structural quantities of interest, obtained from such an analysis, are axial force, bending moments and natural frequency. The dependence of these quantities on the non-dimensional parameters is examined in the context of parametric sensitivity studies. In cases where this dependence is found to be small, the effects of the corresponding non-dimensional parameters are accounted for by the use of analysis factors, the magnitudes of which are obtained from the sensitivity studies carried out. In assessing the magnitudes of these factors a conservative approach is followed, so that frequency is minimised, and the bending moments and axial forces are maximised in magnitude. The use of these analysis factors has the effect of reducing the number of non-dimensional pararneters that need to be considered in analysis even further, thus simplifying the overall approach. The dependence of the structural characteristics of pipe spans on the remaining important non-dimensional pararneters is then examined in full.

5.2 RANGES OF, PARAMETERS

In order to assess the behaviour of a span in a consistent and rational manner, 1t ts necessary to establish the practical ranges of the static and dynamic non-dimensional parameters, in relation t6 subsea pipelines. Each of the parameters developed in Sections 3 and 4 is considered to establish these practica! ranges.

5. 2.1 Linear axial force parameter, ~~

The linear axial force pararneter, ~ •• has been derived in Section 3.3. It takes the form:

" _ T1L2

f'l - El

In order to establish the practica} range of this parameter, it is appropriate to consider the possible limit states of pipeline spans under various loading conditions. This approach should yield the linút values of the critica! span pararneters. When a span is subjected to comprcssion, for example as a result of pressure effects, it behaves similar toa compressed bar, for which the limit state corresponds to Euler type buckling. This can be used to examine practica! ranges of ~ .. as Euler buckling depends on sinúlar variables, such as axial force, span length and pipe flexura! rigidity.

79

When the compressive force is increased and it nears the value of the Euler buclding load, T ,, the displacements and the bending moments become large. In axiallly restrained pipes, non-linear tension may develop which has the effect of reducing the deflections. This effect is illustrated by curves 1 and 11 in Figure 3.3. The non-linear tension will have a stronger effect on a stiff pipe than a flexible one. A limit state criterion can be established by comparing the linear axial force parameter,f31, to the Euler buckling load, T,.

Since, assuming simple supports:

and:

1t2EI T,=u

R - T,L' pi - El

It is possible to express the axial force, T, in the form:

T - [3,T, 1 - 1t2

A suitable criterion is to limit the linear axial force, T¡, to a value less !han the Euler buckling load, T,, because if T,, is greater !han T, the stresses may increase toa leve! which will precipitate collapse.

Therefore, if it is assumed that at the limit:

T1 = T,

then:

[3, = 10

As the boundary conditions may not necessarily correspond to simple supports, a somewhat larger range may be defined for [3,,

-15 < [3¡ < 15

From the above assessment it can be seen that this may be a reasonable range for t3,.

5. 2. 2 Seabed slope parameter, A. The parameter which describes the slopes of the span at the supports is given by

A. - @J!lli! - qL'

Therefore A. depends on the following factors, which need to be considered in order to define its parametric limits:

• the relative angle between the slope of the support and the span, dljl

• the length of span, L • pipe properties, q and El.

From surveys of various pipelines, for example the Frigg Line survey reported in Reference [7], it is apparent that seabed slopes are rarely greater than 2°. An exception is the Norwegian Trench where slope angles may reach 6°. However, it is not practicable to allow for all exceptional cases in the basic analysis. For this reason a 2° relative slope angle is initially taken as the maximum in the range of geometries considered. When slopes are outside this range then it may be necessary to carry out more accurate methods of analysis, as for exarnple described in Reference [17].

Typical, feasible, span lengths will depend on the pipe properties, and can be estimated on . the basis of vortex shedding considerations. Por instance a span of 15m can be critical for small diameter pipe (6-inch), while it may be insignificant for a large diameter pipe (40-inch). Similarly a span of lOOm can be critica! for a large diameter pipe but it is not feasible with a small diameter pipe, which will usual! y follow the seabed profile forming only smaller spans. The critica! span length for a particular pipe size, can, therefore, be obtained on the basis of vortex shedding calculations, as described in Codes of Practice, for example Reference [1]. Values of the typical critica! span lengths, Lcr. are. given in

80

Reference [8] anda graph showing the dependence of La on pipe diameter, D, is shown in Figure 5.1. V alues of critica! span length for a number of diameters are summarised in Table 5.1 and can be used in determining parametric ranges for A..

D

(m)

0.1

0.5

l. O

1.2

Table 5.1 Critica! span lengths

[.,y

(m)

15

34

96

140

[.,y

D

150

68

96

116

The variation of !• with span length and pipe sizes has been calculated for empty and

water filled pipes with 50mm and lOOmm concrete coating. A representative maximum value for A. can then be chosen from the variation in the computed values shown in Table 5.2.

Table 5.2 Varlation of slope parameter

Case Pipe

D/t [.,y El

1.. Dia (m) qL' dllf=1 o dllf=20

1.,=2" 0.1 lO 15 0.3 0.0052 0.0105 0.5 30 34 2.4 0.0419 0.0838

empty 1.2 30 140 0.55 0.0096 0.0192

~=2" 0.1 10 15 0.29 0.0051 0.0101 water 0.5 30 34 1.2 0.0269 0.0419 filled 1.2 30 140 0.13 0.0023 0.0045

1.,=4" 0.1 10 15 0.14 0.0024 0.0049 0.5 30 34 1.2 0.0209 0.0419

empty 1.2 30 140 0.25 0.0044 0.0087

!<,=4" 0.1 10 15 0.13 0.0023 0.0045 water 0.5 30 34 0.72 0.0126 0.0251 filled 1.2 30 140 0.105 0.0018 0.0037

A maximum of A.= 0.04 covers the rnajority of cases for ang1es of 1 o and 2°. The exception in Tab1e 5.2 appears lo be with a slope angle of 2°, an empty pipe with diameter 0.5m and concrete coating of 50mm.

A similar range for A. rnay be obtained by considering the slopes al the end of a simply supported span, 'lfu, given by,

_ 51tgL3

'l'u - 384EI

81

E ~

'-' -' r 1-

"' z w ...J z íi'. (J)

-' <[ u ;:: ¡¡: u

82

1 1

102

101

NOTES

PIN- ENDED STROUHAL NUMBER MODIFIER = 1·0

ADDED MASS COEFFICIENT, Cm= 1

LOGARITHMIC DECREMENT OF STRUCTURAL DAMPING= 0·126

20< .!f < 40

Q-5 < Us < 1 ms-1

10°1-----i----4----t---+--+----0 0·5 1·0

PIPE DI AME TER D( m)

Figure 5.1 Approxlmate dependence of crltlcal span length qn pipe diameter

The corresponding non-dimensional slope parameter A. , is given by "

' - l!f,.EJ - ~ - O 04 "-n - A.L3 - 384 = ·

This value for A., of 0.04, is considered reasonable within working bounds. However, values of A. greater than 0.04 and within the range -0.12 < A. < 0.12 will be investigated in the sensitivity study.

5. 2. 3 Soil stiffness parameter, y

The soil stiffness parameter is defi.ned as

It therefore depends on the following:

• soil stiffness, k,

• length of span, L • pipe properties, EJ.

All of these factors will need to be considered when determining the practica] range of the soil stiffness parameter and examined in the following.

As most available methods of soil survey of the seabed produce mostly qualitative results, it is unrealistic to expect high accuracy when defining values of soil parameters. These are usually given within relative wide ranges, which give an indication of the behaviour of certain types of soil.

Soil Stiffness

Tbe stiffness, k,, of certain typical types of soil is discussed in Section 2.7.7. Values of k. appear to vary bétween IMPa for soft soils and 106 MPa for hard rocks.

Length of Span

Typical critica! span lengths, based on vortex shedding criteria, corresponding to various pipe sizes are given in Section 5.2, and are summarised in Table 5.1.

Range of Soil Stiffness Parameters

Based on the range of span lengths given in Tab1e 5.1 the 1imits of (~;)can be deterrnined

for various pipe sizes. Thus by incorporating the limits of soil stiffness the corresponding ranges of y can be found as shown in Table 5.3.

Table 5.3 Limits of soil stiffness parameters

D k (;;;:¡• kL' (m) D/t (m) El r=a

M in Max ks-106 ks-1012

0.1 10 15 0.082 8.2x104 8.2x108

0.5 30 34 0.0084 8.4x103 8.4x109

1.2 30 140 0.079 7.9x104 7.9x1010

83

Discrete values covering the complete range of y can be obtained by examining Figure 4.4, which was derived from Reference [J4]. These values are JO', JO', JO' and JO'. In cases where soiJ parameters fall within these discrete values, the immediately lower value should be used for conservativeness. Otherwise interpolation may be employed.

5. 2. 4 Characteristic displacement parameter, ~

The characteristic displacement pararneter, ~. is given by:

é, = >'E D

A reasonably practica) range for the displacement of a clamped bearn, y,, is y,~ 0.6 D, thus ~ ~ 0.6. .

5.2.5 Pipe/soil friction parameter, 11 The friction pararneter, 1], is defined by:

D 11 = ~ L

This depends on the coefficient of pipelsoil friction, and pipe diarneter to span length ratio.

Typical values of the coefficient of longitudinal friction, ¡l, are given in Section 2.7.7, TabJe 2.2. These values appear to vary between 0.2 and J.2.

The typical range of pipe diameter to span length ratios is estimated to be between JnO and IIJ50, using the results shown in Table 5.1. Based on these, the practica! range of the friction parameter, 1], can be defined as O.OOJ < 11 < 0.02.

5.3 SENSITIVITY STUDIES

It has been shown in Sections Four and Five that the frequency, bending moments and sag tension are functions ofthe effective axial force, soil stiffness,.characteristic displacement, seabed slopes, and soil friction parameters, so that:

f = f (~"y,é,,A.,l])

Initial studies ha ve shown that the effect of slopes and friction on frequency, f, moments, M, and sag tension, ~,,, is relatively small. This suggests that it may be possible to account for the effects of slope, A., and friction, 11.. parameters by the use of "analysis factors", so that, for example:

Where .PJ.. and .p'l are the analysis factors for slope and friction respectively. Thus by using this method it becomes possible to reduce the number of independent parameters that determine structural response, to 13,. y and ~-

Values for the analysis factQrs can be detennined from a sensitiv.ity analysis. This may be based on parametric studies of the variations in f, M and ~'" with A. and 1]. The results of such sensitivity studies are shown in Figures 5.2 to 5.5. The ranges of variation are shown shaded, with the percentage changes from the standard cases shown on each plot.

The variations are relatively small for frequency and bending moments. Even though the corresponding variations in sag tension appear to be very large, they usually have a reasonabJy smaJI effect on the total axial stresses. This is due to the fact that the sag tension is in most cases a small part of the effective axial force. The analysis factors obtained from the graphs in Figures 5.2 to 5.5 are given in Table 5.4. These have been chosen on the basis that a lower bound is conservative for frequency, while an upper bound is conservative for bending moments and axial forces. Also shown in Table 5.4 are analysis factors to account for approximations in dynamic analysis, discussed in Section 4.3.

84

( lc)'l I·O+-----.----r-----,---

09

o 8 1---t-------t----t--~

0·7

o 6 -t---------J-----t----,¡

0·5

0·3

02-t--~~---,----r---r---t--~

01

E--r~--~-+--r-~-,--+--r--t--r~(-11c)~=0006 01 02 03 OA 05 0·6 07 08 0·9 10 H

i) FREQUENCY

SAGGING

1·5

(:ch=0006

-~L_ _ _J_ __ -t- -1·5"'---~---"---...._-

ii) MOMENT

Figure 5.2 Varlation of frequency and moments with frlctlon parameter

85

6

45%./ ..

V / i '

'f

k( /.

/ v·~~~.l ~·

v/ V •

/

~-~ ~·

~· ·' lL v··

1 . .• / 1/ ....

1 V. /. ~/ L

~ /.'

/ / •

5

4

3

2

¿~V [;.

o o 2 3 4 5 6 7

Figure 5.3 Varlatlon of non-linear tenslon wlth friction paramet8r

86

{ tc) slope 1·1

1·0

0·9

era

0·7

0·6

0-5

0·4

0·3

02

0·1

4o/. /i 1/. ¡,/ l .· .. •·····. ll: ."'

1 lh V 5"t.

/ ~ ,,.

/ h. / 1,¿' /

.,. / -¿.

/t:# ~

lv l/i (fe) flat o o 1 0·2 03 04 05 06 0·7 era 0·9 I·O 1·1 1·2

i) FREQUENCY

HOGGING

1----1-----l-----,fL---+---+---+ (~e \al - ~5 -¡.o 0·5 I·O 1·5 1'

SAGGINGJ

• , ii) MOMENT

Figure 5.4 Variatlon of natural frequency and bending moments wlth slopes

87

o 2 4 5 6

Figure 5..5 Variatfon of non-linear tenslon wtth slopes

88

Table 5.4 Recommended analysis factors

Analysis Factors Minimum Maximum Non-Linear Frequency Bending Moments Axial Tensions

Hogging Sagging Maximum Minimum Ar Am. A.. 1\,¡~ Anlmin

S1oping Sides 0.95 1.15 1.08 2.4 0.68 A.< 0.12

Friction 0.94 1.05 1.06 1.45 0.73 Effects 0.2 < 11 < 1.0

Dynamic 0.95 Analysis Approximated

Recommended 0.85 1.20 1.15 3.5 0.5 Analysis Factor

5.4 NON-DIMENSIONAL DESIGN CURVES The methods of span analysis developed in Sections 3 and 4 are used to develop the non­dimensional, parametric design curves, presented in Figures 5.6 to 5.11. These curves permit the evaluation of the equivalen! tensile stress, cr,, for the purpose of a combined stress yield check, and the evaluation of the span natural frequency of vibration for the purpose of a vortex shedding check.

The design curves in Figures 5.6 to 5.9 show graphically the relationship between the non­linear axial force parameter, 13nt. and the controlling linear axial force parameter, ¡3~, the characteristic displacement pararneter, ~. and the soil friction pararneter, y.

The design curves shown in Figure 5.10 show graphically the relationship between the maximum hogging and sagging bending moment ratios, Mh/Mc and Ms/Mc. and the controlling effective axial force pararneter, p., and the soil stiffness pararneter, y.

The design curves shown in Figure 5.11 show graphically the relationship between the span natural frequency ratio, f/fc. and the controlling effective axial force parameter, 13e. and the soil stiffness pararneter, y.

It can be seen from Figures 5.6 to 5.9 that the non-linear axial force parameter, p,¡ is always tensile, and increases with:

. • increasing characteristic displacement, ~ • decreasing linear axial force, l3n1 • , increasing soil stiffness, y.

The effective axial force pararneter, p., is related to the linear and non-linear axial force pariuneters; ¡3, and 13ru. by the relation be1ow:

~' = lh ~-The eft(:ctive axial force parameter, 13e. is related to the linear and non-linear axial force pararneters; ¡3, and ~.1. by the relation below:

l3o = ~¡ + ~,¡

It can be seen from Figure 5.10 that the hogging and sagging bending moment ratios, Mi/M., and MJM.,, increase in absolute value with:

• decreasing effuctive axial force, ¡3, • decreasing soil stiffness, y.

89

9

8

~ z ..

7 111:

"' ~ "' :E e 6 111: e ....

"' 5 u 111: o ... .J 4 e )( ... 111: 3 ... "' z .J

1 2 z

o z

o

90

SOIL STIFFNESS PARA METER • y •lO 7

~l= -15

~.: -7·5

~l= o

~ = 7·5 l

~l= 15

o 0·1 0·2 0·3 0·4 0·5 0·6 0·7

CHARACTERISTIC OISPLACEMENT PARAMETER, ~=~e

Figure -5.6

Non-linear axial force deslgn curves for· r = 107

_, = . 111:

"' ~ "' :E e 111: e 0..

"' u 111: o ... .J ... )( ... 111: ... "' z .J

SOIL STIFFNESS PARAMETER ,Y•t05

9

8 Pl= -15

-' z .. 7

"' "' -' 1- .. "' :10 6 "' cr

"' "' Pl=-7·5 1-cr "' CL :E

"' cr

5 "' .., cr "' 0.. o ....

"' .J ..,

4 a: cr Pl" O o )( .... cr

.J a: cr cr 3 Pl= 7·5 )(

"' cr z .J Pl= 15 a:

1 2 cr

z "' o z z :::;

o o 0·1 0·2 0·3 0·4 0·5 0·6 0·7

CHARACTERISTIC OISPLACEMENT PARAMETER, ~=~e

Figure 5.7

Non-linear axial force design curves for 1 = 105

91

9

8

..J z .,_

7 lt:

"' 1-

"' 2i 6 .. lt: .. ....

"' 5 u a: o IL

.J 4 .. )( .. lt: 3 .. "' z .J 1 2 z

o z

o o

92

SOIL STIFFNESS PARAMETER ,Y•I04

Ji.=-15

V /¿J

~ V

V 1/ P.: -7·5

/ /' , /

lf" V PL= o

/ ~"' V /

~ V ~ PL= 7·5

/_ / . ~" PL• 15

V / (/: ~ v· V

¿~ ~ ~ ,..,

• 0·.1 0·2 0·3 0·4 0·5 0·6 0·7

CHARACTERISTIC DISPLACEMENT PARAMETER, ~=~e

Figure 5.8

Non-linear axial force deslgn curves for y = 104

-lt:

"' ,_ "' ~ .. lt: .. ....

"' u lt: ¡r .J .. )( .. a: .. "' z :;

9

8

..J z ...

7 11::

"' 1-

"' :E 6 cr 11:: cr ... "' 5 u 11:: o .... ..J 4 cr )(

cr 11:: 3 cr "' z ..J 1

2 z o z

o o

SOIL STIFFNESS PARA METER , Y •10'5

1 ....... 1!¡_=-7·5

1 1 /

~L: Q

~·

1 1/ / ~L: 7· 5

/ ~ ~· ,......... PL= 15

_,~~ ~ ,

' 0·1 0·2 0·3 0·4 0·5 0·6 0·7

CHARACTERISTIC OISPLACEMENT PARAMETER, ~=~e

Figure 5.9

Non-linear axial force deslgn curves for y = 103

..J ...

"' u a:: ~ ..J cr )(

cr

a:: cr "' z ..J

93

lt can be seen from Figure 5.11 that the span natural frequency ratio, f/fc increases with:

• decreasing effective axial force, ~ • decreasing soil stiffness, y.

It can be concluded that conservative upper bound values of bending moment, M, and natural frequency, f, are associated with:

• minimum non-linear axial force parameter, ~nl

• minimum effecti ve axial force parameter, ~' • minimum soil stiffness.

It is importan! to note that the non-linear axial force pararoeter, ~"'' develops only in spans. Th~ non-linear axial force parameter, ~nl. should be included only in such situations.

The analysis factors given in Table 5.4 should be applied to all values of non-linear axial force pararneter, ~'" bending moment ratio, M/M, and natural frequency ratio, f/f, obtained from Figures 5.6 to 5.11.

5.5 SPAN ANALYSIS METHODS

Phase 1 S pan Analysis

The parametric method of span analysis is a complex procedure in tended for the analysis of critica! spans. In order to identify which spans are critica!, a simple, quick and conservative method of span analysis is needed.

Such an analysis, called the Phase 1 Span Analysis, has been developed and is presented, together with worksheets, in draft form in a document for discussion with the industry Reference [74]. The Phase 1 span analysis conservatively assumes the span is simply supported and checks the following failure modes:

• yield • bar buckling • flow induced vibration.

Spans found unacceptable by the criteria applied in the Phase 1 span analysis are considered critica] and should be subject to further more detailed analysis. The Phase 2 span analysis is Considered a suitable more detailed method of analysis for spans identified as critica!.

Phase 2 Span Analysis

The non-dimensional, parametric method of span analysis developed within this report is also presented in Reference [74], which contains a tlow-chart that sub-divides the Phase 2 span assessment into 26 activities. A worksheet is provided for each activity together with guidance notes for the completion of each activity worksheet.

94

u ,. .... ,. 2 1-.. 0:

1-z "' ,. o ,. "' z ¡¡ z

"' .,

u -:::: o l­.. a: ,.. o z

"' " o .. 0: ... ..... .. "' " lr z

"' 1·5 ;!'; O> Y >lo• O> o :<

Y =104

Y= 10'

-15

Y >lo5

.. y =104 z ¡;;

Y= 105 O> .. (/)

EFFECTIVE AXIAL FORCE PARAMETER, Pe

Figure 5.10 Bending moment ratio (M/Mc) design curves

1·1

1 v >1 0 7

1·0

~ 0·9 -~ ~ "\..._ ¡--Y=I05

O·" ./~ "'" .-- '\ """' ;,...o. 7 Y=l 0 4

~ .<s ~ lL: L 0·5 "\.

¿_ ~ ~ Y=I0 3

~ 0·3

-15 -10 -5 o 10 15 COMPRESSION TENSION

EFFECTIVE, AXIAL FORCE PARAMETER, Pe

Figure 5.11 Natural frequency ratio (flfc) design curves

~

rL "' 1-

"' ,. .. 0: .. ... (/) (/)

"' z ... ... ;:: (/)

-' o (/)

0:

"' 1-.. ,. .. 0:

: (/)

"' "' z ... ... 1-

"' -' o "'

95

6. CONCLUSIONS

The Phase 1 span analysis method presented in Reference [74] provides a simple and conservative method of identifying critica! spans.

The Phase 2 span analysis method presented in Reference [74] provides a non-dimensional parameter based method for the analysis of critica! spans.

The static analysis concludes that the static behaviour of a span is govemed by five non­dimensional parameters, namely:

• axial force parameter, J3 • soil stiffness parameter, y • characteristic displacement, ~

• soil friction factor, 11 • seabed slope pararneter, A

The dynamic analysis concludes that the natural frequency, f, of a span is a function of the same ti. ve parameters, and may be calculated from the static solution using the equation below:

where:

(, = natural frequency of the clamped span

Y e = lateral displacement of a clamped span

Yo = lateral displacement of real span.

The use of the non-dimensional design curves is dependent on the value of the non­dimensional parameters falling within the recommended ranges of applicability.

The sensitivity analysis concludes that the soil friction and seabed slope parameters may reasonably be considered constant. These constants are included in the Analysis Factors, A, that are applied to the results determined from the non-dimensional design curves.

The parametric analysis concludes that conservative values of bending moment, M, and natural frequency, f, are associated with:

minimum non-linear axial force parameter, Bnt

minimum effecti ve axial force parameter, Be minimum soil stiffness parameter, y.

The non-linear axial force parameter, Bnt. is present when the span develops through the action of scour. Thus the non-linear axial force, Bnl. should only be included when scour can be shown to be the cause of spanning.

In the event of multiple span interaction, the spans should be treated as simply supported for the purposes of span assessment.

97

A

e

E E.

f fe

G G,

g

H

f¡ Hs H.

K K. K. K.E.

100

NOTATION

cross sectional area of steel pipe

minimum natural frequency analysis factor maximum hogging bending moment analysis factor maximum sagging bending moment analysis factor maximum non-linear axial force analysis factor minimum non-linear axial force analysis factor

water particle acceleration in-line amplitude of oscillation

drag coefficient inertia coefficient lift coefficient added mass coefficient

friction constant

nominal pipe diameter interna! pipe diameter externa! pipe diameter, including coatings

modulus of elasticity of steel modulus of elasticity of soil

flattening of pipe cross section, or ovality

soil friction force limiting friction factor critica! buckling load drag force inertia force lift force maximum combined drag and inertia force

natural frequency of pipeline span characteristic natural frequency of clamped pipe

maximum gap below span shear modulus of soil

gravity

depth of pipe below water surface

height of pipe on barge above water leve! height of breaking wave depth of burial of pipe from soil surface to pipe centreline

second moment of area of steel pipe

curvature critica! buckling curvature coefficient of lateral soil stress at rest kinetic energy

M M.o M, M, M,

N

P.E.

q q q. q, q,

R

r,

T, T,.. T, T, Ttay

T., T, Te T, T. t

""' " t.,

soil spring stiffness soil stiffness soil reaction force

pipe span length characteristic length wave length intermediate support length between multiple spans multiple span lengths

side span length

bending moment characteristic moment for clamped pipe hogging bending moment full plastic moment of pipe section sagging bending moment

added mass from surrounding water mass of concrete coating mass of contents effective mass mass of steel pipe

nonnal reaction force

collapse pressure for local buckling initiation pressure interna! pressure externa! pressure propagation pressure pressure required for full plasticity in the hoop direction

potential energy

lateral force due to pressure effects

submerged weight of pipe

unit of weight of pipe in air horizontalloading of spari lateral pressure load per unit length verticalloading of span

radius of curvature of pipe

interna} radius of pipe

combined axial force minimum tension applied to pipe by laybarge effective axial force linear axial force residuallay tension non-linear axial force pressure axial force thermal axial force Poisson axial force wave period pipe wall thickness corrosion allowance concrete coating thickness externa! corrosion coating thickness

101

u U, u. ü.

u

Ur

X

x*

y

y Y< y, Yo, Y os y, y*

a

y Y•

V v,

102

total velocity of sea water steady current velocity from surge and tidal currents wave induced velocity

maximum wave induced velocity

longitudinal displacement

usage factor

axial co-ordinate non-dimensional axial deflection

norrnalised deflected shape

lateral co-ordinate central displacement of clamped pipe amplítude of pipe vibrations central displacement of span deflected shape during vibration of pipe span non-dimensional lateral deflection

coefficient of therrnal expansion of steel

axial force parameter combined axial force parameter effective axial force parameter linear axial force parameter linear axial force parameter for multiple spans non-linear axial force parameter pressure axial force parameter

soil stiffness parameter soil stiffness parameter for multiple spans

strain non-linear strain bending strain

soil/pipe friction parameter

ambient temperature temperature of pipe contents temperature difference of pipe contents over surrounding water wave phase angle

seabed slope parameter seabed slope parameter atA seabed slope parameter at B

longitudinal friction coefficient

Poisson's ratio for sted Poisson's ratio for soil

characteristic displacement parameter

Po density of concrete p, density of externa\ corrosion coating

1' density of contents pipe p, density of steel p, submerged density of soil p. density of seawater

cr, longitudinal stress cr, axial stress cr, bending stress cr, expansion stress range

<>• Von Mises equivalent stress cr. equivalent tensile stress crH, cry hoop stress cry yield stress

t shear stress

<l>r angle of interna! friction of soil

"'' scabcd slope lj/A pipe slope at A

'1'• pipe slope at B

Óljl relative angle

O) angular velocity

NOTE: 1) The following subscripts are added to the above notation:

h = horizontal v = vertical max maximum mm = minimum a = allowable

103

DEFINITIONS

Longitudinal stress

Normal stress acting parallel to pipe axis.

Hoopstress

Normal stress acting in the circumferential direction.

Pipe bending moment

Bend.ing moment (M) in the pipe cross section as a whole.

Pipe bending stresses

Longitudinal stresses dueto pipe bending moment.

Interna/ pressure

Pressure inside the pipe. May be given as absolute pressure or gauge pressure.

Externa/ pressure

Pressure (immediately) outside the pipe. May be given as absolute pressure or gauge pressure.

lnitiation pressure

Externa! overpressure required to initiate a propagating buckle from an existing local buckle or dent.

Propagation pressure

Externa! overpressure required to initiate a propagating buckle that has been initiated (at a higher pressure).

Design pressure

Maximum interna! operating pressure.

Maximum operating pressure

Maximum pressure to which a piping system will be subjected in operation, which .should include static pressure and pressure required to overcome friction.

Test preSsure

Pressure specified to be applied to a pipe on completion of manufacture and/or on completion of construction. It may also be the pressure specified to be applied to a pipe after appropriate periods in operation.

Mínimum design temperature

Lowest possible steady state temperature which the pipeline system experiences during installation and operation. Environmental as well as operational temperatures are to be considered.

Maximum design temperature

Highest possible steady state temperature which the pipeline system may be exposed to during installation and operation. Environmental as well as operational temperatures are to he considered.

Restrained lines

Pipelines which cannot expand or contrae! in the longitudinal direction due to fixed supports or friction hetween pipe and soil.

104

Unrestrained Unes

Pipelines without substantial axial restraint (Maximum one fixed support and no substantial friction).

Span length

Length of a pipeline without contact with the sea bottom or other supports (= unsupported length).

Laying parameters

Essential parameters affecting the stresses in a pipeline during laying, such as applied tension, stinger curvature, etc.

Nominal wall thickness

The pipe wall thickness that is specified for supply of pipes.

Nominal pipe diameter

The outside pipe diameter to be used in the design calculation.

Submerged weight of pipe

The weight of the pipe, coatings and contents, per unit length, after allowing for buoyancy.

Design Premise

A document listing al! necessary information to carry out design of a pipeline.

Route Survey

Detailed inspection along the proposed pipeline route, performed to provide sufficient data for design and construction.

Seabed Topography

The mapping of the seabed to give sufficient detail of uneveness and features such as spans along the pipeline route.

Vortex Shedding Response

A situation where a free span on a pipeline experiences periodic disturbing loading, due to unsteady fluid flow past the pipe forming vortices. This may lead to oscillations of the pipe normal to its axis.

Vortex Shedding Frequency

The frequency at which vortices are formed, dueto steady fluid flow pasta pipe span.

105

APPENDIX 8

REFERENCES

107

REFERENCES

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14. KIRK, C.L.

108

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109

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74. J.P. KENNY & PARTNERS LTD

112

The Assessment nf Submarine Pipeline Spans - Draft Guidance Notes Discussion Document prepared for the Departrnent ofEnergy, February 1985

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